2294
J. Phys. Chem. 1982, 86, 2294-2304
FEATURE ARTICLE Microemulsions and the Flexibility of OWWater Interfaces P. G. De Gennes” and C. Taupln Coli&e de France, 7523 1 Paris Cedex 05, France (Received: November 30, 198 1: In Final Form: February I, 1982)
The phase diagram of a system of oil + water + surfadant is usually dominated by a variety of regularly organized phases (lamellar,hexagonal, etc.) which are highly viscous. However, in some favorable cases, these organized phases are less stable than a certain “microemulsion”where no periodicity occurs. These microemulsions are much more fluid than the regularly organized phases. They often exist over a broad domain of concentrations. In some limiting cases a microemulsion is simply made of swollen micelles (of oil in water or water in oil). However, various experiments indicate that “bicontinuous”structures also occur. Our aim is to understand why a random structure of this type does not collapse into an ordered phase. The interface saturated by surfactant has a nearly vanishing surface tension; one essential parameter is then the elastic constant K describing the curvature elasticity of the interface. The “persistence length” f K of the interface increases exponentially with K. This should have some important effects. (1)When K is above a certain critical value K , the interfaces tend to stack or, more generally, to build up a periodic, stable, phase. (2) When K is below K, the interface can become extremely wrinkled and the resulting gain in entropy is larger than the loss of energy due to the departure from a periodic array. This case K < K , would correspond to microemulsions. In this picture one major effect of cosurfactants (additives which favor the microemulsion phase) is to increase the flexibility of the layers.
I. Distinct Features of Microemulsions Mixtures of oil and water are naturally unstable, but can be stabilized by addition of suitable “surfactants”. These molecules contain a polar group (soluble in water) and an aliphatic tail (soluble in oil). They optimize their interactions by standing at the oil-water interface and decrease drastically the interfacial energy y. (Similar reductions are observed at the air-water interface (see Figure l).) (1)In most cases this decrease of y stops at some point; if we add a larger amount of surfactant (Figure IC)the value of y stays roughly constant and is definitely nonvanishing. The reason is that, beyond a certain limiting bulk concentration, the added surfactant does not go to the interface but prefers to stay in one of the bulk phases in the form of micelles (Figure IC).With finite but small interfacial tensions y, one can make emulsions of oil droplets in water (O/W) or of water droplets in oil (W/O). Here the droplets are rather large (10 pm) and they are metastable (they coalesce slowly). (2) Some surfactants, however (or some surfactant ’ mixtures: surfactant + “cosurfactant”), have a different behavior; by increasing their bulk concentration, it is possible to reach a state of zero interfacial tension, without being blocked by a previous micelle formation. A system of this sort will tend to increase the total area of interface between oil and water. This leads to highly divided systems; they fall into two classes. (a) In most cases the oil and the water regions form a periodic array based on lamellae, or on rods, or on more complex objects.’+ These arrays or “macrocrystals” have been well characterized by Bragg peaks in low-angle X-ray crystallography, at least for relatively simple cases such (1)P.Ekwall in “Advances in Liquid Crystals”, Vol. 1, G. Brown, Ed., Academic Press, New York, 1971, p 1. (2) V. Luzatti and F. Husson, J. Cell. Biol.,12, 207 (1962). (3) C. Madelmont and R. Perron, Bull. SOC.Chim. Fr., 425 (1974). ( 4 ) G.J. Tiddy, Phys. Rep., 57, I (1980).
0022-365418212086-2294$01.25/0
as the soap-water system^.^^^ Several orders of reflection have been observed in favorable cases. On a macroscopic scale, the periodic structures behave like weak solids, yielding to plastic flow, and giving very high apparent viscosities. This is enough to rule them out for many industrial applications. (b) In a few favorable cases (e.g., for special surfactants) the interfacial sheets do not build up a regular array but are distributed at random. At low concentration of oil (or of water) this occurs in the form of swollen micelles (Figure 2a,c). But at intermediate concentrations the shapes are probably much more complex (Figure 2b).5 In all these cases the system behaves like a transparent fluid of low viscosity; it is called a microemulsion.6 Transparency is a consequence of the size of the oil (or water) regions (which is of order 100 A, much smaller than an optical wavelength). The low viscosity expresses the fluid character of the overall structure; it is a favorable feature for most applications. These observations suggest that two main conditions must be met for the successful achievement of a microemulsion: (i) the surfactants must prefer to remain at the oil-water interface rather than to form separate objects inside one of the bulk phase (asthey do in Figure IC),and (ii) the oil-water interface, saturated with surfactants, must not build up a periodic network; the macrocrystal must melt. Unfortunately, it is very difficult to discuss quantitatively principles (i) and (ii). Our aim in the present paper is first to review some of the basic experimental facts leading to these principles (section 11). Then we analyze (5) The most recent and detailed images of the structure (obtained by freeze etching and electron microscopy) can be found in J. Biais, M. Mercier, P. Bothorel, B. Clin, B. Lalanne, and B. Lemanceau, J. Microsc., 121, 169 (1981). (6) L. M.Prince in “Surfactant Science Series”, Vol. 6, Part I, K. J. Lissant, Ed., Marcel Dekker, New York, 1976, Chapter 3.
0 1982 American Chemical Society
The Journal of Physical Chemistry, Vol. 86, No. 13, 1982 2295
Feature Article
T PC)
’*7 ,+T
log c
amphiphile
pola r head Iipoph i l e t a iI Flgure 1. Interfacial behavior of amphiphiles at the air-aqueous solution interface: (a) the pressue in the dilute adsorbed film decreases progressively the surface tension y; (b) the surface tension y decreases abruptly and the film becomes compact; (c) when mlcelles appear the surfactant/water system is buffered; y remains constant.
/o
I Q OIL
>a)
OIL
,
0
-WATER
-
100 9(
Flgure 3. Phase diagram of the sodium laurate/water system (simplified plot, after Madelmont and Perron, ref 3). The “neat” (N) phase is lameliar, the ”middle phase” (M) hexagonal, and the intermediate (I2) phase is cubic. The shaded areas are two-phase regions. W
( 0 )
I C
1
Flgure 2. Structure of microemulsionsas a function of the water-tc-oil ratio (w/o): (a and c) swollen micelles (respectively w/o > 1; (b) “bicontinuous” structures, first proposed by ScrivenaBB The example shown here can be described as a network of water tubes in an oil matrix. (In this figue the indiiual surfactant molecules are not shown; the surfactant film is represented as a continuous sheet.)
the main properties of a large saturated interface following the early ideas of Schulman. We extend his model, including in particular the entropy of a fluctuating interface (section 111). In section IV we discuss the persistence length of the interface. In section V we arrive at the second principle and conclude that (ii) is satisfied if the interface is very flexible. Finally, in the Appendix we try to relate the flexibility to various types of measurements magnetic birefringence, rheology, etc. The ideas presented in this paper are conjectural and their predictive power is low; we are not able to guess what type of surfactant/cosurfactant mixture will give rise to high flexibilities. Our aim is simply to define certain important parameters, not to predict their actual value. 11. A Selection of Experimental Facts 1. The Ordered Phases. As mentioned in section I, the
most common structure for concentrated systems involving water and/or oil plus surfactants is a periodic structure (“macrocrystal”). We discuss this now in some more detail. a. WaterlSurfactant Systems. Figure 3 gives the phase diagram of the system sodium laurate + water, which is a classic examples3 When the laurate is dilute, we find micelles; but at higher concentrations (around room temperature) we find a hexagonal crystal of rods (middle phase) and a lamellar crystal of bilayers (neat p h a ~ e ) .In ~ (7) The lamellar phase is smectic and is a liquid crystal rather than a crystal. But it is also highly viscous. We do not discriminate (here) between smectica and crystals.
W polar headtai1-v
+
17 +
n+n+n-
W
Figure 4. A naive steric mcdei conelating the shape of the amphiphib to the spontaneous curvature of the interface.
many cases we also find (at low water content) inverted phases with a water sphere, or cylinder, surrounded by aliphatic regions. Above a certain temperature limit, the aliphatic chains in all these systems are liquidlike; we shall be concerned only with this regime, which is the relevant one for our purposes. Note that the macrocrystals are special: ordered and periodic at large scales (100 A) but disordered and liquid like at the atomic level (3 A). What are the stabilizing forces? One essential parameter is the area Z of interface per surfactant molecule. In a zeroth order approximation, Z is roughly constant for all surfactant concentrations. In an improved approximation it has been noted that I: increases regularly with the water content.* Another relevant parameter is the curvature 1/R of the interface counted as positive for a direct micelle and negative for an inverse micelle (here R is the radius of curvature). For ionic surfactants with a small polar head and an aliphatic tail, the curvature is largely controlled by
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The Journal of Physical Chemistry, Vol. 86, No. 13, 1982
parent, nonviscous liquid which has been called a microemulsion by Schulman.15 For some reason, macrocrystals are suppressed. By various methods (electron microscopy,16J8X-ray,16 or neutron scattering,17etc.) we know that the constituent objects are small ( 100 A)-much smaller than the droplets of an ordinary emulsion (10 pm). The main features are the following: a. Area and Curvature. The interfacial area per surfactant X can be inferred from measurements of the droplet size, at least in cases where the droplets are well defined in shape (e.g., spherical with radius R) and reasonably monodisperse. Consider for instance a dilute system of water dropleh in oil, with v dropleta/cm3, giving a volume fraction of water 4w and associated with n, surfactants/ cm3. These parameters are linked by the relations 4aR2 n, = -v (11.1) X 4aR3 4w = -v (11.2) 3 giving 4w/% = Y3ZR (11.3)
-
-
amphiphile cosurfactant /// w a t e r phase \ \ \ oil p h a s e 5. Stnrcture of the Interfacial Rkn: R,, radius of the water core; R,, hydrodvnamlc radius (radsus of a moving droplet as seen in viscosity and centrifugation experiments); R,, corresponds to the discontinuity of chemical compostkn due to the limit of penetration of the oil In the film.
zhr
simple steric constraints. This is illustrated in Figure 4 and discussed by Mitchell and Ninham.8 Let us call V the volume of a surfactant molecule; the ratio 3V/Z defines a natural radius of curvature Ro. This is the radius of a dense, spherical micelle made with p molecules; the total area is 4rRO2= pZ and the total volume is (4r/3)RO3 = p V . However, the surfactant chains have an extended length lo and the micellar structure is acceptable only if lo and Ro are ~omparable.~Similarly for cylinders lo 2V/X and for lamellar structures lo V / 2. Thus by forming the parameter l o X / V one can, to some extent, guess the type of aggregate which will show up preferentially.’O On the whole the discussion of ref 8 helps us to understand why a particular macrocrystal form tends to occur (Le., lamellae vs. rods). But the establishment of a welldefined periodicity between the lamellae, or the rods, depends on the existence of long-range forces between them: attractive (van der Waals) or repulsive (electrostatic, steric, plus the short-range MarEelja repulsion”). In any case we end up with a material which is not liquid and is highly viscous. b. Water Oil + Surfactant. We have but few X-ray or neutron data on systems with oil, water, and surfactant. But from the mechanical properties (also sometimes from observations on the optical birefringence), it usually appears that most of the concentrated regions of the phase diagram again corresponds to macrocrystals.12 2. Microemulsions. If we choose special surfactants6 or preferably if we add to the surfactant a cosurfactant (usually an alcohol ranging from C4to C6),we can obtain, in a broad domain of concentrated mixtures,14 a trans-
-
-
+
(8)D. J. Mitchell and B. W. Ninham, J. Chem. SOC.,Faraday Trans. 2, 77, 601 (1981). (9)This statement is very approximate (the chains do not converge radially toward the center of the micelle) but it gives an upper limit which is not unrealistic for conventional surfactants. (10)The weakest point in this discuseion is the precise definition of b For molecules which em not too long, it may be nearly wrrect to define 10 by the length of the fully extended chain. But for block copolymers, where the chains are often far from full extension, 1, would become a function of Z. (11)S. Marcelja and N. Radic, Chem. Phys. Lett., 42, 129 (1976). (12)Exceptional objects, such 88 the lyotropic nematics made of rodlike (or platelike) micelle^,'^ are not included in OUT discussion. (13)J. Charvolin, A. M. Levelut, and E. T. Samulski, J.Phys. Lett., 40,L 587 (1979).
Thus a measurement of R gives 2 directly. At large R, X tends toward a well-defined limit X*; this appears to be a rather general rule, valid for systems with a single surfactant as well as for the more frequent cases with surfactant + cosurfactant.lg We shall see in section I11 how this relates to the pioneering ideas of Schulman. Another interesting aspect of these data is that they tell us what is the curvature 1 / R of the interface under given constraints (given 4, and n,). Here the situation is quite different from what we had in the water/surfactant systems. One given microemulsion may show a broad spectrum of curvatures depending on 4wand n,. Indeed a number of microemulsions transform without any apparent discontinuity from water/oil to oil/water.20 Thus the average curvature is not a leading feature of their stability. (We shall give a theoretical discussion of curvature effects later in section 111). b. Structure of the Interface. Several detailed structural studies have been performed more recently on microemulsions.17-23 We focus here on the studies of the interfacial film.The composition and thickness of the film can be deduced from hydrodynamic measurements and from neutron scattering. The local state of the surfactant in the film is known by various means, e.g., EPR on spin-labeled amphiphiles, NMR relaxation, fluorescence depolarization. (i) Neutron scattering allows one to perform “variable contrast experimentd”’ which are not feasible with X rays. ~~~
~~
(14)It is important to observe that, in many systems, microemulsions dominate a large fraction of the phase diagram, extending continuously from the oil-rich to the water-rich side. (15)T. P.Hoar and J. H.Schulman,Nature (London),152,102(1943). (16)J. H.Schulman, W. Stockenius,and L. M. Prince, J. Phys. Chem., 63,1677 (1959). (17) M. Dvolaitzky, M. Guyot, M. Lagties, J. P. Lepesant, R. Ober, C. Sauterey, and C. Taupin, J. Chem. Phys., 69,3279 (1978). (18)J. Biais, M. Mercier, P. Bothorel, B. Clin, P. Lalanne, and B. Lemanceau, J. Microsc., 121,10 (1980). (19)See, for instance, H. F. Eicke and J. Rehak, Helu. Chim. Acta, 59, 2883 (1976). (20)B.Lindmann, N.Kamenka, I. Kathopoulis, B. Brun, and P. G . Nilson, J. Phys. Chem., 64,2486(1980);F.Larche, J. Rouvigre, P. Delord, B. Brun, and J. L. Dusewoy, J. Phys. Lett., 411,437 (1980). (21)One difficulty may occur, however; the behaviors of an ionic surfactant in HzO and in D20 are not exactly identical (see, for instance, S. Chiou and D. Shah,J. Colloid Interface Sci., 80,49 (1981)). (22)A. M. Cazabat and D. Langevin, J . Chem. Phys., 74,3148(1981). (23)R. Ober and C. Taupin, J. Phys. Chem., 84,2418 (1980).
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The principle is explained in ref 17; it is based on the difference between the scattering amplitudes of hydrogen and deuterium. From plots of the scattered intensity vs. H/D fraction in the oil (and in the water) one can ultimately extract the composition of the interfacial film. The "film" is defined here as the region for which the scattering amplitude is unaffected by isotopic substitution in the oil (or in the water).21 (ii) Hydrodynamic experiments measure for instance the mobility of water droplets in an oil matrix; this can be done either by sedimentation1' or by photon beat methods.22 What is measured here is a certain "hydrodynamic thickness" of the film (see Figure 5 ) . (iii) Local probes in the interface can also be used; there are many possibilities here (fluorescent dyes, etc.). Some of the most detailed information comes from spin labels, modified surfactant molecules carrying a nitroxide radical on the aliphatic chain.25 From the EPR spectrum of the label one can extract two parameters: an order parameter S measuring the alignment of the label along the normal to the interface, and a correlation time 7, giving some estimate of the motional frequencies in the film.26 The main conclusions from all these sources appear to be following: The cosurfactant adsorbs strongly on the interface. Typically one finds two cosurfactant molecules per surfa~tant.~~ The order parameter S measured by spin labels in the middle of the chain, in the presence of the cosurfactant, is very small (S < 0.1).28329This is to be contrasted with the ordered phases; for instance, in the lamellar phases of amphiphile water, S is of order 0.3-0.4. This points toward a special fluidity of the interface in microemulsions. The same qualitative conclusion is also obtained from NMR studies of correlation times30and by various indirect procedures: e.g., the addition of a cosurfactant often increases the interpenetration between the surface films of two adjacent droplets, and increases the coalescence rates. The interface is not seriously modified when the oil/ water ratio is modified and in particular when structural ~aterloil).~' inversion occurs (oil/water 111. Existing Models 1. The Saturated Interface ( S c h ~ l m a n ) .a. ~ ~Principle. Let us start with a single interface of arbitrary shape separating the oil from the water; the total area of interface is A. It contains a number n, of surfactant molecules, each of them covering an area Z = A/n,. We wish to discuss first a simple view where (i) the surfactant is insoluble in bulk oil or water, (ii) interactions between different portions of the interface are negligible, and (iii) curvature energies are omitted. We are then led to a free energy of the form (111.1) f = fbulk + YowA + ~ E G ( Z )
+
-
(24)H.B.Stuhrmann, J. Appl. Crystallogr., 7, 173 (1974). (25) M. Dvolaitzky and C. Taupin, Nouu. J . Chim., 1, 355 (1977). (26)W.L. Hubbell and H. M. Mc Connell, J.Am. Chem. SOC.,93,314 (1971). (27)However, if the microemulsion is near a limit of stability (high1 salted systems)the ratio cosurfactant/surfactantis often much rduced. (28)M. Dvolaitzky, C. Sauterey, and C. Taupin, Communication presented at the Third International Conference on Surface and Colloid Science, Stockholm, 1979. (29)One interesting exce tion is found with the "constrained"films of unstable microemulsions. (30)A. M. Bellocq, J. Biais, B. Clin, P. Lalanne, and B. Lemanceau, J . Colloid Interface Sci., 70, 524 (1979). (31)M. Dvolaitzky, R. Ober, and C. Taupin, C.R. Acad. Sci. Paris, 296,11, 27 (1981). (32)J. H.Schulman and J. B. Montame. Ann. N.Y. Acad. Sci., 92,366 (1961).
I
where the second term corresponds to the bare (i.e., without surfactant) interface (with an interfacial tension yaw) and G(Z) is a surfactant free energy, depending on the area Z and containing in particular the effect of surfactant/surfactant repulsions. The Langmuir surface pressure of the film is n(z)= -aG/aI: (111.2) and the actual interfacial tension is
Y = Yow - n (111.3) If we minimize (III.l), at fixed n,, with respect to Z we obtain the condition 0 = df/dZ = ns(Yow- n) = n,y (111.4) Thus the system will adopt a well-defined area per surfactant which we call I:*;it is defined by the implicit equation n(Z*)= Yow (111.5) The state Z = Z* will be called the saturated state. It can be reached only if other possible states of the surfactant (such as pure surfactant micelles in water, or in oil) are of higher free energy. (See again Figure ICfor a counter example). In the saturated state we are dealing with a system of zero surface tension, as shown by eq 111.4. The area A is entirely defined by the number of surfactants available A* = n,Z* (111.6) b. The Case of Block Copolymers. The ideas described above are essentially due to S ~ h u l m a nand ~ ~have experienced successive periods of fashion and of rejection. But, in our view, they represent a first, necessary step in the discussion. The second step is to observe that, in many systems, the saturated state cannot be reached because, at surface densities Z-' smaller than (I:*)-',the surfactant may prefer to remain inside one of the bulk phases, either in micelles or in more complicated forms. (Recall that in (a) above we started with the surfactant being insoluble in the bulk.) A semiquantitative study of these points has been carried out for a special class of detergent^:^^ diblock copolymers, with an aliphatic chain and a polar chain welded together (e.g., polyethylene-polyoxyethylene). This type of molecules is an extension (toward high molecular weights) of the more common "nonionic surfactants". From a theoretical point of view, these long-chain objects are attractive because (i) they can be treated by well-known procedures of polymer statistics and (ii) the resulting theoretical laws give definite predictions concerning the dependences on molecular weight. For instance, with a diblock (AB) of NA(NB)monomers in the A(B) portions, one ends up with a formula for Z*(NAJVB). In the simplest case of a symmetric diblock (NA = NB = N) the prediction for large N is of the form
I:*
-
p/11
(111.7)
One interesting feature of the Cantor calculation^^^ is that a (rough) comparison is made between the surfactant at an interface and the surfactant in a micelle (on either side, oil or water). The conclusion (for large N , and NA NB) is that the interfacial situation is favored, even at I: = Z*; the repulsions between neighboring surfactant molecules on the interface are weaker than the repulsions in the case of a micelle.
~~~
(33)R. Cantor, Macromolecules, 14,1186 (1981).
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The Journal of Physical Chemistry, Vol. 86,No. 13, 1982
Thus it appears feasible to comply with principle (i) of section I by using diblock copolymers, provided that one block has a strong preference for oil, and the other for water.34 2. Limitations of the Schulman Argument. The discussion of the saturated interface in paragraph II.la gives a simple feeling for the balance of forces. But it must be refined in practice. We list some of the major factors below. a. Entropy Effects. The interface is very soft and may be bent randomly at large scales (100 A); the associated entropy is very small (since we are dealing typically with one degree of freedom per (lo0A)2) but it is nevertheless relevant, since all other contributions to the free energy are also small. These entropy effects have been mentioned first by Ruckenstein and Chi.% Talmon and Pragel.36 later showed that, in the absence of any other interaction effects, the entropy term may impose certain phase transitions; when the amount of surfactant available is decreased, they predict that the microemulsion separates into two phases (oil rich and water rich). We give a modified version of their argument in section 111.3. b. Curvature energies are weak and do not affect the local properties (e.g., the area per surfactant Z) very much. However they influence the large-scale properties and the phase equilibria. We give a discussion of the two basic curvature parameters (and of their influence) in section 111.4. c. Electrostatic energies play a role in the stability of swollen micelles of ionic surfactants. Some specific longrange effects have been considered in a recent review.37 For droplets larger than the Debye screening length, the electrostatic terms are not singular and may be lumped into the interfacial energy G(Z). They also contribute to the curvature parameters. d. Interactions between different droplets (or other shapes) tend to favor ordered structures or even to promote droplet coalescence. This will be reviewed briefly in section 111.5. 3. Entropy Effects for Flexible Interfaces. We discuss these effects in a model which is related to,but somewhat different from, the original proposal by Talmon and Prager.36 We observe first that the interface must have a certain persistence length tK: (i) it is essentially flat3* at scales smaller than tK(we shall justify this concept and estimate f K later in section IV); (ii) consecutive “pieces of have independent orientations. interface”, with an area tK2, A rough but convenient model is then obtained by dividing all space into consecutive cubes, each of linear size [E Each cube is either filled with oil or with water. The overall proportion of cubes filled with oil (water) is called #, (#w). Two adjacent cubes will have no interface, and no energy, if they are of the same type. But if they are different, we must count a free energy contribution y&’, y being the interfacial tension (111.3). For the moment, we do not assume y = 0. Rather, we say that a given chemical potential p, of the surfactant imposes a certain area per surfactant E,, through the thermodynamic condition (obtained from differentiating (111.1)with respect to n, and using (111.2) and Z, A / n , ) p8 = G(E8) + n(z8)ES (111.8) (34)The case of polyoxyethylene is more complex. POE is soluble in water (at not too high temperatures) but has also a certain compatibility with oil. (35)E. Ruckenstein and 3. Chi, J. Chem. SOC.,Faraday Trans. 2,71, 1690 (1975). (36)Y.Talmon and S. Prager, J. Chem. Phys., 69, 2984 (1978). (37)J. Overbeek, Faraday Discuss., Chem. SOC.,65, 7 (1978). (38)For the moment we assume no spontaneous curvature.
De Gennes and Taupin
1
1 PH
W
0
Flguro 6. Phase diagram in the modified Talmon-Prager model. At high surfactantcontents (low y) the mkoemulslon is stable at all water fractions. At lower surfactant contents, a phase Separation occurs. The present model does not include the spontaneous curvature of the interface associated with Brancroft’s rule. Then (and only then) the plot Is symmetrical.
The result is then a certain ~ ( 2 which ,) depends ultimately on h,.39 We have now reduced the statistics of the interface to a “lattice gas model”. Clearly, the description is very crude; the oil (or water) regions in a microemulsion do not look like an assembly of cubes. But the lattice gas model keeps some essential features of a random surface. Also, the resulting statistical behavior is well-known. When the coupling between adjacent cubes ( y t K 2is ) weaker than kT, or more precisely when yc
= &kT/EK2
(111.9)
(where a = 0.44 for a simple cubic lattice), we expect a single phase with the oil and water mixed down to the scale [ E But when inequality 111.9 is reversed, we may have phase separation. The phase diagram is shown in Figure 6. A number of significant properties emerge from the model and are probably of more general validity: (a) The values of y involved are weak. As shown by (111.9) the range of interest is y k T / t K 2 .The persistence length .$K is expected to be rather large, and thus y should be small, we are not very far from the Schulman criterion. (b) Phase separation occurs not because of specific interactions between the droplets, but purely because of a balance between interfacial entropy and interfacial energy. (c) The region of phase separation corresponds to y > yc, i.e., low surfactant content. Consider for instance a small water fraction I # J ~ in the form of droplets inside an oil matrix. If R is the radius of the droplets (assumed spherical and monodisperse) and Y the number of droplets/cm3, we have (cf. eq 11.1-11.3) v ( ~ T R ~ ) =Z n, [~
-
v(4r/3)R3 = $ J ~ giving R = 3$,/(n,&)
Thus we would expect that a sufficiently low n, (surfactants/cm3) can lead to very large R. But this is never observed! In practice the droplet sizes of all microemulsions remain rather small. A low n, leads to phase separation between the oil-rich microemulsion and a (nearly pure) water phase. (d) In practice one often observes more complex phase diagrams. In particular, certain microemulsions can coexist (39)Alternatively, we could start from a situation with a fixed number of surfactant molecules (assuming that they all lie at the interface) and introduce y as a Lagrange multiplier.
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In the concentration effect, the cosurfactant may migrate (for instance) toward the regions of strong curvature of the interfacial film; this has been discussed by H e l f r i ~ h . ~ ~ ~ ~ ~ It leads to a weaker K value and may facilitate structures which include regions of both positive and negative curvature. (iii) For films of nonionic surfactants, Robbins has considered curvature effects in some We shall give here only a simplified presentation of his ideas, rephrased in the language of eq 111.10. He proceeds in three steps. First, using steric considerations, he estimates the optimal radius R for a spherical droplet. Second, by an independent argument (analyzing the stresses on the two sides of the surfactant film)he estimates K/Ro. (The sign of K / R o is actually related to a classic rule of B a n c r ~ f t ~ ~ for ordinary emulsions;the trend is to have the best solvent of the amphiphile on the outside of the droplet.) In a third step, Robbins writes the free energy Fdof one droplet in the form
simultaneously with an oil phase and a water phase. The above lattice gas model generates two-phase equilibria only. However, the model is highly degenerate. Small perturbations on the structure of the free energy (induced by curvature effects or other corrections) might lead to three-phase equilibria. A first attempt in this direction is described in ref 36. Some difficulties are pointed out in ref 40. 4. Curvature Effects. The Schulman description ignores all energies associated with the curvature of the interface. Indeed, for many problems involving fluid/ fluid interfaces, curvature energies represent only a very minor correction, and the interfacial energy y dominates the behavior. Here, however, we deal with interfaces where y 0, and curvature effects become relevant. a. Rigidity and Spontaneous Curvature. These two basic ingredients have been defined most clearly in a paper by Helfricha41 For a curvature 1/R we expect an energy contribution per unit area of the form K F = y - - +K(111.10) Rc8 2R2 here l/Ro is the spontaneous curvature and can be of either sign (we count Ro as positive when the trend is toward direct micelles). The parameter K has the dimensions of an energy and may be called the rigidity of the interface.42 Equation 111.10 holds only if R and Ro are much larger than the interfacial film thickness L. How can we estimate K and Ro? (i) For ionic surfactants the steric considerations of Ninham and Mitchell8 may give an estimate of the spontaneous curvature. More detailed computer studies on aliphatic chains anchored at a curved interface have been carried out recently in Bordeaux.43 All these studies assume that the interfacial shell occupied by the surfactant tails is free of oil. This may be correct for the relatively short chains of conventional surfactants. (ii) The addition of a cosurfactant may act strongly on l / R o and also on K . For instance, in the (different but related) soap water systems, Charvolin and Mely showed that a certain mixture of C18and Clo soaps could give a cubic phase which is not present with the pure C14 soap.& The cubic phase is believed to be an array of rod portions (with positive curvature) related by branching regions (with negative curvature). More recently, Hendricks&took data on another system and showed the curvature trends induced by an exchange between sulfate polar heads and alcohols. From the theoretical side, we can think of at least two effects of the cosurfactant related to the curvature: a simple wedge effect and a concentration effect. In the simple “wedge effect”, changes in the average parameters E,lo, and V for the mixed interface react on l/Ro. Ninham and Mitchell8 propose that the main role of the cosurfactant is to change Ro by this process. Our opinion is different; many alkanols can adsorb in the interfacial film and modify Rot but only a few are efficient to induce microemulsions. Also, many microemulsions are continuously stable when we change the composition from oil rich ( R > 0) to water rich ( R < 0). Thus the role of the cosurfactant cannot be entirely reduced to its effect on Re
Inserting his estimates for K / R o and for R , he can now predict y; y is the interfacial energy/cm2for a flat interface ( R = a),i.e., y represents the interfacial tension between the w/o microemulsion and pure water. This procedure works remarkably well; both the estimates of R and y are quite good for a number of nonionic surfactants. There are, however, certain ambiguities in the scheme. In particular, it is not clear why the steric arguments of step 1 should give R rather than Ro. Our own preference would be more phenomenological; namely, to take R and y from direct measurements and to derive from these the spontaneous curvature constant R / R o ,using the Robbins’ eq 111.11. All the discussion assumes droplets which are not far from spherical and rather monodisperse. These two conditions often seem to be met in practice. But we should keep in mind that, whenever R > EK, the shapes must be strongly nonspherical. (iv) For diblock copolymers (which are nonionic surfactants with relatively long chains, allowing for rather simple statistical calculations), Cantor has given explicit expressions for Ro and K.33 Consider for instance the following (idealized)case: each chain (A) (B) is in an excellent solvent (oil for A, water for B); also for simplicity assume the same monomer size ( a ) for (A) and (B). Then the condition of weak curvature imposes that N A / N Bbe close to unity
(40) J. Jouffroy, P. Levinson, and P. C. De Gennes, to be published. (41) W. Helfrich, Z. Naturforsch. C, 28, 693 (1973). (42) Alternatively,we shall sometimes define the flexibility of the film by the inverse (R’) of the rigidity. (43) B. Lemaire and P. Bothorel, Macromolecules, 13, 311 (1980). (44) J. Charvolin and B. Mely, Mol. Cryst. Liquid Cryst. Lett., 41,209 (1978). (45) Y. Hendricks and J. Charvolin, J. Phys., 42, 1427 (1981).
(46) W. Helfrich, Phys. Lett. A, 43,409 (1973). A detailed calculation of the rigidity K for a mixture of two types of block copolymers has been performed by one of us (P. G. De Gennes, lectures at College de France, 1982, unpublished). (47) M. L. Robbins, ‘Micellization, Solubilization and Microemulsions”,Vol. 2, K. Mittal, Ed., Plenum, New York, 1977, p 273. (48) W. D. Bancroft and C. W. Tucker, J. Phys. Chem., 31, 1680 (1927).
-+
where the first three terms derive from eq 111.10, while the last term is related to the chemical potential of the inner constituent (e.g., water, if we are dealing with inverse micelles). He then restricts his attention to a microemulsion which is in equilibrium with one solvent (e.g., w/o in equilibrium with water) and argues that X = 0 at this point. He then derives the optimal radius R by minimization of Fd (with X = 0) and reaches the condition
K
y=2ROR
(111.11)
2300
De Gennes and Taupin
The Journal of Physical Chemistty, Vol. 86, No. 13, 1982 NA
= N(l
+ t)
NB = N(1-
a
e)
b
and the parameters in eq 111.8 are predicted to obey the following scaling laws: Ro = L / ( 6 t )
L = 2Na(a2/2*)1/3= (constant)P/'la K = (constant)y&2 = (constant)kTW8/"
(111.12)
(In eq 111.12 we have ignored prefactors containing powers of the dimensionless parameter yo&2/kT, which turns out to be of order unity in many practical cases.49 The essential features are as follows: The thickness of the surfactant layer is nearly proportional to the chain length (varying like P/llat large N). This is remarkable because we are still far from close packing (2* is much larger than A2) as is clear from eq 111.7. The spontaneous curvature (l/Ro) is directly proportional to the dissymmetry (measured by 4. The elastic constant K becomes enormously larger than k T when N is large. (This, as we shall see, may be unfavorable from the microemulsion point of view.) b. Impact of Curvature Energies on the Phase Diagrams. (i) The rigidity parameter K controls the persistence length f K . We shall discuss this in section IV, where we find that fK increases exponentially with rigidity. This in turn may react on phase separation properties, since (as explained in 111.3) one essential parameter is Y/( k Tk2). (ii) The spontaneous curvature also reacts on the phase diagram. In the original Talmon-Prager a very special view was taken, according to which interfaces of finite curvature f l/Ro were equally favored with respect to the flat interface. This may then lead to three-phase equilibria where these three possibilities compete. If we keep the (more realistic) view, according to which only one curvature sign (dictated by the Bancroft rule) is favored, we come to slightly different conclusions. The only case where predictions are available is the case of weak spontaneous curvature (IfK/Rol < 1); the main effect is a distortion of the phase diagrams.* The phase (o/w or w/o) which is privileged by Bancroft's rule increases its domain of existence. For instance, at high salinity, many ionic microemulsions show a phase equilibrium with nearly pure water. In the low salt systems, the Coulomb repulsions between polar heads tend to give a positive curvature (Le., as in direct micelles). This then leads, from our discussion, to a natural phase equilibrium between droplets (of oil in water) and pure oil. On the other hand, in high salt concentration, the Coulomb forces are suppressed, and the steric repulsions between aliphatic tails dominate, favoring negative curvature; then if we have droplets of water in oil, they can coexist with pure water. In nature more complex phase equilibria may in particular, one may find coexistence between three phases: nearly pure oil and water, plus a microemulsion. But the "augmented Talmon-Prager model" of ref 40 does not seem to account for these three-phase equilibria; interaction effects have to be invoked here. 5 . Long-Range Interactions. Up to now we omitted all effects related to possible interactions between different droplets or, more generally, between different portions of the interface. But it is clear that these interactions must (49) Also we replaced the mean field exponents of Cantor by the (slightly different) scaling exponents: see S. Alexander, J. Phys., 38,983 (1977).
,
,--\
t
hydrophilic sequence
Polymeric amphiphile hydrophobic sequence
Flgure 7. Phase separation induced by long-range van der Waals forces. (a) We start with a lamellar phase of a polymeric amphiphile, where the hydrophilic portion is swollen by water, and the hydrophobic portion by oil. (b) van der Waals attractions (e.g., between two neighboring hydrophobic regions) tend to squeeze out the water, but the process stops when the surfactants from these two layers come into contact (steric repulsion).
play a role in most cases (Le., provided that we are not dealing with dilute droplets). This has been approached from two main directions: (a) through a study of ordered phases (lamellar, etc.); (b) through measurements and theories involving the interactions between droplets in disordered phases. a. Stability of the Lamellar Phases. The simplest systems allowing for a study of these interactions are the ordered lamellar phases. For soap/water and lecithin/ water systems, Parsegian assessed the role of the longrange van der Waals forces and of certain repulsion forces (electrostatic, steric). More recently, a (less-refined but more transparent) discussion was given by for the case of interest here, namely, when we have successive layers of oil and water separated by surfactant films. There are, however, some points where we disagree with Huh. (i) He assigns a finite surface tension y to each layer; we tend to believe that y = 0 for any lamellar system of this type (the free energy is stationary with respect to 2 ) . (ii) His analysis of thermal fluctuations in the lamellar phase is open to some doubt. Because of (i) he has stabilizing terms due to interfacial tension which, we think, are not realistic. We shall come back to this fluctuation problem in section V. An improved discussion of thermodynamic stability (against phase separation) has been given in ref 33 for the lamellar phases of oil + water + block copolymer. When the long-range van der Waals attractions are significant, a swollen lamellar phase, containing sheets of pure oil and pure water, should be thermodynamically unstable with respect to a system of "wet bilayers" as shown on Figure 7. In the final state each half of bilayer is still wetted with its preferential solvent (water on the part shown in the figure) but the water film has become thinner (as required by the long-range van der Waals forces). The thinning stops when the two copolymer sheets are put into contact; then steric repulsions between the chains become rapidly dominant. (For charged systems, electrostatic repulsions play a similar role.) One interesting consequence of this discussion is the prediction of three-phase equilibria: oil + water + wet bilayers. The composition corresponding to the wet bilayer is easily obtained from standard statistical arguments on (50) Chun Hu, J . Colloid Interface Sei., 71, 408 (1979).
The Journal of Physical Chemistry, Vol. 86, No. 13, 1982 2301
Feature Article
block copolymers. The changes in this composition which are predicted by the theory when the length of the chains (or the quality of the solvents) is modified, resemble closely what is found experimentally on three phase equilibria involving microemulsion oil water. A future aim for theoretical work should thus be to enlarge the Cantor discussion from lamellar phases on to more disordered systems, such as a “bicontinuous” microemulsion where both the oil and water regions are connected over macroscopic distances. Our approach in section V will not be to try and analyze these complex disordered phases but rather to search for fluctuation instabilities of the corresponding lamellar phases. b. Interactions between Droplets. When two spherical droplets (of radius R) are separated only by a small gap h, the van der Waals attraction V between them is of order V(h) = -AR/(6h) (h < R) (111.13)
complex spacetime correlations. The general idea is that, with y = 0, we have giant fluctuations of the cell shape, which become observable (under phase contrast) with an optical microscope. Our approach to microemulsions in section IV and V will follow similar lines. We start in section IV with a discussion of the fluctuations of a single interface. In section V we modify this discussion to include (as far as we can) the interactions between different pieces of the divided structure. 2. Statistics of a Random Interface. Let us assume now that our interface has a negligible spontaneous curvature (l/Ro 0) and is close to a certain reference plane ( x y ) . The distances between the plane and the interface will be called { ( x y ) . The curvature is then
where A is the Hamaker constant (and is often comparable to k T). Light-scatteringl and neutronz3experiments on microemulsions can give the osmotic compressibility. Assuming a specific model with monodisperse, undeformable droplets, one then estimates the interactions V. In a number of cases V is indeed strong, especially for large droplets as predicted by (111.13). The distance of closest approach h may also be rather small (because the soft interfacial films from the two spheres interpenetrate each other slightly). These effects have been analzyed recently on one typical system (still using an empirical value for h);5zwith surfactant and cosurfactant, the agreement between osmotic data and theoretical estimates for A and h is rather good. Very often the van der Waals attractions between droplets may lead to a liquid-gas transition where each droplet plays a role similar to that of one argon atom in fluid argon.53 This could provide an explanation for certain phase equilibria between two microemulsions which have been observed in systems of well-adjusted sal i n i t p or temperature.55 However, it is not at all sure that this picture is always right: (i) the actual concentration of the minority phase at criticality is often much smaller than predicted by the liquid-gas analogy; (ii) in many cases the denser system of droplets probably becomes bicontinuous, and the description should be refined accordingly-returning to the end of paragraph III.5a.
The free energy (111.10) becomes
+ +
IV. The Role of Flexibility 1. Comparison with Red Blood Cells. In eq 111.10 two fundamental constants are associated with a saturated interface: the spontaneous curvature l/Ro and the curvature elastic constant K. These two parameters have been introduced first, in connection with surfactant films, by Helfrichtl with applications to vesicles and red blood cells. In fact, in one of the existent models for red blood cells,@ the surface area of the cell is assumed to adjust and to minimize the overall free energy; thus, in this model, the surface has y = 0. One major success of the model has been to explain the scintillation of red blood cells and their (51) A. Calje, W. Atgerof, and A. Vrij in ‘Micellization, Solubilization and Microemulsions”, K. Mittal, Ed., Plenum, New York, 1977, p 779. (52) B. Lemaire, D. Roux, and P. Bothorel, J. Phys. Chem., to be published. (53)C. Miller, R. Hwan, W. Benton, and Tomlinson Fort, Jr., J. Colloid Interface Sci.. 61. 554 (1977). See also E. Ruckenstein, Chem. Phys. Lett., 57, 517 (1978). (54) K. Shinoda and H. Kunieda, J. Colloid Interface Sci., 75, 601 (1980). (55) K. Shinoda and S. Friberg, Adu. Colloid Interface Sci., 4, 281 (1975). (56) F. Brochard and J. F. Lennon, J. Phys, 36, 1035 (1975).
-
(IV.1)
where we have gone to two-dimensional Fourier transforms
lq= Jdx dy {by) exp[i(q,x + qyy)l
(IV.3)
We shall be mainly interested in the local orientation of the surface, defined by a unit vector n normal to it ny = -al/ay n, 1 (1v.4) n, = -a{/& For small fluctuations 6n = (nx,ny)we can write 16n,I2 = q 2 l ~ q l 2
(IV.5)
Applying the equipartition theorem to all modes in eq IV.3 we obtain the thermal average of these fluctuations (16n,l2) = kT/(Kqz)
(IV.6)
where Tis the temperature and k is Boltzmann’s constant. We can now look at the angular correlations between two points (0 and r) on the surface e2(r) = (16n(0)- 6n(r)I2)= C2[1 - cos (q-r)](16nq12) P
(IV.7)
where l / a is a high q cutoff-a microscopic length related to the detergent size. Jo(x)is a Bessel function; the factor 1 - Jo(x)is essentially equal to 1 for x >> 1and to 0 for x
