Langmuir 2003, 19, 4573-4581
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Formation and Growth of Anionic Vesicles Followed by Small-Angle Neutron Scattering I. Grillo,*,† E. I. Kats,†,‡ and A. R. Muratov†,§ Institut Laue Langevin, 6 rue Jules Horowitz, F-38042 Grenoble, Cedex 9, France, L. D. Landau Institute for Theoretical Physics, RAS 117940, Kosygina 2, Moscow, Russia, and Institute for Oil and Gas Research, RAS, 117701, Gubkina 3, Moscow, Russia Received October 24, 2002. In Final Form: February 14, 2003 We present the first kinetic small-angle neutron-scattering experiments carried out on the instrument D22 (Institut Laue Langevin, Grenoble) with a stopped-flow apparatus (Bio-Logic company). D22 offers unique features for real-time experiments. The high flux and the large q range covered in only one instrumental configuration combined with the rapid electronics open up possibilities for few hundred millisecond resolution measurements. We have used these technical developments to study the formation and growth of spontaneous vesicles after addition of salts (NaCl, NaBr, KCl, and KBr) in a micellar solution of AOT in D2O from 500 ms to 5 h after mixing. The vesicle radii and the growth rate depend on the salt concentration and decrease with increasing ionic strength. The driving force of the transition is the screening of the electrostatic repulsion between adjacent surfactant headgroups that favors formation of a locally planar bilayer. Assuming that the aggregation is controlled by micelle diffusion, a simple kinetic approach predicts that the average radius increases with the power law R ∝ t1/6, in close agreement with the experimental data.
Introduction Surfactants in water exhibit a rich structural behavior. Stability and microstructure result from the interplay of attractive and repulsive forces between colloids. A small variation in the intermolecular force balance (pH, temperature or ionic strength variation, dilution, mixing, or application of a magnetic field) may induce a reorganization of the system, such as phase separation or a structural phase transition. The steady state of many surfactant systems is now well characterized,1 with kinetic data becoming available due to new experimental techniques. Models on intermediary phases and structures are still under development. The knowledge of the phase evolution and possible intermediate species is crucial to understand and modify the characteristics of the final state. Scattering experiments provide powerful method to explore colloidal structures from the scale of a few angstro¨ms to thousands of angstro¨ms. Time-resolved static and dynamic light-scattering experiments have been developed, and kinetic experiments have only recently been performed on intense synchrotron SAXS diffractometers. Fundamental questions of soft condensed matter,2-4 polymer science, inorganic material growth,5 and †
Institut Laue Langevin. L. D. Landau Institute for Theoretical Physics. § Institute for Oil and Gas Research. ‡
(1) Evans, D.; Wennerstro¨m, H. In The Colloidal Domain where Physics; Chemistry; Biology and Technology Meet; Advances in Interfacial Engineering Series; VCH Publishers: Weinheim, Germany, 1994. Laughlin, R. In The Aqueous Phase Behavior of Surfactant; Surfactant Science Series; Academic Press: New York, 1994. (2) Egelhaaf, S. U.; Schurtenberger, P. Phys. Rev. Lett. 1999, 82, 2804-2807. (3) Egelhaaf, S. U.; Schurtenberger, P.; Morris, J.; Olsson, U.; Wennerstro¨m, H. In ILL Annual Report; Bu¨ttner, H. G., Lelie`vre-Berna, E., Pinet, F., Eds., 1997. Egelhaaf, S. U.; Olsson, U.; Schurtenberger, P.; Morris, J.; Wennerstro¨m, H. Phys. Rev. E 1999, 60, 5681-5684. (4) Schmo¨lzer, S.; Gra¨bner, D.; Gradzielski, M.; Narayanan, T. Phys. Rev. Lett. 2002, 88, 258301-1-258301-4. (5) De Moor, P.; Beelen, T.; Komancheck, B.; Diat, O.; Van Santen, R. J. Phys. Chem B 1997, 101, 11077-11086. Ne´, F.; Testard, F.; Zemb, Th.; Petit, J.-M. ESRF Newslett. 1999, 33, 23-25.
also biological systems6,7 have found answers. Real-time measurements are now also possible using SANS. The new generation of SANS diffractometers, such as D22 at the ILL (Institut Laue Langevin, Grenoble), have very high fluxes at the sample position (up to 108 neutrons/ s/cm2) and allow short acquisition times of the order of a few hundreds of ms.8 A large dynamic q range, up to qmax/ qmin ) 20, can be covered in only one instrument configuration. Moreover, SANS experiments offer the possibility of solvent or molecular labeling to follow only the structural rearrangements of one particularly chosen compound. Vesicles are single bilayer shells and have been subject to increased interest during the past decades. Used as model for biologic membranes, they have found also applications as microreactor or for drug encapsulation and delivery. The challenge in formulation is to find inexpensive stable vesicles with well-defined sizes. Common but energetically costly methods to obtain vesicles are sonication, dialysis, extrusion of lamellar suspensions, or reverse-phase evaporation,1 but in most cases these methods form aggregates that are unstable and highly polydisperse. Spontaneous formation of vesicles is widely described in the literature: a nonexhaustive list is found in refs 9-22, with systems generally composed of two (6) Roessle, M.; Manakova, E.; Lauer, I.; Nawroth, T.; Gebhardt, R.; Narayanan, T.; Heumann, H. ESRF Newslett. 1999, 33, 10-11. (7) Pe´rez, J.; Defrenne, S.; Witz, J.; Vachette, P. Cell. Mol. Biol. 2000, 46, 937-948. (8) http://www.ill.fr/YellowBook/D22/ and http://www.ill.fr/YellowBook/D22/D22_info/. (9) Viseu, M. I.; Edwards, K.; Campos, C. S.; Costa, S. M. B. Langmuir 2000, 16, 2105-2114. (10) Kaler, E. W.; Murthy, A. K.; Rodrigez, B. E.; Zasadzinski, J. A. N. Science 1989, 245, 1371-1374. (11) Bergstro¨m, M. Langmuir 1996, 12, 2454-2463. Bergstro¨m, M. Langmuir 2001, 17, 7675-7686. (12) Marques, E. F.; Regev, O.; Khan, A.; Da Grac¸ a Miguel M.; Lindman, B J. Phys. Chem. B 1998, 102, 6746-6758. Marques, E. F.; Regev, O.; Khan, A.; Da Grac¸ a Miguel M.; Lindman, B J. Phys. Chem. B 1999, 103, 8353-8363. (13) McKelvey, C. A.; Kaler, E. W.; Zasadzinski, J. A. N.; Coldren, B.; Jung, H.-T. Langmuir 2000, 16, 8285-8290. (14) Yuet, P. K.; Blankschtein, D. Langmuir 1996, 12, 3819-3827. (15) Edwards, K.; Almgren, M. Langmuir 1992, 8, 824-832.
10.1021/la0208732 CCC: $25.00 © 2003 American Chemical Society Published on Web 04/24/2003
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Figure 1. Schematic view of the path of the solutions filling the three syringes through the delay lines and mixing chambers up to the observation cell.
surfactants (cationic/cationic,9 catanionic,10-14 nonionic/ ionic mixtures,15 or addition of cosurfactants in ionic mixtures16-18). These spontaneous vesicles are thermodynamically more stable, and theories based on molecular packing and bending energy approaches have been developed.19-22 Experimental and Method Stopped-Flow Apparatus. The general interest of a stoppedflow apparatus is to control the mixing of several solutions within a short time window (10-90 ms) and to know precisely the delay between the time of mixing and the beginning of observation. The SFM-3 from the Bio-Logic company (Bio-Logic, 1 rue de l’Europe, 38640 Claix. WWW: http://www.bio-logic.fr/rapidkinetic/.) handles three stainless steel syringes each of 20 mL driven by independent stepping motors and two mixing chambers. The vertical syringes are filled from the top of the apparatus. This geometry simplifies the evacuation of air bubbles that may form in the syringes during filling. A schematic picture of the liquid path is shown in Figure 1. The first mixing chamber is situated after syringes 1 and 2, followed by a delay line whose volume determines the evolving time between the two mixing chambers tET. The second mixing chamber at the end of the delay line mixes the solution (1+2) with the solution 3. The final mixture (1 + 2 + 3) is stopped in the observation cell for measurement. By definition, the dead time tDT is the time between the mixer 2 and the middle of the cell. The minimal time to begin the observation after mixing is tET + tDT and is on the order of 50200 ms depending upon the flow rate and the way of use of the syringes. The observation head has been specially designed for neutronscattering applications. A rectangular Hellma cell of 1 mm path and 0.2 mL has been adapted to the stopped-flow geometry. The solution is filled from bottom to top of the cell.8 A 6 × 9 mm2 aperture placed at the end of the collimation guide sets the size of the direct beam at the sample position. The sequence of mixing defines volumes and times. The minimal volume is 20 µL per syringe, and the maximal flow rate is 2 mL/s to avoid turbulence in the cell. Due to the large volume of the observation cell and its geometry, at least 600 µL of solution (3 times the cell volume) is necessary to empty completely the old solution and replace it by a fresh one. (16) Bergstro¨m, M.; Eriksson, J. C. Langmuir 1996, 12, 624-635. (17) Herve´, P.; Roux, D.; Bellocq, A.-M.; Nallet, F.; Gulik-Krzzywicki, T. J. Phys. II France 1993, 3, 1255-1270. (18) Gradzielski, M.; Mu¨ller, M.; Bergmeier, M.; Hoffmann, H.; Hoinkis, E. J. Phys. Chem. B 1999, 103, 1416-1424; Gradzielski, M.; Bergmeier, M.; Hoffmann, H.; Mu¨ller, M.; Grillo, I. J. Phys. Chem. B 1999, 103, 11594-11597. (19) Safran, S. A.; Pincus, P.; Andelman, D. Science 1990, 248, 354356. (20) Safran, S. A.; Pincus, P. A.; Andelman, D.; MacKintosh, F. C. Phys. Rev. A 1991, 43, 1071-1078. (21) Yuet, P. K.; Blankschtein, D. Langmuir 1996, 12, 2; 3802-3818. (22) Jung, H. T.; Coldren, B.; Zasadzinski, J. A.; Iampietro, D. J.; Kaler, E. W. PNAS 2001, 98, 1353-1357.
Grillo et al.
Figure 2. Dilute part of the AOT phase diagram according to ref 27. L1: isotropic micellar phase. LR: lamellar phase. L3: sponge phase. The arrows show the compositions of the samples investigated. At any time of the sequence of mixing, the stopped-flow electronics can send if required a TTL (0-5 V) signal to the instrument workstation starting the sequence of acquisition. A sequence of acquisition defines the number of spectra and the time of each one. Time (s) is converted into ticks (1 tick ) 90.9 ns) allowing a precise time control. A new electronics card especially designed for rapid kinetics experiments can store temporarily up to 450 spectra, with a maximal time preset per run of 391 s. Physical System. We used sodium bis(2-ethyl hexyl)sulfosuccinate, usually called AOT. The general formula is C20H37O4SO3-‚Na+. AOT was purchased from Fluka and used as received. The density is 1.13, molar mass 444.57 g/mol, and molecular volume vm ) 650 Å3. The critical micellar concentration (cmc) at pH ) 8-9 determined by surface tension measurements is 6.3 × 10-4 mol/L.23 Using Gibbs law, we calculate an area per polar head of 86 Å2 at the air/liquid interface, by explicitly counting one surfactant plus its counterion (supposed to be 100% bound) as one molecule.24 The phase diagram of AOT in water or in brine (NaCl solutions) has been studied for more than 25 years.25-27 The dilute part of the phase diagram according to ref 27 is shown in Figure 2. The mass and volume fractions of AOT are w and Φ, respectively. In H2O, the following phases are formed. Between the cmc (6.3 × 10-4 mol/L, i.e., w ) 0.028%23) and 1.4%, a single micellar phase is formed. Between w ) 1.4% and 10%, micelles are in equilibrium with the lamellar phase at its maximum swelling. Between w ) 10% and 17%, the existence of a single-phase lamellar domain is not clear. Frances et al. have suggested the presence of a spherulite phase.28 Above w ) 17%, a single-phase LR exists. We have measured by SAXS the bilayer thickness and found δ ) 19.7 Å29 and an area per headgroup of ah ) vm/δ ) 67 Å2. According to refs 25-27, the addition of NaCl induces an equilibrium between micelles and a swollen LR phase. For a preliminary experiment, samples with 0% < w < 1% and salt concentrations cs lower than 0.17 mol/L were prepared. One day after, clear suspensions were obtained and typical scattering curves of vesicles (instead of swollen LR phase coexisting with micelles) were measured. This transition has been once reported with choline chloride salts where large objects of 1000-2000 Å diameters are formed.30 Mother solutions of AOT at 0.5 wt % and 1 wt % in D2O (35 times cmc) were prepared. We worked with four salts NaCl, NaBr, KCl, and KBr dissolved in H2O. The concentrations were 0.08, (23) Grillo, I. Insertion de particules anisotropes dans des phases lamellaires tensioactives. The`se de l’universite´ Paris XI, Paris 1998. (24) An, S. W.; Lu, J. R.; Thommas, R. K.; Penfold, J. Langmuir 1996, 12, 2446-2453. (25) Fontell, K. J. Colloid Interface Sci. 1973, 44, 156-164 and 318329. (26) Skouri, M.; Marignan, J.; May, R. Colloid. Polym. Sci. 1991, 269, 929-937. (27) Balinov, B.; Olsson, U.; So¨derman, O. J. Phys. Chem. 1991, 95, 5931-5936. (28) Frances, E. I.; Hart, T. J. J. Colloid Interface Sci. 1983, 94, 1-13. (29) Grillo, I.; Levitz, P.; Zemb, Th. Langmuir 2000, 16, 4830-4839. (30) Murthy, A. K.; Kaler, E. W.; Zasadzindski, J. A. N. J. Colloid Interface Sci. 1991, 45, 598-600.
Formation and Growth of Anionic Vesicles
Langmuir, Vol. 19, No. 11, 2003 4575 Table 1. Surfactant Parameters and Scattering Length Densities of AOT Monomers, H2O, and D2O AOT monomer volume (Å3) δT (Å) (in lamellar state23) δH (Å) (in lamellar state23) FT (cm-2) FH (cm-2) Faverage (cm-2)
Figure 3. Schematic representation of a vesicle. 0.17, 0.34, and 0.68 mol/L (NaCl only). The 0.5 wt % AOT solution was only mixed with NaCl 0.17 mol/L. The AOT and salt solutions were filled in syringes 1 and 3, respectively. Syringe 2 was filled with pure H2O and used to rinse the quartz cell. 900 µL of AOT and 300 µL of brine were mixed during 600 ms. Thus 14 samples were studied: their compositions were 0.75 wt % AOT with 0.021, 0.042, 0.085, and 0.17 mol/L (NaCl only) of salt and 0.375 wt % AOT with 0.042 mol/L of NaCl. The solvent was composed of 25% of H2O and 75% of D2O in volume. The dilution line and final compositions are shown with arrows in Figure 2. By working with syringes 1 and 3, the evolving time tET was zero (see Figure 1). The acquisitions were started just at the end of the mixing sequence, i.e., 90 ms after the beginning of mixing tDT. SANS Experiments. For static experiments two configurations were used (λ ) 8 Å, detector offset of 390 mm, D ) 17.6 m, Coll ) 17.6 m and D ) 4 m, Coll ) 5.6 m) to cover a large q range from 6.2 × 10-3 to 0.2 Å-1. Kinetic measurements were carried out at 6 Å, the peak flux of the cold source. During the first 740 s after mixing, a “middle angle” configuration was set with a sample-to-detector distance D ) 5 m and a collimation at 5.6 m (9.8 × 10-3 < q < 0.14 Å-1). For the long time evolution, the “small-angle” configuration, D ) 8 m and Coll ) 8 m (6.1 × 10-3 < q < 8.7 × 10-2 Å-1), was chosen to follow the formation of larger aggregates. Raw data were corrected for electronic background and the empty cell and were normalized by water using standard ILL software. Around 3500 spectra were collected in total. Time Scale. During the first 776 s after the beginning of the mixing, we chose a geometric series for the time sequence
tn ) an-1t1,
Tn )
1 - an t 1-a 1
(1)
where tn is the acquisition time for the frame n, Tn is the accumulated time after mixing. t1 ) 500 ms and a ) 1.1. Fiftythree frames were measured for a total time of around 776 s. The mixing process and the acquisition sequences were repeated 10 times to have adequate statistics. The quartz cell was rinsed with H2O between two injections. The raw data corresponding to the same time window in the acquisition sequence were automatically summed and stored on the detector electronics memory card so that good statistics were obtained (from 2 × 105 counts to 3 × 106 total counts on the whole detector). The long time evolution, during 5 h after mixing, was followed by 2 min measurements between 776 s and 1 h and then 3 min acquisitions until the end of the experiment. The average time after mixing TAMn for the nth runs is
Tn - Tn-1 + tDT, 2 tDT ) 90 ms (2)
654 8.1 1.9 -3.84 × 109 5.65 × 1010 6.37 × 109 H2O
(Å3)
vm FH2O (cm-2)
30 -5.59 × 109 D2O
(Å3)
vm FD2O (cm-2)
30 6.38 × 1010
Fsolvent (cm-2) (75 vol %D2O/25 vol %H2O)
4.64 × 1010
vesicle, and Fs and Fv are the scattering length densities of solvent and surfactant, respectively (Table 1). Kc is the constant contrast and increases with volume fraction and vesicle radius. Kc defines scattered intensity at q ) 0. We use the standard form factor F2(q,rext) of a spherical shell,31
F(q,rext) )
[
sin (qrext) - qrext cos(qrext) 3 V Vext - Vi ext (qrext)3 Vi
]
sin (qri) - qri cos(qri) (qri)3
(4)
where Vi and Vext are the inner and external volumes of radius ri and rext, respectively. The bilayer thickness δ ) rext - ri is taken to be 19.7 Å as the bilayer thickness measured in the lamellar state.29 δ value may be calculated from the minimum of the function eq 4 at qδ ) 4.493, i.e., q ) 0.228 Å-1 in our case. This value is at the upper limit of the experimental q range window where the scattered signal is close to the background and the minimum is not visible on the experimental curves. The experimental determined intensity suffers from a smearing of eq 3 by the size distribution of scatterers and by the instrumental resolution. The two contributions are detailed in the Appendix, and the final equation for modeling can be written as
Imodel(q) )
∫ R(q,∆q,q′)∫ ∞
0
∞
0
KcG(rext,σ,r′) F2(r′,q′) dr′ dq′ + IB (5)
The instrumental resolution R(q,∆q,q′) can be represented as a Gaussian function. The polydispersity G(rext, σ,r′) is characterized by a log-normal distribution, and IB represents a flat background. Agreement between the model and the experimental points can be estimated from a χ2 test:
χ2 )
∑[(I
model(q)
- I exp(q))/E(q)]2/(N - 2)
(6)
q
(3)
where N is the number of points in the experimental curve (ca. 80 points per instrumental setting on the D22 detector) and E(q) the statistical error of the intensity. The scattering curves are fitted using an in-house Fortran program. We only fit the form factor since the structure factor is assumed to be equal to 1. Kc is adjusted for the intensity scaling, while the parameter IB is fixed. The unknown parameters are rext and σ. The program varies rext by 0.1 Å steps and σ by 0.005 Å steps between given minimal and maximal approximate values. χ2 (eq 6) is evaluated for each iteration. The parameters giving the smallest χ2 are retained. We estimate the relative error on the radius rext at 5% for relatively low polydispersity (σ < 0.22). For higher σ values, we observe a strong smearing and a close to q-2 slope in log-log
where Φ is the volume fraction of vesicles, V is the volume of one
(31) Pedersen, J. S. Adv. Colloid Interface Sci. 1997, 70, 171-210.
TAM1 )
t1 + tDT 2
and
TAMn )
Scattering Model. Vesicles of external radius rext and inner radius ri may be schematically represented as in Figure 3. Scattering from our dilute solution of vesicles in the presence of salt does not exhibit any correlation peak such that the structure factor S(q) equals 1 in the q range investigated. The scattered intensity in absolute scale from a monodisperse suspension of spherical vesicles without interaction can be written as
I(q)(cm-1) ) ΦV(Fs - Fv)2F2(q,rext) ) KcF2(q,rext)
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Grillo et al. Table 2. Radius of the Vesicles and Standard Mean Deviation of the Log-Normal Size Distribution Law 24 h after Sample Preparation NaCl (mol/L) rext24h
(Å)
σ
0.021
0.042
0.085
0.17
396 0.22
364 0.22
316 0.22
282 0.22
NaBr (mol/L) 24h
Figure 4. SANS curves 24 h after the sample preparation. (9) AOT 1 wt % in D2O,(b) AOT 0.75 wt % in a 75 vol % D2O/25 vol % H2O solvent, (]) AOT 0.75 wt % with 0.042 mol/L NaCl, in a 75 vol % D2O/25 vol % H2O solvent, (s) best fit obtained for rext ) 364 Å and σ ) 0.22. representation of the scattering data. In this case, the fitting procedure is less convergent and the error on the radius is at least 10%. It should be noted that in the proposed scattering model we only consider scattering from the vesicles and neglect the signal coming from the micelles still present in the sample as explained in the following section. On one hand, the q ranges where the scattering from the two kinds of objects can be seen are different (Figure 4). For time-resolved measurements, the maximum q is on the order of 0.1 Å-1 such that scattering contribution from the micelles cannot be seen. Furthermore, when calculating the constant contrast Kc (eq 3), even in the case where less than 1% of micelles have been transformed into vesicles, the scattering coming from the vesicles is on the order of 10-100 times larger than that from the micelles. Thus micelle scattering represents a relatively small background contribution.
rext σ
(Å)
0.021
0.042
0.085
no vesicle formed no vesicle formed
553 0.18
380 0.25
KCl (mol/L) rext24h (Å) σ
0.021
0.042
0.085
439 0.21
399 0.21
315 0.22
KBr (mol/L) rext24h (Å) σ
0.021
0.042
0.085
523 0.23
485 0.25
458 0.25
Results Steady State 24 h after Sample Preparation. The stock solution of AOT in D2O was diluted with the salt solutions in H2O, as described in the Experimental Section. Samples were prepared 24 h before the experiment and mechanically stirred. Figure 4 shows the scattering curves of AOT in D2O (1 wt %), AOT in 75 vol %D2O/25 vol % H2O (0.75 wt %), and AOT (0.75 wt %) with NaCl 0.042 mol/L also in D2O/H2O. Without salt, one observes typical scattering patterns from small interacting spherical micelles. The decrease in intensity for w ) 0.75% is due to the dilution and the difference of scattering length densities of the solvents (D2O and 75 vol %D2O/25 vol %H2O). Using a spherical model for the form factor smeared by a log-normal law for the size distribution of the vesicles (eq 5), a radius of 16.5 Å and a standard mean deviation σ ) 0.15 are calculated. That implies an aggregation number of 29 monomers in good agreement with ref 32 and a surface per headgroup of 118 Å2. The micellar radius remains unchanged after dilution. Addition of salt changes drastically the pattern. We observe a strong increase of intensity at low q values, characteristic of larger objects. Between 4 × 10-3 and 10-2 Å-1, Figure 4 shows the signature of spherical particles: the intensity of the plateau is related to the volume and concentration of scatterers (eq 3) while the bump provides information on the radius and polydispersity. Between 10-2 and 0.4 Å-1, the q-2 slope is the signature of a flat bilayer. The data are fitted with eq 5 for the different samples. Radii and standard mean deviations σ are reported in Table 2. One has to be cautious with these results since stirring introduces energy and promotes
vesicle formation. Nevertheless we observe two tendencies. First, an increase of salt concentration decreases the vesicle radius. That behavior has been reported previously in ref 30 after addition of choline chloride in AOT solutions and also for cationic vesicles of DDAOH vesicles in the presence of NaOH.33 Second, NaBr and KBr induce globally the formation of larger vesicles. The samples kept in rest have been remeasured 1 week after the preparation. The oscillation is much less visible, but that q-2 remains over three decades in q suggests that the vesicles are in a metastable state and evolves toward open bilayers, a precursor phase of the lamellar phase formed at higher surfactant concentrations. Real-Time Measurements. Figure 5 shows the time evolution of the scattering curves during the first 776 s after mixing for the sample 0.75 wt % AOT, NaCl 0.042 mol/L. The transition from micelle to vesicle has already occurred after TAM1 ) 340 ms. The oscillation shifts toward lower q values, and the intensity of the plateau increases as growth continues. For this system and time scale investigated we do not observe the appearance of intermediate structures, such as wormlike micelles, as it is observed in other systems.2,4
(32) Kitahara, A.; Kobayashi, T.; Tachibana, T. J. Phys. Chem. 1962, 83, 3, 7132-7136.
(33) Brady, J. E.; Evans, D. F.; Warr, G. G.; Grieser, F.; Ninham, B. W. J. Phys. Chem. 1986, 90, 1853-1859.
Figure 5. Time evolution of the small-angle neutron-scattering intensity I in absolute scale (cm-1) until 776 s after mixing. AOT 0.75 wt % with 0.042 mol/L NaCl.
Formation and Growth of Anionic Vesicles
Figure 6. Example of data fitting. (4) Experimental points. (‚‚) Form factor (eq 4). (s) Best fit taking into account instrument resolution and polydispersity (eq 5). In the inset the log-normal function describing the size distribution of the vesicles. The spectra is the 30th of the series, concentration in NaCl is 0.17 mol/L. The acquisition time is t30 ) 7.93 s and repeated 10 times; the time after mixing is TAM30 ) 78.4 s. The best χ2 equals 1.2 for rext ) 105.0 Å and σ ) 0.20.
The vesicle stability in time varies with the nature and concentration of salt. Vesicles are still present 5 h after dilution with NaCl for all the concentrations studied. For some concentrations of NaBr, KBr, and KCl (interrupted curves in Figure 7), the typical scattering pattern of vesicles is replaced by a q-2 slope only 1 or 2 h after the mixing. This indicates that the vesicles have evolved into “floppy” bilayers. An example of data fitting is presented in Figure 6, and the size distribution of the vesicles is drawn in insert. The scattering curve in absolute unit is the 30th of the series; NaCl concentration after mixing is 0.17 mol/L. The counting time is t30 ) 7.93 s, and the measurement results from the sum of 10 frames equivalent to a total acquisition time of 79.3 s. The time after mixing is here TAM30 ) 78.4 s. After radial averaging, the 79.3 s of acquisition gives a good statistic as shown by the small error bars. Equations 4 and 5 are used to fit the data and are represented by dotted and full lines, respectively. The best-fitting parameters are obtained with rext ) 105.0 Å and σ ) 0.20 giving χ2 ) 1.2. The time evolutions of rext for the different salts and concentrations are plotted in Figure 7a (NaCl), 7b (NaBr), 7c (KCl), and 7d (KBr). The general features are •The speed of the growth decreases by increasing the amount of salt but is independent of the AOT concentration. •The polydispersity of the vesicles strongly increases with time and the system evolves toward equilibrium between open bilayers and micelles. •The rate of growth depends slightly on the nature of salt. From the faster to the slower growth, one finds the following sequence: NaBr, NaCl, KBr, KCl. Discussion A central question is what is at the origin of the transition from small micelles to vesicles and the mechanism of growth? Formation and stability of spontaneous vesicles has been discussed widely in the literature.19-22 Two major approaches have been developed: the curvature-elasticity approach and the molecular approach. The first theory describes the vesicle bilayer by its spontaneous curvature and its bending elasticity.20,34,35 The finite size (34) Helfrich, W. Z. Naturforsch. 1973, 28c, 693-703. (35) Helfrich, W. Z. J. Phys. 1986, 47, 321-329.
Langmuir, Vol. 19, No. 11, 2003 4577
of the aggregates depends therefore on the interplay between the two quantities, although this approach is only applicable to large vesicles. A molecular approach, based on geometric-packing arguments, has been used to describe the shape of aggregates.36 More recently, Yuet at al.21 have developed a molecular-thermodynamic model where particular attention is paid to the calculation of the free-energy change of vesicle formation. From these studies, the following and general conclusions emerge: •Vesicles composed of a single surfactant are generally unstable to either lamellar or micellar phases.21 Indeed, the curvature of the inner ci and outer cext layers have the same magnitude but opposite signs. The curvature energy per unit area at the midplane between the inner and outer layers is fc ) 2K[(c + cext)2 + (c - ci)2], with K an effective bending constant related to the bending rigidity κ and the Gaussian curvature modulus κj constant by 2K ) 2κ + κj. The lowest curvature energy is found for c ) 0, i.e., a flat bilayer. Small vesicles, with sizes comparable to the molecular size, should be considered separately since the inner and outer layers may have highly different curvature and number of monomers. In a dilute condition, the entropy of mixing of a vesicle phase can be much higher than that of the lamellar phase. The gain in entropy can match the frustration of the bending energy. •In the case of two surfactants, the interaction between the two species (for example between the headgroups in a catanionic mixture) is crucial to stabilizing the vesicles. The surfactant monomer concentration is different in the inner and outer layer, allowing each surfactant to be close to its packing parameter and spontaneous curvature. The energy penalty of one layer is thus prevented. In the present study where only one surfactant is present, the addition of salt screens the electrostatic forces and, therefore, reduces the repulsive interaction between the aggregates and also between the headgroups of the adjacent monomers. Within the first 500 ms after mixing, the packing parameter p increases from a value close to 0.33 (spherical micelle) to 1 (planar bilayer). The surface per headgroup and the number of aggregation are relevant parameters of the surfactant bilayer evolving during the formation and growth of vesicles. We assume that there is no significant difference between the inner and outer interfaces when the radius is much larger than the bilayer thickness (this assumption has been verified in AOT microemulsions in ref 37). The number N of monomers in the shell is equal to the volume VN of the shell of radius rext divided by the monomer volume vm. The average surface per headgroup ah is the total surface of the object divided by the number of monomers.
N)
2vm 4π 4π (rext3 - ri3), ah ) [rext2 + ri2] ≈ 3vm N δ
(7)
ah decreases from 86 to 67 Å2 in the first hundred milliseconds after mixing and then remains constant at around 66.5 Å2. It is convenient to introduce the average radius R ) (rext + ri)/2. The last equation becomes 4πR2 ) Nah/2 and shows that the number of aggregation increases with the square of the radius. It varies from 29 monomers for the micelles to 5 × 104 monomers for the largest vesicles. (36) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525-1568. Israelachvili, J. N. In Intermolecular & Surface Forces, 2nd ed.; Academic Press: New York, 1991. (37) Nave, S.; Eastoe, J.; Heenan, R. K.; Steytler, D.; Grillo I. Langmuir 2000, 16, 8741-8748.
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Figure 7. Growth of the vesicle radius with time for the different salts and concentrations studied. The full lines are the best fits using eq 17. (a) AOT 0.75% and NaCl (b) 0.021 mol/L, (0) 0.042 mol/L, ([) 0.085 mol/L, (4) 0.17 mol/L, (- -) AOT 0.375% and NaCl 0.042 mol/L. (b) AOT 0.75% and NaBr (0) 0.042 mol/L, ([) 0.085 mol/L. (c) AOT 0.75% and KCl (b) 0.021 mol/L, (0) 0.042 mol/L, ([) 0.085 mol/L. (d) AOT 0.75% and KBr (b) 0.021 mol/L, (0) 0.042 mol/L, ([) 0.085 mol/L.
Thermodynamic Model. Structural changes in any system may be able to be considered in terms of free energies.38 In this section we delineate the necessary thermodynamics and kinetics that underline a description a self-assembly. Our aim is to illustrate the essential physics by a simple (but yet nontrivial) model with predictions that can be directly tested experimentally. Our theoretical consideration closely follows (with a natural modification for our case) the model proposed recently by Olsson and Wennerstro¨m39 for vesicle ripening by amphiphile monomers diffusion. The thermodynamic theory literally speaking is limited to description of the stages before vesicle formation and growth. To obtain information about the growth rate kinetic analysis must be used. We are interested in two principle steps of the process. First, salt added to AOT micellar solution nucleates vesicles. Free ions produced by the dissociation of the salt molecules screen the repulsive electrostatic interactions between the micelles in solution and allow vesicle aggregation. This kind of scenario is well appreciated for colloidal systems (see, for example, ref 35). This first step is rather fast, and as a result, a number of vesicles appears. The step is over when the chemical potential µm of a micelle becomes close to the chemical potential µv of the same number of amphiphilic molecules of a vesicle. Vesicles that appear at this first stage are typically small. Our data show that the number of micelles required to produce a vesicle is about 25-50. (38) Landau, L. D.; Lifshits, E. M. Course Theoretical Physics, Physical Kinetics, Vol. 10; Pergamon Press: New York, 1981. (39) Olsson, U.; Wennerstro¨m, H. J. Phys. Chem. 2002, 106, 51355138.
Once the formation of new vesicles has mostly stopped vesicle may still grow. Since our experimental observations clearly demonstrate that we are dealing with almost monodisperse systems of vesicles (and in order to keep our model in the most simple form), we consider a system of noninteracting monodisperse vesicles in a micellar solution. For a vesicle to grow it is necessary to fill the interior with water; it requires a certain amount of water to overcome an energy barrier when crossing the bilayer structure. In this case the processes of vesicle growth can be understood as a reaction,
M1 + MK f MK+1
(8)
where the micelle M1 is added to a vesicle MK (K is the number of micelles in the vesicle) to form a vesicle MK+1. At equilibrium, the free energy change is zero, i.e.,
∆FK - µm ) 0
(9)
where ∆FK ) FK+1 - FK. FK is the free energy of a vesicle with the micelle aggregation number K and can be written as
FK ) Nµv + EK - kBTSK
(10)
where EK accounts for curvature energy34 and SK is the
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entropy of mixing. From eqs 9 and 10 it can be shown that
µm - µv ) ∆EK - kBT(SK+1 - SK)
(11)
If µm - µv is negative the aggregates are energetically favored promoting the transition from micellar solution to aggregated states. Unfortunately, little more can be said from such a general statement without the explicit and detailed form of µm - µv. The choice of this explicit form is dictated by the principle of minimal requirements. In what follows we regard the vesicles as an ideal gas such that the mixing entropy can be omitted from eq 10 as its effect is smaller than the energy change. Despite these strictly erroneous assumptions, the model correctly predicts the characteristic time scale and behavior in the problem and this sin of omission can be easily relaxed when more detailed information concerning vesicle and micelle distributions and interactions becomes available. The harmonic contribution to the curvature energy is well-known34 and can be written as
E0 ) 8πκ + 4πκj
(12)
Thus this harmonic contribution does not depend on the aggregation number and therefore does not produce any “driving force” for the vesicle growth. To obtain the driving force one should consider anharmonic contributions to the curvature elasticity. The next nonvanishing order term to eq 12 in the case of spherical vesicles is the fourth order over the curvature and can be represented by34,35
κan
Ean ) 4π
R2
)
32π2κan ahN
(13)
where κan is a phenomenological anharmonic elastic modulus. For a system of vesicles where the vesicle-vesicle interactions are negligible, the rate of change of the vesicle sizes is determined by the diffusion flux of micelles. The rate of change depends certainly on two factors: the thermodynamic driving force and the vesicle size distribution function. The former follows directly from eq 13 while the latter can be calculated using a standard kinetic equation approach.38 For our particular experimental case (monodisperse system of vesicles), we can restrict ourselves to one vesicle placed in a solution containing a concentration c∞ of micelles. Using eq 13 the kinetic equation for a number N of surfactant molecules on the vesicle surface can be written as
dN δEN ∝ dt δN
(14)
In the spirit of the phenomenological Landau approach, the relaxation rate is determined by δE/δN.38 Thus using eq 13, eq 14 becomes
dN κan ah ∝ 2 2 dt R R
(15)
Since dN/dt is proportional to the micelle concentration c∞ and recalling that N ) 8πR2/ah, we arrive at the following rate equation 2 dR γc∞ah κan 1 ) dt T R5
(16)
where γ is a kinetic coefficient. The solution to eq 16 has
Figure 8. Evolution of the radius in function of time in loglog representation to show the power law (eq 17). (0) NaCl 0.042 mol/L and AOT 0.75%, (s) best fit for rext ) 74.54t0.154.
a t1/6 scaling behavior, i.e.,
R ) At1/6
(17)
Before proceeding further, one central question remains, namely, whether the vesicle growth is limited by diffusion or by aggregation reaction processes. A characteristic kinetic energy of a micelle can be estimated as kBT, where kB is the Boltzman constant and T is the temperature. If the electrostatic potential barrier Vb > kBT, the aggregation is reaction-limited and its rate would be proportional to exp(-Vb/kBT). The potential barrier Vb on its own is proportional to kd2, where kd ) x8πLBcs and is the inverse Debye radius and LB is the Bjerrum length, i.e., to the salt concentration in solution. Thus, for a reaction-limited case we should observe an exponential dependence of the vesicle growth rate on salt concentration which is not observed in our experiments. Furthermore, for a reaction-limited aggregation the growth rate should be proportional to the size of the aggregate such that the aggregation rate should increase with time. Our experimental data suggest therefore that we should assume Vb < kBT (i.e., not reaction-limited), and the aggregation rate is determined by the micelle diffusion. In fact, eq 16 has the same form for the both mechanisms, but the kinetic coefficient γ is determined either by the micelle diffusion or by the reaction rate. As demonstrated above, vesicle growth process resembles the standard coalescence stage of kinetics for the first order phase transitions. Vesicle growth is determined mainly by the diffusion flux of micelles, but they can also grow as a result of the fusion processes, similar to standard Lifshits-Slyozov theory38 leading to the R ∝ t1/3 growth rate. Other mechanisms of the vesicle growth have been discussed in the literature. In the case of surface tension dominating growth,40 one finds R ∝ t1/2, and for bending thermal fluctuation mechanisms also proposed in ref 39, one finds R ∝ t1/4. On the other hand only anharmonic bending mechanisms lead to R ∝ t1/6, in agreement with our experimental data. This is consistent with the accepted picture that the vesicle surface tension is small, and thermal fluctuational contributions to the energy are also relatively weak. Example data for AOT 0.75% and NaCl 0.042 mol/L is presented in Figure 8 on a log-log scale. The multiplying factors A and power-law exponents are summarized in (40) Zhdanov, V. P.; Kazemo, B. Langmuir 2000, 16, 7352-7354.
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Table 3. Power Law R ) AtB According Equation 17 Describing the Average Increase of the Vesicle Radius as a Function of Time AOT w ) 0.375% NaCl (mol/L) 0.042 A B
61.69 0.169 AOT w ) 0.75% NaCl (mol/L)
A B
0.021
0.042
0.085
0.17
78.41 0.159
74.54 0.154
56.01 0.160
51.41 0.159
NaBr (mol/L) A B
0.021
0.042
0.085
-
74.79 0.181
54.89 0.182
Figure 9. ([) A values (obtained with eq 17 by fixing B at 1/6) in function of NaCl concentration cs. (s) Evaluation of the trend with a function A ) A1/cs1/6; A1 ) 39.04.
approximation, electrostatic potentials inside and outside the spherical vesicle are
φin )
KCl (mol/L) A B
0.021
0.042
0.085
64.13 0.168
56.61 0.181
45.76 0.177
KBr (mol/L) A B
0.021
0.042
0.085
68.79 0.173
61.78 0.181
51.59 0.183
Small deviations to the power law are observed after several hours of growth. These coincide with the increase of polydispersity (σ > 0.3) and the difficulty to obtain high fit quality to the experimental data. Moreover, the large objects are separated by only few angstro¨ms and the bilayers might come into contact and fuse, in particular because the repulsive electrostatic forces are screened by salt. In this case and in addition to micelle diffusion, an Oswald ripening mechanism is also involved in the growth. The rate of the vesicle growth depends on the salt concentration. From eq 16 we see that the growth rate is directly proportional to γκanah2. Our experimental data show that the area per polar head is almost constant during the growth and does not depend on the salt concentration. If aggregation is diffusion-limited we can neglect the dependence of the kinetic coefficient γ on the salt concentration. For the reaction-limited case such dependence is very strong, as discussed above. Therefore, the observed dependence on the salt concentration could appear from anharmonic bending modulus κan. Further insight into the physical significance of this dependence can be obtained by considering a more specific microscopic model for vesicle constructions. To examine it we consider a spherical bilayer membrane with its internal radius ri and external radius rext, and assume an equal surface charge density σs on the both surfaces. In the Debye
sinh (kdr) r kdri cosh(kdri) - sinh (kdri)
φout )
4πrext2σs exp(kd(rext - r)) kdrext + 1 r
(18)
Here r is the distance from the vesicle center and is the dielectric constant of the solution. Electrostatic energy of the spherical vesicle is
E) Table 3 for all the salts and concentrations. The powerlaw exponents B belong to the range 5.5 < B-1 < 6.5. For a model with such a small physical input, our model shows good agreement with experimental data in particular with the NaCl series. For other salts the growth is slightly faster.
4πri2σs
(
rext3 (4πσs)2 ri3 + 2 kdri coth(kdri) - 1 kdrext + 1
)
(19)
Taking into account that kdR , 1 and δ/R , 1 and expanding the right-hand side of eq 19 over these small parameters, we arrive at the equation
E)
(
2kdδ2 - 4δ 2π2σs2 δ3 8R2 + + + ... kd kd k R2 d
)
(20)
The first term on the right-hand side of eq 20 determines the electrostatic contribution to the surface tension: this exists also for the micelles and these contributions determine the chemical potentials µm and µv. The second term in eq 20 gives the electrostatic contribution to the harmonic bending modulus depending on kd but not on R and therefore does not create a driving force for the growth. The last term determines the electrostatic part of the anharmonic bending modulus κan and is proportional to kd-2. In other words, the electrostatic part of κan and the rate of growth vary as the inverse of salt concentration. For the NaCl series, the growth laws R(t) have been fitted again with eq 17 by fixing the power coefficient B at 1/6 as required by the model. The new A values (slightly lower from those obtained previously) are reported in Figure 9 in function of the salt concentration cs. Within the experimental error, it is indeed possible to represent the behavior by a function A ) A1/cs1/6, a satisfactory result considering the simple approach. Conclusion We have demonstrated the possibility of combining realtime measurements with few hundred millisecond time resolution and SANS on high-flux spectrometers, such as D22 at the ILL. A stopped-flow device controlled precisely
Formation and Growth of Anionic Vesicles
the volume and time of mixing and synchronized the end of mixing with the beginning of data acquisition. We were interested in the formation and growth of vesicles induced by addition of salt solution (NaCl, NaBr, KCl, and KBr) in micellar solution of AOT. The driving force for the transformation of some micelles to vesicles is the screening of the electrostatic repulsion between adjacent headgroups that favors formation of a locally flat bilayer. This step is rather fast (
