Macroemulsion Stability: The Oriented Wedge Theory Revisited

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Macroemulsion Stability: The Oriented Wedge Theory Revisited Alexey Kabalnov*,1 and Håkan Wennerstro¨m Physical Chemistry 1, Chemical Center, University of Lund, Lund, S-2100, Sweden Received May 9, 1995. In Final Form: August 30, 1995X The correspondence between the equilibrium phase behavior of oil-water-surfactant mixtures and the macroemulsion type and stability is examined. Both the phase behavior and emulsion stability are argued to be dependent on the bending elasticity of the surfactant monolayer at the oil-water interface. At positive spontaneous curvatures, O/W emulsions are stable; at negative spontaneous curvatures W/O emulsions are stable, whereas, in the balanced state of the surfactant film, an emulsion break usually occurs for a wide variety of systems. To explain the effect of the monolayer bending properties on the macroemulsion stability, the thermally activated rupture of emulsion films is theoretically studied. We consider emulsion films covered by saturated surfactant monolayers, with strong lateral interactions among the adsorbed surfactant molecules. The monolayer at the edge of a nucleation hole in the emulsion film is strongly curved; the bending energy penalty involved leads to a strong dependence of the coalescence barrier on the sign and the absolute value of the monolayer spontaneous curvature. By contrast with earlier hole nucleation theories, the emulsion film thickness is allowed to vary, in order to minimize the free energy of the nucleation hole. At large positive spontaneous curvatures, H0, the oil-water-oil (O/W/O) films are stable, with a coalescence barrier, in a first approximation, proportional to the bending modulus κ. On the other hand, W/O/W films break without a barrier. Conversely, for large negative values of H0, W/O/W films are stable, while O/W/O films break without a barrier. In the vicinity of the balanced state, a very steep change in film stability with H0 is predicted. The model reproduces the macroemulsion stability sequence: O/W emulsion-emulsion breakage-W/O emulsion, observed in polyethoxylated nonionic surfactant-oil-water mixtures with increasing temperature. The macroemulsion break is predicted to occur at the balanced (PIT) point, as has been observed experimentally by Shinoda et al. The hole nucleation in multilamellar-stabilized films is shown to be drastically suppressed, in agreement with the experimental findings of Friberg et al. For rigid surfactant monolayers (κ ∼ 100 kT) and multilamellar-stabilized systems, a region of stable multiple emulsions is predicted, while it is prohibited for flexible (κ ∼1 kT) monolayers. Possible extensions of the model to other systems are discussed.

1. Introduction Whether the system formed on mixing oil, water, and surfactant will be oil or water continuous is a central problem of emulsion science. It was realized at the early stages of emulsion research that the volume fractions of oil and water were not that important and that the emulsion type and stability were determined primarily by the nature of the surfactant. There are three cornerstones guiding practical emulsion formulation which address this problem: the Bancroft rule ,2,3 Griffin HLB scale ,4,5 and Shinoda phase inversion temperature (PIT) concept .6 According to Bancroft,2,3 the phase in which the surfactant is predominantly dissolved tends to be the continuous phase: water-soluble surfactants (e.g., sodium oleate) tend to stabilize oil-in-water (O/W) emulsions, while oil-soluble surfactants (e.g. calcium dioleate) stabilize water-in-oil (W/O) emulsions. Griffin4,5 suggested an empirical hydrophilic-lipophilic balance (HLB) scale which characterizes the tendency of emulsifiers to form O/W and W/O emulsions. Surfactants with low HLB values (∼4) tend to stabilize W/O emulsions, while those with high HLB values (∼20) stabilize O/W emulsions. Surfactants with intermediate HLB values (∼10) are usually ineffective stabilizers of either O/W or W/O X Abstract published in Advance ACS Abstracts, November 15, 1995.

(1) Current address: Alliance Pharmaceutical Corp., 3040 Science Park Rd., San Diego, CA, 92121. (2) Bancroft, W. D. J. Phys. Chem. 1913, 17, 501. (3) Bancroft, W. D. J. Phys. Chem. 1915, 19, 275. (4) Griffin, W. C. J. Soc. Cosmet. Chem. 1949, 1, 311. (5) Griffin, W. C. J. Soc. Cosmet. Chem. 1954, 5, 249. (6) Shinoda, K.; Friberg, S. Emulsions and Solubilization; John Wiley & Sons: New York-Chichester-Brisbane-Toronto-Singapore, 1986; p 174.

0743-7463/96/2412-0276$12.00/0

emulsions. Later Davies 7-9 suggested a simple group contribution method to evaluate the HLB of different surfactants from their molecular structure. The HLB scale does not take into account, however, the effects of temperature and the nature of the oil on emulsion stability. These effects are naturally incorporated into Shinoda’s PIT concept ,6,10-14 which correlates (macro)emulsion stability with the phase behavior of oil-water-polethoxylated nonionic surfactant (O-W-S) mixtures. It is well-known that increasing the temperature induces a Winsor I-Winsor III-Winsor II sequence of the microemulsion phase equilibrium shown in Figure 1. According to Shinoda et al., O/W macroemulsions are stable in the Winsor I region at temperatures ca. 20 °C below the “phase inversion” (or in a more modern terminology, “balanced”) temperature of the O-W-S microemulsion phase diagram. At this temperature, micellar solution coexists with excess oil. Conversely, W/O emulsions are stable ∼20 °C above the PIT, where the system is in the Winsor II region, and excess water coexists with inverse micelles. In the vicinity of the PIT point (Winsor III region), where oil, water, and bicontinuous microemulsion phases coexist in a three-phase equilibrium, neither emulsion is stable; see Figure 1. (7) Davies, J. T. In Proc. 2nd Int. Cong. Surf. Activity; London, 1957; p 417. (8) Davies, J. T.; Rideal, E. K. Interfacial Phenomena; Academic: New York-San Francisco-London, 1963; p 129. (9) Davies, J. T. In Progress in Surface Science; Danielli, J. F., Parkhurst, K. G. A., Riddford, A. C., Eds.; Academic: New York, 1964; Vol. 2, p 129. (10) Shinoda, K.; Saito, H. J. Colloid Interface Sci. 1968, 26, 70. (11) Shinoda, K.; Saito, H. J. Colloid Interface Sci. 1969, 30, 258. (12) Shinoda, K.; Sagitani, H. J. Colloid Interface Sci. 1978, 64, 68. (13) Saito, H.; Shinoda, K. J. Colloid Interface Sci. 1970, 32, 647. (14) Davis, H. T. Colloids Surf., A 1994, 91, 9.

© 1996 American Chemical Society

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the following Hookean equation:17

W)

∫

∫

κ (2H0 - H1 - H2)2 dA + κj H1H2 dA (1) 2

Here H1 and H2 are the reciprocal principal radii of curvature of the monolayer, H0 is the spontaneous curvature, κ and κj are the bending and saddle-splay moduli, and dA is the surface area of a patch on the neutral surface. According to the sign convention, the curvature of O/W droplets is considered positive. The second integral, due to the Gauss-Bonnet theorem, can be reduced to the following form

∫H1H2 dA ) 2πχ

(2)

Bancroft’s rule, Griffin’s HLB scale, and Shinoda’s PIT concepts are strongly related and, to some degree, can be reduced to one another. Although all of them are in a good agreement with experiment, the PIT approach is certainly the most accurate of the three. Indeed, numerous exceptions to Bancroft’s rule and Griffin’s HLB scale have been noted .14-16 On the other hand, Shinoda’s PIT concept works even in circumstances when Griffin’s HLB scale and Bancroft’s rule fail: the authors of this paper are not aware of a single exception to the PIT rule, provided the surfactant concentration is above the critical micelle concentration (cmc), and the volume fractions of the emulsified phases are not extreme. It is hard to believe that the correlation between O-W-S phase behavior and macroemulsion stability is purely coincidental; the same physical parameters may in fact control both. In this paper, we argue that properties of note are the surfactant monolayer bending parameters: the spontaneous curvature and bending and saddle splay moduli. In particular, the value and sign of the monolayer spontaneous curvature control the type (O/W or W/O) and stability of the emulsion formed. According to Helfrich, the bending energy of a surfactant monolayer can be represented by

where χ is the Euler characteristic of the surface, which is a topological invariant proportional to the difference between the number of disjoint surfaces and the number of handles. The bending and saddle-splay moduli have a dimensionality of energy and cover the range from ∼1 kT (“flexible” monolayers) to ∼10-100 kT (“rigid” monolayers). Experimental and theoretical estimates of these parameters are now available; for recent reviews, see refs 19 and 18 and 19, respectively. For a person involved in the emulsion stability field, the thesis that the monolayer bending elasticity affects macroemulsion stability may sound rather controversial. While it is generally accepted that microemulsion phase behavior is indeed strongly dependent on the monolayer bending properties, the significance of these parameters for macroemulsion stability has been doubted for many years. The correlation between the monolayer spontaneous curvature (in terms of the “two interfacial tensions” on the oil and water sides of the surfactant film) and the emulsion type emerged in Bancroft’s classical paper.3 The idea was later developed by Harkins and Langmuir 20,21 and termed the “oriented wedge” theory. According to the authors, surfactants with bulky alkyl chains and small head groups stabilize W/O emulsions due to packing constraints at the oil-water interface of an emulsion drop, while single-tailed surfactants with big polar heads stabilize O/W emulsions for similar reasons (Figure 2a). Although the oriented wedge theory has been used successfully as an empirical correlation, it stands on a rather shaky physical basis. According to Harkins and Langmuir, the monolayer bending energy affects the free energy of the emulsion droplets themselves. Consider the spontaneous curvature of the surfactant monolayer, which is controlled by the “shape” of the surfactant molecules. For unbalanced surfactants, the radius of curvature has an order of ∼10-7 cm and is several orders of magnitude smaller than the radius of the emulsion droplets themselves. As a result, the monolayer “frustration” energy at the interface of both O/W and W/O drops is essentially the same: for the monolayer, both the O/W and W/O interfaces are essentially planar on a molecular scale. The oriented wedge theory was subjected to criticism on this basis by Hildebrand in the early 1940s, and is now considered as being of merely historical interest.6,8 In this paper, we, in a way, revisit the oriented wedge concept. We argue that the emulsion type and stability are indeed determined by the sign and absolute value of the monolayer spontaneous curvature. By considering

(15) Smith, D. H.; Covatch, G. C.; Lim, K. H. J. Phys. Chem. 1991, 95, 1463. The authors also claim that they have observed a violation of the PIT rule. The surfactant concentration in their study was below the critical micelle concentration; the PIT behavior however is limited to the above-cmc concentrations; see discussion below. (16) Binks, B. P. Langmuir 1993, 9, 25.

(17) Helfrich, W. Z. Naturforsch. 1973, 28c, 693. (18) Helfrich, W. J. Phys. Condens. Matter 1994, 6, A79 (19) Lekkerkerker, H. N. W. Physica A 1989, 159, 319. (20) Harkins, W. D.; Davies, E. C. H.; Clark, G. L. J. Am. Chem. Soc. 1917, 39, 541. (21) Langmuir, I. J. Am. Chem. Soc. 1917, 39, 1848.

Figure 1. Schematic representation of the nonionic microemulsion phase equilibria and macroemulsion stability pattern found in the vicinity of the balanced state. (upper row) Increasing temperature induces a transition from Winsor I (equilibrium of spherical micelles with excess oil) through Winsor III (threephase equilibrium of a bicontinuous microemulsion, excess oil and excess water) to Winsor II (two-phase equilibrium of inverse micelles with excess water). (middle row) Appearance of equilibrium 1:1 oil-to-water samples with increasing temperature. (lower row) Appearance of nonequilibrium macroemulsion samples. At low temperatures, a stable O/W emulsion is formed. Increasing temperature makes the O/W emulsion unstable, and it ultimately breaks in the balanced state. Further increases in temperature stabilize a W/O emulsion. All macroemulsion layers have a milky color, in contrast with the bicontinuous microemulsion phase, which is bluish.

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Figure 2. “Oriented wedge” theory. (a) Schematic representation of the oriented wedge theory, as presented by Harkins and Langmuir. According to these authors, the monolayer covering emulsion droplets have different frustration energies, which favor one emulsion type to another. Note that the picture shows the macroscopic emulsion droplets and not the surfactant micelles. (b) Our version of the “oriented wedge” theory. The monolayer bending energy affects not the free energy of the droplets themselves but the free energy of the coalescence “transition state”sthe nucleation pore in the bilayer. (b, top) monolayer spontaneous curvature fits that of the hole edge and the rupture occurs without a barrier. (b, bottom) monolayer at the edge of the hole nucleus is “frustrated” and the hole nucleation is suppressed. Arrows indicate the frustration stress in the bent monolayer.

several O-W-S systems, we show that the correlation between phase behavior and macroemulsion stability indeed holds with amazing accuracy. We emphasize, however, that the physical basis of this correlation is different from what was previously believed.20,21 We argue that the monolayer bending properties, in particular the spontaneous curvature, control the rates of coalescence of W/O and O/W emulsions via the values of the corresponding coalescence energy barriers. To create a passage in an emulsion film, one has to pay a monolayer bending energy penalty, because the surfactant monolayer must be strongly curved at the edge of the passage (Figure 2b). This penalty is dramatically different between W/O/W and O/W/O lamellae. This determines which emulsion type, W/O or O/W, becomes more stable and dominates over time. To justify our approach we stress the following: (i) We limit ourselves to the case of “stable” emulsions formed in an excess of surfactant above the cmc, whose lifetimes are on the order of hours or more. The short-term rupture (occurring at low surfactant concentrations) may be dependent on film thinning hydrodynamics and hydrodynamic stability,23 while the long-term stability is controlled predominantly by hole nucleation in emulsion

Kabalnov and Wennerstro¨ m

films. It is the latter process that is the object of study in this paper. (ii) The main difference of our approach from earlier hole nucleation theories, 24-31 is in examining the film rupture on a molecular level. The previous hole nucleation studies analyzed film rupture in terms of two phenomenological parameters: the interfacial tension of the film and the linear tension of the hole nucleus. There have been no attempts to explain Bancroft’s rule or the PIT concept on the basis of these theories. To explain the PIT, one can argue that the hole linear tension differs between O/W/O and W/O/W films (which is in a way correct). Without the molecular curvature model in hand, however, this approach does not provide very much insight into the problem. Our approach is rather close (although not identical) to that of Chernomordik et al. 32,33 who examined hole nucleation in phospholipid bilayers. (iii) We assume the coalescence process to be film-rupturecontrolled, which corresponds to the coalescence in creamed emulsion layers and high internal phase ratio emulsions. The other limiting case of the aggregation controlled coalescence has been considered in detail in DLVO and other classical theories of colloidal stability. The paper has the following outline. In the first part, by analyzing experimental data found in literature, we show empirically that the macroemulsion stability pattern is a function of the monolayer spontaneous curvature for a wide variety of surfactant systems. Whenever possible, we make direct comparisons between the macroemulsion stability behavior and the O-W-S phase diagrams. In the second part, we briefly review previous attempts to interpret the Bancroft rule and PIT behavior. In the third part, we propose a simple theory of emulsion stability/ inversion at the balanced point within the monolayer curvature approach. We believe that this model provides a theoretical/molecular basis for the observed correlation between spontaneous curvature and emulsion stability. 2. Spontaneous Curvature-Emulsion Stability Correlation 2.1. Flexible Balanced Monolayers (Microemulsion Limit). In this section, we discuss the macroemulsion stability observed for systems which in their equilibrium state form balanced, or nearly balanced, microemulsions. The Winsor III region and adjacent domains are our main interest (Figure 1). For simplicity, we start with microemulsion systems formed by polyethoxylated nonionic surfactants. These systems have an inherent advantage in that the typical microemulsion behavior can be realized in simple three-component systems. The trends outlined below are very general and (22) Hildebrand, J. H. J. Phys. Chem. 1941, 45, 1303. (23) Thin Liquid Films: Fundamentals and Applications; Ivanov, I. B., Ed.; Marcel Dekker: New York, 1988; Vol. 29, p 1126. (24) Derjaguin, B. V.; Gutop, Y. V. Kolloid Zh. 1962, 24, 431. (25) Derjaguin, B. V.; Prokhorov, A. V. J. Colloid Interface Sci. 1981, 81, 108. (26) Prokhorov, A. V.; Derjaguin, B. V. J. Colloid Interface Sci. 1988, 125, 111. (27) Kashchiev, D.; Exerowa, D. J. Colloid Interface Sci. 1980, 77, 501. (28) Kashchiev, D. Colloid Polym. Sci. 1987, 265, 436. (29) Kashchiev, D.; Exerowa, D. Biochim. Biophys. Acta 1983, 732, 133. (30) Exerowa, D.; Kashchiev, D.; Platikanov, D. Adv. Colloid Interface Sci. 1992, 40, 201. (31) Chizmadzhev, Y. A.; Pastushenko, V. F. In Thin Liquid Films: Fundamentals and Applications; Ivanov, I. B., Ed.; Marcel Dekker: New York-Basel, 1988; Vol. 29; p 1059. (32) Chernomordik, L. V.; Kozlov, M. M.; Melikyan, G. B.; Abidor, I. G.; Markin, V. S.; Chizmadzhev, Y. A. Biochim. Biophys. Acta 1985, 812, 643. (33) Chernomordik, L. V.; Melikyan, G. B.; Chizmadzhev, Y. A. Biochim. Biophys. Acta 1987, 906, 309.


Figure 3. Macroemulsion stability pattern as a function of temperature. At low temperatures, W/O emulsions are stable and there is no free oil separation. Note that the W-phase layer at the bottom left corner of the figure is due to emulsion creaming, and not due to coalescence. Near the Winsor III region (the narrow region between ca. 75 and 78 °C), the O/W emulsion starts to coalesce; see unfilled circles at the top of the plot. Increasing temperature produces a stable W/O cream. Reprinted, with permission from ref 11. Copyright 1969 Academic Press.

apply to any microemulsion systems in the vicinity of Winsor I-Winsor III-Winsor II phase transitions. The microemulsion phase equilibrium of oil-waterpolyethoxylated nonionic surfactant three component mixtures is very complex; it follows, however, a very general pattern for a wide variety of systems. 34-36 Detailed theoretical description of the phase behavior remains controversial. It is generally accepted, however, that the classic Winsor I-Winsor III-Winsor II sequence observed in these microemulsions with increasing temperature is due to changes in the surfactant monolayer spontaneous curvature from positive (Winsor I), through zero (Winsor III), to negative values (Winsor II). The changes in spontaneous curvature are induced by temperature-driven dehydration of the polyethoxylated surfactant headgroups.35,36. Consider now the nonequilibrium macroemulsion systems. With some effort, coexistent phases of Winsor I, Winsor II, and Winsor III equilibria can be emulsified in one another (Figures 1 and 3). The pattern of macroemulsion stability correlates strongly with the microemulsion phase behavior.6 We start from the balanced state where the spontaneous curvature is equal to zero and the bicontinuous microemulsion phase (m) (D-phase in terms of Shinoda and Friberg) cosolubilizes equal volumes of water and oil. At low surfactant concentrations, the m-phase coexists with excess oil (L2) and water (L1) in the Winsor III three-phase equilibrium region. In the balanced state, the three phases cannot be effectively emulsified in each other; the emulsions are very unstable and break within several seconds after shaking. A slight decrease in temperature (∼0.5 °C) produces, however, a notable change in the phase behavior and in the observed emulsion stability. The bicontinuous microemulsion phase becomes water-rich (say, 2:1 in water-to-oil by volume). The coexistent phases can be now emulsified in each other with a much greater degree of success. After gentle shaking, one finds some amount of oil emulsified in water, which rapidly forms a white cream layer at the top of the lower (water) phase. Water, however, does not emulsify in oil (Figures 1 and 3). The most important feature is that the “cream” stability is strongly temperature dependent. Having no stability (34) Kahlweit, M.; Strey, R. Angew. Chem., Int. Ed. Engl. 1985, 24, 654. (35) Strey, R. Colloid Polym. Sci. 1994, 272, 1005. (36) Olsson, U.; Wennerstro¨m, H. Adv. Colloid Interface Sci. 1994, 49, 113.

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at the balanced temperature (PIT), the emulsion is stable for 1 day at ∆T ) -0.5 °C and for ∼1 month at ∆T ) -2 °C. Decreasing the temperature further drives the system outside the Winsor III region into the Winsor I region. This, however, has little effect on the macroemulsion stability trend. The cream of the oil phase which coexists with a micellar solution gets more and more stable as the temperature is decreased. At ∆T ) -20 °C, the cream, according to Shinoda et al., reaches its maximum stability. Decreasing temperature further does not affect the emulsion stability substantially, although, some decrease is noted.6 It is well-known that the O-W-S ternary phase diagram has a plane of symmetry originating from the S corner, and the phase diagram is symmetric with respect to O T W and ∆T T -∆T inversion.36 A peculiar feature of the macroemulsion stability is that it is also oil-water symmetric. Increasing the temperature above the PIT reproduces essentially the same picture of emulsion stability, except that now water is emulsified in oil. At ∆T ) 20 °C, a very stable W/O macroemulsion at the bottom of the oil phase is formed (Figure 3). Up to this point, we were concerned with threecomponent O-W-S systems. It is known that the microemulsion phase behavior can be “tuned” not only by changes in temperature but also by adding different “cosolvents” or “cosurfactants”. 34,37,38 Some of these additives merely produce temperature shifts in the phase diagrams, leaving the general picture of the phase behavior unaltered. For instance, the balanced point of the n-C8H18-C10E5-water system is at ∼45 °C, while that of the n-C8H18-C10E5-10% NaCl brine is at ∼28 °C. Otherwise, the phase diagrams are essentially the same. 39 The observed effects on the phase behavior produced by these additives are attributed to changes in the monolayer spontaneous curvature. Thus, the addition of inorganic salts dehydrates the surfactant poly(oxyethylene) brush due to depletion and osmotic pressure effects, thereby bending the monolayer toward water. 40 Substituting a short-chain hydrocarbon for a long-chain one decreases the degree of hydrocarbon penetration into the surfactant alkyl chain brush, thereby curving the monolayer away from water.35,40-42 Adding a cosurfactant with a small “packing parameter” produces a positive shift in the spontaneous curvature. The most important observation made by Shinoda and Friberg is that the changes in microemulsion phase behavior induced by additives lead to a similar shift in the macroemulsion stability profile. In other words, the effects of inorganic salts, hydrophilic polymers, etc. on macroemulsion stability do not need to be treated in a special manner but, on a phenomenological level, are naturally incorporated into the emulsion stability/spontaneous curvature correlation. To summarize, we conclude that the stability of O/W and W/O emulsions with polyethoxylated nonionic surfactants is a unique function of the monolayer spontaneous curvature (H0). At large positive values of H0, O/W emulsions are stable. Decreasing H0 deteriorates O/W emulsion stability, and it breaks at H0 ) 0. Decreasing H0 further leads to an emulsion inversion: W/O emulsions become stable, the stability increasing with decreases in H0. (37) Kahlweit, M.; Lessner, E.; Strey, R. J. Phys. Chem. 1984, 88, 1937. (38) Kahlweit, M.; Strey, R.; Haase, D. J. Phys. Chem. 1985, 89, 163. (39) Kahlweit, M.; Strey, R.; Busse, G. Phys. Rev. E 1993, 47, 4197. (40) Kabalnov, A.; Olsson, U.; Wennerstro¨m, H. J. Phys. Chem. 1995, submitted. (41) Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1981, 77, 601. (42) Aveyard, R.; Binks, B. P.; Fletcher, P. D. I.; MacNab, J. R. Langmuir 1995, 11, 2515.

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Figure 4. Variation of macroemulsion conductivity (filled circles) and interfacial tension (unfilled circles) vs salinity of n-heptane-AOT-water-NaCl system. High conductivity values correspond to the O/W emulsion, and low values to W/O. Note that the macroemulsion inversion point at ∼0.07 M NaCl almost exactly matches the interfacial tension minimum, corresponding to the balanced state (∼0.05 M). Reprinted with permission from ref 48. Copyright 1993 Elsevier.

Originally, the PIT concept was proposed for polyethoxylated nonionic surfactants. Soon it was found, however, that the same trends are observed with ionic microemulsions. 43-48 Ionic surfactants usually do not form microemulsions by themselves and necessitate the addition of cosurfactants (e.g. medium chain alcohols) to make the surfactant monolayer flexible and balanced. As an additional tool to “tune” the microemulsion phase behavior, one may use inorganic salts. These additives decrease the monolayer spontaneous curvature by screening the electrostatic repulsions between the surfactant polar heads, which enables one to observe the classical Winsor I-Winsor III-Winsor II sequence. 49 At low salinities, the systems are in the Winsor I region and O/W emulsions are favored; the lower the salinity, the more stable the emulsions. At high salinities, the systems are in the Winsor II region and W/O emulsions are favored. At intermediate salinities, the systems are in the Winsor III region and neither O/W or W/O emulsions are stable (Figure 4). Let us return back to the experiment in which the oil phase of the Winsor III equilibrium has been emulsified in the lower water phase. We want to emphasize that the macroemulsion particles are relatively big and creaming is rapid. After a few minutes, a cream layer is formed, which may be stable for hours, days, or months, depending on how far the temperature is from the balanced point. On this time scale, the emulsion stability is controlled (43) Baldauf, L. M.; Schechter, R. S.; Wade, W. H.; Graciaa, A. J. Colloid Interface Sci. 1982, 85, 187. (44) Salager, J. L.; Loaiza-Maldonado, I.; Minana-Perez, M.; Silva, F. J. Dispersion Sci. Technol. 1982, 3, 279. (45) Salager, J. L.; Lopez-Castellanos, G.; Minana-Perez, M.; Parra, C. J. Dispersion Sci. Technol. 1991, 12, 59. (46) Salager, J. L. In Encyclopedia of Emulsion Technology; Becher, P., Ed.; Marcel Dekker: New York and Basel, 1983; Vol. 3; p 159. (47) Aveyard, R.; Binks, B. P.; Fletcher, P. D. I.; Ye, X.; Lu, J. R. In Emulsions: A Fundamental and Practical Approach; Sjo¨blom, J., Ed.; Kluwer: Amsterdam, 1992; p 97. (48) Binks, B. P. Colloids Surf., A 1993, 71, 167. (49) Temperature effects on the phase behavior of ionics are less pronounced and have an opposite sign.


Figure 5. Schematic phase diagram of a balanced nonionic microemulsion. In the Winsor III (L1 + L2 + m) region, the macroemulsions are unstable. In the three-phase equilibrium regions of m + L1 + LR and m + L2 + LR (darkened triangles), the emulsions are very stable even in the balanced state due to multilamellar stabilization.

neither by film thinning hydrodynamics nor the particle aggregation rate; the emulsion droplets are closely packed. Obviously, it is the film rupture that controls the overall rate of the process. Up to now, we have discussed how emulsion stability depends on temperature, i.e., how it changes along the temperature (Y axis) of the “Kahlweit fish”.34,35 The other important feature of the macroemulsion stability is its dependence upon the surfactant concentration (X-axis). Increasing surfactant concentration at the balanced temperature above the Winsor III range leads to the appearance of a lamellar liquid crystal phase, LR, which cosolubilizes considerable amounts of oil and water and coexists with excess oil and a bicontinuous microemulsion phase or with excess water and a bicontinuous microemulsion phase in the three-phase equilibria, Figure 5. One can find multilamellar-covered droplets in these systems, which are known to have a remarkable stability, even very close to the balanced state. 6,50-52 2.2. Rigid Balanced Monolayers (Liquid Crystalline Limit). Typical microemulsion phase behavior is observed when the surfactant monolayer is flexible and has a bending modulus on the order of 1 kT. For rigid balanced monolayers having a bending modulus of ∼100 kT, liquid crystalline phases are usually favored over disordered ones. For instance, instead of a three-phase equilibrium of excess oil, excess water, and a bicontinuous microemulsion phase (Winsor III region), a three-phase equilibrium of oil, water, and lamellar phase swollen in both solvents is observed. This type of phase equilibrium is exhibited, for instance, by the n-dodecane-waterDDAB (didodecyldimethylammonium bromide) system;53 see Figure 6. Adding salts (NaBr, Na2SO4) to this system produces a negative shift in the monolayer spontaneous curvature. At low salinities, the phase equilibrium pattern (50) Friberg, S.; Rydhag, L. Colloid Polym. Sci. 1971, 244, 233. (51) Friberg, S.; Jansson, P. O.; Cederberg, E. J. Colloid Interface Sci. 1976, 55, 614. (52) Friberg, S. E. In Emulsions - A Fundamental and Practical Approach; Sjo¨blom, J., Ed.; Kluwer Academic Publishers: Amsterdam, 1992; p 1. (53) Skurtveit, R.; Sjo¨blom, J.; Bouwstra, J.; Gooris, G.; Selle, M. H. J. Colloid Interface Sci. 1992, 152, 205.


Figure 6. Schematic representation of the phase equilibrium of the n-dodecane-water-DDAB-Na2SO4 system: (a) 0.504 mM Na2SO4 system; (b) 4.88 mM Na2SO4. Increasing salinity induces a transition from the L1-LR-L2 three phase equilibrium shown in (a) to the L1-HII-L2 equilibrium shown in (b). The three-phase triangles are darkened. The phases of the first equilibrium form a stable O/W emulsion, whereas the phases of the second form a stable W/O emulsion. Reprinted with permission (not to scale and with simplifications) from ref. 53. Copyright 1992 Academic Press.

does not change significantly and the “frustration” bending energy remains frozen into a lamellar structure. Above a certain salt concentration, the frustration becomes too great, and a three-phase equilibrium of oil (L2), water (L1), and a reverse hexagonal phase, HII, is observed. While the L1 phase is still essentially pure water, the L2 phase becomes substantially enriched in water and surfactant and has a bicontinuous structure. The coexistent phases of these equilibria can be emulsified in each other. An interesting observation made by Skurtveit et al.53 is that the phase behavior correlates well with the observed macroemulsion stability pattern. At low salinities, where the monolayer spontaneous curvature is positive, and the W-O-LR phase equilibrium is observed, very stable O/W emulsions are formed. At intermediate salinities, O/W emulsions become unstable. At high salinities, where the W-O-HII phase equilibrium becomes favored, very stable W/O emulsions are observed. Thus, the stability trend fits rather nicely the O/Wemulsion break-W/O sequence, observed for flexible monolayer systems (microemulsions) on decreasing the spontaneous curvature. Phospholipid-stabilized emulsions represent another example of systems stabilized by a rigid monolayer. Fat emulsions stabilized by phospholipids have a considerable practical significance because of their use in the food industry 54 and in medicine as intravenous parenteral nutrition and drug delivery systems. 55 In addition, there are continuing efforts to prepare fluorocarbon-in-water emulsions for local oxygen transport. 56 The main phospholipid components phosphatidylcholine (PC) and phosphatidylethanolamine (PE) do not bear electric charge at neutral pH. Their hydrophobic tails represent C16 - C18 aliphatic acyl or alkoxy chains. The spontaneous curvature of typical natural phospholipids at the oil-water interface is close to zero and the monolayer is nearly balanced; fine adjustment of the spontaneous curvature can be attained by changing the phospholipid polar headgroup, the length and degree of saturation of the acyl chain, and the type of hydrocarbon oil. 57 Thus, phosphatidylcholines typically have a higher spontaneous curvature than phosphatidylethanolamines.57 For the (54) Becher, P. Emulsions: Theory and Practice, 2nd ed.; R. E. Krieger Publishing Co.: Malabar, FL, 1985; p 440. (55) Davis, S. S.; Hadgraft, J.; Palin, K. J. In Encyclopedia of Emulsion Technology; Becher, P., Ed.; Marcel Dekker: New York and Basel, 1983; Vol. 2, p 159. (56) Riess, J. G. Vox Sang. 1991, 61, 225. (57) Seddon, J. M. Biochim. Biophys. Acta 1990, 1031, 1.

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same polar headgroup, the spontaneous curvature increases with the degree of saturation of the hydrocarbon chain and with a shortening of the acyl chain length.57 The spontaneous curvature of phospholipids at the n-alkane-water interface increases with the alkane chain length due to reduced penetration of the long-chain alkanes into the surfactant “brush”. 57-60 The phase equilibrium of phospholipid-water-triglyceride ternary mixtures is reported to involve a threephase equilibrium of oil, water, and lamellar phase.51 If n-alkanes are used as the oil component, the coexistence of inverse hexagonal phase with isotropic phases is also reported.58 Schurtenberger et al. 61 found inverse micelles in the oil corner of the phase diagram of phospholipidalkane-water mixtures. The complete ternary phase diagrams of phospholipid-water-alkane mixtures have yet to be determined. There are some indications that the macroemulsion stability of phospholipid systems is very sensitive to the monolayer spontaneous curvature. Early experiments in this field 62,63 are very confusing and plagued by artifacts. 64 Recent experiments of Handa et al. 66 on triglyceridein-water emulsions showed that triglyceride emulsions in saline stabilized by PC were stable, whereas those stabilized by PE (with the same acyl chain lengths) were not; the emulsions broke into triglyceride and saline ca. 30 s after preparation. Similar data are reported by Ishii 67 who found the stability series LPC (lysophosphatidylcholine) > PC . PE for analogous systems. These data support the emulsion stability/spontaneous curvature correlation, because one may expect the spontaneous curvature of the phospholipid monolayers to decrease in the same sequence.57 The dependence of phospholipid-stabilized emulsion stability on the nature of the emulsified hydrocarbon has not attracted very much attention; it can however shed some light on the emulsion stability behavior. In our recent study, 68 the phospholipid-stabilized emulsions of n-alkanes and water were studied. The oil-to-water ratio was kept at 1:1 by volume at a moderate surfactant concentration (3%). Whereas the emulsions of hexadecane were relatively stable and had an O/W type, the emulsion stability rapidly deteriorated with decreasing hydrocarbon chain length. For instance, an emulsion of n-octane separated most of its oil within several days of standing. Most strikingly, the emulsion of n-hexane had a W/O type and was very stable! In other words, the typical trend stable O/W emulsion-emulsion break-stable W/O emulsion is observed in n-alkane-water-phospholipid systems with decreasing alkane chain length, i.e., in the same direction as the spontaneous curvature is expected to decrease. It is interesting to note that O/W emulsions of short-chain fluorocarbons (e.g., perfluorooctane) prepared under similar conditions are very stable to coales(58) Sjo¨lund, M.; Lindblo¨m, G.; Rilfors, L.; Arvidson, G. Biophys. J. 1987, 52, 145. (59) Sjo¨lund, M.; Rilfors, L.; Lindblo¨m, G. Biochemistry 1989, 28, 1323. (60) Siegel, D. P.; Banschbach, J.; Yeagle, P. L. Biochemistry 1989, 28, 5010. (61) Schurtenberger, P.; Peng, Q.; Leser, M. E.; Luisi, P. J. Colloid Interface Sci. 1993, 156, 43 and references therein. (62) Hansrani, P. K. Thesis, University of Nottingham, 1980. (63) Davis, S. S.; Hansrani, P. J. Colloid Interface Sci. 1985, 108, 285. (64) Phospholipids have a very low molecular solubility in water. Special care must be taken to avoid artifacts due to very slow adsorption of phospholipids at the oil-water interface, if the adsorption is accomplished from the water phase; for a detailed discussion, see ref 65. (65) Kabalnov, A.; Weers, J.; Arlauskas, R.; Tarara, T. Langmuir 1995 11, 2966. (66) Handa, T.; Saito, H.; Miyajima, K. Biochemistry 1990, 29, 2884. (67) Ishii, F. J. Abura Kagaku 1992, 41, 787. (68) Kabalnov, A. Manuscript in preparation.

282


cence. Due to the mutual phobicity of hydrocarbons and fluorocarbons, the degree of fluorocarbon penetration into the hydrocarbon acyl chains is low. Thus, the spontaneous curvature does not get a negative increment which would favor W/O emulsion stability. To conclude this section, there are indications that the typical sequence O/W emulsion-break-W/O emulsion is observed on decreasing the spontaneous curvature of rigid monolayers in the vicinity of the balanced state. The bicontinuous microemulsion phase formation per se does not seem to be a prerequisite for the emulsion break/ inversion observed with microemulsion-forming surfactants. Rather, it is the zero value of the spontaneous curvature that is important. The experimental data are, however, very limited and the conclusions are tentative. 3. Previous Interpretations of the PIT and Bancroft Rule 3.1. General Considerations. Under the action of gravity, most macroemulsions cream (except for the rare instance in which the densities of the dispersed and continuous phases match). Emulsion coalescence in the cream layer is a multistage process. First, the emulsion particles approach one another. Initially, the film dividing the particles is curved; it then flattens due to hydrodynamic interactions. The film then thins due to suction from the Plateau borders by Laplace pressure/gravity effects. Sometimes, a “dimple” in the center of the film is formed, i.e., the central part of the film gets thicker than its periphery. Then, the dimple gradually disappears and the film becomes flat parallel. When the film thickness reaches the range of molecular forces, the film drains, faster if the forces are attractive and slower if the forces are repulsive. In the latter case, the film may reach an equilibrium, provided the molecular repulsive forces counterbalance the capillary pressure. Finally, external disturbances or thermal fluctuations ultimately produce film rupture. There have been several attempts to interpret the PIT in the past, each of them addressing a specific stage of the coalescence process. They can be loosely classified into the following groups: (i) “surfactant mass flow” theories; (ii) “interlamellar/interdroplet force” theories; (iii) “passage formation” theories. In the following section, we consider each of these briefly relative to the PIT experimental data outlined above. 3.2. Surfactant Mass Flow Theories. This group of theories/qualitative ideas is directed to explain the Bancroft rule rather than the PIT. In 1941, Hildebrand theoretically examined the stability of equilibrium emulsion films as a function of fluctuations in thickness.22 Consider a water film, separating two oil droplets. A fluctuation in this film creates an oil-water interface, which locally has a lower surfactant concentration and a higher interfacial tension. Hence, the film automatically strengthens itself wherever a break is threatened; this is the so-called Gibbs elasticity effect. The Gibbs elasticity is suppressed by the surfactant diffusion from the bulk phases to the threatened point of the film. Hildebrand notes that “the rupture of a film separating two droplets can be resisted by a larger rise in interfacial tension at the threatened point, if the reserve emulsifying agent is dissolved in the liquid forming the film, i.e., the external phase, owing to the lower rate of adsorption in that case.” Indeed, if the film thickness (2b) is considerably smaller than its radius (R), the surfactant mass flow from the oil phase is larger than from the water phase, by roughly the geometric factor 2b/R; see Figure 7. Therefore, if the surfactant is dissolved in the oil, the Gibbs elasticity is suppressed, and the film separating the two oil droplets is unstable. Conversely, the oil lamella is unstable if the


Figure 7. Schematic representation of the surfactant mass flow (arrows) to an emulsion film defect.

surfactant is predominantly dissolved in the water phase. This explains Bancroft’s rule. Another group of studies 23,69,70 considers the rate of thinning of emulsion films. According to the hydrodynamic approach, film thinning kinetics is governed to a large extent by a feedback loop of surfactant diffusion to the interface. A liquid flow in the film depletes surfactant from the adsorption layer and creates an interfacial tension gradient opposing flow. This gradient is canceled by the gradual diffusion of surfactant from the bulk. Again, as in Hildebrand’s geometric arguments, the diffusion rate is substantially dependent upon where the surfactant is dissolved. If the surfactant is dissolved in the droplet phase, the thinning is fast; if it is dissolved in the film, it is slow.69,70 Therefore, if the emulsion lifetime is controlled by film thinning kinetics, Bancroft’s rule is predicted. In addition to these two approaches, there are several other interpretations of Bancroft’s rule based on Gibbs elasticity/Marangoni effect arguments; see, e.g., refs 71 and 72. In all cases, the Bancroft rule is physically reproduced as the ratio of the surfactant mass flows from the film and from the emulsion droplets. However attractive these interpretations may seem, a closer inspection reveals that they eventually fail with balanced systems. In a vast majority of nonionic microemulsions within the Winsor III region, the Bancroft rule fails, although the PIT concept holds. To prove this, we start by examining the correspondence between the Bancroft rule and the PIT concept. The ideas presented here are due to Binks.16 (i) Consider a polyethoxylated nonionic surfactant-oilwater system. Far below the balanced temperature in the Winsor I region, macroemulsions have an O/W type. Because the surfactant forms normal micelles, it is predominantly dissolved in the water phase. In addition to micellar solubility, there is some molecular solubility of the surfactant in the oil and water. This is a minor factor with respect to the micellar solubility, however, and can be neglected. Obviously, the PIT and Bancroft rule’s agree with each other in this region. The same applies to the Winsor II region, where the macroemulsions have a W/O type and the surfactant solubility is strongly biased toward oil due to the presence of reverse micelles. Consider now the Winsor III region. We remove the surfactant middle phase and emulsify the oil and water in each other. There are no micelles in either the water or oil phases, and it is the molecular solubility of surfactant in these phases that is important. The molecular solubility of all balanced polyethoxylated nonionic surfactants is heavily biased toward the oil. For example, the molecular solubility of C12E5 in heptane is on the order of 1%, while (69) Ivanov, I. B. Pure Appl. Chem. 1980, 52, 1241. (70) Zapryanov, Z.; Malhotra, A. K.; Aderangi, N.; Wasan, D. T. Int. J. Multiphase Flow 1983, 9, 105. (71) Lyklema, J. Colloids Surf. A 1994, 91, 25. (72) Walstra, P. Chem. Eng. News 1993, 48, 333.


that in water is ∼0.001%.16 Bancroft’s rule predicts, therefore, that W/O emulsions must be favored at a few degrees above and a few degrees below the balanced temperature, which is not observed in experiment. The Bancroft rule and the mass flow arguments, therefore, fail within the Winsor III region. (ii) Consider a microemulsion system far from the balanced point, where the Bancroft rule holds. The question arises: can one reduce the mass flow arguments to the Bancroft rule so straightforwardly? Far from the balanced point, the Bancroft rule is observed due to the surfactant solubility in micellar form. The surfactant mass flows, which are considered in the theories outlined above, are, however, not simply proportional to the sum of the molecular and micellar solubilities weighted by their respective diffusion coefficients. The matter is more complex because micelles, as a rule, cannot merge directly with the oil-water interface, and a preliminary stage of micellar dissociation to monomers is necessary .73-75 This makes the micellar contribution to the mass flow comparable to that of the molecular solution.75 (iii) If the hydrodynamic interpretations based on film thinning kinetics are examined,69,70 it is apparent that the time scale is too short to explain the PIT phenomena. Typical film drainage times are on the order of several minutes, whereas typical emulsion lifetimes are that short only very close to the balanced temperature. Deviations of only ∼0.2 °C from the balanced point lead to much longer lifetimes, where emulsion stability is most probably controlled by the film rupture rate. 3.3. “Forces” and Emulsion Stability. Although no explicit attempts to explain macroemulsion inversion at the PIT by changes in interdroplet interactions have been made, there is a strong opinion that all colloidal stability phenomena can be reduced to “disjoining pressure” effects. One can think of two possible mechanisms by which interparticle interactions affect emulsion stability. (i) Repulsive (electrostatic or steric) forces between emulsion particles prevent their aggregation, which is a necessary step prior to coalescence. These arguments are not applicable here, because our particular case involves the emulsion “cream” with particles closely packed together. (ii) Forces may affect the stability of an emulsion film. Consider an equilibrium emulsion film, with repulsive interlamellar forces counterbalanced by suction through the Plateau boarders. For this film to break, one arguably must pay the work-against-repulsive-forces penalty. This opinion is widespread in the literature; for a review of pro- and counterarguments in application to biological membrane fusion see ref 76. It is difficult to argue against this idea on a physical basis because the detailed mechanism of emulsion film rupture is unknown. Our objective here is to show that the experimental PIT behavior does not always coincide with the trend in interlamellar forces. In previous sections we have shown that the emulsion stability trend exactly matches the microemulsion phase behavior in the vicinity of the PIT. It is generally accepted, that the phase behavior is driven by the monolayer bending (73) Lucassen, J. Faraday Discuss. Chem. Soc. 1976, 59, 76. (74) Johner, A.; Joanny, J. F. Macromolecules 1990, 23, 5299. (75) Increasing the concentration of micelles leads to the crossover to a smaller diffusion path d ∼ a/(3φ)1/2, where φ is the volume fraction of micelles and a is the micellar radius. In this regime, the adsorption rate is proportional to the square root of the micellar concentration. The driving force of the mass transfer is still equal to the molecular solubility gradient (i.e., cmc) and is not affected by the concentration of micelles (A. Kabalnov, J. Weers, and H. Wennerstro¨m, manuscript in preparation). (76) Leikin, S. L.; Kozlov, M. M.; Chernomordik, L. V.; Markin, V. S.; Chizmadzhev, Y. A. J. Theor. Biol. 1987, 129, 411.

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elasticity .35,36,77 The latter, to some degree, correlates with the interactions between planar monolayers but generally has a different physical origin. The exact coincidence between the microemulsion phase behavior and macroemulsion stability for a wide variety of systems (ionic and nonionic) which exhibit repulsive forces of a different nature allows one to conclude that the film stability is most probably driven by the bending elasticity itself and not by the forces. Typical examples, where the role of forces is questionable, are presented below. (i) Consider the dependence of emulsion stability on the nature of the emulsified hydrocarbon. For instance, at 45 °C, the C16H34-C12E5-H2O system is below the PIT temperature, and a very stable O/W emulsion is formed. On the other hand, at the same temperature the C10H22C12E5-H2O system is above the PIT, and a very stable W/O emulsion is formed. This observation is quite difficult to explain simply by “force” arguments. The van der Waals forces across emulsion lamellae are in both cases essentially the same. It is also difficult to see why the steric repulsions between the surfactant tails must be so strongly dependent on the oil chain length. (ii) In the vicinity of the balanced state of nonionic microemulsions, a drastic change in emulsion stability occurs within a 1-2 °C temperature range. For instance, a stable O/W emulsion is formed in the C10H22-C12E5-H2O system at 37 °C and a stable W/O emulsion at 39 °C. Although water indeed becomes a worse solvent for the poly(oxyethylene) brush with increasing temperature, and decreases in the steric repulsions are also expected, it is unclear why such a drastic change must happen over such a narrow temperature range. (iii) For the PIT behavior to be observed, the surfactant concentration must be above the cmc.16,47,48 Below the cmc, the emulsion stability trends are much more irregular and the typical O/W-emulsion breakW/O sequence is not observed. It is difficult to explain why this would occur from force arguments. 3.4. Passage Formation Penalty. There have been two attempts to explain emulsion inversion at the PIT by considering the emulsion film rupture rate. The first one6 can be traced back to the paper of Davies;9 the other is due to Hazlett and Schechter. 78 Davies 7-9 developed a general emulsion stability theory, which covers a wide variety of emulsion stability phenomena. For various ionic and nonionic surfactants, it predicts which type of emulsion, O/W or W/O is stable. In the late 1950s, the theory served as a basis for developing the additive HLB scheme.7 Moreover, it tackles the effect of electrolytes on emulsion type and stability and explains the emulsion break/ inversion observed in surfactant mixtures. Davies considers the following contributions to the coalescence barrier energy of nonionics: (i) The work of dehydration of the surfactant polar headgroups estimated as W2 ) θ∑Eh where Eh is the energy of dehydration per polar group and θ is the fraction of interface covered; (ii) the energy penalty due to direct contact of the surfactant hydrocarbon tails with water, evaluated as W3 ) 2mθU, where U is the free energy of hydration of one -CH2 group, ∼300 cal/mol, m is the number of carbon atoms in the surfactant chain, and the coefficient 2 comes “because water has to bridge the gap across 2m -CH2- chains”. There are many unclear points about how these free energy penalties are estimated. First, they refer to a single surfactant molecule (the barrier energy is then directly reduced to the thermal energy kT). This choice is rather arbitrary. The size of the critical hole nucleus may be (77) Safran, S. A. In Structure and Dynamics of Strongly Interacting Colloids and Supramolecular Aggregates in Solution; Chen, S., Huang, J. S., Tartaglia, P., Eds.; Kluwer Academic Publishers: Dordrecht, the Netherlands, 1992. (78) Hazlett, R. D.; Schechter, R. S. Colloid Surf. 1988, 29, 53.

284



Figure 8. Hazlett-Schechter percolation mechanism of coalescence.

much larger than a single surfactant molecule. Second, when a water passage is formed in an oil lamella, the unfavorable oil-water contact occurs regardless of the surfactant present. One may expect, therefore, that the hydrophobic energy penalty will not depend on the surfactant chain length and the fraction of interface covered but rather on the thickness of the lamella. Finally, it is unclear why the direct contact of oil and water at the passage must occur at all. The energy penalty of this transition state is very high and Nature most likely would choose another easier way. For example, the surfactant monolayer at the passage may curve to “heal” the naked interface. The hypothesis of Hazlett and Schechter was specifically designed to explain the PIT phenomenon, in particular the deemulsification observed in the balanced state. The authors convincingly show that the inversion cannot be adequately explained by DLVO arguments and look for an alternative mechanism. According to their ideas, the rupture of an emulsion film can be mediated with the help of micelles. Consider two oil droplets in an aqueous micellar solution. If the volume fraction of micelles is high enough, they can “short circuit” the oil droplets by a percolation mechanism (Figure 8). A volume fraction of micelles of ∼0.1-0.2 is believed to be sufficient to promote coalescence. The deemulsification in the balanced state is explained “by a higher volume fraction of micelles in this state”. Putting aside the fact that the middle phase of a Winsor III equilibrium has a bicontinuous, and not micellar structure, these arguments are quite paradoxical. Indeed, macroemulsion systems having micelles in the continuous phase are known to be stable, while those having them dissolved in the disperse phase are not. On the other hand, the volume fraction of micelles per se does not seem to play any role in emulsion stability. For instance, O/W emulsions stabilized by typical soaps (e.g. sodium dodecyl sulfate) do not break even if the micelle volume fraction reaches 0.2. In addition, micelles are usually depleted from the interface of emulsion droplets.79 In general, percolation arguments have sense if there is no energy barrier for the micelles to merge with each other or with emulsion droplets. This is most probably not the case, however, since each of these fusion acts involves a passage formation in the surfactant film and is related to a free energy penalty. The simultaneous fusion of several micelles and emulsion droplets in most cases has a negligible probability; see section 4.2. 4. Hole Nucleation Theory 4.1. Nucleation Pore in a Monolayer-Covered Emulsion Film. Our own interpretation of macroemulsion inversion at the PIT is based on thermally activated hole nucleation in emulsion films. We do not consider here the alternative hydrodynamic instability mechanism for film rupture proposed by Sheludko and Vrij80,81 and later developed by others.82,83 This mechanism is attrac(79) Aronson, M. Langmuir 1989, 5, 494. (80) Sheludko, A. D. Adv. Colloid Interface Sci. 1967, 1, 391. (81) Vrij, A. Discuss Faraday Soc. 1966, 42, 23. (82) Ruckenstein, E.; Jain, R. K. J. Chem. Soc., Faraday Trans. 2 1974, 70, 132.

Figure 9. Hole nucleation in a liquid film. The total surface area as a function of the hole radius shows a maximum at a ) 0.57b, if the hole propagation occurs at a constant film thickness.84

tive-force-driven, and its time scale is usually much shorter than the one observed in PIT systems (by “PIT systems” we mean any nearly-balanced microemulsion system where the coexistent phases are emulsified in each other). We assume that the emulsion films are stabilized by repulsive electrostatic or steric forces, with a positive second derivative of the interaction potential. This rules out the possibility of force-driven rupture. Consider a flat-parallel O/W/O liquid film with a hole in it; the edge of the hole is rounded (Figure 9). The surfactant monolayer, covering the emulsion film, is assumed to be in a thermodynamic equilibrium with the micellar solution in the bulk. The free energy penalty of forming a hole can be evaluated as the difference in the surface tension integrals over the film surface area before and after forming the hole:84

W)

∫

∫

σ(x,y,z) dA - σ(x,y,z) dA (flat film) (film with hole)

(3)

The emulsion film rupture is driven by reducing the surface area of the planar part of the emulsion film. On the other hand, the edge of the nucleation hole creates an extra surface area and, therefore, a free energy penalty. The energy barrier of nucleation comes from the interplay of the free energy penalty at the edge of the hole and the free energy gain at the planar part. Up to this point, our model essentially coinsides with de Vries theory.84 This theory further assumes that the interfacial tension of the edge of the hole is equal to that of the planar part of the film. If the hole propagation occurs at a constant film thickness, the total surface area of the film shows a maximum at a certain hole radius (Figure 9). The coalescence barrier energy then can be regarded as a product of this excess surface area and the interfacial tension.84 Although our model is analogous to that of de Vries, there are two major differences between them. First, we assume that the surface tension at the edge is different (83) Maldarelli, C.; Jain, R. K. In Thin Liquid Films: Fundamentals and Applications; Ivanov, I. B., Ed.; Marcel Dekker: New York-Basel, 1988; Vol. 29, p 497. (84) de Vries, A. J. Recl. Trav. Chim. Pays-Bas 1958, 77, 383.


Langmuir, Vol. 12, No. 2, 1996 285

from that of a planar film, because the monolayer “frustration” at the edge is not the same as that in a planar monolayer:

σcurved ) σplanar + 2κ(H - H0)2 - 2κH02

(3a)

evaluated in explicit form:

{

W3 ) 2πκ 2πH0a +

2(a + b)2 bxa(a + 2b)

arctan

xa +a 2b +

}

2(π - 4)bH0 - 4 Here σplanar and σcurved are the interfacial tensions of the curved and planar films, respectively and H is the mean curvature of the curved patch. To elucidate this equation, consider an emulsion film patch, having a mean curvature H and being in thermodynamic equilibrium with the micellar solution. Compare the state of monolayers, covering the interface of the micelles and the emulsion film patch. In the micellar solution, the monolayer has a third dimension to its disposal and can acquire the curvature which is close to the spontaneous curvature H0 (for a moment, we assume the saddle splay modulus is equal to zero). On the other hand, the monolayer, covering the emulsion film patch is constrained to a certain curvature and is, in general, frustrated, the more so the higher the difference between H and H0. Therefore, the “curvature component” of the surfactant chemical potential is bigger in the emulsion film than in the micelle. Since both monolayers are in a thermodynamic equilibrium with each other, this increment of the chemical potential must be canceled by the reduced lateral pressure of the monolayer, covering the emulsion film. This results in the dependence of the interfacial tension of emulsion films on the monolayer spontaneous curvature H0 and the actual (geometrical) mean curvature of the film H; the closer those values, the lower the interfacial tension. The full derivation of eq 3a is given in the Appendix. The free energy of the hole W, eq 3, can be presented as the sum of four terms. (i) The first interfacial tension term is equal to the increment of the surface area for the planar part of the film multiplied by the interfacial tension of the planar monolayer:

W1 ) -2πσ(a + b)2

(4)

(ii) The second and third terms refer to the surface of revolution formed by revolving a semicircle around the symmetry axis. The W2 term is equal to its surface area multiplied by the interfacial tension of the planar monolayer:

W2 ) 2πσ[πb(a + b) - 2b2]

(5)

The third “bending energy” term accounts for the extra (positive or negative) interfacial tension of the surface of revolution with respect to the planar state (for detailed arguments, see Appendix):

∫

κ W3 ) [ (2H0 - H1 - H2)2 - 4H02] dA 2

(6)

The principal radii of curvature at a given point of the surface of revolution are equal to H1 ) -1/b (the “meridianal” curvature) and H2 ) (sin θ)/x (the “parallel” curvature), where x is the distance from a given point on the surface of revolution to the symmetry axis and φ is the angle between the vertical and the vector connecting the center of the semicircle and the point on the surface of revolution (Figure 9). The signs of H1 and H2 refer to the passage in an O/W/O film; for a W/O/W film, the signs must be reversed. The 4H02 term accounts for the bending energy of the planar monolayer. The integral (6) can be

(7)

Again, for W/O/W films, the sign of the terms, proportional to H0 must be reversed. Equation 7 was first derived by Kozlov and Markin 85 in reference to “stalk” formation in biological membranes. It was rederived by Nanavati et al. 86 (iii) The fourth term is the Gaussian curvature contribution, accounting for the fact that fusion of two emulsion droplets reduces the Euler characteristic of the monolayer by 2:

Wg ) -4πκj

(8)

Note that this contribution does not depend on the film orientation and is the same for O/W/O and W/O/W films. The total free energy is equal to the sum of eqs 4, 5, 7, and 8

W ) W1 + W2 + W3 + Wg

(9)

The interfacial tension of a planar emulsion film shows a deep minimum in the balanced state of a surfactant monolayer, which can be explained by the monolayer frustration arguments;35,87-90 see also Appendix. We expand the interfacial tension of the planar film in series versus the monolayer spontaneous curvature in the vicinity of the balanced state:

σ ) σ0 + 2H02λ

(10)

The linear in curvature term is lacking due to symmetry; the coefficient of 2 instead of 1/2 is introduced for convenience; see below. It is possible to show that, within the Safran model of spherical micelles,91 λ ) κ2/(κ + κj/2) and σ0 ) 0; see Appendix. In the simplest case of zero saddle splay modulus, λ ) κ. The experimental values of the coefficients are λ ∼ kT and σ0 ∼ 10-3-10-4 dyn/cm for typical microemulsion systems, see, e.g., refs 35 and 8890. The second difference of our model from de Vries theory84 is that both the hole radius and the film thickness are allowed to vary. At an arbitrary hole radius a, the film adjusts its thickness b to a value at which the hole free energy has a minimum. We do not consider any interaction forces penalty which might be involved in this adjustment. This may be a reasonable approximation because, as will be shown later, the optimal thickness is usually on the order of hundreds of angstroms, a distance at which the interactions are weak. At any fixed spon(85) Kozlov, M. M.; Markin, V. S. Biofizika 1983, 28, 242. Due to geometrical similarity, the bending energy of the “stalk” joining two bilayers and the hole spanning one bilayer can be presented by essentially the same bending energy terms. (86) Nanavati, C.; Markin, V. S.; Oberhauser, A. F.; Fernandez, J. M. Biophys. J. 1992, 63, 1118. There is an inconsistency in the literature about the absolute value and sign of the spontaneous curvature in eq 1. In our paper, we follow the same notation as, e.g., in ref 35. Note that in the paper of Nanavati et al. the spontaneous curvature is a factor of 2 larger and has the opposite sign, in comparison with our notation. In addition, the third term on the right hand side of eq 7 contains an erratic multiplier (π/2 - 4) instead of (π - 4). Our equation is consistent with the original Kozlov and Markin paper.85 (87) de Gennes, P. G.; Taupin, C. J. Phys. Chem. 1982, 86, 2294. (88) Guest, D.; Langevin, D. J. Colloid Interface Sci. 1985, 112, 208. (89) Binks, B. P.; Meunier, J.; Abillon, O.; Langevin, D. Langmuir 1989, 5, 415. (90) Aveyard, R.; Binks, B.; Lawless, T.; Mead, J. Can. J. Chem. 1988, 66, 3031. (91) Safran, S. A. Phys. Rev. A 1991, 43, 2903.

286



taneous curvature, the σ value is determined by eq 10, and the free energy (9) becomes a unique function of the hole radius a and the film half-thickness b, W(a,b). Note that, rigorously speaking, one cannot discriminate the linear- and quadratic-in-a terms in the free energy, and one cannot, therefore, treat the hole nucleation problem in terms of the linear and surface tensions, as it is done in refs 24-31. The latter approach is valid as an approximation if the hole radius a is much bigger than the film half-thickness b; see, e.g., ref 32. We call this limit the “big hole” approximation (BHA). Its applicability is discussed in section 4.3. Equations 4, 5, 7, 8, and 10 are not very transparent. Before proceeding with numerical calculations, we outline the physical scenario of hole nucleation. Consider an O/W/O film with H0 , 0, or a W/O/W film with H0 . 0 (we define these films as “inside-out” films). At any arbitrary (in absolute value) spontaneous curvature, the film can adjust its thickness in such a manner that the spontaneous curvature of the monolayer H0 will fit the mean curvature of the hole edge, viz. H ) (H1 + H2)/2 ≈ H0. At this optimal thickness, the surface tension of the hole edge is very low and stays low for any arbitrary hole radius a. This means that the film breaks without a significant energy barrier. Consider now an O/W/O film with H0 . 0, or a W/O/W film with H0 , 0 (we call these films “normal”). In this case, the mean curvature of the monolayer at the edge of the hole H and the spontaneous curvature H0 have opposite signs. Thus, there is always a monolayer frustration energy involved in the hole formation. The frustration can be reduced by increasing the film thickness: in this case, H f 0 and the difference between H and H0 decreases. The decrease in the Hookean (H -H0)2 term is, however, traded for an increase in the total surface area of the edge. As a result, at any given hole radius, the free energy must have a minimum at some finite (“optimal”) bmin value. Now, allow the radius of the hole to increase and keep the film thickness variable at the “optimal” value b ) bmin(a). Increasing the hole radius increases the interfacial free energy by reducing the surface area of the planar part, which is, loosely speaking, ∼a2. This is offset somewhat by an increase in the surface area of the edge ∼a. As in traditional 2-D nucleation theories, the squared-in-radius term overweighs the linear-in-radius term and the W(a,bmin(a)) curve must have a maximum at a ) amax. Hence, the free energy surface W(a,b) of the “normal” film must have a saddle point: a maximum versus a and a minimum versus b. The problem of finding the energy barrier against coalescence is then reduced to determining the saddle point of this surface:

∂W ) 0; ∂a

∂2W >0 ∂a2

∂W ) 0; ∂b

∂2W 0 and W/O/W films at H0 < 0 “normal” and W/O/W films at H0 > 0 and O/W/O films at H0 < 0 “inside-out” films. Nomenclature σ, σplanar ) the interfacial tension of a planar film σcurved ) the interfacial tension of a curved film σ0 ) the interfacial tension at zero spontaneous curvature λ ) the second expansion-in series coefficient of the interfacial tension vs spontaneous curvature γ ) the linear tension of the hole a ) the hole radius b ) the film half-thickness κ ) the bending modulus κj ) the saddle splay modulus H0 ) the spontaneous curvature H h 0 ) H0 (κ/(κ + κj/2) H1, H2 ) the principal curvatures of the monolayer H ) (H1 + H2)/2, the mean curvature A ) the film surface area As ) the area per surfactant molecule W ) the film free energy amax ) the hole radius at which the free energy reaches maximum bmin ) the film thickness, at which the free energy reaches minimum µs ) the surfactant chemical potential Π ) the surfactant interfacial pressure f ) the film rupture frequency per cm2 f0 ) the film rupture preexponent N ) the number of bilayers F1, F2, F3, F4, f1, f2, f3 ) the scaling numerical coefficients

Acknowledgment. The idea of the paper has emerged after discussions with Ulf Olsson, Jeff Weers, and Tom Tarara. We are thankful to Jeff Weers and Leo Trevino for comments on the manuscript. The paper was technically written during the stay of A.K. with Alliance Pharmaceutical Corp., San Diego, CA. 6. Appendix Interfacial Tension of Planar and Curved Films, Covered with Saturated Surfactant Monolayers. The oil vs water interfacial tension shows a deep minimum in the balanced state in both ionic and nonionic microemulsion systems see, e.g., refs 35 and 88-90. The same trend is observed in phospholipid-oil-water systems.65 This behavior was explained by de Gennes and Taupin87 and others88,89,35 by monolayer frustration arguments. Consider a planar monolayer at the oil-water interface in equilibrium with spherical micelles. The planar monolayer is frustrated with respect to the micellar state because the mean curvature of the former is fixed at zero. On the other hand, the monolayer covering the interface of the swollen micelles has a third dimension at its disposal and can more efficiently minimize its bending energy. Safran91 has shown that, under neglect of entropy of mixing term, the micellar free energy is minimized at the micelle

Langmuir, Vol. 12, No. 2, 1996 291

radius r ) 1/H0, where

κ κ + κj/2

H h 0 ) H0

(34)

The bending energy at this minimum equals

(

h 20κj Wmic ) H

κj 2κ

)

+1 A

(35)

where A is the total surface area of the monolayer, covering the surface of the micelles. We assume the surfactant area-per-molecule As to be constant. The total surface area is therefore equal to ns As where ns is the number of surfactant molecules in the system. On the other hand, the surfactant monolayer forming the emulsion film has zero mean and Gaussian curvatures and its bending energy is equal to

Wfilm ) 2H02κ A

(36)

The free energy gain on forming a micelle is, therefore, equal to

∆W ) Wfilm - Wmic ) 2H h 02(κ + κj/2)A

(37)

Alternatively, eq 37 may be written in terms of the surfactant chemical potential (the free energy per molecule)

∆µs ) 2H h 02(κ + κj/2)As

(38)

The curvature component of the surfactant chemical potential is, therefore, higher in the emulsion film than in the micelle. Since the chemical potentials of the surfactant molecule in the micelle and the emulsion film must be equal, this increment must be canceled by the reduced interfacial pressure in the emulsion film. Assuming the monolayer to be incompressible, one gets

∆µs ) As∆Π

(39)

where ∆Π is the increment of the surface pressure. Since the monolayer forming the micelle is under zero tension due to the cancellation theorem,87 the O/W emulsion film has an interfacial tension of35,89

σ ) -∆Π ) 2H h 02(κ + κj/2) ) 2H02

κ2 κ + κj/2

(40)

According to this model, the interfacial tension must increase quadratically with spontaneous curvature, which is indeed observed in the experiment.35 However, the interfacial tension does not vanish in the balanced state, as is predicted by eq 40. The failure of eq 40 in the balanced state is not surprising, because the treatment is not applicable to balanced bicontinuous microemulsions. Now, assume that the emulsion film is curved to a mean curvature H ) (H1 + H2)/2. If the interfacial tension of the planar film is known, the frustration free energy can be estimated with the planar monolayer as a reference point. This leads to the following expression:

σcurved ) σplanar + 2κ(H - H0)2 - 2κH02 + Gaussian term (41) Note that this equation is valid even in the balanced state (Winsor III equilibrium), because all the unknown free energy terms of the bicontinuous microemulsion phase

292


are incorporated into the first term on the right hand side. Consider now the fourth Gaussian curvature term of the curved monolayer (the Gaussian curvature term of the planar monolayer is equal to zero). For a hole in an emulsion film, the Euler characteristic of the monolayer gets an increment of -2, and a Gaussian curvature penalty, 2κjχ, is involved (eq 8). Note that this penalty is not dependent on the size of the hole and film orientation (i.e. it is the same for O/W/O and W/O/W films). It,


therefore, does not contribute to the interfacial tension (insofar as it is not proportional to the film area) but contributes to the coalescence barrier (eq 8, Wg term). The first term on the right hand side of eq 41 equals the sum of W1 and W2 terms. On the other hand, the sum of the second and the third term on the right hand side of eq 41 is equal to W3 term. LA950359E

Macroemulsion Stability: The Oriented Wedge Theory Revisited

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