J . Phys. Chem. 1992, 96,4174-4187
4174
confirm all hypotheses proposed for the reaction mechanism on the basis of experimental investigations and supply valuable further details at the molecular level. Acknowledgment. Financial support for this work by the Austrian "Fonds zur Forderung der wissenschaftlichen Forschung"
(Projects 7393-CHE and 8475-MOB) is gratefully acknowledged. Thanks are due to the Computer Centre of the University of Innsbruck for supplying generous computer time on the Convex 220. Registry No. NaCI, 7647-14-5; CuC12,7447-39-4; H20, 7732-18-5.
FEATURE ARTICLE Sponge Phases in Surfactant Solutions D. ROUX,*C. Codon, Centre de Recherche Paul Pascal, Avenue du Dr Schweitzer, 33600 Pessac. France
and M. E. Cates Cavendish Laboratory, Madingley Road, Cambridge, CB3 OHE, UK (Received: August 12, 1991, In Final Form: January 22, 1992)
In surfactant solutions the amphiphilic molecules can self-assemble reversibly into a variety of structures. In many cases the most stable structural unit is a bilayer which, although locally flat, tends to wander entropically at large distances. This can result in a stable isotropic phase, known as L3, which consists of a spongelike "random surface" of bilayer that divides space into two interpenetrating solvent labyrinths. This phase has unusual thermodynamics: in particular, it can support a second-order phase transition of Ising type from a symmetric state, in which the two solvent regions have equivalent statistics, to an asymmetric state in which they differ. In this article we survey the present understanding of this phase in relation to the theory of elasticity for two-dimensional fluid films. We then focus more closely on the thermodynamics of the system, which we develop within the framework of Landau-Ginzburg model with two coupled order parameters. This reproduces correctly several interesting features of the phase diagram and also explains the very unusual correlation functions of the sponge phase that are measured in static light scattering. We present evidence for a line of second-order phase transitions between symmetric and asymmetric states in a sponge system and show that the correlation functions are different on the two sides of the transition, in accordance with theory. It is argued that sponge phases may provide an ideal testing ground for theories of tricritical points and higher order critical behavior.
1. Introduction
Due to their amphiphilic character, surfactant molecules in dilute solution (either in water or in oil) tend to self-assemble reversibly into large aggregates. Roughly speaking, one may distinguish between spherical, cylindrical, and lamellar shapes. Spherical micelles are naturally limited in size due to packing constraints (there must not be a hole in the middle) whereas cylindrical micelles and bilayers can in principle become extremely large compared to a molecular size in at least one direction. These supermolecular assemblies can organize themselves on a larger scale, with either long-range order (liquid-crystal phases) or only short-range correlation (liquid isotropic phases). Examples of liquid crystalline phases can be found involving finitesize micelles packed on a three-dimensional cubic lattice, infinite cylinders packed on a two-dimensional hexagonal lattice, or infinite lamellae packed on an one-dimensional lattice (the lyotropic smectic A phase).] It is also possible to have cubic phases in which the structural unit is a continuous bilayer in the form of a periodic, saddle-shaped ~ u r f a c e . ~There , ~ are many other cases involving, for example, nematic order in phases of (relatively small) micellar disks or r0ds.~3 (1) Ekwald, P. In Advances in Liquid Crystals; Brown, G. H., Ed.; Academic Press: New York, 1975. ( 2 ) Charvolin, J. J . Chim. Phys. 1983, I , 80. (3) Helfrich, W. Physics of Defects; Les Houches X X X V ; North Holland Publishing Co.: New York, 1981; p 716.
0022-365419212096-4174$03.00/0
In isotropic phases, definite identification of the local structural unit is sometimes more difficult to achieve. However, it is quite clear experimentally that elongated cylindrical aggregates can arise; these may be either relatively short and fairly rigid (for example, in charged systems at low ionic strength), or extremely long and semiflexible, which gives rise to interesting polymerlike behavior.6-'0 The sphere to rod transformation is not a phase transition but a continuous process that occurs upon changing the surfactant (or sometimes salt) concentration. Concerning the existence of two-dimensional aggregates in the liquid state, the experimental situation is less clear. However, there is now very strong evidence that the so-called L3phase of surfactant solutions, which occurs widely in materials that exhibit a dilute lyotropic smectic A phase, is based on a' local bilayer as the structural unit. Various different models for the larger scale structure have been proposed, among the earliest of these being a phase of disks." From a simple argument that the length of (4) Yu,L. J.; Saupe, A. Phys. Reo. Leu. 1980, 45, 1000. ( 5 ) Boden, N.; Corne, S. A,; Jolley, K.W. J. Phys. Chem.1987,91,4092. (6) Cates, M. E.; Candau, S.J. J . Phys. Condens. Matter 1990, 2,6869. (7) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Carey, M. C. J. Phys. Chem. 1983, 87, 1264. (8) Porte, G.; Appell, J.; Poggi, Y. J . Phys. Chem. 1980, 84, 3105. (9) Candau, S.J.; Hirsh, E.; Zana, R. J. Colloid Interface Sci. 1985, 105, 521.
(10) Cates, M. E. J . Phys.
Fr. 1988, 49, 1593.
0 1992 American Chemical Society
Feature Article
The Journal of Physical Chemistry, Vol. 96, No. 11, 1992 4175
a
Figure 1. Two-dimensional cut of long-range organization for surfaces embedded in 3-Dspace. (a) Lamellar phase in which the membranes are packed in a one-dimensionally long-range ordered phase. (b) Liquid phase of spherical vesicles. (c) A liquid phase of connected random bilayers.
the edge (presumed unfavorable in free energy) of a disk grows linearly with its radius, it is easy to argue12that dilute phases of large isolated disks are highly unstable (although disks with aspect ratio not too far from unity can certainly be observed5). On the other hand, there can be no similar objection to a phase of closed ~esic1es.l~However, in what follows we shall present evidence that in many L3 phases a dilute system of bilayer-forming surfactant satisfies the need to avoid edges in a different way: by making a continuous web of bilayer in a spongelike structure. (See Figure 1.) We shall be concerned mainly with dilute systems in which the thickness of the bilayer is small compared to the characteristic length scale of the sponge structure. In this case (assuming there are no strong Coulombic interactions) a description can be based on the theory of continuum elasticity of two-dimensional fluid films embedded in three-dimensional space.14J5 The detailed microscopic composition of the bilayer (which may in practice contain small amounts of additional components such as alcohol as well as just surfactant) is then represented by a small number of elastic constants. Through this approach, connection can be made with a large body of work on random surfaces, much of which has been motivated by analogous problems that arise in the physics of elementary parti~1es.l~We outline the basis of this approach in section 2 and use it further in sections 5 and 6 to discuss scaling laws and a lattice-based model. In section 3 we give some fundamental arguments coneeming the symmetry of the sponge phase, which can be different from that of an isotropic liquid. These ideas are developed further in sections 7 and 8 where we develop a Landau4inzburg model with two fluctuating order parameters. Section 4 reviews some experimental evidence for the presence of a continuous web of bilayer in the sponge phase, and in section 9 we present data on light scattering and phase transition behavior that confirm some of the fundamental ideas that emerge from the Landau-Ginzburg approach. 2. Elasticity of Membranes As a preliminary to our discussion of the statistical physics of sponge phases, we briefly review the continuum elasticity theory for a bilayer. Within the harmonic approximation the elastic free energy of a curved surface, whose microscopic structure is symmetric about the midplane, can be described asI4
1
E = S [ p + i P + a K dS
(1)
where p is proportional to the chemical potential of the surfactant, H is the curvature (H = l/R1 + 1/R2, where R1 and R2 are the two principal radii of curvature of the surface), and K = 1/(R1R2) is the Gaussian curvature. The parameter K is the mean bending constant and R is the Gaussian bending constant. We have assumed that the area of bilayer per surfactant molecule is a constant; hence the surfactant concentration is proportional to the ~~~~~
~~~
(1 1) Nilsson, P. G.; Lindman, B. J . Phys. Chem. 1988,88, 4764. (12) Ben-Shaul, A.; Szleifer, I.; Gelbart, W. M. J. Chem. Phys. 1985,83, 3597. (1 3) Israelashvili, J.; Mitchell, D. G.; Ninham, B. J. Chem. Soc., Faraday Trans. 1976, 72, 1525. (14) Helfrich, W. Z . Naturforsch. 1973, 28c, 693. (15) David, F. In Statistical Mechanics of Membranes and Surfaces; Nelson, D., Piran, T., Weinberg, S., Eds.; World Scientific: Singapore, 1989.
b a Figure 2. Typical membrane organization: handle (a) or isolated component (sphere, b).
surface area of bilayer per unit volume. It can be shown (Gauss-Bonnet theorem) that15
S K d S = 47r(nc - nh)
(2)
where n, is the number of disjoint pieces or components and nh is the number of handles of the surface (Figure 2). Consequently, the Gaussian curvature is a function only of the topology. For stability, the mean curvature rigidity K must be positive, and R must obey -2K < R < 0. Within this range, if the local conditions of the bilayer are changed so as to make R less negative, structures with many handles, such as the sponge phase, will be relatively favored in free energy compared to a phase of vesicles or lamellae. The rigidities K and R are both expressed in units of energy and this allows us to define two separate regimes concerning the statistical physics of membranes. In order to understand the effect of thermal fluctuations, one has to compare K to kBT ( k Bis the Boltzmann constant and T the temperature). When K >> kBT, the membrane will be basically rigid and thermal fluctuations will have little effect on the shape of the membrane; on the other hand, when K kBT (flexible surfaces) thermal fluctuations will become important. This idea can be made more quantitative by calculating the persistence length of a bilayer, [k, which is defined as the length scale over which thermal fluctuations lead to a significant departure from flatness. The result is an exponential function of the mean curvature rigidity16
-
tk = ae4*d3kBT
(3) where a is a molecular length. This special form arises from the two-dimensional character of the fluctuations about a flat film. A bilayer is expected to become thermally crumpled for lengths larger than [k. Thermal fluctuations are also responsible for effects on lengths smaller than the persistence length. For a system of stacked membranes (lamellar phase), Helfrich has shown17 that a longrange repulsive interaction comes from the restriction of the undulations of the membrane (compared to a free membrane) resulting from steric exclusion by neighbors in the stack. In a regime where the distance d between membranes is smaller than the persistence length, the entropy lost per unit area associated with this restriction is
-F= - -3a2 - ~ B ~T B T A 128 K &
(4)
where the prefactor depends somewhat on the approximations made.” Experimental evidence for such an interaction has been recently demonstrated in systems of flexible membranes in the lamellar state.18-22 The peculiarity of the systems studied is the large range of distances over which the lamellar phase is stable. Because of the long-range repulsive interaction, these systems can (16) de Gennes, P. G.; Taupin, C. J. Phys. Chem. 1982,86,2294. (17) Helfrich, W. Z . Naturforsch. 1978, H a , 305. (18) Safinya, C. R.; ROUX,D.; Smith, G. S.; Sinha, S. K.; Dimon, P.; Clark, N . A.; Bellocq, A. M. Phys. Rev. Lett. 1986, 57, 2718. (19) ROUX,D.; Safinya, C. J. Phys. (Fr.) 1988,49, 307. (20) Nallet, F.; ROUX,D.; Prost, J. J. Phys. (Fr.) 1989, 50, 3147. (21) Safinya, C. R.; Sirota, E.; Roux, D.; Smith, G. Phys. Rev. Lett. 1989, 62, 1134. (22) Bassereau, P,; Marignan, J.; Porte, G. J. Phys. (Fr.) 1987,48,673.
Roux et al.
4176 The Journal of Physical Chemistry, Vol. 96, No. I I , I992
scam
Figure 3. Typical defects that are not allowed in a sponge phase. The cost in bending energy of such defects is assumed to be prohibitive, because the surfactant film is bent on a molecular length scale (instead of a much larger one of order the “pore size” or persistence length).
be swollen by a large factor on adding solvent, and the repeat distance d of the smectic order can vary from a molecular length (50 A) to very large distances (1000-10000 A),23allowing a quantitative check of both the d dependence and the amplitude in eq 4. This provides good evidence that the harmonic elastic description of membranes is appropriate for dilute phases of surfactant bilayers. Although it can exist over a large range of dilutions (repeat distances), a lamellar phase stabilized by undulation interactions must eventually melt, on further dilution, into an isotropic liquid phase. It may melt into a micellar phase through a strongly first-order phase transition, but the more interesting possibility is to melt into an isotropic phase (the L3 phase) in which the local structural unit remains a bilayer. Below we present evidence that this is usually a sponge phase. In this scenario,24the sponge phase is typically very dilute (at most a few percent bilayer by volume) and is stabilized mainly by entropic effects which melt the lamellar phase when its layer spacing is too large compared to the persistence length. Note that the sponge has a characteristic length scale or pore size, 4, which depends on volume fraction 4 as 4 6/4, which is the same scaling as in the lamellar phase, 6 being the membrane thickness. E~perimentally,~~ however, it is often possible to make a stable L3 phase at rather high concentrations, though usually only in a very narrow range of composition (for example, by adding just the right amount of alcohol, which incorporates into the bilayer as a cosurfactant). A plausible explanation is that this “fine tuning” is needed to arrange for the effective Gaussian rigidity R to be very close to (but, for stability, less than) zero.26 This allows handles of small mean curvature (R,= -R2) to form at almost no cost in curvature energy. In this case, the sponge phase can be thought of as a disordered analogue of a cubic minimal surface
-
3. Structure and Symmetry of the Sponge Phase
We shall assume the L3 phase is a sponge phase, as defined by the following properties: (a) the structure is made up locally of bilayer, with a spatial distribution that is uniform at large distances; (b) there is neither long-range positional nor orientational ordering; (c) the free energy cost of edges or seams in the bilayer is considered as prohibitive and therefore such defects are taken to be absent (Figure 3). Of these, the first two assumptions are no longer controversial. In contrast, the last assumption is obviously more questionable, and indeed many other models of the L3 phase have been proposed (23) Larche, F.; Appell, J.; Porte, G.; Bassereau, P.; Marignan, J. Phys. Rev. Lett. 1986, 56, 1700. (24) Cates, M. E.; Roux, D.; Andelman, D.; Milner, S.; Safran, S. Europhys. Lett. 1988,5,733; Erratum Zbid. 1988, 7,94. Cates, M. E. Proceedings of NATO ASI on Phase Transitions in Sop Condensed Matter; Sherrington, D., Riste, T., Eds.; Plenum: New York, in press. (25) Porte, G.; Marignan, J.; Bassereau, P.; May, R. J. Phys. (Fr.) 1988, 49, 511. (26) Porte, G.; Appell, J.; Bassereau, P.; Marignan, J. J. Phys. (Fr.) 1989, 50, 1335. (27) Anderson, D. M.; Wennerstrom, H.; Olsson, U. J. Phys. Chem. 1989, 93, 4243.
a
b
Figure 4. Lattice construction of a random phase of bilayers (sponge phase). The lines represent the interface between domains of a given spin orientation. In the model they are assumed to be surfactant bilayers separating two identical media (water, for example) labeled I (inside) and 0 (outside).
which are at odds with it. In particular, it rules out a priori the possibility of a phase of large disks.” Ultimately we shall appeal to the success of the model in explaining a wide range of experimental data, particularly small-angle light scattering, as justification that this assumption is appropriate. Of course, it cannot be strictly true that there are no tear or seam defects whatever in a macroscopic sample of L3 since these defects can be made at finite energy cost. The real question is whether they are numerous enough to change the conclusions of a simple model in which they are omitted. This important issue has been addressed in detail recently by Huse and Leibler, using a mapping onto a lattice gauge theory.28 Their conclusion was that the main predictions of the simplified model (in particular the phase transition that it predicts between a symmetric and an asymmetric sponge phase under dilution) are not altered by the presence of a finite line energy for tear and seam defects, so long as this remains above some critical threshold. Above the threshold, all the line defects occur as finite closed loops and they do not disrupt the large-distance behavior of the system. In contrast, at the critical line energy, the edges become infinite and the special character of the system is destroyed. With this reassurance, we generally ignore such defects in the rest of this paper, which permits a much simpler mathematical’description. It is instructive at this point to introduce a lattice model that generates a geometrical arrangement of the bilayer consistent with our assumption^.^^ (The model, discussed further in section 6, corresponds to a slight modification of spin models used to described bicontinuous microemulsion phases.29) We introduce a three-dimensional jattice of Ising spins (Figure 4), each of which can occupy two states (up or down). Assuming an arrangement of spins with only short-range correlation, one can define everywhere the interface between “up” and “down” domains. Next, the bilayer is imagined to be draped over this interface (smoothing out any sharp corners that may arise). We can then remove the spins and keep the bilayer; as shown in Figure 4, the resulting surface has no defects and divides space into two continuous regions of identical material (solvent), which we shall arbitrarily designate as I (inside) and 0 (outside). Notice that, starting from the interface, one can reconstruct the entire spin state uniquely as soon as that of any single spin is given. (Simply count whether an odd or even number of interfaces occur between this spin and any other to determine the latter’s state.) This applies even within any disconnected droplets of (say) I in 0 that happen to be present; it is a result of a basic topological theorem that any surface without edges in three dimensions divides space in two. Since the labeling of I and 0 is arbitrary, any Hamiltonian we use to describe the system must be invariant under a global interchange of I and 0. This powerful exact symmetry derives from the local symmetry of the bilayer itself (about its midplane); there is no telling one side of the interface from the other. This holds not only for the harmonic elasticity Hamiltonian (eq 1) but also for more general forms. Thus the I and 0 domains should be (28) Huse, D. A.; Leibler, S. Phys. Rev. Lett. 1991, 66, 437. (29) Andelman, D.; Cates, M. E.; ROUX,D.; Safran, S. J. Chem. Phys. 1987, 87, 7229. See also Phys. Rev. Lett. 1986, 57, 491.
The Journal of Physical Chemistry, Vol. 96, NO. 11, 1992 4177
Feature Article statistically identical unless this global symmetry is spontaneously broken. The spontaneous breaking of the symmetry can result in a secondsrder phase transition, whose significance in the context of random surface models was first emphasized by Huse and Leibler.30 We refer to phase of unbroken symmetry as a symmetric sponge. If the symmetry is broken, we have an asymmetric sponge phase; both types appear to exist in nature. The symmetry is most easily presented in terms of spins and the lattice model, but it applies quite generally to any phase that satisfies the three assumptions a-c detailed above. It is of exactly the same character as the symmetry in the king model of a ferromagnet, where up and down spin domains are statistically equivalent above the Curie point, but not below, when there is a spontaneous symmetry breaking (magnetization). We reiterate that, according to the recent work of Huse and Leibler,28 the Ising-like symmetry survives even when the interfaces contain finite defects, although its mathematical description becomes more complicated. Formally, the same Ising-like symmetry breaking is of course present at a binary (liquid-gas) critical point: however, there is a crucial difference of principle in the present system. Specifically, even for a binary system (pure surfactant solvent) the existence of a bilayer that divides space imparts an exact underlying symmetry at all compositions and temperatures for which the sponge phase exists. This means that, rather than an isolated critical point, there can easily arise an entire line of second-order critical points, corresponding to the continuous breaking of the king like symmetry, on the two-dimensional (concentration/temperature) phase diagram for a binary system at fured pressure. This is most unusual for liquid systems (although not unique, arising for example in 3He/4He mixtures at low temperature). Although this has not yet been found for a true binary system, below we shall present evidence for an equivalent line of critical points in a multicomponent system under variation of two composition variables at fixed temperature and pressure. A corollary is that tricritical points and other higher order critical behavior should be unusually easy to study in sponge phases, as discussed further in section 7.
+
4. Experimental Evidence for Bilayer Continuity Apart from the light scattering data presented in section 9 below, there is substantial experimental evidence for the existence of a (symmetric) sponge phase as defined above. Both oil- and water-rich sponge phases have been described. They are found in the vicinity of swollen lamellar phases stabilized by undulation forces. The first reported phasesz5*"were quaternary systems but even binary (surfactant plus water) systems have now been shown to exhibit a sponge phase.32 The experimental facts leading to a geometrical description of a structure of connected bilayers are based on four measurement techniques: transport properties measured using conductivity and self-diffusion; structural determination using neutron scattering; and freeze-fracture electron microscopy. The most spectacular conductivity measurements are found in an oil-rich sponge phase.31 This particular phase is stable in a surfactant concentration range from about 20% down to about 2%. The solvent is a mixture of dodecane and pentanol and is an insulator. The membrane is an inverted bilayer, slightly swollen with water, which can be considered as a conductor (the water is filled with charges coming from the surfactant counterions). Measurements of the reduced conductivity (compared to the expected conductivity for the same total volume of pure water + charges) show a high value (around 0.65) all along the dilution path, even for the most dilute samples. A comparison with the conductivity of a nearby droplet phase (inverted micelles) for the same surfactant concentration shows order of magnitude differences in the reduced conductivity (which is of the order 10-'-104 (30) Huse, D.; Leibler, S. J . Phys. (Fr.) 1988, 49, 605. (31) Gazeau, D.; Bellocq, A. M.; Roux, D.; Zemb, T. Europhys. Letf. 1989, 9, 447.
(32) Strey, R.; Shomaker, R.; Roux, D.; Nallet, F.; Olsson, U. J . Chem. SOC.,Faraday Trans. 1990,86, 2253.
for the droplet phase). This indicates that the bilayers are connected and that continuous water paths through the system are present. The value of 0.65 instead of 1 (expected, for example, in a perfect lamellar phase along the direction parallel to the layers) is easily explained in terms of the tortuosity of the conduction path through a spongelike membrane structure.' In a water-rich sponge, the conductor is the solvent and the insulator is the membrane; in this case the reduced conductivity is found to be about 213 of the conductivity of the pure s o l ~ e n t . ~ ~ ~ ~ ~ Again, the ratio of 213 can be interpreted as related to the tortuous path through a conducting medium filled with randomly distributed insulating surface elements. More recent experiment^^^ have generalized these results to diffusion of surfactant. Other evidence for the sponge structure has been obtained by neutron scattering experiment^.^^,^^.^^ The scattering intensity can be divided into two regimes: the large wave vector (large q) regime where all the data fall on a universal curve (which is the form factor of a flat bilayer, of the same thickness as measured in the nearby lamellar phase) and the small-q regime where correlations between pieces of bilayer lead to a broad peak. The large-q regime crosses over from a 1/ q 2 behavior for q < 2 r / d ( d is the bilayer thickness), which is expected for the scattering of a flat bilayer with random orientation, to a 1/ q 4 behavior at larger q (q > 2 r / d ) characteristic of the scattering from a thin interface. The position of the broad peak due to bilayer-bilayer correlations gives useful structural information. In a lamellar phase the quasi-Bragg peak due to the long-range order of the bilayers can be exactly located in q space (neglecting logarithmic corrections due to the undulations) as q = 2 r / d with d = 814 (4 is the volume fraction of membrane). For a random distribution of connected bilayers, one expects d = y 614 with y a number larger than 1 (the lattice model leads to y = 1.5). The peak position for the sponge phase indeed shows a 114 behavior with y = 1.4-1.6 depending on the system.z5~31~35 Recently, some very interesting direct images of the structure of the sponge phase have been obtained using freeze-fracture electron microscopy measurement~.~~ The structure indeed looks spongelike, and the images are strikingly similar to those obtained for bicontinuous micro emulsion^?^ with the difference that instead of two types of solvents (oil and water) there is, in the L3 phase, only one solvent present.
5. Scaling Laws for Dilution of a Symmetric Sponge The symmetry arguments given in section 3 , and developed further in sections 7 and 8 below, should apply generally for systems containing continuous bilayers: they make no direct appeal to the ideas of elasticity as laid out earlier in section 2. However, if we assume a symmetric sponge structure for the L! phase, the elasticity theory proves extremely powerful in allowing the formulation of scaling laws. These describe how the thermodynamics of the phase should vary with dilution and temperature. Such laws were first discussed by Huse and Leibler for cubic phases30 and later developed much more fully for the sponge phase by Porte and colleague^.^^^^^ The scaling laws stem from the observation that the elastic Hamiltonian (1) is invariant under the mathematical operation of dilation. For if, starting from a bilayer or membrane in a certain state, all its spatial coordinates are magnified or "dilated" by a factor A, there is generated a new membrane whose spatial extent and thickness are both increased by this factor. The membrane's surface area is obviously increased by a factor A2, and if it occupies a three-dimensional volume V (with a uniform density a t large scales) as occurs in either a sponge phase or a lyotropic smectic A phase, the volume has been increased by a factor A3. Hence the volume fraction 4 is the same as before, (33) Anderson, D. M.; Wennerstrom, H. J . Phys. Chem. 1990,94, 8643. (34) Balinov, B.; Olsson, U.;Sderman, 0.J . Phys. Chem. 1991, 95, 5931. (35) Skoury, M.; Marignan, J.; May, R. Colloid Polym. Sci., in press. (36) Strey, R.; Jahn, W.; Porte, G.; Bassereau, P. Lungmuir 1990, 6, 1635. (37) Jahn, W.; Strey, R. J . Phys. Chem. 1988, 92, 2294. (38) Porte, G.; Skouri, M.; Appell, J.; Marignan, J.; Delsanti, M.: Billard, I.; Debauvais, F. J . Phys. (Fr.), in press.
Roux et al.
4178 The Journal of Physical Chemistry, Vol. 96, No. 11, 1992 90
xh
/
I
80
70
Figure 5. Expansion of a typical configuration of the sponge phase. The expansion corresponds to a factor X in the three directions.
although the surface area per unit volume is decreased by a factor X (Figure 5 ) . The crucial observation now26is that the elastic energy of the membrane, as determined by eq 1, is completely unaltered by this transformation (as can be checked by a simple dimensional analysis). The probability of any microstate of the first system is the same as that in the second; hence so is the stable equilibrium state. This provides us with a powerful law of corresponding states which says that the thermodynamic state of two membrane systems that have the same elastic constants and volume fraction but different film thicknesses is identical. Notice that the law does not apply if the surface has many separate components, since each of these carries a translational entropy that depends on the molecular size of a solvent and is not invariant under the scaling. However, it appears experimentally (see below) that the law holds very well in both the symmetric sponge phase and the lyotropic smectic A phase. Since our two systems have different thicknesses of bilayer (but must have the same elastic constants K and n), the law as it stands cannot be applied directly to experiment in a simple way: for although bilayer thickness may be varied in some systems (for example, by adding a second solvent to swell the layers), this presumably leads to a shift in the elastic constants at the same time. Much more useful would be a law that related states of the same bilayer at different dilutions. Fortunately, this can be obtained from the previous argument, if we further assume that two systems of bilayer, with the same elastic constants and same surface area, but different thicknesses have the same thermodynamics. In this case, the diluted system is no different from one that has been diluted by a factor A, except that the volume fraction 4 in the first case is reduced to $/A in the second. Taking this argument at face value, we can draw two “conclusions”: (a) Since the energy of each microstate is unchanged, the free energy of the system is unchanged, so that the free energy density F has been decreased by a factor X3. In terms of volume fraction, this shows that F 0: d3. (b) No phase transition, either from smectic to sponge or from one sponge state to another, can be brought about by a change in concentration alone. The equilibrium state of the system at a low concentration is that produced simply by dilating a more concentrated state, which cannot change its symmetry. In view of the second conclusion, which contradicts the experimental existence of lyotropic phases (Le., phases whose state of order is controlled by concentration) we should not be surprised to find that the scaling argument, in its simplest form, is not quite correct. The reason is that, in equating the statistics of two bilayer systems with different thicknesses, we have ignored an important entropy contribution from thermal undulations of the bilayer at short distances. The spectrum of undulations is cut off at a higher wave vector for a thinner bilayer, and the entropy difference that this involves in fact depends on the state of curvature of the film.39940 This effect can be partially understood in terms of a “renormalization” of the elastic constants of the film, which then depend on the ratio [/a, where 4 0: 6/4 is the characteristic length scale of the sponge and 6 is the bilayer thickness. Hence the effective elastic constants depend on the volume fraction 4 of (39) Helfrich, W. J . Phys. (Fr.) 1985, 46, 1263. (40) Peliti, L.; Leibler, S. Phys. Rev. Lett. 1986, 54, 1690.
60 k
0.1 0 2
05
1
2
5
C,,E, (wt
10 20
50 100
Yo)
Figure 6. Semilog representation of the CI,E5/water phase diagram (from ref 32). Note the quasilinear phase boundary between the sponge phase and the dilute phase L, (except in the very dilute regime).
s u r f a ~ t a n t . ~ ~The $ ~ renormalizations ~*~~ of K and si can be calculated p e r t ~ r b a t i v e l y and , ~ ~ lead ~ ~ to the following structure for the free energy of the sponge phase where A and B depend only on dimensionless combinations of the unrenormalized elastic constants and kBT. Exactly the same form, but with different coefficients, is expected for the smectic phase since the scaling argument applies to it equally well. In both cases, if undulations are neglected, the coefficient of the logarithm is zero. This modified scaling behavior now allows phase transitions to be driven by concentration (due to the different A and B coefficients in the different phases), and moreover for a binary system predicts a distinctive dependence of the phase diagram on temperature. Specifically, the boundaries between different phases (which must encompass curves along which the free energies are equal, and should correspond closely to such lines if miscibility gaps are narrow) should appear linear when plotted on a T, log 4 representation. This applies when the concentrations are not too small so that the perturbation theory is applicable; in this regime the basic trend is well confirmed experimentally as shown in Figure 6. In fact, the scaling law ( 5 ) can also be confirmed in several more direct such as measurement of the osmotic compressibility as discussed in section 9. 6. Lattice Model In section 3 we used a lattice model to illustrate the unusual symmetry of the sponge phase. Such a model has also been used to estimate the free energy of the phase;24743>44 we outline the approach here since it helps motivate the more powerful (and abstract) Landau-Ginzburg theory presented in section 7. Assuming a structure such as that shown in Figure 4, we can argue that the free energy has two contributions: the entropy of the membrane (arising from its random spatial arrangement) and the elastic free energy. Since the 1/0 symmetry of the Hamiltonian may break spontaneously, the ratio of volume fractions of inside and outside reactions (I/O ratio) need not be unity, in general. A realistic model should allow the system to choose for itself a ratio between I and 0. This can be taken into account in the lattice model by introducing a parameter $ for the volume (41) Golubovic, L.; Lubensky, T. Europhys. Lett. 1989, 10, 513. (42) ROUX,D.; Cates, M. E.; Olsson, U.; Ball, R. G.; Nallet, F.; Bellocq, A. M. Europhys. Lett. 1990, 1 1 , 229. (43) Roux, D.; Cates, M. E. In Dynamics and Patterns in Complex Fluids; Onuki, A., Kawasaki, K., Eds.; Springer Proceedings in Physics Vol. 52; Springer: Berlin, 1990; p 19. (44) Cates, M. E.; Roux, D. J . Phys. Condens. Matter 1990, 2, SA339.
The Journal of Physical Chemistry, Vol. 96, No. 11, 1992 4179
Feature Article fraction of (say) I, in which case (1 - $) is the volume fraction of 0. The free energy can be written as a function of variables $ and 4 (the surfactant volume fraction) and then it can be minimized with respect to $. For simplicity we assume a random mixing approximation in which the spins are randomly up (probability $) or down (1 - $) a t each site independently. The free energy contribution from the mean curvature rigidity K can be estimated by examining only four-spin configurations (at the level of random mixing24*29*41). This leads to an elastic energy density
where 5 is the cell size, which can be directly related to the surfactant concentration 4, simply by counting the amount of interface in the system: (7) Here X is a geometrical parameter defining the amount of curvature per corner for the lattice model. If we assume that curvature energy of an isolated lattice “cube” should, realistically, reduce to that of a sphere, then X = 8lr. In eq 6, ~ ( 5 )is the bending constant, which is taken to have the scale-dependent form
as derived from perturbation theory. There is also a contribution from the Gaussian rigidity ii. This can be estimated by studying eight-spin contribution^.^^,^^ The simplest reasonable approximation seems to be the following:43
The Gaussian curvature is then negative for $ = 1/2, indicating, as expected, that in the symmetrical sponge phase the number of handles is larger than the number of disconnected components. The entropic contribution to F is dominated by the entropy of mixing of the I and 0 domains and can be easily estimated in the random mixing approximation for two types of object (I and 0, or upspins and down-spins) on a lattice:
The variational free energy density F($,4) is obtained by summing eqs 6, 9, and 10. It may now be minimized over $ to obtain the free energy F(4)as a function of surfactant volume fraction, from which a phase diagram can be calculated in the usual way. This features a symmetric sponge phase ($ = 1/2) at moderate and high volume fractions but an asymmetric one at low volume fractions. It is straightforward to confirm that for the symmetric state the resulting free energy has precisely the scaling form in eq 5 . However, this is no longer true when the symmetry is broken; ultimately, at low volume fractions $ becomes very small and the model predicts a phase of small vesicles. This is consistent with the expectation that translational entropy is the dominant factor at low enough concentrations, although in practice a simple micellar phase could arise instead. To complete the picture we require a comparison with the smectic (lamellar) phase. The lamellar free energy is easily estimated, either separately or within a version of the lattice model,4i and in any case has the scaling form of eq 5 . Figure 7 shows some typical phase diagrams with the lamellar phase included. An important feature of the phase diagrams is the presence of a phase transition between a symmetric sponge and an asymmetric sponge or vesicle phase. This phase transition can be either first order (7a) or second order (7b).24,41The transition can be provoked by tuning the Gaussian rigidity iz, as well as by (45) Kleinert, H. J . Chem. Phys. 1987, 86, 3565.
0
+
Figure 7. Typical phase diagrams calculated with the free energy given by eqs 6-9 and 10. L is a lamellar phase, S the symmetric sponge phase, and A an asymmetric sponge or (when very dilute) a phase of small vesicles. The phase transition between the symmetric sponge and the asymmetrical sponge can be either first order (left) or second order (dashed line, right).
raising temperature, reducing K , or decreasing the concentrati0n.4~ We should note that the elastic part of the free energy generates an effective interaction between domains (or spins) which can be attractive. When the free energy F ( 4 ) is constructed, this can lead to regions of liquid-liquid coexistence between phases of different surfactant concentration. In other words, the elasticity of the surfactant bilayer can mediate an effective attraction between surfactant molecules. This is complicated by the presence (and spontaneous breaking) of inside/outside symmetry, which alone would be equivalent to an Ising-like model. It is clear that lattice models of this type, although useful in clarifying basic concepts of symmetry, and establishing trends on variation of the elastic parameters, are necessarily based on a sequence of ad hoc approximations in both the proposal of a trial structure and the subsequent estimate and minimization of the free energy. For this reason it is useful to have a more general phenomenological description on which more reliable (though less specific) predictions concerning the phase behavior can be based. The most obvious candidate for a more general picture is a Landau-Ginzburg approach, which is amenable to analysis at various levels of sophistication. In what follows, we give a detailed account of a Landau-Ginzburg mode142-44*46 that is suitable for describing binary (or pseudobinary) systems of bilayer-forming surfactants, close to the symmetric/asymmetric phase transition. This model allows us to calculate not only the different basic types of possible phase diagrams43*”but also gives expressions for correlation function^^^,^^ that are quite distinctive in character and can be checked experimentally.
7. Landau Expansion and Phase Diagrams A main feature of the sponge phase, which can be deduced either from the lattice model Cjust described) or more generally from the symmetry arguments of section 3, is that the appropriate Landau expansion of the free energy depends on not one but two order parameters. One corresponds to the volume fraction of surfactant, and the other corresponds to the special Ising-like symmetry arising from the fact that the bilayer divides space. As usual we choose, as the concentration order parameter, the difference between the real volume fraction of membrane and a certain reference value: P=4-4* (1 1) We shall see below that 4* corresponds to the volume fraction at a special point of very high symmetry in the space of Landau coefficients, in fact a double critical end point (DCE).50 (46) Coulon, C.; Roux, D.; B e l l q , A. M. Phys. Rev.Lett. 1991.66, 1709. Coulon, C.; Roux, D.; Cates, M. E. Phys. Reu. Lett. 1991, 67, 3194. (47) Blume, M.; Emery, V. J.; Griffiths, R. B. Phys. Rev. A 1971,4, 1071. (48) Experimentally one is also normally working at fixed pressure rather than fixed volume, but these are equivalent so long as one considers the overall system to be incompressible, which is a very good approximation. (49) Kincaid, J. M.; Cohen, E. G.D. Phys. Rep. 1975, 22, 57. (50) This point could, more logically, be referred to as a tetracritical point but unfortunately that name is used to mean something different. However this point has been refered as either bicritical end point4gor double critical end point by: Yokoi, C. S. 0.; Coutinho-Fillho, M. D.; Salinas, S. R.Phys. Rev.B 1981, 24, 4047, which seems to us more appropriate.
Roux et al.
4180 The Journal of Physical Chemistry, Vol. 96, No. 11, 1992
For our second order parameter, which describes the breaking of the 1 / 0 symmetry, we choose to define where [I] and [O] denote the volume fractions of the regions labeled (arbitrarily) as “inside” and “outside”. An equally good definition (perhaps more appealing, especially if one wishes to include the possibility of defects in the surfactant film) could have been made in terms of the mean curuature ( 1/ R , + 1/ R 2 )of the infinite component of the bilayer: this has a global average that vanishes in the symmetric state but is nonzero in an asymmetric one. With either definition, the order parameter is arbitrarily up to a sign, and so can only enter the Landau expansion in even powers. This statement derives directly from the underlying 1/0 symmetry of the Hamiltonian for the system. 7.1. The Landau Expansion. Our starting point is the following Landau expansion of the thermodynamic potential, @ = f - pp, wheref is the (Helmoltz) free energy and p the chemical potential a l A @ = -pp -p2 7 4 -12 -l 114 + l (13) 2 4 2 4
+
+
+
+
value of 7 for which f is minimized at given p. The common tangent construction may now be applied to the minimized function Ap). Analytically, if there is a two-phase coexistence (meaning that the second derivative is equal to zero somewhere), the common tangent construction corresponds to demanding the equality of both the chemical potential and the osmotic pressure:
($), ( =
with p , and p2 referring to the respective concentrations of p in phase 1 and phase 2. On minimizing f ( p , o ) with respect to q, we simply obtain q,(p)
y?
This potential corresponds to an expansion around the point where both q and p are zero, which as we shall see is a DCE. Because in the theory both of the expansion coefficients a and A can change sign, we have included fourth-order terms in p and q to ensure thermodynamic stability; the quartic coefficients have been chosen positive and arbitrarily set to 1/4 without losing generality (since the two order parameters can be rescaled to make this the case). Although q can only enter in even powers, the p order parameter is a concentration and both even and odd powers of it are allowed. However, the third order term in p has been omitted because it can always be eliminated by redefining correctly the zero of p, and the remaining Landau coefficients p, etc4’ An important choice in defining our Landau expansion concerns the coupling between p and q . We have here included only the lowest order coupling term allowed by symmetry (p$) with a coefficient that we have set to 1/2 without loss of generality. (Any negative sign of this coefficient can be absorbed into a sign change for p; setting the remaining value to 1/2 corresponds to a choice of units for the thermodynamic potential itself). We do not include any explicit term in q6,but in fact its inclusion would make almost no difference, because such a term is already generated from within the model (as can be confirmed by minimizing eq 13 over p ) . The same applies to various higher order couplings between q and p that could be included, but at the cost of greatly increased mathematical complexity. As written in eq 13, this free energy is very similar to a Landau expansion for the Blume-Emery-Griffiths (BEG) model, which was originally invented to describe the superfluid/normal fluid phase transition in a mixture of 3He/4He.47It is helpful in the present context to derive the possible topologies of phase diagram directly from eq 13, although many of the same results can be found in the literature relating to the BEG model itself. From the thermodynamic potential (13) one can proceed analytically to calculate phase diagrams in two ways. Either @Cp,q) is minimized at a given chemical potential p, and the phase diagram is drawn in (a,A,p) space, or, by using the common tangent construction, the phase diagram can be calculated in (a,A,p) space (for which the linear term pp is no longer relevant). The second method turns out to be slightly simpler and also corresponds more closely to the experimental situation, where one has control over the surfactant concentration rather than chemical potential. It is, of course, always possible to go from one representation to the other, using a Legendre transform.48 We may therefore consider instead the following free energy expression: a l A l l -p4 + -q2 -t -q4 + -pq2 f(p,?) = -p2 (14) 2 4 2 4 2 from which the phase diagram is found by the following method: f(p,q) is first minimized with respect to variation in q, thus defining a function of one variablef(p) = f(p,qm(p)), where q&) is the
$)2
qm(P)
with
p*
=0
= (P* - P)”’
p
(16a)
1 p*
if P
(16b)
P*
= -A. This leads to the minimized free energy f(p) = (a/2)p2
f(p)
if
= (a/2)p2
+ p4/4
+ p4/4 - (p*
for p L p*
- ~ ) ~ / 4 for p
< p*
(17a) (17b)
The nonanalytic behavior at p* corresponds to a second-order transition between a symmetric state ( p 1 p * , qm(p) = 0) and a state where the symmetry of the phase has been spontaneously broken (p < p*, q&) # 0). As we shall see below, this continuous transition is physically realized for some parameter values, although for others the point p* lies inside a two-phase region and remains inaccessible. We will not go into details concerning the rest of the calculation, which corresponds to solving eqs 15 in the region where spinodal lines exist. The calculations are mainly analytical but require at some points a straightforward numerical treatment. In fact, a good qualitative feeling for the phase diagrams can be obtained just in plotting the functionflp) (eqs 17) and looking for common tangents. We leave this to the reader, describing below the results obtained from the full calculation. 7.2. Classification of Phase Diagrams. In total, five types of phase diagram are obtained in the (a,p) plane, depending only on the location of p* with respect to the spinodal curves (at which the curvature of the free energy changes sign). The phase diagrams are drawn in Figure 8; it should be obvious that we can pass smoothly from one to another by varying p * . We can also represent the same phase diagrams in the (a,p plane) and this is done in Figure 9. In fact the complete evolution of the twodimensional phase diagrams can be represented by a figure inscribed in the (a,p*,p) or (a,p*,p) volume. A sketch of such a representation in (a,p*,p) is shown in Figure 10 and discussed later on. We first describe in turn the five types of phase diagram shown in Figure 8. When p* is positive, we get a phase diagram of type a where for large values of a (a > a*, = - p* 2, a line of second order critical points (critical line, or CL) separates the symmetric phase ( p > p * ) from the asymmetric one ( p < p*). This line ends on a critical end point (CEP) at the meeting between a first-order line and the CL. This first-order transition line ends on an ordinary critical point (CP) at p = 0, a = which corresponds to a liquid/liquid type critical point between two asymmetric sponge phases. For a < a * , , the first order line separates a symmetric phase from an asymmetric one. When p* is exactly zero we obtain a phase diagram of type b, containing a double critical end point (DCE) at p = 0 and a, = At this point the shape of the coexisting line goes as a ( p P * ) ~on one side and a ( p - p * ) l I 2 on the other side. There is no longer a separate ordinary critical point. When p* is smaller than zero but larger than -1/4, a phase diagram of type c is found. Here the symmetric phase is separated from the asymmetric phase by a second order line for a > a*2 = ‘I2- 3p* and by a first order one for a < a**. The point at
"'IA
The Journal of Physical Chemistry, Vol. 96, No. 11, 1992 4181
Feature Article
0.61
A
1
Sponge
0.3
0.1
-0.6 -0.4-0.2 0
0.2
0.3
01
0.2
0,
p
-0.4-0.2 0
p
Sponge
C 0.1
0
t
-0.21
a
-o.2 C _ _E P Y
-0.1 I
1
I
-0.4-0.2 0
1
0.2 0.47
b
c
d
e
17
-0.3
0 0.2 0.4 0.6
-0.6 -0.4-0.2
Figure 8. Phase diagrams obtained with the Landau free energy given by eq 13. These diagrams are plotted in a mixed field/concentration representation and correspond to different values of the parameter A (a, b, c, d, e correspond respectively to A = -0.2, 0,0.2, 0.35,0.5). The symmetric sponge to asymmetric sponge transition can be either first order (solid lines) or second order (dashed lines). T is a tricritical point, Tr is a triple point, CEP is a critical end point, DCE is the double critical end point, and C a liquid-gas-type critical point.
~W-IIIW=IHYUIIII*
.......... l " "
I
line of critical points line of tricritical points line of triple points line of critical endpoints
Figure 10. Summary of the three dimensional representation of the Landau phase diagram (a,p,A space). A is a double critical end point,
B is a critical end point, and C is a tricritical end point.
,A''\+
0.2
0.08 0.12
0.3
0.1
0.2
0
p
"T
-0.12-o.oao.04
o
8
@ -0.1 O[
\
c,
-0.3-0.4 -
-0.1 -0.21
1
I
-0.08
1
I
a
k
\ , , -0.04 0 0.04 p
Figure 9. Typical phase diagram obtained in a field/field representation, a, b, c, d, and e correspond to the same values of A as in Figure 8.
which the first-order line becomes second order is a tricritical point (TCP) of the symmetric type. When p* is smaller than -1/4 but larger than -1/2, in addition to the previous structure, a first-order line between two symmetric phases appears. This ends in a critical point at a = 0, p = 0 (type d). Finally, when p* is smaller than -1/2 there is no longer a tricritical point, and we get a phase diagram of type e where the second-order line ends on a CEP. This completes the mean field (Landau) calculation of our model. We observe that even a very simple free energy leads to a very rich structure in the three-dimensional parameter space (a,&), as shown in Figure 10. Predictions include a DCE, a line of tricritical point (TCP), a line of triple points (TP), a tricritical end point (TCEP). Coming back to the starting free energy (eq 13), we can now identify the point around which the free energy has been expanded as the double critical end p0int.4~At this point, minimization of eq 13 over p gives a Landau expansion for in even powers of r) (the 1/0 order parameter) in which the first
nonvanishing coefficient is that of 98.50 This high order critical point in fact has an "upper critical space dimension" of 8/3.44 This is the dimension above which critical point fluctuations can be correctly treated using mean-field theory. Therefore, a mean-field treatment as presented above is expected to describe exactly the critical properties very close to the DCE in three-dimensional systems. Similarly, mean-field theory is subject only to weak logarithmic corrections along the line of tricritical points. For this reason one should not be surprised to find experimentally a very narrow Ginzburg interval, and a wide region displaying mean-field like behavior, in sponge systems that happen to be near the DCE or tricritical line in (a,A,p) space. In fact, all the results presented above do not differ from the classical mean-field treatment of the BEG Indeed, a Landau expension of the original BEG model leads to exactly the same free energy (eq 13). The Occurrence of a line of tricritical points in the three-dimensional space of (a&) is specially notable. It means that, even in a binary system of surfactant + solvent, there is a finite probability of finding a tricritical point on the ($,T) phase diagram without any special adjustment of other thermodynamic variables (see Figures 8 and 9). All the phase diagrams so far obtained with microscopic models ~-~l belong to one or other of the types obtained a b ~ v e . ~ More generally, since (after the rescalings described in connection with eq 13) our Landau expansion has only two independent coefficients, and since the elastic description of bilayers also has two independent elastic constants, it should be possible in principle to span all the possible phase diagrams simply by varying K and R. Of course, the precise relation between these elastic parameters and those of the Landau expansion cannot be calculated without an exact statistical treatment of multiply-connected bilayers which obviously we lack at present. In practice, sponge phases often contain several chemical species, which allows K and R to be varied by tuning concentrations of cosurfactant, salt, etc.. These systems may therefore provide an ideal experimental testing ground for theories of higher order critical behavior (see section 9). 8. Correlation Functions We now study the correlation functions that one can expect to measure by small-angle light scattering in sponge phases. Our treatment is based on Landau Ginzburg theory, in which suitable gradient terms are added to the series expansion (13) of the thermodynamic potential. Experimentally, the observable fluc-
4182 The Journal of Physical Chemistry, Vol. 96, No. 11, 1992
tuations are those of the surfactant density, since there is no probe able to couple to the 1/0order parameter, q, directly. However, as we shall see, there are some very interesting indirect effects of the q fluctuations on the density-density correlation functions for surfactant .jl We assume throughout this section that a Gaussian treatment of the fluctuations is adequate. If (as argued in the previous section) the Ginzburg interval is liable to be fairly narrow, this should be reasonably accurate as one approaches the critical 1/0 line in a binary or pseudobinary phase diagram, so long as one does not come extremely close to it. (For completeness, however, we briefly discuss critical fluctuation effects in section 10 below.) The approximation should be even better near any tricritical or higher order critical point. Adding to eq 13 the most general form for gradient terms at harmonic order, the thermodynamic potential reads as follows @ = -pp
+
a -p2 2
+
l -12 2
l
1
+ -4 q4 + Y P t 2
+7 7P
PPI2
- + + T IV1I2
Roux et al. where the terms in q come from the correlations in the locally biased mean density (induced by long-range correlations in the Ising-like order parameter, q ) and the remaining term coma from local fluctuations about this mean. The term in square brackets is usually referred to as the energy-energy correlation function. Since we assume Gaussian fluctuations for q, we may use the standard results for averages of products of Gaussian random variables to obtain ( v 2 ( r )V 2 ( O ) ) = ( 1 2 ) 2+ 2(.?(r) from which we can at once calculate the density-density correlator for surfactant as
whose Fourier transform is
7s
I(q) = ycq(?q-ap)
2
(18)
where we now allow the order parameters q and p to be fluctuating functions of spatial position. The p4 term has been dropped since we are working within a Gaussian approximation. A nonzero coefficient of q4 is retained, but we shall assume below that the effective quartic coefficient for q (after the p field has been integrated out) is again small. 8.1. A Special Case. Before giving a full treatment, we present results for a special choice of parameters that can be solved exactly This illustrates some of the most within the Gaussian interesting features which can be confirmed by the more general treatment that follows later. We assume for simplicity that (7) is zero which corresponds to a symmetric sponge phase (although it is a straightforward exercise to perform a similar calculation for the asymmetric state). The soluble case corresponds to setting y p = yc = 0 in eq 13, giving a 1 (19a) @ = -PP + TP2 + ;Pa2 + Ho(11)
A’ X 7 4 -112 -74 pql2 2 4 2 where A’ = A p / a , and X = ( 1 - ( a / 2 ) ) . For small enough X (Gaussian approximation) the fluctuations of q are exactly calculable and take the usual Ornstein-Zernicke form e-rltn (70%) = (21) 4 v 7 O(q)
=
+
+
+
where $:, = y,,/A‘. As mentioned above, this behavior cannot be measured directly in scattering experiments, since there is no contrast between I and 0 regions which contain identical solvent. However, the large length scale for q fluctuations has an interesting effect on the surfactant density-density correlation function g(r) = (6P(O) 6 P W ) (22) where 6 p ( x ) = p ( x ) - ( p ) . This correlation function is calculated by noting from eq 19 that the probability distribution for p ( x ) is Gaussian with mean p / a - q ( ~ ) ~ /and 2 a variance l / a . We obtain at once42
arctan (q€n/2) qE7/2
where the constants C1and C2 obey 167ray:
€7 c, = cz = -
1 6=a2y,Z’
€7
The function Z(q) (the Fourier transform of the density correlator) is precisely what one measures in static light scattering experiments. We expect our analysis to apply in the regime of small-angle scattering, corresponding to large spatial distances over which the correlations are dominated by fluctuations in the underlying Ising order parameter q . Obviously the formula 25 should not apply for high q, corresponding to the characteristic pore scale ,$ of the sponge or even shorter distances. The validity of this simple treatment depends on the assumption that y p and y7 in eq 18 are both negligible; this is unlikely to be true close to a tricritical point where the intrinsic fluctuations of the surfactant density become large. (This is treated in the full calculation that follows below.) However, it should apply near the critical line and away from the tricritical region, so long as one remains in the regime of small (Gaussian) fluctuations. The behavior predicted in eq 25 is quite characteristic, involving a dependence Z(q) l / q + constant (at large qt,,), rather than the usual (Ornstein-Zernicke) result Z(q) l / q 2 constant, which would be expected if the order parameter, q , responsible for long-range correlations, was directly observable. (The latter form is, however, recovered in an asymmetric sponge phase as can readily be checked by a calculation that exactly parallels the one just presented for the symmetric case.) In fact, the experimental observation of l / q scattering (see section 9) preceded the theoretical development of the model; the key step of recognizing this 1 / q scattering as the signature of the energy-energy correlator was made by Ball.42 8.2. Perturbation Expansion. We now present a different calculation of the density correlation function, which is based on perturbation theory, and allows us to relax our assumption that the gradient terms other than ( V V )in~ eq 18 have coefficient zero. The resulting formula is therefore more general: in particular, it seems to be necessary to account for the data in systems close to a tricritical point (see section 9 below). We first describe the symmetric case. The estimate of the scattering function Z(q) by means of a perturbation technique in given in Appendix A. We obtain
-
In this limit, there is no gradient coupling between the surfactant density p ( x ) at different spatial positions x. Hence the probability distribution for p ( x ) depends only on the local value of q ( x ) . This means that an exact integral over the p order parameter can be performed. We obtain thereby a potential function
1
c1 c2 +
-
+
(26) By the same perturbative approach one may compute the scattering function for an asymmetric sponge phase. Details are also given in Appendix A. We find
The Journal of Physical Chemistry, Vol. 96, No. 11, 1992 4183
Feature Article
pentanol
These formulas have been used to fit light scattering data near a continuous symmetric to asymmetric sponge transition, where, as detailed in the next section, the simplified version given in eq 25 does not give an adequate fit. Since they were derived by a perturbation theory, the results given above must be of limited validity. Firstly, their validity is restricted to outside the critical domain (within which, non-Gaussian fluctuations dominate the critical behavior). This means that the formulas cannot be used too close to the second-order line. However, since the upper critical dimension is three for the tricritical point, they are expected to be basically correct whenever tricritical fluctuations are dominant. This also applies, of course, to the neighborhood of the double critical end point. Second, the results in principle require a convergence of the perturbation expansion, which implies a small coupling between v2 and p. [We have earlier chosen units so that this coupling constant is of order unity, in which case its smallness must be compared with the other Landau expansion coefficients.] In practice, however, we have some confidence in the present results, particularly that for the symmetric sponge, since it agrees with other estimates of the density correlation function obtained nonpertubatively for special parameter values. Indeed, for i$ = 0 ( y p = 0) and yc= 0 the result reduces correctly to that presented above in eq 25.42 A second special case, yc/2 = can also be solved exactly in direct space within the Gaussian approximation, and gives
€2,
w/s = 1.so
dodecane L3
90 SDS
I
1.5
2.
2.5 % Pentanol
Figure 11. (a, top) Phase diagram of a oil-rich sponge phase (L3).Note that the sponge phase, when diluted with oil, ends with a first-order transition. (b, bottom) Phase diagram of a water-rich sponge. The sponge phase is separated from an asymmetric phase by a maximum turbidity line (MTL) corresponding to a second-order symmetric/asymmetric transition. I s-‘
which also agrees with eq 26. This suggests that the domain of validity of the formula eq 26 may, for practical purposes, be rather large. This does depend on exprcssion A4b being used rather than (A4a), even though both are in principle equivalent to the order of accuracy employed. [Actually this is quite a common situation in perturbation theory, applying for example to the usual Omstein-Zernicke calculation, where the form I(q) 1/(1 t2q2)is frequently obeyed much more accurately than I(q) (1 - t2q2), although these are formally equivalent to the derived order in 4.1 N
-+
9. Experimental Results on Light Scattering The theoretical derivation of the correlation functions presented above was directly motivated by the need to explain experimental data.42-44,46Since we are interested in large wavelength fluctuations (larger than the typical pore size of the sponge) static light scattering is a very appropriate technique to study the correlation functions. As we will show, analysis of the light scattering data allows us to demonstrate the existence of two order parameters. This is despite the fact that no probe couples linearly to the inside/outside order parameter ( q ) , which therefore cannot be directly measured. However, as explained in the previous paragraph, light scattering (or X-ray/neutron scattering) does measure the surfactant density-density correlation function; and we predicted above that the presence of the special coupling term p q 2 leads to nontrivial forms for the correlation function, which in certain cases may strongly deviate from a “classical” (OmsteinZernicke) behavior. In fact two different experimental situations have been analyzed, both of which strongly support the description proposed above. In a first series of experiments an oil-rich sponge phase has been studied as a function of the membrane concentration (varying the cell size t). In this particular system (we will call it the “oil” dilution in what follows) the sponge phase undergoes a first-order transition to a phase of practically pure solvent (Figure 1
a.u.
7
2 1
0.
0.001
0.002
0.003 q
I
A”
Figure 12. Plot of the inverse scattering intensity against 9 for a sponge phase at different concentration. The dotted line is a fit using a normal Ornstein-Zernike formula, whereas the full lines are fits using q 25.
In a second series of experiments, a water-rich system (called “water” dilution in what follows) has been studied. This system exhibits a second-order phase transition between a symmetric sponge and an asymmetric state (Figure llb).46 Moreover, in the oil dilution the measurement of the osmotic compressibility (I(q=O)) follows precisely a functional form that derives from the scaling law for the free energy (eq 5 ) on the assumption that the bilayers can be described with an elastic HamiltoniannMSince this work, other systems have been studied and analyzed within this framework and more complete data related to the dynamics has also been p r e ~ e n t e d . ~ ~ For the case of oil Figure 12 shows typical observations for the inverse of the scattered intensity l / I ( q ) as function of the wave vector transfer q. The observed q dependence is in striking contrast to a classical Omstein-&mike form (dotted line in Figure 12) usually expected for density-density fluctuations at small wave vectors. In fact, an analysis using the correlation function derived from Landau-Ginzburg theory using two coupled order parameters (eq 25) leads to an excellent fit of the experi-
Roux et al.
4184 The Journal of Physical Chemistry, Vol. 96, No. 11, 1992
loo000 0
5
%$10
5
0
$76
10
B 0.0
' '
-2.0
I
-1.0
-1.5
-0.5
Log ($1
Figure 13. Plot of the inverse of the (reduced) scattering intensity at q = 0 for a sponge phase varying the volume fraction. The linear behavior is due to logarithmic correction of the compressibility (see eqs 5 and 29).
mental data (solid lines Figure 12). The fact that the simplest expression works so well indicates that the correlation length for the order parameter p is very small. This is consistent with the observation that we are relatively far from any type of second order phase transition. We can extract three interesting parameters from the experimental fits: the correlation length for the In/Out order parameter (&); an amplitude ratio of the arctangent to the constant term in eq 25; and the osmotic compressibility x = Z(q=O). The correlation length f , is several times larger than the cell size and the amplitude ratio is of the order of 2/3. This ratio is a measure of the relative weight of local surfactant density fluctuations (which are always present) as opposed to the induced fluctuations, brought about through coupling to the 1/0 order parameters, which are correlated over long length scales. These two results provide strong evidence that the long wavelength surfactant density fluctuations are indeed governed by the underlying In/Out order parameter 7. The measurement of the osmotic compressibility Z(0) as a function of the membrane volume fraction $ provides a direct check for the functional form of the free energy. Indeed, we found that Z(0) varies approximately as 1/$, but with systematic logarithmic corrections as shown in Figure 13. Using either scaling arguments or microscopic models, we have seen that the free energy is expected to vary as 4 3 ( A B log $) (eq 5 ) . The zero wave vector scattering function can then be easily calculated as
+
I(0) a (4 log (4/4*))-l
(29)
as observed experimentally ($* is a parameter depending on A and E ) , These data provide a direct evidence for the need to take into account logarithmic corrections to simple elasticity theory. Precisely such corrections are predicted to arise from thermal undulation effect^,^^,^ as described in section 5 above. The second system studied (water dilution) leads to some more general and complementary results. Indeed, instead of a first-order transition between symmetric and asymmetric state as previously described, this system shows a continuous transition (Figure 1lb),& whose signature is the existence of a maximum turbidity line (MTL) separating two isotropic liquid phases. On one side (higher concentration of membranes) the phase is a symmetric sponge and on the other side an asymmetric one. Light scattering data have been obtained in each phase approaching the transition line. To analyze the q dependence of the scattering intensity, the complete function Z(q) has been used (eq 26 or 27 depending on the nature of the phase, symmetric or asymmetric). Approaching the MTL both the 1 / 0 correlation length and the osmotic compressibility can be measured. Figure 14 shows the experimental behavior of both quantities as a function of the surfactant volume fraction for two different paths crossing the MTL a t different locations ( A or B in Figure 1lb). Divergences of both the correlation length and the osmotic compressibility at the MTL (located a t = +,) demonstrate clearly the existence of a secondorder transition. Moreover, two critical exponents can be measured. Here we only discuss the divergence of Z(0) which is less sensitive to the details of the fit than the correlation length. We
+
loo0 0.01
01
1
10
@-$c
Figure 14. (a, top) Light scattering results for dilution lines A and B (see Figure 1 Ib). I(0) is extrapolated intensity at q = 0; t7is the correlation length for the order parameter 7. Open and solid symbols stand for the S and A fits, respectively. (b, bottom) log-log plot of &I(O) versus @ 4, for the dilution line B. The straight line has a slope of 0.5.
have to take into account that the regular part of the osmotic compressibility, which varies as 1/4. Consequently, in order to get the critical exponent for the compressibility, we have plotted &Z(O) as a function of the reduced variable ($ - &) (Figure 14b). The critical exponent measured that way is of the order of 0.5. This value requires some more precise analysis,5s and further experiments are necessary to confirm these results. In what follows, we will compare this preliminary value to theoretical predictions. 10. Critical Exponents
The experimental existence of a line of continuous phase transitions in a phase diagram plotted as a function of only two degrees of freedom is strong evidence for the existence of a s p i a l symmetry in sponge phases as discussed in earlier sections. It is interesting in this context to ask whether the measured exponent (as mentioned above, our best estimate for the susceptibility exponent based on a fit where the regular l/$ dependence is corrected for, is 0.5) of the divergence of the susceptibility can be explained. We discuss here only the approach from the symmetric side, since this has been measured more accurately, although our arguments can easily be reworked for the asymmetric case. The Gaussian fluctuation picture, with which we have derived the correlation functions, should be reliable to calculate the critical exponents only when quartic terms are negligible. The resulting ( p - pc)-1/2,Z(0) ( p - pc)'/2 (where eq 26 exponents are E has been used). These can be relied upon if one undergoes the transition via a path through the DCE,and to within logarithmic corrections for a path through a tricritical point (Le., through the line of tricritical points shown in Figure 10). However, for any approach through a path on the (4, i") phase diagram which cuts the line of second-order transitions (Le., through the continuous surface of such transitions in Figure 10) the results are not asymptotically correct. Nonetheless, they may be useful in fitting the mean-field behavior that should arise close to the transition but outside the Ginzburg interval. This interval can be very narrow for a system that is nearly tricritical and the resulting mean-field region may be wide enough to measure the mean-field (or, more
-
-
The Journal of Physical Chemistry, Vol. 96, No. 11, 1992 4185
Feature Article accurately speaking, tricritical) exponents. In contrast, within the Ginzburg interval, the exponents will differ because of non-Gaussian critical fluctuations. The correlation length has the usual divergence, [ (p - pc)-' with v = 0.63 for the Ising model in three dimensions. The surfactant osmotic compressibility Z(O), as measured in light scattering, 7 ~ ) function,52 continues to be dominated by the ( ~ ~correlation although this can no longer be calculated within a Gaussian approximation. In fact, this correlation function is known to diverge as ( p - p J W , where a = 0.1 1 is the specific heat exponent for the Ising model. These results concern divergences that arise as one approaches the transition by varying chemical potential p at fmed temperature; the same exponents hold if these two parameters are interchanged. In practice, in experimental systems the control parameters are not (p, T ) but either (p, T), which applies in the case of a binary system, or two densities (as was the case with the data presented in the previous section). Since the relation between densities and chemical potentials is singular at the phase transition, this leads to a shift of exponents, usually called Fisher ren~rmalization.~' With Gaussian (Le., tricritical) exponents, we find p - pc (p - pc)1/2.Thus, when expressed in terms of the surfactant density, the mean-field critical behavior becomes
-
-
5
-
(P
- Pc)-l
I(0) ( P - fJc1-I (30) with the same exponents of unity expected if instead the transition is approached by varying temperature at fmed surfactant density. The renormalization effect was proved by Fisher using general thermodynamic arguments. The effect is a very large one near a tricritical point, and its origin is best illustrated by an exact calculation which can be made for our Landau-Ginzburg model with a special choice of the coefficients. This is presented in Appendix B. For systems within the Ginzburg interval, Fisher renormalization gives only a small effect,5' p - pc (p - pc)l/(*-u), so that we expect in this region:
5
N
I(0)
(p
-
- pc)-4(W
( p , - pc)-a/(l-a)
-
- pc)4.71 ( p - pc)4"2
(p
(31) A comparison of (29) and (30) leaves unexplained the observation of the value 0.5 for the compressibility exponent described in section 9. The exponent should be 1 in mean field theory (eq 29) or 0.12 in the critical region (eq 30) whereas the observation lies between the two. It is possible that this value reflects a crossover between Gaussian critical fluctuations as the Ginzburg interval is entered." It suggests that the tricritical line is nearby in the parameter space for this system, though more detailed experiments are needed to decide whether it actually cuts the experimental phase diagram shown in Figure 11, N
N
11. Conclusions
We have described how surfactant molecules in solution can self-assemble to give phases of fluctuating surfaces. In particular, sponge phases are phases of connected membranes randomly distributed in space. These membranes divide space into two volumes (named respectively inside and outside) and it can be argued from a theoretical viewpoint that a phase transition between a symmetric state (for which "inside" and "outside* domains are statistically equivalent) and an asymmetric one (in which inside and outside domains differ) should be observed. This phase (51) Fisher, M. E. Phys. Rev. 1968, 176, 257. (52) Ramaswamy, S.,private communication. See also: Pfeuty, P. Phys. Rev. Lefr. (Comments section), in press. (53) Note that a Gaussian factorization of the ( q 2 v 2 ) correlator gives an incorrect exponent of 27 - dv rather than the correct value of a. This factorization was used inappropriately in ref 44. (54) Note that the specific heat exponent itself is renormalized from a to -u/(l - a);the divergence in specific heat g a s away. However, the same
does not apply to the "energy energy" correlation function ($$), which diverges with renormalized exponent a / ( l - a ) as expected from the behavior of p - pc ( p - ps)l/(l-u). For an explanation of the anomalous behavior of the specific heat itself, the reader is referred to the original paper of Fisher.51
-
transition corresponds to a breaking of an exact Ising-like symmetry which is a special property of membrane systems in three dimensions. Consequently, a real system made up of membranes dispersed in a solvent should be described with two order parameters: the membrane concentration and the order parameter related to the inside/outside symmetry (for example the amount of Inside respecting the total volume). A microscopic lattice model based on a competition between the entropy of mixing of the membranes and the curvature energy leads naturally to the expected phase transition. More generally, a Landau expansion based on the two coupled order parameters allows us to describe the different phase diagrams expected and predicts the existence of high order critical behavior (tricritical line and double critical endpoints). Moreover, correlation functions can be calculated using a Landau-Ginzburg approach. The main result is the prediction of nontrivial correlation functions (differingstrongly from the usual Ornstein-Zernike form), which have now been observed quite unambiguously in experiments. A careful comparison between theory and experiments has been made and as expected both fmt-order and secondsrder transition can be observed in different systems. Tricritical fluctuations have also probably been observed even if the exact values of the observed critical exponents is yet an open problem. The osmotic compressibility of symmetric sponge phases has a characteristic dependence on surfactant volume fraction which can be explained by a scaling theory based on an elastic Hamiltonian for the constituent bilayers. The logarithmic corrections that arise from undulation theory are experimentally observable. The conjunction of both the theoretical and experimental a p proaches described here leads to a coherent description of systems of membranes demonstrating the richness of two-dimensional systems embedded in three dimensional space. The special symmetry present in these systems should allow various types of higher order critical behavior to be observed and studied in relatively simple (binary or pseudobinary) liquid systems. Acknowledgment. Parts of this work has been done in collaboration with many colleagues; we would like to specially thank D. Andelman, R. Ball, A. M. Bellocq, D. Gazeau, S.Milner, F. Nallet, U. Olsson, S.Safran, and R. Strey. We also have benefited from many discussions with D. Anderson, F. David, R. Granek, K. Kawazaki, T. Lubensky, S.Leibler, P. Pfeuty, G. Porte, J. Prost, S.Ramaswamy, C. Safinya, M. Schick, and H. Wennerstrom.
Appendix A In this appendix, we estimate the function g(r) by means of a perturbation technique. (In fact a nonperturbative method has recently been developed which gives very similar results.55 This also allows one to predict dynamic light scattering signals, which do not concern us in the present paper.) The first step is to separate the potential into a Gaussian part a0which is diagonal in Fourier space, and an interaction term Go which is not. In terms of bp(r) = p(r) - ( p ) , the Gaussian contribution reads in q space
where 6pk and
are the Fourier transforms of 6p(r) and q(r) and
ii = a
+ 3(p)2
2=A + (p)
The interaction term comes from fourth order terms and from the coupling between the two order parameters p and 7 . The latter contribution reads
Since we want to discuss specifically the influence of this coupling keeping only lowest order terms in the free energy, we (55) Granek, R.; Cates, M. E., to be published.
4186 The Journal of Physical Chemistry, Vol. 96, No. 11, 1992
ignore fourth-order terms in the following. Within the framework of a perturbation theory we may write
Roux et al. and the only term left after this procedure is the one Q = q. Finally, with the help of the usual formula for averages of products of Gaussian random variables ( 1 6 ~ ~=1 ~~ () ~I ~ P , I ~ ) ~ ~
Statistical averages over this probability distribution for the fluctuating coupled order parameters v and p can be computed exactly since they just require calculation of Gaussian integrals. Moreover, the term linear in 9,(being linear in 6 p ) gives no contribution to the correlation functions after averaging, and so keeping only the quadratic term we arrive at the effective probability distribution
The required density correlation function then reads in q space
one obtains (16P,12)
= (lSPql2)o + (l~P,lz)owq)
(A6a)
Y(q) = B2c~kq(lVk1z1Vk-,12)o
(A6b)
k
To be general in the calculation of Y(q),we assume q # 0 which implies
B2 Y(q) =
(1
+ YCi.ij)(2 + Ycq2)
d3k
( A + T,,k2)(A+ yq(Z-i)2)
j-J
This integral can be estimated by the following steps. (1) Introduction of spherical coordinates leads to L
(A4a) where the integration is made over all variables bp, and 7., The index 0 means an average using the unperturbed statistical weight e-pA*0. Since we work to quadratic order in the perturbation, we may alternatively write
(lb,I2) =
(I6p,l2)o
P2
+ 5((lb,l2@:)o -
(A4b)
( ~ ~ P ~ I ~ ) O ( @ C ~ ) O )
Although the two forms are in principle equivalent (to the level of accuracy at which we work) we argue below that the second one is more satisfactory as it agrees closely with nonperturbative results available for special values of the parameters (such as that presented in section 8.1). We thus keep the form eq A4b in the following. In the above formula, the first term on the right is simply the well-known Omstein-Zemike term obtained when the interaction term between and p is ignored. The correction represents the effect of 7 fluctuations on the density correlation function for surfactant. Inserting the expression A2 of the coupling term, we obtain
However, the only terms left after a Gaussian integration over the unperturbed distribution correspond to Q’ = -Q and k’ 5 -,k or k’ = k - Q. The first channel gives a_ erefactor (L + y$Q) while the second one leads to (1 + y,k.Q)(l - y c ( k - Q).Q). Adding the two contributions with equal weights allows us to write the coupling term in the following equivalent form: : 9
1 -c(1+ Y ~ i * & ) ( 2-k
(2) Integration over q now gives
r,QZ)laP,(21~k12hk-e12
k,Q
b2(2 + ycq2) 16a2
A+ T,(k + ‘4)2 1n(A+y7(k-q)2)B=
yc
k2 dk
, f G v
(A + y,(k2 + 42))
+
(3) By separating B into two parts ( A yqk2and yqq2)it can be shown that the first contribution just cancels with the last term of Y(q),and the remaining contribution is
(4) Introducing the variables
5, = ( 7 , / 4 ” 2 ,
x = 5$,
Y = &+7
we finally obtain
=
~rke16pe121Vk121Vk-Q12
(A5b)
k.e
Thus we obtain (l6Pg12@,2)0 - (16Ps12)o(@:)o
=
(E rk,e(ISp4121Spe121Vk121Vk-et2)okJ2 ( 16pq12)o$rk.p( 16pe121Vk121Vk-p12)o)
This average is now to be taken under the statistical weight of the Gaussian free energy a0.Since a0is separable over p and 7, the average can be taken independently over the two variables,
which can also be written
(;
Y(q) =
1
+ fq2)2A-2&-3 arctan ( q t q / 2 )
(A7)
and by substitution into (A6a) we finally obtain the following approximation for the scattering function I ( q ) :
Feature Article
The Journal of Physical Chemistry, Vol. 96, No. I I, 1992 4187 start from eq 19 in that section:
(A8) We now give the main steps of the calculation for the asymmetric case. The mean value of a has now a nonzero value and we should write 7 = ( a ) 67. Then, the coupling term can be decomposed into three terms:
+
a, = 1/26P(a)2 + 6P(11)61 + Y26P 6a2
649)
The first term renormalizes p and only the latter two terms, from now on labeled a,, and ac2,are relevant for the present calculation. Note that the third term is similar to the one considered on the symmetric side. The perturbation technique then requires the calculation of 92.This generates three contributions, namely
92 = @,I2
+ ac22+ 2ac,ac2
(A 10) The last term which gives cubic terms in 67 does not contribute after averaging and we are left with the sum of the contributions coming from (PCl2and @c22. The same arguments can be extended considering the last gradient term in (18) which also gives two contributions. Finally, the result is similar to (A6a) with Y(q) = Y,(q)+ Y2W (All) where Y2(q)is the symmetric function given by (A6b) and (A7) while Yl(q)is calculated by the same technique as Y2(q).One obtains
First note that a path that passes through the tricritical line at right angles may be taken by setting X = 1 - 1/2a= 0 and then varying either p (chemical potential) or A (a temperature like variable). The effective potential for is then (eq 20)
The critical point may be approached varying p, with critical value p, = -A/2. Obviously the usual Gaussian model exponents apply; I(0) [, ( p - pc)-’/2. Likewise, at fixed p, we can vary the “temperature” A , giving I(0) E, ( A - Ac)-1/2, where A, = 2pc. We now ask what happens if the density p is used as a control variable rather than chemical potential. For given ~ ( x )we , can solve (Bla) exactly to find the probability distribution for p ( x ) , which is Gaussian with mean 2p - ~ ( x and ) ~variance 2. [We have chosen X = 0 so that a = 1/2 in eq 19.1 The global average P = ( ~ ( obeys 4)
- -
- -
P
= 2p- (v2)
(B3)
where
where CI and C2 are constants and the asymptotic form at small + ( A / 2 ) has been taken. Hence,
p
p = 2p - CI
Finally, the asymmetric formula reads
+ C2(p + (A/2))II2
(B5)
We see a t once that, close to the transition at p = pc = -A/2, the surfactant density varies as P - Pc
(‘413) As expected the new asymmetric contribution cancels when the order parameter ( a ) vanishes. On the other hand, since it originates from the lowest order term in a,, it should be dominant not too close to the critical line on the asymmetric side. Therefore, a simplified asymmetric formula reads
-
(P - Pc)’/2
(B6)
This leads directly to the Fisher renormalization effect described in the main text. If we instead wish to vary temperature A at fixed density p, we must approach along a path p(A) such that p remains constant. We find from (B5) that the required path has (introducing further order unity constants as needed) p
= C, - C4(A’)1/2
(B7)
+
where A ’ = A 2p is the distance from the transition. Rearranging this, we obtain A - A, = A’ + 2C4(A’)1/2 (A14) with K’= K 1 ( q ) 2 . This simplified formula has been satisfactory used to fit experimental results in the asymmetric phase not to close the S / A phase t r a n ~ i t i o n . ~ ~
Appendix B For the special case of the parameters considered in section 8.1, the Fisher renormalization effect can be derived explicitly. We
(B8)
where Ac(p)is the critical value of A . Close to the transition, A - A, varies as so that the exponents governing the divergences in [ and I(0) are ihcreased by a factor 2 (just as they were for an approach at fixed A and varying p ) . In fact, as shown generally by F i ~ h e r ,the ~ ’ renormalization factor that arises when p not p is a control parameter is the same whether one varies T or p , and is given by 1/(1 - a). [Here positive cy is assumed.] Since a,the specific heat exponent, is for a Gaussian (Le., tricritical) point, our explicit results accord with, and illustrate, this general effect.54
