Langmuir 1991, 7, 1864-1866
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Bicontinuous to Lamellar Transitions in L3 Phases and Microemulsions S. A. Safrant Exxon Research and Engineering, Corporate Research Science Laboratories, Annandale, New Jersey 08801 Submitted to Symposium Chairman December 1 , 1989. Received November 15, 1990 Surfactant monolayers at water-oil interfaces and amphiphilic bilayers in either water or oil exhibit large-scale,isotropicstructures in the neighborhood of ordered,lamellar phases. A model for the structure, based upon a treatment of bicontinuous microemulsions, is used to compare the phase stability of the lamellar and bicontinuous L3 phases.
I. Introduction While early models of microemulsions focused on droplet phases (e.g., water drops surrounded by surfactant monolayers in an oil-continuous medium), recent attention has focused on the bicontinuous phases in these systems.’ These phases are isotropic with no long-range lamellar ordering. However, both scattering experiments and electron microscopy indicate that the characteristic length scales can be much longer than molecular sizes; in some cases, the structure can be described as a bicontinuous system of water and oil domains with surfactant monolayers at the interfaces. Isotropic phases with long length scales are also observed in surfactant-water or surfactantoil systems. These phases, commonly termed L3 phases,2 occur near ordered, lamellar phases and consist of surfactant bilayer^.^ In addition, these phases are often flow birefringent.2-3 In this paper, a model for these bicontinuous phases is reviewed. The free energy is discussed in terms of the entropy of the random surface and the curvature elasticity of the surfactant film. Thermal fluctuations are shown to drive a phase transition from the lamellar to bicontinuous (or L3)phases as the system is diluted and the surfactant volume fraction is decreased. In addition, it is shown that these transitions can also be driven by an increase in the saddle-splay elastic constant, since the bicontinuous phase has negative Gaussian curvature. The model is consistent with recent experimental studies of the L3 phases which can occur when alamellar phase is diluted2Jor if the elastic properties of the surfactant film are altered by addition of co~urfactant.~ The analogy between the bicontinuous microemulsions and the L3 phases of surfactant bilayers is appropriate when the spontaneous curvature of the surfactant monolayer in the microemulsion is zero. This implies that the surfactant monolayer is Ybalancednand has no tendency + Present address: Department of Polymer Research, Weizmann Institute of Science, Rehovot, Israel 76100. (1) Safran, S. A,; Roux, D.; Catas, M.; Andelman, D. Phys. Rev.Lett. 1986,57,491; In Surfactants in Solution: Modern Aspects, ed. Mittal, K., Ed.;Plenum: New York, in press. Andelman, David; Catas, M.; Roux, D.; Safran, S. A. J.Chem. Phys. 19897,87,7229. Andelman, D.; Safran, S.A.; Roux, D.; Cates, M. Langmuir 1988, 4, 802. (2) Benton, W. J.; Millner, C. J. J. Phys. Chem. 1983,87,4981. Nilsson, P. G.; Lindman, B. J.Phys. Chem. 1984,88,4764, and see references in ref 6. Gazeau, D.; Bellocq, A. M.; Roux, D.; Zemb, T. Europhys. Lett. 1989. 9. ~ ,. . ~ - ,447.
to bend toward either the oil or the water domains: in what follows, only this case is considered. The curvature elasticit? of such a film has contributions from both the average and Gaussian curvatures of the film. The bending energy, Fb,is written
F,,= K
1dS(c, +
c2)2
+ K 1dSclc2
(1)
The local curvatures are c1 and c2 and d S is the surface integral. The first term accounts for the energy cost of changing the average curvature from zero, and the second is related to the Gaussian curvature, C ~ C Z ,with a saddlesplay modulus K. The Gauss-Bonnet theorem implies that the term proportional to depends only on the topology, since
where x is a topological constant (equal to unity for spheres), proportional to the difference between the number of disjoint surfaces and the number of handles. 11. Interface Model and Free Energy The surfactant bilayer is modeled as a continuous film which is an incompressible, two-dimensional fluid. The hydrophobic interactions between the inside of the bilayer (e.g., the hydrocarbon chains of the surfactant molecules in the case of water-continuous systems) and the external phase require that the film have no edges or seams; the film is composed of either closed or infinite sections. For the bilayer system, the film divides space into an inside and an outside (see Figure l),which are chemically identical and physically indistinguishable; for the microemulsions the inside and outside correspond to the distinguishable water and oil domains on either side of the surfactant monolayer. The structure of the bicontinuous phase is modeled with a coarse-grained, cubic lattice construction used originally to model bicontinuous microemulsions and extended to treat L3 phases in ref 6. The lattice size is 5 and each cell is associated with the inside or outside phase (water or oil in the case of the microemulsion) with probability rC, and 1- ,)I respectively. At the interface between the inside and outside (water or oil) phases are the bilayer sheets (monolayer films). Conservation of surfactant relates the length scale $, to the volume fraction of surfactant, 9 so
~~
(3) Bellcoq, A. M.; Roux, D. In Microemulsions; ed. Friberg, S., Bothorel, P., Eds.; CRC Press: Boca Ratan, FL 1987. (4) Porte, C.; Marignan, J.; Bassereau, P.; May, R. J. Phys. (Paris) 1988,49,511. Porte, G.; Appell, J.; Bassereau, P.; Marignan, J. J. Phys. (Paris) 1988, 49, 511.
0743-7463/91/2407-l864$02.50/0
( 5 ) Helfrich, W. 2.Naturforsch., A: Phys., Phys. Chem., Kosmophys. 1973,28,693. (6) Cates, M. E.; Roux, D.; Andelman, D.; Milner, S. T.; Safran, S. A. Europhys. Lett. 1988,5,733. Roux,D.; Catas, M. E. Proceedings of the
4th Nishinomyia Yulcawa Symposium; Springer-Verlag, in preea.
0 1991 American Chemical Society
Bicontinuous to Lamellar Transitions
Langmuir, Vol. 7, No. 9, 1991 1865
that
-t-
d = 6 a W - $)/t
(3) For the microemulsion, $ is determined by the water/oil concentrations; for the bilayer L3 phases, $ is determined below by minimizing the free energy. The free energy per unit volume, f, includes the entropy of mixing of the inside and outside regions (water and oil regions for the microemulsion), and the free energy of bending of the surfactant film. The free energy per unit volume, fa, associated with the entropy of mixing is
+
f, = ~ ( 1 / [ ~log ) [ $+ (1- $1 log (1 - $11
(4) The bending energy, f b , is estimated by noting that the average curvature is of order f 1 and that the number of interfaces is proportional to $(1- $), so that (5)
The total free energy per unit volume f = f b + f,. Here, tis a number of order unity which accounts for the details of the bending geometry (e.g., if all bends were sections of spheres, I = 1). Equation 5 does not account for the saddle-splay term in the bending energy, which will be discussed in section IV. 111. Lamellar to Bicontinuous Transition For the L3 phase, the free energy is minimized with respect to $. For small values of the surfactant volume fraction 6,the minimum is at values of # */2. These phases consist of films with more outside than inside (for $ < l / 2 ) and correspond, in the limit of small $, to dilute phases of vesicles.6 These phases are stabilized by the entropy of mixing, which offsets the curvature energy cost. For larger values of the surfactant volume fraction, the minimum free energy state has $ = l / 2 , corresponding to a bicontinuous, spongelike state with equal amounts of inside and outside. In the microemulsion case, this is the 'middle phase" microemulsion, which can coexist with both water and oil phases. For the bilayer, L3 phases, the analogous coexistence is with the vesicle phases. The detailed nature of the vesicle sponge transition is discussed in ref 7. The free energy of the symmetric sponge phase with $ = l / z can be written f = gad3,where g, is a constant that depends only on the ratio KIT. The free energy of the lamellar phase has been discussed by Helfrichs in terms of the steric repulsion of the surfactant sheets. This repulsion limits the meandering entropy of these sheets and raises the free energy9 (per unit area) by an amount proportional to T ( T / K ) / d 2where , d l / d is the spacing between sheets. The free energy per unit volume, fl, is therefore, fl = g143, where as above,gl is a function of KIT. Thus, transitions from lamellar to sponge can only occur if changes in the bending modulus decreaseg, to the extent thatg, < gl; when this happens, at sufficiently small values of K / T , the lamellar phase melts into a bicontinuous structure. However, this simple model implies that there is no transition as a function of dilution (decreasingd)-in contradiction to experiment.2~3 The transition from lamellar to a spongelike, symmetric phase as a function of dilution can be understood if the
+
-
-
(7) Row, D.; Cabs, M. E.; Olseon, U.; Ball, R. C.; Nallet, F.; Bellocq, A. M.Europhys. Lett. 1990,1I,229. (8) Helfrich,W. Z . Naturforsch., A: Phys., Phys. Chem.,Kosmophys. 1978,33,305. (9) Golubovic, L.; Lubensky, T. C. Phys. Reu. A: Gen. Phys. 1990,41, 4343.
Figure 1. Coarse grained lattice construction for random bicontinouos phases. The shaded and unshaded regions represent the inside and outside phases, respectively, and the dark solid line indicates the surfactant film. renormalization of the bending constant, K,by thermal fluctuations is taken into account. The effect of these fluctuations is to reduce the free energy cost of large scale bends, since the film is spontaneously bent at scales comparable to its intrinsic persistence length,1° [K u X exp[4rK/aT], where a is a molecular size and a is a modeldependent parameter of order unity. For nearly flat sheets, the effective" bending constant K([)is written
-
The microscopic or bare bending modulus is KO; it is renormalized to smaller values for bends on length scales comparable to the persistence length, &, although the perturbation theory which leads to eq 6 is quantitatively correct only for values of K([) KO. Including the renormalization of K in the calculation of the free energy of the sponge phase implies that f = g,@ log (d/d*), where d* is related to the persistence length, [K, by eq 1. At some value of 4 < +*, the log term allows the sponge phase to have a smaller free energy than the lamellar phase. This argument as well as detailed calculations of the phase equilibria explains the observed transition from lamellar to bicontinuous sponge as a function of dilution in both L3 and microemulsion systems.
-
IV. Saddle-Splay Transitions Phase transitions from lamellar to bicontinuous L3 phases occur not only as a function of dilution but also with changes in the properties of the surfactant film. Such changes can occur in systems where the film is composed of a surfactant and a cosurfactant, usually an alcohol. Porte and co-workershave observed a transition from an ordered, lamellar phases to an isotropic,L3 phase as the surfactantalcohol ratio is varied and have interpreted4 these transitions in terms of the development of saddlelike bicontinuous surfaces. While ordered, bicontinuous, saddle structures have sometimes been observed, the experiments on the L3 pha~es2f.~ indicate that there is no long-range order of the surfactant interfaces. The model discussed above focuses on the competition between the entropy of mixing and the bending energy in stabilizing the disordered, bicon(10)de Gennee, P. G.; Taupin, C. J. Phys. Chem. 1982,86,2294. (11)Helfrich, W.J. Phys. (Paris)1986,46,1263; J. Purys. (Paris) 1987, 48,285. Peliti, L.; Leibler, S. Phys. Rev. Lett. 1986,54,1690.Foerater, D.Phys. Lett. A 1986,114,115.Kleinert,H.Phys.Lett.A 1986,114,263. Golubovic, L.; Lubensky, T. C. Phys. Rev. B Condens. Matter 1989,39, 12110 (1989).
1866 Langmuir, Vol. 7,No. 9,1991 tinuous phases. The bending free energy can be modified to include the effects of the saddle-splay energy so that the effects of changes of the film properties can be predicted. Equations 1and 2 show that the saddle splay energy is proportional to the product of the elastic constant, K,and the average Gaussian curvature. Calculations based on a continuum model of random interfaces12as well as coarsegrained lattice appro~imationsl~ indicate that the average Gaussian curvature of a random, bicontinuous, spongelike structure can be negative. At small values of the volume fraction of either the inside or outside phase (water or oil for the microemulsion), J, S 0.1 or 1- J, 5 0.1, the structure is dropletlike and the average Gaussian curvature is positive. In the entire range of II. between these values, the average Gaussian curvature term is negative. Changes in the film properties such as the addition of alcohol etc. can result in larger values of R, which would tend to stabilize the spongelike phases with respect to the lamellar phases, where the average Gaussian curvature is zero. (12)Berk, N. F. Phys. Rev. Lett. 1987, 58, 2817. Teubner, M. Unpublished. (13) Roux, D., Cates, M. To be submitted for publication.
-
Safran
The lamellar sponge transition induced by changes in K should be only weakly dependent on the surfactant volume fraction. In summary, a coarse-grained lattice model of spongelike, random, structures can be used to understand the transition from the lamellar to the L3 or bicontinuous phase. These transitions can occur as (i) the bare bending constant, KO,is decreased, thus decreasing the energy cost of the bends in the random phase, (ii) the system is diluted, thus increasing the length scale of the sponge phase and reducing the effective bending constant, K(& and (iii) the value of K is increased, favoring the sponge phase which has a saddlelike topology. The transitions induced by dilution should be only weakly 6 dependent, since for nearly flat membranes, the renormalization of the bending constant depends only logarithmically on 6. These trends are consistent with the qualitative experimental observations on L3 phases.
Acknowledgment. This work is based on collaborations with D. Andelman, M. Cates, S. Milner, and D. Roux. The author is also grateful to G. Porte and C. Safhya for useful discussions.
