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Articles Phase Behavior of Dilute Aqueous Solutions of Lipid-Surfactant Mixtures: Effects of Finite Size of Micelles Y. Roth, E. Opatowski, D. Lichtenberg, and M. M. Kozlov* Department of Physiology and Pharmacology, Sackler Faculty of Medicine, Tel Aviv University, Ramat-Aviv 69978, Israel Received July 22, 1999 We consider the phase behavior of an aqueous solution of a mixture of two amphiphiles having different structural properties: a phospholipid, which tends to self-assemble into extended flat bilayers forming closed vesicles, and a surfactant, which self-organizes into small, strongly curved micelles. In a mixture of the two amphiphiles, the two kinds of aggregates, mixed vesicles, and the mixed threadlike micelles can be formed in coexistence with aqueous solution of surfactant monomers. The composition-induced transition between the micelles and the vesicles is generally assumed1 to be determined solely by the surfactant-to-lipid ratio within the aggregates and to be independent of the aqueous concentrations of the amphiphiles per se. While this approach described successfully the experimental phase boundaries in the range of relatively high lipid concentrations L, it demonstrated an internal contradiction upon extrapolation to low L. The goal of the present study is to develop a model free of this self-inconsistency. For that we consider the effects of the finite size of the threadlike micelles on the phase behavior of the system. Specifically, we treat the vesicular phase and the aqueous solution of the surfactant monomers using the previous approach but present a different thermodynamic description of the micelles as built of cylindrical parts and inhomogeneous end-caps, analogous to Gibbs treatment of interfaces. Based on this description, and accounting for the translational entropy of the micelles, we determine the chemical potentials of the lipid and surfactant in the micellar phase. Equalizing the chemical potentials of each amphiphilic compound in all phases available for it, we find the conditions of coexistence of different phases and, by so doing, determine the phase diagram. We show that the proposed approach describes self-consistently the existing experimental data. Furthermore, we use our analysis to make several testable predictions and perform the experiments, which verify these predictions and support the conclusions of the study.
Introduction Amphiphiles in aqueous solutions undergo self-assembly.2-9 This phenomenon is driven by the dual hydrophilichydrophobic nature of an amphiphilic molecule made of a polar head and a hydrophobic moiety, which in most cases is composed of one or two hydrocarbon chains. When the aqueous concentration of an amphiphile exceeds a certain characteristic value called the critical micelle concentration, cmc, the amphiphilic molecules self-assemble into aggregates in which the hydrophobic moieties are shielded from the aqueous medium by the polar heads. The shapes of the aggregates are diverse and determined (1) Lichtenberg, D. Biochim. Biophys. Acta 1985, 821, 470. (2) Tanford, C. The Hydrophobic Effect- Formation of Micelles and Biological Membranes, 2nd ed.; John Wiley and Sons: New York, 1980. (3) Hoffmann, H.; Ulbricht, W. The Formation of Micelles. Thermodynamic Data for Biochemistry and Biotechnology; Hans-Jurgen Hinz, ed.; Springer-Verlag: Berlin, 1986; Chapter 12. (4) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. Biochim. Biophys. Acta 1977, 470, 185. (5) Israelachvili, J. N.; Marcelja, S.; Horn, R. G. Quarterly Reviews of Biophysics 1980, 13, 121. (6) Lasic, D. D. Liposomes: from Physics to Applications; Elsevier Science BV: Amsterdam, 1993. (7) Barenholz, Y., Lasic, D. D., Eds. Handbook of Non-Medical Applications of Liposomes; CRC Press: New York, 1996. (8) Lichtenberg, D. In Handbook of Non-Medical Applications of Liposomes; Barenholz, Y., Lasic, D. D., Eds.; CRC Press: New York, 1996; p 199. (9) Shiloach, A.; Blankschtein, D. Langmuir 1998, 14, 7166.
by the molecular structure of the constituting amphiphiles.10 In the present study we consider only three common types of amphiphilic aggregates, namely, the extended amphiphilic bilayers, which have a nearly flat shape and form closed vesicles, and two kinds of strongly curved aggregates, the cylindrical and spherical micelles. While the self-assembly of individual amphiphiles has been determined in many amphiphilic solutions,11-16 much less is known about mixtures of amphiphilic compounds. Of special interest are the mixtures of two amphiphiles displaying very different properties, namely, of a surfactant such as octylglucoside (OG) that forms small spherical micelles and the phospholipid phosphatidylcholine (PC), which self-assembles into flat bilayers and serves as a structural basis for cell membranes.13-15 Investigation of such systems is strongly stimulated by their applications in biotechnology, where solubilization of cell membranes by addition of a micelle-forming surfactant into the (10) Ben-Shaul, A.; Szleifer, I.; Gelbart, W. M. J. Chem. Phys 1985, 83, 3597. (11) Langevin, D., Meunier, J., Boccara, N., Eds. Physics of Amphiphilic Layers; Springer: Berlin, 1987. (12) Gelbart, W. M., Ben-Shaul, A., Roux, D., Eds. Micelles, Membranes, Microemulsions and Monolayers; Springer: New York, 1994. (13) Pidgeon, C.; Hunt, C. A. J. Pharm. Sci 1981, 70, 173. (14) Lichtenberg, D.; Markello, T. J. Pharm. Sci 1984, 73, 122. (15) Pidgeon, C.; McNeely, S.; Schmidt, T.; Johnson, J. E. Biochemistry 1987, 26, 17. (16) Zoeller, N.; Blankschtein, D. Langmuir 1998, 14, 7155.
10.1021/la990984+ CCC: $19.00 © 2000 American Chemical Society Published on Web 12/22/1999
Phase Behavior of Lipid-Surfactant Mixtures
extracellular medium is the most commonly used method of extraction and concentration of membrane proteins.17-20 The structural behavior of mixed aqueous solutions of phospholipids and surfactants has been addressed in many experimental studies,21-29 which revealed several qualitatively common features exhibited by different mixtures: (i) A stepwise addition of small portions of surfactant to lipid bilayers (liposomes) first results in the insertion of a part of the surfactant molecules into the bilayers while the rest of the surfactant remains in the aqueous solution in nonaggregated (monomeric) form. (ii) When the total concentration of surfactant reaches a certain characteristic value, which depends on the phospholipid concentration, the mixed bilayers become saturated and no longer accommodate additional surfactant. Further addition of surfactant results in the onset of formation of mixed micelles. In contrast to the spherical micelles formed by pure surfactant, those formed from the saturated mixed bilayers are cylindrical and very long compared to their cross-section so that they are called threadlike micelles.25-29 (iii) Further addition of surfactant results in growing amount of mixed micelles formed at the expense of the mixed vesicles whose number decreases. (iv) At a second characteristic concentration of surfactant, micellization is completed; i.e., the only type of aggregates remaining in the system are the mixed micelles. Continuous addition of surfactant to the solution results in reduction of the micellar lengths, while the type of aggregates does not change anymore.30 This behavior is presented by a phase diagram expressed in terms of the total aqueous concentrations of lipid L and detergent D. A typical phase diagram is shown in Figure 1 for mixtures of the phospholipid PC and the surfactant OG.29 It consists of three parts: the lower part corresponds to the phase of mixed vesicles, the upper part represents the phase of the mixed threadlike micelles, and the intermediate part describes the range of coexistence of the mixed vesicles and micelles. The phase boundaries between the three distinct regions are represented in Figure 1 by the experimental points obtained by various methods including ITC (isothermal titration calorimetry), QLS (quasielastic light scattering), and others.29 A remarkable feature of these results is that in a first approximation the two phase boundaries can be described by straight lines. This observation resulted in a phenomenological concept (17) Helenius, A.; Simons, K. Biochim. Biophys. Acta 1975, 415, 29. (18) Klausner, R. D.; Renswoude, J. V.; Blumenthal, R.; Rivnay, B. Reconstitution of Membrane Receptors. Molecular and Chemical Characterization of Membrane Receptors; Alen R. Liss, Inc.: New York, 1984; Chapter 9. (19) Racker, E. Reconstitution of Transporters, Receptors and Pathological States; Academic Press: Orlando, FL, 1985. (20) Walter, A. Molecular and Cellular Biochemistry 1990, 97, 145. (21) Lichtenberg, D. In Biomembranes: Physical Aspects; Shinitzky, M., Ed.; VCH Balaban Publishers: Weinheim, 1993; Chapter 3. (22) Small, D. M. The Physical Chemistry of Lipids; Plenum: New York, 1986. (23) Schurtenberger, P.; Mazer, N. A.; Kanzig, W. J. Phys. Chem. 1985, 89, 1042. (24) Fromherz, P.; Rocker, C.; Ruppel, D. Faraday Discuss. Chem. Soc. 1986, 81, 39. (25) Walter, A.; Vinson, P. K.; Kaplun, A.; Talmon, Y. Biophys. J. 1991, 60, 1315. (26) Almog, S.; Kushnir, T.; Nir, S.; Lichtenberg, D. Biochemistry 1986, 25, 2597. (27) Ollivon, M.; Eidelman, O.; Blumenthal, R.; Walter, A. Biochemistry 1988, 27, 1695. (28) Vinson, P. K.; Talmon, Y.; Walter, A. Biophys. J. 1989, 56, 669. (29) Opatowski, E.; Kozlov, M. M.; Lichtenberg, D. Biophys. J. 1997, 73, 1448. (30) Almog, S.; Litman, B. J.; Winley, W.; Cohen, J.; Wachtel, E. J.; Barenholz, Y.; Ben-Shaul, A.; Lichtenberg, D. Biochemistry 1990, 29, 4582.
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Figure 1. Phase diagram for aqueous mixtures of PC (phosphatidylcholine) and OG (octyl glucoside) from Opatowski et al. (1997). The slopes are RSOL ) 3.1 and RSAT ) 1.6, and the intercepts are DSOL ) 15.9 and DSAT ) 15.5.
of the system,1 according to which the phase state of the mixture as well as the aqueous concentration of the surfactant monomers Dw are determined solely by one variable: the ratio R between the concentrations of surfactant DA and lipid LA inside the aggregates,
R)
DA LA
(1)
Within this model, the lower phase boundary corresponding to the bilayers saturated with respect to accommodation of the surfactant is described by a constant ratio R ) RSAT and is given by the equation
+ RSATL D ) DSAT w
(2)
is the aqueous concentration of the surfacwhere DSAT w tant monomers corresponding to RSAT. Similarly, the upper phase boundary corresponding to the mixed micelles just formed from the saturated bilayers is characterized by the constant ratio R ) RSOL and is given by the equation
+ RSOLL D ) DSOL w
(3)
representing the aqueous concentration of the with DSOL w surfactant monomers, corresponding to RSOL. According to (2) and (3), the compositions of the mixed bilayers RSAT and the mixed micelles RSOL corresponding to transition between these two kinds of aggregates are determined by the slopes of the straight lines representing the two phase boundaries. The related aqueous concenand DSOL are trations of the surfactant monomers DSAT w w given by the intercepts of the two phase boundaries. A thermodynamic background of this phenomenological model can be easily understood.31-32 For that we define three possible subsystems constituting the mixture: (i) mixed vesicles (considered separately from the aqueous solution) indicated by a superscript “b”, (ii) mixed micelles (also considered separately from the water) indicated by the superscript “m”, and (iii) the aqueous solution of the surfactant monomers indicated by the superscript “w”. It is assumed that each of these subsystems is a real (31) Andelman, D.; Kozlov, M. M.; Lichtenberg, D. Europhys. Lett. 1994, 25, 231. (32) Fattal, D.; Andelman, D.; Ben-Shaul, A. Langmuir 1995, 11, 1154.
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thermodynamic phase, which means that the chemical potentials of the surfactant, µD, and of the lipid, µL, in each of the subsystems are determined only by the intensive thermodynamic variables of this subsystem. The intensive variables characterizing the compositions of the three subsystems are the detergent-to-lipid ratios Rb and Rm for the vesicles and the micelles, respectively, and the concentration Dw for the aqueous solution of the surfactant monomers. This consideration means that the vesicles are regarded as one extended closed bilayer with composition Rb, while all the micelles are represented by just one threadlike micelle with composition Rm, which is so long that the inhomogeneity in its structure related to the two endcaps can be neglected. All the effects related to separation of the bilayer and the micelle into smaller aggregates and their repartitioning in the available water volume are neglected. These assumptions result in several desirable relationships. In the lower part of the phase diagram mixed vesicles coexist with surfactant monomers so that the chemical potential of surfactant is equal for these two subsystems, w b b µw D(D ) ) µD(R )
(4)
Solution of eq 4 determines the function Dw (Rb) for the vesicular phase. Analogously, equality of the chemical potential of surfactant in the micellar phase w m m µw D(D ) ) µD (R )
(5)
yields the relationship Dw (Rm) for the upper part of the phase diagram. And, finally, consideration of the conditions of coexistence between mixed vesicles and micelles corresponding to the composition induced transition between them requires the equality of the surfactant chemical potential in all three subsystems, b m w b m µw D(D ) ) µD(R ) ) µD (R )
(6)
while the chemical potential of lipid must be equal in the vesicles and micelles (it is commonly assumed for simplicity that lipid is not solvable in water),
eqs 2 and 3. All the experimentally determined phase boundaries, which relate to a limited range of lipid concentrations L, were approximated by straight lines. The intercepts of these lines to L ) 0 were assumed to represent DSAT and DSOL w w . The results of most of these and DSAT have different studies indicated that DSOL w w and DSAT was small values. The difference between DSOL w w compared to their absolute values, but reproducible, DSOL w being always larger than DSAT w . For example, for the = 16 mixture of PC and OG, illustrated in Figure 1, DSOL w = 15.5 mM. These differences are inconmM and DSAT w sistent with the above thermodynamic background, leading to the conclusion that the aqueous concentration of surfactant has to be fixed and equal Dw* in the whole region of the vesicles-micelles coexistence including the phase boundaries. In other words, the model above predicts and DSAT to be equal DSOL ) DSAT ) Dw*. the values DSOL w w w w In terms of the phase diagram (Figure 1) this means that upon extrapolation of the straight lines representing the phase boundaries to L ) 0, these lines should intersect the axis D in the common point Dw*. The goal of the present work is to further develop the treatment of aqueous mixtures of lipids and surfactants in order to resolve the contradiction of the previous concept. For this purpose we remove one of the main assumptions of the previous model describing the micelles as a separate thermodynamic phase. We explicitly account for the finite size of the threadlike micelles, their repartitioning in the water volume, and the effects of the end-caps. We demonstrate that this consideration leads to a different physical picture. The chemical potentials in the micellar phase depend not only on the surfactant-to-lipid ratio Rm but also on the absolute aqueous concentrations Lm and Dm of the amphiphiles constituting the micelles. We show that the phase boundaries deviate from straight lines and that this deviation becomes pronounced in the range of low lipid concentrations where the micelles become relatively short. We also demonstrate that the nonstraight real shape of the phase boundaries is the reason for the and DSAT observed in apparent difference between DSOL w w the previous studies. The purely thermodynamic approach we suggest does not use any model assumptions about the micellar structure and energetics. Statement of the Problem
This provides one with three equations whose solution, Dw*, Rb*) RSAT, and Rm*) RSOL, represents the compositions of the three phases in the range of coexistence between them. Thus, the assumptions above indeed explain the major features of the phenomenological model.1 They predict the constant ratios RSAT and RSOL determining the phase boundaries of the system and the relationships Dw (R) in the regions of single phases. However, one important issue, related to the aqueous concentrations of the surfactant monomers at the phase and DSOL boundaries, DSAT w w , still poses a challenge. These values have been determined from measurements by different groups using various experimental methods and investigating different combinations of amphiphilic compounds.27,29,33-34 Treatment of the data resulting in DSAT and DSOL was common for all studies and based on w w
We consider an aqueous solution of a mixture of lipid and surfactant. These amphiphiles self-assemble in the form of closed vesicles and/or threadlike micelles. We assume that the lipid exists only in the aggregated form, whereas the surfactant repartitions between the aggregates and the aqueous solution of monomers. According to the experimental results, we consider the following possible states of the system: (i) mixed lipid-surfactant vesicles in equilibrium with surfactant monomers, (ii) mixed surfactant-lipid micelles in equilibrium with surfactant monomers, and (iii) coexistence of mixed vesicles, mixed micelles, and surfactant monomers in the range of phase transition between the two types of aggregates. The goal of the present study is to determine the phase diagram of the mixture accounting for the effects of the finite size of the threadlike micelles. We will derive the equations for the phase boundaries in terms of the total aqueous concentrations of lipid L and surfactant D and determine the changes of the surfactant-to-lipid ratio of
(33) Paternostre, M.; Meyer, O.; Grabielle-Madelmont, C.; Lensieur, S.; Ghanam, M.; Ollivon, M. Biophys. J. 1995, 69, 2476.
(34) de la Maza, A.; Coderch, L.; Gonzalez, P.; Parra, J. L. J. Controlled Release 1998, 52, 159.
m µbL(Rb) ) µm L (R )
(7)
Phase Behavior of Lipid-Surfactant Mixtures
Figure 2. Definition of reference micelle. (a) Illustration of a real threadlike micelle. (b) Illustration of a corresponding reference micelle with the same detergent/lipid ratio as in the cylindrical part of the real micelle. Detergent and lipid molecules are schematically represented by filled and empty tips, respectively.
vesicles, Rv, and micelles, Rm, and of the aqueous concentration of the surfactant monomers, Dw, in the range of coexistence (iii). Similarly to our previous studies, we use the thermodynamic approach, according to which in equilibrium the chemical potential of each component is equal in all the available phases. Hence, the range of coexistence is determined by the equations of the type (6) and (7). We first present a detailed description of the micelles, allowing us to account explicitly for their finite size. Based on this description we derive the expressions for the chemical potentials of the surfactant and lipid in the micelles, as well as in the two other phases. Finally, we insert the expressions for the chemical potentials into (6) and (7), solve the resulting equations, and determine the phase diagram of the system. Mixed Micelles A threadlike micelle consists of a cylindrical part and two end-caps (Figure 2a). According to the common view, an end-cap has two major features. First, it is characterized by an energy which is required to create an end-cap from the cylindrical part of the micelle. A physical process resulting in creation of two end-caps is scission of a threadlike micelle, thus the energy 2 is called the scission energy.35 Second, the components of a mixed micelle are expected to repartition between the end-cap and the cylindrical part. A force driving this repartitioning can be qualitatively understood in terms of the effective shapes of amphiphilic molecules.36-38 A surfactant molecule tends to adopt a strongly curved shape matching that of an endcap. In contrast, the lipid molecules have a tendency to retain a less curved shape. As a result, in a threadlike micelle the surfactant is supposed to preferably partition into the end-caps at the expense of the lipid. The extent of this partitioning is limited by the entropy of mixing of the two amphiphiles.31 The detailed structure of an end-cap is unknown and, therefore, a theoretical determination of the end-cap energy as well as the partitioning of the components requires model assumptions.39 In this study we describe the end-caps in analogy to the Gibbs description of interfaces,40 avoiding any specific model. To clarify this consideration, we present it in several stages. First we consider one micelle and introduce the notions of a reference micelle and of end-cap excesses of the compo(35) Cates, M. E.; Candau, S. J. J. Phys. Condens. Matter 1990, 2, 6869. (36) Helfrich, W. Z. Naturforsch. C 1973, 28, 693. (37) Gruner, S. M. Proc. Natl. Acad. Sci. U.S.A. 1985, 82, 3665. (38) Gruner, S. M. J. Phys. Chem. 1989, 93, 7562. (39) Kozlov, M. M.; Lichtenberg, D.; Andelman, D. Phys. Chem. B 1997, 101, 6600. (40) Gibbs, J. W. On the Equilibrium of Heterogeneous Substances. In The Scientific Papers of J. Willard Gibbs; Dover Publications: New York; Vol. I, 1878.
Langmuir, Vol. 16, No. 5, 2000 2055
nents. Next we consider an ensemble of similar micelles and define the excess free energy of the end-caps. Then we account for the translational entropy of the micelles and derive the chemical potentials of the components in the micellar phase necessary for determination of the phase diagram. Reference Micelle. Consider one threadlike micelle (Figure 2a) consisting of nD surfactant molecules and nL lipid molecules so that the total aggregation number is n ) nD + nL. We assume that the micelle is sufficiently long so that its middle part has a cylindrical shape. This cylindrical part is homogeneous along the micellar axis; i.e., its properties are not influenced by the end-caps. The local composition of the cylindrical part can be characterized by the surfactant-to-lipid ratio (1), which in this case is determined as a ratio between the numbers of surfactant δnD and lipid δnL molecules constituting any homogeneous element of the cylindrical part, Rnc ) δnD/δnL. Drawing an analogy between the threadlike micelle and a system of two nonmixing volume phases separated by a transition region,40 the cylindrical part of the micelle is the counterpart of a homogeneous volume phase, while an endcap is analogous to the transition region. Since the structure of an end-cap is unknown (Figure 2a), we define for each real micelle an imaginary reference micelle (Figure 2b), which consists of ncL lipid and ncD surfactant molecules and has the shape, the surfactantto-lipid ratio Rnc ) ncD/ncL, and all the intensive thermodynamic properties constant along its whole surface and equal to those of the cylindrical part of the real micelle. This is analogous to the introduction by Gibbs of a dividing surface inside a transition region between two volume phases and extension of each phase with its bulk properties unchanged up to the dividing surface. We still have to fix the aggregation number of the reference micelle, nc ) ncD + ncL, in analogy to fixing the location of the Gibbs dividing surface inside the interface. This requires one additional condition.40 For this condition we set the total number nc to be equal to the aggregation number of the real micelle, nc ) n, meaning that
ncL + ncD ) nL + nD
(8)
Note, however, that the amounts of the two amphiphiles in the reference micelle, ncL and ncD, considered separately, can be different from the corresponding numbers nL and nD in the real micelle. The differences
2gD ) nD - ncD,
2gL ) nL - ncL
(9)
characterize partitioning of the two components between the end-caps and the cylindrical part of the real micelle. Hence gD and gL, which are analogous to the Gibbs surface excesses (adsorption) of the components, will be called the end-cap excesses of detergent and lipid, respectively. According to our definition of the reference micelle, the end-cap excesses (8) and (9) are related by gD ) -gL ) g, which means that if an end-cap is enriched in surfactant it is depleted of lipid to the same extent. End-Caps Free Energy. We now consider an ensemble of similar real micelles of aggregation number n and the corresponding ensemble of the same amount of the reference micelles. The aqueous volume concentration of these micelles is denoted by Mn. In the following we consider the free energy of the system calculated per unit volume of the aqueous solution. A part of the free energy of the ensemble, which does not account for the effects of partitioning of the micelles in the aqueous solution, will
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be called the standard free energy and indicated by superscript “0 ”. This value for the real micelles, F0n, is different from the standard free energy of the reference e micelles F0c n . The difference Fn between them constitutes the excess free energy of the end-caps,
Fen ) F0n - F0c n
(10)
Recalling the similarity to the Gibbs theory, F0c n is analogous to the energy of a volume phase, while the endcaps free energy, Fen, is the counterpart of the Gibbs energy of interface. Following this analogy, the role of the area A of the Gibbs interface is played in our system by the number of the micelles Mn, while the total adsorption of the substances on the interface40 is analogous to the total end-caps excesses of the surfactant, ΓD ) Mn2g ) Γ, and of the lipid, ΓL ) -Mn2g ) -Γ. The resulting equation for the differential of the end-caps free energy is
dFen ) 2dMn + µeDdΓD + µeLdΓL
(11)
where µeD and µeL are the end-cap chemical potentials of the two amphiphiles. The first term in (11) determines the energy corresponding to the change of the number of the micelles, dMn, at fixed total end-cap excesses of the components dΓD ) dΓL ) 0. Hence, the scission energy defined by
∂Fen ∂MnΓD,ΓL)const
2 )
(13)
The end-cap energy is a first-order homogeneous function of the extensive variables Mn, Γ so that
Fen ) 2Mn + (µeD - µeL)Γ
(14)
It can be shown by a standard procedure that the intensive thermodynamic functions of the end-caps satisfy an equation analogous to the Gibbs equation of adsorption
d ) -g - µeL)
d(µeD
(15)
The reference micelles represent, by definition, a homogeneous thermodynamic system. The energy of this system can be presented as
Fcn ) µ0DncDMn + µ0LncLMn
(16)
with the chemical potentials of the components µ0D(Rnc ) and µ0L(Rnc ) being functions of the surfactant-to-lipid ratio Rnc and satisfying the equation
dµ0L/dµ0D ) -Rnc
µeD ) µ0D(Rnc ), µeL ) µ0L(Rnc )
(18)
This assumption together with (15) determines the scission energy 2(Rnc ) and the end-cap excess of the surfactant, g( Rnc ), as functions of the surfactant-to-lipid ratio Rnc of the cylindrical part of the micelle. Accounting for these relationships along with (9), (10), and (14)-(16), the standard free energy of the considered ensemble of the micelles is given by
F0n ) µ0D(Rnc )Dn + µ0L(Rnc )Ln + 2(Rnc )Mn
(19)
Here Dn ) nDMn and Ln ) nLMn are, respectively, the volume concentrations of the surfactant and lipid molecules constituting the ensemble of micelles. We will call µ0D and µ0L the standard chemical potentials of the components in the micelles. Chemical Potentials in the Micellar Phase. The total free energy of the ensemble of similar micelles with the aggregation number n consists of the standard part (19) and a contribution due to partitioning of the micelles in the aqueous volume. We assume that the solution of the micelles is diluted so that the translational entropy of the micelles can be presented by an expression for an ideal solution. Adding this term to the free energy (19) gives
(12)
is a counterpart of the Gibbs surface tension. The chemical potentials µeD and µeL determine the excess energy dFen related to the changes of the end-cap excess of surfactant dΓD and lipid dΓL, respectively, provided that the number of micelles is fixed, dMn ) 0. Accounting for the definitions above, (11) can be written as
dFen ) 2dMn + (µeD - µeL)dΓ
the two components are equal in the cylindrical part and in the end-caps,
(17)
We will assume that each real micelle is in internal equilibrium, which means that the chemical potentials of
Mn Fn ) µ0DDn + µ0LLn + 2Mn + kTMn ln eW
(20)
where e is the base of the natural logarithms, k is the Boltzmann constant, T the is absolute temperature, and W is the volume concentration of water molecules (=55 M), which greatly exceeds the concentration of the micelles, W . Mn. The chemical potentials of the surfactant and lipid in the ensemble of the micelles are determined by µnL ) ∂Fn/ ∂Dn and µnL ) ∂Fn/∂Ln, respectively. Differentiation of (20) accounting for (9), (15), and (17) and for the relationship Mnn ) Dn + Ln results in
[ [
( )] ( )]
µnD ) µ0D(Rnc ) +
Mn 1 2(Rnc ) + kTln n W
(21)
µnL ) µ0L(Rnc ) +
Mn 1 2(Rnc ) + kT ln n W
(22)
The whole aqueous micellar solution consists of micelles of different aggregation numbers. The ensemble of similar micelles of a specific n considered above represents a subsystem of the whole system. The distribution of the micelles according to their aggregation numbers is given by a set {Mn}, which determines the concentration of the micelles as a function of n. We will assume that all the micelles in the solution are in equilibrium with respect to exchange of the amphiphilic molecules between them. This means that the chemical potentials (21) and (22) have to be equal for all the micelles, independent of the aggregation number n. This condition results in two major conclusions. First, along with the relationship µnD - µ0D(Rnc ) ) µnL - µ0L( Rnc ) following from (21) and (22), it requires that the composition of the cylindrical part of a micelle does not depend on its aggregation number n, so that we can drop out the corresponding superscript, Rnc ) Rc. And, second,
Phase Behavior of Lipid-Surfactant Mixtures
Langmuir, Vol. 16, No. 5, 2000 2057
Rc, Rm, and Lm
the distribution of the micelles is given by
[
] ( )
2(Rc) n exp kT n j
Mn ) W exp -
(23)
where n j is a constant equal to the average aggregation number, which remains to be determined. Expression (23) has a form of a standard distribution obtained for onecomponent micelles.35 The only difference is that the scission energy in (23) depends on the composition of the cylindrical part of the micelles, Rc. This simple low is the result of the assumption of thermodynamic equilibrium between all the micelles. Taking into account a possibility of fluctuations of micellar composition yields a different kind of distribution.41 The value n j can be found from the conditions of a given total concentration Dm + Lm of amphiphilic molecules in the micellar phase, ∞
nMn ) Dm + Lm ∑ n
(24)
min
where nmin is the minimal possible aggregation number. j , inserting (23) Assuming that nmin is much smaller than n into (24) and replacing the summation in (24) by integration from zero to infinity, we obtain
(
Dm + Lm n j) M0
)
1/2
(25)
[
]
2(Rc) kT
(26)
0 µm D ) µD(Rc) -
kT n j
(27)
0 µm L ) µL(Rc) -
kT n j
(28)
Expressions (25)-(28) represent the chemical potentials as functions of the surfactant-to-lipid ratio, Rc, characterizing the cylindrical part of a a threadlike micelle, and the total amphiphile concentration (24) in the micellar phase. m Derivation of the phase diagram requires µm D and µL to be expressed in terms of the total surfactant-to-lipid ratio in the micellar phase Rm ) Dm/Lm, which differs from Rc because of the contribution of the end-caps. Using the definition (9) of the end-caps excesses and summing over all the micelles we obtain for the total concentration Dm of surfactant molecules in the micellar phase
Rc
∞
∑
1 + Rcnmin
∞
nMn + 2g(Rc)
(30)
Rc ) Rm - (1 + Rm)3/22g(Rm)(M0/Lm)1/2
(31)
Inserting (31) into (27) and (28), accounting for (25), and performing expansion of the resultant equations, we obtain in our approximation for the chemical potentials in the micellar phase
[
0 m m 2 m µm D ) µD(R ) - kT + (1 + R ) 2g(R ) ×
[
]
∂µ0D(Rm) ∂Rm
( )
M0 1 m 1/2 L (1 + R ) m
0 m m 2 m µm L ) µL(R ) - kT + (1 + R ) 2g(R ) ×
]
∂µ0L(Rm)
1/2
( )
M0 1 m L (1 + R ) m
(32)
1/2
(33)
Chemical Potentials in the Vesicular Phase and in the Aqueous Solution of Surfactant Monomers
Introducing (23) into (21) and (22), we obtain the following expressions for the chemical potential of the surfactant m µm D and the lipid µL in the micellar phase
Dm )
]
1/2
The second term in (30) can be presented as g(Rc)/n j . We consider only those cases where the average aggregation number (25) is sufficiently large, n j . 1, and we will account for the first nonvanishing order in its inverse value 1/n j , 1. In this approximation, the solution of the eq 30 is
∂Rm
where
M0 ) W exp -
[
Rc M0 Rm 1 ) + 2g(Rc) m m 1 + Rc 1+R 1 + R Lm
∑ Mn
(29)
nmin
Replacing summation in (29) by integration within the j , and dividing the resultant assumption that nmin , n equation by the total concentration of the lipid molecules in the micellar phase Lm, we obtain an equation relating (41) Ben-Shaul, A.; Rorman, D. H.; Hartland, G. V.; Gelbart, W. M. J. Phys. Chem. 1986, 90, 5277.
These two phases are described similarly to the previous studies.31-32 Since the radius of the vesicles observed in the mixed lipid-surfactant systems is of the order of 100 j ν of the amphiphilic molecules nm,25,27-28,33 the number n in an average vesicle is very large. The contributions to the chemical potentials related to repartitioning of the vesicles in the aqueous volume are proportional to 1/n j ν, similar to (27) and (28). We will therefore neglect these contributions and describe the vesicles as one homogeneous bilayer constituted of surfactant and lipid of concentrations Dν and Lν, respectively. Hence, the chemical potentials in the vesicular phase, µνL (Rν) and µνD (Rν) are assumed to be determined by the surfactant-to-lipid ratio Rν ) Dν/Lν and to be independent of the total concentrations of the amphiphilic molecules Dν and Lν. The aqueous solution of surfactant monomers of concentration Dw is assumed to be sufficiently diluted so that their chemical potential is given by an expression for an ideal solution w w 0 µw D(D ) ) µ1 + kT ln(D /W)
(34)
where µ01 is the standard part of the chemical potential accounting for the interactions of a monomer with the surrounding medium. Phase Diagram We are now in a position to describe the conditions of coexistence between the different phases and, hence, to determine the phase boundaries indicating the composition-induced phase transition between the mixed micelles and vesicles. Equality of the chemical potential of each amphiphile in the vesicular and micellar phases results in the equations for the surfactant-to-lipid ratios Rv and
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Rm of the two phases corresponding to their coexistence. Taking into account (32) and (33), these equations are
[
µνD(Rν) ) µ0D(Rm) - kT + (1 + Rm)22g(Rm) ×
]
∂µ0D(Rm) ∂Rm
( )
M0 1 m 1/2 L (1 + R ) m
µνL(Rν) ) µ0L(Rm) -
[
1/2
(35)
]( )
∂µ0L(Rm) M0 kT + (1 + Rm)22g(Rm) (1 + Rm)1/2 Lm
1/2
(36)
Next, we seek for solutions of (35) and (36) in the form Rm m ) Rm j ) and Rν ) RνD + Rν1(1/n j ) + ... representing 0 + R1 (1/n the expansions order by order in the inverse average aggregation number of the micelles 1/n j , 1. ν To obtain the zero order solutions Rm 0 and R0, corresponding to infinitely long micelles, we neglect the second terms in the right sides of (35) and (36),
µνD(Rν0) ) µ0D(Rm 0)
(37)
µνL(Rν0) ) µ0L(Rm 0)
(38)
In our model-free description we do not know the form of the functions entering (37) and (38) and, hence, we cannot solve these equations explicitly. However, an important general conclusion is that the solutions of (37) and (38) do not depend on the total concentrations L, D of the components of the two phases. This corresponds to the previous description of the system1 mentioned in detail in the Introduction. Therefore, we will use for these solutions SOL. The the previous notations, Rν0 ) RSAT and Rm 0 ) R first-order solutions of (35) and (36) are the simplest approximation accounting for the finite size of the micelles. Simple algebra, using (17) and the analogous relation for the bilayers, dµνL/dµνD ) -Rν, yields
Rm ) RSOL -
Am (Lm)1/2
(39)
where
[
(1 + RSAT)
(1 + R
Dw ) D/w -
Aw (Lm)1/2
(43)
where
D/w ) W exp
[
]
µ0D(RSOL) - µ01 kT
(44)
and
(1 + RSOL)1/2 Aw ) D/wM0(RSOL)1/2 SOL (R - RSAT)
) (R
SOL
-R
SAT
∂µ0D(RSOL)/kT
)
∂Rm
-
]
2g(RSOL)(1 + RSOL)3/2 (40)
and
Rν ) RSAT -
Aν (Lm)1/2
(1 + RSOL)1/2
Aν ) M0(RSOL)1/2 (R
SOL
-R
SAT
)
∂µνD(RSAT)/kT ∂Rν
(42)
(47)
Accounting for (46) and (39)-(43), the relationship between the total concentrations of surfactant D and lipid L within the region of coexistence can then be expressed by
D ) D/w + [(RRSOL) + (1 - R)RSAT]L -
where
(46)
where Rm, Rν, and Dw are given by (39), (41), and (43), respectively. To express the degree of transition from vesicles to micelles within the range of coexistence, we define the ratio R between the concentration Lm of the lipid in the micellar form and the total lipid concentration, R ) Lm/L, so that
Lm ) RL, Lν ) (1 - R)L
(41)
(45)
The first contribution D/w in (43) is constant in the whole range of coexistence and corresponds to the previous description1 neglecting the effects of the finite size of the micelles. The second term in (43) accounts for these effects and describes the change of the monomer concentration Dw with the concentration of lipid molecules in the micellar phase Lm. As in the case of (39) and (41), this correction to (44) is small but may become considerable for low Lm. We can now describe the whole range of coexistence in terms of the total aqueous concentrations of surfactant D and lipid L. The surfactant concentration D can be presented as
D ) Dm + Dν + Dw ) RmLm + RνLν + Dw
Am ) M0(RSOL)1/2 ×
SOL 1/2
According to (39) and (41), the effects of the finite size of the micelles result in dependence of the compositions Rm and Rν of the coexisting mixed micelles and vesicles on the total concentration of the amphiphilic compounds that constitute the micelles, which, in turn, is a function of Lm. This dependence has been neglected in the previous description.1 The corresponding contributions to (39) and (41) are usually small compared to RSOL and RSAT, since they are proportional to 1/n j . On the other hand, these contributions may become considerable at low lipid concentrations in the micellar phase Lm, when the micelles are relatively short. It is now possible to determine the concentration of the surfactant monomers Dw in the range of coexistence. Equality of the chemical potentials of surfactant in the ν w v aggregated and the monomeric forms µw D (D ) ) µD (R ) accounting for (34), (35), and (39)-(42) results in the following approximate expression
Aw
R L1/2 L1/2 [Aν(1 - R) + AmR] 1/2 (48) R 1/2
The first and second contributions in (48) correspond to
Phase Behavior of Lipid-Surfactant Mixtures
Langmuir, Vol. 16, No. 5, 2000 2059
the previous model1 in which the effects of the finite size of the micelles were neglected, yielding a phase diagram with phase boundaries represented by straight lines. The two last terms in (48), which are nonlinear in the lipid concentration L, account in a first approximation for the finite size of the micelles and related effects. As noted above, these contributions are significant only in the range of small lipid concentrations L. In this range, the third contribution, which is proportional to Aw and related, according to (43), to the concentration of monomers Dw, is expected to become larger than the fourth one, which accounts for the corrections to the surfactant-to-lipid ratios, Rm and Rν, in the aggregates. Hence, to illustrate the major features of the present description, we further simplify (48) to the form
D ) D/w + [RRSOL + (1 - R)RSAT]L -
Aw 1/2
R L1/2
(49)
The upper phase boundary corresponding to complete transformation of the mixed vesicles into the mixed micelles is characterized by R ) 1 and, accounting for (49), is described by the equation
D ) D/w + RSOLL -
Aw L1/2
(50)
At the lower phase boundary most of the lipid is in the vesicular phase and, hence, R has to be small. Note, however, that within our description, R is not allowed to be too small as the approximation used is valid only as long as the two last terms in (48) are much smaller than the precedent ones. In physical terms this means that our approach is valid only when the micelles are still sufficiently long, which requires that a certain nonvanishing fraction of the lipid, Lm ) RL, resides in micelles. Assuming Rmin , 1 in (49), the equation for the low phase boundary is
D ) D/w + RSATL -
Aw (RminL)1/2
(51)
The first two terms in (50) and (51) neglect the effects of the finite size of the micelles and correspond to the previous model of the phase diagram.1 Indeed, for the upper phase boundary (50) they represent a straight line with a slope RSOL and intercept D/w, while for the lower phase boundary (51) they describe a straight line with a slope RSAT and the same intercept D/w. Let us emphasize again that this approximation does not explain the difference between and DSAT observed experimenthe intercepts DSOL w w tally.27,29,33-34 The third terms in (50) and (51) account for the finite size of the micelles and, thus, determine deviations of the phase boundaries from the straight lines. For high lipid concentrations L these corrections can be neglected, while for low L they influence considerably the shape of the phase boundaries. We suggest that this deviation of the phase boundaries from straight lines results in the apparent difference between the intercepts DSOL and DSAT w w . This can be illustrated by comparison of the phase boundaries (50) and (51) with the experimental results obtained for mixtures of the phospholipid PC and the surfactant OG. The parameters that determine the functions (50) and (51), taking into account (45) and (26), are D/w, RSAT, RSOL, and 2(RSOL). The first three parameters are the compositions of the coexisting phases as
Figure 3. Phase diagram described by the present model. The upper and lower phase boundaries are determined by (50) and (51), respectively, at Rmin ) 0.31, RSAT ) 1.55, RSOL ) 3.5, D/w ) 15.9 mM, and Aw ) 0.25 (mM)3/2. According to (45), the latter value corresponds to the scission energy of 2 ) 19.5kT. The experimental points are taken from our previous study (Opatowski et al., 1997).
obtained in the approximation of the previous model.1 Hence, according to our previous results,29 their values have to be close to D/w = 15.9, RSAT = 1.6, and RSOL = 3.1. The value of the scission energy, as known for purely surfactant micelles,35 is expected to be of the order of 20kT. The results of fitting the previous experimental points29 to (50) and (51) are presented in Figure 3. The parameters of the theoretical phase boundaries in Figure 3 are D/w ) 15.9 mM, RSAT ) 1.55, RSOL ) 3.5, and 2(RSOL) ) 19.5 kT in good agreement with the estimations above. This supports our approach of describing the phase boundaries by the curves (50) and (51), which accounts for the apparent difference between the intercepts. Discussion and Experimental Verification We presented a thermodynamic description of phase behavior of a dilute aqueous solution of a mixture of two amphiphiles, the one, lipid, tending to self-assemble into extended flat bilayers and the second, surfactant, forming strongly curved micelles. The approach we proposed accounts for the effects of the finite sizes of the micelles formed in the system. This is unlike the previous model,1 which had an internal contradiction presented in detail in the Introduction. The present approach succeeds to describe self-consistently the measured phase boundaries as illustrated in Figure 3. Our description is an extension of the previous model,30 taking into account the contributions of the first nonvanishing order in the inverse average aggregation number 1/n j of the mixed micelles in the range of their equilibrium with the mixed vesicles. For a total lipid concentration L this means that the previous approach is related to a range of high L, where the micelles are so long that the contributions to the thermodynamic values proportional
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Figure 4. Phase diagram for PC/OG mixtures including experimental points at low lipid concentrations. The upper and lower phase boundaries are fitted by the same curves as in Figure 3.
to 1/n j can be neglected. The present corrected model describes a wider range of lipid concentrations, L, including the moderately low concentrations, where the average aggregation number n j becomes sufficiently small to render significant those contributions to the thermodynamic values that are proportional to 1/n j . Yet, it is important to stress that we still neglect the contributions of the higher orders in 1/n j , which means that our results are valid only when L does not become too small. Several predictions testable experimentally can be drawn from the theoretical results. First, the phase boundaries have to deviate from straight lines in the range of relatively low lipid concentrations and to adopt in this range convex shapes as illustrated in Figure 3. We have checked this prediction by extending the measurements of the phase boundaries in mixtures of PC and OG to lower concentrations than those studied previously, using the calorimetric measurements, as previously described.29 The experimental points presented in Figure 4 are those obtained in the present study and in the previous measurements. The theoretically derived phase boundaries depicted in Figure 4 by the solid lines are in a good agreement with the experimental results, supporting the prediction of our model. The range of the lipid concentrations where a considerable deviation of the phase boundaries from the straight lines should be expected can be estimated as follows. The slopes of the upper and lower phase boundaries, as predicted by (50) and (51), respectively, are presented by
1 Aw dD ) RSOL + dL 2 L3/2
(52)
dD 1 Aw ) RSAT + dL 2 R1/2 L3/2
(53)
min
The curvature of the phase boundaries becomes considerable when the values of the second contribution in (52)
Figure 5. Phase diagram for PC/HG mixtures including experimental points at low lipid concentrations. The upper and lower theoretical phase boundaries are determined by (50) and (51), respectively at Rmin ) 0.35, RSAT ) 1.3, RSOL ) 3.2, D/w ) 52.5 mM, and Aw ) 1.65 (mM)3/2. According to (45), the latter value corresponds to the scission energy of 2 ) 18.0kT.
and (53) become comparable to the major slopes RSOL and RSAT. Hence, the lipid concentration has to be lower than a characteristic value, L*, which, taking into account (52), (53), (45), and (26), is given for the upper and lower phase boundaries by
L/SOL )
and
L/SAT )
[
] [
[
] [
/ SOL 1/2 1 Dw (1 + R ) W1/2 2 RSOL (RSOL - RSAT)
2/3
/ SOL 1/2 1 Dw (1 + R ) W1/2 2 RSAT (RSOL - RSAT) R1/2 min
2/3
]
SOL 2 (R ) exp 3 kT
(54)
]
SOL 2 (R ) exp 3 kT
(55)
respectively. The characteristic values (54) and (55) depend exponentially on the scission energy 2. This implies that for a surfactant providing a smaller scission energy 2, the deviation of the phase boundaries from straight lines have to be observed in a broader range beginning already at higher lipid concentrations L* and extending toward the low concentrations, L e L*. According to our qualitative model,39 the scission energy should be smaller for a surfactant with higher spontaneous curvature corresponding to shorter hydrocarbon chains. This prediction can be verified by comparison of the phase boundaries in mixtures of the same lipid with surfactants of different chain lengths. We determined by calorimetric method29 the phase boundaries in mixtures of PC with the surfactant heptylglugoside (HG), whose hydrocarbon chain (consisting of seven CH2 groups) is shorter than that of OG. Figure 5 depicts the experimental data points and the fitted theoretical phase boundary curves given by (50) and (51).
Phase Behavior of Lipid-Surfactant Mixtures
Langmuir, Vol. 16, No. 5, 2000 2061
Comparison of Figures 5 and 4 shows that, in accordance with our predictions, a considerable deviation of the phase boundaries from the straight lines in the case of HG corresponds to lipid concentrations of about L/SOL = 0.5 mM and L/SAT = 1 mM, while for OG L/SOL = 0.1 mM and L/SAT = 0.3 mM. The scission energy 2 was obtained for each of the mixtures from the fitting parameter Aw, according to (45) and (26). The resulting scission energy for the PC/HG micelles is 2 ) 18kT, while for the PC/OG micelles 2 ) 19.5 kT. These values can be considered only as qualitative estimations. Another prediction can be made for the effective intercepts of the phase boundaries, obtained by treating them as straight lines and extrapolating these lines to zero lipid concentration L ) 0. Let us denote by L ˜ * the minimal lipid concentration available in the measurements. Based on (50) and (51), description of the experimental points by the straight lines for L g L ˜ means that the upper and lower phase boundaries are presented by
D ) D/w + R ˜ SOLL -
Aw L ˜ 1/2
(56)
and
D ) D/w + R ˜ SATL -
Aw R1/2 ˜ 1/2 minL
(57)
˜ SAT are the effective slopes. The intercepts where R ˜ SOL and R ˜ 1/2 and DSAT of the lines (56) and (57) are DSOL ) D/w - Aw/L / 1/2 1/2 ) Dw - Aw/(RminL ˜ ) so that the difference between them, accounting for (26) and (45) is
DSOL - DSAT ) D/w (1 + RSOL)1/2 L ˜ 1/2 (RSOL - RSAT)
(
W1/2
1
R1/2 min
) [
]
(RSOL) ) (58) kT
- 1 exp -
Similarly to L*, the difference between the intercepts (58) exponentially depends on the scission energy. Hence, a measurement of DSOL - DSAT for surfactants with various chain lengths providing different is another way to check this prediction. Comparison of the difference between the intercepts obtained for the two surfactant, according to Figures 4 and 5, gives DSOL - DSAT equal to 1.2 mM for HG and 0.4 mM for OG in agreement with our predictions. And, finally, a direct verification of our approach would be a measurement of the compositions of the mixed micelles and vesicles, Rm and Rν, as well as of the aqueous concentration of the surfactant monomers Dw in the range of coexistence as functions of the total lipid concentration L, and comparison of the results with the dependences described by (39)-(45). This study is a matter of future experimentation. In conclusion, accounting for the finite size of the surfactant-phospholipid threadlike mixed micelles results in nonlinear phase boundaries separating different ranges of the phase diagram. This theoretical result is consistent with all the published data. Moreover, the experimental verification of the specific theoretical predictions, performed in the present study, provided additional support for the model. Acknowledgment. Support from the Israel Academy of Sciences and Humanitiesscenters of Excellence Program is gratefully acknowledged. LA990984+
