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Electrostatic Interactions in Ionic Microemulsions Ingemar Carlsson, Andrew Fogden, and Håkan Wennerstro¨m* Physical Chemistry 1, Center for Chemistry and Chemical Engineering, University of Lund, P.O. Box 124, Lund S-221 00, Sweden Received December 3, 1997. In Final Form: April 6, 1999 We have analyzed the role of electrostatic interactions in ionic microemulsions. Using a recently developed formalism (Daicic, J.; Fogden, A.; Carlsson, I.; Wennerstro¨m, H.; Jo¨nsson, B. Phys. Rev. E 1996, 54, 39843998), we have been able to qualitatively, and in some cases quantitatively, account for a number of experimental observations. We account for (i) the change in the relative stability of lamellar and microemulsion phases as the salt content is varied, (ii) the tilted appearance of the “fish” plot for ionic systems, (iii) the sequence of phases observed as the temperature is changed, (iv) the relative location of the Winsor three-phase area and the lamellar phase, (v) the interrelation between salt content, temperature, and surfactant concentration for the location of balanced conditions, and (vi) the lack of symmetry with respect to an exchange of water and oil at balanced conditions. The overall conclusion is that the Helfrich curvature energy formalism can be successfully adopted for the description of ionic microemulsions, provided that one describes the electrostatic effects using the Poisson-Boltzmann equation at finite film concentrations.
1. Introduction In a microemulsion a monolayer surfactant film separates oil and water domains in a thermodynamically stable isotropic solution. The film can form a wide variety of structures of differing geometry and topology, ranging from oil in water spherical droplets, elongated aggregates, bicontinuous systems, to water in oil droplets. The dominant parameter determining the structure is the spontaneous curvature of the film.1 Even for chemically rather different systems one observes a generic sequence of phases and aggregate structures. Of particular importance for the description of a microemulsion system is the so-called balanced conditions where the film has zero spontaneous curvature. In this state the film curves equally readily toward the water regions as toward the oil ones, and as a result the solution has a bicontinuous structure. To reach the balanced conditions in an experimental situation one has to be able to tune some intensive variable. For microemulsions based on nonionic surfactants of the alkyl oligoethylene oxide type the spontaneous curvature is very temperature sensitive and it is possible to cover the whole range from oil in water to water in oil droplets over an easily accessible temperature range. With ionic surfactants this problem is less readily solved. First, one has to find a surfactant that forms a film with a spontaneous curvature reasonably close to zero. In the pioneering studies2 this was accomplished by combining a micelle-forming ionic surfactant and cosurfactant alcohol like pentanol or hexanol. Later, double chain ionic surfactants were used to avoid the complications of controlling the mixing ratio between surfactant and cosurfactant. To “fine-tune” the spontaneous curvature, electrolyte is added,3 which affects the electrostatic interactions in general and thus also the spontaneous * To whom correspondence should be addressed. Email:
[email protected]. (1) Olsson, U.; Wennerstro¨m, H. Adv. Colloid Interface Sci. 1994, 49, 113-146. (2) Schulman, J. H.; Stoeckenius, W.; Prince, L. M. J. Phys. Chem. 1959, 63, 1677. (3) Kahlweit, M.; Strey, R.; Firman, P.; Haase, D. Langmuir 1985, 1, 281.
curvature of a charged film. An additional finding was that the electrolyte also was necessary to suppress the formation of a lamellar phase at balanced conditions. For the nonionic systems the phase behavior shows some remarkable symmetry properties.4 Temperature changes T0 + ∆T f T0 - ∆T, where T0 is the temperature of the balanced film, combined with composition changes R f 1 - R, where R is the volume fraction oil/(oil + water), leads to a new system with the same phase and the same mesoscopic structure except that oil and water have exchanged positions and the film curvature has changed sign. This symmetry property follows from the assumption that the spontaneous curvature of the film is the dominant parameter determining the structure in the system and that it varies linearly with temperature. For ionic systems the symmetry properties are qualitatively the same, but there are obvious quantitative deviations. Figure 1 illustrates this for a so-called “fish plot” showing the observed phases at a constant ratio of oil to water (R ≈ 0.5) and with surfactant concentration and temperature as the independent variables. For the nonionic system of Figure 1a5 the diagram is symmetric about the line T ) T0, which is a manifestation of the symmetry property described above, since for R ) 0.5 one has R ) 1 - R. For the ionic AOT system of Figure 1b,6 on the other hand, the microemulsion single-phase and the Winsor three-phase areas are tilted relative to temperature axes. It has been pointed out that this difference between the nonionic and the ionic systems is an expected effect of electrostatic interactions.4,7 There are some additional qualitative differences between the ionic and nonionic systems. For the latter, the oil in water droplet microemulsions are found at temperatures lower than the balanced one, demonstrating that the spontaneous curvature is decreasing with in(4) Evans, F.; Wennerstro¨m, H. The Colloidal Domain-where Physics, Chemistry, Biology and Technology Meet, 1st ed.; VCH Publishers: New York, 1994. (5) Kahlweit, M.; Strey, R.; Firmin, P. J. Phys. Chem. 1986, 90, 671677. (6) Kahlweit, M.; Strey, R. J. Phys. Chem. 1986, 90, 5239-5244. (7) Lekkerkerker, H. N. W.; Kegel, W. K.; Overbeek, J. T. G. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 206-217.
10.1021/la971333l CCC: $18.00 © 1999 American Chemical Society Published on Web 07/09/1999
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1b, the lamellar phase occurs at higher temperatures than that where the microemulsion is most stable. When experimentally finding a bicontinuous microemulsion with an ionic surfactant, one can either work at a fixed temperature and vary the electrolyte concentration to locate the balanced point or fix the electrolyte content of the aqueous phase and vary the temperature. The general observation is that the temperature at the balanced point increases with increasing electrolyte content. This effect should also be caused by the electrostatic interactions and their temperature dependence. In this paper we aim at describing these effects in ionic microemulsions that can qualitatively be ascribed to electrostatic interactions. We do this in a quantitative formalism based on a combination of the Helfrich bending free energy and a description of the electrostatics using the Poisson-Boltzmann (PB) equation. Previously we have shown that the nonlinear PB equation, combined with the cell model of liquids, is adequate to describe the effect of electrostatics on rather intricate phase equilibria8-11 in systems with phases containing aggregates of simple geometries. Similarly, the bending energy concept has been successfully used to describe phase equilibria in systems of bicontinuous microemulsions12,13 and the structurally related sponge phases.14,15 Recently, Fogden, Daicic, and co-workers16-19 have developed a formalism for determining the electrostatic contributions to the parameters in the bending free energy that incorporates the essence of the cell model and allows for an analysis of relative phase stability. The current paper can be seen as an illustration of the usefulness of this approach to microemulsion systems. The paper is organized so that in section 2 we summarize the theory for the description of the electrostatic contributions to the bending free energy. Then we address the qualitative questions raised in the Introduction and substantiate with explicit calculations. This is followed by a brief concluding section. 2. Theory The curvature free energy has turned out to be a very useful concept, for both the general understanding and the quantitative description of microemulsions. When the area density of curvature free energy for the monolayer, gc, is expanded to second order, one obtains the Helfrich form20
gc ) 2κ(H - H0)2 + κjK Figure 1. (a) Fish cut for the C12E5/n-tetradecane/Water system from ref 5 for a mixture of constant 1:1 oil/water volume ratio. The dashed line marks an area where a lamellar phase is present. (b) Fish cut for the AOT/n-decane/water/NaCl system from ref 6 for a mixture of constant 1:1 oil/brine weight ratio and ) 0.58 wt % of NaCl in the water. Markers are experimental determinations of phase boundaries, while the solid and dashed lines represent interpolated boundaries.
creasing temperature (if taken as positive when curved toward oil). For the ionic systems the trend is the reverse, so the oil in water droplets are found in the high temperature range for the few systems studied. Again this has qualitatively been attributed to the special entropic character of electrostatic interactions in an aqueous medium.4 Another obvious difference is that for the nonionic systems the lamellar phase has its maximum swelling at the same temperature as for the microemulsion, while for the ionic system, as can be seen in Figure
(1)
(8) Jo¨nsson, B.; Persson, P. K. T. J. Colloid Interface Sci. 1987, 115, 507-512. (9) Jo¨nsson, B.; Wennerstro¨m, H. J. Colloid Interface Sci. 1981, 80, 482-496. (10) Rajagopalan, V.; Bagger-Jo¨rgensen, H.; Fukuda, K.; Olsson, U.; Jo¨nsson, B. Langmuir 1996, 12, 2939-2946. (11) Jo¨nsson, B.; Wennerstro¨m, H. J. Phys. Chem. 1987, 91, 338352. (12) Daicic, J.; Olsson, U.; Wennerstro¨m, H. Langmuir 1995, 11, 2451-2458. (13) Andelman, D.; Cates, M. E.; Roux, D.; Safran, S. A. J. Chem. Phys. 1987, 87, 7229-7141. (14) Daicic, J.; Olsson, U.; Wennerstro¨m, H.; Jerke, G.; Schurtenberger, P. J. Phys. II 1995, 5, 199-215. (15) Anderson, D.; Wennerstro¨m, H.; Olsson, U. J. Phys. Chem. 1989, 93, 4243-4253. (16) Fogden, A.; Daicic, J. Colloids Surf. A, in press. (17) Fogden, A.; Daicic, J.; Kidane, A. J. Phys. II 1997, 7, 229-248. (18) Fogden, A.; Daicic, J.; Mitchell, D. J.; Ninham, B. W. Physica A 1996, 234, 167-188. (19) Daicic, J.; Fogden, A.; Carlsson, I.; Wennerstro¨m, H.; Jo¨nsson, B. Phys. Rev. E 1996, 54, 3984-3998. (20) Helfrich, W. Z. Naturforsch. 1973, 28c, 693-703.
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where H, H0, and K denote the mean, spontaneous mean, and Gaussian curvatures, respectively. κ is the bending rigidity modulus and is the saddle splay modulus. The parameters H0, κ, κj and depend on intensive thermodynamic variables such as temperature, pressure, and composition of the solvents, but in the original formulation they were regarded as independent of the concentration of surfactant film. However, for charged films there is an electrostatic interaction, which can be long ranged, introducing to H0, κ, and κj a dependence on both salt and surfactant concentration. For the purposes of this study we express gc as the sum of an electrostatic and a nonelectrostatic part (with the concentrations only explicitly entering the former contribution). Thus the coefficients of H, H2, and K in eq 1 (excluding factors of 2 and 4) become
κH0 ) ∆1el + ∆1nonel
(2)
κ ) κel + κnonel
(3)
κj ) κjel + κjnonel
(4)
The two contributions to κH0 are labeled ∆1, indicating that they originate from the first-order (linear) term in the curvature expansion and avoiding confusion from the fact that H0 alone only has a physical meaning once the sum of both contributions has been minimized. There is a considerable amount of literature on the electrostatic contributions to the bending moduli for the limiting case of a single monolayer or bilayer in excess electrolyte,21-23 which accounts for the salt concentration dependence at sufficiently high brine dilutions. These studies can serve as a useful starting point, but for many problems it is also essential to properly describe the surfactant concentration dependence, away from the highly diluted regime, where interactions between adjacent electrical double layers take place. As we will see, this effect is significant over the concentration ranges in the systems studied here. The generalized theory of Daicic et al.19 extracted the electrostatic contribution, within the nonlinear PB description, through a perturbation around the planar state, imposing a small bend in spherical and cylindrical geometries. They provided results valid for both finite salt and surfactant concentrations, which, in a condensed form, can be written as
or ∆1el ) 2 κel ) -
or 2
∫0d
0
∫0
d0
[
dXX 2
κjel ) - or
dXX
(dψ dX)
2
(5)
dψ dψ ∂ dψ -X dX ∂H dX dX
∫0d
0
2
( ) ( )] dψ dXX ( ) dX 2
2
(6)
formulas are expressed analytically in terms of the solution for the planar reference state. The appearance of ∂(dψ/ dX)∂H indicates that κ will depend on external constraints, since the partial derivative is dependent upon which variables are held constant during bending, in contrast to the X derivative for which the choice of constant variables is uniquely defined by the planar reference state. Three cases were considered with increasing degree of bending energy penalty for the system: (1) fixed water and salt chemical potentials; (2) fixed salt chemical potential and amount of water; (3) fixed amounts of water and ions. The third case represents a natural choice for the present study, since we consider a one-phase region; hence there is no reservoir or other phase for water or ions to escape to during bending. The paper we now have reviewed deals with spherical and cylindrical bending; other studies considered sinusoidally rippled planes.16-18 These three choices of geometry were found to give consistent results provided the bending energy is the leading-order term. This geometrical invariance is of course important when we now bring the results over to the bicontinuous phase, which is disordered and hence complex in its global geometry. 3. Bicontinuous Microemulsion versus Lamellar Phase For a film system where interactions are of short range, as when the surfactant is nonionic, the free energy, G, per volume, V, of both bicontinuous microemulsion and lamellar phases can be found, by a scaling argument,13 to be proportional to the cube of the volume fraction of surfactant, Φ (here equal to the bilayer volume fraction), so that
G kBT ) 3 a Φ3 V l
Here kB is Boltzmann’s constant, T is absolute temperature, and ls is the monolayer thickness. The dimensionless coefficient a depends on the nature of the phase and on the values of the parameters Ho, κ, and κj. For a lamellar phase Helfrich derived24
aLR )
(21) Lekkerkerker, H. N. W. Physica A 1989, 159, 319-328. (22) Lekkerkerker, H. N. W. Physica A 1990, 167, 384-394. (23) Mitchell, D. J.; Ninham, B. W. Langmuir 1989, 5, 1121-1123.
3π2 kBT 1024 κ
(9)
and the coefficient is independent of κj due to the simple topology in the lamellar state. For the microemulsion there is both an entropic and a topology-dependent term. The former provides a negative contribution to a, and thus to G, while the topological term should be positive for microemulsions. We are not aware of any quantitatively reliable derivation of the entropic term, but in the high κ limit it should be of the same character as in eq 9, giving
(7)
Here o is the permittivity of vacuum, r is the dielectric constant of water, do is the distance to the midplane between two charged parallel planar surfaces, and ψ is the electrostatic potential at some distance X from either charged surface in the intervening salt solution. A brief recount of the PB solution for the planar geometry and a complete substitution of the two functionals dψ/dX and ∂(dψ/dx)/∂H are given in the ref 19. Hence all three
(8)
s
aµE ≈ -constant1
kBT κj - constant2 κ kBT
(10)
where we expect κj to be negative in the present case.10 A problem with this description is that the concentration dependence of the free energy is the same for the microemulsion and the lamellar phase, where one of the two will be the most stable in the whole concentration range. In ref 12 we argued that when aµE < aLR the microemulsion will be most stable in the low concentration range but that at increased concentrations higher-order (24) Helfrich, W. Z. Naturforsch. 1978, 33a, 305-315.
Electrostatic Interactions in Ionic Microemulsions
terms appear in the expression for the free energy in eq 8. These are more significant for the bicontinuous structure than for the planar one, which leads to a transition to a lamellar state at these higher concentrations. However, when aLR < aµE only the lamellar phase will appear. When we now consider a system where the surfactant film is charged, the description changes in two important respects. First, the parameters Ho, κ, and κj become concentration dependent through the long-range character of the electrostatic interactions, although we now have a formalism that quantitatively accounts for this. Second, there also emerges a contribution to the free energy from the electrostatic interactions that is independent of the nature of the phase but which does not obey the Φ3 scaling law. This is the electrostatic free energy of a reference planar lamellar state and its value depends only on the surfactant-to-water ratio when the oil domains are considered free of ionic species. An important consequence of this term is that the nice symmetry in the R value is broken, so that one should not expect a symmetric phase diagram as in ref 13 at balanced conditions. Consequently in Figure 1a the temperature where the microemulsion extends to the lowest concentrations can be identified as the balanced temperature, while for the ionic system of Figure 1b this cannot be done with full confidence. However, below we will also adopt this criterion for the ionic systems, remembering though that it is only an approximation and may introduce errors in quantitative applications. There is a substantial electrostatic contribution to the bending elastic constants κ and κj. The lower the salt content and the lower the surfactant concentration, the higher the magnitude of the elastic coefficients, since the charge-charge interactions make the films stiffer. Figure 2 shows κel and κjel (calculated from eqs 6 and 7) as a function of the 1:1 salt concentration, both for a film at high dilution (0.1% AOT) and for a film concentration corresponding to 12% AOT. The electrostatic contribution to both quantities is of order kBT (for T ) 298 K) at low electrolyte and decreases in magnitude with increasing electrolyte content, with a stronger salt dependence for κ than for κj in the 12% case. Also, the κj magnitudes are smaller at the finite film concentration than at infinite dilution, while for κ the situation is reversed for salt concentrations above ∼0.05 wt %, the relative difference being smaller for κ than for κj. We can use the values of Figure 2 to investigate how the relative stability of lamellar and microemulsion phases develops with increasing salt content. For the lamellar phase a decrease in κ leads to an increase in free energy (see eq 9), while for the microemulsion the corresponding free energy term has the opposite sign so there the free energy is decreased. Additionally κj decreases in magnitude which also results in a decrease of aµE in eq 10. We should thus expect an increased stability of the microemulsion phase relative to the lamellar as the electrolyte concentration is increased. The argument is qualitative in nature, but the steep rise in the elastic constants at low electrolyte concentrations makes it very reasonable that only lamellar phases are observed in this limit. 4. Temperature and Concentration Dependence of the Electrostatic Contributions As pointed out in the Introduction, the temperature dependence of the structure in a microemulsion is opposite for the nonionic system of Figure 1a and the ionic of Figure 1b. Having an independent quantitative description of the electrostatic contribution, we can analyze its tem-
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Figure 2. Calculated curves for κel with the constraints of a fixed amount of water and ions (solid lines), and κjel (dashed lines), at 12 wt % AOT and 0.1 wt % AOT. At the low AOT concentration the elestic constants increase drastically as the salt concentration decreases. System parameters for AOT:14 monolayer thickness ls ) 9.5 Å; area per headgroup, ao ) 67 Å2; T ) 298 K.
perature dependence. In general the electrostatic free energy can be seen as a contribution from two parts:25 one “energetic” from the ion-ion interactions and one from the entropy of the ions. However, due to the temperature dependence of the dielectric permittivity of water, even the “energetic” part is, from a strictly thermodynamic point of view, also mainly entropic. Thus increasing the temperature increases the magnitude of the electrostatic terms nearly proportional to the relative change in temperature. To illustrate the magnitude of the purely electrostatic temperature effects on the spontaneous curvature, Figure 3 shows the calculated curve H0 ) 0 for the system with 1 wt % of salt in the water ( ) 1), under the assumption that the nonelectrostatic contribution is temperature independent. H0 is arbitrarily chosen to be zero at 22 °C for vanishing surfactant concentration. The higher the film (surfactant) concentration, the higher the temperature at which H0 equals zero. The graph also shows two lines at constant finite values of H0, equal to plus or minus 10-3 Å-1. The upper curve is for a positive spontaneous curvature corresponding to the formation of oil in water droplets, while the lower curve is for a negative spontaneous curvature consistent with a water-in-oil system. These calculations indeed confirm the experimentally observed trend in structural change with temperature changes in ionic systems. The balanced point where the spontaneous curvature is zero provides the condition for optimum stability of bicontinuous microemulsions. In the ionic systems there is an electrostatic interaction across the aqueous regions and this affects the spontaneous curvature, which consequently becomes concentration dependent. There is in (25) Evans, F.; Wennerstro¨m, H. In The Colloidal Domain; VCH Publishers: New York, 1994; Chapter 3.
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Figure 3. Calculated curves of constant spontaneous curvature, H0 ) 0 (dashed line); film curved toward oil, H0 ) 10-3 Å-1 (upper solid line); film curved toward water, H0 ) -10-3 Å-1 (lower solid line). In all cases ) 1.0 wt % of NaCl in the water. H0 is chosen to be zero at 22 °C at infinite dilution. System parameters are the same as those cited previously for AOT (see caption for Figure 2).
fact two different mechanisms that contribute to the concentration dependence of H0. In addition to the direct interaction there is also an effect which is related to changes in the chemical potential of salt in the electrolyte. Experimentally one controls the average concentration of electrolyte in the aqueous region. In the presence of the charged surface of the surfactant film, this electrolyte will be depleted from a zone of order the Debye length close to the film. Thus the electrolyte concentration increases somewhat in the remaining aqueous region and the chemical potential is slightly higher than that for a homogeneous aqueous solution of the same average concentration. This effect operates even at concentrations where the characteristic size of the aqueous domains greatly exceeds the Debye length. In Figure 3 we see that the curve H0 ) 0 slopes toward higher temperatures with increasing film concentration. One might naively think that the shorter the distance between the surfaces the larger the electrostatic contribution to the spontaneous curvature. The calculations show that in fact the opposite is the case. It is a general rule that the higher the concentration, the smaller the electrostatic contribution to the free energy, and the effect on spontaneous curvature is no exception. The fact that the lines of constant H0 tilt even at the lowest concentrations is due to the variation of electrolyte chemical potential discussed above. On the basis of the results of Figure 3 we can qualitatively understand additionally two of the features that are different between nonionic and ionic systems. In Figure 1b the body of the “fish” tilts relative to the concentration axis so that both the upper and lower borders are displaced to higher temperatures. Clearly this is due to the fact that the lines of constant spontaneous curvature tilt in this same direction. That this is the relevant interpretation is more apparent when one considers the location of the lamellar phase. As seen in Figure 1b, this
Carlsson et al.
Figure 4. Fish cut for the AOT/n-tetradecane/water/NaCl system for a mixture of constant 1:1 oil/brine weight ratio. Solid lines and markers are experimental determinations of phase boundaries for three different weight percents of salt in water (in decreasing order for ) 1, ) 0.58, and ) 0.4). Data are taken from ref 6. Dashed lines are calculated curves for H0 ) 0 under the asumption that H0 is zero at the temperature at which the microemulsion extends to the lowest concentration. These points are labeled A, B, and C for ) 1, ) 0.58, and ) 0.4, respectively, and they were used to determine the parameters of eq 12: β1 ) 6.22 × 10-15 J/m K and β2 ) 1.62 × 10-17 J/m K2. System parameters for AOT (see caption for Figure 2).
is found at temperatures substantially higher than that where the Winsor three-phase equilibrium occurs. Both the three-phase equilibrium and the lamellar phase have optimal conditions at H0 ) 0 and that they are observed at different temperatures we interpret as due to the combined temperature/concentration dependence of the electrostatic contribution to the spontaneous curvature demonstrated in Figure 3. 5. Electrolyte Dependence of the Spontaneous Curvature The experimentally simplest and conceptually most direct way to effect electrostatic interactions is to vary the electrolyte concentration in the bulk medium. For ionic systems, microemulsions can be prepared over a range of salt concentrations provided that one also varies the temperature. Figure 4 shows a “fish” plot for three different electrolyte concentrations in the AOT-water-decane system. In addition to the tilt of the "fish”, which we have already discussed, we see that the microemulsion range moves to higher temperatures the higher the electrolyte content. On the basis of the discussion in the previous section, we should expect this trend. Adding electrolyte weakens the importance of the electrostatic interactions, but we concluded above that these, on the other hand, increase in importance at elevated temperatures. Thus a temperature increase can compensate the effect of an increased salt content. Up to this stage we have mainly discussed qualitative effects and showed that the formalism explains a number of experimentally well-established trends. To what extent
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is it also quantitatively accurate? The basic intrinsic parameter that determines the phase equilibria in Figure 4 is the spontaneous curvature of surfactant film. We have a method to determine how the electrostatic contribution to H0 varies with external parameters. This would allow us to calculate lines H0 ) 0 for the different salt concentrations, provided we can devise a method to account for effects of a nonelectrostatic origin. There are two basic problems. As for the nonionic surfactants, there is surely a temperature effect in the nonelectrostatic contribution to the spontaneous curvature in ionic systems, at least from the chain packing effect. Second, and more problematically, addition of salt also affects the packing in the film and the area per molecule should become slightly smaller. A double chain surfactant like AOT generally has highly squeezed chains and there is no room for substantial changes in the lateral packing, but by the same token, small changes can have substantial consequences for the spontaneous curvature. In the Helfrich expression (eq 1), the coefficient of the term linear in the mean curvature is κH0, and this is the primary quantity obtained in the calculations of the electrostatic contribution to the curvature free energy. To arrive at a quantitative description let us consider κH0 ) ∆1(T,,γ), where is the weight percent of salt in water and γ is the total weight percent of surfactant. For the nonionic systems of the type shown in Figure 1a, it is well established that the temperature at which the microemulsion extends to the lowest concentration represents the balanced temperature where H0 ) 0. We adopt the same view for the ionic systems of Figure 4, remembering that in this case it might only be approximately true. From eq 2 and the condition H0 ) 0 we have that
∆1nonel ) -∆1el
(11)
at all three points, A, B, and C, in Figure 4. If we assume a fixed area a0 ) 67 Å2 per surfactant in the film, we can directly calculate ∆1el from eq 5 and thus obtain the balancing ∆1nonel at A, B, and C from eq 11. It is our basic assumption that the and γ dependence of ∆1 is due to the electrostatic effects, so the variation in ∆1nonel should come from the temperature dependence alone. An expansion to second order about the temperature TA at point A yields
∆1nonel(T) ) ∆1nonel(TA) + (T - TA)β1 + (T - TA)2 β2 (12) where β1 and β2 are adjustable coefficients. When these have been determined by fitting to the points A, B, and C (see Figure 4 caption for the values), we then calculate the line ∆1(T,,γ) ) 0 for all three systems. The dashed lines in Figure 4 show the resulting curves. For the systems of ) 0.58 and ) 1.0 wt % of NaCl in water, the calculated lines fall in the microemulsion region. Moreover, the line for the 0.58% system runs straight through the lamellar phase region, substantiating the discussion of the previous section. For the system with the lowest salt content, the calculated H0 ) 0 curve is just outside the one-phase area,
indicating a slightly too strong concentration dependence of the calculated spontaneous curvature. We have no definite rationalization of this discrepancy since it can be caused by the approximation that the area per molecule remains constant or can be due to the fact that the chosen reference point may not fully correspond to zero spontaneous curvature at low salt concentrations where the electrostatic interactions are substantial, resulting in a considerable asymmetry with respect to the exchange R f 1 - R. 6. Conclusions We have applied a recently developed formalism that accounts for the concentration dependence of the parameters entering the Helfrich expression for the bending free energy, to explain a number of qualitative observations made for microemulsion systems based on ionic surfactants. The results can be summarized as follows: (i) For systems with no, or small amounts of, additional electrolyte the lamellar phase tends to be more stable than the microemulsion since there are substantial electrostatic contributions to the elastic constants κ and κj. As the electrolyte content is increased, these contributions decrease in magnitude, and this has the combined effect of increasing the free energy of the lamellar phase and decreasing it for the microemulsion. (ii) For ionic systems the “fish” plot shows a tilted appearance for the microemulsion and the three-phase areas. This is caused by the fact that the spontaneous curvature decreases with increasing film concentration due to a combined electrolyte depletion and electrostatic interaction effect. (iii) With increasing temperature one observes, for ionic systems, a sequence of water in oil, to bicontinuous, to oil in water structures, which is the opposite of the trend observed for nonionic surfactants. This is shown to be due to the fact that the electrostatic contributions increase in magnitude with increasing temperature and these work to favor a positive curvature of the film. (iv) In ionic systems the lamellar phase occurs at higher temperatures than the optimal Winsor three-phase equilibrium for systems of equal amounts of oil and water. This is due to a combined effect of a spontaneous curvature that decreases with increasing concentration and increases with increasing temperature. (v) For a given surfactant, the higher the salt concentration, the higher the temperature where the bicontinuous microemulsion occurs. This is due to the combined effect of a decreasing spontaneous curvature with increasing salt concentration and the temperature effect already described. In addition, explicit calculations show the model could account quantitatively for this interplay between temperature and concentration effects in two cases out of three. (vi) We have also pointed out that for ionic systems at balanced conditions there is no longer symmetry with respect to an exchange between oil and water (R f 1 R) due to the electrostatic interaction across the aqueous regions. LA971333L
