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9 Future Opportunities in Thermodynamics
9 Future Opportunities in Thermodynamics H. Ted Davis Department of Chemical Engineering & Materials Science University of Minnesota Minneapolis, Minnesota
I. Introduction My job of discussing Keith Gubbins’ paper [ 11 would be easier if I disagreed with his summary of the current status and future opportunities in the use of thermodynamics in chemical engineering. Unfortunately for me, his is a fine paper, and it touches on most of the important areas that are likely to prosper during the next couple of decades. However, there are some areas that I believe he underemphasizes or whose importance he underestimates. I agree with Gubbins that the future advancement in our understanding will come through the interplay of molecular theory, computer simulation, and experimentation (Fig. 1). The rapidly increasing speed and memory capacity of computers enable the simulation of more realistic molecular systems and the solution of more rigorous molecular theories. The next two decades will witness a substantial increase in the number of chemical engineers using computer-aided molecular theory and molecular dynamics as routine tools for elucidation of thermodynamic mechanisms and for prediction of properties. Chemical engineers already have a considerable presence in the area. Two of the more significant developments in computer simulations of the last decade have been accomplished by young 169 ADVANCE8 IN CHEMICAL ENGINEERING, VOL. 16
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Figure 1. Developments in thermodynamics.
chemical engineers. One is the Gibbs ensemble Monte Carlo method, created by A. Panagiotopoulos [2], which is especially potent in simulating multiphase equilibria of mixtures. The other is the excluded volume map, a technique invented by G. Deitrick [3] to reduce by orders of magnitude the time required to compute chemical potential in molecular dynamics simulations. An example drawn from Deitrick’s work (Fig. 2) shows the chemical potential and the pressure of a Lennard-Jones fluid computed from molecular dynamics. The variance about the computed mean values is indicated in the figure by the small dots in the circles, which serve only to locate the dots. A test of the “thermodynamic goodness” of the molecular dynamics result is to compute the chemical potential from the simulated pressure by integrating the Gibbs-Duhem equation. The results of the test are also shown in Fig. 2. The point of the example is that accurate and affordable molecular simulations of thermodynamic, dynamic, and transport behavior of dense fluids can now be done. Currently, one can simulate realistic water, electrolytic solutions, and small polyatomic molecular fluids. Even some of the properties of micellar solutions and liquid crystals can be captured by idealized models [4,51.
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In the future, experimentation will play an increasingly important role in understanding thermodynamic behavior. Current and emerging instrumentation will enable study of structure and dynamics on the molecular or colloidal scale (angstroms to hundreds of nanometers). With the aid of the rotating anode and synchrotron radiation sources and sophisticated computeraided data analysis, x-ray scattering now provides an accurate noninvasive probe of molecular-level organization. At national and international facilities, neutron scattering competes with x-ray scattering in structural analysis and provides dynamical data on molecular rearrangement not accessible by xrays. Another modem tool, the video-enhanced light microscope, can be used to study structure and transformation kinetics on the scale of tenths to tens of microns. The transmission electron microscope, extended by fast-freeze cold-stage technology, enables photography and examination by eye or computer-aided image analysis of molecular organization on a scale ranging from a few nanometers to tens or even hundreds of nanometers. Scanning electron microscopy spans the gap between transmission electron microscopy and ordinary light microscopy. The recently invented scanning
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tunneling microscopy has begun to yield detailed images of the atomic topography of biological membranes and proteins. The turns in the helical structure of DNA are clearly visible in several recent publications of scanning tunneling micrographs [ 6 ] .This remarkable microscope can be operated on a laboratory bench without the aid of clean room heroics and can be open to the atmosphere-in fact, it can be operated under liquid. The emerging atomic force microscope, which does not require proximity of an imaged object to a conducting substrate, could be even more powerful than the scanning tunneling microscope. Modern spectroscopies, such as fast Fourier transform infrared spectroscopy, picosecond laser spectroscopy, x-ray photoemission spectroscopy, electron spin resonance, and nuclear magnetic resonance, based on improved instrumentation and high-speed computer hardware and sophisticated software, provide much information concerning molecular structure and dynamics. For example, with the aid of computer-controlled magnetic field pulsing sequences, nuclear magnetic resonance will give a fingerprint spectrum, which identifies the phase of the subject system, and magnetic relaxation rates, from which local molecular rotational behavior can be deduced and tracer diffusivities of every component of the material can be determined simultaneously. Furthermore, NMR imaging can be used as a noninvasive tool to map out the microstructure of fine composites, biological organisms, and polymeric materials. The surface force apparatus is now being used routinely to study the equation of state of solutions confined between opposed, molecularly thin solid films. The apparatus is also used in one laboratory to study electrochemistry of thin films at electrodes a few nanometers thick and in a few other laboratories to study the behavior of molecularly thin films subjected to shear and flow [7]. An important point is often overlooked with regard to the role of experimentation. It is not unusual in thermodynamics to think of experiment as the servant of theory and to consider theory as that which is derived from the equations of thermodynamics or a statistical mechanics-based hierarchy of mechanics (e.g., classical, relativistic, quantum). In my opinion, experimental thermodynamics has a more proactive role to play in establishing the conceptual basis on which theoretical models will be built. For example, it has been observed that certain patterns of phase behavior are generic, functions of field variables (intensive thermodynamic variables which are the same in all coexisting phases at equilibrium). Examples include the law of corresponding states, behavior near a critical point, and the sequence of phases that appear in scans of temperature, activity of a solute, carbon number in a homologous series, and the like. The canonization of generic behavior or patterns shared by a large class of thermodynamic systems is
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indeed theory. So is the concept that the atomics of a solid sit at threedimensionally periodic sites. If the behavior can also be predicted from model equations of state or statistical mechanics, so much the better. In his paper, Gubbins identifies the corresponding states theory, perturbation and cluster expansion theory, and density functional theory as those of particular interest to chemical engineers. I agree that these are important theories for future growth in chemical engineering. However, during the next decade or so, the best one can hope to accomplish with these theories is handling molecular solutions of not too complex polar, polyatomic molecules. Microstructural materials such as micellar solutions, liquid crystals, microemulsions, protein solutions, and block copolymer glasses will require approaches not emphasized in Gubbins’ paper. One of these approaches is lattice theory, which, contrary to his assertion, is likely to be very useful in understanding and predicting properties of microstructured materials. Another approach neglected by Gubbins is what I have chosen to call the elemental structures model. In what follows, I will present the phase behavior of oil, water, and surfactant solutions as a case study to illustrate the special problems posed by microstructured fluids and to indicate how lattice and elemental structures models useful to chemical engineers might be developed.
11. Microstructured Fluids: A Primer A surfactant molecule is an amphiphile, which means it has a hydrophilic (water-soluble) moiety and a hydrophobic (water-insoluble) moiety separable by a mathematical surface. The hydrophobic “tails” of the most common surfactants are hydrocarbons. Fluorocarbon and perfluorocarbon tails are, however, not unusual. Because of the hydrophobic tail, a surfactant resists forming a molecular solution in water. The molecules will tend to migrate to any water-vapor interface available or, at sufficiently high concentration, the surfactant molecules will spontaneously aggregate into association colloids, i.e., into micelles or liquid crystals. Because of the hydrophific head, a surfactant (with a hydrocarbon tail) will behave similarly when placed in oil or when put in solution with oil and water mixtures. Some common surfactants are sodium or potassium salts of long-chained fatty acids (soaps), sodium ethyl sulfates and sulfonates (detergents), alkyl polyethoxy alcohols, alkyl ammonium halides, and lecithins or phospholipids. The special property of surfactants in solution is that they associate into a monolayer or sheetlike structure with the water-soluble moieties (hydrophilic heads) on one side of the sheet and water-insoluble moieties on the other side [8]. These sheetlike structures provide the building blocks for a rich variety of fluid microstructures, zhich, depending on thermodynamic
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conditions, assemble spontaneously. Several known examples of surfactant fluid microstructures are illustrated in Fig. 3. At an air-water interface, a monolayer forms with heads lying down and tails up (toward air), whereas at an air-hydrocarbon interface the monolayer lies with tails down. By closing on the tail side, the sheetlike structure can be dispersed in aqueous solutions as spherical, rodlike, or disklike micelles (Fig. 3). Closure on the head side forms the corresponding inverted micelles in oil. Oil added to a micellar solution is incorporated into the interior of the micelle to form a swollen micellar solution. Thus, surfactant acts to solubilize substantial amounts of oil into aqueous solution. Similarly, a swollen inverted micellar solution enables significant solubilization of water in oil. Micellar solutions are isotropic microstructured fluids which form under certain conditions. At other conditions, liquid crystals periodic in at least one dimension can form. The lamellar liquid crystal phase consists of periodically stacked bilayers (a pair of opposed monolayers). The sheetlike surfactant structures can curl into long rods (closing on either the head or tail side) with parallel axes arrayed in a periodic hexagonal or rectangular spacing to form a hexagonal or a rectangular liquid crystal. Spherical micelles or inverted micelles whose centers are periodically distributed on a lattice of cubic symmetry form a cubic liquid crystal. In addition to the geometrical order associated with the relative arrangements of the surfactant fluid microstructures, the sheetlike assembly of the surfactant imposes a topological order. The two sides of the sheetlike monolayer are different, water-rich fluids always lying on the head side and oil-rich fluids always lying on the tail side. Swollen micellar solutions are water-continuous and oil-discontinuous; i.e., the water-rich region continuously spans a sample of the solution whereas the oil-rich regions are disjoint and lie inside surfaces formed by surfactant monolayers closed on their tail side. Swollen inverted micellar solution, on the other hand, is oil-continuous and water-discontinuous. Similarly, the cubic liquid crystals described above are water-continuous if it is micelles that are periodically arrayed, and oil-continuous if it is inverted micelles that are periodically arrayed. As shown in Fig. 3, topologically different microstructures are possible, namely, bicontinuous microstructures. In these microstructures, the sheetlike surfactant structure separates a sample-spanning water-rich region from a sample-spanning oil-rich region. As indicated in Fig. 3, bicontinuous microstructures occur in liquid crystalline phases (bicontinuous cubic) and in isotropic phases (some microemulsions having substantial amounts of oil and water). The existence of bicontinuous microstructures was postulated by Scriven in 1976 [9] and after several years of controversy has now been experimentally verified.
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Proof of the geometric symmetries of the liquid crystals shown in Fig. 3 has been ambiguously established by x-ray scattering. However, unequivocal demonstration of the bicontinuity of some cubic liquid crystals and certain microemulsions requires more than x-ray scattering. The microemulsions story is especially interesting since it represents a case of potential engineering application, namely enhanced oil recovery, driving basic scientific discovery. The triangular phase diagram shown in Fig, 3 is but a single projection out of multivariable parameter space. By varying a field variable-temperature, pressure, activity of a fourth component, carbon number of a homologous series of the oil or surfactant, etc.-a triangular prismatic phase diagram results. The generic pattern of microemulsion phase splits as a function field variable, which has been identified by the Minnesota group [ 101 and by Kahlweit, Strey, and co-workers [ 1 1 1 , is illustrated in Fig. 4. Shown there is temperature-composition diagram indicating the isotropic phase equilibria that occur in mixtures of pentaethylene glycol dodecyl ether (C,2E5), water, and octane. At low temperatures two phases coexist, the lower
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phase (in a test tube) being a water-rich microemulsion. As temperature is decreased, a third phase appears at a critical end point (CEP) at the waterrich end of the two-phase tie line. With continued increase in temperature, the middle phase microemulsion increases in oil concentration until the middle phase disappears at a CEP at the oil-rich end. This pattern of phase behavior has been designated by Knickerbocker et al. [lo] as a &3,2 sequence: denotes a two-phase system in which the microemulsion is the lower, water-rich phase; 3 denotes a three-phase system in which the microemulsion is the middle phase and contains appreciable amounts of oil and water; and 5 denotes a two-phase system in which the microemulsion is the upper, oil-rich phase. The special significance of the &3,2 microemulsion sequence to petroleum recovery is that, under certain conditions (of temperature, pH, choice of surfactant, addition of cosurfactant, salt activity, etc.), a middle phase microemulsion has ultralow tension against both oil-rich and water-rich phases [12]. In the example illustrated in Fig. 5 , a 2,3,2 sequence of phase splits can be generated as a function of salt activity in mixtures of oil, brine, and ionic surfactants. The ultralow-tension microemulsion is obtained by selecting a particular surfactant and adding a small amount of a low-molecular-weight alcohol-all of which amount to locating the right region of field variable parameter space to find the desired pattern of behavior.
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The sequence shown in Fig. 5 has played a prominent role in resolving the question of the microstructure of the mid-range microemulsion. At low salinity, the microemulsion has a low oil concentration and thus is a water-continuous swollen micellar solution. At high salinity, the microemulsion has a low water concentration and is an oil-continuous swollen inverted micellar solution. Somewhere between low and high salinities, the microemulsion undergoes a transition from a water-continuous to an oilcontinuous phase. This could happen by sudden inversion of micellar droplets or through the occurrence of bicontinuous microstructure [ 131 (Fig. 6). The sequence of structures hypothesized in Fig. 6 would be accompanied by very different diffusion behavior (Fig. 7). In the droplet transition case, the selfdiffusion coefficient of water would be high until inversion to oil-continuous microemulsion, after which water and surfactant would diffuse together as a relatively slow swollen inverted micelle. Oil and surfactant diffusivities would be low and equal in the water-continuous region and oil diffusivity would be high in the oil-continuous region. Alternatively, if the transition is through bicontinuous microemulsions, there will be an intermediate salinity region in which oil and water diffusivities are comparable while the surfactant diffusivity is lower than either, since surfactant is constrained to move along the sheetlike surface layers separating oil-rich and waterrich regions. Using pulsed field gradient spin echo NMR, Guering and Lindman [ 141 and, independently, Clarkson et aE. [ 151 measured the self-diffusion coefficients of the components of microemulsions of sodium dodecyl sulfate (SDS), toluene, butanol, and NaCl brine. The results (Fig. 8) establish unequivocally the existence of bicontinous microemulsion. It appears that the role of increasing salinity is to change the mean curvature of the surfactant sheetlike structure from a value favoring closure on the oil-rich regions (swollen inverted micelles). In between, in bicontinous microemulsion having comparable amounts of oil and water, the preferred mean curvature must be near zero. The globular-to-bicontinuous transition of a microemulsion along the &3,2 sequence of phase splits is of special interest because of its relationship to enhanced oil recovery technology. However, the microstructural transition is not controlled by phase transitions. If one considers a constant oil/water plane in the prismatic phase diagram given in Fig. 4,the resulting temperature-surfactant composition phase diagram [ 151 (Fig. 9) has a one-phase (10) corridor lying between a liquid crystalline phase (La)to the right and two- and three-phase regions to the left. The watedoctane weight percentages are fixed at 60/40 in Fig. 9. This corresponds to about equal volumes of water and octane. Thus, along the corridor, the states A, B, ..., A' represent a progression of one-phase microemulsion of about equal water and
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Figure 6. Schematic diagram comparing the droplet inversion transition with the bicontinuous transition of microemulsions in the 2.3.2 sequence of phase splits. Reprinted with permission from Kaler and Prager [ 131.
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Figure 7. Schematic diagram comparing the behavior of self-diffusion coefficients of oil (Do),water ( D J , and surfactant (D,) expected for the droplet inversion transition and the bicontinuous transition of microemulsiondepicted in Fig. 6 .
oil volumes. Diffusivities (Fig. 10) measured along this corridor by NMR show plainly that, between G and €1,a transition from globular to bicontinous microstructure occurs [ 161. Although NMR results provide perhaps the most convincing evidence of the bicontinuous structure of some microemulsions, many other techniques support their existence. These techniques include electrical conductimetry, x-ray and neutron scattering, quasielastic light scattering, and electron
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Toluene 10-
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Figure 8. Self-diffusion coefficients of the components of a microemulsion of sodium dodecyl sulfate (SDS), butanol, toluene, and NaCl brine. Vertical lines denote 2,3 and 3.5 phase transitions. Reprinted with permission from P. Guering and B. Lindman, Langrnuir 1,464 (1985) [14]. Copyright 1985 American Chemical Society.
microscopy. For example, a tantalum-tungsten replica of a fracture surface of frozen microemulsion viewed in the transmission electron microscope [17] shows a cross section consistent with a structure of intertwined water and oil-continuous regions (Fig. l l is a TEM micrograph of C,?E,, water, octane microemulsion at state J in Fig. 9). In the globular regime, TEM micrographs reveal disjoint spherical structures quite different from the structures seen in Fig. 11 [ 161.
111. Microstructured Fluids: Theory and Simulation At its most satisfying level, a statistical thermodynamic theory would begin by specifying realistic interaction potentials for the molecular components of a complex mixture and from these potentials the thermodynamic functions and phase behavior would be predicted without further approximation. For the next decade or so, there is little hope to accomplish such a theory for microstructured fluids. However, predictive theories can be obtained with the aid of elemental structures models. Also, lattice models
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provide an affordable means to probe, primarily by computer simulation, the connection between thermodynamic behavior and molecular interactions. The ideas underlying elemental structures models are to establish microstructures experimentally, to compute free energies and chemical potentials from models based on these structures, and to use the chemical potentials to construct phase diagrams. Jonsson and Wennerstrom have used this approach to predict the phase diagrams of water, hydrocarbon, and ionic surfactant mixtures [ 181. In their model, they assume the surfactant resides in sheetlike structures with heads on one side and tails on the other side of the sheet. They consider five structures: spheres, inverted (reversed) spheres, cylinders, inverted cylinders, and layers (lamellar). These structures are indicated in Fig. 12. Nonpolar regions (tails and oil) are crosshatched. For these elemental structures, Jonsson and Wennerstrom include in the free energy contributions from the electrical double layer on the water
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side of the surfactant sheets, the entropy of mixing of micellar aggregates and of the molecular components in aqueous and oleic regions, and the surface energy of the surfactant sheets. Figure 13 compares the observed ternary phase diagram of water, octanol, and potassium caprate with that calculated by Jonsson and Wennerstrom. The agreement, though not quantitative in every detail, is remarkable given the complexity of the phase behavior. The agreement is certainly good enough to encourage future development of theory along these lines for other mixtures. Wennerstrom and Jonsson have in fact extended their analysis to a few other systems. In solutions of water and surfactant, the surfactant monolayers can join, tail side against tail side, to form bilayers, which form lamellar liquid crystals whose bilayers are planar and are arrayed periodically in the direction normal to the bilayer surface. The bilayer thickens upon addition of oil, and the distance between bilayers can be changed by adding salts or other solutes. In the oil-free case, the hydrocarbon tails can be fluidlike (La) lamellar liquid crystal or can be solidlike (Lp) lamellar liquid crystal. There also occurs another phase, Pp, called the modulated or rippled phase, in which the bilayer thickness varies chaotically in place of the lamellae. Assuming lamellar liquid crystalline symmetry, Goldstein and Leibler [ 191 have constructed a Hamiltonian in which (1) the intrabilayer energy is calculated
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Figure 11. Freeze-fracture transmission electron micrograph of the microemulsion denoted as sample J in Fig. 9. Reprinted with permission from W. Jahn and R. Strey,J. Phys. Chem. 92,2294 (1988) [17]. Copyright 1988 American Chemical Society.
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C. d. e. Figure 12. Schematic representation of the five elemental structures used by Jonsson and Wennerstrom [ 181: (a) spherical, (b) cylindrical, (c) lamellar, (d) inverted cylindrical, and (e) inverted spherical. Nonpolar regions are crosshatched.
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Figure 13. (a) Experimental ternary phase diagrams of water, octanol, and potassium
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from a Landau-Ginzburg Hamiltonion accounting for the energy of thickening, stretching, and bending of the bilayer and (2) the interbilayer interactions are approximated by the van der Waals attractive potential plus the hydration repulsion potential. The hydration potential has been found empirically to be an exponential function of the distance between adjacent
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3 04 06 0 02 04 06 I-+ ’-4 Figure 14. Predicted (Goldstein and Leibler [ 191)temperature-compositionphase diagrams of lamellar phases in water and phospholipid mixtures. La and Lp are periodic liquid crystals and P is a modulated or rippled lamellar phase. a2 is a model parameter that sca es as temperature.fis the volume fraction of phospholipid.Reprinted with permission from Goldstein and Leibler [ 191.
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bilayers. Goldstein and Leibler’s elemental structures model yields a phase diagram (Fig. 14) similar to those observed for mixtures of water and phospholipids, which are the well-known surfactants in biological membranes. Many of the special properties of microemulsions have been predicted with the aid of the following elemental structures model advanced by Talmon and Prager [20]: (1) oil- and water-rich domains are randomly interspersed in space, (2) the surfactant lies entirely in monolayers separating the oiland water-rich domains, and (3) the free energy arises from the entropy of interspersion of the domains and from the energy of curvature of the surfactant layers. Talmon and Prager used a random subdivision of space, namely the Voronoi polyhedral tessellation, to provide a means of randomly interspersing oil and water domains. In later variations of the Talmon-Prager model, de Gennes and Taupin [21], Widom [22], and Safran et al. [23] used the simpler cubic tessellation of space to model the random interspersion. Qualitatively, the choice of tessellation does not appear to make much difference; from the micrograph shown in Fig. 11, it is clear that no polyhedral tessellation will capture the detailed geometry of a microemulsion, although it might adequately account for the bicontinuous topology. The transition from globular to bicontinuous microemulsion is predicted by the Talmon model, and the electrical conductivity observed in a 33,2 salinity scan is quantitatively predicted by that model. The 2,3,2 sequence of
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W
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Figure 15. Monte Carlo simulation of the phase diagram of the amphiphile H,T,, water, and oil modeled on a cubic lattice. Reprinted with permission from Larson
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phase splits and the low-tension behavior are qualitatively accounted for by various versions of the model. Recent versions of the model [22, 231 also focus attention on the importance of the preferred curvature and bending energy of the surfactant sheets for the patterns of phase behavior of surfactant microstructured fluids. As I stated earlier, lattice models, aided by computer simulations, are likely to be very useful in predicting patterns of thermodynamic behavior and in understanding molecular mechanisms in microstructured fluids. Perhaps the simplest example of such a lattice model is the one employed recently by Larson [24] to investigate patterns of liquid crystal formation by Monte Carlo simulation. He assumes that the molecules occupy sites of a cubic lattice. Water and oil molecules occupy single sites in the lattice. The surfactant molecules (HmTn)consist of a string of rn head segments and n tail segments. A surfactant molecule occupies a sequence of adjacent sites, either nearest neighbor or diagonal nearest neighbor. In terms of pair interaction energies, water and a head segment are equivalent, and oil and a tail segment are equivalent. A simulated phase diagram for H4T4 is shown in Fig. 15. For this symmetric surfactant (equal lengths of head and tail moieties), no cubic-phase liquid crystals are observed. Nevertheless, a surprisingly complex phase diagram is found, with many of the features depicted in Fig. 3. Larson found the bicontinuous liquid crystal phase with the asymmetric surfactant H,T,.
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(0)
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Figure 16. A lattice model accounting for the bending energy of surfactant layers. Filled circles denote the head group and unfilled circles the tail group of the surfactant. Reprinted with permission from Widom [25].
Widom has recently formulated a lattice model that takes into account the amphiphilic nature of surfactant and introduces molecular interactions that explicitly affect curvature of surfactant sheetlike structures [25]. The three configurations of adjacent surfactant molecules shown in Fig. 16 have different energies of interaction. In addition to these interactions, the model allows pair interactions among water, oil, surfactant head groups, and surfactant tail groups. Taking advantage of the relationship between Widom’s model and an equivalent Ising model, Dawson et al. [26] have computed the phase diagram for oil, water, surfactant solutions from known results for the Ising model and from mean field theory. They find lamellar liquid crystalline phases, micellar solutions, and bicontinuous microemulsion phases. For the model microemulsion, Dawson [27] has predicted phases exhibiting ultralow tensions against oil-rich and water-rich phases having compositions very different from that of the microemulsion. This is in agreement with experiment. Further mean field theory and computer simulations based on this model should be fruitful. There are presently several groups around the world conducting molecular dynamics simulations of micellization and liquid crystallization of more or less realistic models of water, hydrocarbon, and surfactants. The memory and speed of a supercomputer required to produce reliably equilibrated microstructures constitute a challenge not yet met, in my opinion. By taking advantage of identified or hypothesized elemental structures one can, however, hope to learn a great deal about the dynamics and stability of the various identified microstructures.
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16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.
K. Colton, ed.), p. 125. Academic Press, San Diego, Calif., 1991 (Adv. Chem. Eng. 16). Panagiotopoulos, A. Z., Mol. Phys. 61,8 13 (1987). Deitrick, G. L., Scriven, L. E., and Davis, H. T., J. Chem. Phys. 90,2370 (1 989). Jonsson, B., Edholm, O., and Teleman, O., J. Chem. Phys. 85,2259 (1986). Egberts, E., and Berendsen, H. J. C., J. Chem. Phys. 89,3718 (1988). Lee, G., Arscott, P. G., Bloomfield, V. A., and Evans, D. F., Science 244,475 (1989). Israelachvili, J. N., McGuiggan, P. M., and Homola, A. M., Science 241,795 (1988); Van Alsten, J., and Granick, S., Phys. Rev. Lett. 61, 2570 (1988). Davis, H. T., Bodet, J. F., Scriven, L. E., and Miller, W. G., in Physics of Amphiphilic Layers (J. Meunier, D. Langevin, and N. Boccara, eds.), p. 3 10. Springer-Verlag, Berlin, 1987. Scriven, L. E., Nature 283, 123 (1976). Knickerbocker, B. M., Pesheck, C. V., Scriven, L. E., and Davis, H. T.,J. Phys. Chem. 83, 1984 (1979); J . Phys. Chem. 86,393 (1982). Kahlweit, M., Strey, R., Haase, D., Kunieda, H., Schemling, T., Faulhaber, B., Borkovec, M., Eicke, H.-F., Busse, G., Eggers, F., Funck, Th., Richmann, H., Magid, L., Soderman, O., Stilbs, P., Winkler, J., Dittrrich, A., and Jahn, W., J. Colloid fnterface Sci. 118,436 (1987). Healy, R. N., Reed, R. L., and Stenmark, D. G., Soc. Pet. Eng. J . 16, 147 (1976). Kaler, E. W., and Prager, S., J. Colloid Interface Sci. 86, 359 (1982). Guering, P., and Lindman, B., Langmuir 1,464 (1985). Clarkson, M. T., Beaglehole, D., and Callaghan, P. T., Phys. Rev. Lett. 54, 1722 (1985). Bodet, J. F., Bellare, J. R., Davis, H. T., Scriven, L. E., and Miller, W. G., J. Phys. Chem. 92, 1898 (1988). Jahn, W., and Strey, R.,J. Phys. Chem. 92,2294 (1988). Jonsson, B., and Wennerstrom, H., J. Phys. Chem. 91, 338 (1987);Wennerstrom, H., and Jonsson, B., J. Phys. France 49, 1033 (1988). Goldstein, R. E., and Leibler, S., Phys. Rev. Lett. 61,2213 (1988). Talmon, Y., and Prager, S., J. Chem. Phys. 69,2984 (1978). de Gennes, P. G., and Taupin, C., J. Phys. Chem. 86,2294 (1982). Widom, B., J. Chem. Phys. 84,6943 (1986). Andelman, D., Cates, M. E., Roux, D., and Safran, S. A., J. Chem. Phys. 87, 7229 ( 1 987). Larson, R. G., J. Chem. Phys. 89, 1642 (1988). Widom, B., J.Chem. Phys. 84,6943 (1986). Dawson, K. A., Lipkin, M. D., and Widom, B., J. Chem. Phys. 88,5149 (1988). Dawson, K. A,, Phys. Rev. A 35, 1766 (1987).
I. Introduction My job of discussing Keith Gubbins’ paper [ 11 would be easier if I disagreed with his summary of the current status and future opportunities in the use of thermodynamics in chemical engineering. Unfortunately for me, his is a fine paper, and it touches on most of the important areas that are likely to prosper during the next couple of decades. However, there are some areas that I believe he underemphasizes or whose importance he underestimates. I agree with Gubbins that the future advancement in our understanding will come through the interplay of molecular theory, computer simulation, and experimentation (Fig. 1). The rapidly increasing speed and memory capacity of computers enable the simulation of more realistic molecular systems and the solution of more rigorous molecular theories. The next two decades will witness a substantial increase in the number of chemical engineers using computer-aided molecular theory and molecular dynamics as routine tools for elucidation of thermodynamic mechanisms and for prediction of properties. Chemical engineers already have a considerable presence in the area. Two of the more significant developments in computer simulations of the last decade have been accomplished by young 169 ADVANCE8 IN CHEMICAL ENGINEERING, VOL. 16
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Figure 1. Developments in thermodynamics.
chemical engineers. One is the Gibbs ensemble Monte Carlo method, created by A. Panagiotopoulos [2], which is especially potent in simulating multiphase equilibria of mixtures. The other is the excluded volume map, a technique invented by G. Deitrick [3] to reduce by orders of magnitude the time required to compute chemical potential in molecular dynamics simulations. An example drawn from Deitrick’s work (Fig. 2) shows the chemical potential and the pressure of a Lennard-Jones fluid computed from molecular dynamics. The variance about the computed mean values is indicated in the figure by the small dots in the circles, which serve only to locate the dots. A test of the “thermodynamic goodness” of the molecular dynamics result is to compute the chemical potential from the simulated pressure by integrating the Gibbs-Duhem equation. The results of the test are also shown in Fig. 2. The point of the example is that accurate and affordable molecular simulations of thermodynamic, dynamic, and transport behavior of dense fluids can now be done. Currently, one can simulate realistic water, electrolytic solutions, and small polyatomic molecular fluids. Even some of the properties of micellar solutions and liquid crystals can be captured by idealized models [4,51.
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In the future, experimentation will play an increasingly important role in understanding thermodynamic behavior. Current and emerging instrumentation will enable study of structure and dynamics on the molecular or colloidal scale (angstroms to hundreds of nanometers). With the aid of the rotating anode and synchrotron radiation sources and sophisticated computeraided data analysis, x-ray scattering now provides an accurate noninvasive probe of molecular-level organization. At national and international facilities, neutron scattering competes with x-ray scattering in structural analysis and provides dynamical data on molecular rearrangement not accessible by xrays. Another modem tool, the video-enhanced light microscope, can be used to study structure and transformation kinetics on the scale of tenths to tens of microns. The transmission electron microscope, extended by fast-freeze cold-stage technology, enables photography and examination by eye or computer-aided image analysis of molecular organization on a scale ranging from a few nanometers to tens or even hundreds of nanometers. Scanning electron microscopy spans the gap between transmission electron microscopy and ordinary light microscopy. The recently invented scanning
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tunneling microscopy has begun to yield detailed images of the atomic topography of biological membranes and proteins. The turns in the helical structure of DNA are clearly visible in several recent publications of scanning tunneling micrographs [ 6 ] .This remarkable microscope can be operated on a laboratory bench without the aid of clean room heroics and can be open to the atmosphere-in fact, it can be operated under liquid. The emerging atomic force microscope, which does not require proximity of an imaged object to a conducting substrate, could be even more powerful than the scanning tunneling microscope. Modern spectroscopies, such as fast Fourier transform infrared spectroscopy, picosecond laser spectroscopy, x-ray photoemission spectroscopy, electron spin resonance, and nuclear magnetic resonance, based on improved instrumentation and high-speed computer hardware and sophisticated software, provide much information concerning molecular structure and dynamics. For example, with the aid of computer-controlled magnetic field pulsing sequences, nuclear magnetic resonance will give a fingerprint spectrum, which identifies the phase of the subject system, and magnetic relaxation rates, from which local molecular rotational behavior can be deduced and tracer diffusivities of every component of the material can be determined simultaneously. Furthermore, NMR imaging can be used as a noninvasive tool to map out the microstructure of fine composites, biological organisms, and polymeric materials. The surface force apparatus is now being used routinely to study the equation of state of solutions confined between opposed, molecularly thin solid films. The apparatus is also used in one laboratory to study electrochemistry of thin films at electrodes a few nanometers thick and in a few other laboratories to study the behavior of molecularly thin films subjected to shear and flow [7]. An important point is often overlooked with regard to the role of experimentation. It is not unusual in thermodynamics to think of experiment as the servant of theory and to consider theory as that which is derived from the equations of thermodynamics or a statistical mechanics-based hierarchy of mechanics (e.g., classical, relativistic, quantum). In my opinion, experimental thermodynamics has a more proactive role to play in establishing the conceptual basis on which theoretical models will be built. For example, it has been observed that certain patterns of phase behavior are generic, functions of field variables (intensive thermodynamic variables which are the same in all coexisting phases at equilibrium). Examples include the law of corresponding states, behavior near a critical point, and the sequence of phases that appear in scans of temperature, activity of a solute, carbon number in a homologous series, and the like. The canonization of generic behavior or patterns shared by a large class of thermodynamic systems is
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indeed theory. So is the concept that the atomics of a solid sit at threedimensionally periodic sites. If the behavior can also be predicted from model equations of state or statistical mechanics, so much the better. In his paper, Gubbins identifies the corresponding states theory, perturbation and cluster expansion theory, and density functional theory as those of particular interest to chemical engineers. I agree that these are important theories for future growth in chemical engineering. However, during the next decade or so, the best one can hope to accomplish with these theories is handling molecular solutions of not too complex polar, polyatomic molecules. Microstructural materials such as micellar solutions, liquid crystals, microemulsions, protein solutions, and block copolymer glasses will require approaches not emphasized in Gubbins’ paper. One of these approaches is lattice theory, which, contrary to his assertion, is likely to be very useful in understanding and predicting properties of microstructured materials. Another approach neglected by Gubbins is what I have chosen to call the elemental structures model. In what follows, I will present the phase behavior of oil, water, and surfactant solutions as a case study to illustrate the special problems posed by microstructured fluids and to indicate how lattice and elemental structures models useful to chemical engineers might be developed.
11. Microstructured Fluids: A Primer A surfactant molecule is an amphiphile, which means it has a hydrophilic (water-soluble) moiety and a hydrophobic (water-insoluble) moiety separable by a mathematical surface. The hydrophobic “tails” of the most common surfactants are hydrocarbons. Fluorocarbon and perfluorocarbon tails are, however, not unusual. Because of the hydrophobic tail, a surfactant resists forming a molecular solution in water. The molecules will tend to migrate to any water-vapor interface available or, at sufficiently high concentration, the surfactant molecules will spontaneously aggregate into association colloids, i.e., into micelles or liquid crystals. Because of the hydrophific head, a surfactant (with a hydrocarbon tail) will behave similarly when placed in oil or when put in solution with oil and water mixtures. Some common surfactants are sodium or potassium salts of long-chained fatty acids (soaps), sodium ethyl sulfates and sulfonates (detergents), alkyl polyethoxy alcohols, alkyl ammonium halides, and lecithins or phospholipids. The special property of surfactants in solution is that they associate into a monolayer or sheetlike structure with the water-soluble moieties (hydrophilic heads) on one side of the sheet and water-insoluble moieties on the other side [8]. These sheetlike structures provide the building blocks for a rich variety of fluid microstructures, zhich, depending on thermodynamic
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conditions, assemble spontaneously. Several known examples of surfactant fluid microstructures are illustrated in Fig. 3. At an air-water interface, a monolayer forms with heads lying down and tails up (toward air), whereas at an air-hydrocarbon interface the monolayer lies with tails down. By closing on the tail side, the sheetlike structure can be dispersed in aqueous solutions as spherical, rodlike, or disklike micelles (Fig. 3). Closure on the head side forms the corresponding inverted micelles in oil. Oil added to a micellar solution is incorporated into the interior of the micelle to form a swollen micellar solution. Thus, surfactant acts to solubilize substantial amounts of oil into aqueous solution. Similarly, a swollen inverted micellar solution enables significant solubilization of water in oil. Micellar solutions are isotropic microstructured fluids which form under certain conditions. At other conditions, liquid crystals periodic in at least one dimension can form. The lamellar liquid crystal phase consists of periodically stacked bilayers (a pair of opposed monolayers). The sheetlike surfactant structures can curl into long rods (closing on either the head or tail side) with parallel axes arrayed in a periodic hexagonal or rectangular spacing to form a hexagonal or a rectangular liquid crystal. Spherical micelles or inverted micelles whose centers are periodically distributed on a lattice of cubic symmetry form a cubic liquid crystal. In addition to the geometrical order associated with the relative arrangements of the surfactant fluid microstructures, the sheetlike assembly of the surfactant imposes a topological order. The two sides of the sheetlike monolayer are different, water-rich fluids always lying on the head side and oil-rich fluids always lying on the tail side. Swollen micellar solutions are water-continuous and oil-discontinuous; i.e., the water-rich region continuously spans a sample of the solution whereas the oil-rich regions are disjoint and lie inside surfaces formed by surfactant monolayers closed on their tail side. Swollen inverted micellar solution, on the other hand, is oil-continuous and water-discontinuous. Similarly, the cubic liquid crystals described above are water-continuous if it is micelles that are periodically arrayed, and oil-continuous if it is inverted micelles that are periodically arrayed. As shown in Fig. 3, topologically different microstructures are possible, namely, bicontinuous microstructures. In these microstructures, the sheetlike surfactant structure separates a sample-spanning water-rich region from a sample-spanning oil-rich region. As indicated in Fig. 3, bicontinuous microstructures occur in liquid crystalline phases (bicontinuous cubic) and in isotropic phases (some microemulsions having substantial amounts of oil and water). The existence of bicontinuous microstructures was postulated by Scriven in 1976 [9] and after several years of controversy has now been experimentally verified.
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Proof of the geometric symmetries of the liquid crystals shown in Fig. 3 has been ambiguously established by x-ray scattering. However, unequivocal demonstration of the bicontinuity of some cubic liquid crystals and certain microemulsions requires more than x-ray scattering. The microemulsions story is especially interesting since it represents a case of potential engineering application, namely enhanced oil recovery, driving basic scientific discovery. The triangular phase diagram shown in Fig, 3 is but a single projection out of multivariable parameter space. By varying a field variable-temperature, pressure, activity of a fourth component, carbon number of a homologous series of the oil or surfactant, etc.-a triangular prismatic phase diagram results. The generic pattern of microemulsion phase splits as a function field variable, which has been identified by the Minnesota group [ 101 and by Kahlweit, Strey, and co-workers [ 1 1 1 , is illustrated in Fig. 4. Shown there is temperature-composition diagram indicating the isotropic phase equilibria that occur in mixtures of pentaethylene glycol dodecyl ether (C,2E5), water, and octane. At low temperatures two phases coexist, the lower
H20IAJ
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Figure 4. Generic phase prism of a nonionic surfactant, water, and oil system. Reprinted with permission from Kahlweit er al. [ 1 11.
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phase (in a test tube) being a water-rich microemulsion. As temperature is decreased, a third phase appears at a critical end point (CEP) at the waterrich end of the two-phase tie line. With continued increase in temperature, the middle phase microemulsion increases in oil concentration until the middle phase disappears at a CEP at the oil-rich end. This pattern of phase behavior has been designated by Knickerbocker et al. [lo] as a &3,2 sequence: denotes a two-phase system in which the microemulsion is the lower, water-rich phase; 3 denotes a three-phase system in which the microemulsion is the middle phase and contains appreciable amounts of oil and water; and 5 denotes a two-phase system in which the microemulsion is the upper, oil-rich phase. The special significance of the &3,2 microemulsion sequence to petroleum recovery is that, under certain conditions (of temperature, pH, choice of surfactant, addition of cosurfactant, salt activity, etc.), a middle phase microemulsion has ultralow tension against both oil-rich and water-rich phases [12]. In the example illustrated in Fig. 5 , a 2,3,2 sequence of phase splits can be generated as a function of salt activity in mixtures of oil, brine, and ionic surfactants. The ultralow-tension microemulsion is obtained by selecting a particular surfactant and adding a small amount of a low-molecular-weight alcohol-all of which amount to locating the right region of field variable parameter space to find the desired pattern of behavior.
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The sequence shown in Fig. 5 has played a prominent role in resolving the question of the microstructure of the mid-range microemulsion. At low salinity, the microemulsion has a low oil concentration and thus is a water-continuous swollen micellar solution. At high salinity, the microemulsion has a low water concentration and is an oil-continuous swollen inverted micellar solution. Somewhere between low and high salinities, the microemulsion undergoes a transition from a water-continuous to an oilcontinuous phase. This could happen by sudden inversion of micellar droplets or through the occurrence of bicontinuous microstructure [ 131 (Fig. 6). The sequence of structures hypothesized in Fig. 6 would be accompanied by very different diffusion behavior (Fig. 7). In the droplet transition case, the selfdiffusion coefficient of water would be high until inversion to oil-continuous microemulsion, after which water and surfactant would diffuse together as a relatively slow swollen inverted micelle. Oil and surfactant diffusivities would be low and equal in the water-continuous region and oil diffusivity would be high in the oil-continuous region. Alternatively, if the transition is through bicontinuous microemulsions, there will be an intermediate salinity region in which oil and water diffusivities are comparable while the surfactant diffusivity is lower than either, since surfactant is constrained to move along the sheetlike surface layers separating oil-rich and waterrich regions. Using pulsed field gradient spin echo NMR, Guering and Lindman [ 141 and, independently, Clarkson et aE. [ 151 measured the self-diffusion coefficients of the components of microemulsions of sodium dodecyl sulfate (SDS), toluene, butanol, and NaCl brine. The results (Fig. 8) establish unequivocally the existence of bicontinous microemulsion. It appears that the role of increasing salinity is to change the mean curvature of the surfactant sheetlike structure from a value favoring closure on the oil-rich regions (swollen inverted micelles). In between, in bicontinous microemulsion having comparable amounts of oil and water, the preferred mean curvature must be near zero. The globular-to-bicontinuous transition of a microemulsion along the &3,2 sequence of phase splits is of special interest because of its relationship to enhanced oil recovery technology. However, the microstructural transition is not controlled by phase transitions. If one considers a constant oil/water plane in the prismatic phase diagram given in Fig. 4,the resulting temperature-surfactant composition phase diagram [ 151 (Fig. 9) has a one-phase (10) corridor lying between a liquid crystalline phase (La)to the right and two- and three-phase regions to the left. The watedoctane weight percentages are fixed at 60/40 in Fig. 9. This corresponds to about equal volumes of water and octane. Thus, along the corridor, the states A, B, ..., A' represent a progression of one-phase microemulsion of about equal water and
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Figure 6. Schematic diagram comparing the droplet inversion transition with the bicontinuous transition of microemulsions in the 2.3.2 sequence of phase splits. Reprinted with permission from Kaler and Prager [ 131.
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SALINITY
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Figure 7. Schematic diagram comparing the behavior of self-diffusion coefficients of oil (Do),water ( D J , and surfactant (D,) expected for the droplet inversion transition and the bicontinuous transition of microemulsiondepicted in Fig. 6 .
oil volumes. Diffusivities (Fig. 10) measured along this corridor by NMR show plainly that, between G and €1,a transition from globular to bicontinous microstructure occurs [ 161. Although NMR results provide perhaps the most convincing evidence of the bicontinuous structure of some microemulsions, many other techniques support their existence. These techniques include electrical conductimetry, x-ray and neutron scattering, quasielastic light scattering, and electron
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Toluene 10-
Butanol Butanol
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Figure 8. Self-diffusion coefficients of the components of a microemulsion of sodium dodecyl sulfate (SDS), butanol, toluene, and NaCl brine. Vertical lines denote 2,3 and 3.5 phase transitions. Reprinted with permission from P. Guering and B. Lindman, Langrnuir 1,464 (1985) [14]. Copyright 1985 American Chemical Society.
microscopy. For example, a tantalum-tungsten replica of a fracture surface of frozen microemulsion viewed in the transmission electron microscope [17] shows a cross section consistent with a structure of intertwined water and oil-continuous regions (Fig. l l is a TEM micrograph of C,?E,, water, octane microemulsion at state J in Fig. 9). In the globular regime, TEM micrographs reveal disjoint spherical structures quite different from the structures seen in Fig. 11 [ 161.
111. Microstructured Fluids: Theory and Simulation At its most satisfying level, a statistical thermodynamic theory would begin by specifying realistic interaction potentials for the molecular components of a complex mixture and from these potentials the thermodynamic functions and phase behavior would be predicted without further approximation. For the next decade or so, there is little hope to accomplish such a theory for microstructured fluids. However, predictive theories can be obtained with the aid of elemental structures models. Also, lattice models
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Figure 9. Temperature-compositionphase diagram of the system pentaethylene glycol ether (C,2E5), water, and n-octane. The weight ratio of water to octane is fixed at 60 to 40. Diffusivities were measured in samples denoted by filled circles. Phase diagram determinedby Kahlweit et al. Reprinted with permission from Kahlweit et al. [l 11.
provide an affordable means to probe, primarily by computer simulation, the connection between thermodynamic behavior and molecular interactions. The ideas underlying elemental structures models are to establish microstructures experimentally, to compute free energies and chemical potentials from models based on these structures, and to use the chemical potentials to construct phase diagrams. Jonsson and Wennerstrom have used this approach to predict the phase diagrams of water, hydrocarbon, and ionic surfactant mixtures [ 181. In their model, they assume the surfactant resides in sheetlike structures with heads on one side and tails on the other side of the sheet. They consider five structures: spheres, inverted (reversed) spheres, cylinders, inverted cylinders, and layers (lamellar). These structures are indicated in Fig. 12. Nonpolar regions (tails and oil) are crosshatched. For these elemental structures, Jonsson and Wennerstrom include in the free energy contributions from the electrical double layer on the water
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Mixture # Figure 10. NMR diffusivities for the components of the samples denoted in Fig. 9 by tilled circles. The samples are microemulsions lying along the one-phase comdor in Fig. 9. Reprinted with permission from Bodet er af.[16].
side of the surfactant sheets, the entropy of mixing of micellar aggregates and of the molecular components in aqueous and oleic regions, and the surface energy of the surfactant sheets. Figure 13 compares the observed ternary phase diagram of water, octanol, and potassium caprate with that calculated by Jonsson and Wennerstrom. The agreement, though not quantitative in every detail, is remarkable given the complexity of the phase behavior. The agreement is certainly good enough to encourage future development of theory along these lines for other mixtures. Wennerstrom and Jonsson have in fact extended their analysis to a few other systems. In solutions of water and surfactant, the surfactant monolayers can join, tail side against tail side, to form bilayers, which form lamellar liquid crystals whose bilayers are planar and are arrayed periodically in the direction normal to the bilayer surface. The bilayer thickens upon addition of oil, and the distance between bilayers can be changed by adding salts or other solutes. In the oil-free case, the hydrocarbon tails can be fluidlike (La) lamellar liquid crystal or can be solidlike (Lp) lamellar liquid crystal. There also occurs another phase, Pp, called the modulated or rippled phase, in which the bilayer thickness varies chaotically in place of the lamellae. Assuming lamellar liquid crystalline symmetry, Goldstein and Leibler [ 191 have constructed a Hamiltonian in which (1) the intrabilayer energy is calculated
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Figure 11. Freeze-fracture transmission electron micrograph of the microemulsion denoted as sample J in Fig. 9. Reprinted with permission from W. Jahn and R. Strey,J. Phys. Chem. 92,2294 (1988) [17]. Copyright 1988 American Chemical Society.
a.
C. d. e. Figure 12. Schematic representation of the five elemental structures used by Jonsson and Wennerstrom [ 181: (a) spherical, (b) cylindrical, (c) lamellar, (d) inverted cylindrical, and (e) inverted spherical. Nonpolar regions are crosshatched.
b.
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Figure 13. (a) Experimental ternary phase diagrams of water, octanol, and potassium
caprate at 20°C. (b) Phase diagram predicted by Jonsson and Wennerstrom [18] using an elemental structures model. Reprinted with permission from B. Jonsson and H. Wennerstrom, J . Phys. Chem. 91,338 (1987) [ 181. Copyright 1987 American Chemical Society.
from a Landau-Ginzburg Hamiltonion accounting for the energy of thickening, stretching, and bending of the bilayer and (2) the interbilayer interactions are approximated by the van der Waals attractive potential plus the hydration repulsion potential. The hydration potential has been found empirically to be an exponential function of the distance between adjacent
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o,/H 4
3 04 06 0 02 04 06 I-+ ’-4 Figure 14. Predicted (Goldstein and Leibler [ 191)temperature-compositionphase diagrams of lamellar phases in water and phospholipid mixtures. La and Lp are periodic liquid crystals and P is a modulated or rippled lamellar phase. a2 is a model parameter that sca es as temperature.fis the volume fraction of phospholipid.Reprinted with permission from Goldstein and Leibler [ 191.
0
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bilayers. Goldstein and Leibler’s elemental structures model yields a phase diagram (Fig. 14) similar to those observed for mixtures of water and phospholipids, which are the well-known surfactants in biological membranes. Many of the special properties of microemulsions have been predicted with the aid of the following elemental structures model advanced by Talmon and Prager [20]: (1) oil- and water-rich domains are randomly interspersed in space, (2) the surfactant lies entirely in monolayers separating the oiland water-rich domains, and (3) the free energy arises from the entropy of interspersion of the domains and from the energy of curvature of the surfactant layers. Talmon and Prager used a random subdivision of space, namely the Voronoi polyhedral tessellation, to provide a means of randomly interspersing oil and water domains. In later variations of the Talmon-Prager model, de Gennes and Taupin [21], Widom [22], and Safran et al. [23] used the simpler cubic tessellation of space to model the random interspersion. Qualitatively, the choice of tessellation does not appear to make much difference; from the micrograph shown in Fig. 11, it is clear that no polyhedral tessellation will capture the detailed geometry of a microemulsion, although it might adequately account for the bicontinuous topology. The transition from globular to bicontinuous microemulsion is predicted by the Talmon model, and the electrical conductivity observed in a 33,2 salinity scan is quantitatively predicted by that model. The 2,3,2 sequence of
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W
O
Figure 15. Monte Carlo simulation of the phase diagram of the amphiphile H,T,, water, and oil modeled on a cubic lattice. Reprinted with permission from Larson
WI.
phase splits and the low-tension behavior are qualitatively accounted for by various versions of the model. Recent versions of the model [22, 231 also focus attention on the importance of the preferred curvature and bending energy of the surfactant sheets for the patterns of phase behavior of surfactant microstructured fluids. As I stated earlier, lattice models, aided by computer simulations, are likely to be very useful in predicting patterns of thermodynamic behavior and in understanding molecular mechanisms in microstructured fluids. Perhaps the simplest example of such a lattice model is the one employed recently by Larson [24] to investigate patterns of liquid crystal formation by Monte Carlo simulation. He assumes that the molecules occupy sites of a cubic lattice. Water and oil molecules occupy single sites in the lattice. The surfactant molecules (HmTn)consist of a string of rn head segments and n tail segments. A surfactant molecule occupies a sequence of adjacent sites, either nearest neighbor or diagonal nearest neighbor. In terms of pair interaction energies, water and a head segment are equivalent, and oil and a tail segment are equivalent. A simulated phase diagram for H4T4 is shown in Fig. 15. For this symmetric surfactant (equal lengths of head and tail moieties), no cubic-phase liquid crystals are observed. Nevertheless, a surprisingly complex phase diagram is found, with many of the features depicted in Fig. 3. Larson found the bicontinuous liquid crystal phase with the asymmetric surfactant H,T,.
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(0)
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Figure 16. A lattice model accounting for the bending energy of surfactant layers. Filled circles denote the head group and unfilled circles the tail group of the surfactant. Reprinted with permission from Widom [25].
Widom has recently formulated a lattice model that takes into account the amphiphilic nature of surfactant and introduces molecular interactions that explicitly affect curvature of surfactant sheetlike structures [25]. The three configurations of adjacent surfactant molecules shown in Fig. 16 have different energies of interaction. In addition to these interactions, the model allows pair interactions among water, oil, surfactant head groups, and surfactant tail groups. Taking advantage of the relationship between Widom’s model and an equivalent Ising model, Dawson et al. [26] have computed the phase diagram for oil, water, surfactant solutions from known results for the Ising model and from mean field theory. They find lamellar liquid crystalline phases, micellar solutions, and bicontinuous microemulsion phases. For the model microemulsion, Dawson [27] has predicted phases exhibiting ultralow tensions against oil-rich and water-rich phases having compositions very different from that of the microemulsion. This is in agreement with experiment. Further mean field theory and computer simulations based on this model should be fruitful. There are presently several groups around the world conducting molecular dynamics simulations of micellization and liquid crystallization of more or less realistic models of water, hydrocarbon, and surfactants. The memory and speed of a supercomputer required to produce reliably equilibrated microstructures constitute a challenge not yet met, in my opinion. By taking advantage of identified or hypothesized elemental structures one can, however, hope to learn a great deal about the dynamics and stability of the various identified microstructures.
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References 1. Gubbins, K. E., in Perspectives in Chemical Engineering: Research and Education (C.
2. 3. 4. 5. 6. 7.
8. 9. 10. 11.
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