1300
Langmuir 1992,8, 1300-1306
Minimal Surfaces and Winsor I11 Microemulsions Stig Ljunggren' and Jan Christer Eriksson' Department of Physical Chemistry, Royal Institute of Technology, S-100 44 Stockholm, Sweden Received July 11,1991. I n Final Form: January 2, 1992 Employing a generalized Young-Laplace equation for a surfactant-loaded oiVwater interface of mean curvature H and Gaussian curvature K and invoking the conventional Helfrich bending free energy expression, we analyze what interfacial configurations are compatible with mechanical and physicochemical equilibrium in Winsor type microemulsions. Our treatment is based on the notion that the Laplace pressure must be zero for an equilibrium aggregate when the microemulsion phase coexists with an excess is different from zero, spherical and rod-shaped equibulk phase. When the spontaneous curvature, Ho, librium droplets are possible. On the other hand, for HO= 0, we find that extended minimal surfaces with H = 0 can exist if the saddle splay constant K, is positive. In order to attain stability of minimal surface structures with respect to the size variations, it is necessary to include a quadratic term in the Gaussian curvature in the free energy expression. The stability of these structures with respect to shape deformations is ale0 studied. The theoretical scheme outlined applies equally well to minimal surface structures formed by thin symmetric bilayers. Introduction In recent years much attention has been devoted to infinite periodic minimal surfaces as possible structures of cubic liquid crystalline phases and, in disordered forms, of bicontinuous microemulsions and the so-called L3 It appears that behind such models lurks the assumption that minimal surfaces would possess a (constrained) minimum of the overall bending free energy, although, in the original application to soap films bounded by frames, it is, of course, stretching (rather than bending) free energy that is minimized. Yet, such an assumption does not seem unreasonable but no clear-cut proof has been presented so far, except for the case of symmetric, nonextensional bilayers.6 The purpose of this paper is primarily to provide a thermodynamic rationale for applying the geometrical concept of minimal surfaces to weakly interacting, surfactant-loaded oil/water interfaces with very low (but nonzero) curvature-dependent interfacial tension in order to model middle phase (Winsor 111) microemulsions. However, the general argument applies to symmetric bilayers as well. Expressions for t h e Laplace Pressure Let us consider a Winsor type of microemulsion (I, 11, or 111) which coexists with either one or two excess oil/ water bulk phases. At the outset we assume that any one of these microemulsion phases can be regarded, in essence, as an assembly of small (but regular) oil/water two-phase systems with surfactant-loaded interfaces of variable shape having very low, curvature-dependent interfacial tension. Neglecting, for the time being, the influence of fluctuations, it is evident that thermodynamic equilibrium requires that the free energy of such an oil/water system should be a t a minimum. However, since this minimization has to be performed a t constant chemical potentials, pi, the appro(1) Scriven, L. E. Nature (London) 1976,266, 123. (2) Hyde, S. T.; Andereson, S.; Larseon, K. Z. Kristallogr. 1986,174, 237. (3) Lareeon, K. J.Phys. Chem. 1989,93, 7304. (4) Charvolin, J.; Sadoc, J. F. J. Phys. (Paria) 1987,48,1559. (5) Hyde, S. T. J. Phys. Chem. 1989,93,1458. (6) Anderson, D.; WennerstrBm, H. J.Phys. Chem. 1989, 93, 4243. (7) Porte, G.; AppeU, P.; Bwereau, P.; Marignan, J. J. Phys. (Paris) 1989,50, 1335. (8) H u e , D. A,; Leibler, S.J. Phys. (Paris) 1988, 49, 605.
priate free energy to be minimized is not just the Helmholtz free energy, F, but rather the grand %potential, defined as Q(T,V,Pi) = F - Cinipi, i.e. as a Legendre transformation of F. As we shall discuss further below, one finds that minimizing the &potential leads to a minimum condition in the form of an expression for the pressure drop A p across the interface. Setting Ap = 0 then ensures that the dispersed droplets/aggregates are also in physicochemical equilibrium with the corresponding excess bulk phase. The basic reason for this is that two solutions with the same composition can have the same chemical potentials only when they are subject to the same pressure. Alternatively, one could, of course, minimize the Helmholtz free energy of the entire system (including the coexisting bulk phases). However, using the simple Ap = 0 condition as a physicochemical equilibrium condition is much more convenient in this context and leads to the same result. Moreover, it should be mentioned that there is also an additional equilibrium condition to be imposed for a microemulsion bulk phase which will be discussed in the section Aggregation Equilibrium to follow. This condition statesthat the overall Q-potentialof such a phase must be equal to the standard value, -pV. From the surface-thermodynamic developments made earlier by Buff and M e l r o ~ eBoruvka ,~ and Neumann,lo and, more recently, Markin et al.," the results indicate that the Laplace pressure due to an interface of mean curvature H and Gaussian curvature K is given by the expression Ap
2Hy - C,(&
-K) -2CaK-
l/2VS2C, - KVs**(V,C,) (1) where Vs2 is the Laplace-Beltrami operator in the surface and Vs* = bijriVj (where biJ = -nt-rj is the second fundamental tensor of the surface) is a special operator introduced by Weatherburn (cf. ref 29). The coefficients C1 and C2 are defined as follows (at constant temperature, T, and chemical potentials, pi) (9) Melrose, J. C. Ind. Eng. Chem. 1968, 60, 53. (10) Boruvka, L.; Neumann, A. W. J. Chem. Phys. 1977,66,5464. (11) Markin, V. S.; Koalov, M. M.; Leikin, S. L. J . Chem. Soc.,Faraday Trans. 2 1988,84, 1149.
0743-746319212408-1300$03.00/0 0 1992 American Chemical Society
Minimal Surfaces and Winsor 111 Microemulsions
Langmuir, Vol. 8, No. 5, 1992 1301
c, = ( W d l l ) K c, = ( W d K ) ,
(2) where y is the interfacial tension or, to be more precise, the grand %potential (Kramer's function) of the interface per unit area. (In other words, we do not attribute any mechanical significance to y.) Note that eq 1 holds for an arbitrary dividing surface, although y and the coefficients C1 and C2will have different values, depending, of course, on the particular choice of the dividing surface condition. In this context, we normally assume that the dividing surface is located at the hydrocarbon/water contact. For a uniformly curved interface, C1 and CZdo not vary with the surface coordinates and, hence, the last two terms in eq 1 drop out. The corresponding expression for the Laplace pressure Ap 2Hy - C1(21P- K) - 2 C 8 K (3) was contained in the well-known paper by Melrosegfrom 1968 as discussed by Markin et al.ll Now, in analogy with Helfrich's expression for the bending free energy,', for a surfactant-loaded oil/water interface with low interfacial tension, as a first approximation, we write = yo + 2k,(H - H0), + K$
(5)
c, = fi,
(6) and the following expression for the Laplace pressure emerges from eqs 1 and 2 Ap = myo- 4k,(H - Ho)(H2+ HHo - K ) - 2k,V:H
(7)
that should constitute rather a satisfactory approximation, at least for curvatures of similar order as Ho.Accordingly, Ap is independent of the Gaussian curvature (or saddle splay) constant, K,. It is worth observing that the experimentally accessible case of a planar interface with H = K = 0 and y = 7. that is given by the expression
(8) 7-= 7 0 + 2k$: serves as a reference here. In other words, when assuming eq 4 to hold we are actually considering changes of the interfacial tension, relative to an experimentally measured value of y., which are caused by curvature variations. Using eq 8, we can obtain an alternative form of the expression for the Laplace pressure Ap = W y ,
- 4kC[H(H2- K ) + Hdc] - 2k,V:H
Ap = -4k$&
(9)
As will be further discussed below, Ou-Yang and Helfrichlghave recently made a detailed theory of the overall energy of a vesicle membrane which resulted in essentially the same expression for Ap as the above eq 7. Their socalled shape equation (that is a mechanical equilibrium
y
= yo + 2k@
+ fi$
(11)
and the relevant expression for the Laplace pressure becomes Ap = W [ y o- 2k,(lP - K)]
(12)
An expression for the Laplace pressure like eq 7,9, or 12 is nothing else than a minimum condition for the grand thermodynamic potential, Q = $$sy dS - PiVi - PeVe, of the two-phase system in question when the interfacial tension obeys the usual Helfrich equation. Here, subscript i denotes the internal phase and subscript e the external phase. We repeat that the reason why this thermodynamic potential is on stage rather than just the Helmholtz free energy is that we are dealing with an open system in the thermodynamic sense where physicochemical equilibrium prevails. Aggregation Equilibrium The second general condition of paramount significance, which must be satisfied in order for an interface to exist within the realm of a macroscopic bulk phase at equilibrium, is that the overall grand &potential of such a phase must necessarily be equal to the standard value for a bulk phase, -p V . Otherwise expressed, some additional mechanisms must operate which counterbalance s2 + PiVi + PeVe (that for a regular interfacial system equals $3, S denoting the interfacial area) for each one of the small two-phase systems. For Winsor I and I1 droplet type of microemulsionswithpi = p athe surface free energy, 47rR27, expended to form a droplet of radius R is counterbalanced by negative, entropy-dominated free energies due to dispersing the droplet in the surrounding oil or water medium and to shape fluctuations.14-16 For infinite structures with noninteracting internal interfaces that may form when Ap = 0 and HO= 0, we shall have to demand, however, that the overall excess surface free energy $Jsy d S becomes equal to zero since there are no dispersion or fluctuation free energies to be separately invoked in this caw. Interestingly, minimal surfaces offer novel possibilitiesto realize this condition for the important (14) Overbeek,J. T.G.;Verhoeckx,G.J.;DeBruyn,P. L.;Lekkerkerker,
H.N.W. J . Colloid Interface Sci. 1987,129, 422.
(15) Borcovic, M.; Eicke, H.-F.;Ricka, J. J. Colloidlnterjace Sci. 1989,
131, 366.
(12) Helfrich, W. 2.Naturforsch. 1973, C28, 693. (13) Ou-Yang, 2.; Helfrich, W. Phys. Reu. A 1989, 39, 5280.
(10)
Thus, when physicochemical equilibrium imposes a constraint of zero pressure difference between the two subphases and k , # 0, it is a necessary requirement that HO = 0 if minimal surface structures are to exist, since K is generally different from zero (except at the flat points, of course, which is of no relevance in this connection). For interfaces which deviate only to a minor extent from minimal surfaces, the mean curvature,H, is always a small quantity, irrespective of the length scale of a representative patch, whereas the Gaussian curvature, K, may vary considerably. For this class of interfaces we can write, assuming HOto be equal to zero
(4)
where yo = y ( H = Ho, K = 0) and HOis the spontaneous curvature and where k , and K, are two constants relating to variations of the mean and Gaussian curvatures, respectively. Here we anticipate that eq 4 refers to curvature variations occurring at constant T and pi (i.e. to an interface that is actually open in the thermodynamic sense) and that the same dividing surface condition is employed as in eq 1. Hence we get Cl = 4k,(H - Ho)
condition) also includes the term, 2k,Vs2H, which is obviously different from zero when the mean curvature varies along the interface. For interfaces, which are actually minimal surfaces, the mean curvature, H, is zero everywhere by definition, and eq 7 reduces to
(16) Eriksson, J. C.;Ljunggren, S. h o g . Colloid Polym. Sci. 1990,82,
41.
1302 Langmuir, Vol. 8, No. 5, 1992
Ljunggren and Eriksson
case when yo is somewhat larger than zero. Note that, according to eq 8, yo is equal to the interfacial tension ym of a planar, surfactant-loaded oil-water interface at the phase inversion where HO= 0. The two conditions discussed above are actually the counterparts of (i) the phase and (ii) chemical equilibrium conditions introduced by Hill in his treatment of aggregation of surfactants to form ordinary micelles." Winsor Microemulsions Let us now again consider Winsor I (o/w + excess oil phase), Winsor I1 (w/o + excess water phase), and Winsor I11 (middle phase + excess oil and water phases) microemulsions and implement the condition Ap = 0 as a physicochemical equilibrium condition. Implicitly, we then assume that we can neglect the interactions among the interfaces, i.e. that bulk properties pertain in the central parts of the dispersed phase and that (Gibbsian) surface thermodynamics constitutes a valid basis when accounting for the interfacial properties. In the Winsor I and I1cases we have well-foundedreasons to assume HO # 0 and spherical geometry ( K = 112) at equilibrium, and from eq 9 we then derive that (13) Hence, in these regimes, spherical microemulsion droplets of certain equilibrium sizes, Reqs = l/Hqs, are predicted, about which, however, fairly large size and shape fluctuations occur (cf.ref 16). Accordingto the stability analysis made by Ou-Yang and Helfrich13for the spherical case, we only need to spell out the rather self-evident condition y 1:0 in order to secure that eq 13 represents a fully stable equilibrium corresponding to minimization of the 52potential of a small droplet system. Similarly, in the case of cylindrical geometry, from eq 9 we get
He: = h(ym/2kc)1'z where further analysis shows that 7 0 must be equal to zero (and, hence, ym= 2k&02, resulting in Hqc = Ho)in order to allow the formation of infinitely long cylindrical aggregates. However, elongated aggregates of restricted length with a cylindrical or a string-of-bead-like contour form when yo is somewhat larger than zero by may Of a size-fluctuation mechanism to that yielding rod-shaped surfactant micelles.'* In line with a vast amount of experimental evidence, we can generally associate the three-phase Winsor II1 regime with zero spontaneous curvature, iae*Ho = According to eq 7 we then find that the condition Ap = 0 can be written
It is anticipated that minimal surface structures based on single surfactant-loaded (monolayer) oil/water interfaces would generally accommodate about equal volumes of the oil and water phases in separate labyrinth network channels. By now it is, in fact, rather firmly established that bicontinuous microemulsions (whether single-phase or middle-phase) usually fulfill such a volume condit i ~ n . 'Moreover, ~ ~ ~ ~ it is known experimentally that ymof the flat oil/water interface mostly attains some ultralow minimum value (1O61O6 N m-l) in the center of the middle phase region where Ho = 0, and, hence, ym= yo (cf. ref 21). However, ymis always larger than zero. We note in passing that in this range with extremely low ym-values,the flat, macroscopic interface becomes very rough as a result of large thermal fluctuations,22 and consequently, we have to assume that thermal fluctuations modulate the structures formed by curved, internal interfaces in a corresponding fashion. The above means that for each unit cell (subscript u) of the minimal surface structure there will always be a positive "stretching" free energy amounting to YmS,, S, denoting the corresponding interfacial area. Still, the excess of overall interfacial free energy associated with such a unit cell which is characterized by H = 0, Ho = 0, and K I O , as obtained from eq 11, i.e.
may well become zero, as is definitely required for the formation of an extended internal interface within a macroscopic bulk phase. Noting that ym> 0, it is seen that K, must be positive in order to obtain Qua = 0 since it follows from the Gauss-Bonnet theorem that JJ& d S = 27rxuEwhere the Euler characteristic per unit cell xuE is less than zero. Hence, we obtain the followingadditional condition for the genesis of a minimal surface structure formed by noninteracting, surfactant-loaded oil/water interfaces 52;
+
= ymsu 27rR,XuE = 0
Accordingly, when h, > 0,this condition implies that the "stretching" free energy ymSUis counterbalanced by the free energy, gainedI byo.forming a doubly "bending" curved interface with 27rhcXuE, = and
It is evident, however, that the condition expressed by eq 17 does not correspond to a stable equilibrium. This is 80 because, according to this equation, nus does not have a minimum with respect to S, variations at the point where the two terms balance. To attain this necessary feature we shall have to assume that some (repulsive) mechanisms enter into the picture in such a way that the average value of might be written
H [ y 0- 2k,(H2 - K ) ]- k,VszH = 0 showing that minimal surfaces, which by definition are characterized by H = 0, K I0, are compatible with physicochemical equilibrium, irrespective of the values of yo and k,. In other words, a periodic minimal surface structure is justified thermodynamically from the point of view of the phase equilibrium condition as a model of the auerage structure of a Winsor I11 microemulsion. Equally possible in this respect is, of course, a lamellar structure with H = K = 0. (17) Hill, T.L. Thermodynamics of Small Systems; Benjamin: New York, 1963, 1964. (18) Eriksson, J. C.;Ljunggren, S. J. Colloid Interface Sci. 1991,145, 224.
where C, stands for a positive constant. As a matter of fact, such an expression results for a minimal surface upon extending the Helfrich expression, eq 11, to include a quadratic term in K and noting that ( K 2 ) / ( K )is2 a fixed number somewhat larger than 1. Apart from additional repulsive monolayer effects of pure bending at large interaction forces may become important in this context. Hence, in order for (y)S, to become zero and simultanteously to be at a minimum, we find that the condition
w,
(19) Billman, J. F.; Kaler, E. W. Langmuir 1990,6,611. (20) Kahlweit, M.;Strey, R.; Busse, C. J. Phys. Chem. 1990,94,3881. (21) Aveyard, R.;Binks, B. P.; Fletcher, P. D. I . Langmuir 1989, 5, 1210. (22) Langevin, D.Acc. Chem. Res. 1988,21, 255.
Minimal Surfaces and Winsor 111 Microemulsions
has to be fulfilled, where ymbecomes determined by y m = Equation 19 differs from eq 17 above by a factor 2. Inserting typical numerical values, h, = 10-2l J, xuE= -4 (valid for a Schwarz P-type minimal surface structure), and ym= 0.5 X J m-2, yields S, = 2.51 X 105 A2, which does seem to be of the right order of magnitude. When h, < 0 the stretchinglbending free energy compensation as described above cannot occur and the main alternative left would be that yo = 7. is reduced to zero by attractive surface/surface interactions resulting in the formation of a lamellar phase. Above we have assumed that the interfacial tension y (i.e. the local interfacial excess free energy density) varies with the curvilinear coordinates on the surface between positive and negative values. At the flat points where K = 0, y equals y... This raises the secondary question as to how the mechanical equilibrium is realized in the lateral directions along an interface of variable curvature, a problem that has been treated rather recently by Neogi et al.23 and by Kozlov and Markin,%who, however, arrived at differing results. Moreover, their equilibrium conditions are at variance with statements made in the classical paper by Boruvka and Neumann.lo A major stumbling-block in this connection is, of course, that the Gibbs notion of a surface of tension, where the interfacial tension is supposed to act mechanically, cannot be retained for an interface with nonuniform curvature across which the pressure drop is zero. A full analysis of this difficult issue would, however, be beyond the scope of the present paper. The stability of the minimal surface interfacial system with respect to shape deformations is investigated in the Appendix. There we conclude that a special interest has to be attached to the state of the external surface of the whole lattice of unit cells. In order to ensure that the lattice is stable with respect to shape deformations, it is necessary that the outermost parts of the lattice next to the external surface have a special structure such that certain mathematical boundary conditions are satisfied. The ultimate purpose of this special interfacial structure is to serve as a replacement for the missing framework that is known to be indispensable in order to span up minimal surfaces formed by ordinary soap films. Symmetric Bilayers Forming Minimal Surface Structures
Even when the spontaneous curvature, Ho,of an interfacial monolayer is somewhat different from zero, there is a chance that minimal surfaces may form with symmetric bilayers rather than monolayers as structureforming elements. For symmetric bilayers it is natural to employ the middle surface as dividing surface. We assume the ordinary Helfrich expression, eq 4, to hold for each of the constituent monolayers at constant T and pi. It is intuitively clear that Hobof the bilayer may be zero even if HOof the monolayers is somewhat different from zero. However, the stretching free energy, yob, will tend to increase with the absolute value of Ho. As a matter of fact, upon modeling the bilayer as a twin interface composed of two monolayers, each obeying eq 4, one easily verifies that an expression of the same form as (23) Neogi, P.; Muyngseo, Kim; Friberg, S. E. J.Phys. Chem.1987,91, 605.
(24) Kozlov, M. M.; Markin, V. S. J. Chem. SOC.,Faraday Trans. 2
1989,85, 211.
Langmuir, Vol. 8, No. 5, 1992 1303
eq 11 holds true for the resulting bilayer tension. Moreover, by means of introducing parallel surfaces located at the distance 5 on each side of the middle surface which is characterized by the curvatures H and K, and computing the mean curvatures of the constituent monolayers to the first order in 5 using well-known expressions from differential geometry, we derive the following relations :y
= 27,
+ 4k$:
(20)
= 27,
k,b = 2kc
(21)
K,b = 2(Rc+ 4kPO5)
(22)
where superscript b denotes a bilayer property and 25 stands for the distance between the hydrocarbon/water contact planes. The sign convention adopted is such that HO> 0 for the water-in-oil situation and HO< 0 for the oil-in-water situation. These relations are in line with what Porte et al.’ and Helfrich and R e n n ~ c h u hhave ~~ derived earlier. The corresponding expression for the Laplace pressure becomes
W[Y:- 2k,b(H2- K)]- 2k,bV:H
=0
(23)
which is essentially the same as in the Winsor I11 case. Thus, symmetric bilayers can also form extended minimal surface structures with H = 0, even when HO# 0 for each of the monolayers that constitute the bilayer. In analogy with eq 19 this would require, however, that h,b > 0, which, in turn, according to eq 22, requires (K, + 4ka05) for the constituent monolayers to be positive. The La (“sponge”) phase may probably be rationalized on this basis, at least when the bilayers involved are comparatively thin (cf. refs 6 and 7). Note that the bilayers we have in mind here can be of two kinds, either with the head groups pointing toward a very thin, central water film and the hydrocarbon tails facing comparatively large oil phase regions or vice versa. In these two cases, by definition we have 5 < 0 and 5 > 0, respectively. Thus, when HO> 0, there is a positive contribution to K,b, equal to 4kJI05,for an oil-swollen surfactant bilayer of the usual kind, as well as for an inverse bilayer with a central water film when HO< 0. This yields indirect support to the rather natural idea expressed by Anderson and Wennerstrom6that a spontaneous curvature HOtoward the more abundant solvent correlates with the formation of the corresponding L3 phase. Incidentally, for symmetric bilayers forming a minimal surface structure with H = 0, it follows in a rather elementary way from the differential geometric expressions for H and K of a parallel surface that K is equal for the two hydrocarbon/water interfaces symmetrically located at equal distances from the middle surface. This fact was actually utilized by Charvolin and Sadoc4as a motivation why symmetric bilayers may take the form of minimal surfaces. However, our own motivation presented above which is based on the generalized Y oung-Laplace equation and the Helfrich expression, relies on a physical rather than a geometrical argument. Comments on the Related Theory of Vesicle Membranes Presented by Ou-Yang and Helfrich In the case of a geometrically closed vesicle membrane, Ou-Yang and Helfrich13recently obtained an expression for Ap by minimizing what they call the “shape energy”, Ik,JJs(H - HOP dS, subject to constraints of constant volume and surface area. Using our nomenclature, their (25) Helfrich, W.; Rennschuh, H. Colloq. Phys. 1990, 51, C7-189.
1304 Langmuir, Vol. 8, No. 5, 1992
expression is as follows
+
Ap = 2xH - 4k,(H - Ho)(H2 HHo- K ) - 2k,Vs2H (24) The term 2xH arises due to the imposed restriction of a constant surface area, A being a Lagrange multiplier. In spite of the formal similarities with the expression given by eq 7, the approach taken by Ou-Yang and Helfrich is different from ours, since, primarily, they treat the vesicle membrane as a system that is closed in the thermodynamic sense (i.e. it does not exchange surfactant molecules with the bulk solution) and that has a constant overall surface area. In our own approach, which emphasizes the long-term thermodynamic stability, the interfacial system is always considered to be open in the thermodynamic sense and there is no requirement, whatsoever, for the interfacial area to remain constant. Thus, the two curvature-related terms in eq 4 are only formally equivalent to the expression of Helfrich for the bending energy of .a membrane that does not exchange molecules with the environment, and it is reasonable to expect that the absolute values of k, and k, in eq 4 are much smaller than the corresponding values in Helfrich’s theory of bending elasticity proper, which, on the other hand, might be expected to be a reasonable model for more rapid shape deformations.26 In spite of these conceptual differences there is no objection to using the mathematical machinery of OuYang and Helfrich in order to minimize the SZ-potential of a fully equilibrated and geometrically closed (small) two-phase system defined by
where y is given by eq 4, with no restriction as to variations of the interfacial area but keeping the overall volume constant. We then obtain Ap = 2Hy0- 4k,(H - Ho)(H2+ HHO - K)- 2k,VlH (25)
Ljunggren and Erikaaon
One realizes that for regions where H and K vary significantly, a change of the dividing surface condition does not correspond to constant $. Thus, the generalized Young-Laplace equation (1)derived, in essence, by Boruvkaand NeumannlOby minimizing the Q-potentialis more general in the sense that it holds even for interfaces with nonuniform curvature.
Conclusion In conclusion, we have shown that a theoretical framework for treating microemulsions in a unified manner may be obtained by combining surface thermodynamics in a general form with a Helfrich type of expression for the free energy of bending an interface. It is valid in the weak interaction limit and it covers droplet as well as bicontinuous oillwater equilibrium structures. In particular, it can account for the existence of extended interfacial structures in the form of minimal surfaces characterized by zero mean and spontaneous curvatures (Winsor I11 microemulsions). Finally, we remark that in the present context, minimal surfaces serve to minimize the grand Q-potential, as is likewise the case for ordinary soap films. For infinite interfaces with k, > 0 this results in a certain equilibrium value of the interfacial area per unit cell such that the ‘stretching” free energy becomes counterbalanced by a negative overall saddle-splay free energy. The role of the saddle-splay constant, k,,has been subject to much discussion re~ently.~’In view of the above considerations the sign of k, appears to be of a crucial importance for the formation of bicontinuous microemulsions.
Appendix There are two additional problems to which this Appendix is devoted. The first problem is whether the Ap-expression given by eq 7 derived by Ou-Yang and Helfrich13to be a minimum condition for the overall bending free energy of a geometrically closed surface can really be applied to the extended minimal surface structures discussed in this article, as was tacitly assumed above. The second problem, likewise related to the boundary conditions, is whether the solution obtained by setting the first variation of the Q-potentialequal to zero actually represents a stable state. For this latter purpose, we have to evaluate the second variation of the &potential and show that it is positive with respect to any arbitrary deformation, $. This is referred to as shape stability. In their minimization of the free energy of bending OuYang and Helfrich derived the following expression for the first variation of the mean curvature (apart from a trivial difference in sign convention)
which is the same as eq 7 but which differs from the expression of eq 24 by the absence of the term with the Lagrange multiplier, A. Instead a novel, rather resembling term, 2Hy0,appears. However, yo stands for the interfacial tension of a cylindrical interface of the spontaneous curvature HOand it cannot be identified right off with the Lagrange multiplier, A. In the paper by Ou-Yang and Helfrich, the deformed surface is constructed as the locus of points displaced a distance $ in the direction of the outward surface normal. Thus, a parallel displacement would correspond to a constant value of $. However, more general deformations are also considered where $ is not constant as in a parallel deformation but depends on the curvilinear coordinates of the surface. It is, in fact, the derivatives of $ with respect to these coordinates in the expression for H of the deformed surface that give rise to the term containing Vs2Hin the expression for Ap. On the other hand, the original derivation by Melrose9 of a generalized Laplace expression is based on a fictitious parallel displacement of the dividing surface without changing the thermodynamic state of the interfacial system, and it does not yield the two last terms of eq 1. A closer analysis reveals that this is due to the restriction to parallel displacements of the dividing surfaces in Melrose’s approach, resulting in a A p expression, eq 3, that is valid only for surface patches with constant curvature.
which, when combined with eq 4, results in the following expression for the first variation of Q
(26) Szleifer,I.;Kramer, D.; Ben-Shaul, A.; Gelbart, W. M.; Safran, S. A. J. Chem. Phys. 1990, 92, 6800.
(27) Safran, S. A. In Modern Amphiphilic Physics; Ben-Shad, A,, Gelbart, W., Roux, D., Eds. In press.
6‘”H $(K- W2) - l/p$g%i$j Using the identity V,2$ = giivi$j
(26)
(27)
where Vs2 is the Laplace-Beltrami operator, this may be written 6‘l’H = $(K - W )- ’/2V:$
(28)
Minimal Surfaces and Winsor 111 Microemulsions
6%
= JJs$[Ap - 2y&
Langmuir, Vol. 8, No. 5, 1992 1305
+ 4kc(H- H,)X
(p+ HH,- K)] d S + 2kcJJS(H - Ho)V:$
d S (29)
where the term with Ap has ita origin in the terms -piVi - p,Ve in the expression for Q, since 6"'(-piVi
- p,V,)
= -@i
- p,)6"'Vi
= -Ap 6"'Vi
This, in turn, may be written where
and because pi and pe must be constant a t equilibrium and using the fact that and The last integral in eq 29 is actually the origin of the term containing Vs2H in the final expression for Ap. In their derivation of the expression for the first variation of the bending free energy Ou-Yang and Helfrich de facto used the following "partial integration" procedure
+
6"'QZ = ' / 2 r o ~ ~ s [ 2 W (Vs$)21 2 dS
(36)
Denoting, temporarily, x = Vs2$ and y = $, the integrand of 6(2)Q1 can be written as a quadratic form g = ax2
+ 2bxy + cy2
(37)
where
J c [ ( H - Ho)Vs$
- $VsHl.m
a = 1/2kc
ds (30)
although they expressed themselves in component formalism. Here, C is the curve of intersection between the outer boundary surface delimiting the interfacial system and the interface itself, m is a unit vector, tangent to the interface but at right angles to the curve C, and ds is an infinitesimal element of the curve C. In the case of a geometrically closed surface, however, the curve C will reduce to a single point and the integral over C vanishes, which is the case treated by Helfrich. Now, turning to a minimal surface structure composed Ho)Vs$ of a large number of unit cells, we note that [(H- $VsH] is the same on the common boundary surface of two adjacent cells, but m will have different signs. In a lattice of unit cells the second integral on the right-hand side of the above equation will, therefore, cancel on the internal surfaces between adjacent cells; Le. we can extend the integrals to comprise the whole of the periodic lattice and the integral over C should thus be referred to the curve of intersection of the minimal surface with the external surface of the whole lattice. In addition to the last terms in eq 30 there are a number of other terms, also in the form of integrals over C, which are related to the free energy of the dividing lines, as was shown by Boruvka and Neumann."J Consequently, we shall have to impose the condition that the external surface of the whole lattice is organized in such a way that the integrals over C vanish or that they are negligible since the length of the curve C is proportional to the external surface of the lattice while the total area of the interface is proportional to the volume of the lattice. The necessary condition for shape stability of the minimal surface is that the second variation of the Qpotential of the two-phase system in question, 8 W , is larger than zero. For the case where H = HO= 0, we have, using the formalism of Ou-Yang and Helfrich, and neglecting, as before, the free energy of the dividing lines
b = -k$
(38)
c = 2k$
Easy manipulation shows that g = '/&,(x 10 (39) i.e. the integrand g is positive semidefinite which means that it never attains negative values in any part of the x / y plane, and consequently, 6(2)Q1 > 0. NitscheZ8has shown that the integral 6 W 2 is always positive if $ vanishes on the boundary of the minimal surface. In his proof, Nitache introduces an arbitrary, nowhere vanishing function 4 that satisfies the differential equation
VE:
- 2K4 = 0
(40)
Using this equation and the identity
42[vs($/E)12= (VS$)' + ($'/E)V:E
- v~*[($~/[)V,[l
(41) he obtained the following expression for the integrand of 6W2
(Vs$I2 + 2K$' 42[vs($//f)12 + vs*[(rt2/Dvs41 (42) where the first term on the right-hand side is always positive. By use of the surface divergence the following expression for the integral of the second term in the integrand of 6(2)Q~emerges
JJsVs~[($2/€)~s€ldS = Jc($2/€)Vs4-m
ds -
2JJsW2/4)HVs&n dS (43)
Employing the identity
where the integration is to be performed over the whole lattice. The second integral on the right-hand side vanishes because Vs[ is always orthogonal to the surface normal, n. As far as the first integral on the right-hand side is concerned, Nitache assumes that $ vanishes on the boundary of the minimal surface, and the integral over C consequently vanishes. From the discussion in connection with eq 30 it is clear that this integral should also vanish or be negligible for a periodic minimal surface structure. We conclude that 6(2)Q2, and consequently also 6 W , is
(VS$)2 = g'i'$.$. 8 J (32) where Vs is the surface gradient operator, as well as eq 27, the expression for the second variation of s2 becomes
(28) Nitsche, J. C. C. Vorlesungen aber Minimalflhchen; Springer-Verlag: Berlin 1975, sections 103 and 104. (29) Weatherburn, C. E.DifferentialGeometry of Three Dimensions; Cambridge University Press: Cambridge, 1927.
6%
= JJs[$2(rJ
+ 2k$) 2k$$(giVi$j)
+ '/zy,#$#j -
+ l/zkc(g'Vi$j)zl
dS (31)
Ljunggren and Eriksson
1306 Langmuir, Vol. 8, No. 5,1992
positive if the integral over C vanishes on the external surface, which means that the structure is stable. It is easy to see that the vanishing of the integral over C is essential if 6(2)Q2 is to be positive. As an example, in the simple case where the external boundary surface cuts through the minimal surface structure and where $ is constant everywhere, i.e. also on the external surface, we find that 6(2)Q2 = 2$2$$& dS, which is always negative but, in this case, the negative value of 6(2)Q2 may be compensated by a larger positive value of 6(2)Q1provided that the bending constant k , is large enough. However, there are other $ functions with $ # 0 on the external surface which may cause global instability as can be inferred if the integrand of eq 33 is written as a quadratic form in $, Vs2$, and V&
+
h = ax2 2bxy + cy2 + dz2 (44) where x and y are defined as above and where z = VS$, and the coefficients are a = lJ2k,
b = -k$ c = 2kF2
(45)
+ y&
d = 1/2y0
A study of the principal minors, D1> 0, D2, D3 < 0, shows that this form is neither positive nor positive semidefinite
which means that there exist $-functions for which the integrand is predominantly negative, and consequently 6(2)Q < 0. However, Nitsche’s proof implies that 6(2)Q > 0 whenever the integral over C vanishes on the boundary of the lattice. It is, indeed, remarkable (thoughnot entirely surprising) that this is true irrespective of the size of the lattice. In addition, it should be remembered that Nitache’s stability condition (that the integral over C vanishes on the boundary) is a sufficient, rather than a necessary, one and that there may perhaps exist somewhat less stringent boundary conditions, particularly when k, is large. We shall refrain, however, from speculating about the possible nature of these conditions at the present time. As Nitache also pointed out, the energy minimum in the minimal surface configuration is rather shallow, which should lead to large thermal fluctuations, particularly when the interfacial tension yo has a very low value. However, as compared with a soap film with low tension, our case is more favorable due to the inclusion of the bending free energy. In addition, the higher order terms in the expression for the bending free energy that were necessary in order to ensure size stability are likely to provide additional stabilization of the shape of the minimal surface. A complete analysis of the stability properties involving such higher order terms would be extremely tedious, however, and will not be attempted in the present context.
