0 Copyright 1993 American Chemical Society
The ACS Journal of
Surfaces and Colloids FEBRUARY 1993 VOLUME 9, NUMBER 2
Letters On the Flexible Surface Model of Sponge Phases and Microemulsions HBkan Wennerstrom: and Ulf Olsson* Division of Physical Chemistry 1, Chemical Center, Lund University, P.O. Box 124, S-221 00 Lund, Sweden Received September 8, 1992. In Final Form: November 17,1992
We investigate the flexiblesurface model of sponge phases and microemulsions where the local curvature (free)energy, expanded to second order, leads to a total free energy per unit volume that is proportional to the arealvolume (AIV)ratio to the third power. This is a general result applicableto lamellar phases, sponge phases, and balanced microemulsions. A number of conclusions follow from this simple relation, the most significantbeing the following: (i) For the majority of sponge or L3 phases we conclude that the coefficient for the ( A l W term in the free energy is negative. To account for the stability of the phase at finite concentration,one has to introducefourth-orderterms in the curvature energy. (ii)It is suggested that also for a balanced microemulsion the ( A l V term is negative, and this provides a simple and direct explanationof the existence of the three-phase triangle with a microemulsion in equilibriumwith almost pure oil and water phases. Introduction In a number of amphiphile-solvent systems a selfassembly process leads to the formation of an amphiphile film of macroscopic extension. In a bicontinuous microemulsion a monolayer film separates an oil and a water domain, in a lamellar phase bilayers alternate solvent layers,and in an L3or a sponge phase an amphiphilebilayer separates two domains of the same solvent, be it water or oil. It has been found the flexible surface model, using the curvature energy concept,' is very useful in the theoretical analysis of these intriguing self-assemblystructures. The basis of the model is to picture the monolayer or bilayer as a geometrical surface and for each configuration of the surface assign a curvature (free) energy G, obtained as a surface ( 2 ) integration of a local curvature free energy density gc: Gc = $ p g c
The total free energy is then obtained by a summation (1) Helfrich, W. 2.Naturforsch. 1973,28c, 603.
over allballowedsurfaceconfigurationsusingthe Boltzmann factor exp{-G J k B T J as a weight factor. It is customaryto expand the local curvature free energy density to second order in the curvatures. For a symmetrical bilayer and a balanced microemulsion, where the spontaneous curvature is zero, the expression for g,, to second order in the local curvatures, is1t2 g,(H,K) = 21& + KK (2) where H is the mean and K the Gaussian curvature. The two expansion coefficienta are called the bending rigidity K and the saddle splay constant K. Equation 2 has been used in a number of discussion^^-^^ of free energiesand phase equilibria in amphiphilesolvent (2) Wennerstrcm, H.; Anderson, D.M. In Statistical Mechanics and Differential Geometry of Micro-Structured Materials; Friedman, A., Nitache, J. C. C., David, H. T., Eds.; Springer Verlag: Berlin, in press. (3) Helfrich, W. 2.Naturforsch. 1978,33a, 305. (4)deGennes, P. G.; Taupin, C. J. Phys. Chem. 1982,86,2294. (5)Turkevich, L. A.; Safran, S. A.; Pincua, P. A. In Surfactants in Solution; Mittal, K. L., Bothorel, P., Eds.; Plenum Press: New York, 1986; Vol. 6,p 1177. (6) Safran, S. A. Phys. Reu. A 1991,43, 2903. (7) Andelman, D.; Cates,M. E.;Roux, D.; Safran, S. A. J.Chem. Phys. 1987,87, 7229.
0743-7463/93/2409-0365$04.00/00 1993 American Chemical Society
Letter8
366 Langmuir, Vol. 9,No. 2, 1993
systems starting with Helfrich's work on the undulation forces. In each specific application one has to introduce constraints on the allowed surface configurations. Typically one disallows edges, holes, intersections, and bifurcations. Sometimes there is also a constraint on the topology of the surfaces as,for example, in the description of lamellar phases. In any case it is a demandingproblem to make a proper Boltzmann-averaged summation over the possible surface configurations to obtain the partition function. There exists a simple exact scaling relation for the free energy, resulting from the scale invariance of G,. This relation was first recognized by Porte et al.,12 who also addressed its consequences on phase behavior and structural and dynamical properties of the sponge phase.14J5 In this letter we follow up on the discussion of the flexible surface model and in particular on ita application to the sponge phase and its possible relevance for balanced microemulsions. Scaling Properties of the Free Energy An important question for the interpretation of the phase behavior and the thermodynamicsin general is how the free energy varies with the concentration. In the flexible surface model the concentration is measured by the surface area A per volume V. Compare the partition functions for two different A/ V ratios. Pick one allowed configuration at one of the concentrations. By a simple length scale change we obtain an allowed configuration relevant for the other configuration. Now, both of the coefficients K and K have the dimensionality of an energy and are invariant under a change of the length scale. Thus, the curvature energies of the two corresponding configurations are the same,and they enter the partition function with the same Boltzmann weight factor. This implies that solving the partition function is one and the same problem for all concentrations, provided there are no concentrationdependent constraints. The free energy changes with concentration only through a length scale change. By a simple dimensionality argument we find that the total free energy per unit volume is
n3= a(kBT/L3)a3
GI v = akBT(A/
(3)
where a(K/kBT,i/kBT)is a function of the reduced elastic constants. The second equality comes from assigning a thickness, L, to the physical film and introducing CP as its volume fraction. The dependence of the free energy on the elastic constants is contained in the function a,which also depends on the applied constraints. In all ita innocence, eq 3 contains some significant predictions and it can be used for a fruitful analysisof a number of currently discussed problems. We see from eq 3 that the free energy varies monotonically with the concentration. This implies that concentration changes do not trigger phase changes in the model. The phase with the lowest value of a is the most stable at all concentrations. Similarly when the amphiphilechem~
(8)Golubovic, L.; Lubensky, T. C. Phys. Reu. A 1990, 41,4343. (9) Huse, D. A.; Leibler, S. Phys. Reu. Lett. 1991, 66,437. (10) Huse, D. A.; Leibler, S. J. Phys. (Paris) 1988, 49, 605.
(11) Cates, M. E.; Row, D.; Andelman, D.; Milner, S. T.; Safran, S. A. Europhys. Lett. 1988,5,733. (12) Porte,G.; Appell, J.;Bassereau, P.;Marignan, L. J.Phys. (Paris)
ical potential is changed at constant values of the elastic parameters, this implies a simple concentration change and there are no phase transitions within the model. A simple illustration of the implicationsof eq 3 is found when it is applied to fluctuating membranes in a lamellar liquid crystal. It follows directly from the a3dependence of the free energy density that the solvent chemical potential also varies as @3. In terms of a force distance curve this means that the so-called undulation force varies as the inverse cube of the separation. In the original derivation3 Helfrich obtained an explicit expression for / ~ . work has the factor a(K/kgT) = ( 2 r 2 / 9 ) k ~ T Later confirmed the basic result although the values of the numerical coefficients are still under debate. Sofar we have neglected any constraints on the principal curvatures or, similarly, on the undulation spectrum of the surface. In this case, the only length entering the problem is VIA and an exact scaling relationship, G/V = a@3, is obtained. However, in this case the surface crumplesleand the structural size diverges. Usually this problem is solved by fixing the number of bending degrees of freedom per area through introducing a cutoff at some length scale a. This is typically chosen as the square root of the area per molecule, ag, or as the bilayer/monolayer thickness, L. In real amphiphilic systems, however, we see from direct imaging, scattering, and spectroscopic experimentsthat strong bending fluctuations do not occur on small length scales although more crumpled configurations are allowed by simple molecular geometry. Instead it is our interpretation that these crumpled configurations do not occur because they are energetically unfavorable. The curvature energy expression in eq 2 is obtained as a series expansion to second order around the planar state. There is no theoretical reason to believe that a truncation at second order is sufficient in the whole range a-1> c > %-I, where c is a principal curvature. Introducing higher order elastic energy terms has the effect of decreasing the mean amplitude of the short wavelength thermal excitations. The surface becomes stiffer on a short length scale, which will reduce the tendency for the surface to crumple. For a symmetrical bilayer there are three fourth-order terms containing H4, K2, and W K . For the monolayer at zero spontaneousmean curvature there is in addition two third-order terms, 113 and HK. The coefficients of the fourth-order terms have to be positive to ensure stability. It is not yet clear whether a soft damping of the undulation modes by higher order terms in the curvature free energy is sufficient to prevent the surface from crumpling or if a short wavelengthcutoff is still necessary. We will in the following disregard the possible necessity of a short wavelength cutoff. We expect, however, the effects of such a cutoff to decrease with dilution, and if present, the approximation that the cutoff length a = 0 corresponds to describing the asymptotic behavior of the system at high dilution. With the higher order tems included, the free energy per unit volume no longer scales as (A/V)3. Assuming that the effect of the extraterms is small,which is expected ) ' / ~ ,~ ( 4is) a fourth order when VIA >> ( ~ ( 4 ) l ( k ~ T )where elastic constant, we can apply statistical mechanical perturbation theory using the harmonic curvature energy as the reference system. Since
1989,50,1335. (13) Anderson, D.; Wennerstrdm, H.; Olsson, U. J.Phys. Chem. 1989,
__
9.5'. - -, 424.1 - -.
(14) Porte,G.; Delsant, M.; Billard, I.; Skouri,M.; Appell, J.; Marignan, J.; Debeauvais, F. J. Phys. II 1991, 1 , 1101. (15) Skouri, M.; Marignan, J.; Appell, J.; Porte,G. J. Phys. II 1991, 1 , 1121.
(HmKn), a (A/V)m+2n we find that (16) Helfrich, W. J. Phys. (Paris) 1986,46, 1263.
(4)
Letters
Langmuir, Vol. 9, No.2, 1993 367
G/V
+
+
(kgT/L3)(d3 @Os ...) (5) where @ a 44) > 0. Thus, we see that the higher order elastic coefficients contribute more to the free energy the higher the concentrations. There is a general consensusthat G/ V is dominated by a a3term for lamellar and sponge phases and microemulsions. However, to account for concentration-induced transitions between these phases, one has to introduce correction terms. A commonly adopted view is that free energy density of the sponge phase and of the balanced microemulsionshould have a logarithmic correction term resulting in G / V 0: + b2 In a), where bl and bz are ~ o n s t a n t s . ~ ~This J ~ Jcorrection ~ increases with dilution while it vanishes at 9 = 1. In our analysis we find a different behavior. Similar to the case of the lamellar phase, we find that due to the fiinite size of the monolayer/ bilayer film there is a correction to the O3 term which increaseswith the concentration and which is expected to be small when the f i b thickness is small compared to VIA. In fact, we expect G / V a a3to be the asymptotic behavior at high dilution, also in the presence of a short wavelength cutoff. The logarithmic correction has been explained in the literature to arise from the renormalizations of the elastic c0nstantsl1J4J7where the In 9 dependence is associated with the identification of a long wavelength cutoff of thermal undulations at a length ,A, a 1/9.The renormalized bending constants measure the rigidity of a base surface around which the real surface undulates.l6 While the rigidity of the base surface, which has a reduced area compared to the real surface, is reduced due to thermal undulations, the rigidity of the real surface is measured by the bare elastic constants. Hence, when the curvature free energy is analyzed using renormalized elastic constants, a proper identification of a base surface is important. In the present derivation the curvature free energy is evaluated at the real surface, Le., in the amphiphile monolayer/bilayer film, and the renormalized elastic constants do not enter the problem. In recent lattice models of microemulsions7 and sponge phases" a renormalized bending constant was used to evaluate the curvature energy,resultingin a logarithmiccorrectionterm in the free energy density. An assumption in those models is that the base surface has to bend its mean curvature to conform to the bicontinuous structure. We have argued along another traditi0n,l3J%~~ identifying the structure with an infiite,multiply connected,dividingsurfacewhich locally is saddle-like. In this case we see a problem in identifyinga suitable base surface for an analysis in terms of renormalized elastic constants. If this is chosen as a minimal surface,there is no mean but a Gaussiancurvature energy associatedwith the base surface. The same problem arises in a lamellar phase where the relevant base surface is a plane with zero mean and Gaussian curvature everywhere. Sponge Phase The topology of the surface is an essential structural aspect in the flexible surface model. The Euler characteristic, XE, is an extensive property, and it follows from varies only the argument leading to eq 3 that XE/(A/V)~ with K/kgT and K/kgT. If we constrain the surface to be simply connected, the possible values of the XE density (17) Rous, D.; Codon, C.; Cates, M. E. J.Phys. Chem. 1992,96,4174. (18) Anderson, D. M.; WennerstrBm, H.J. Phys. Chem. 1990,94,8683. (19) Scriven, L. E. Nature 1976,263, 123. (20) Ljunggren, S.; Eriksson, J. C. Langmuir 1992, 8, 1300.
vary from zero to highly negative. The higher the value of K, the energeticallymore favorable, or less unfavorable, it is to generate tubular contacts between surface patches and the lower the XE density. There is an important entropic contribution to the free energy from the fluctuations in the local topology. This contribution is larger the higher the concentration. At some point one reaches conditions when a(K/kgT,i/kgT) changes sign as K/kBT is increased. What are the consequences? With a negative a the free energy, and the chemical potential of the amphiphile, decreases with increasing concentration, at high dilution. This signifies a thermodynamicallyunstable condition. At higher concentrations the @a5 term ensures local stability. Once again we see the need for a higher order term to obtain a physically acceptable solution. Let us now turn to the experimental observations. In all binary systems, where an L3 or sponge phase has been located, it is found that it exists at finite concentrations in equilibriumwith a very dilute sol~tion.'~ Consequently, if we analyze the system within the flexible surface mode, we conclude that the coefficient a is negative over the majority of the temperature range where the phase is found. This is necessary to reconcile the observed nonswellingof the sponge phase with the result of eq 3, which should apply at sufficiently high dilution. A second conclusion is that we have to introduce higher order terms in the curvature energy to account for the stability of the sponge phase at finite concentrations. An equilibrium with pure water occurs for the sponge or L3 phase at 9 = (-a/2j3)1'2 (6) We have previously suggested a particular model accountingfor the nonswellingof the spongephase,13while we here present a general proof that the CP3 term is necessarily negative if one wants to describe the phase behavior within the framework of the flexible surface model. In another context we will report explicit calculations of phase equilibria for these systems. One should also note that for a few systems the L3 phase seems to changestructural properties at high dilution,possibly from infinite to finite aggregates, and our general scaling arguments have to be modified under such conditions.
Bicontinuous Microemulsions from Topology Fluctuations
In a balanced microemulsionthe spontaneouscurvature of the monolayer amphiphile film is zero. In this case, eq 2 is the relative expression for the curvature energy density to second order. A balanced microemulsion is bicontinuous, and the film has a topology similar to that found for the bilayer in a sponge phase. When dilutedwith an equal mixture of oil and water, there is a separation into a threephase equilibrium: microemulsion-wateril. It is usuallystated4J that the microemulsionphase earns a major part of its stability relative to a lamellar phase from fluctuations in the mean curvature and that it is favored by a low value of K. In view of the previous discussion it seems that the crucial parameter is i rather than K. In the microemulsionthere are fluctuations in the local topology that are not present in the lamellar phase. The energy cost for those fluctuations is measured by the saddle splay constant K rather than the bending rigidity K. For the microemulsionswe expect that ilkgTis negative but small so that there are large thermal topology excitations.
368 Langmuir, Vol. 9, No. 2, 1993
Strey et recently found in a small-angle neutron scattering experimentthat the structure factor of the film in a balanced microemuleion was within the experimental uncertainty the same as for the bilayer in a comparable sponge phase. For the sponge phase we concluded that LY was negative, and when the structure of the film is the same it seemsa reasonableconjecturethat Q is alsonegative for a balanced microemulsion. This would provide a simple answer to the much-debated question of the mechanistic source of the three-phase e q ~ i l i b r i u m . ~The ~ - ~micro~ emulsion is simply unstable at high dilutions where the a3 term dominatesthe free energy. A system prepared at high dilution would spontaneously expel excess solvent and contract to a higher concentration. Here higher order terms in the free energy would yield internal stability. The expelled solvent clearly separates into an oil and a water phase.
Conclusions Although we are much inspired by recent theoretical work on sponge phases, there are some important differences between the views expressed above and those previously presented in the literature. For the sponge phase we see the importantcontributions to the free energy as arising from the entropy of the connected structur_eand the elastic energy associated with it as measured by K. This differs from the view of Cates et del1 but followsthe argument of Porte et al>z However, the latter authors conclude that the coefficient a! is positive because otherwise there is an instability. We point out that this instability is actually observed for an overwhelming majority of phases. It leads to a contraction of the structure to a finite volume fraction due to the presence of higher order terms in the elastic energy. For binary systems, this behavior is apparent; see ref 13 for a list of such systems. For ternary and quaternary systems the (21) Strey, R.; Winkler, J.; Magid, L. J. Phys. Chem. 1991,95, 7502. (22) Jouffroy, J.; Levinson, P.; deGennes, P. G. J. Phys. (Paris) 1982, 43. 1241. ~. (23) Safran, S. A.; Rous, D.; Cates, M. E.; Andelman, D. Phys. Rev. Lett. 1986, 57, 491. (24) Widom, B. J . Chem. Phys. 1984,82, 1030.
Letters
instability is somewhat masked by the presence of several components. Particularly for charged systems it is found that the sponge phase swells monotonically upon dilution with a given, unique, electrolyte solution. However, due to the electrostatic contribution to the monolayer spontaneous curvature, K for the bilayer changes considerably on such a dilution path. The stability of bicontinuous microemulsions, under balanced conditions, is usually attributed to free energy contributions from mean curvature fluctuations and influenced by the renormalization of K on larger length scales. We argue that fluctuations, associated with a change in topology and influenced by the magnitude of the elastic constant K, make a more crucial contribution to the free energy. The basic difference between alamellar phase and the microemulsiodsponge phase is not the mean curvature fluctuations but the excitation of the difference curvature modes? leading to a change in topology. When these modes are excited we have the peculiar feature that the entropy increaseswith increasingconcentration, which potentially can result in an instability at high dilutions. These qualitative differences in the views on the most essential contributions to the free energy also lead to some quantitative differencesfor the concentrationdependence of the free energy. A logarithmic correction to the a3term in G / V for sponge phases and balanced microemulsions, as suggested by other authors,7J1J4J7is not found in the present analysis. Rather, we obtain G/V a: a3 as the asymptotic behavior at infinite dilution, similar to the case of the lamellar phase. The particular features of the sponge phase and ita relation to the lamellar phase can be understood further when the mean curvature energy of the two oppositely oriented monolayers that carry a spontaneous mean curvature13is analyzed. However, analyzing the problem in terms of a single dividing surface has pointed toward some interesting similarities between sponge phases and bicontinuous microemulsions.
Acknowledgment. This work was supported by the Swedish Natural Science Research Council (NFR).
