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Model for Formation and Growth of Vesicles in Mixed Anionic/Cationic (SOS/CTAB) Surfactant Systems Akihisa Shioi† and T. Alan Hatton* Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Received March 18, 2002. In Final Form: June 21, 2002 The dynamics of vesicle formation and growth following the mixing of cationic cetyltrimethylammonium bromide (CTAB) and anionic sodium octyl sulfate (SOS) surfactant solutions were determined using static and dynamic light scattering. The results are well-described by a two-stage model in which initial nonequilibrium vesicles are generated rapidly following the mixing of the surfactant solutions and the subsequent vesicle growth is by slower vesicle fusion. The formation of the initial vesicles is based on a balance between the unfavorable edge energy of disklike aggregates and the bending energy required to form spherical structures. The sizes of the initial vesicles correlated well with the equilibrium sizes determined after 2 months, in accordance with the model predictions. The growth of both the disklike bilayers (bilayer-bilayer fusion) before their transition to the initial vesicles and of the nonequilibrium vesicles (vesicle-vesicle coalescence) was modeled using Smoluchowski’s theory for colloid coagulation. Estimates of the coalescence rate constants showed that the fusion of bilayer disks is at least an order of magnitude faster than the coalescence of nonequilibrium vesicles, with rate constants of ∼200 and ∼5-10 (Ms)-1, respectively.
Introduction Mixtures of anionic and cationic surfactants exhibit rich microstructured phase behavior in aqueous solution,1 and aggregate structures such as spherical and rodlike micelles, vesicles, lamellar phases, and precipitates have all been observed depending on the concentrations of the two surfactants in solution.2 The potential applications of these microstructured fluids have led to a strong interest in their overall properties, with the result that ternary phase diagrams are now available for a number of aqueous mixtures of anionic and cationic surfactants, including the sodium octyl sulfate (SOS)/cetyltrimethylammonium bromide (CTAB),1 SDS/DTAB,2 and SDBS/CTAT3 systems. However, while the range of equilibrium microstructures spanning single-component micelles to vesicles has been mapped satisfactorily in many cases, the dynamics of these transitions are only poorly understood. We are concerned here with systems in which oppositely charged surfactants are mixed to fall within the vesicle region of the phase diagram. Vesicles, in particular, have received much attention lately, and an understanding of the dynamic behavior of these vesicle systems could be important if they are to be exploited effectively in applications such as models for biological membranes, agents for fragrance and flavor encapsulation, selective separations aids, vehicles for controlled release applications, and microreactors for the preparation of inorganic nanoparticles, for instance. Dynamic light scattering studies indicate that aggregates larger than the original micelles in the surfactant solutions prior to mixing grow slowly with time, while the time dependency of the light †
Present address: Department of Chemistry and Chemical Engineering, Engineering, Yamagata University, Jonan 4-3-16, Yonezawa 992-8510, Japan. (1) Yatcilla, M.; Herrington, K. L.; Brasher, L. L.; Kaler, E. W.; Chiruvolu, S.; Zasadzinski, J. A. J. Phys. Chem. 1996, 100, 5874. (2) Herrington, K. L.; Kaler, E. W.; Miller, D. D.; Zasadzinski, J. A.; Chiruvolu, S. J. Phys. Chem. 1993, 97, 13792. (3) Kaler, E. W.; Murthy, A. K.; Rodriguez, B.; Zasadzinski, J. A. Science 1989, 245, 1371.
scattering intensity also depends on the initial compositions of the two surfactant solutions prior to mixing. The aggregate growth begins almost immediately after the mixing of the two solutions, but it can take months for the vesicles to reach their equilibrium sizes.1 Similar behavior has been observed for micelle/vesicle transitions triggered by the addition of salts or ionic surfactants to other micellar solutions.4-6 It has been suggested that the time dependency of the scattering intensity and the aggregate sizes in the SOS/ CTAB systems is due to the formation of nonequilibrium vesicles during the early stages of the process and to the subsequent growth by coalescence of these vesicles until their final equilibrium size is reached. Since fusion of the bilayer is needed for this coalescence, it takes a very long time to form the equilibrium vesicles. The relationship between the scattering intensity and the aggregate size suggests the presence of such nonequilibrium vesicles,7 although the reasons for their formation and mechanisms for their growth have not yet been elucidated. We draw on a series of experimental and theoretical observations reported on other systems in recent years to propose here a simple model that captures both the early formation of small, nonequilibrium vesicles and the initial stages of the growth of these vesicles in the SOS/CTAB system. NMR studies8 show that cationic and anionic surfactants can be tightly bound by electrostatic interactions to form “complexes” composed of oppositely charged surfactants. The resulting decrease in the average area occupied by a headgroup induces the transition from spherical (or cylindrical) aggregates to disklike objects, in (4) Farquhar, K. D.; Misran, M.; Robinson, B. H.; Steytler, D. C.; Morini, P.; Garrett, P. R.; Holzwarth, J. F. J. Phys.: Condens. Matter 1996, 8, 9397. (5) Brinkmann, U.; Neumann, E.; Robinson, B. H. J. Chem. Soc., Faraday Trans. 1998, 94, 1281. (6) Samuel, E. C.; Zhang, Z.; Friberg, S. E.; Patel, R. Langmuir 1998, 14, 590. (7) O’Connor, A. J.; Hatton, T. A.; Bose, A. Langmuir 1997, 13, 6931. (8) Salkar, R. A.; Murkesh, D.; Samant, S. D.; Manohar, C. Langmuir 1998, 14, 3778.
10.1021/la020268z CCC: $22.00 © 2002 American Chemical Society Published on Web 08/20/2002
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accordance with the well-known packing structure criterion of Israelachvili et al.9 Such disklike precursors for the vesicles have been observed in transmission electron microscopy (TEM) studies on phospholipid systems10-12 and in cryo-TEM studies on the SOS/CTAB system.13 The addition of salt to screen electrostatic repulsion between the surfactant headgroups in sodium 6-phenyltridecanesulfonate micelles was found to induce spontaneous vesiculation,4 again likely due to changes in the packing factor. Clearly, then, the formation of bilayers as the result of a modification of the effective headgroup area (i.e., the packing structure factor) must be an important initial step in the formation of the vesicles, followed by the folding over of these bilayers to form the closed vesicle structures. The bending of the floppy bilayer to form a closed vesicle is opposed to some extent by the imposition of an elastic bending energy penalty as the curvature is increased. The floppy bilayer has an unstable edge, however, and the fine balance between the edge energy and the bending energy essentially determines the shape of the surfactant assembly. In some phospholipid systems, it has been shown that surfactants that control the activity of the edge of disklike bilayers can change the stable sizes of the aggregates and the onset of the micelle/vesicle transitions.10,11 It has been suggested that in order to diminish the total edge energy of the aggregates, the floppy bilayers coalesce with each other until the aggregates reach a critical size at which the effect of the unstable edge overcomes the effect of the bending elastic energy. Therefore, vesicles with sizes smaller than the thermodynamically favored equilibrium size are formed despite the unfavorable bending energy. In other studies, it has been observed that small vesicles bud off from larger giant lipid vesicles such as biological cells,14 a phenomenon which can also be attributed to a balance between the edge and elastic energies of the bud.15,16 The nonequilibrium vesicles thus formed no longer have unstable edges, but the unfavorable bending energy remains. When two vesicles collide, some are able to overcome an energy barrier to fusion, and a larger vesicle will be formed at the expense of the smaller vesicles participating in the fusion event. The properties of surfactant membranes, for example, the rigidity, probably dominate the rate process for this fusion, as suggested by Chizmadzhev et al.17 We draw on these experimental and theoretical insights to propose a model for the formation of nonequilibrium vesicles and their subsequent growth in the SOS/CTAB system and use static and dynamic light scattering to provide experimental measurements of the aggregate growth. The model correlates well the initial sizes of the nonequilibrium vesicles with their equilibrium sizes and captures the time dependency of the growth of the vesicle sizes and their size distributions. (9) Israelachvili, J. N.; Mitchell, D. J.; Ninham, R. W. J. Chem. Soc., Faraday Trans. 1 1976, 72, 1525. (10) Fromherz, P.; Ruppel, D. FEBS Lett. 1985, 179, 155 (11) Fromherz, P.; Rocker, C.; Ruppel, D. Faraday Discuss. Chem. Soc. 1986, 81, 39. (12) Edwards, K.; Gustafsson, J.; Almgren, M.; Karlsson, G. J. Colloid Interface Sci. 1993, 161, 299. (13) Xia, Y.; Goldmints, I.; Johnson, P. W.; Hatton, T. A.; Bose, A. Langmuir 2002, 18, 3822. (14) Alberts, B.; Bray, D.; Lewis, J.; Raff, M.; Roberts, K.; Watson, J. D. Molecular Biology of the Cell; Garland Publishing: New York, 1988. (15) Lipowsky, R. J. Phys. II France 1992, 2, 1825. (16) Lipowsky, R. Physica A 1993, 194, 114. (17) Chizmadzhev, Y. A.; Cohen, F. S.; Shcherbakov, A.; Zimmerberg, J. Biophys. J. 1995, 69, 2489.
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Figure 1. Schematic of the intermediate states during the formation and growth of vesicles in the SOS/CTAB system. (1) SOS monomers enter the CTAB micelles to form nonspherical mixed micelles that grow by coalescence to form (2) floppy bilayer membranes that then curve on themselves to form (3) closed bilayer vesicles of initial diameter D0. (4) Subsequent vesicle growth is by coalescence.
Materials and Methods Materials. Sodium octyl sulfate (99%) obtained from Lancaster Synthesis, Inc., and hexadecyltrimethylammonium bromide (CTAB, >99%) from Sigma Chemical Co. were used without further purification. Surfactant solutions were prepared on a weight basis using Milli-Q water. Light Scattering. Light scattering techniques were used to monitor the changes in aggregate properties with time following the mixing by hand shaking of equal volumes of the two surfactant feed solutions. Dynamic light scattering measurements were performed on a Brookhaven model BI-200SM instrument (Brookhaven Instrument Corp.) with a Lexel 8W argon ion laser (Lexel Laser, Inc., Freemont, CA) operating at 514 nm. The data were analyzed by the cumulant and the inverse Laplace transformation methods using a BI-9000AT digital correlator (Brookhaven Instrument Corp.). Time-resolved light scattering intensities at 276 nm were determined on a QuantaMaster luminescence spectrometer (Photon Technology International Inc., PTI) at a scattering angle of 90°. Each measurement was completed within 60 s. All time-resolved measurements were made during the first few hours after the mixing of the surfactant solutions. The temperature was maintained at 25 °C by a recirculating water bath. The final, equilibrium sizes of the vesicles were determined after the solution mixtures had aged for 2 months.
Model for Growth of Nonequilibrium Vesicles The formation and growth of vesicles following the mixing of the two surfactant solutions are assumed to follow the sequence of events shown in Figure 1.7 During the early stages of the growth process, mixed micelles grow into floppy bilayers that then fold over to form vesicles of initial size D0, a process which can take up to tens of minutes to occur. Once formed, the vesicles grow by coalescence until they attain their equilibrium size distribution. In the sections that follow, we first propose a model for the formation of the initial nonequilibrium vesicles and then suggest a closed-form description of vesicle growth assuming that a monodisperse population of these initial vesicles forms rapidly, that is, we neglect the finite growth rates of the floppy bilayers. In the final section, we relax this last assumption to incorporate also the finite growth rate of the prevesicular aggregates, or floppy bilayers.
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Figure 2. Cryo-TEM photograph of an SOS/CTAB solution 8 min after mixing. A mixture of closed vesicles, floppy bilayers, and spherical caps is clearly evident. As the bilayers and spherical caps grow, they fold over to form closed vesicle structures.
Formation of Nonequilibrium Vesicles. We assume that the size of the initial vesicles, which themselves are in a nonequilibrium state, is determined by the thermodynamic stability of the growing floppy bilayers. Prior to mixing, the CTAB in solution is almost entirely incorporated in spherical micelles (very low critical micelle concentration (cmc)), while the SOS is present primarily as individual surfactant molecules in solution, since the cmc for SOS is high, at about 3 wt %.1 Upon mixing of these two solutions, the SOS molecules penetrate the CTAB micelles to form mixed micellar aggregates in which the close association of the anionic and cationic surfactants causes a decrease in the average area occupied by the surfactant headgroups, yielding a packing factor of close to unity. This change in packing effectiveness of the surfactants favors the formation of floppy bilayer-type structures which characteristically have high edge energies owing to the unfavorable packing constraints there. These edge energies can be minimized either by growth of the floppy bilayers to reduce the overall edge length relative to the membrane area or by bending of the bilayers to form spherical caplike structures with smaller peripheries, which eventually close up on themselves to form vesicles. This mechanism for vesicle formation has been proposed by Lasic18-20 and Fromherz et al.10,11,21 Bilayer membranes and disklike aggregates have been shown to be precursors for vesicle formation as shown by TEM for some phospholipid systems10-12 and recently for the SOS/ CTAB system discussed here; Figure 2 shows a typical cryo-TEM photograph in which the curved disks can readily be identified.13 The critical condition for the bilayer to form the vesicle can be formulated using the theory of Lipowsky15,16 for shape transformations of bilayer membranes. Figure 3 shows a schematic representation of the floppy bilayer, which we model as a spherical cap. The bending energy for this bilayer with surface area πL2, curvature C, and bending energy κ is given by
Ebend ) 2πκ(LC - LCsp)2
(1)
where Csp is the spontaneous or equilibrium curvature for the bilayer. For the spherical cap geometry shown in Figure 3, the edge energy is
Eedge ) λ2πLx1 - (LC/2)
2
(2)
where λ denotes the edge energy per unit length. The (18) Lasic, D. D. Biochim. Biophys. Acta 1982, 692, 501. (19) Lasic, D. D. J. Theor. Biol. 1987, 124, 35. (20) Lasic, D. D. Biochem. J. 1988, 256, 1. (21) Fromherz, P. Chem. Phys. Lett. 1983, 94, 259.
Figure 3. Model for the initial formation of vesicles. The edge energy of a floppy bilayer of area L2 can be reduced by curving the bilayer to reduce the edge length, but this is at the expense of gaining elastic curvature energy. At some critical bilayer area, Lcr2, the curvature energy associated with a closed vesicular structure will be less than the edge energy and the vesicle will form spontaneously.
Figure 4. The bilayer energy as a function of the curvature parameter (LC) and the bilayer size (L). As the bilayer grows (L increases), the energy barrier to vesicle formation disappears and the vesicle will form spontaneously. The structures associated with different values of the curvature parameter are shown.
total energy is given by the sum of these two contributions and is, in dimensionless form,
E′ ) (Ebend + Eedge)/2πκ ) [LC - Φ(L/ξ)]2 + (L/ξ)x1 - (LC/2)2 (3) where ξ ) κ/λ and Φ ) Cspξ. The total energy, E′, is shown in Figure 4 as a function of curvature and membrane area for a fixed value of Φ. For small aggregates, there is clearly a maximum between C ) 0 (representing a flat membrane) and C ) 2/L (a closed vesicle with no edge) which acts as an energy barrier for vesiculation, inhibiting vesicle formation. The floppy bilayer grows in area to diminish the total edge energy for the aggregate, resulting in an increase in the length of its unstable edge. At a certain critical membrane area, the energy barrier vanishes and the floppy bilayer closes spontaneously to form a vesicle. The critical condition is given by15
( ) ∂E′ ∂(LC)
)0
and
Φ,L/ξ
( ) ∂2E′ ∂(LC)2
)0
(4)
Φ,L/ξ
From eqs 3 and 4, we obtain
Lcr ) 8ξ/[1 + (4Cspξ)2/3]3/2
(5)
where Lcr represents L at the critical condition and is the
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Figure 5. The correlation of final vesicle size with the initial vesicle size. The curve has a slope of unity, in accord with theoretical predictions, and the intercept yields the ratio of the edge to bending energy parameters. Results are those of Yatcilla et al. (ref 1) (open symbols) and this work (closed symbols). The two triangle points are outliers.
diameter of the initial vesicle, that is, D0 ) Lcr. The spontaneous curvature, Csp, is equal to 2/Deq where Deq represents the equilibrium size of the vesicles. When ξ approaches infinity (a rigid bilayer), the initial vesicles of the equilibrium size form immediately, and there is no further growth. For a finite value of ξ, however, eq 5 gives D0 < Deq, an inequality that is consistent with the experimental results. Equation 5 can be rewritten as
D0-2/3 ) Deq-2/3 + (8ξ)-2/3
(6)
which relationship can be verified experimentally using dynamic light scattering results as shown in Figure 5, where D0 is the size determined 500 s after the mixing of the surfactant solutions and Deq is measured 2 months after this mixing. The data of Yatcilla et al.1 for the hydrodynamic radii of initial and aged vesicles in the SOS/ CTAB systems are also shown in Figure 5. Equation 6 is well satisfied for both sets of data (excluding the two outliers shown by triangles). The value of ξ calculated from the intercept of Figure 5 is 6.9 nm, which is close to the approximately 10 nm estimated for this parameter for phospholipid membranes where κ and λ are known to be on the order of 10-19 J and 10-20 J/nm, respectively.20 The agreement between the model predictions and the experimental results is encouraging and, bolstered by the credible value obtained for the ξ parameter, lends support to this suggested mechanism for initial vesicle formation. Model for Growth of Nonequilibrium Vesicles. Mechanisms proposed for vesicle growth following the formation of the initial nonequilibrium vesicles are generally based on either the fusion model, in which two vesicles coalesce to create a larger one, or on Ostwald ripening, in which larger vesicles grow at the expense of smaller vesicles having higher curvature energies. In this latter case, monomeric surfactants released by the smaller vesicles by desorption and/or by complete dissolution of these vesicles are incorporated by the larger vesicles, which grow until they reach their final equilibrium distribution. The bimodal distribution of vesicle sizes dictated by the Ostwald ripening process was not detected in our light scattering studies, and we have therefore rejected this model in favor of the vesicle fusion model in our analysis below. The coalescence process has many features in common with the coagulation of colloidal particles, differing primarily in that while colloidal particles form fractal objects on coagulation and the individual particles retain
their identities even when in their coagulated state,22 the fusion of any two vesicles necessarily results in their loss of identity as the new vesicle is created. In colloidal coagulation, the aggregates can grow to be infinite in size after sufficient time, although this is not always the case, while the vesicles have an upper size limit as reflected in their equilibrium size distribution. The time required to attain the equilibrium vesicle size is a few months,1 which is considerably longer than the time range of at most a few hours investigated here. The present work, therefore, treats the very early stages of vesicle growth, and the effect of equilibrium size constraints on vesicle growth is neglected in our analysis. We assume that at time t ) 0 vesicles of a given size are formed and that through the growth process we obtain n-mers that contain n of these initial vesicles. The well-known equation of Smoluchowski for colloid coagulation can then be used to describe the vesicle coalescence and growth processes:23
dNn dt
∞
∑ j)1
) -Nn
Kn,jNj +
1n-1
∑ Kn-j,jNn-jNj
(7)
2 j)1
where Nn denotes the number density of aggregates comprised of a number n of the initial vesicles, that is, of n-mers, and Kn,m represents the rate constant for the coagulation between n-mers and m-mers. For the present purposes, we assume Kn,m to be a constant, K, independent of n and m, which permits eq 7 to be solved analytically to yield23
(KN0t)n-1 Nn ) N0 (1 + KN t)n+1
(8)
0
Here, N0 denotes the number density of initial vesicles formed just after the mixing of the two surfactant solutions. We assume that the aggregates corresponding to the n-mer are also vesicles with surface areas nA0, where A0 represents the surface area of an initial vesicle. Then, conservation of the interfacial area provides, to a first approximation,
Dn ) D0n1/2
(9)
where Dn represents the diameter of the n-mer vesicles. The light scattering intensity In from the n-mer vesicles is, therefore,
(KN0t)n-1 n2 In ∝ Nn(Dn2)2 ) N0D04 (1 + KN0t)n+1
(10)
This intensity distribution as a function of vesicle size can be compared directly with that obtained from dynamic light scattering experiments. The mean diameter DZ of vesicles based on the scattering intensity distribution (Zaverage) is ∞
DZ )
/
∞
(KN0t)n-1
∞
∑ DnIn n)1 ∑ In ) D0 n)1 ∑ n)1
n+1
(1 + KN0t) ∞
∑ n)1
/
n5/2
(KN0t)n-1 (1 + KN0t)n+1
n2 (11)
The light scattering intensity from the solution is (22) Witten, T. A.; Cates, M. E. Science 1986, 232, 1607. (23) von Smoluchowski, M. Phys. Z. 1916, 17, 557, 585.
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Assuming Csp ) 0 for the order estimation of Echange, we obtain Echange = 8πκ. The bending energy, κ, for bilayer membranes is of the order of 10-19 J, which gives Echange = 610kBT, a value so large that vesicle breakage to form two smaller vesicles is highly unlikely. Interactions between the vesicles have similarly been ignored in the above developments. Since the diameters of the vesicles are much smaller than the wavelength of the incident light, the effect of these interactions on the light scattering intensities is reflected in S(0), the structure factor at zero scattering angle. If the nonequilibrium vesicles behave as hard spheres and S(0) can be calculated from the equilibrium equation of state, then according to the Carnahan-Starling equation,
S(0) ) (1 - φ0)4/(1 + 4φ0 + 4φ02 - 4φ03 + φ04)
Figure 6. Comparison of model fits with experimental measurements of (a) vesicle diameter and (b) scattered intensity as a function of time since mixing 1.22% CTAB and 2.77% SOS surfactant solutions. Note that the model assumes that initial vesicles of size D0 are formed immediately; the poor model fits at short times are attributed to the finite formation rate of initial vesicles which is not incorporated in this model.
obtained from ∞
I(t) ∝
∞
∑ In ∝ N0D0 n)1 ∑ 4
n)1
(KN0t)n-1
n2
n+1
(1 + KN0t)
(12)
The time-dependent mean vesicle size for one surfactant mixture is shown with the model fit (eq 11) in Figure 6a. The model captures the experimental growth in vesicle size reasonably well, except for the first few hundred seconds, and the parameters extracted from this fit were used to predict the intensity variations (Figure 6b) and the evolution of the vesicle size distribution (Figure 7) over the same time period. It is clear that the model also captures the intensity variation trends well except at the very short times, where the scattering intensities are overpredicted. It is likely that over this initial period it is not the growth of vesicles that is monitored but rather the growth of the vesicle precursors, that is, of the mixed micelles and floppy disks that eventually fold over to form the closed vesicle structures. The kinetics of these early processes is substantially different from that of the vesicle growth process and therefore is not captured here; we discuss this point later. In eq 7, the scission of a vesicle to form two vesicles of smaller sizes is not taken into account. The change in the bending elastic energy of vesicles upon coalescence is approximated as
Echange ) 4πκA[(C - Csp)2 - ((1/x2)C - Csp)2]
(13)
where C and A represent the curvature and the interfacial area, respectively, of the vesicle bilayer before the coalescence, and Csp denotes the spontaneous curvature.
(14)
The volume fraction φ0 is the sum of the water and surfactant volume fractions. If all of the surfactant molecules are in vesicles and the average area occupied by a headgroup is about 0.34 nm2, which is estimated from surface tension data,1 the volume fraction of water entrapped in the vesicles is φ ) (ΣNS/πD2)(πD3/6), where Σ and NS are the headgroup area and surfactant concentration, respectively. The range of φ0 values observed in our experiments was 0.03-0.1, corresponding to S(0) values ranging from 0.8 to 0.5. These results overestimate the interaction effects, however, as we have not taken into account the fluctuations in the vesicle shape about the average spherical geometry, nor the attractive interactions between the vesicles which increase S(0) and may compensate for the decrease in S(0) due to the hard-sphere interactions such that S(0) is close to unity. In any event, it is not the absolute value of S(0) that is important but rather its relative variation during the course of an experiment; if S(0) can be assumed to be constant during any particular experiment, then it will cancel out whenever ratios of sums of intensities are calculated. For any given set of experimental conditions, less than a 2-fold change in vesicle volume fraction was observed during the measurement period, leading to at most a 15% change in S(0). Consequently, at the current level of analysis, we are justified in ignoring the effects of these interactions on the analysis of vesicle growth rates. Vesicle Growth Model Including Vesicle Formation Process In the above discussions, we have assumed two stages for the vesicle evolution: the initial vesicle formation, which is well described by eq 6, followed by the growth of these vesicles by coalescence and fusion according to eq 7. In the growth model, it was implicitly assumed that the initial vesicle formation was rapid and completed almost instantaneously in the very early phases of the vesicle growth process. It is clear from the experimental measurements of the size evolution in these systems that the pure growth stage of the vesicles only begins after several (10-15) minutes. Thus, for a more complete description of the growth processes, it is necessary to account for the early stages during which the growth of the floppy bilayers, their transition over to closed initial vesicles, and the growth of these vesicles by fusion occur simultaneously. Thus, we assume that the floppy bilayers grow by coalescence and fusion until they reach a critical size, at which point they spontaneously fold over to form initial vesicles, which then grow by fusion with other vesicles and perhaps floppy bilayers in solution. A population balance for aggregates with a continuous size distribution
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Figure 7. Evolution of the vesicle size distribution. The solid symbols were estimated using the inverse Laplace transformation method, and solid bars represent model calculations using parameters estimated from the fits to the mean size.
yields the equation
dN(s,t) ∞ ) -N(s,t) 0 K(s,s′) N(s′,t) ds′ + dt 1 s K(s,s′) N(s - s′,t) N(s′,t) ds′ (15) 2 0
∫ ∫
where N(s,t) is the concentration of surfactant aggregates with interfacial area s at time t. K(s,s′) is the rate constant for the coalescence of s- and s′-aggregates, for which we assume the size dependency to be
K(s,s′) )
{
Kfloppy for s,s′ < scr Kves for s > scr or s′ > scr
(16)
Here
scr ) πLcr2
(17)
is the critical area for vesicle formation, such that the aggregates are vesicles for s > scr and floppy bilayers for s < scr. We estimate Lcr using eq 5 with Csp ) 2/Deq and ξ ) 6.9 nm. When both aggregates to be fused are floppy bilayers, the fusion rate constant is Kfloppy. On the other hand, K ) Kves when at least one of the two aggregates is a vesicle. In general, as we shall see later, Kves is smaller than Kfloppy. To compare model predictions with experimental observations, the scattering intensity from a solution containing floppy bilayers and vesicles was calculated using
I(t) ∝
∫0∞ s2N(s,t) ds
(18)
while the mean diameter of aggregates was approximated by
Dmean ) (
∫0s
cr
s2N(s,t)x4s/π ds +
∫s∞ s2N(s,t)xs/π ds)/∫0∞ s2N(s,t) ds cr
(19)
where N(s,t) is the number concentration of aggregates of area s at time t provided by the solution to eq 15. The two
Figure 8. Comparison of model fits with experimental measurements of (a) vesicle diameter and (b) scattered intensity as a function of time since mixing 1.22% CTAB and 2.77% SOS surfactant solutions. Note that the model fits at short times are satisfactory owing to allowance for the finite rate of formation of initial vesicles in the model.
parameters Kfloppy and Kves were adjusted to ensure good agreement between the experimental and predicted intensity variations with time and then used in eq 19 to predict the time-dependent growth of the aggregate size without further parameter adjustment. Figure 8 shows examples of the calculated scattering intensity and the predicted mean diameter following mixing of the two surfactant solutions. The Runge-Kutta algorithm was used in the solution of eq 15, with initial condition N(s0,0) ) ΣNS/s0 where s0 is the area of the bilayer generated immediately after the mixing of the SOS and CTAB solutions, that is, of the initially formed mixed micelles. The results were not particularly sensitive to s0, which we took to be 20 nm2, independent of solution
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respectively, and T is temperature. Since Khard is about 7.4 × 109 (Ms)-1 at 298 K in water, one fusion of vesicles takes place about every 109 collisions. Overall Process of Mixed Micelle Formation and Vesicle Evolution
Figure 9. Rate constant for the continuous distribution model as a function of the total surfactant concentration. The floppy bilayer fusion rates are more than an order of magnitude greater than the vesicle coalescence rates.
concentrations. The model is able to capture the initial steep rise in the scattering intensity, corresponding to the rapid growth of the floppy bilayers, which was not incorporated in the discrete model used earlier. The scattering intensity is more sensitive to the presence of these nonvesicular aggregates than is the mean aggregate size, although once the vesicles have formed, their growth can be reasonably described by the discrete model for which simple analytical expressions can be obtained. Figure 9 shows the parameters Kfloppy and Kves that were extracted from the model fits to the experimental scattering intensity measurements as functions of the total surfactant concentration. The vesicle growth rate, reflected in the rate constant Kves, is smaller than the growth rate, Kfloppy, of the prevesicular aggregates by at least an order of magnitude. It would seem that Kfloppy is almost independent of the total surfactant concentration, while Kves decreases slightly with surfactant concentration. This dependence of the rate constants on the total surfactant composition of the solution compositions cannot be interpreted in the absence of knowledge of the bilayer composition, which can be different from the bulk surfactant composition in the system.13,24 It has been suggested that the bilayer is almost equimolar in the anionic and cationic surfactants, with all of the cationic surfactant and perhaps a slight excess of the anionic SOS surfactant present in the membrane, the remaining SOS being dispersed molecularly in the bulk solutions. This slight negative charge on the surfaces of the vesicles is not expected to provide a significant barrier to coalescence, as when two vesicles approach, the excess charges on their surfaces will repel each other and move away from the approach points. Thus the approach points on the vesicles will be free of charge, and it is possible that the line tension change caused by the induced gradient in the surface concentration could lead to instability in the bilayer that would facilitate coalescence. The differences in the rates during the two stages of the process can be reasonably explained in terms of the relative ease with which two aggregates can fuse. For the bilayers, this fusion is simply an attachment of one aggregate to another with a resultant reduction in unfavorable edge energy. For vesicles to coalesce, however, it is necessary for the two closed bilayers to open before fusing to form a single bilayer, which provides an additional energy barrier to the coalescence process; this may be facilitated by redistribution of the charges in the bilayer as described above. As seen in Figure 9, Kves is in the range of 1-10 (Ms)-1. For a hard-sphere fluid, the bimolecular collision factor is given by Khard ) 8000RT/(3η) (Ms)-1 where R and η denote the gas law constant and solvent viscosity, (24) Brasher, L. L.; Kaler, E. W. Langmuir 1996, 12, 6270.
The discussion thus far has focused on the formation and growth of the floppy bilayers and the vesicles and has not addressed the issues of what happens in the very short time frame following the mixing of the solutions. The change in scattered light intensity following the rapid mixing of two surfactant solutions in a stopped flow system was reported earlier by O’Connor et al.7 Prior to the mixing of the solutions, the sample cell contained an older sample, which was flushed out during the mixing process and replaced with fresh mixture. The stopped flow dead time was about 4 ms. In the absence of growth of the micelles, the scattering intensity would have remained at the base level indicated for a CTAB solution alone. O’Connor et al.7 showed, however, that the initial intensity depended on the surfactant concentrations, consistent with there being some initial process faster than 4 ms leading to the growth of scattering objects. As no processes with time constants in the millisecond to second range were detected, the initial process of mixed micelle formation must have been over in the dead time of the apparatus. We conclude that during this stage of the process, the SOS is taken up by the CTAB micelles to form mixed micelles of, or at least close to, the final equilibrium composition (it has been shown using different approaches that the composition ratio is approximately equimolar); these mixed micelles may be either rod or spheroidally shaped. It is also reasonable to assume that the number of micelles remains unchanged from the initial CTAB micelle concentration, as coalescence of these micelles can be assumed to be quite slow. The next stage of the process is the coalescence of these micelles to form larger disklike lamellar structures, which grow until they are no longer favored energetically and form closed, vesicle-like structures; this process has a time constant on the order of minutes. The vesicles themselves grow by fusion as discussed above, over periods of hours and days. Thus the process of mixed micelle initiation, precursor formation, and subsequent vesicle formation and growth is a multiscale process with well-separated time scales. When two vesicles fuse, the surface area is conserved, and barring the direct encapsulation of additional water during the fusion process, the total volume is less than that of a more energetically favorable spherical vesicle. Phospholipid bilayers are permeable to water, and therefore it is reasonable to assume that the SOS/CTAB bilayers will similarly be permeable such that the enclosed water volume can increase by simple permeation through the membrane. The time constant for this process can be estimated as PA/V, where P is the bilayer permeability, which we take to be on the order of 3 × 10-4 cm/s based on known values for phospholipid blayers,25 and A and V are the vesicle area and volume, respectively. For 30 nm vesicles, this time constant is on the order of milliseconds to tens of milliseconds, and thus our assumption that the vesicles immediately take on their spherical shape on formation is acceptable. Conclusions We have discussed the formation and subsequent growth of vesicles in mixed anionic/cationic surfactant (25) Walter, A.; Gucknecht, J. J. Membr. Biol. 1986, 90, 207.
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systems in terms of a combination of Lipowsky’s membrane mechanics model to describe vesicle formation and the Smoluchowski formulation for colloid coagulation to capture the dynamics of the subsequent vesicle growth. During the early stages of the process, mixed micelles grow to form floppy bilayer disks with large edge energies. As the disks grow, they distort to form spherical caplike structures to minimize the edge energy at the expense of curvature energy. Beyond a certain size, however, these disks cannot sustain the large edge energies relative to the decreasing curvature energy and fold over on themselves to eliminate the edges by forming small, closed nonequilibrium vesicles; this process takes place shortly (within several minutes) after the mixing of SOS and CTAB solutions. The subsequent vesicle growth was captured in terms of a population balance analysis in which the vesicles were assumed to grow by coalescence and fusion. The vesicle formation model captures well the predicted relationship between the sizes of the initial vesicles and the equilibrium vesicles obtained after 2 months incubation and provides an estimate of the relative
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magnitudes of the two energy contributions to the disk energetics. The initial time-dependent growth of the floppy bilayer disks and the subsequent development of the vesicle size and size distributions with time were both described well by the vesicle growth model. The rate constant for the vesicle coalescence is very low compared to the bimolecular collision constant for hard-sphere fluids, consistent with membrane fusion being the controlling factor in vesicle growth. The initial growth of the floppy bilayers prior to vesicle formation, by contrast, is significantly faster, with rate constants at least an order of magnitude larger than for vesicle fusion processes. Acknowledgment. We thank Paul Johnston and Arijit Bose of the University of Rhode Island for providing the cryo-TEM images. A.S. is grateful to the Japan Society for the Promotion of Science (JSPS) for fellowship support. Partial support of this work was provided by the Singapore-MIT Alliance (SMA). LA020268Z
