Langmuir 1991, 7, 1849-1854
1849
b'ymp osta Structure Factor for Microemulsions with Finite Spontaneous Curvature P. Chandra*s+and S. A. Safrant Corporate Research Science Laboratories, Exxon Research and Engineering Company, Annandale, New Jersey 08801 Submitted to Symposium Chairman January 8, 1990. Received November 15, 1990 We calculate the structure factor for microemulsions with finite spontaneous curvature by using a statistical mechanicsmodel of the interface ensemble. The morphologicaltransition between hard spheres and the random bicontinuous phase is treated in a unified fashion, and the implications for experiment are discussed.
I. Introduction A microemulsion' is an equilibrium mixture of water, oil, and surfactant, where the domains of water and oil (sizes 1oQ-1OOO A) are separated by surfactant monolayers.2 The structure of the system can be tuned with salinity or temperature, for example, to yield droplet, wormlike cylinder, ordered lamellar, and random bicontinuous phases.3 Microemulsions, like other self-assembling systems, have the special property that their individual and collective energies are on the same scale; for example, globular interactions can lead to individual globular deformations, and likewise the local curvature energy can induce correlations on macroscopic length scales. Experimentally4 and theoreticall? the morphological evolution of a microemulsion from the droplet to the bicontinuous phases can result from a change in its spontaneous curvature,w reflecting the surfactant film's preference for bending toward water or oil. Small angle scattering experiments probe the mean domain size of the microemulsion; extensive studies have been performed in both the globular1° and b i c o n t i n u ~ u s ~ regimes. ~-'~ The
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t Present address: NEC Research Institute, Inc., 4 Independence Way, Princeton, NJ 08540. Present address: Department of Polymer Science, Weizmann Institute, Rehovot 76100, Israel. (1) For an introduction to this subject see Langevin, D.; Meunier, J.; Cazabat A.-M. Recherche 1986,16,720. (2) Meunier, J. J . Phys. Lett. 1985,46, L-1005. (3) For a general survey see (i) Mittal, K., Lindman, B., Eds. Surfactants in Solution; Plenum: New York, 1984,1987. (ii) Safran, S. A., Clark, N. A., Eds., Physics of Complex Fluids; Wiley: New York, 1987. (4) Chen, S.-H.; Chang, S.-L.; Strey, R.; Kahlweit, M. Preprint. (5) Talmon, Y.; Prager, S. J . Chem. Phys. 1978,69, 2984; J. Chem. Phys. 1982, 76, 1535. (6) deGeMes,P.G.;Taupin,C.J.Phys. Chem. 1982,86,2!294. Jouffroy, J.; Levinson, P.; de Gennes, P. G . J. Phys. (Park) 1982,43, 1241. (7) Widom, B. J . Chem. Phys. 1984,81, 1030. (8) Safran, S. A.; Roux, D.; C a b , M. E.; Andelman, D. Phys. Reo. Lett. 1986,57,491;In Surfactants in Solution: Modern Aspects, Mittal, K., Ed.;Plenum: New York, in prese. Andelman, D.; Cates, M. E.; Row, D.; Safran, S. A. J. Chem. Phys. 1987,87,7229. (9) Huse, D.; Leibler, S. J . Phys. (Paris) 1988, 49, 605. (10) E.g.: Ober,R., Taupin, C. J. Phys. Chem. 1980,84,2418. Cazabat, A. M.; Langevin, D. J . Chem. Phys. 1981, 74, 3148. Huang, J. S.; Safran, S. A.; Kim, M. W.; Great, G. S.;Kotlarchyk, M.; Quirke, N. Phys. Reu. Lett. 1983, 53, 592. Lemaire, B.; Botherel, P.; Roux, D. J. Chem. Phys. 1983,87, 1023.
experimental structure factors have been fitted reasonably well by geometrical however these approaches assume particular scattering forms or microstructures that are only suitable for a narrow range of experimental parameters. For example, in the dilute limit with finite spontaneous curvature, the microemulsion is adequately represented by interacting hard spheres.1° The random bicontinuous structure occurs for comparable oil and water volume fractions and zero spontaneous curvature; here, the calculated scattering from a disordered lamellar system fits the experimental results quite s u ~ c e s s f u l l y . ~The ~J~ aim of the present work is to treat the morphological transition from hard spheres to the random bicontinuous phase in a unified fashion, using an approach that correctly models the thermodynamics of the problem in each limit. Our calculated structure factor provides a sensitive test of our underlying theoretical assumptions, because it must reproduce important local effects. At a fundamental level, one would like to relate the physics on molecular lengthsl8 to the large-scale structure of the microemulsion. Such microscopic models must produce structural organization with domain sizes much larger than that of the molecular length. This approach is useful for the study of periodic, ordered, or nearly ideal structures where the correlations are effectively infinite.18 However, the presence of random, bicontinuous phases with no long-range order requires "exact" solution of the microscopic model, which at this time is not feasible. An alternative phenomenological approachH is to focus on the physics of the surfactant film separating water and oil domains, in order to understand the local morphology (11) Auvray, L.; Cotton, J. P.; Ober, R.; Taupin, C. J . Phys. (Park) 1984,45,913. de Geyer,A.;Tabony,J. InPhysics of Amphiphilic Layers; Meunier,J., Langevin, D., Boccara,N.,Eds.; Springer-Verlag: New York, 1988. (12) Kaler, E. W.; Benner, K. E.; Davis, H. T.;Scriven,L. E. J. Chem. Phys. 1983, 79, 5673. (13) Berk, N. F. Phys. Reu. Lett. 1987,58,2718. Teubner, M.; Strey, R. J. Chem. Phys. 1987,87, 3195. Lichterfeld, F.; Schmeling,T.; Strey, R. J.Phys. Chem. 1986,90,5762. (14) Chen,S. H.;Chang,S.L.;Strey, R. To be submitted for publication. (15) Ninham, B. W.; Barnes, I. S.; Hyde, S. T.; Deian, P. J.; Zemb, T. N. Europhys. Lett. 1987,4,5651. (16) Wheeler, J. C.; Widom, B. J. Am. Chem. SOC.1968,90,3064. Widom, B. J. Chem. Phys. 1986,84,6943. Schick, M.; Shih, W. H. Phys. Reu. B Condens. Matter 1986,34,1797; Phys. Rev. Lett. 1987,69,1205. Chen, K.; Ebner, C.; Jayprakash, C.; Pandit, R. J. Phys. C Solid State Phys. 1987,20, L361. Gompper, G.;Schick, M.; Shih, W. H. Phys. Reu. Lett. 1987,59,1205. Gompper, G.;Schick, M. Phys. Reu. Lett. 1989,62, 1647.
0 1991 American Chemical Society
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1850 Langmuir, Vol. 7,No. 9,1991
in the disordered phases. Here the surfactant interface is treated as a flexible, incompressible sheet between two continuum liquids. The lowest energy fluctuations of these two-dimensional fluid membranes are out-of-plane curvature deformations. For a two-dimensional interface, thermal fluctuations will also play an important role. The free energy in this approach includes both entropy and bending energy terms; the competition between these two contributions determines the domain structure. Here, we extend a coarse-grained interfacial model of microemulsions and calculate the scattering structure factor S(q) for an ensemble of surfactant membranes as a function of the film spontaneous curvature. We show that this approach successfully treats the morphological transition between hard spheres and coalescent "bicontinuous" structures in the dilute limit; this is the first step toward constructing a general structure factor for arbitrary volume fraction and general spontaneous curvature. We begin with a brief description of the extended thermodynamic model in section 11. The bending energy as a source of correlations between water and oil domains is discussed in section 111. These effects should be most dramatic in the dilute limit when the system has a preferred length scale, the spontaneous radius of curvature. In section IV we describe the schematics of the structure factor calculation, and we end with results (section V) and implications for experiment (section VI). 11. The Model The continuum description of random, microemulsion phases began with the work of Talmon and Prager: and has been further developed by de Gennes and co-workers,6 and by Widom.' Andelman, Cates, ROUX,and Safrane (ACRS) have extended this approach to include thermal fluctuations of the surfactant film.g More recently, Golubovic and Lubenskyl' have incorporated steric entropy into their analysis, in order to treat bicontinuous and periodic phases in a more unified fashion. The ACRS model has been described in detail elsewhere; here we discuss its extension to the case of finite spontaneous curvature.18 In this approach, the microemulsion is treated as an ensemble of random, fluid membranes. Because the area per surfactant is assumed constant, the surface free energy per unit volume, f , has only two contributions: one, fs, arising from the entropy, and the other, fc, coming from the curvature energy. Ideally, one would like to calculate the entropy of the ensemble of surfactant films directly, though issues of shape, size and length distribution, and self-avoidance make this intractable at the present time. Instead, we associate the film entropy with the entropy of the bulk material in its vicinity; this we can then calculate by treating the bulk twocomponent fluid in the standard lattice gas approximation. In our calculation, space is divided into cubes of size with occupation probability for water {oil)determined by the Ising spin variables si at site i, where Si = 1 and Si = 0 signify water and oil, respectively. In particular, the average probability, ( s i ) , for a cell to contain water is
1
d = d, + id*
considered (Figure 1);la the expectation values of these Ising variables coarse-grainover the microscopic energetics and are functions of the bending modulus and the spontaneous curvature. Because we are interested in fluctuations on length scales greater than the lattice spacing, these "mesoscopic" Ising variables are set to their mean field (random mixing) values in the present treatment. If we assume that all the surfactant resides a t incompressible interfaces between water and oil domains, then the lattice size 5 is determined by the expression
., where A is the interfacial area per cube and u is the film thickness. For the case of zero spontaneous curvature and large bending modulus
(3) within the random mixing approximation where z = 6 is the cubic lattice coordination number. We emphasize that the presence of the interfacial variables discussed above results in different expressions for 5 as a function of bending modulus and spontaneous curvature even a t the random mixing level;18 similarly these additional Ising variables lead to equilibrium phase behavior somewhat modified from that predicted by the original ACRS model.'* Within this approach, the interfacial entropy is now treated as the entropy of mixing oil and water domains, and in this instance (large bending modulus and finite spontaneous curvature) the associated free energy is
T f, = +d log (4) + (1- 44 log (1 - $11 (4) E3 The curvature energy, Fb,of a single membrane with
local curvatures c1 and c2 such that c1, c2 l/2; however for x < a distinct correlation peak occurs. This nontrivial dependence of the structure factor on the spontaneous curvature arises from the microscopic energetics of the problem, specifically the presence of the interfacial Ising variables,'s and does not emerge from the original version of the ACRS model. Because S(q) is dominated by the [1111 direction, the powder-averaged structure factor should show a marked peak near q- = 3 % r / ( for fixed dilute volume fraction and decreasing spontaneous curvature; this effect should be quite dramatic in the dilute phase, where normally one expects the structure factor to be quite featureless.
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1854 Langmuir, Vol. 7, No. 9, 1991
VI. Conclusions Starting from an extended interfacial model for microemulsions that is thermodynamic in origin, we have calculated structure factors for general volume fraction 4 and spontaneous curvature coal8 Without explicitly including any interfacial interactions, this approach predicts water-oil correlations solely due to the presence of the surface bending energy. These effects should be most dramatic in the dilute phase with varying CO, where the energetic and concentration-dependent length scales may 1 (emulsification be mismatched. We find that for x failure%)the two lengths are equal, and the structure factor is featureless, aside from a lattice or hard sphere peak. In the region l / 2 < x < 1the concentration constraints demand the formation of dopains smaller than the spontaneous radius of curvature, PO; however the microscopicenergetics of the system do not allow for the formation of nonspherical interface, and a thus a droplet structure with radius 5 < po forms. However, as x l / 2 , the spheres can coalesce using the flat surfactant film available, thereby attempting tg compensate for the mismatch between their energetic ( p ) and concentration-dependent (5) length scales. Specifically we find that S(q) has a peak at
sition from spheres to bicontinuous structures in a unified manner. We also note that this may not be the whole story, particularly in the dilute case. Specifically, our model predicts monodisperse spheres in the region before the onset of coalescence;this is the result of allowing only one energetic length scale in this approach. Furthermore, rather than fusing, the droplets may undergo shape f l u ~ t u a t i o n s , 2 ~possibility *~~a not accessible from a coarsegrained lattice model. Experimentally, attractive interactions between dilute spherical droplets have been observed in many globular microemulsions1° that are too large to be due to pure van der Waals effects. These systemshave been modeled quite successfullyby hard spheres with a short-range potential;% here we emphasize that this is a q = 0 attraction that leads to macroscopic phase behavior. In contrast, the curvature interaction that we are proposing will indeed be repulsive at q = 0; the interesting effects will occur at finite wavevector. At fixed dilute volume fraction, the hard sphere peak will not change as a function of bending modulus or spontaneous curvature; our model predicts the development of a correlation peak as the droplets fuse to create nascent bicontinuous structures. By this point, the experimental trends are fav0rab1e;~J~ specific tests of these predictions will help guide the future theoretical developments in this field.
where x [ / i o . The idea that curvature effects can lead to apparent interaction effects has been treated in continuum theories;27~2~ however in these cases isolated interfaces are considered. We emphasize that here we are observing similar effects for an ensemble of surfactant films; this is the first time that a thermodynamic model of microemulsions has been able to treat the morphological tran-
Acknowledgment. We are grateful to M. E. Cates, s. T. Milner, D. ROUS,and Z. G. Wang for many helpful discussions. This research was supported in part by the National Science Foundation under Grant No. PHY8217853, supplemented by funds from the National Aeronautics and Space Administration, at the University of California at Santa Barbara.
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(27) Safran,S. A. J. Chem. Phys. 1983, 78, 2073. (28) Auvray, L. J. Phys. Lett. 1985, 46, L163.
(29) Huang, J. S.;Safran,S. A.; Kim, M. W.; Great, G. S.; Kotlarchyk, M.; Quirke, N. Phys. Rev. Lett. 1983,53, 592.
