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- http://ufdc.ufl.edu/AA00004726/00001
## Material Information- Title:
- The kinetics of surface-mediated phase separation in the quasi-binary mixture of guaiacol-glycerol-water
- Creator:
- Shi, Qingbiao, 1963-
- Publication Date:
- 1994
- Language:
- English
- Physical Description:
- viii, 120 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Binary fluids ( jstor )
Critical temperature ( jstor ) Kinetics ( jstor ) Light scattering ( jstor ) Metastable atoms ( jstor ) Molecules ( jstor ) Phase diagrams ( jstor ) Silanes ( jstor ) Water temperature ( jstor ) Wetting ( jstor ) Dissertations, Academic -- Physics -- UF Physics thesis Ph.D Separation (Technology) ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1994.
- Bibliography:
- Includes bibliographical references (leaves 116-119).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Qingbiao Shi.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 030582204 ( ALEPH )
31920348 ( OCLC ) AKF8489 ( NOTIS )
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THE KINETICS OF SURFACE-MEDIATED PHASE SEPARATION IN THE QUASI-BINARY MIXTURE OF GUAIACOL-GLYCEROL-WATER By QINGBIAO SHI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1994 THE KINETICS OF SURFACE-MEDIATED PHASE SEPARATION IN THE QUASI-BINARY MIXTURE OF GUAIACOL-GLYCEROL-WATER By QINGBIAO SHI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1994 Dedicated to my grandmother FENG-HWA FU ACKNOWLEDGMENTS First of all, I would like to thank my supervisor Andy Cumming for his efforts and patience throughout the whole period that I worked on the project. I was very much in a limbo when he took me on board. Starting from the ground up, I learnt from him not only the basic light scattering techniques and the setups, but also how to be an experimentalist. Furthermore, he was a constant source of ideas and encouragement which were essential to push this project through. Also, I want to express my deep thanks to Wade Robinson, our resident engineer in the lab, who had helped a great deal in putting the lab together and in keeping it running smoothly. He had to endure the almost constant pestering from me and other students to get this or that done, sometimes even just to finish other's dirty work. Without his patience and help, our lab would not have been running. Thanks are also in order to my fellow students in the lab, especially to James Ellis Teer, who has done all the silane treatments of the windows, and to Bill Rippard, who did most of the microscopy studies for the project. There are some other people in the University of Florida I would like to thank: first to the other members on my supervisory committee, Jim Dufty, John Klauder, Neil Sullivan and Randy Duran, for patiently sitting through both my oral exam and then thesis defense, and providing insightful suggestions and comments; Chuck Hooper, who was always ready to help if needed; and the fellow graduate students in the physics department, especially Mike Jones and Laddawan Rumsuwan, for their help throughout the years. On a more personal side, I want to express of deep gratitude to Richard Trogdon and Suzy Spencer, for their friendship and help, especially when I first came to this country and everything was foreign and you were an alien. Their m patience and warmheartedness were indispensable for me to get over the initial "culture shock" and the language barrier. Heartfelt thanks go to Rob and Cynthia, for their friendship over the years, even after they had moved away. Their enthusiasm and encouragement are deeply appreciated, especially when I felt down and out. Finally, to a special friend, C. J., I want to say thanks. IV TABLE OF CONTENTS gage ACKNOWLEDGEMENTS iii ABSTRACT vii CHAPTERS 1 INTRODUCTION 1 1.1 Phase Diagram 4 1.2 Mean Field Theory of Metastability and Unstability 9 1.3 Cahn's Linear Theory of Spinodal Decomposition 14 1.4 Later Stage Coarsening 20 1.4.1. Droplet Coalescence 21 1.4.2. Lifshitz-Slyozov Theory 24 1.4.3. Hydrodynamic Growth Mode in Concentrated Mixture 26 1.4.4. Scaling Hypothesis for Structure Function at the Later Stage 29 1.5 Experimental Results 33 1.6 Wetting Phenomena and Phase Separation 36 2 LIGHT SCATTERING METHOD: CONCEPTS 42 3 APPARATUS 48 3.1 Light Scattering Apparatus 49 3.2 Temperature Regulation and Quench System 57 4 EXPERIMENTAL PROCEDURE 65 4.1 Sample Preparation 66 4.2 Phase Diagram of Guaiacol-Glycerol-Water Mixture 67 4.3 Sample Cell and Carrier 70 4.4 Data Acquisition and Processing 73 4.5 Treatment of Glass Surface with Trichlorosilanes 75 5 RESULTS AND DISCUSSIONS 81 5.1 Data Analysis 81 5.2 Phase Separation Kinetics 94 5.3 Gravitational Effects 100 5.4 Surface Treatments and Their Effects 103 v 5.5 Discussions 109 6 CONCLUSIONS 112 APPENDIX SAMPLE CELL CARRIER DESIGN 114 REFERENCES 116 BIOGRAPHICAL SKETCH 120 vi Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE KINETICS OF SURFACE-MEDIATED PHASE SEPARATION IN THE QUASI-BINARY MIXTURE OF GUAIACOL-GLYCEROL-WATER By Qingbiao Shi April, 1994 Chairman: Dr. Andrew W. Cumming Major Department: Physics The kinetics of phase separation in a quasi-binary mixture of guaiacol- glycerol-water has been investigated using a time-resolved light scattering method following a sudden quench into the thermodynamically unstable state of the mixture. The mixture was confined between two optically transparent windows with thickness less than 1 mm. In addition to the common growth mode in the bulk, where the average domain size, L, grows as t1/3 (where t is time), a novel growth mode was observed that the domain size L grows as ft, where b increases from 1.1 to about 1.5, depending on the quench depth. As the same novel mode had been observed earlier in a polymer mixture, it is thus established that it is generic to all classes of binary fluids. Experimental results from both the light scattering and video-microscopy suggested that the novel growth mode was closely associated with wetting phenomena of the two separating fluid phases near the glass surfaces. It was also dependent on the properties of the glass surface, which could be altered with a self-assembled monolayer of silane molecules. This experimental Vll evidence confirmed that the novel mode of phase separation was surface mediated. vin CHAPTER 1 INTRODUCTION The phenomena of phase-separation have been observed in a wide range of systems with two or more components. It has been an increasingly important process in material manufacturing and processing. Therefore comprehensive understanding is needed in both physical and chemical aspects of the phenomena. Over the last 30 years, experimental investigations have been conducted in alloys [Gaulin et al., 1987] and solid solutions [Hono and Hirano, 1987], simple binary solutions [Goldburg, 1983] and glass forming mixtures [Tomozava et al., 1970], and more recently, in liquid-polymer solutions [Sasaki and Hashimoto, 1984] and polymer-polymer blends [Hashimoto, 1988; Bates and Wiltzius, 1989], etc. On the theoretical side, efforts have been concentrated on elucidating the growth mechanisms that govern the kinetics of phase separation, especially those growth kinetics that are universal, depending only the general properties of the phase transition, e.g., the symmetries of the system and the order parameter, but not on specifics of the system. Significant progress has been achieved so far in our understanding of the kinetics of phase separation, both in experiment and theory, especially in the last two decades. In classical theories, phase separation proceeds either through spinodal decomposition or through nucleation and growth [see, e.g., Gunton et al., 1983]. The former leads to the decay of an thermodynamically unstable state, through long wavelength fluctuations of infinitesimal amplitude. The latter is 1 2 due to the instability against localized (droplet-like) fluctuations with finite amplitude, and that leads to the decay of a metastable state. This distinction is clear in the mean field theory, as the unstable and the metastable states are separated by a so-called spinodal line in the phase diagram. Although later development has shown that the spinodal line is not well defined and the transition is not clear cut, it is still very helpful to distinguish these two from a thermodynamic point of view. This chapter is a brief summary of the theoretical models, and a general comparison of their results with the experiments. As a number of excellent and extensive reviews have been written in the field [e.g. Gunton et al., 1983], here I will only discuss the key elements, and all the discussions will be concentrated on binary fluids from now on. First the phase diagram is presented along with the equilibrium properties, its general features can be readily understood in terms of some elementary thermodynamic arguments. And then I will discuss the thermodynamic unstability and metastability in terms of mean-field theory. Next Cahn's linear theory of spinodal decomposition is described, from which we can clearly see the difference of the early stage of decay from the unstable or metastable state. Some nonlinear modification is also mentioned, and it will be helpful later on to understand how the domain size is measured in a scattering experiment. Section 1.4 is a description on three key growth modes at later stages that have been clearly identified so far: droplet coalescence, evaporation and condensation [Lifshitz and Slyozov, 1961] and the hydrodynamic mode due to E. Siggia [Siggia, 1979]. Also in the same section, the dynamic scaling theory is discussed [Lebowitz et al., 1982], And then a general comparison between experiment and theory is presented in section 1.5. Finally, we review the recent experimental observations and a general discussion on the wetting phenomena, which provided the motivations for the present work. Temperature 3 1-PHASE REGION (Miscible) FIG. 1.1 The coexistence (solid) line and the spinodal line (dashed) in classical theory (mean field theory) for binary fluid. Two types of quenches are differentiated: (1) into the metastable state which is bounded by the coexistence and spinodal lines; (2) into unstable state bounded by the spino dal curve. In the first case the system is quenched into the metastable state by the depth ST, which corresponds to an initial supersaturation <5C= C1 CA. The phase separation eventually results in two phases, a major phase with concentration CA, and a minor phase of CB. The volume ratio of the two phases is given by the ratio 5C/(AC-5C), where AC = CB- CA. 4 1.1 Phase Diagram A generic phase diagram of binary fluids is shown in Fig. 1.1. The coexistence curve, also known as the binodal, divides it into two regions: the miscible region (one phase region or disordered phase) and the immiscible region (two-phase region or ordered phase). For binary fluids, the phase transition between the miscible phase and the immiscible phase is of second order, with the critical point at temperature Tc and critical concentration Q- It is well known that the equilibrium properties of the fluids around the critical point belong to the universality class of the 3-d Ising model, where the properties around the critical point can be determined through: the correlation length Â£ = <^o| t \ ~v, (1.1a) where the critical exponent v = 0.63, and t= (T Tc)/Tc, the reduced temperature, and the isothermal compressibility Kj ~ 111 ~y- (1-lb) with the corresponding exponent y= 1.3. All the other critical exponents can be determined once two of them are known. The qualitative features of the phase diagram can be readily understood in terms of the free energy F, the internal energy U and the entropy S: F=U-TS where T is the temperature in Kelvin. The equilibrium state has minimal free energy for given external conditions. For most fluids, the interactions between 5 the molecules are the van der Waals force, which is an attractive force due to the fluctuating induced dipole moments of the molecules. In most cases, the attractive force between the same species of molecules is much stronger than that of different species. Then the internal energy U will be minimized when the same species of molecules stay together, thereby favoring the phase separation. On the other hand, the entropy term has the temperature as a prefactor, so it is more important at high temperature. The entropy term will dominate over the internal energy above a certain temperature, the system will try to maximize its entropy and thus lower the free energy, then the fluids are miscible. At the other end, when the temperature is low, then the entropy term has minimal effects on total free energy F, and the internal energy term is more important, which is typically minimized when the like molecules stay together, so the fluids will try to keep away from each other. This explains the immiscibility at low temperature. Most binary fluid systems have a similar phase diagram as Fig. 1.1, which is known as the upper critical solution temperature (UCST) type. There are other classes of mixtures whose phase diagrams are quite different; Fig. 1.2 lists several other common types. The opposite type to UCST is called the lower critical solution temperature (LCST) and behaves as shown in Fig. 1.2b, which has the immiscible phase at high temperature. This kind of behavior is due to some specific interactions between the different species of molecules in the mixture, e.g., hydrogen bonding. Some binary fluids can even show both UCST and LCST behavior, as described in Fig. 1.2c and 1.2d. Here I just want to mention particularly the closed-loop type of phase diagram as Fig. 1.2c, which shows a miscibility gap between the two critical temperature and Tc/, because the system involved in this study Guaiacol-Glycerol-Water (GGW) shows this type of phase behavior. Temperature Temperature 6 a B d) CL, g a> H Concentration Concentration Vh 4-* a> a 6 ai H Concentration Concentration FIG. 1.2. Four types of phase diagrams of binary fluids are shown here schematically. The shaded area and beyond are the two-phase regions bounded by the coexistence curve, (a) is the most common, espcially if the fluids are nonpolar; All the other types usually involve some types of specific interaction, e.g., hydrogen bonding between the molecules. The phase boundary is determined by the delicate balance of energy and entropy due to these interactions among the fluid molecules. 7 The closed-loop phase diagram, also known as a reentrant phase diagram, is due to the hydrogen bonding among the unlike species of molecules. One of the peculiarities of the hydrogen bond is that it is highly oriented; the bond will break if it sways beyond 10 degrees away from its optimal orientation axis. Due to this feature, hydrogen bonding effectively freezes the orientational degrees of freedom of the molecules, and hence decreases the compositional entropy of the mixture. At low temperature, the entropic contribution to the free energy will be minimal, so the hydrogen bonding between the unlikes strongly favors mixing. The close loop phase diagram can be understood as follows: at high temperature the fluids are miscible and the entropy is high. In this phase, the molecules mix together and are oriented randomly with respect to each other, and the entropy is maximized. As the temperature is lowered, internal energy due to van der Waals attraction becomes more important. At some point, it will dominate over the entropy contribution to the free energy and the fluids become immiscible, because the like molecules stay together which lowers internal energy. Now if we lower the temperature even further, the hydrogen bonding comes into play. And the mixture becomes miscible again by forming hydrogen bonds between the unlike molecules if the temperature is low enough, so the fluids reenter the miscible phase. The close loop type of phase diagram can be elegantly fitted with the Walker-Vause (WV) model [Walker and Vause, 1980; Walker and Vause, 1983], which is an Ising-like lattice model, but each spin site can assume q different states. The Hamiltonian H of the model is H= 1 [K1(l-8ss)5Ci0. + K2(l-5Ssi)(l-50i0)] 11 1 1(1.2) 8 where the summation is over the nearest neighbor lattice sites. There aretwo sets of variables: sj and aj, the spin variable sf can be +1 (spin up) or -1 (spin down), which can also be designated as molecule species A or B respectively. The other site variable, cr/, can take values 1, 2, 3, q, which designates different orientational states of the molecules. If the two neighboring A and B molecules are in the same orientation state, i.e. = <7g, then the energy is Kj/ if not, the energy will be K.2- If we let Kj < fy, the Hamiltonian will simulate approximately the case of hydrogen bonding between the neighboring sites when they are in the same CT-state, and the number q is a measure of directionality of the hydrogen bond. The phase diagrams of many binary fluid systems that exhibit the close loop have been fitted successfully by the WV model [Walker and Vause, 1983; Johnston, 1983], including the GGW mixture used in this study. From the molecular structures of guaiacol (2-Methoxyphenol) and glycerol (1,2,3- Propanetriol) and Water (Fig. 1.3), we can see the presence of multiple hydroxyl groups -OH, and the hydrogen bonding is a dominating factor in the mixing of the components. Strictly speaking, the system is a quasi-binary system, with a small amount of H2O used in the mixture (less than 6% of the glycerol mass). Without the presence of this small amount of water, the guaiacol and glycerol will be miscible in any respective amounts. Only when a small amount of water is added to the mixture, a miscibility gap opens up in the phase diagram. The size of the immiscibility loop depends on the amount of water added. As it is well known, glycerol is highly hygroscopic; it will readily absorb the moisture from the surrounding air if the container is left open. On the other hand, the solubility of guaiacol in water is rather low; 70 ~ 80 cm3 of water can only dissolve 1 g of guaiacol at room temperature. Therefore, most of the water 9 present will dissolve in glycerol, and glycerol-water effectively forms one component in the quasi-binary mixture, with guaiacol as the other . OCH3 Guaiacol (2-Methoxyphenol) CH2 OH CH OH CH2 OH Glycerol (1,2,3-Propanetriol) FIG.1.3 The molecular structures of guaiacol and glycerol. Due to the presence of the hydroxyl group OH, hydrogen bonds can form between these groups of neighboring molecules. Water is also a strong hydrogen-bonding molecule. 1. 2. Mean Field Theory of Metastabilitv and Unstabilitv Let us go back to Fig. 1.1 and explain how a phase separation experiment is done in general. Assume a binary mixture is made up of two fluids, A and B, and the concentration of A is Cj. Initially, the fluid mixture is set at temperature T which is close to the coexistence curve but in the one-phase region. The fluid is miscible and is optically homogeneous. The temperature is suddenly and quickly changed to Tf, 10 inside the two-phase region. The sudden change in temperature is called a quench. Once inside the two-phase region, the fluid is either unstable or metastable. It will demix into the final equilibrium (stable) phases, which will have concentrations Ca and Cg that are at the coexistence curve at the temperature Tf. The final state of the fluid mixture is phase separated, and it consists of 2 phases of fluids: one is A-rich (concentration Ca), the other A-poor or B-rich (concentration Cg). To study the kinetics of phase-separating fluids, various experimental techniques can be used to monitor the growth of the domains after a quench. The phase separation process starts as soon as the fluid is inside the two- phase region of the phase diagram. It is a nonequilibrium process because it concerns the transition from an unstable or metastable state back to equilibrium. Phase-separation kinetics is the study of this process of how the initially homogeneous fluid mixture demixes into heterogeneous phases. Some technical jargon should be made clear here. The temperature difference AT = I Tf- T, I is called the quench step, but a more relevant quantity is <5T = Tco Tf, the quench depth (see Fig. 1.1); Tco is the temperature at which the fluid first crosses the coexistence curve. If the fluid is of critical concentration Cc, then Tc0 = Tc, the critical temperature, as shown in Fig. 1.1 by quench (2). In the mean-field-approximation picture, the two-phase region is further divided into two regions: unstable and metastable. The boundary line between the two is shown as the dashed line in Fig. 1.1. This line is the so-called spinodal line. The fundamental assumption in the mean-field approximation is the existence of the free energy F as an analytic function of thermodynamic variables temperature T and concentration C, even inside the two-phase region. This may not to be true in general, because these are not equilibrium states inside the two- phase region. But as a starting point, the mean field picture is still very useful in 11 understanding the phase transition, and even some general features in the phase separation. The mean field theory starts with the Ginzburg-Landau-like free energy functional F (C, T), where C is the concentration in the case of binary fluids, and T is the temperature. In general, an order parameter y/ corresponding to the phase transition will take the place of the concentration C, e.g., the magnetization M in the case of ferromagnetic-paramagnetic transition. The Ginzburg-Landau free energy functional F(C,T) (1.3) where f (C,T) = hi (C C0)2 + Ui(C C0)4 2 4 (1.4) and a = A(T-TC), A > 0, k, u are positive constants. The actual thermodynamic equilibrium state is determined through the minimization of F: = 0, as the equilibrium means the lowest energy state possible, therefore most stable. Notice that a changes from positive to negative as T is varied from the upper side of Tc. Then the shape of f(C,T) as a function of C changes as T crosses Tc: At T > Tc, F has only one minimum at C = Co; (see Fig. 1.4 for details). At T < Tc, Co is no longer an absolute minimum, instead two minima appear at (1.5) The coexistence curve (binodal line) is determined by the equation: 12 P = and we can introduce a susceptibility (1.6) (1.7) and the line defined by x = 0 is the spinodal line. Inside the two-phase region of the phase diagram (see Fig. 1.1), the area between the binodal and spinodal line corresponding to the metastable state, since with x >0 ,a small fluctuation of the concentration CO) will increase the energy, an energy barrier which needs to overcome before the system can decompose and phase separation can occur. In contrast, the area inside the spinodal line corresponds to the unstable state, because with x < 0 any spontaneous small fluctuation will drive the system to lower free energy, and the system can phase separate without having to overcome an energy barrier. The phase separation process is thus the decay of the unstable or metastable state into equilibrium states. Due to the difference in metastability and unstability, the phase separation proceeds differently in general. If the fluid is quenched into the unstable region of its phase diagram, the phase separation is said to proceed via spinodal decomposition. On the other hand, it proceeds via nucleation and growth after being quenched into the metastable state. From the phase diagram, we can see that the unstable region usually has the initial concentration fairly close to the critical concentration Cc, while the metastable part corresponds to concentration further away from Q, for off-critical quenches. Free Energy Density f(C,T) 13 Concentration FIG. 1.4 The Ginzburg-Landau form of the free energy density f(C, T) (eq. (4)) at tempertures around critical temperature Tc. At T > Tc,f has only one minimum at Cg. But at T < Tc, double minima are present, indicating two phases. The coexistence curve (binodal line) can be derived from f(C,T) by eq.(1.6) and the spinodal line by the susceptibility X = 0 (see eq.(1.7)). 14 It is important to determine the validity of the mean-field theory. Remember the basic assumption in the theory is that the fluctuations in the system is small compared to the mean value (statistical average). In many cases, it is not valid around the critical point of a phase transition, where the fluctuations are significant. Furthermore, the existence of a free-energy functional in a nonequilibrium state, as inside the two-phase region, is somewhat in doubt. Therefore the validity of the mean field treatment is uncertain, especially near the spinodal line. In fact, it has been shown using the renormalization group method that the spinodal line shifts as a function of the renormalization size L [Langer, 1974; Binder et al., 1978; Kaski et al., 1983]. It is now generally accepted that the spinodal line is not sharp, and the transition from metastable and unstable states is gradual. However, despite all these problems, the mean field theory still provides a simple and elegant picture about the phase transition and its dynamics. 1. 3 Cahn's Linear Theory of Spinodal Decomposition Cahn's treatment of spinodal decomposition is one of the pioneering steps toward the understanding of the general phenomena of the dynamics of phase transition. In a series of papers [Cahn, 1961; 1962; 1966; 1968], Cahn outlined a linearized theory of spinodal decomposition, and concluded an exponential growth of the order parameter occurs at the early stage following the quench into the unstable region of the phase diagram. In the case of phase separation in binary fluids, the order parameter is the local concentration C(r), which is conserved (the molecules of each species do not simply disappear.), in contrast to some other systems where the order parameter is not conserved. A simple case 15 with nonconserved order parameter is the Ising-like spin system, where spins are allowed to flip. This system can be made to simulate the binary fluid system when the spin flipping is prohibited (so-called spin-exchange models), then the number of up-spins or down-spins is conserved. The linearized theory can be generally regarded to describe the phase separation as a diffusion process. The diffusion constant D is proportional to which is the susceptibility, as defined in eq. (1.7). Inside the unstable region, where ^ < 0, D is negative, the diffusion is along the concentration gradient, therefore achieving phase separation. Conservation of the order parameter is expressed in the continuity equation: (1.8) where j is the interdiffusion current (1.9) j = M V/i (r), and M is the mobility, and /i(r) is the local chemical potential, which can be related to the free energy F : [i (r) = 8C(r) (1.10) So if we use the Ginzburg-Landau free energy functional F shown in (1.3), we get the following equation ^^- = MV2{-iC V2C + ^ } (1.11) 16 Cahn linearized this nonlinear equation about the average concentration Co, and obtained where du( r) dt = -M K V2 + u( r) (1.12) u(r) = C(r) C0 (1.13) 2 Note in the long wavelength limit, the term K V can be ignored. Then we have a diffusion equation, with a diffusion constant D = MÂ¡^-\ (1.14) \dC2 Co Fourier transform eq.(l-12) respect to space leads to the following = CO (k) u (k) dt here u (k) is the Fourier component of u(r) and oik) = -MKk2(k2 + Â£-1 1 ) \5C2 jco (1.15) (1.16) and the shape of co as a function of k is shown in Fig. 1.5. Inside the spinodal region where x is negative, co is positive for k < kc where (1.17) 17 So the long wavelength fluctuations will grow exponentially in the spinodal region. Notice co reaches its maximum at km = kc/i2 . The experimentally more relevant quantity is the structure function S (k, t) = \u (k)fj, which is the Fourier transform of the two-point correlation function. It can be measured by the scattering methods commonly used in physics (X-ray, light, etc.). Cahn's theory thus predicted an exponential growth S (k,t) = S (k,0) exp( 2 oXk) t) (1.18) Therefore at the initial stage, the spinodal decomposition should undergo an exponential growth in time in the scattering intensity for k < kc, and the intensity profile has a peak at km which does not change during this stage of growth. Due to the fast growth, the peak wave number km will soon dominate over the other length scale. As a result, km characterizes the typical size of the domain seen in spinodal decomposition experiments. The linearization can only hold at the very early stage of spinodal decomposition when the fluctuation amplitude is small, and beyond that nonlinear effects will become important. This exponential-growth stage passes too quickly to be observed experimentally in most of the physical systems such as binary fluids, with the exception of the polymer blends, where due to its high viscosity, the dynamics is slowed down tremendously. Here it is convenient to point out the difference in dynamics if the order parameter y/ is not conserved. The corresponding equation to eq.(1.12) can be 2 obtained by replacing M V in eq. (12) by M, co(k) 18 FIG. 1.5 The growth rate as predicted by the linear theory for (a) a unstable quench, (b) a metastable quench. In the linear theory, the peak which has maximum rate is wave number km, which doesn't change with time. But if nonlinear effect are incorporated, km will shift toward smaller k as time proceeds. dt = -M K V2+(-^-) Co vKr) (1.19) and the corresponding rate of growth 19 oXk) = M (1.20) Here the maximal growth occurs at k = 0 instead of at some wave number km- An important extension to Cahn's linear theory was made by Cook [Cook, 1970]. He observed it was necessary to add a noise term to eq. (1.12) to have a correct statistical description of the dynamics in alloys. The noise arises from the random thermal motion of the atoms or molecules in the system. Within the context of linear theory, the noise term does not affect the major results we have discussed so far. Attempts have been made to include the nonlinear effects. Immediately we encounter an equation that includes higher order correlation functions, and it leads to the typical hierarchy of coupled equations. So some kind of truncation approximation has to be employed. One of the simplest schemes is due to Langer, Bar-on and Miller (LBM) [Langer et al., 1975], which assumes the odd order-n two-point functions are zero, and if we only keep the first higher order correlation function S^(k), which is related to the normal, or second order, two- point correlation function S(k) (= S2(k)) as follows S4a) = 3 (m 2) s(k) (1.21) with (1.22) then the correction to the linear theory is the replacement of Co by 20 (1.23) Consequently kc will decrease as (u 2(f)) increases with time, (see eq. (1.17)), and the peak km in the scattering profile will shift toward to lower k, indicating growth of the domain structure. This coarsening behavior is qualitatively accurate for the early stage of spinodal decomposition. Another effect that is specific to fluids is hydrodynamic motion, which has not been included into consideration so far. Efforts have been made to take it into account [Kawasaki and Ohta, 1978], and it brought significant modification to the LBM results. But its agreement with experimental data is still only qualitative. Hydrodynamics will have a significant effect at the later stage of coarsening, and we will discuss it in the next section. 1.4 Later Stage Coarsening It is difficult to develop a theory for the later stage, that is the coarsening process of phase separation. The main difficulties stem from the nature of nonequilibrium and nonlinearity in the phase separation process itself. So far our understanding at the later stage of phase separation remains qualitative. Most of the theories attempt to find universal growth laws and mechanisms in some simple scenarios. But in real systems there is no way to make these distinctions, many of these mechanisms are at work at the same time. The following discussion in section 1.4.3 is largely based on the paper by Siggia [Siggia, 1979], who discussed the key coarsening mechanisms responsible at the later stage of phase separation, especially the hydrodynamic effects. 21 The division of the phase separation process into early and later stages is somewhat vague. Qualitatively the later stage starts after domains of two distinct phases have formed, i.e., the concentrations of the domains have reached Ca and Cg, respectively, so the driving force is mainly derived from the minimization of the domain wall, namely the reduction of the interface area and thickness. 1.4.1 Droplet Coalescence Droplet coalescence, also known as coagulation, is the recombination process of small droplets (usually of liquids) upon encounter. As a result, larger droplets form as more and more small ones bump into each other and recombine into larger ones. It is readily observed in colloids and aerosols, in which aggregates form. In binary fluids, we can imagine the minor phase may form a suspension of small droplets in the medium of the majority phase at later stage, (in the case of Fig. 1.1, the majority phase is the A-rich phase with concentration Ca, and the minor phase is B-rich with concentration Cg). Small droplets have larger surface-to-volume ratio compared to larger ones, so the energy can be reduced by recombining into large droplets. Consider a droplet with radius Ri, fixed at the origin and surrounded in a mist of droplets with radius R2, with number density n0. The motion of the droplets are Brownian, i. e., diffusive. And the number of droplets that collide into the drop in the origin and thus recombine can be calculated through the diffusion equation: = nli ir? \ dt r23r| dr I (1.24) 22 with the boundary conditions n =n0 n = 0 n =n0 The number of collisions per unit time as for r>Ri+R.2 at t = 0 for r = Ri + R2 t > 0, (1.25) for r > can be calculated (see, e.g., Levich, 1962) I = 4 n (Ri + R2 ) D2n0, (1.26) where D2 is the diffusion constant of droplets with radius R2, which are related through the Einstein relation D2 = ^ (1.27) 6 n pvR2 where p is the density, v is the kinetic viscosity of the medium, and ks is the Boltzmann constant. In reality the droplet at the origin is moving as well; then the number of collision per unit time per volume is 11,2 = 4 n (Rj + R2 ) (D2 + Di)nin0, (1.28) 23 Assuming that droplets are approximately monodisperse at a given moment, then the number density will decrease due to the recombination as = -16 nDRn2 dt (1.29) where we have set Dj = D2 = D, and Ri = R2 =R. From eq. (1.27), (D R) is independent of the time, we can find R3= 12 (DR) vt (1.30) where v is the volume fraction of the droplets. We get from this simple model of droplet coalescence that the droplet size grows as tJl3 approximately. In the above simplistic droplet coalescence model, the droplets are treated as free particles and the medium is static, and all the interactions between the droplets are neglected. The reality is that both medium and the droplets are fluids, and they are undergoing hydrodynamic motion. At the level of droplets, they have to squeeze aside the liquid along the way upon approach. We can imagine that will slow down the recombination rate of droplets. In his paper [Siggia, 1979], Siggia took into account both the effects of the hydrodynamics and the van der Waals forces between the droplets. He found the correction to eq. (1.30) as following 3 (DR) v t log(R) (1.31) So at the later time R still grows as t1!3 asymptotically. 24 In the above argument we have explicitly assumed the spherical droplets, which means that the above coalescence model can only apply when the volume fraction of the minor phase v is small, so the minor phase forms the isolated droplets. In this picture, the process is distinctively nucleation and growth. However, it has been argued [Voorhees and Glicksman, 1984; Huse, 1986] that the R ~ fll3 law still applies, when the morphology is far from spherical, as in the case of spinodal decomposition. 1.4.2. Lifshitz Slvozov Theory As mentioned before, the Lifshitz-Slyozov (LS) theory [Lifshitz and Slyozov, 1961] is one of the few well established results in the field of phase transition kinetics. The change from metastable to stable phases occurs as a result of fluctuations, which form a new phase or nuclei out of the original homogeneous medium. There are two main factors contributing to the energy of these nuclei, and thus determine their stability. They are the free energy of the nuclei and the surface energy due to the creation of the interface. The former is negative in the metastable state, because the nuclei usually have energy of the equilibrium state, and it is proportional to the volume of each nucleus. On the order hand, the creation of an interface always costs energy. It is positive and proportional to the surface area of the individual nucleus. The competition between the two terms results that those nuclei smaller than a critical radius Rcr unstable, due to their lower volume-to-surface area ratio, and thus tend to evaporate, while those larger than RCr are stable and thus grow larger. The larger nuclei grow at the expense of the smaller ones, thus Lifshitz-Slyozov theory is also known as the evaporation-condensation mechanism. 25 The coarsening process is still governed by diffusion. In the limit of small supersaturation, the diffusion gradient will be small. And we can use the steady state approximation to the diffusion equation V2 C = 0 (1.32) with appropriate boundary conditions. From the solution, one can obtain the growth rate equation (see, e.g., Lifshitz and Pitaevskii, 1981) dJL D iC(i-RcA (1.33) dt R c R 1 where 8C(t) = C(t) is the supersaturation, and AC= Cg C^. If the radius of the droplet R > Rcr, then dR/dr > 0, it grows. And if R < Rcr, then dR/dt <0, it shrinks and eventually evaporates. The critical radius Rcr is related to the supersaturation through Rcr(t)= cc/SC(t) (1.34) where a is a constant proportional to the surface tension a The distribution function f(R,t) satisfies the continuity equation (1-35) where v(R) is the radial velocity, given by the eq. (1.34), and f(R)dR is the number density of nuclei with radius between R and R + dR. And finally, we have the conservation of the solute 26 ^0.'..y.(O=4jrR3N (136) AC 5C(t) 3 where <5Co is the initial supersaturation, N is the number of the nuclei droplet per unit volume and ^-R 3 N is the average volume of the nuclei.. Lifshitz and Slyozov were able to obtain the asymptotic solution to the above coupled equations. Besides the distribution function f(R,t), they found the growth law R \t) = Da t 9 (1.37) This LS theory is only strictly true at small supersaturation, when the volume ratio of the minor phase to the major phase is small. With increased volume ratio, significant deviation from LS theory has been found. Hence, unlike the previous model of droplet coalescence, LS mode of growth is specific to the decay of metastable state, i. e., to the nucleation and growth. 1.4.3. Hydrodynamic Growth Mode in Concentrated Mixture [E. Siggia, 19791 The observation of phase separation of binary fluids reveals two types of distinct morphologies at later stage. One type is usually seen in systems with small volume ratio, so there is clearly a minor phase and a major phase. In this case, the minor phase grows as isolated spherical droplets in the medium of the major phase. However, in the systems where the volume ratio of the two phases is comparable, as in spinodal decomposition of a near critical concentration, a 27 different type of morphology emerges: the structure of a jumbled network of interpenetrating tubes. An idealization of the structure is two interconnecting 3- d networks of A and B phases, like those that have been clearly identified in the amphiphilic systems. This change of morphology is gradual as the volume ratio is increased. At the later stage of growth, we expect that two distinct phases have formed that and the interface between the two phases is well defined and has a definite surface tension, cr. In the concentrated mixture, the two phases exhibit the morphology of an interpenetrating tube-like network. Siggia has pointed out that the tube-like structure is unstable against pinching. This can be readily understood from the Laplace formula (1.38) where p is the pressure difference between two sides of the tube, and rj and are the principal radii of curvature at a given point on the surface of the tube. As shown in Fig. 1.6, when a tube is pinched at a given point, the pressure p at the point will increase due to the decrease of r's, that will push the fluid away from the point of pinching, and decrease the radius even further. Siggia proposed a novel mechanism of growth due to the instability of the tube-like structure in binary fluids. Image a long wavelength disturbance along the axis of a tube of radius R, where the wavelength / R. According to eq. (1.38), it will lead a pressure gradient ~ o/R l along the tube axis, that will tend to transfer the liquid from the neck to the bulges. The corresponding average velocity due to the Poiseuille flow is 28 v ~ 0.1 o R/l rÂ¡ (1.39) Take this as the rate of growth dR/dt, then R ~ (0.1 a/rÂ¡) t (1.40) The prefactor 0.1 is only based on rough estimations, and r] is the shear viscosity of the fluid. As a results of the hydrodynamic flow, the coarsening (a) (b) (c) FIG. 1.6 The tube-like structure is unstable against the pinching, which will lead to the eventual breakdown of the tube, (a) shows a section of tube which forms the interface between the two phases; (b) As the tube is pinched, the pressure inside the tube at the pinching site increases according to eq. (38), which consequently induce a pressure gradient that push the fluid away from the pinching site (as indicated by the arrows), and that further shrinks the tube, and eventually the tube breaks down, as shown in (c). 29 induces the growth to be linear with time. It is not essential to assume tube-like structure in the above argument. At the concentrated mixture, any kind of nonflat interface may induce this type of coarsening. The above mechanism is observed in the spinodal decomposition of binary fluid. Up to now, the general consensus on the process is that early on the diffusive mode dominates, i.e., the domain grows as fl/3, when the interface may not yet be well defined and the hydrodynamic growth mechanism by surface tension is not possible. Gradually there is a crossover from the diffusive growth to the faster hydrodynamic growth regime, and domain size grows as t. In the nucleation and growth process, the surface tension does not come into play, and consequently the growth should be diffusive for all times. 1.4.4. Scaling Hypothesis for Structure Function at the Later Stage The renormalization group has brought a revolutionary leap to our understanding of critical phenomena and phase transitions. At the basic level, it is recognized that the correlation length | is divergent at the critical point. As a result around the critical point, the only relevant length scale is Â£, and at length scales L the physical system is scale invariant. Or simply put, the system looks self-similar at any scale L exactly at critical point, where Â£ - >. Then the thermodynamic functions scale around the critical point with respect to j. And these functions only depend on some very general properties of the Hamiltonian of the system, like the symmetry of the system and the space dimensions of the system and order parameter, but not on the details of the interactions in the system. This enables us to classify phase transitions into different broad universality classes, in which all the systems in the same class have the same properties around the critical point. 30 So far the quantities involved are the equilibrium quantities such as the heat capacity, susceptibility, etc. It is tempting to extend the same type of argument into the dynamic aspect of the phase transitions. But so far it has proven to be much more difficult. The dynamic properties of system seem to be more system specific then the static quantities around the critical point. However, it is still desirable to apply similar concepts to the dynamic aspects of the criticality, which is the so-called dynamic scaling hypothesis. The most important function in the kinetics of phase-separating is the 2- point structure function S(k,t). The basic idea of the dynamic scaling hypothesis in this case is as follows: After an initial period of transient time following the quench, a characteristic length scale l(t) is well established which represents the average size of the domain structure. The l(t) is the only relevant length in the problem, and it plays the similar role as the correlation length Â£ in the critical phenomena. Then the structure function S(k,t) is scale invariant if all its lengths are scaled relative to l(t). The definition of the characteristic length scale l(t) may vary. For example, in experiment where S(k,t) is measured, it is convenient to choose the inverse of the peak position, k^ft). In the case of computer simulation studies, kj'1 is usually used, where kj is the first moment of the structure function S(k,t). In this study, we will stick to the peak position km-J(t). Let us first introduce a normalized structure function S(k,t): S(k,t) S(k,t) 5>2s(Jt,o k so that ^ k 2S(k,t) is independent of time. k 31 Then the scaling hypothesis assumes the renormalized structure function has the following form S(k,t)= k^F(k/km) (1.41) where d is the dimensionality of the system. The functional form of F(x) is universal to some degree. In the phenomenological theories of Furukawa [Furukawa, 1977; 1978; 1979; 1985.], a family of functions F(x) is proposed (1 + y/2) x2 y! 2 +x2+r (1.42) where the constant yis related to the dimension d: (1) Critical quench ,y= 2d; (2) Off-critical quench, y= d+1. As a result of the scaling hypothesis (1.42), the time t only enters the structure function through km(t). Otherwise the structure function S(k,t) doesn't depend on time explicitly. In addition, if we let k= km in equation (1.42), we will have S(km) ~kj (1.43) If plotting on log-log axes, it is a straight line with slope -d. Another straightforward result from eq. (1.41), if it is true, is that the ratio of the second moment of the structure function, kzft), to the square of the first moment kj(t), r(t) = kht) (1.44) 32 should be independent of time t. The scaling behavior was noticed first in the computer simulations of phase separation process in Ising-like lattice models, where the structure function satisfies (1.42) at later stage. In the simple binary fluids, it has been shown experimentally that it holds approximately. But the function form F(x) can vary with temperature, quench depth, and concentrations. We should only take (1.42) as a first order approximation, as various type of corrections may be present in real systems. I have summarized the key mechanisms related to the phase separation in the binary fluids. Rather than attempting to give an exhaustive list of the results, I chose to concentrate on several key elements and well established models instead. In terms of results, I have ignored most of the contributions from the computer simulation studies, except just to say that most of the results are consistent with the experiments. Our understanding of the phase separation process is still very limited, e.g., we are unable to calculate some of the most important and basic quantities like the correlation functions and thus the structure factor, which can be directly measured experimentally. The intrinsic problems about the process are its nonlinearity, which we still do not know how to deal with analytically, and the fact that it is a nonequilibrium process. Most of our current knowledge in this field is qualitative, restricted to the certain modes of growth laws. It is not surprising to see that new phenomena still being discovered over the years. The phenomena of spinodal decomposition and nucleation and growth are far richer and complex than the current models have suggested. 33 1.5 Experimental Results As I have mentioned, the phase separation phenomena have been studied in a variety of systems. Among these, the most detailed studies have been conducted in alloys, binary fluids and polymer blends. Here we will concentrate on the results from binary fluids, and furthermore limit our discussion mostly to the spinodal decomposition process in critically quenched systems. The binary fluid systems used most in the phase separation experiments are 2, 6 lutidine water (L-W), and isobutyric acid water (I-W), as pioneered by Goldburg and coworkers [Chou and Goldburg, 1979; 1981], and Knobler and coworkers [Wong and Knobler, 1978; 1979; 1981; Knobler and Wong, 1981]. Later the French group headed by Beysens [Guenoun et al., 1987] used a quasi-binary system, deuterated cyclohexane (D-cyclohexane) cyclohexane methanol, which has the advantage of density matching between methanol and the D-cyclohexane -cyclohexane mixture of the right proportion, thereby minimizing the effect of gravity. All three groups used time-resolved light scattering to probe the evolution of the phase separation. I will discuss the details of the techniques later. Following a temperature or pressure quench of the system into the unstable state, the mixture will turn milky and thus starts to scatter light strongly. Because fluids are isotropic, the scattered light forms a ring, known as the spinodal ring, which brightens and decreases in its diameter as time proceeds. Experimentally the scattered light intensity profile I(cj,t) is measured, where cÂ¡, the momentum transfer, is a function of the scattering angle 6. We can determine the ring diameter c\m(t) and the peak intensity l(cjmrt). When the multiple scattering effect is small and can be ignored, is proportional to the structure function S(q,t), 34 then cÂ¡m-1 represents the average length scale of the structures growing in the phase separating mixture, and therefore we can directly compare with theories. If the quench depthes are well within the critical region, it is natural to use the scaled variables k and t, k = q Â£ and r = Df/Â£2 (1.45) where Â£ and D are the correlation length and diffusion coefficient, respectively. In these dimensionless variables, the evolution of km vs. r falls remarkably onto a single curve for quenches of different depths. Experimental results from different binary systems showed excellent agreement [Goldburg, 1983]. In the two opposite limits km~ 1 and Jtml the growth can be approximated as a power law: km =Az~b and the kinetic exponents are b = 0.30.1, for 0.6 > k,> 0.3, b = 1.1 0.1 for 0.1 >ltm> 0.08. There is a gentle crossover between the two limits. This is just what the theories predicted for the coalescence process. In the beginning the phase separation is diffusive, and the length scale L grows as L ~ t1/3. At later stage when interfaces are well developed, the interfacial tension will induce the hydrodynamic flow, and that will speed up the coarsening process, and the 35 growth law approaches L ~ t, just as pointed out by Siggia. However, it should be pointed out that the agreements are only qualitative, there are large discrepancies in the estimation of the constant A between the theories and experiments. In the simple binary fluid systems, the linear regime in early stage as predicted by Cahn has not been observed experimentally. Only in the polymer blends do we have convincing evidence of the existence of this linear regime, where the structure function grows exponentially with time [Bates and Wiltzius, 1989; Hashimoto, 1988;]. Evidently the high viscosity of the polymeric systems slows down the dynamics of phase separation significantly, which makes it feasible to study the early stage of phase separation. A more difficult comparison between experiments and theories is in peak intensity behavior and the structure function S(krt). Apart from linear Cahn theory, there are no satisfactory calculations up to now which can be used to fit the experimental results, especially at the later stage. However, most of the experimental results were tested to see if they satisfy the dynamic scaling hypothesis (eq. (1.42)). (see Â§ 1.4.4). Define a normalized structure function S(k,t) in terms of the scattering intensity as S(k,t) = Kk,t) The range of integration covers most of the ring area, where the peak in I(k,t) is centered. The S(k,t) defined in such way was shown to follow roughly the scaling hypothesis (eq. (1.42)), especially at the later stage when r> 10, with the scaling 36 function F(q/qm(t)) exhibiting no time dependence. But generally the function form F(x) is believed not to be universal, it shows some dependence on the concentrations of the samples, and even on the quench depth. Later experiments [Hashimoto et al., 1986a; 1986b; Izumitani et al., 1990; Takenaka et al., 1990] on polymer blends exhibit similar behaviors. From the scaling hypothesis (eq. (1.42)), the peak intensity I(km,t)~ S(km,t) = kj F(l) where d=3 for bulk samples. At later stage, km ~ r'1 for critical mixtures, and therefore I(km, x) ~ r3. This power law was confirmed in experimentally. The exponential growth predicted in Cahn's theory was not observed, the LBM theory and extensions due to Kawasaki and Ohta, who included hydrodynamic interaction, have some qualitative features similar to the experimental observation, but did not fit the data properly. In general, it can be said that those results are consistent with the theoretical results although there might still be intricacies that need to be clarified. 1.6 Wetting Phenomena and Phase Separation Wetting is a phenomenon associated with the interfaces between different phases of matter, of which the most familiar is the contact angle, e.g., a sessile liquid drop on a solid substrate (see Fig 1.7). At equilibrium the contact angle 6 is determined by the surface tensions a of different phases through the Young equation: 37 cos e =vs-OLS' glv (1.46) where the subscript V, L, S stand for the three phases, vapor, liquid and solid (substrate) respectively. Fig. 1.7. Partial and complete wetting: (a) Partial wetting, the contact angle 6 is finite, the liquid phase form isolated droplets on solid substrate; (b) Complete wetting, angle 6=0. the liquid phase form a continuous layer of film on the substrate, separating the vapor and the substrate. The similar phenomena can be observed in binary fluids when they are fully phase separated. Now instead of the vapor and the liquid phase, we have two liquid phases a and /}, in contact with a solid substrate, which is usually the container wall. The Young equation still applies, and the contact angle 6 can be measured at the edge of the meniscus as shown in Fig. 1.8. From the Young equation, we can see that |as" tfps| ^ ap ' (1.47) for any finite contact angle 6, where the subscript s stands for the substrate or solid phase. This case is usually called the partial wetting. If the equality holds, i. e., |as aps| = aap, (1-48) then the contact angle must be zero. In this case, it is more energetically favorable to have the two phases with the highest surface tension physically separated by a layer of the third phase, as shown in Fig. 1.8b. This case is known as complete wetting. To be more concrete, we assume cas > aps, then the Â¡5 phase will be sandwiched between the a phase and the substrate, and we will say that the /? phase wets the substrate. Fig. 1.8 Wetting in binary fluids with the container wall; (a) Partial wetting in fully phase separated binary fluids, the finite contact angle 6 can be measure at the edge of the meniscus; (b) Complete wetting: a layer of phase (5 completely covers the substrate, insulating phase a from it. 39 The interests in wetting phenomena increased significantly after Cahn [Cahn, 1977] pointed out that there is always a phase transition from partial to complete wetting upon the approach to a critical point. His argument was simple and intuitive, based on the thermodynamics of the phase transition. Upon approach to the critical point inside the two-phase region, the interfacial tension a,ap vanishes as the inverse square of the bulk correlation length Â£ i.e.f Oaf) ~ ~f2v 0-49) with v = 0.63 is a universal exponent, and the t is the reduced temperature. On the other hand the difference in surface tension Ous Ops should vanish as the order parameter close to the substrate (in the case of vapor and liquid, the density difference between the two phases), i. e., it goes to zero as t^, with = 0.8. Then according to Young's equation (1.46), the contact angle cos 6 ~ t ^'2v ~ t'05, (1.50) thus it will reach unity as the critical point is approached. That point marks the transition from partial to complete wetting, and it is known as the wetting temperature Tw. In the case of binary fluids, in the two phase regions of the phase diagram, there is a temperature Tw that marks the boundary. In the last decade we have seen remarkable advances in our understanding about the wetting phenomena, especially on the wetting transition. Most of the effects have been concentrated on the equilibrium properties. But gradually the dynamic aspect of the wetting effect started to catch certain attentions among physicists, especially in the topics of liquid or polymer spreading [Chen and Wada, 1989; Heslot et al., 1989]or dewetting 40 [Redon et al., 1991] on solid substrates The study of the effect of wetting on the dynamics of phase separation started relatively recently. Because usually Tw is at least several degrees Celsius away from the critical point, the final state in the quench experiments will be within the complete wetting regime. As a result, there is a complete wetting layer of the preferentially adsorbed phase next to the solid substrate (usually the container wall). It is of great interest to understand the growth of this wetting layer. Early on it was pointed out theoretically [Lipowsky and Huse 1986] that the thickness l(t) of the wetting layer will grow as a power law at the late stage of the phase separation if it is diffusion limited, i.e., l(t) ~fb, (1.51) where b is equal to 1/8 and 1/10 respectively for nonretarded and retarded van der Waals force. It is much slower than the coarsening process, e.g., Lifshitz - Slyozov theory, largely because after the initial wetting layer formed on the substrate, it leaves a depletion zone right next to the wetting layer. Any further growth of the wetting layer will have to diffuse through this zone, which act like an energy barrier in slowing down the process. The type of slow growth law was confirmed in some computer simulations, even though it hasn't been observed experimentally in binary fluid systems. More recently Guenoun, Beysens and Robert [Guenoun et al., 1990] undertook optical microscopy study on the morphology and domain growth in a binary fluid mixture undergoing spinodal decomposition near the container wall. Besides observing the morphology changes due to the wetting effect, they found the wetting layer thickness / grows with time roughly like t1/2. But that result should be taken with certain caution, as it is well known that the kinetic exponent extracted from microscopic observation is unreliable, (see, e. g., Chou, 41 1979). More recently, light scattering was used by Wiltzius and Cumming [Wiltzius and Cumming, 1991] on a quenched binary polymer mixture, there they discovered a new "fast" mode where the length scale L grows as f3/2 with time. This fast mode was attributed to the wetting effect of the container wall on the fluids. But up to the moment we have still no theoretical understanding that can explain the nature of this fast growth process. Our work had been motivated by that of Wiltzius and Cumming. First of all, we sought to establish the generality of the fast mode by studying a simple low-molecular-weight binary system, guaiacol-glycerol-water (GGW), and we gave a detailed characterization of the fast mode and its relations to the other physical quantities of the system. From there, we hoped to gain a deeper understanding of the phenomena which might shed light on certain aspects of phase separation processes, especially on the role of wetting. Also through these experiments, we hoped to provide interested theorists enough experimental input and incentive to develop a comprehensive theoretical model. CHAPTER 2 LIGHT SCATTERING METHOD: CONCEPTS Light scattering has been widely used nowadays in almost every discipline of natural science, particularly in physics and chemistry, where it has become one of the basic tools in studying the structures and the dynamics of many systems. Over the past century, light scattering method had enjoyed a steady development starting with the classical work of Lord Rayleigh and others. The advent of lasers as a primary light source brought a revolution to this field, during which the dynamic light scattering method was invented and widely applied to a whole range of systems. Nowadays laser light scattering, along with X-ray and neutron scattering, are the standard tools for studying the structures of various materials in science and engineering. As light scattering is the most convenient and inexpensive among the scattering methods, it is more widely available and used. Over the years, there have been excellent reviews and books published on the subject of light scattering [Berne & Pcora 1976; Chu, 1974; Chu, 1991; Kerker, 1969; van der Hulst, 1957], where its principles have been well elucidated and its applications to a varieties of physical and chemical systems outlined. Therefore, here in this chapter I will limit myself to a brief review the basic concepts and some of the common notations involving light scattering. Furthermore, in relevance to our experiments we will confine our discussion here largely to elastic light scattering. 42 43 Fig. 2.1. Scattering geometry. lg, Is, and It are incident, scattered and transmitted light intensity respectively, 6 is the scattering angle, ko and ks are respectively the incident and scattered wave vectors, whose magnitudes are 2;r/As, where As = Ao/n, with Ao and n being the wavelength in vacuo and the refractive index of the scattering medium. Fig. 2.1 shows the typical light-scattering geometry, where the incident light beam impinges on the scattering medium (shaded area). Most of the light will transmit through the medium without any scattering, which is the transmitted light with intensity It, and a small portion of light is scattered that has the intensity Is at angle 6. In this geometry we have ignored variations with respect to the azimuthal angle, which is appropriate for systems which are isotropic, like most liquids. Experimentally we can measure not only the intensities and the direction (the scattering angle), but also the frequency (energy) changes of the scattered photons, as there can be energy exchange between the scattered photon and the scattering medium. Whereas the scattered intensity can be related to the structures of particles or 44 inhomogeneities in the medium, the optical spectra reveal the dynamical motions. In the experiments we are concerned with here, we are interested mostly in the structures of domains and their growth, therefore experimentally we only measure the scattered light intensity Is and its profile, i. e., its variation over the scattering angle 6. This is known as elastic light scattering, where we are not concerned with the change of energy of the scattered photons, only their numbers (intensity). Equivalently we assume that the photon energy (frequency) isn't changed from the scattering, or the photon scatters elastically. In contrast, those light scattering experiments which also probe the energy (frequency) change of the photons are generally called dynamic light scattering. I will limit the discussion here only to elastic scattering. We want to measure the variation of the scattered light Is(6, t). Conventionally the more relevant quantity is the momentum transfer q which is defined as (2.1) q = ks ko, where and ks are ko are the wave vectors of the scattered and the incident photon, respectively, as shown in Fig. 2.1. Because the magnitudes of ks and ko are the same, and equal to 2/r/As, the magnitude of q is simply q = ^ sin(&) = 4ft n sin(@~) (2.2). Therefore, q is a function of the scattering angle 6, but it has the dimensional of inverse length. In isotropic medium, the scattered intensity Is only depends on the magnitudes of q, i. e., q. 45 The structures and dynamics of a system are characterized by its structure factor. The dynamic structure factor S(q,iu) is defined as the Fourier transform of the time-dependent correlation function G(r, t), which is (2.3) where (pit, t) is the local order parameter, and for binary fluids, (p is the local concentration C of one of the molecular species. The angular brackets denotes an equilibrium ensemble average, and <5
(2.4)
Fig. 4.1 A schematic phase diagram of GGW mixtures with different |

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KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORtTSK\ 5DQGROSK 6 'XUDQ $VVLVWDQW 3URIHVVRU RI &KHPLVWU\ PAGE 130 7KLV GLVVHUWDWLRQ ZDV VXEPLWWHG WR WKH *UDGXDWH )DFXOW\ RI WKH 'HSDUWPHQW RI 3K\VLFV LQ WKH &ROOHJH RI /LEHUDO $UWV DQG 6FLHQFHV DQG WR WKH *UDGXDWH 6FKRRO DQG ZDV DFFHSWHG DV SDUWLDO IXOILOOPHQW RI WKH UHTXLUHPHQWV IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ $SULO 'HDQ *UDGXDWH 6FKRRO PAGE 131 /' + D 81,9(56,7< 2) )/25,'$ THE KINETICS OF SURFACE-MEDIATED PHASE SEPARATION IN THE QUASI-BINARY MIXTURE OF GUAIACOL-GLYCEROL-WATER By QINGBIAO SHI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1994 Dedicated to my grandmother FENG-HWA FU ACKNOWLEDGMENTS First of all, I would like to thank my supervisor Andy Cumming for his efforts and patience throughout the whole period that I worked on the project. I was very much in a limbo when he took me on board. Starting from the ground up, I learnt from him not only the basic light scattering techniques and the setups, but also how to be an experimentalist. Furthermore, he was a constant source of ideas and encouragement which were essential to push this project through. Also, I want to express my deep thanks to Wade Robinson, our resident engineer in the lab, who had helped a great deal in putting the lab together and in keeping it running smoothly. He had to endure the almost constant pestering from me and other students to get this or that done, sometimes even just to finish other's dirty work. Without his patience and help, our lab would not have been running. Thanks are also in order to my fellow students in the lab, especially to James Ellis Teer, who has done all the silane treatments of the windows, and to Bill Rippard, who did most of the microscopy studies for the project. There are some other people in the University of Florida I would like to thank: first to the other members on my supervisory committee, Jim Dufty, John Klauder, Neil Sullivan and Randy Duran, for patiently sitting through both my oral exam and then thesis defense, and providing insightful suggestions and comments; Chuck Hooper, who was always ready to help if needed; and the fellow graduate students in the physics department, especially Mike Jones and Laddawan Rumsuwan, for their help throughout the years. On a more personal side, I want to express of deep gratitude to Richard Trogdon and Suzy Spencer, for their friendship and help, especially when I first came to this country and everything was foreign and you were an alien. Their m patience and warmheartedness were indispensable for me to get over the initial "culture shock" and the language barrier. Heartfelt thanks go to Rob and Cynthia, for their friendship over the years, even after they had moved away. Their enthusiasm and encouragement are deeply appreciated, especially when I felt down and out. Finally, to a special friend, C. J., I want to say thanks. IV TABLE OF CONTENTS page ACKNOWLEDGEMENTS iii ABSTRACT vii CHAPTERS 1 INTRODUCTION 1 1.1 Phase Diagram 4 1.2 Mean Field Theory of Metastability and Unstability 9 1.3 Cahn's Linear Theory of Spinodal Decomposition 14 1.4 Later Stage Coarsening 20 1.4.1. Droplet Coalescence 21 1.4.2. Lifshitz-Slyozov Theory 24 1.4.3. Hydrodynamic Growth Mode in Concentrated Mixture 26 1.4.4. Scaling Hypothesis for Structure Function at the Later Stage 29 1.5 Experimental Results 33 1.6 Wetting Phenomena and Phase Separation 36 2 LIGHT SCATTERING METHOD: CONCEPTS 42 3 APPARATUS 48 3.1 Light Scattering Apparatus 49 3.2 Temperature Regulation and Quench System 57 4 EXPERIMENTAL PROCEDURE 65 4.1 Sample Preparation 66 4.2 Phase Diagram of Guaiacol-Glycerol-Water Mixture 67 4.3 Sample Cell and Carrier 70 4.4 Data Acquisition and Processing 73 4.5 Treatment of Glass Surface with Trichlorosilanes 75 5 RESULTS AND DISCUSSIONS 81 5.1 Data Analysis 81 5.2 Phase Separation Kinetics 94 5.3 Gravitational Effects 100 5.4 Surface Treatments and Their Effects 103 v 5.5 Discussions 109 6 CONCLUSIONS 112 APPENDIX SAMPLE CELL CARRIER DESIGN 114 REFERENCES 116 BIOGRAPHICAL SKETCH 120 vi Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE KINETICS OF SURFACE-MEDIATED PHASE SEPARATION IN THE QUASI-BINARY MIXTURE OF GUAIACOL-GLYCEROL-WATER By Qingbiao Shi April, 1994 Chairman: Dr. Andrew W. Cumming Major Department: Physics The kinetics of phase separation in a quasi-binary mixture of guaiacol- glycerol-water has been investigated using a time-resolved light scattering method following a sudden quench into the thermodynamically unstable state of the mixture. The mixture was confined between two optically transparent windows with thickness less than 1 mm. In addition to the common growth mode in the bulk, where the average domain size, L, grows as t1/3 (where t is time), a novel growth mode was observed that the domain size L grows as ft, where b increases from 1.1 to about 1.5, depending on the quench depth. As the same novel mode had been observed earlier in a polymer mixture, it is thus established that it is generic to all classes of binary fluids. Experimental results from both the light scattering and video-microscopy suggested that the novel growth mode was closely associated with wetting phenomena of the two separating fluid phases near the glass surfaces. It was also dependent on the properties of the glass surface, which could be altered with a self-assembled monolayer of silane molecules. This experimental Vll evidence confirmed that the novel mode of phase separation was surface mediated. vin CHAPTER 1 INTRODUCTION The phenomena of phase-separation have been observed in a wide range of systems with two or more components. It has been an increasingly important process in material manufacturing and processing. Therefore comprehensive understanding is needed in both physical and chemical aspects of the phenomena. Over the last 30 years, experimental investigations have been conducted in alloys [Gaulin et al., 1987] and solid solutions [Hono and Hirano, 1987], simple binary solutions [Goldburg, 1983] and glass forming mixtures [Tomozava et al., 1970], and more recently, in liquid-polymer solutions [Sasaki and Hashimoto, 1984] and polymer-polymer blends [Hashimoto, 1988; Bates and Wiltzius, 1989], etc. On the theoretical side, efforts have been concentrated on elucidating the growth mechanisms that govern the kinetics of phase separation, especially those growth kinetics that are universal, depending only the general properties of the phase transition, e.g., the symmetries of the system and the order parameter, but not on specifics of the system. Significant progress has been achieved so far in our understanding of the kinetics of phase separation, both in experiment and theory, especially in the last two decades. In classical theories, phase separation proceeds either through spinodal decomposition or through nucleation and growth [see, e.g., Gunton et al., 1983]. The former leads to the decay of an thermodynamically unstable state, through long wavelength fluctuations of infinitesimal amplitude. The latter is 1 2 due to the instability against localized (droplet-like) fluctuations with finite amplitude, and that leads to the decay of a metastable state. This distinction is clear in the mean field theory, as the unstable and the metastable states are separated by a so-called spinodal line in the phase diagram. Although later development has shown that the spinodal line is not well defined and the transition is not clear cut, it is still very helpful to distinguish these two from a thermodynamic point of view. This chapter is a brief summary of the theoretical models, and a general comparison of their results with the experiments. As a number of excellent and extensive reviews have been written in the field [e.g. Gunton et al., 1983], here I will only discuss the key elements, and all the discussions will be concentrated on binary fluids from now on. First the phase diagram is presented along with the equilibrium properties, its general features can be readily understood in terms of some elementary thermodynamic arguments. And then I will discuss the thermodynamic unstability and metastability in terms of mean-field theory. Next Cahn's linear theory of spinodal decomposition is described, from which we can clearly see the difference of the early stage of decay from the unstable or metastable state. Some nonlinear modification is also mentioned, and it will be helpful later on to understand how the domain size is measured in a scattering experiment. Section 1.4 is a description on three key growth modes at later stages that have been clearly identified so far: droplet coalescence, evaporation and condensation [Lifshitz and Slyozov, 1961] and the hydrodynamic mode due to E. Siggia [Siggia, 1979]. Also in the same section, the dynamic scaling theory is discussed [Lebowitz et al., 1982], And then a general comparison between experiment and theory is presented in section 1.5. Finally, we review the recent experimental observations and a general discussion on the wetting phenomena, which provided the motivations for the present work. Temperature 3 1-PHASE REGION (Miscible) FIG. 1.1 The coexistence (solid) line and the spinodal line (dashed) in classical theory (mean field theory) for binary fluid. Two types of quenches are differentiated: (1) into the metastable state which is bounded by the coexistence and spinodal lines; (2) into unstable state bounded by the spinoÂ¬ dal curve. In the first case the system is quenched into the metastable state by the depth ST, which corresponds to an initial supersaturation <5C= C1 - CA. The phase separation eventually results in two phases, a major phase with concentration CA, and a minor phase of CB. The volume ratio of the two phases is given by the ratio 5C/(AC-5C), where AC = CB- CA. 4 1.1 Phase Diagram A generic phase diagram of binary fluids is shown in Fig. 1.1. The coexistence curve, also known as the binodal, divides it into two regions: the miscible region (one phase region or disordered phase) and the immiscible region (two-phase region or ordered phase). For binary fluids, the phase transition between the miscible phase and the immiscible phase is of second order, with the critical point at temperature Tc and critical concentration Q- It is well known that the equilibrium properties of the fluids around the critical point belong to the universality class of the 3-d Ising model, where the properties around the critical point can be determined through: the correlation length Â£ = <^o| t \ ~v, (1.1a) where the critical exponent v = 0.63, and t= (T - Tc)/Tc, the reduced temperature, and the isothermal compressibility Kj ~ 111 ~y- (1-lb) with the corresponding exponent y= 1.3. All the other critical exponents can be determined once two of them are known. The qualitative features of the phase diagram can be readily understood in terms of the free energy F, the internal energy U and the entropy S: F=U-TS where T is the temperature in Kelvin. The equilibrium state has minimal free energy for given external conditions. For most fluids, the interactions between 5 the molecules are the van der Waals force, which is an attractive force due to the fluctuating induced dipole moments of the molecules. In most cases, the attractive force between the same species of molecules is much stronger than that of different species. Then the internal energy U will be minimized when the same species of molecules stay together, thereby favoring the phase separation. On the other hand, the entropy term has the temperature as a prefactor, so it is more important at high temperature. The entropy term will dominate over the internal energy above a certain temperature, the system will try to maximize its entropy and thus lower the free energy, then the fluids are miscible. At the other end, when the temperature is low, then the entropy term has minimal effects on total free energy F, and the internal energy term is more important, which is typically minimized when the like molecules stay together, so the fluids will try to keep away from each other. This explains the immiscibility at low temperature. Most binary fluid systems have a similar phase diagram as Fig. 1.1, which is known as the upper critical solution temperature (UCST) type. There are other classes of mixtures whose phase diagrams are quite different; Fig. 1.2 lists several other common types. The opposite type to UCST is called the lower critical solution temperature (LCST) and behaves as shown in Fig. 1.2b, which has the immiscible phase at high temperature. This kind of behavior is due to some specific interactions between the different species of molecules in the mixture, e.g., hydrogen bonding. Some binary fluids can even show both UCST and LCST behavior, as described in Fig. 1.2c and 1.2d. Here I just want to mention particularly the closed-loop type of phase diagram as Fig. 1.2c, which shows a miscibility gap between the two critical temperature and TcÂ¡, because the system involved in this study â€” Guaiacol-Glycerol-Water (GGW) shows this type of phase behavior. Temperature Temperature 6 aÂ» B QJ CL, g a> H Concentration Concentration Vh 4-Â» a> a s ai H Concentration Concentration FIG. 1.2. Four types of phase diagrams of binary fluids are shown here schematically. The shaded area and beyond are the two-phase regions bounded by the coexistence curve, (a) is the most common, espcially if the fluids are nonpolar; All the other types usually involve some types of specific interaction, e.g., hydrogen bonding between the molecules. The phase boundary is determined by the delicate balance of energy and entropy due to these interactions among the fluid molecules. 7 The closed-loop phase diagram, also known as a reentrant phase diagram, is due to the hydrogen bonding among the unlike species of molecules. One of the peculiarities of the hydrogen bond is that it is highly oriented; the bond will break if it sways beyond 10 degrees away from its optimal orientation axis. Due to this feature, hydrogen bonding effectively freezes the orientational degrees of freedom of the molecules, and hence decreases the compositional entropy of the mixture. At low temperature, the entropic contribution to the free energy will be minimal, so the hydrogen bonding between the unlikes strongly favors mixing. The close loop phase diagram can be understood as follows: at high temperature the fluids are miscible and the entropy is high. In this phase, the molecules mix together and are oriented randomly with respect to each other, and the entropy is maximized. As the temperature is lowered, internal energy due to van der Waals attraction becomes more important. At some point, it will dominate over the entropy contribution to the free energy and the fluids become immiscible, because the like molecules stay together which lowers internal energy. Now if we lower the temperature even further, the hydrogen bonding comes into play. And the mixture becomes miscible again by forming hydrogen bonds between the unlike molecules if the temperature is low enough, so the fluids reenter the miscible phase. The close loop type of phase diagram can be elegantly fitted with the Walker-Vause (WV) model [Walker and Vause, 1980; Walker and Vause, 1983], which is an Ising-like lattice model, but each spin site can assume q different states. The Hamiltonian H of the model is h= i [^(i-yVi^2(i-y(i-w] 11 1 1(1.2) 8 where the summation is over the nearest neighbor lattice sites. There aretwo sets of variables: sf and aj, the spin variable sf can be +1 (spin up) or -1 (spin down), which can also be designated as molecule species A or B respectively. The other site variable, cr/, can take values 1, 2, 3, q, which designates different orientational states of the molecules. If the two neighboring A and B molecules are in the same orientation state, i.e. = 0b, then the energy is Kjf if not, the energy will be K.2- If we let Kj < fy, the Hamiltonian will simulate approximately the case of hydrogen bonding between the neighboring sites when they are in the same CT-state, and the number q is a measure of directionality of the hydrogen bond. The phase diagrams of many binary fluid systems that exhibit the close loop have been fitted successfully by the WV model [Walker and Vause, 1983; Johnston, 1983], including the GGW mixture used in this study. From the molecular structures of guaiacol (2-Methoxyphenol) and glycerol (1,2,3- Propanetriol) and Water (Fig. 1.3), we can see the presence of multiple hydroxyl groups -OH, and the hydrogen bonding is a dominating factor in the mixing of the components. Strictly speaking, the system is a quasi-binary system, with a small amount of H2O used in the mixture (less than 6% of the glycerol mass). Without the presence of this small amount of water, the guaiacol and glycerol will be miscible in any respective amounts. Only when a small amount of water is added to the mixture, a miscibility gap opens up in the phase diagram. The size of the immiscibility loop depends on the amount of water added. As it is well known, glycerol is highly hygroscopic; it will readily absorb the moisture from the surrounding air if the container is left open. On the other hand, the solubility of guaiacol in water is rather low; 70 ~ 80 cm3 of water can only dissolve 1 g of guaiacol at room temperature. Therefore, most of the water 9 present will dissolve in glycerol, and glycerol-water effectively forms one component in the quasi-binary mixture, with guaiacol as the other . OCH3 Guaiacol (2-Methoxyphenol) CH2 OH CH OH CH2 OH Glycerol (1,2,3-Propanetriol) FIG.1.3 The molecular structures of guaiacol and glycerol. Due to the presence of the hydroxyl group - OH, hydrogen bonds can form between these groups of neighboring molecules. Water is also a strong hydrogen-bonding molecule. 1. 2. Mean Field Theory of Metastabilitv and Unstabilitv Let us go back to Fig. 1.1 and explain how a phase separation experiment is done in general. Assume a binary mixture is made up of two fluids, A and B, and the concentration of A is Cj. Initially, the fluid mixture is set at temperature Tâ€ž which is close to the coexistence curve but in the one-phase region. The fluid is miscible and is optically homogeneous. The temperature is suddenly and quickly changed to Tf, 10 inside the two-phase region. The sudden change in temperature is called a quench. Once inside the two-phase region, the fluid is either unstable or metastable. It will demix into the final equilibrium (stable) phases, which will have concentrations Ca and Cg that are at the coexistence curve at the temperature Tf. The final state of the fluid mixture is phase separated, and it consists of 2 phases of fluids: one is A-rich (concentration Ca), the other A-poor or B-rich (concentration Cg). To study the kinetics of phase-separating fluids, various experimental techniques can be used to monitor the growth of the domains after a quench. The phase separation process starts as soon as the fluid is inside the two- phase region of the phase diagram. It is a nonequilibrium process because it concerns the transition from an unstable or metastable state back to equilibrium. Phase-separation kinetics is the study of this process of how the initially homogeneous fluid mixture demixes into heterogeneous phases. Some technical jargon should be made clear here. The temperature difference AT = I Tf- T, I is called the quench step, but a more relevant quantity is <5T = Tco - Tf, the quench depth (see Fig. 1.1); Tco is the temperature at which the fluid first crosses the coexistence curve. If the fluid is of critical concentration Cc, then Tc0 = Tc, the critical temperature, as shown in Fig. 1.1 by quench (2). In the mean-field-approximation picture, the two-phase region is further divided into two regions: unstable and metastable. The boundary line between the two is shown as the dashed line in Fig. 1.1. This line is the so-called spinodal line. The fundamental assumption in the mean-field approximation is the existence of the free energy F as an analytic function of thermodynamic variables temperature T and concentration C, even inside the two-phase region. This may not to be true in general, because these are not equilibrium states inside the two- phase region. But as a starting point, the mean field picture is still very useful in 11 understanding the phase transition, and even some general features in the phase separation. The mean field theory starts with the Ginzburg-Landau-like free energy functional F (C, T), where C is the concentration in the case of binary fluids, and T is the temperature. In general, an order parameter y/ corresponding to the phase transition will take the place of the concentration C, e.g., the magnetization M in the case of ferromagnetic-paramagnetic transition. The Ginzburg-Landau free energy functional F(C,T) (1.3) where f (C,T) = hi (C - C0)2 + ^(C - C0)4 2 4 (1.4) and a = A(T-TC), A > 0, k, u are positive constants. The actual thermodynamic equilibrium state is determined through the minimization of F: = 0, as the equilibrium means the lowest energy state possible, therefore most stable. Notice that a changes from positive to negative as T is varied from the upper side of Tc. Then the shape of f(C,T) as a function of C changes as T crosses Tc: At T > Tc, F has only one minimum at C = Co; (see Fig. 1.4 for details). At T < Tc, Co is no longer an absolute minimum, instead two minima appear at (1.5) The coexistence curve (binodal line) is determined by the equation: 12 P = and we can introduce a susceptibility X'. (1.6) (1.7) and the line defined by ^ = 0 is the spinodal line. Inside the two-phase region of the phase diagram (see Fig. 1.1), the area between the binodal and spinodal line corresponding to the metastable state, since with x >0 ,a small fluctuation of the concentration CO) will increase the energy, an energy barrier which needs to overcome before the system can decompose and phase separation can occur. In contrast, the area inside the spinodal line corresponds to the unstable state, because with x < 0 any spontaneous small fluctuation will drive the system to lower free energy, and the system can phase separate without having to overcome an energy barrier. The phase separation process is thus the decay of the unstable or metastable state into equilibrium states. Due to the difference in metastability and unstability, the phase separation proceeds differently in general. If the fluid is quenched into the unstable region of its phase diagram, the phase separation is said to proceed via spinodal decomposition. On the other hand, it proceeds via nucleation and growth after being quenched into the metastable state. From the phase diagram, we can see that the unstable region usually has the initial concentration fairly close to the critical concentration Cc, while the metastable part corresponds to concentration further away from Q, for off-critical quenches. Free Energy Density f(C,T) 13 Concentration FIG. 1.4 The Ginzburg-Landau form of the free energy density f(C, T) (eq. (4)) at tempertures around critical temperature Tc. At T > Tc,f has only one minimum at Cg. But at T < Tc, double minima are present, indicating two phases. The coexistence curve (binodal line) can be derived from f(C,T) by eq.(1.6) and the spinodal line by the susceptibility X = 0 (see eq.(1.7)). 14 It is important to determine the validity of the mean-field theory. Remember the basic assumption in the theory is that the fluctuations in the system is small compared to the mean value (statistical average). In many cases, it is not valid around the critical point of a phase transition, where the fluctuations are significant. Furthermore, the existence of a free-energy functional in a nonequilibrium state, as inside the two-phase region, is somewhat in doubt. Therefore the validity of the mean field treatment is uncertain, especially near the spinodal line. In fact, it has been shown using the renormalization group method that the spinodal line shifts as a function of the renormalization size L [Langer, 1974; Binder et al., 1978; Kaski et al., 1983]. It is now generally accepted that the spinodal line is not sharp, and the transition from metastable and unstable states is gradual. However, despite all these problems, the mean field theory still provides a simple and elegant picture about the phase transition and its dynamics. 1. 3 Cahn's Linear Theory of Spinodal Decomposition Cahn's treatment of spinodal decomposition is one of the pioneering steps toward the understanding of the general phenomena of the dynamics of phase transition. In a series of papers [Cahn, 1961; 1962; 1966; 1968], Cahn outlined a linearized theory of spinodal decomposition, and concluded an exponential growth of the order parameter occurs at the early stage following the quench into the unstable region of the phase diagram. In the case of phase separation in binary fluids, the order parameter is the local concentration C(r), which is conserved (the molecules of each species do not simply disappear.), in contrast to some other systems where the order parameter is not conserved. A simple case 15 with nonconserved order parameter is the Ising-like spin system, where spins are allowed to flip. This system can be made to simulate the binary fluid system when the spin flipping is prohibited (so-called spin-exchange models), then the number of up-spins or down-spins is conserved. The linearized theory can be generally regarded to describe the phase separation as a diffusion process. The diffusion constant D is proportional to which is the susceptibility, as defined in eq. (1.7). Inside the unstable region, where ^ < 0, D is negative, the diffusion is along the concentration gradient, therefore achieving phase separation. Conservation of the order parameter is expressed in the continuity equation: (1.8) where j is the interdiffusion current (1.9) j = - M V/i (r), and M is the mobility, and /i(r) is the local chemical potential, which can be related to the free energy F : [i (r) = 8C(r) (1.10) So if we use the Ginzburg-Landau free energy functional F shown in (1.3), we get the following equation ^^- = MV2{-iC V2C + ^ } (1.11) 16 Cahn linearized this nonlinear equation about the average concentration Co, and obtained where du( r) dt = -M K V2 + u( r) (1.12) u(r) = C(r) - C0 (1.13) 2 Note in the long wavelength limit, the term X V can be ignored. Then we have a diffusion equation, with a diffusion constant D = M(-^-| (1.14) \dC2 I Co Fourier transform eq.(l-12) respect to space leads to the following = co(k) u (k) dt (1.15) here u (k) is the Fourier component of u(r) and oik) = -MKk2(k2 + |-^| ) \5C2 jo, (1.16) and the shape of co as a function of k is shown in Fig. 1.5. Inside the spinodal region where x is negative, co is positive for k < kc where (1.17) 17 So the long wavelength fluctuations will grow exponentially in the spinodal region. Notice co reaches its maximum at km = kc\Ã2 . The experimentally more relevant quantity is the structure function S (k, t) = \u (k)fj, which is the Fourier transform of the two-point correlation function. It can be measured by the scattering methods commonly used in physics (X-ray, light, etc.). Cahn's theory thus predicted an exponential growth S (k,t) = S (k,0) exp( 2 cXk) t) (1.18) Therefore at the initial stage, the spinodal decomposition should undergo an exponential growth in time in the scattering intensity for k < kc, and the intensity profile has a peak at km > which does not change during this stage of growth. Due to the fast growth, the peak wave number km will soon dominate over the other length scale. As a result, km characterizes the typical size of the domain seen in spinodal decomposition experiments. The linearization can only hold at the very early stage of spinodal decomposition when the fluctuation amplitude is small, and beyond that nonlinear effects will become important. This exponential-growth stage passes too quickly to be observed experimentally in most of the physical systems such as binary fluids, with the exception of the polymer blends, where due to its high viscosity, the dynamics is slowed down tremendously. Here it is convenient to point out the difference in dynamics if the order parameter y/ is not conserved. The corresponding equation to eq.(1.12) can be 2 obtained by replacing M V in eq. (12) by - M, co(k) 18 FIG. 1.5 The growth rate as predicted by the linear theory for (a) a unstable quench, (b) a metastable quench. In the linear theory, the peak which has maximum rate is wave number km, which doesn't change with time. But if nonlinear effect are incorporated, km will shift toward smaller k as time proceeds. M<) dt = -M K V2+(-^-) Co iiKr) (1.19) and the corresponding rate of growth 19 oXk) = - M (1.20) Here the maximal growth occurs at k = 0 instead of at some wave number km- An important extension to Cahn's linear theory was made by Cook [Cook, 1970]. He observed it was necessary to add a noise term to eq. (1.12) to have a correct statistical description of the dynamics in alloys. The noise arises from the random thermal motion of the atoms or molecules in the system. Within the context of linear theory, the noise term does not affect the major results we have discussed so far. Attempts have been made to include the nonlinear effects. Immediately we encounter an equation that includes higher order correlation functions, and it leads to the typical hierarchy of coupled equations. So some kind of truncation approximation has to be employed. One of the simplest schemes is due to Langer, Bar-on and Miller (LBM) [Langer et al., 1975], which assumes the odd order-n two-point functions are zero, and if we only keep the first higher order correlation function S^(k), which is related to the normal, or second order, two- point correlation function S(k) (= S2(k)) as follows S4(fc) = 3 (m 2) s(k) (1.21) with (1.22) then the correction to the linear theory is the replacement of Co by 20 (1.23) Consequently kc will decrease as (u 2(f)) increases with time, (see eq. (1.17)), and the peak km in the scattering profile will shift toward to lower k, indicating growth of the domain structure. This coarsening behavior is qualitatively accurate for the early stage of spinodal decomposition. Another effect that is specific to fluids is hydrodynamic motion, which has not been included into consideration so far. Efforts have been made to take it into account [Kawasaki and Ohta, 1978], and it brought significant modification to the LBM results. But its agreement with experimental data is still only qualitative. Hydrodynamics will have a significant effect at the later stage of coarsening, and we will discuss it in the next section. 1.4 Later Stage Coarsening It is difficult to develop a theory for the later stage, that is the coarsening process of phase separation. The main difficulties stem from the nature of nonequilibrium and nonlinearity in the phase separation process itself. So far our understanding at the later stage of phase separation remains qualitative. Most of the theories attempt to find universal growth laws and mechanisms in some simple scenarios. But in real systems there is no way to make these distinctions, many of these mechanisms are at work at the same time. The following discussion in section 1.4.3 is largely based on the paper by Siggia [Siggia, 1979], who discussed the key coarsening mechanisms responsible at the later stage of phase separation, especially the hydrodynamic effects. 21 The division of the phase separation process into early and later stages is somewhat vague. Qualitatively the later stage starts after domains of two distinct phases have formed, i.e., the concentrations of the domains have reached Ca and Cg, respectively, so the driving force is mainly derived from the minimization of the domain wall, namely the reduction of the interface area and thickness. 1.4.1 Droplet Coalescence Droplet coalescence, also known as coagulation, is the recombination process of small droplets (usually of liquids) upon encounter. As a result, larger droplets form as more and more small ones bump into each other and recombine into larger ones. It is readily observed in colloids and aerosols, in which aggregates form. In binary fluids, we can imagine the minor phase may form a suspension of small droplets in the medium of the majority phase at later stage, (in the case of Fig. 1.1, the majority phase is the A-rich phase with concentration Ca, and the minor phase is B-rich with concentration Cg). Small droplets have larger surface-to-volume ratio compared to larger ones, so the energy can be reduced by recombining into large droplets. Consider a droplet with radius Ri, fixed at the origin and surrounded in a mist of droplets with radius R2, with number density n0. The motion of the droplets are Brownian, i. e., diffusive. And the number of droplets that collide into the drop in the origin and thus recombine can be calculated through the diffusion equation: â€” = nli ir? dt r23r| dr I (1.24) 22 with the boundary conditions n =n0 n = 0 n =n0 The number of collisions per unit time as for r>Ri+R.2 at t = 0 for r = Ri + R2 , t > 0, (1.25) for r â€”> can be calculated (see, e.g., Levich, 1962) / = 4tt(R1+R2 )D2n0, (1.26) where D2 is the diffusion constant of droplets with radius R2, which are related through the Einstein relation D2 = â€”^â€” (1.27) 6 n pvR2 where p is the density, v is the kinetic viscosity of the medium, and ks is the Boltzmann constant. In reality the droplet at the origin is moving as well; then the number of collision per unit time per volume is h,2 =4tt(Ri+R2 ) (D2 + Di)nin0, (1.28) 23 Assuming that droplets are approximately monodisperse at a given moment, then the number density will decrease due to the recombination as = -16 nDRn2 dt (1.29) where we have set Dj = D2 = D, and Ri = R2 =R. From eq. (1.27), (D R) is independent of the time, we can find R3= 12 (DR) vt (1.30) where v is the volume fraction of the droplets. We get from this simple model of droplet coalescence that the droplet size grows as tJl3 approximately. In the above simplistic droplet coalescence model, the droplets are treated as free particles and the medium is static, and all the interactions between the droplets are neglected. The reality is that both medium and the droplets are fluids, and they are undergoing hydrodynamic motion. At the level of droplets, they have to squeeze aside the liquid along the way upon approach. We can imagine that will slow down the recombination rate of droplets. In his paper [Siggia, 1979], Siggia took into account both the effects of the hydrodynamics and the van der Waals forces between the droplets. He found the correction to eq. (1.30) as following 3 (DR) v t log(R) (1.31) So at the later time R still grows as t1!3 asymptotically. 24 In the above argument we have explicitly assumed the spherical droplets, which means that the above coalescence model can only apply when the volume fraction of the minor phase v is small, so the minor phase forms the isolated droplets. In this picture, the process is distinctively nucleation and growth. However, it has been argued [Voorhees and Glicksman, 1984; Huse, 1986] that the R ~ t1/3 law still applies, when the morphology is far from spherical, as in the case of spinodal decomposition. 1.4.2. Lifshitz - Slvozov Theory As mentioned before, the Lifshitz-Slyozov (LS) theory [Lifshitz and Slyozov, 1961] is one of the few well established results in the field of phase transition kinetics. The change from metastable to stable phases occurs as a result of fluctuations, which form a new phase or nuclei out of the original homogeneous medium. There are two main factors contributing to the energy of these nuclei, and thus determine their stability. They are the free energy of the nuclei and the surface energy due to the creation of the interface. The former is negative in the metastable state, because the nuclei usually have energy of the equilibrium state, and it is proportional to the volume of each nucleus. On the order hand, the creation of an interface always costs energy. It is positive and proportional to the surface area of the individual nucleus. The competition between the two terms results that those nuclei smaller than a critical radius Rcr unstable, due to their lower volume-to-surface area ratio, and thus tend to evaporate, while those larger than RCr are stable and thus grow larger. The larger nuclei grow at the expense of the smaller ones, thus Lifshitz-Slyozov theory is also known as the evaporation-condensation mechanism. 25 The coarsening process is still governed by diffusion. In the limit of small supersaturation, the diffusion gradient will be small. And we can use the steady state approximation to the diffusion equation V2 C = 0 (1.32) with appropriate boundary conditions. From the solution, one can obtain the growth rate equation (see, e.g., Lifshitz and Pitaevskii, 1981) dJL = D iC(i-RcA (1.33) dt R ac R 1 where 8C(t) = C(t) - is the supersaturation, and AC= Cg - Ca- If the radius of the droplet R > Rcr, then dR/dr > 0, it grows. And if R < Rcr, then dR/dt <0, it shrinks and eventually evaporates. The critical radius Rcr is related to the supersaturation through Rcr(t)= a/SC(t) (1.34) where a is a constant proportional to the surface tension o. The distribution function f(R,t) satisfies the continuity equation ^r= -Ã©lv(R)f(R>] <135) where v(R) is the radial velocity, given by the eq. (1.34), and f(R)dR is the number density of nuclei with radius between R and R + dR. And finally, we have the conservation of the solute 26 go.'..y.(0=4jrR3N (136) AC - 5C(t) 3 where <5Co is the initial supersaturation, N is the number of the nuclei droplet per unit volume and ^-R 3 N is the average volume of the nuclei.. Lifshitz and Slyozov were able to obtain the asymptotic solution to the above coupled equations. Besides the distribution function f(R,t), they found the growth law R \t) = - Da t 9 (1.37) This LS theory is only strictly true at small supersaturation, when the volume ratio of the minor phase to the major phase is small. With increased volume ratio, significant deviation from LS theory has been found. Hence, unlike the previous model of droplet coalescence, LS mode of growth is specific to the decay of metastable state, i. e., to the nucleation and growth. 1.4.3. Hydrodynamic Growth Mode in Concentrated Mixture [E. Siggia, 19791 The observation of phase separation of binary fluids reveals two types of distinct morphologies at later stage. One type is usually seen in systems with small volume ratio, so there is clearly a minor phase and a major phase. In this case, the minor phase grows as isolated spherical droplets in the medium of the major phase. However, in the systems where the volume ratio of the two phases is comparable, as in spinodal decomposition of a near critical concentration, a 27 different type of morphology emerges: the structure of a jumbled network of interpenetrating tubes. An idealization of the structure is two interconnecting 3- d networks of A and B phases, like those that have been clearly identified in the amphiphilic systems. This change of morphology is gradual as the volume ratio is increased. At the later stage of growth, we expect that two distinct phases have formed that and the interface between the two phases is well defined and has a definite surface tension, cr. In the concentrated mixture, the two phases exhibit the morphology of an interpenetrating tube-like network. Siggia has pointed out that the tube-like structure is unstable against pinching. This can be readily understood from the Laplace formula P=CT(1+1) (1.38) where p is the pressure difference between two sides of the tube, and rj and r2 are the principal radii of curvature at a given point on the surface of the tube. As shown in Fig. 1.6, when a tube is pinched at a given point, the pressure p at the point will increase due to the decrease of r's, that will push the fluid away from the point of pinching, and decrease the radius even further. Siggia proposed a novel mechanism of growth due to the instability of the tube-like structure in binary fluids. Image a long wavelength disturbance along the axis of a tube of radius R, where the wavelength / Â» R. According to eq. (1.38), it will lead a pressure gradient ~ velocity due to the Poiseuille flow is 28 v ~ 0.1 cr R/l rÂ¡ (1.39) Take this as the rate of growth dR/dt, then R ~ (0.1 a/rÂ¡) t (1.40) The prefactor 0.1 is only based on rough estimations, and 77 is the shear viscosity of the fluid. As a results of the hydrodynamic flow, the coarsening (a) (b) (c) FIG. 1.6 The tube-like structure is unstable against the pinching, which will lead to the eventual breakdown of the tube, (a) shows a section of tube which forms the interface between the two phases; (b) As the tube is pinched, the pressure inside the tube at the pinching site increases according to eq. (38), which consequently induce a pressure gradient that push the fluid away from the pinching site (as indicated by the arrows), and that further shrinks the tube, and eventually the tube breaks down, as shown in (c). 29 induces the growth to be linear with time. It is not essential to assume tube-like structure in the above argument. At the concentrated mixture, any kind of nonflat interface may induce this type of coarsening. The above mechanism is observed in the spinodal decomposition of binary fluid. Up to now, the general consensus on the process is that early on the diffusive mode dominates, i.e., the domain grows as fl/3, when the interface may not yet be well defined and the hydrodynamic growth mechanism by surface tension is not possible. Gradually there is a crossover from the diffusive growth to the faster hydrodynamic growth regime, and domain size grows as t. In the nucleation and growth process, the surface tension does not come into play, and consequently the growth should be diffusive for all times. 1.4.4. Scaling Hypothesis for Structure Function at the Later Stage The renormalization group has brought a revolutionary leap to our understanding of critical phenomena and phase transitions. At the basic level, it is recognized that the correlation length Â¿j is divergent at the critical point. As a result around the critical point, the only relevant length scale is and at length scales L Â« the physical system is scale invariant. Or simply put, the system looks self-similar at any scale L exactly at critical point, where Â£ -Â» Â«>. Then the thermodynamic functions scale around the critical point with respect to Â¿j. And these functions only depend on some very general properties of the Hamiltonian of the system, like the symmetry of the system and the space dimensions of the system and order parameter, but not on the details of the interactions in the system. This enables us to classify phase transitions into different broad universality classes, in which all the systems in the same class have the same properties around the critical point. 30 So far the quantities involved are the equilibrium quantities such as the heat capacity, susceptibility, etc. It is tempting to extend the same type of argument into the dynamic aspect of the phase transitions. But so far it has proven to be much more difficult. The dynamic properties of system seem to be more system specific then the static quantities around the critical point. However, it is still desirable to apply similar concepts to the dynamic aspects of the criticality, which is the so-called dynamic scaling hypothesis. The most important function in the kinetics of phase-separating is the 2- point structure function S(k,t). The basic idea of the dynamic scaling hypothesis in this case is as follows: After an initial period of transient time following the quench, a characteristic length scale l(t) is well established which represents the average size of the domain structure. The l(t) is the only relevant length in the problem, and it plays the similar role as the correlation length Â£ in the critical phenomena. Then the structure function S(k,t) is scale invariant if all its lengths are scaled relative to l(t). The definition of the characteristic length scale l(t) may vary. For example, in experiment where S(k,t) is measured, it is convenient to choose the inverse of the peak position, km-J(t). In the case of computer simulation studies, ki1 is usually used, where kj is the first moment of the structure function S(k,t). In this study, we will stick to the peak position km-J(t). Let us first introduce a normalized structure function S(k,t): S(k,t) S(k,t) 5>2s(Jt,o k so that ^ k 2S(k,t) is independent of time. k 31 Then the scaling hypothesis assumes the renormalized structure function has the following form S(k,t)= kÂ¿F(k/km) (1.41) where d is the dimensionality of the system. The functional form of F(x) is universal to some degree. In the phenomenological theories of Furukawa [Furukawa, 1977; 1978; 1979; 1985.], a family of functions F(x) is proposed (1 + y/2) x2 y! 2 +x2+r (1.42) where the constant yis related to the dimension d: (1) Critical quench ,y= 2d; (2) Off-critical quench, y= d+1. As a result of the scaling hypothesis (1.42), the time t only enters the structure function through km(t). Otherwise the structure function S(k,t) doesn't depend on time explicitly. In addition, if we let k= km in equation (1.42), we will have S(km) ~ktt? (1.43) If plotting on log-log axes, it is a straight line with slope -d. Another straightforward result from eq. (1.41), if it is true, is that the ratio of the second moment of the structure function, k2(t), to the square of the first moment kj(t), r(t) = k2(t) kht) (1.44) 32 should be independent of time t. The scaling behavior was noticed first in the computer simulations of phase separation process in Ising-like lattice models, where the structure function satisfies (1.42) at later stage. In the simple binary fluids, it has been shown experimentally that it holds approximately. But the function form F(x) can vary with temperature, quench depth, and concentrations. We should only take (1.42) as a first order approximation, as various type of corrections may be present in real systems. I have summarized the key mechanisms related to the phase separation in the binary fluids. Rather than attempting to give an exhaustive list of the results, I chose to concentrate on several key elements and well established models instead. In terms of results, I have ignored most of the contributions from the computer simulation studies, except just to say that most of the results are consistent with the experiments. Our understanding of the phase separation process is still very limited, e.g., we are unable to calculate some of the most important and basic quantities like the correlation functions and thus the structure factor, which can be directly measured experimentally. The intrinsic problems about the process are its nonlinearity, which we still do not know how to deal with analytically, and the fact that it is a nonequilibrium process. Most of our current knowledge in this field is qualitative, restricted to the certain modes of growth laws. It is not surprising to see that new phenomena still being discovered over the years. The phenomena of spinodal decomposition and nucleation and growth are far richer and complex than the current models have suggested. 33 1.5 Experimental Results As I have mentioned, the phase separation phenomena have been studied in a variety of systems. Among these, the most detailed studies have been conducted in alloys, binary fluids and polymer blends. Here we will concentrate on the results from binary fluids, and furthermore limit our discussion mostly to the spinodal decomposition process in critically quenched systems. The binary fluid systems used most in the phase separation experiments are 2, 6 lutidine - water (L-W), and isobutyric acid - water (I-W), as pioneered by Goldburg and coworkers [Chou and Goldburg, 1979; 1981], and Knobler and coworkers [Wong and Knobler, 1978; 1979; 1981; Knobler and Wong, 1981]. Later the French group headed by Beysens [Guenoun et al., 1987] used a quasi-binary system, deuterated cyclohexane (D-cyclohexane) - cyclohexane - methanol, which has the advantage of density matching between methanol and the D-cyclohexane -cyclohexane mixture of the right proportion, thereby minimizing the effect of gravity. All three groups used time-resolved light scattering to probe the evolution of the phase separation. I will discuss the details of the techniques later. Following a temperature or pressure quench of the system into the unstable state, the mixture will turn milky and thus starts to scatter light strongly. Because fluids are isotropic, the scattered light forms a ring, known as the spinodal ring, which brightens and decreases in its diameter as time proceeds. Experimentally the scattered light intensity profile I(cj,t) is measured, where cÂ¡, the momentum transfer, is a function of the scattering angle 6. We can determine the ring diameter qm(t) and the peak intensity l(qmrt). When the multiple scattering effect is small and can be ignored, I(q,t) is proportional to the structure function S(q,t), 34 then cÂ¡m-1 represents the average length scale of the structures growing in the phase separating mixture, and therefore we can directly compare with theories. If the quench depthes are well within the critical region, it is natural to use the scaled variables k and t, k = q Â£ and t = Dt/%2 (1.45) where Â£ and D are the correlation length and diffusion coefficient, respectively. In these dimensionless variables, the evolution of km vs. r falls remarkably onto a single curve for quenches of different depths. Experimental results from different binary systems showed excellent agreement [Goldburg, 1983]. In the two opposite limits km~ 1 and kmÂ«l the growth can be approximated as a power law: km =Az~b and the kinetic exponents are b = 0.3Â±0.1, for 0.6 > k,â€ž> 0.3, b = 1.1 Â±0.1 for 0.1 >ltm> 0.08. There is a gentle crossover between the two limits. This is just what the theories predicted for the coalescence process. In the beginning the phase separation is diffusive, and the length scale L grows as L ~ t1/3. At later stage when interfaces are well developed, the interfacial tension will induce the hydrodynamic flow, and that will speed up the coarsening process, and the 35 growth law approaches L ~ t, just as pointed out by Siggia. However, it should be pointed out that the agreements are only qualitative, there are large discrepancies in the estimation of the constant A between the theories and experiments. In the simple binary fluid systems, the linear regime in early stage as predicted by Cahn has not been observed experimentally. Only in the polymer blends do we have convincing evidence of the existence of this linear regime, where the structure function grows exponentially with time [Bates and Wiltzius, 1989; Hashimoto, 1988;]. Evidently the high viscosity of the polymeric systems slows down the dynamics of phase separation significantly, which makes it feasible to study the early stage of phase separation. A more difficult comparison between experiments and theories is in peak intensity behavior and the structure function S(krt). Apart from linear Cahn theory, there are no satisfactory calculations up to now which can be used to fit the experimental results, especially at the later stage. However, most of the experimental results were tested to see if they satisfy the dynamic scaling hypothesis (eq. (1.42)). (see Â§ 1.4.4). Define a normalized structure function S(k,t) in terms of the scattering intensity as S(k,t) = Kk,t) The range of integration covers most of the ring area, where the peak in I(k,t) is centered. The S(k,t) defined in such way was shown to follow roughly the scaling hypothesis (eq. (1.42)), especially at the later stage when r> 10, with the scaling 36 function F(q/qm(t)) exhibiting no time dependence. But generally the function form Fix) is believed not to be universal, it shows some dependence on the concentrations of the samples, and even on the quench depth. Later experiments [Hashimoto et al., 1986a; 1986b; Izumitani et al., 1990; Takenaka et al., 1990] on polymer blends exhibit similar behaviors. From the scaling hypothesis (eq. (1.42)), the peak intensity I(km,t) ~ S(kmit) = kj F(l) where d=3 for bulk samples. At later stage, km ~ r'1 for critical mixtures, and therefore I(km, x) ~ r3. This power law was confirmed in experimentally. The exponential growth predicted in Cahn's theory was not observed, the LBM theory and extensions due to Kawasaki and Ohta, who included hydrodynamic interaction, have some qualitative features similar to the experimental observation, but did not fit the data properly. In general, it can be said that those results are consistent with the theoretical results although there might still be intricacies that need to be clarified. 1.6 Wetting Phenomena and Phase Separation Wetting is a phenomenon associated with the interfaces between different phases of matter, of which the most familiar is the contact angle, e.g., a sessile liquid drop on a solid substrate (see Fig 1.7). At equilibrium the contact angle 6 is determined by the surface tensions a of different phases through the Young equation: 37 cos e =Â°vs-OLS' glv (1.46) where the subscript V, L, S stand for the three phases, vapor, liquid and solid (substrate) respectively. Fig. 1.7. Partial and complete wetting: (a) Partial wetting, the contact angle 6 is finite, the liquid phase form isolated droplets on solid substrate; (b) Complete wetting, angle 6=0. the liquid phase form a continuous layer of film on the substrate, separating the vapor and the substrate. The similar phenomena can be observed in binary fluids when they are fully phase separated. Now instead of the vapor and the liquid phase, we have two liquid phases a and /}, in contact with a solid substrate, which is usually the container wall. The Young equation still applies, and the contact angle 6 can be measured at the edge of the meniscus as shown in Fig. 1.8. From the Young equation, we can see that |aÂ«s " CTfJsl ^ ^ap ' (1.47) for any finite contact angle 6, where the subscript s stands for the substrate or solid phase. This case is usually called the partial wetting. If the equality holds, i. e., |Â°as - <5ps| = aap, (1-48) then the contact angle must be zero. In this case, it is more energetically favorable to have the two phases with the highest surface tension physically separated by a layer of the third phase, as shown in Fig. 1.8b. This case is known as complete wetting. To be more concrete, we assume cas > aps, then the Â¡5 phase will be sandwiched between the a phase and the substrate, and we will say that the /? phase wets the substrate. Fig. 1.8 Wetting in binary fluids with the container wall; (a) Partial wetting in fully phase separated binary fluids, the finite contact angle 6 can be measure at the edge of the meniscus; (b) Complete wetting: a layer of phase /J completely covers the substrate, insulating phase a from it. 39 The interests in wetting phenomena increased significantly after Cahn [Cahn, 1977] pointed out that there is always a phase transition from partial to complete wetting upon the approach to a critical point. His argument was simple and intuitive, based on the thermodynamics of the phase transition. Upon approach to the critical point inside the two-phase region, the interfacial tension oap vanishes as the inverse square of the bulk correlation length Â£ i.e., (rap ~ ~f2v , 0-49) with v = 0.63 is a universal exponent, and the t is the reduced temperature. On the other hand the difference in surface tension Gas - Gps should vanish as the order parameter close to the substrate (in the case of vapor and liquid, the density difference between the two phases), i. e., it goes to zero as t^, with = 0.8. Then according to Young's equation (1.46), the contact angle cos 6 ~ t h'2v ~ t â€˜Â°'5, (1.50) thus it will reach unity as the critical point is approached. That point marks the transition from partial to complete wetting, and it is known as the wetting temperature Tw. In the case of binary fluids, in the two phase regions of the phase diagram, there is a temperature Tw that marks the boundary. In the last decade we have seen remarkable advances in our understanding about the wetting phenomena, especially on the wetting transition. Most of the effects have been concentrated on the equilibrium properties. But gradually the dynamic aspect of the wetting effect started to catch certain attentions among physicists, especially in the topics of liquid or polymer spreading [Chen and Wada, 1989; Heslot et al., 1989]or dewetting 40 [Redon et al., 1991] on solid substrates . The study of the effect of wetting on the dynamics of phase separation started relatively recently. Because usually Tw is at least several degrees Celsius away from the critical point, the final state in the quench experiments will be within the complete wetting regime. As a result, there is a complete wetting layer of the preferentially adsorbed phase next to the solid substrate (usually the container wall). It is of great interest to understand the growth of this wetting layer. Early on it was pointed out theoretically [Lipowsky and Huse 1986] that the thickness l(t) of the wetting layer will grow as a power law at the late stage of the phase separation if it is diffusion limited, i.e., l(t) ~ t b, (1.51) where b is equal to 1/8 and 1/10 respectively for nonretarded and retarded van der Waals force. It is much slower than the coarsening process, e.g., Lifshitz - Slyozov theory, largely because after the initial wetting layer formed on the substrate, it leaves a depletion zone right next to the wetting layer. Any further growth of the wetting layer will have to diffuse through this zone, which act like an energy barrier in slowing down the process. The type of slow growth law was confirmed in some computer simulations, even though it hasn't been observed experimentally in binary fluid systems. More recently Guenoun, Beysens and Robert [Guenoun et al., 1990] undertook optical microscopy study on the morphology and domain growth in a binary fluid mixture undergoing spinodal decomposition near the container wall. Besides observing the morphology changes due to the wetting effect, they found the wetting layer thickness / grows with time roughly like t1/2. But that result should be taken with certain caution, as it is well known that the kinetic exponent extracted from microscopic observation is unreliable, (see, e. g., Chou, 41 1979). More recently, light scattering was used by Wiltzius and Cumming [Wiltzius and Cumming, 1991] on a quenched binary polymer mixture, there they discovered a new "fast" mode where the length scale L grows as f3/2 with time. This fast mode was attributed to the wetting effect of the container wall on the fluids. But up to the moment we have still no theoretical understanding that can explain the nature of this fast growth process. Our work had been motivated by that of Wiltzius and Cumming. First of all, we sought to establish the generality of the fast mode by studying a simple low-molecular-weight binary system, guaiacol-glycerol-water (GGW), and we gave a detailed characterization of the fast mode and its relations to the other physical quantities of the system. From there, we hoped to gain a deeper understanding of the phenomena which might shed light on certain aspects of phase separation processes, especially on the role of wetting. Also through these experiments, we hoped to provide interested theorists enough experimental input and incentive to develop a comprehensive theoretical model. CHAPTER 2 LIGHT SCATTERING METHOD: CONCEPTS Light scattering has been widely used nowadays in almost every discipline of natural science, particularly in physics and chemistry, where it has become one of the basic tools in studying the structures and the dynamics of many systems. Over the past century, light scattering method had enjoyed a steady development starting with the classical work of Lord Rayleigh and others. The advent of lasers as a primary light source brought a revolution to this field, during which the dynamic light scattering method was invented and widely applied to a whole range of systems. Nowadays laser light scattering, along with X-ray and neutron scattering, are the standard tools for studying the structures of various materials in science and engineering. As light scattering is the most convenient and inexpensive among the scattering methods, it is more widely available and used. Over the years, there have been excellent reviews and books published on the subject of light scattering [Berne & PÃ©cora 1976; Chu, 1974; Chu, 1991; Kerker, 1969; van der Hulst, 1957], where its principles have been well elucidated and its applications to a varieties of physical and chemical systems outlined. Therefore, here in this chapter I will limit myself to a brief review the basic concepts and some of the common notations involving light scattering. Furthermore, in relevance to our experiments we will confine our discussion here largely to elastic light scattering. 42 43 Fig. 2.1. Scattering geometry. Iq, Is, and It are incident, scattered and transmitted light intensity respectively, 6 is the scattering angle, ko and ks are respectively the incident and scattered wave vectors, whose magnitudes are 2;r/As, where As = Ao/n, with Ao and n being the wavelength in vacuo and the refractive index of the scattering medium. Fig. 2.1 shows the typical light-scattering geometry, where the incident light beam impinges on the scattering medium (shaded area). Most of the light will transmit through the medium without any scattering, which is the transmitted light with intensity It, and a small portion of light is scattered that has the intensity Is at angle 6. In this geometry we have ignored variations with respect to the azimuthal angle, which is appropriate for systems which are isotropic, like most liquids. Experimentally we can measure not only the intensities and the direction (the scattering angle), but also the frequency (energy) changes of the scattered photons, as there can be energy exchange between the scattered photon and the scattering medium. Whereas the scattered intensity can be related to the structures of particles or 44 inhomogeneities in the medium, the optical spectra reveal the dynamical motions. In the experiments we are concerned with here, we are interested mostly in the structures of domains and their growth, therefore experimentally we only measure the scattered light intensity Is and its profile, i. e., its variation over the scattering angle 6. This is known as elastic light scattering, where we are not concerned with the change of energy of the scattered photons, only their numbers (intensity). Equivalently we assume that the photon energy (frequency) isn't changed from the scattering, or the photon scatters elastically. In contrast, those light scattering experiments which also probe the energy (frequency) change of the photons are generally called dynamic light scattering. I will limit the discussion here only to elastic scattering. We want to measure the variation of the scattered light Is(6, t). Conventionally the more relevant quantity is the momentum transfer q which is defined as (2.1) q = ks - ko, where and ks are ko are the wave vectors of the scattered and the incident photon, respectively, as shown in Fig. 2.1. Because the magnitudes of ks and ko are the same, and equal to 2/r/As, the magnitude of q is simply cj = ^ sin(&) = 4ft n sin(@~) (2.2). Therefore, q is a function of the scattering angle 6, but it has the dimensional of inverse length. In isotropic medium, the scattered intensity Is only depends on the magnitudes of q, i. e., c\. 45 The structures and dynamics of a system are characterized by its structure factor. The dynamic structure factor S(q,iu) is defined as the Fourier transform of the time-dependent correlation function G(r, t), which is (2.3) where (p{r, t) is the local order parameter, and for binary fluids, (p is the local concentration C of one of the molecular species. The angular brackets denotes an equilibrium ensemble average, and <5
(2.4)
Fig. 4.1 A schematic phase diagram of GGW mixtures with different |