834
Langmuir 1990,6, 834-838
The Unbinding Transition and Lamellar Phase-Lamellar Phase Coexistence HAkan Wennerstromt Chemical Engineering and Materials Science Department, University of Minnesota, 151 Amundson Hall, 421 Washington Avenue S.E., Minneapolis, Minnesota 55455 Received January 17,1989.I n Final Form: October 23, 1989 The unbinding transition predicted by Lipowsky and Leibler (Lipowsky, R.; Leibler, S. Phys. Rev. Lett. 1986,56, 2541) involving a continous change from a bound state with two membranes in van der Walls contact to an unbound state with essentially independently distributed membranes is examined. A mean field treatment of the phase equilibria is presented, and on this level it is shown that the system behaves in a way analogous to a fluid showing a first-order liquid-gas transition below a critical pressure, with a coexistance between a bound and an unbound region. Furthermore, the fluctuations occurring at the unbinding transition are analyzed. It is concluded, by use of renormalization group theory, that the fluctuations leading to a continuous transition are localized but of large amplitude. For two bilayers, restricted not to form stacks, there is not sufficient cooperativity in the system to give a first-order transition. However, for stacked membranes, as for example in lyotropic liquid crystals, such large amplitude fluctuations cannot develop independently in adjacent layers, and it is concluded that the transition in such a case is first order, that the mean field calculations are qualitatively correct, and that there is no continous unbinding transition. These conclusions are found to be consistent with the experimental observations of lamellar phase-lamellar phase coexistence and also with the observation that a continuous unbinding transition remains to be reported for a liquid crystalline system. A phase diagram for the binary system lecithin/water is calculated, and it is pointed out that a current controversy concerning the limited or unlimited swelling in this system might be resolved by the presence of a coexistence between a bound and a highly swollen phase. Introduction In a series of recent papers, Leibler and Lipowsky'-* have argued, using renormalization group calculations, that in lamellar systems such as lyotropic lamellar liquid crystals there exists an "unbinding transition". One characteristic feature of this phase transition is that the system goes continuously from a bound state, i.e., one of a finite lamellar separation, to an unbound state of infinite swelling when an intensive parameter, for example, the Hamaker constant W, is varied a t constant osmotic pressure, If,equal to zero. The transition occurs in the presence of an attractive van der Waals force, Le., a t a nonzero value for W. The basic driving force for the transition to the unbound state is provided by the higher entropy associated with the lamellar bending modes in the absence of interlamellar contacts. This free energy contribution was calculated by Helfrich in 1978,' and Helfrich has indeed proposed the existence of an unbound state5' but without going into details concerning the nature of the transition from a bound to an unbound state. In several recent papers,'-'' we have argued, on the basis of mean field type calculations, that under certain
'
Permanent address: Div. Physical Chemistry 1, Chemical Center, P.O. Box 124,5-22100 Lund, Sweden. (1) Lipowsky, R.; Leibler, S. Phys. Rev. Lett. 1986,56, 2641. (2) Leibler, S.; Lipowsky, R. Phys. Rev. B 1987,35, 7004. (3) Lipowsky, R.; Leibler, S. in Physics of Amphiphilic Layers; Meunier, J., Langevin, D., Boccara, N., Eds.; Springer Proceedings in Physics 21, Springer: Berlin 1987; p 98. (4) Leibler, S.; Lipowsky, R. Phys. Rev. Lett. 1987,58, 1796. (5) Helfrich, W. Z. Naturjorsch. 1978,33A, 305. (6) Harbrich, W.; Helfrich, W. Chem. Phys. Lipids 1984, 36, 39. (7) Harbrich, W.; Helfrich, W. Chem. Scr. 1985, 25, 131. (8) Guldbrand, L.; Jonsson, B., Wennerstrom, H., Linse, P. J . Chem. Phys. 1984,80, 2221. (9) Khan, A.; Jonsson, B., Wennerstrom, H. J. Phys. Chem. 1985,89, 5180. (10) Jonsson, B.; Person, P. J . Colloid Interface Sci. 1987, 125,507.
0743-7463/90/2406-0834$02.50/0
circumstances, which include those considered by Leibler and Lipowsky, one should have a lamellar phase-lamellar phase coexistence, i.e., a first-order rather than a continuous transition. Moreover, such a coexistence has been observed experimentally for a number of system^.^*^^-'^ The conditions under which the unbinding transition is predicted to occur are quite analogous to the phase separation between coagulated colloidal particles and a dilute particle solution. The prototype for such a phase change is the liquid-gas transition. In that case, mean field theories provide a reasonable description except at the critical point. These considerations raise the question: what makes the unbinding transition different than a liquid-gas transition from a physical or mechanistic point of view? In the present paper, we try to penetrate this question accepting the formal results obtained in ref 1.
Unbinding versus the Liquid-Gas Transition. In a system consisting of finite colloidal particles in a solvent, there is a competition between the van der Waals attraction, which tends to aggregate the particles, and the entropy of mixing, which acts to disperse the particles. Depending on the relative size of the van der Waals term and the temperature, the system either swells continously on adding solvent or there is a region of coexistence between a concentrated and a dilute solution. This is quite analogous to the behavior of a one-component fluid above and below the critical point. In a lamellar system, there is a similar competition between energetic and entropic factors. The entropy of (11) Wennerstrom, H. in Physics of Amphiphilic Layers; Meunier, J., Langevin, D., Boccara, N., Eds.; Springer Proceedings in Physics 21, Springer: Berlin, 1987; p 171. (12) Fontell, K.; Ceglie, A.; Lindman, B.; Ninham, B. Acta Chem. Scand. 1986, A40, 247. (13) Vincent, J. M.; Skoulios, A. Acta Crystallogr. 1966,20,432,447. (14) Larsson, K.; Krog, N. Chem. Phys. Lipids 1973, 10, 177.
0 1990 American Chemical Society
Langmuir, Vol. 6, No. 4, 1990 835
Unbinding Transition and Lamellar Phase Coexistence mixing term present for particles of finite size is replaced by the entropy of a fluctuating membrane in the lamellar system. However, there is an important difference in that the entropic contribution to the chemical potential diverges logarithmically for finite particles, while for the membrane system it decreases asymptotically to a finite value. Thus the van der Waals attraction can in the latter case be large enough to prevent swelling, even when the system is exposed to a pure solvent. For finite colloidal particles, dissolution always occurs in principle at infinite dilution. The question is now what sequence of phases occurs in a lamellar system as the relative strength of the attractive interaction is decreased, either by an increase in temperature or by a decrease in the Hamaker constant. Two scenarios are possible: One alternative is that on increasing the temperature the system is first in the bound state in equilibrium with excess solvent. Then there is a first-order transition leading to the coexistence of a concentrated, bound state and a swollen, unbound state. On further increasing the temperature the difference between the two coexisting phases decreases and a critical point is reached. Above this critical point, the system swells continously. Below, we will argue that this is the sequence one should expect in systems forming lamellar liquid crystals. The second alternative is that the bound state swells with a continous phase transition into the unbound swollen state. In this case, there is never a coexistence between two lamellar states. This is the sequence found by Lipowsky and Leibler from explicit calculations on two interacting membranes. Mean Field Potential and Phase Equilibria. Consider as in ref 1a pair of surfaces (bilayers, membranes). Relative to a reference x y plane, each point on surface 1 and surface 2 is displaced by a distance L,(x,y) and L,(x,y). The separation between the surfaces is then
L(X,Y) = L,(X,Y) - L,(X,Y) (1) where surface 1 is closest to the reference plane. The quantity L is not in general uniquely defined, due to folding, but we will ignore this complication' presently. The two bilayers interact through van der Waals and other surface forces.I5 They have an internal bending rigidity, and there is an entropy associated with the bending modes. Lipowsky and Leibler' refer to a particular mean field treatment of the problem. In their description, the entropy of the fluctuating membrane is totally ignored, and evidently two laterally macroscopic surfaces will bind to one another as long as there is any attractive interaction. It is perfectly possible to formulate a mean field theory that also takes into account the entropy of the fluctuating membrane in some approximate way. Clearly this is essential if one wants to retain a possible analogy with the liquid-gas transition. Helfrich5 showed that asymptotically this entropy term conbined with the bending (free) energy leads to an effective potential
0
5
3
10
L (nlm)
c
5
3
c 10-
2
!
?
*
p
w 5-
4!
Y
I
I 10
I
5
3
J
L (nlm)
Figure 1. Effective potential V,, as in e 4 versus membrane separation L. Parameters: A = 0.2 J/m9, X, = 0.3 nm, W = J, k = 1.5 x 6x J! (a, top) T = 270; (b, bottom) T = 300 K.
ity. At short range, the membrane fluctuations are relatively unimportant, and for a zwitterionic phospholipid system one has an effective potential, VSR,of the form
provided that direct interactions are more short ranged. Here, kT is the thermal energy and K is the bending rigid-
v,, = VH(L)+ V J L ) + c (3) where V, is due to the repulsive hydration potential and V , is the van der Waals term. To avoid confusion, one should note that the potentials in eqs 2 and 3 have both entropic and energetic components. From this formal point of view, there is no qualitative difference between the effective potentials in eqs 2 and 3. The constant C in eq 3 is added to give the potentials in eqs 2 and 3 a common reference point. Having the potentials in the two regimes and assuming the existence of a V ( ( L ) ) , which can only be valid in a mean field sense, the intermediate range potential can be estimated with different degrees of sophistication. The simplest is clearly to assume additivit of eqs 2 and 3 for all L. As has been pointed O U ~ , ~ * ' ' this ~ ~ is not quantitatively accurate. However, it does give a mean field potential of a qualitatively correct nature, i.e., long-range repulsion and short-range attraction. The relevant points concerning the nature of the phase transition we address in this paper rely on these qualitative features and are independent of the more quantitative aspects of the effective potentials shown in Figure 1. They are obtained from the expression
(15) Israelachvili,J. Intermolecular and surface forces;Academic Presa: New York, 1985. (16) Rand, R. P. Ann. Reo. Biophys. Bioeng. 1981, I O , 277.
(17) Sornette, D.; Ostrowsky,N. Chem. Scr. 1985,25,108. (18) Evans, E.; Parsegian, V. A. Proc. Natl. Acad. Sci. U.S.A. 1986, 83, 7132.
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836 Langmuir, Vol. 6, No. 4, 1990
Wennerstrom
where 6 is the thickness of a bilayer, AH the decay length of the hydration force, and A H its amplitude as in ref 1. By taking the derivative of the potential, we obtain the force per unit area, or pressure, shown in Figure 2. The pressure versus distance curve is quite analogous to the pressure versus volume curve used in elementary treatments of the liquid-gas transition. It can readily be showng that the Maxwell equal area construction also applies to the force curves in Figure 2, and one would on the basis of this mean field treatment conclude that, for certain values of the parameters in eq 4, there is a firstorder transition between a bound state with ( L ) small and a swollen state of larger ( L ) at a particular value of the (osmotic) pressure of the solvent. In the renormalization group calculation of ref 1, the osmotic pressure was chosen as that of the pure solvent, Le., II = 0. It was found that for a given hydration force and a given bending rigidity a continuous increase in the average separation was found as the Hamaker constant W approached a critical nonzero value W,. The authors give a few examples of quantitative determinations of W,. Using eq 4 and the parameters given in ref 1, we calculate the critical Hamaker constant, where the mean field theory gives a first-order phase transition a t II = 0. The results given in Table I show that this W,, is of the same magnitude as W, calculated by using a numerical renormalization group approach. This quantitative similarity suggests that there is also a mechanistic similarity between the phase transitions calculated by use of the two approaches. One fundamental problem with the mean field description based on the effective potential V,, of eq 4 is that Veff strictly does not exist in the metastable and unstable regions in analogy with what is well established for these states in the van der Waals description of a liquidgas transition.” Nevertheless, for the latter case, the mean field description has proved to be of considerable values both for semiquantitative calculations and even more for the conceptual understanding of the phenomenon. There seems to be no good a priori reason that the same is not true for the unbinding transition. Indeed, such an approach has been adopted by Fischer2’ in discussing corresponding wetting problems. To us it is important to establish when and why the mean field description fails qualitatively for lamellar lyotropic liquid crystals. In several cases, we have observed typical “mean field” behavior, while the continuous unbinding transition so far remains unobserved. Obviously, in the mean field theory the effects of fluctuations are poorly treated. In the following subsection, we will try to establish the nature of the fluctuations that lead to the continuous character of the unbinding transition. Unbinding Fluctuations and the Diverging Order Parameter. The unbinding transition at II = 0 has one characteristic feature that makes it qualitatively different from the first-order or continuous transition occurring at II # 0. The order parameter L diverges at II = 0 but remains finite for II > 0. To see that this has non(19) Binder, K. Rep. Prog. Phys. 1987,50, 783. (20) Fisher, M. E. J. Chem. SOC.,Faraday Trans. 2 1986,82, 1569.
I 0
I 1
1.o 2.0
I
I
I
I
9.0
5.7
6.3 4.1
a At ll = 0 using eq 4 compared to WCRGfor the continuous unbinding transition parameters A , = 0.2 J/m2, AH = 0.3 nm, and
7’ = 298 K.
trivial consequences in a mean field description, we digress for a moment and introduce a complication into the model. Consider a system where the membrane contains two amphiphiles with, for example, two different alkyl chain lengths in the lipids. Consider one as the main component and assume that the other is an impurity. Being molecularly different, the two components will have a somewhat different preference for the bound and the unbound state. The prediction of the mean field theory would then be that for II = 0 there is a coexistencebetween the bound and the unbound state over a finite range of the parameter W. Just inside this two-phase region, the amount of membranes in the unbound state is very small, and yet ( L ) is infinite, averaged over the two phases. If we use ( L ) as an index to determine whether there is a pure first-order transition or a two-phase coexistence, the conclusion would be incorrect. ( L ) jumps discontinuously when entering the two-phase region when II = 0 while for n > 0 there is only a discontinuity in d ( L ) / d W . Clearly, it is necessary to be somewhat cautious when one interprets the physical significance of a certain vdue of ( L ) for a system with II = 0. Small sections of mem-
Langmuir, Vol. 6, No. 4, 1990 837
Unbinding Transition and Lamellar Phase Coexistence
L
J
-Bound
Regions
X
Figure 3. Schematic illustration of a cut in a balloon type fluctuation out of the bound state for two membranes: L,separation between membranes; X ,distance along membranes. branes at very large separations can clearly give large contributions to (L). In order to obtain a conceptual understanding of the continuous unbinding transition, it is essential to establish what type of fluctuations dominate in producing the gradual increase in ( L ) . In contrast to the continuous gas-liquid transition a t the critical point, where the density fluctuations have small amplitude and long wavelengths, we conjucture that in the unbinding transition the dominant contributions to (L) come from very large amplitude localized fluctuations in the intermembrane spacing. This conjecture is clearly supported by the observations on the effect of an impurity discussed above. The mean field analysis shows a first-order transition with a jump in ( L ) from a small value to an infinite one. If the continuous transition bears any resemblance to the mean field transition, one is expecting local fluctuations with the characteristic of one of the phases in a bulk of the other phase. Clearly, the membranes have a choice between a bound state of low "energy" and entropy and an unbound state of higher "energyn and higher "entropy" (the quotation marks indicate that the potentials in eq 4 are really a free energy rather than an energy), and intermediate distances provide a bad compromise. In the mean field treatment, this drives the transition to be first order, but the qualitative feature of two favored separated regions in configuration space remains valid even outside a mean field approximation. Consider a large-scale fluctuation in the bound state as illustrated in Figure 3. Due to the unfavorable conditions at distances slightly longer than the equilibrium separation, membranes tend to avoid these distances. Consequently, the most favorable shape at a given (large) area is that of a wrinkled balloon. Once in the unbound region, there is still a driving force for moving to larger values of L, but this is counteracted by the fact that the "balloon" is attached to the bound state. The free energy cost of increasing the area by dA is
-
dG [CL-2,, + AV(W)]dA (6) where the first term in the brackets comes from eq 2 and AV( W)is the value of the effective potential at the minimum. The distance L , represents the maximum value of L in the balloon. Being a constrained system, the added area will preferentially appear at this distance. If the bubble grows in two dimensions, one expects that LZmax A. It is revealing to compare the differential free energy increase in eq '6 with the corresponding differential free energy for liqiud-gas transition in a two-dimensional sys-
-
X
Figure 4. Balloon type defect in a multilayer liquid crystal.
tem. In such a case, a line tension yL will develop and
+
dG = (Ap yL/(4rA)lI2)dA (7) where Ap is the chemical potential difference in the two phases and corresponds to the term AV in eq 6. Due to the slow (A"') decrease of the line tension term with increasing A, the fluctuations are cooperative enough to drive the transition first order. In eq 6, the area-dependent term decreases more rapidly (A-' assuming L2,A), and large fluctuations can more readily develop. In fact, A-' dependence is typical for a one-dimensional system, and a proper phase transition is not expected to develop. Consequently, on the basis of these arguments we expect a continuous growth of ( L ) as W W,. This gives further support to the conjecture that the unbinding transition is indeed due to localized large amplitude fluctuations.
-
-
Transitions in Stacked Bilayers In accordance with the original renormalization group calculation of the unbinding transition,' the discussion has so far been concentrated on the behavior of two on average parallel bilayers. Furthermore, multiple stacking of these layers has been disallowed by assuming L = L,(x,y) - L,(x,y) to be a reasonably well-defined quantity. If one relaxes this constraint, van der Waals forces will lead to multiple stacks in the bound state as observed in chloroplasts. In lyotropic liquid crystals, one has multiple stacking. This observation raises the question of whether or not stacking will have an effect on the nature of the unbinding transition. Leibler and Lipowsky' argue qualitatively that this is not the case. Based on previous analysis of the fluctuations in the unbinding transition, we arrive at different conclusions. The asymptotic expression (eq 2) for the undulation force is derived on the basis that one can neglect the correlation in the undulation of two neighboring bilayers. At large separations, (L), this is valid even for stacked bilayers, but at short range the assumption is problematic. This complication is particularly apparent when we consider a local large fluctuation (a balloon) in the bound state, as in Figure 4. In a stacked system, such a fluctuation can only develop if it is correlated between many layers. This means that a fluctuation in one bilayer develops more easily if one has already developed in a neighboring one. This implies that a qualitatively new cooperation mechanism is operating in the stacked system. Such cooperations tend to make transitions first order. It is more difficult to develop a large-scale fluctuation in
Wennerstrom
030 Langmuir, Vol. 6, No. 4, 1990
320
-i
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I
/Cr't'ca'
t
Po'n'
Unbound TWOPhase Coexistence
I
2.30
I I
i
Bound
50
I
I 75
100
% Water
Figure 5. Calculated binary partial phase diagram for the system egg lecithin-water. Calculation is based on eq 4 and current estimates of constants A H = 0.2 J/m2, AH = 0.3 nm, 6 = 4 nm, W = 6 X lo-** J, k = 1.5 X J. The calculation is not quantitatively reliable due both to the uncertainties in the values of the parameters and to the approximate nature of eq 4. The coexistence region covers an interval of 35% in K or in W.
the three-dimensional stacked system than in a pair of bilayers. It involves formation of a boundary between bound and unbound regions in three dimensions, and one would expect that a free energy expression analogous to the ones in eqs 6 and 7 would involve a surface free energy term. The prediction is then that the "unbinding transition" is first order in a stacked bilayer systems.
Relation to Experimental Observations Recently, there have been several reports of the observation of a coexistence of two lamellar and there also exist a few scattered reports of older observations of this phen~menon.'~.'~ In all these cases, the phase transition is first order in nature, in accordance with the mean field arguments presented above. This as such is, however, not definite proof since one can always argue that the nature of the interbilayer forces is different in these particular cases. There is one interesting case where the considerations presented above could shed light on a controversial question. For egg lecithin, Parsegian, Rand, and coworkersl6P2lfind, in accordance with an established tradition, that there is a maximum swelling in pure water to approximately 40% (w/w) water. Helfrich and coworkers?' on the other hand, maintain that by allowing the system to swell in a large excess of water they reach an unbound state. If one has that W = W,,- t, the mean field theory predicts that there is a coexistence of a bound state and a highly swollen one. This is illustrated more clearly in Figure 5, which shows a calculated phase diagram based on present estimates of the coefficients that enter the effective potential (eq 4). (21)Le Neveu, D. M.; Rand,R. P.; Parsegian, V. A. Nature 1976, 259, 601.
With the experimental methods normally used, lipid plus a finite amount of water, one would hardly be able to see the (small) fraction of lipid in the unbound state. In the procedure used by Helfrich and co-workers, on the other hand, where the bilayers are directly observed in a microscope, the highly swollen state is monitored specifically. Highly swollen lamellar phases have been observed in a number of surfactant ~ystems,2~-~' and the existence of the repulsive term as in eq 2 has been convincingly demonstrated, although electrostatic forces can also play a role under certain circumstances.26 In these systems, there exist no reports of an unbinding transition. One problem is that these are multicomponent systems, and the "impurity problem" briefly discussed above complicates the issue.
Conclusions In the present paper, we have developed a mechanistic understanding of the so-called unbinding transition, predicted by Leibler and Lipowski, in the swelling of lamellar structures. The heuristic arguments have been based on a mean field effective potential for the interbilayer interaction. The mean field phase equilibria have been calculated and used as a reference for discussion of the effect of fluctuations. Without questioning any of the quantitative results obtained in ref 1, we have arrived at a somewhat different interpretation of the behavior of the system. The main conclusions are as follows: (i) The continuous increase in the mean separation ( L )of two membranes as the Hamaker constant approaches a critical value W, is caused by a localized large-scale fluctuation from a bound state. Up to very close to W,, the main fraction of the bilayers is still in the bound state. (ii) The continuous unbinding transition only occurs if multiple stacking of membranes is disallowed. With stacking as in lamellar lyotropic crystals, it is expected that the transition is first order, and the phase behavior is quite analogous to a liquid-gas transition both from a thermodynamic and a mechanistic point of view. Whether this also applies to the behavior at the critical point remains an open question. Acknowledgment. I thank the University of Minnesota for a George T. Piercy Distinguished Visiting Professorship. Stimulating discussions with Ted Davis, Fenne11 Evans, and Evan Evans are gratefully acknowledged. (22)Safhya, C. R.;Roux, D.; Smith, G. S.; Sihna, S. K.; Dimon, P.; Clark, N. A.; Bellocq, A. M. Phys. Reo. Lett. 1986,57, 2718. (23)Roux,D.;Safinya, C. R. In Physics of Amphiphilic Layers; Meunier, J., Langevin, D., Boccara, N., Eds.; Springer Proceedings in Phyaics 21,Springer: Berlin, 1987;p 138. (24)Porte, G.;Bassereau, P.; Marignan, J.; May, R. In Physics of Amphiphilic Layers; Meunier, J., Langevin, D., Boccara, N., Springer Proceedings in Phsics 21,Springer: Berlin, 1987;p 145. (25)Larche, F.C.;El Qebbaj, S.; Marignan, J. J. Phys. Chem. 1986, 90,707. (26)Wennerstrom, H.Colloid Polym. Sci. 1987, 274, 63.
