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Intermicellar Interactions May Induce Anomalous Size Behavior in Micelles Carrying out Bulky Heads with Multiple Spatial Arrangements P. Brocca, L. Cantu`, M. Corti, E. Del Favero, and A. Raudino* Department of Medical Chemistry, Biochemistry and Biotechnologies, L.I.T.A., V.le F.lli CerVi 93, 20090 Segrate, Italy ReceiVed October 20, 2006. In Final Form: December 14, 2006 We report experimental and theoretical results on the concentration dependence of the micellar size of GM1 and GM1acetyl gangliosides, five-sugar-headed anionic glycolipids. Contrary to one of the mainstays of colloid science, that the aggregation number of amphiphile aggregates grows with concentration, an anomalous region is found at intermediate concentrations, where a sharp decrease of the aggregation number occurs. Experiments were performed by small-angle X-ray and neutron scattering (SAXS and SANS). Two models are discussed, reproducing the observed behavior of either GM1acetyl or GM1. The first one is a conventional picture of interacting micelles where a reduction in the molecular surface area, leading to an increase of the aggregate dimension, is paid to reduce intermicellar interactions: it foresees a monotonous increase of the aggregation number with concentration. The second one accounts for a conformational bistability of the bulky headgroups of GM1, modifying the amphiphilic molecular surface area and protrusion from the aggregate surface, and contributing to the inter- and intramicellar interaction balance. Energy minimization leads to a complex behavior of the aggregation number, which is consistent with the anomalous behavior of GM1.
1. Introduction One of the firmest dogmas in colloid science is that the aggregation number of amphiphile aggregates grows with the concentration. Large self-assembled structures enable longer interaggregate distances, which, in turn, reduce the repulsion among the amphiphile headgroups. Particularly interesting is the case of ionic surfactants like, for instance, sodium alkyl sulfates (SAS) where the coulomb repulsion is expected to be particularly strong. At low concentration, but above the critical micellar concentration, only spherical micelles are observed. On raising the surfactant concentration, the SAS micellar radius increases,1 rapidly reaching a value corresponding to the maximum elongation of the surfactant’s hydrocarbon chain. Any further increase of the size of the aggregate violates the packing constraint and must be followed by a change of the aggregate final geometry. Indeed, what one observes is a transition toward cylindrical aggregates, a structure that can virtually accommodate an infinite number of monomers.2 The process sketched above evidences a monotonous increase of the aggregation number N with concentration (hereafter expressed as volume fraction φ), both in the narrow region where the micelles basically maintain their spherical shape and in other geometrical rearrangements (cylinders, cubic phases, lamellae) at higher concentration. The above behavior has been recently confirmed by molecular dynamics computer simulations.3,4 In this paper, we will show, both theoretically and experimentally, that, under particular conditions, considerable deviation from the ideal behavior can be observed. The most intriguing * Dept. of Chemical Sciences, V. le Andrea Doria 6, University of Catania, 95125, Catania, Italy. Tel. ++39 0957385078. E-mail araudino@ dipchi.unict.it. (1) Quina, F. H.; Nassar, P. M.; Bonilha, J. B. S.; Bales, B. L. J. Phys. Chem. 1995, 99, 17028 (and refs therein). (2) Mazer, N. A.; Benedek, G. B.; Carey, M. C. J. Phys. Chem. 1976, 80, 1075. (3) Nelson, P. H.; Rutledge, G. C.; Hatton, T. A. J. Chem. Phys. 1997, 107, 10777. (4) Lazaridis, T.; Mallik, B.; Chen, Y. J. Phys. Chem. B 2005, 109, 15098.
effect is a lowering of the micellar radius in rather concentrated micellar solutions just before entering a cubic phase. Furthermore, the anomalous behavior is restricted to amphiphiles with bulky headgroups that may assume several spatial arrangements (or conformations). We selected an important family of glycolipids bearing unusually large headgroups: the gangliosides. The present experimental results refer to two gangliosides, namely, GM1 (the structure of which is shown in Figure 1) and GM1acetyl (identical to GM1 but with different hydrocarbon chains). When dissolved in aqueous solution, both gangliosides selfaggregate in micelles in a wide range of concentration, from the very dilute region (0.001 volume fraction) to the semidilute and concentrated region (above 0.25 volume fraction), where micelles enter a liquid crystalline phase. As ganglioside headgroups contain a sialic acid, micelles are partially dissociated, thus undergoing electrostatic repulsive interparticle interactions. In the very dilute region, interactions can be suppressed by increasing the ionic strength of the solution, to obtain micellar parameters in the noninteracting case, as reported in Table 1 (data from refs 5 and 6). For GM1 micelles, Table 1 reports two different values for size and aggregation number N, due to different spatial arrangements GM1 headgroups can assume in the aggregate, with different values for the interfacial area A. This is shown pictorially in Figure 2. Extensive work on GM1 aqueous solutions has shown that this bistability is due to important cooperative behaviors in the hydrophilic layer of the aggregate7-11 and that (5) Cantu`, L.; Corti, M.; Del Favero, E.; Digirolamo, E.; Sonnino, S.; Tettamanti, G. Chem. Phys. Lipids 1996, 79, 137. (6) Sonnino, S.; Cantu`, L.; Corti, M.; Acquotti, D.; Kirschner, G.; Tettamanti, G. Chem. Phys. Lipids 1990, 56, 49. (7) Cantu`, L.; Corti, M.; Del Favero, E.; Digirolamo, E.; Raudino, A. J. Phys. II 1996, 6, 1067. (8) Cantu`, L.; Corti, M.; Del Favero, E.; Muller, E.; Raudino, A.; Sonnino, S. Langmuir 1999, 15, 4975. (9) Cantu`, L.; Corti, M.; Del Favero, E.; Raudino, A. Langmuir 2000, 16, 8903.
10.1021/la0630864 CCC: $37.00 © 2007 American Chemical Society Published on Web 02/07/2007
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2. Materials and Methods
Figure 1. GM1 ganglioside structure.
Figure 2. Pictorial sketch of the different spatial arrangements GM1 headgroups can assume in the aggregate. Vpol and Vapol are the GM1 molecule polar (headgroups) and apolar (lipid chains) volumes, and Ao is the area per headgroup at the interface between the polar and the apolar region. The R (β) arrangement corresponds to a tighter (looser) packing in a bigger (smaller) micelle with a thicker (shallower) hydrophilic shell. Table 1. Micellar Parameters for Natural GM1, in the Two Different Packing Arrangements, and GM1acetyla N
Rh (Å)
A0 (Å2)
GM1 301 205 58.7 52.8 95.4 99.5 GM1acetyl 76 34 64.8
Vpol (Å3) Vapol (Å3) 1065 1065
1000 566
a Aggregation number, N; hydrodynamic radius, Rh; area per headgroup at the interface, A0; polar and apolar volumes of the ganglioside molecule Vpol and Vapol.
the cooperative transition between two different packing states of the ganglioside headgroups is driven, for example, by temperature. The physical mechanism for the interconversion between the two spatial arrangements is not known yet, but it is supposed to be connected to a different degree of hydration. In the present work, we study the characteristic behavior of GM1 and GM1acetyl gangliosides in the semidilute range by smallangle X-ray and neutron scattering techniques. The experimental data will be rationalized by two different models. The first one is a conventional picture of interacting micelles where the parameter to be optimized is the amphiphile surface area: it foresees a monotonous increase of the aggregation number with concentration. The second one is more pertinent to the investigated system. It accounts for a conformational transition between different spatial arrangements of the bulky headgroups. Since the two conformations have different steric hindrances, their energy is coupled to the amphiphile surface area. Energy minimization leads to a complex behavior of the aggregation number, which is consistent with our experimental findings.
The following gangliosides (see Figure 1) were used: GM1 in the natural form, that is, extracted and purified as described in ref 12, and the semisynthetic derivative GM1acetyl, obtained from natural GM1 by substitution of the C18 fatty acid hydrophobic chain with a much shorter C2 chain.6 Both gangliosides were prepared in the form of sodium salts and dissolved in pure water. When necessary, the concentration of ganglioside solutions was assessed by standard colorimetric methods and thin layer chromatography (TLC). Ganglioside samples for small-angle X-ray scattering (SAXS) measurements were prepared in a wide range of concentration, from the very dilute (volume fraction φ < 0.001) to the semidilute range (0.07 < φ < 0.25), put in glass capillaries (2 mm optical path length, 0.2 mm wall thickness; VitroCom, Inc., NJ), and then sealed. Samples in the concentrated region (φ > 0.25) were put in 1 mm optical path length capillaries. SAXS experiments were performed on the D22 instrument at LURE (Orsay, France) and on the ID02 instrument at the ESRF high-brilliance synchrotron facility (Grenoble, France) in the range of momentum transfer q ) 0.015-0.3 Å-1 (wavelength of the incident beam λ ) 1 Å). Small-angle neutron scattering (SANS) experiments were performed on the PAXE instrument at LLB Orphe`e (Saclay, France) in the range of momentum transfer q ) 0.0014-0.15 Å-1 (λ ) 8 Å). Ganglioside solutions were prepared in D2O in the semidilute range of concentration (0.025 < φ < 0.21) and put in quartz cells. For both SAXS and SANS, several spectra relative to the empty cells and the solvent were taken, at all the measuring temperatures, carefully compared and subtracted. Spectra report the scattered radiation intensity I as a function of the momentum transfer q. Analysis has been carried out under the hypothesis that I(q) can be factored as I(q) ≈ P(q) × S(q), where S(q) is the static structure factor and P(q) is the form factor of the micelle. S(q) accounts for interparticle interactions, while P(q) is connected to the dimension, homogeneity, and shape of the particles.
3. Theoretical Basis 3.1. Closely Packed Spherical Micelles Made Up of “Simple” Charged Amphiphiles. In a hard-sphere suspension, the average distance R among spheres, given the size and the volume fraction φ, is uniquely defined, and at fixed φ, the smaller the particles, the higher their number, the closer their average distance. On the contrary, if particles are self-aggregating objects, like micelles, R is not uniquely defined for a given φ. In fact, the size of the particles depends on the interfacial area A required by each amphiphile, which may change in response to environmental conditions. For charged amphiphiles, for example, on raising the concentration, the repulsion among nearby micelles becomes progressively stronger: A must change in order to minimize the intermicellar repulsion. This behavior can be easily understood by calculating the total energy in the simple case of a collection of spherical charged micelles. Consider a spherical micelle of radius R and aggregation number N. By invoking substantial incompressibility of the inner hydrocarbon core, NA ) 4πR2 and NV ) 4/3 πR3, with V being the amphiphile molecular volume. The equations can be rearranged as follows:
N ) 36π
V2 A3
R)
3V A
(1)
A key contribution to the total energy arises from the interactions (10) Cantu`, L.; Corti, M.; Del Favero, E.; Raudino, A. Curr. Opin. Colloid Interface Sci. 2000, 5, 13. (11) Brocca, P.; Cantu`, L.; Corti, M.; Del Favero, E.; Raudino, A. Physica A 2002, 304, 177.
(12) Tettamanti, G.; Bonali, F.; Marchesini, S.; Zambotti, V. Biochim. Biophys. Acta 1973, 296, 160.
Anomalous Size BehaVior in Micelles
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with NA being the micelle surface and P(D) the force per unit area acting on it. If curvature effects are ignored, P(D) can be approximated as13
P(D) ≈ 2o
Figure 3. Radial distribution function g(r) around a central micelle set at r ) 0. The dashed line is the ideal distribution function approaching the amphiphile concentration φ at large r. The full line approximately reproduces the case of a lattice-like arrangement of the surrounding micelles.
among different self-assembled objects. In order to calculate the mutual interactions, we have to express the intermicellar distance R as a function of the number M of micelles and amphiphile concentration φ. Define the volume fraction of amphiphile, φ, and solvent, 1 - φ, as φ ) VA(VS + VA)-1 and 1 - φ ) VA(VS + VA)-1, with VA and VS being the amphiphile and solvent volumes, respectively. Clearly, VA ) MNV, where NV is the micelle volume. On the other hand, a concentrated micellar suspension can be depicted as a lattice of M identical cells, with each of them containing one micelle in the center. The cell volume is Vcell ) 1/M(VS + VA). From the above expressions, Vcell ) NV/φ. By symmetry, the distance between the centers of two adjacent cubic cells equates the cell edge R. Since Vcell ) R3, we find R (NV/φ)1/3; inserting this result into eq 1, one obtains
R)
V 36π 1/3 A φ
( )
(2)
The above equation is exact for a face-centered cubic lattice. Different geometries simply rescale the functional dependence of R on φ. Once R has been obtained, we make a step further by calculating the energy of interaction between a micelle set at the center of a radial coordinate system and the surrounding ones. In dilute micellar suspensions, the radial distribution function g(r) approaches the dashed line of Figure 3: g(r) ) 0 for r e R and g(r) ) φ for r > R. In concentrated systems, we may approximate the distribution function g(r) by concentric amphiphile shells of thickness near the micelle diameter 2R: the probability of finding a micelle is zero below a critical distance D > R (Figure 3, full line). This solidlike picture is more realistic at high micellar concentration. The thickness D is rather arbitrary; we made a reasonable choice by setting D as the shortest distance between the surfaces of near-neighbor micelles in a lattice of spacing R; hence, D ≈ R - 2R. From eqs 1 and 2
D)
6V π 1/3 -1 A 6φ
[( )
]
(3)
The total energy of interaction is written as
ETOT INT ) NA
∫D
∞
P(D) dD
(4)
(ZekT) K 2
2
(5)
where and o are the dielectric permittivities of solvent and free space, respectively, k is the Boltzmann constant, T is the absolute temperature, e is the electron charge, and Z is the valence of the counterion. K is a parameter with the dimension of an inverse decay length satisfying the transcendental equation13 -2(kT/ Ze)K tan(KD/2) ) σ/o, where σ is the surface charge density. Such an equation allows us to determine K in terms of σ and D. Simple results are obtained in the limit of either small or high σ values. Since we are dealing with charged micelles, we restrict the analysis to high σ, yielding K f π/D. In that case, the pressure, eq 5, becomes P(D) ≈ D-2. Dividing by N, we obtain from eqs 3 and 4 the interaction energy per amphiphile
πkT 2 A A2 ≈Γ Ze D f(φ)
( )
EINT ≈ 2o
(6)
where f(φ) ≡ [(π/6φ)1/3 - 1] and Γ ≡ (o/3V)(πkT/Ze)2. It is worth noting that at high surface charge density σ, the present case, the electrostatic energy is independent of the micelle charge. At low densities, instead, the energy increases with σ (formula not shown). The total energy of the system is obtained by adding the intermicellar energy (eq 6) to the micellar self-energy EMIC. The simplest expression for EMIC reads14 EMIC ≈ γA + C/A, where γ is the water-micelle interfacial tension and C > 0 is the repulsion among the heads
ETOT ) EMIC + EINT ) γA +
A2 C +Γ A f(φ)
(7)
Above, but still near, the critical micellar concentration (cmc), the amphiphile concentration is extremely low, and micelles essentially behave as independent objects. Hence, at φ f cmc ≈ 0, the last term of eq 7 drops out, and minimization of eq 7 yields limφf0 Ao ) (C/γ)1/2. We expand eq 7 around Ao, a procedure permissible because the intramicellar forces are stronger than the intermicellar ones. After minimization, we obtain A/Ao ≈ [1 - 2ΓAo/γf(φ)]1/2. Introducing A into eq 1, the aggregation number N as a function of the amphiphile volume fraction φ reads
(
N ≈ No 1 -
2ΓAo γf(φ)
)
-3/2
(8)
with No being the aggregation number of the isolated micelle. In passing, we mention that numerical minimization of eq 7 yields results similar to those obtained by eq 8. It must be recalled, however, that eq 8 is valid when R ) 3V/A e lMAX (with lMAX being the maximum length of the amphiphile). When the micelle size grows above lMAX, the energy sharply increases, and the system evolves toward different geometrical structures (e.g., the sphere f cylinder f lamella transition). 3.2. Densely Packed Spherical Micelles Made Up of “Bistable” Amphiphiles. In this section, we generalize the above model to the case of interacting spherical micelles made up of (13) Briscoe, W. H.; Attard, P. J. Chem. Phys. 2002, 117, 5452. (14) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1995; Chapter 16.
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amphiphiles that may exist in two different spatial arrangements. Henceforth, we will call these aggregated structures bistable. Such a situation is rather common in bilayer-type aggregates, where the hydrophobic tails of the amphiphiles give rise to either an ordered (gel phase, Lβ) or disordered (fluid phase, LR) structure, with this latter being a superposition of many disordered conformations of comparable energy.15 Less common is the situation where bistability lies in the headgroups. As we report in the Introduction, an important class of glycolipids, the gangliosides, exhibit a complex bistable behavior that involves conformational transitions both in the hydrocarbon core (like the LR f Lβ transition of phospholipid bilayers) and in the headgroup region. The presence of two (or more) energy minima in the conformation of the headgroups introduces additional phenomena. An interesting effect is that conformational transitions occurring at the surface of the aggregates are particularly sensitive to force variations occurring in the aqueous phase, such as, for instance, to the presence of nearby charged lamellae.9,10 In that respect, the bistability of the inner hydrophobic chains is much less responsive, unless of a strong coupling between the inner and outer regions of the selfaggregate. The studied molecules, gangliosides, carry out bulky and charged oligosaccharide heads, which may pack in aggregates with different lateral hindrances. In fact, molecular modeling and NMR data16-18 suggest that, along with some rigid and “undeformable” regions, flexible spots are present where different allowed torsion angles result in molecular arrangements of comparable energy but different packing hindrances. Two different arrangements are shown in Figure 2, R and β, with the latter corresponding to stronger repulsive contribution. Let PR and Pβ be the concentrations of the R and β headgroup arrangements and -1 < η < 1 be the excess of concentration from the statistical value 1/2; then, PR ) 1/2(1 - η) and Pβ ) 1/2(1 + η). The total energy of a lattice of interacting micelles can be written as the sum of three terms: the lateral interactions within the isolated micelle (EMIC), the conformational energy of the headgroups (ECONF), and the energy of interaction among different micelles (EINT): ETOT ) EMIC + ECONF + EINT. By a straightforward generalization of the model developed in section 3.1, we write down the micellar self-energy per amphiphile, EMIC ≈ γA + C/A, as
1 EMIC ) γA + (WRRPR2 + WββPβ2 + 2WRβPRPβ) A ) γA +
1 C 1 + ∆η + Jη2 A 2
(
)
Besides the interactions described above, depending on the lateral area, the amphiphile headgroups have their own conformational energy, ECONF, that depends on the R/β ratio but does not change with the amphiphile surface area
ECONF ) kT(PR log PR + Pβ log Pβ) + PRhR + Pβhβ 1 ) const + kT[(1 - η) log(1 - η) + 2 (1 + η) log(1 + η)] + ηh
(10)
where k is the Boltzmann’s constant and T the absolute temperature. The first term describes the entropy gain on mixing together R and β monomers, while h ≡ 1/2(hR - hβ) is the energy difference between the two arrangements in the isolated amphiphile molecule. Finally, the conformation-dependent interaction energy between nearby micelles, EINT, can be described by eq 6. This formula, however, must be implemented in order to consider the modulation of the intermicellar repulsion through the different head arrangements. Indeed, in micelles carrying out charged and bulky heads, monomers in the R arrangement protrude toward the aqueous medium more than those lying in the β arrangement where the head is nearly parallel to the micellar surface. Then, the effectiVe intermicellar distance Deff differs from D (eq 3): Deff ) D + 2dη, where d > 0 measures the variation in the distance between the heads’ center of charge and the micelle surface on going from R to β arrangement. Setting the reference state at η ) 0, we find that Deff < D for a micelle lying in the R arrangement (η ) -1), while Deff > D when the heads are in the β arrangement (η ) +1). Such a generalization slightly modifies eq 6 for the micellar interaction energy
EINT ≈ 2o
πkT 2 A A2 ≈Γ Ze Deff f(φ) + χηA
( )
(11)
where χ ≡ d/3V. Adding together the three contributions eqs 9-11 and minimizing the total energy by ∂ETOT/∂A ) 0 and ∂ETOT/∂η ) 0, one obtains
γ-
2f(φ) + χηA 1 C 1 + ∆η + Jη2 + ΓA )0 2 2 A [f(φ) + χηA]2
(
)
(12a)
kT 1+η C log + (∆ + Jη) + 2 1-η A (9)
where γA accounts for the usual interfacial tension effect, while the term in parenthesis describes the mean-field interactions among nearest amphiphiles lying either in R or β arrangement. This latter effect can be described by introducing an arrangementdependent repulsion energy Wij > 0, which depends on the particular conformation pair we are dealing with. The parameters ∆ and J are related to Wij by the relationships C ≡ 1/4(WRR + Wββ + 2WRβ), ∆ ≡ (WRR - Wββ)/2C, and J ≡ (WRR + Wββ 2WRβ)/2C; it can easily be shown19 that ∆ g 0, while no restriction can be put in principle on J: a negative J value means cooperative interactions among the heads, whereas in the case of conformationindependent repulsion energy, ∆ ) J ) 0. (15) Cevc, G.; Marsh, D. Phospholipid Bilayers; Wiley: New York, 1987; Chapter 11. (16) Rodgers, J. C.; Portoghese, P. S. Biopolymers 1994, 34 (10), 1311. (17) Bock, K. FIDIA Res. Ser. 1986, 6, 47. (18) Sonnino, S.; Cantu`, L.; Corti, M.; Acquotti, D.; Venerando, B. Chem. Phys. Lipids 1994, 71, 21 (and refs therein).
h - Γχ
A3 ) 0 (12b) [f(φ) + χηA]2
The system of strongly nonlinear algebraic eqs 12 can be solved numerically; simple expressions are obtained in the relevant cases listed below. a. Low Surfactant Concentration (φ φcrit
TC(φcrit) > T
(17)
Equation 17 gives the variation of N with concentration φ through the function f(φ) (eq 6). The first term on the right-hand side foresees a slow, monotonous increase of N with φ, as in the equations obtained at low and high φ (eqs 13 and 14). In addition, a new decreasing function of φ appears, with a sharp discontinuity at φ ) φcrit. This term, lacking in regular micelles, is related to the heads cooperative transition under the field of the surrounding micelles. In the opposite case TC(φ) < TC, the phase transition disappears, and the variation of N becomes softer; the overall trend, however, is similar to that discussed above. In that case, analytical formulas are cumbersome; it is better to numerically solve eqs 12 as done in the next section. 3.3. Predicted Trends and Numerical Evaluations. The above estimates enable one to grasp the physics of the process but are inadequate to give quantitative results. They are obtained by numerically solving eqs 12a,b and combining the results with eq 1. Reasonable values for the parameters entering in the equations are as follows. The interfacial tension γ ≈ 20‚10-3 J‚m-2,21 the temperature T ) 298 K, the univalent couterion charge Ze ≈ 1.6‚10-19 C, and the vacuum and water dielectric permittivities o ≈ 8.85‚10-12 and ≈ 78, respectively. The GM1 molecular volume was 2.0‚10-27 m3 as deduced from molecular weight and density data.22 Next, we calculated the -2/3 -2/3 - NMAX )/2 dimensionless interaction parameters: ∆ ) (NMIN -2/3 -2/3 -2/3 -2/3 -2/3 No and J ) (NMIN + NMAX- 2No )/No defined in section 3.2, where NMAX and NMIN are the surface areas when all the heads are in the R or β arrangement, while No is the aggregation number of an ideal micelle containing the same amount of R and β conformations. In dilute micellar solutions, most of the heads are in the R conformation; therefore, NMAX approaches the aggregation number of the isolated micelle: NMAX f N|φf0 ≈ 300. The cooperativity parameter J is extremely sensitive to the numerical value of No, turning from positive (in the case of ideal 1/2 mixing, Nideal o ) (NMAXNMIN) ) to negative (in the case of preferential interactions among heads in the same spatial arrangement). In the following, we will consider J and No adjustable parameters. According to NMR, molecular modeling, and theoretical16-18 findings in isolated gangliosides, the energy difference between R and β arrangements is extremely small in comparison with the intra- and intermicellar forces. Recent molecular dynamics simulations23 show that, in phospholipid micelles doped with a small amount of GM1, where the strong repulsion among the bulky ganglioside heads is abolished, the preferred conformation is the β one. Since NMR spectra of either GM1 in solution or GM1 dissolved in phospholipid micelles are similar, one may guess that this is a property of the isolated ganglioside. In order to take into account this tiny conformational preference, we set h ≈ 1 Kcal/mol ≈ -2kT. Finally, d measures the mean expansion of the micelle charged corona accompanying the R f β packing transition. From the GM1 molecular structure, d should be on the order of a few angstroms; we set d ) 1.5 × 10-10 m in all calculations. Using the above estimates, we numerically calculated24 from eqs 12a,b the conformational population η (Figure 4A) and the aggregation number N (Figure 4B) as a function of the amphiphile volume fraction φ.
(20) Landau, L. D.; Lifsits, E. M. Statistical Physics; Pergamon Press: New York, 1985; Chapter 14.
(21) Jonsson, B.; Wennerstrom, H. J. Colloid Interface Sci. 1981, 80, 482. (22) Corti, M.; Cantu`, L.; Salina, P. AdV. Colloid Interface Sci. 1991, 36, 153. (23) Vasudevan, S. V.; Balaji, P. V. J. Mol. Struct. 2002, 583, 215. (24) Wolfram, S. Matematica: a System for Doing Mathematica by Computer; Addison-Wesley: New York, 1996; Chapter 1.5.
N(φ) ≈ NMAX 1 -
2ΓAMIN γf(φ)
(13)
3 where NMAX ) 36πV2/AMIN is the aggregation number when all the heads are in R arrangement and f(φ) ≡ (π/6φ)1/3 - 1. Equation 13 shows a modest monotonous increase of N with amphiphile concentration φ. b. High Surfactant Concentration. In concentrated micellar suspensions, the intermicellar forces are strong, then the bulky heads lie in the less protruding β arrangement (η ≈ +1) and the head surface area is near its maximum value AMAX. Following the same procedure outlined above, we expand eqs 12 in powers of A and η around AMAX and η ) +1, obtaining
(
N(φ) ≈ NMIN 1 -
)
2ΓAMAX γf(φ)
-3/2
(14)
3 where NMIN ) 36πV2/AMAX is the aggregation number when all the heads are in the β arrangement. Equations 13 and 14 are similar, with both showing a monotonous increase of the Nth concentration. c. Intermediate Surfactant Concentration. At intermediate surfactant concentration, the situation is far more interesting. Here, lateral interactions within the micelle (favoring the R arrangement) are comparable with the intermicellar forces (favoring the β arrangement). Therefore, large variations of conformational population η associated to small variations of the intermicellar repulsion are expected. Expanding eq 12b in powers around A0 and η ) 0 up to the cubic terms, we find
3 η3 - η [Tc(φ) - T] + H(φ) ) 0 T
(15)
where Tc(φ) ≡ -(Ao/k)[γJ + 2Γχ2A3of -3(φ)], and H(φ) ≡ (3/ kT)[γAο∆ + h - ΓχA3of -2(φ)]. Equation 15 is nothing but the Landau-Ginzburg equation for a cooperative transition under a field H.20 Here, H measures the difference between intra- and intermicellar interactions, with these latter depending on the surfactant concentration φ. Therefore, on rising φ, the field H(φ) turns from positive to negative. Near the turning point, a first-order transition from the R to β arrangement holds, provided TC(φ) > T. The critical concentration φcrit required to induce the transition is found by setting H(φcrit) ) 0, yielding
[ (
ΓχAo3 π φcrit ) 1 + 6 γAο∆ + h
)]
1/2 -3
(16)
The sharp variation of the headgroups population at φ f φcrit is ≈ 2(3/Τ)1/2 [TC(φcrit) - T]1/2. The accompanying variation of the surface area (eq 12a) and aggregation number N (eq 1) is calculated to the leading terms
[
N(φ) ≈ No 1 -
2ΓAo γf(φ)
]
-3/2
+ ∆η(φ)
where
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Figure 5. SAXS spectra relative to GM1 micellar solutions (R arrangement) at different ganglioside volume fractions: 0.001, 0.004, 0.04, 0.08, 0.12, 0.22, and 0.25 (identified by decreasing lowest q intensity) in the low q region. All of the curves show a minimum in the same position.
Figure 4. Panel A. Calculated headgroup conformational population η of a bistable spherical micelle as a function of the amphiphile volume fraction φ. The two curves correspond to different values for the dimensionless cooperativity parameter J, defined in the text, J ) -0.1 (a) and J ) 0.0 (b). Panel B. Calculated mean aggregation number N vs φ. Dashed curve: ideal micelle. Full curves: bistable micelles, for different values of J, as in panel A.
In the absence of any rearrangement of the headgroup hindrance, the ideal case η ) 0, the increasing concentration produces larger micelles (Figure 4B, dashed line). More intriguing is the case of bistable micelles where the heads may change their packing arrangement. We observe three different regimes: • At low micelle concentration, the mutual repulsion is small; the aggregation number is near that of the isolated micelle. • At intermediate concentration, the head conformational population varies to minimize the repulsion among the micelles. The number of those monomers protruding in water decreases, while the number of shallower heads increases as shown in Figure 4A. The outcome is the increase of the surface area and the decrease of the aggregation number. This behavior is more evident in the case of nonideal mixing between the different headgroup arrangements. Near the ideal mixing case, the decrease of N with φ is continuous (Figure 4B, curve b); on the contrary, when like-like interactions are stronger than the like-dislike ones, the variation of N is wider and occurs through a first-order phase transition (curve a). • Finally, at high concentration, the intermicellar repulsion is so strong that basically only the shallower β arrangement is present. Any further reduction of the intermicellar distance no longer changes the head populations. Hence, bistable micelles behave again like ideal ones, with their size increasing with concentration. The aggregation number, however, is smaller than in the dilute regime.
4. Experimental Results 4.1. GM1 Behavior. Figure 5 shows the evolution of the SAXS spectra of GM1 micellar solutions on increasing concentration. Repulsive interactions among micelles become more and more important, and the structure factor contribution S(q) to the total intensity I(q) ≈ P(q) × S(q) becomes clearly visible. P(q) is the form factor, carrying information about the micellar size. For a nearly spherical particle, P(q) has a pronounced
Figure 6. SAXS spectra relative to GM1 in the two different packing arrangements: state R (thin line) and state β (thick line), in the noninteracting case (0.001 ganglioside volume fraction, in 100 mM NaCl solution).
minimum at a q value depending on its radius. A displacement of qmin toward higher values indicates a reduction in particle dimension, and vice versa. As an example, Figure 6 reports SAXS I(q)’s relative to GM1 micelles in the two different packing arrangements, recalled in Table 1 and Figure 2, in the noninteracting case (very dilute, high ionic strength). In this limit, S(q) ) 1 at all q values and I(q) is simply proportional to P(q). The displacement of qmin corresponding to a difference of 10% in radius (see Table 1) is clearly visible (as well as the variation in the depth of the minimum of I(q) corresponding to an increased sphericity of the micelle). From Figure 5, then, it can be seen that all over the L1 phase the GM1 micelle dimension does not change, nor does the aggregation number, from the very dilute region up to 0.25 volume fraction. An independent confirmation of this result comes from an estimate of the GM1 intermicellar distance as concentration is varied, as extracted from neutron scattering (SANS) data in the same range. SANS spectra are less structured than SAXS ones, due to the difference in the contrast profile of the micelle as seen by the two radiations, resulting in a smoother SANS P(q).25 From the SANS I(q), the S(q) peak position can be extracted for each GM1 concentration. The position of the peak qmax reflects a characteristic length (d ) 2π/qmax) that scales with the distance (25) Cantu`, L.; Corti, M.; Del Favero, E.; Dubois, M.; Zemb, T. J. Phys. Chem. B 1998, 102, 5737.
Anomalous Size BehaVior in Micelles
Figure 7. Panel A. SANS spectra of GM1 micellar solutions (R arrangement) at different ganglioside volume fractions: 0.024, 0.08, 0.12, 0.16, and 0.20 (identified by structure factor peak moving from left to right). Panel B. Swelling behavior of the characteristic lengths d, corresponding to the scaling of the structure factor peak position qmax: log-log plot of φ vs d. The fitting line corresponds to a slope s ) -3. For comparison, a dashed line with slope s ) -2 is also reported. Panel C. SANS spectra relative to GM1 micelles in the two different packing arrangements: state R (open symbols) at φ ) 0.12 and state β (full symbols) at φ ) 0.08. In the two cases, the micelle number concentration is the same.
among the interacting neighboring micelles in solution. Figure 7A reports SANS spectra relative to GM1 micellar solutions in the one-decade range 0.024-0.2 volume fraction at 25 °C in the R-packing arrangement. In Figure 7B, the corresponding d values are reported versus the lipid volume fraction φ in a log-log scale. In this representation, the slope s of the fitting line corresponds to the swelling exponent of the collection of particles, φ ) ds. The observed swelling behavior is three-dimensional (slope -3 in the figure), typical for globular objects of fixed dimension. We then again conclude that GM1 micelles keep the same dimension and aggregation number over the entire concentration range. It is interesting to note, as shown in Figure 7C, that the peak position of the φ ) 0.08 solution of β-packed GM1 micelles coincides with the one relative to the φ ) 0.12 solution of R-packed micelles, as expected on the basis of their aggregation numbers (see Table 1). At higher GM1 volume fraction, SAXS measurements at 25 °C indicate the presence of liquid-crystalline cubic ordering. From the positions (relative and absolute) of all the observed diffraction peaks in the spectra, the symmetry and the dimension of the lattice parameter can be calculated. From 0.27 to 0.4 volume fractions, the Pm3n lattice for disconnected micelles has been detected.26 In this region, the mean aggregation number, as calculated according to the Fontell model for Pm3n,27 is dramatically lower than in the L1 phase. The sharpness of the aggregation number transition is evident in Figure 8 where the behavior of N ) N(φ) relative to GM1 micelles is reported in the entire investigated concentration range. Moreover, after the sharp transition, a slight increase in the aggregation number is observed as concentration is raised within the cubic phase. 4.2. GM1acetyl Behavior. Table 1 reports the micellar parameters for GM1acetyl. It can be seen that GM1acetyl does not display any multiple spatial arrangement. Figure 9 reports the aggregation number, N(φ), of GM1acetyl as a function of volume fraction φ in the range 0.0007-0.26, as obtained from SAXS results. The aggregation number and the size of the micelles increase as expected and usually found in micellar systems of charged amphiphiles.1 The same results are confirmed by the swelling behavior of the characteristic length d, as obtained by SANS measurements (see Figure 10). The experimental results (26) Boretta, M.; Cantu`, L.; Corti, M.; Del Favero, E. Physica A 1997, 236, 162. (27) Fontell, K.; Fox, K. K.; Hansson, F. Mol. Cryst. Liq. Cryst. Lett. 1985, 1, 9.
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Figure 8. Micellar aggregation number (state R) as a function of GM1 volume fraction φ. Open diamond from ref 25. Dashed lines are drawn to guide the eye.
Figure 9. Micellar aggregation number for GM1acetyl as a function of ganglioside volume fraction φ. Dashed lines are drawn to guide the eye.
Figure 10. Log-log plot of φ vs characteristic length d for GM1acetyl micellar solution. Full line (slope s ) -3) indicates a threedimensional swelling of objects of constant size.
lie on a straight line with slope s ) -3, corresponding to a three-dimensional swelling/crowding of objects of constant size, only in the low concentration range, φ < 0.076. Above, the characteristic distance decreases less than expected, as happens if the size of the micelles increases with concentration.
5. General Discussion and Conclusion The presented theoretical model and experimental results fit well with one another. Looking at Figure 8, we may easily identify the three regimes foreseen by our model: the dilute and the concentrate regions, where the headgroup bistability does not play a crucial role on increasing concentration at constant temperature, and the anomalous region at intermediate concentration, where a sharp decrease of the aggregation number related to the headgroups bistability is found. The curves have the same shape as the theoretical ones reported in Figure 4B. If one bears in mind the oversimplified nature of the model, the agreement between theory and experiments is more than satisfactory. On the other hand, in very small micelles (GM1acetyl), due to the high curvature, the bulky heads of the gangliosides are likely to also be accommodated in the wider-packing, slightly more favored β conformation. The concentration-dependent field induced by the surrounding micelles, which favors the shallow β conformation too, therefore has little effect on the conformational population variation of the micelle surface. GM1acetyl
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then behaves like ideal micelles. Indeed, the experimental data of Figure 9 relative to GM1acetyl micelles practically show the ideal behavior predicted by the dashed curve of Figure 4B. It is surprising how well the rough model we have developed also reproduces the position of the aggregation number transition. Using either eq 16 or the more accurate numerical data reported in Figure 4B, we find that the transition occurs at φcrit ≈ 0.330.34, a value in good agreement with the experimental findings φexp crit ≈ 0.27 of Figure 8. On the basis of the combined theoretical and experimental findings, we may conclude that the anomalous behavior of the micellar size with concentration observed in ganglioside assemblies reflects the conformational adaptability (bistability) typical of polysaccharide chains. Other structures that may adopt different conformations, associated to different intermicellar potentials, can, in principle, give rise to comparable behavior. The anomalous variation of N is related to the micellar concentration, with the exact spatial arrangement of the nearest micelles playing only a modest role. Indeed for GM1, the sharp drop of N occurs at φ ) 0.27; at this concentration, the system enters into the Pm3n cubic phase. However, another ganglioside, GT1b, carrying a bulkier head than GM1 and currently being investigated in our laboratory, shows the drop of N at φ ≈ 0.24, just before the micelles reach the crystalline ordered phase (φ ≈ 0.26) (data not reported). Theoretical and experimental curves are in good agreement. The main deficiency of the model is the smooth increase of the aggregation number at low concentration, just before the sharp drop of N at a critical concentration (Figure 4B, curve a), while the experimental data show a flatter curve at low φ (Figure 8). The theoretical behavior follows from the assumption of a spherical shape. In that case, the only way to relieve the intermicellar repulsion is the variation of the aggregation number. Real GM1 micelles are elliptical.28 At low concentration, they can modify their mutual orientation in order to lower the repulsion
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forces (a mechanism which requires a low energy cost), while at higher concentration, the only way to do this is variation of the aggregation number. The use of a spherical approximation does not allow any orientational rearrangement, with the only considered mechanism being the variation of the aggregation number. This simplified model overestimates the variation of N at low concentration. The anomalous concentration behavior of the aggregation number is strongly modulated by the cooperativity among the gangliosides headgroups: the higher the cooperativity, the sharper the drop of the aggregation number (see Figures 4B and 8). The cooperativity of the GM1 ganglioside saccharidic surface in noninteracting micelles has already been proven by investigating the behavior of mixed micelles where the bulky GM1 is diluted by smaller phospholipid molecules.29 According to the classical picture of self-assemblies, the aggregation number of spheroidal micelles cannot grow beyond a critical value dictated by the maximum extension of the hydrocarbon chains. When the critical size has been reached, micelles adopt a different geometry. The anomalous behavior of the aggregation number N investigated here ensures a smoother growth of N with concentration φ (or even a decrease in a narrow concentration range). Therefore, spherical micelles, either randomly distributed (L1) or ordered in a three-dimensional array (Pm3n cubic phase) may exist over a wide concentration range, where regular surfactants totally change their aggregation state. Again, this peculiar behavior is the result of ganglioside headgroup bistability. Acknowledgment. We thank the beamline responsibles at ESRF (ID02), LURE (D22), and LLB (PAXE) for their precious technical support. LA0630864 (28) Corti, M.; Boretta, M.; Cantu`, L.; Del Favero, E.; Lesieur, P. J. Mol. Struct. 1996, 383, 91. (29) Cantu`, L.; Corti, M.; Del Favero, E.; Raudino, A. J. Phys.: Condens. Matter 2000, 12, A321.
