Detailed Structure of Hairy Mixed Micelles Formed by

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Detailed Structure of Hairy Mixed Micelles Formed by Phosphatidylcholine and PEGylated Phospholipids in Aqueous Media Lise Arleth,*,† Beena Ashok,‡ Hayat Onyuksel,‡ Pappannan Thiyagarajan,§ Jaby Jacob,§,| and Rex P. Hjelm*,† Manuel Lujan Jr. Neutron Scattering Center, MS H805, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, Department of Biopharmaceutical Sciences, College of Pharmacy, University of Illinois at Chicago, 833 South Wood Street, Chicago, Illinois 60612, and Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439 Received September 29, 2004. In Final Form: January 17, 2005 Aqueous dispersions of mixed egg yolk phosphatidylcholine (PC) and poly(ethylene glycol) (PEG) modified distearoyl phosphatidylethanolamine (DSPE) were investigated with the purpose of determining shape, size, and conformation of the formed mixed micelles. The samples were prepared at a range of DSPEPEG to PC molar ratios ([DSPEPEG/PC] from 100:0 to 30:70) and with, respectively, DSPEPEG2000 and DSPEPEG5000, where 2000 and 5000 refer to the molar masses of the PEG chains. Particle shape and internal structure were studied using small-angle X-ray scattering (SAXS) and small-angle neutron scattering (SANS). The contrast of the micelles is different for X-rays and neutrons, and by combining SANS and SAXS, complementary information about the micelle structure was obtained. The detailed structure of the micelles was determined in a self-consistent way by fitting a model for the micelles to SANS and SAXS data simultaneously. In general, a model for the micelles with a hydrophobic core, surrounded by a dense hydrophilic layer that is again surrounded by a corona of PEG chains in the form of Gaussian random coils attached to the outer surface, is in good agreement with the scattering data. At high DSPEPEG contents, nearly spherical micelles are formed. As the PC content increases the micelles elongate, and at a DSPEPEG/PC ratio of 30:70, rodlike micelles longer than 1000 Å are formed. We demonstrate that by mixing DSPEPEG and PC a considerable latitude in controlling the particle shape is obtained. Our results indicate that the PEG chains in the corona are in a relatively unperturbed Gaussian random coil conformation even though the chains are far above the coil-coil overlap concentration and, therefore, interpenetrating. This observation in combination with the observed growth behavior questions that the “mushroom-brush” transition is the single dominating factor for determining the particle shape as assumed in previous theoretical work (Hristova, K.; Needham, D. Macromolecules 1995, 28, 991-1002).

I. Introduction Hairy micelles formed by poly(ethylene glycol) (PEG) modified phospholipid or block copolymers containing PEG have recently attracted a great deal of interest due to the applications as lipid assisted drug-delivery systems.2-7 The hydrophobic core of the micelles allows for solubilization of hydrophobic drugs. The extremely low critical micellar concentration of these systems make the drugloaded micelles stable upon the fast dilution that takes place when the drug is injected in the blood vessels. In addition, the PEG corona that forms around the micelles * Corresponding authors. E-mail: [email protected] (L.A.); hjelm@ lanl.gov (R.P.H.). Present address (L.A.): Department of Natural Sciences, Royal Veterinary and Agricultural Unviersity, Bu¨lowsvej 17, 1870 Frederiksberg, Denmark. † Los Alamos National Laboratory. ‡ University of Illinois at Chicago. § Argonne National Laboratory. | Present address: Amgen, Inc., One Amgen Center Drive, Thousand Oaks, CA 91320-1799. (1) Hristova, K.; Needham, D.; Macromolecules 1995, 28, 991-1002. (2) Otsuka, H., Nagasaki, Y.; Kazunori, K. Adv. Drug Delivery Rev. 2003, 55, 403-419. (3) Lin, W.-H., Juang, L.-W.; Lin, C.-C. Pharm. Res. 2003, 20 (4), 668-673. (4) Krishnadas, A.; Rubinstein, I.; O ¨ nyu¨ksel, H. Pharm. Res. 2003, 20 (2), 297-302. (5) Burt, H. M.; Zhang, X.; Toleikis, P.; Embree, L.; Hunter, W. L. Colloids Surf., B 1999, 16, 161-171. (6) Riley, T.; Govender, T.; Stonik, S.; Xiong, C. D.; Garnett, M. C.; Illum, L.; Davis, S. S. Colloids Surf., B 1999, 16, 147-159. (7) Torchilin, V. P. J. Controlled Release 2001, 73, 137-172.

provides a shield against attacks from naturally occurring enzymes, which would otherwise quickly destabilize the micelles. The PEGylated micelles can be regarded as the micellar counterpart to the PEGylated vesicles known as “stealth liposomes” which are already used in several medical products. Because of the large pharmaceutical interest in the “stealth liposomes” most of the published research on mixed phospholipid/PEGylated phospholipid systems has been conducted in the part of the phase diagram where the PEGylated vesicles form. The studies have been less focused on the series of morphological changes that the micelles undergo as a function of PEGylated phospholipid/ phospholipid mixing ratio. However, the theoretical work of Hristova and Needham1 and the experimental cryo-transmission electron microscopy (cryo-TEM) work of Edwards et al. from 19978 showed that a series of morphological transformations from spherical micelles,1,8,9 through rodlike micelles,1,8 disk-shaped aggregates,8 bilayer vesicles,1,8 and planar bilayers/lamellar structures1 will take place as the PEGylated phospholipid/phospholipid molar ratio decreases. The order of the appearance of the different aggregate morphologies is qualitatively in agreement with the predictions from the critical packing parameter theory (8) Edwards, K.; Johnsson, M.; Karlsson, G.; Silvander, M. Biophys. J. 1997, 73, 258-266. (9) Needham, D.; McIntosh, T.; Lasic, D. Biochim. Biophys. Acta 1992, 40, 1108.

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for surfactant systems by Israelachvili et al.10,11 It is, therefore, tempting to explain the morphological transformations as being driven by a change in the spontaneous curvature of the system; the PEGylated phospholipids have a large intrinsic curvature, form spherical or nearly spherical micelles, and consequently have a critical packing parameter near 1/3, while the unmodified phospholipids have a small or zero curvature, form planar or vesicle structures, and have a critical packing parameter close to unity.10,11 Assuming that the critical packing parameter is a combination of those of the two lipids, mixtures of the two types of molecules should give rise to critical packing parameters ranging from 1/3 to unity. This would indeed result in the above-mentioned series of morphologies. In previous studies of PEGylated micelles and their properties as drug-delivery vehicles, the focus has been on the use of small spherical micelles (see, e.g., refs 2-7). However, as both the micelle shape and the micelle size affect the ability of the micelles to diffuse into tissues, the extra latitude in controlling the diffusion that is obtained with the rodlike micelles could also play a role for the final efficacy of the drug-delivery system. In the present work we investigated the phase behavior of the mixed egg yolk phosphatidylcholine (PC) and PEG modified distearoyl phosphatidylethanolamine (DSPE) system in the part of the phase diagram where micelles form. We determined, for the first time, the detailed structure of the polymer modified micelles formed in mixed DSPEPEG/PC systems and did so in a region of the phase diagram where a transition from nearly spherical micelles to long cylindrical micelles takes place. As a part of the project we investigated the extent to which the size and shape of the micelles can be controlled by the DSPEPEG/ PC mixing ratio. We used DSPE conjugated with either 2000 or 5000 Da PEG, where the numbers refer to the molar masses of the PEG chains. This way we also had the opportunity to study how the size and conformation of the PEG corona affects the micelle size and shape in a phospolipid based system. These experimental results will provide interesting comparisons and feedback to theoretical predictions of the phase behavior for exactly the same system that has previously been published.1 Our main experimental techniques were small-angle neutron scattering (SANS) and small-angle X-ray scattering (SAXS). By analyzing our small-angle scattering data with the analytical scattering form factors for ellipsoids and cylinders with attached polymer chains in a Gaussian random coil conformation,12 we determined the size, shape, and internal structure of the micelles. In these studies we were able to perform SAXS and SANS on the same samples within a few days of each other. By fitting the SAXS and SANS data simultaneously with a molecularly constrained model for the micelles, a very detailed structural characterization could be obtained with high certainty. A comparable approach was used previously13 when Pedersen et al. made simultaneous fits to a series of SANS contrast variation data on a polystyrenepolyisoprene (PS-PI) diblock copolymer system and found that the micelle structure could be described by a short cylindrical core with attached polymer chains in a Gaussian random coil conformation. However, polydispersity (10) Israelachvili, J.; Mitchell, D. J.; Ninham, B. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525-1568. (11) Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1991. (12) Pedersen, J. S. J. Appl. Crystallogr. 2000, 33, 637-640. (13) Pedersen, J. S.; Hamley, I. W.; Yeol Ryu, C.; Lodge, T. P. Macromolecules 2000, 33, 542-550.

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effects and a significant scattering contribution from the solvent-swollen PI core gave rise to a smearing of their SANS and SAXS data.13 Such effects are not observed in the DSPEPEG/PC system, and, especially, our SAXS data are rich in features. Our work, therefore, complements ref 13 as a critical experimental test and application of the analytical scattering form factors for ellipsoids and cylinders with attached polymers in a Gaussian random coil conformation.12 In this article we present our analysis and discuss our results for the relation between the corona structure and micelle shape in light of the classical “mushroom-brush” picture of systems with polymer chains attached to the surface.14,15 II. Experimental Section Egg yolk PC was purchased from LIPOID GmbH (Ludwigshafen, Germany). The product, being a mixture of phospholipids, had a PC content of 98% according to the manufacturer. DSPE conjugated with 2000 or 5000 Da PEG was obtained from Avanti Polar Lipids, Inc. (Alabaster, AL). The molecular structures of egg yolk PC and DSPEPEG are shown in Figure 1. Notice that EYPC is actually PC with a mixture of oleyl and palmitoyl fatty acids. DSPEPEG and PC at molar ratios of 100:0, 85:15, 75:25, 60: 40, 50:50, and 30:70 were dissolved in methanol and coprecipitated under a vacuum using a rotary evaporator to form dry films and dried overnight to remove traces of solvent. The films were then rehydrated to a 1 mM lipid concentration with isotonic 0.01 M N-2-hydroxyethylpiperazine-N′-2-ethanesulfonic acid (HEPES) buffer at pH 7.4 and equilibrated in the dark under argon at room temperature. D2O was used for the buffered solutions. The SAXS and SANS measurements were performed within 5 days of each other. This scheduling allowed for measuring the same samples, first in the SAXS instrument and then in the SANS instrument, thus, minimizing issues of sample reproducibility. SAXS experiments were performed at the 12-ID, BESSRCCAT beamline at the Advanced Photon Source, Argonne National Laboratory.16 Incident 12 keV X-rays were used, and the scattering data were measured on a two-dimensional chargecoupled device (CCD) detector. Within a single setting, this gave a q range [q ) (4π sin θ)/λ, where 2θ is the scattering angle and λ is the wavelength of the X-ray beam] from 0.0065 to 0.25 Å-1. The resolution effects are negligible on this instrument and, therefore, not taken into account in the data analysis. The data were radially averaged using the standard software of the facility and rebinned into 200 points evenly distributed on a log q scale. The absolute scale of the scattering intensity in terms of differential scattering cross section per unit volume (cm-1) was determined by measuring the incoherent scattering from pure water. The incoherent scattering from pure water is related to the isothermal compressibility of water and the temperature. At 293 K the constant scattering intensity of water is 1.632 × 10-2 cm-1 (see, e.g., ref 17). The accuracy of this absolute scale calibration is better than 10%. SANS experiments were performed on the time-of-flight smallangle neutron diffractometer at the Intense Pulsed Neutron Source at Argonne National Laboratory.18 The data were reduced to differential cross section per unit volume (cm-1) over a q domain from 0.005 to 0.8 Å-1. In the present experiment, the signalto-noise ratio was only sufficiently good from q ) 0.008 Å-1 to q ) 0.20 Å-1. The absolute scale is determined by using a polymer melt sample and a silica sample as secondary standards. The accuracy is better than 10%. The width of the resolution function, (14) Alexander, S. J. Phys. 1977, 38 (8), 983-987. (15) de Gennes, P. G. Macromolecules 1980, 13 (5), 1069-1075. (16) Seifert, S.; Winans, R. E.; Tiede, D. M.; Thiyagarajan, P. J. Appl. Cryst. 2000, 33 (3), 782-784. (17) Orthaber, D.; Bergmann, A.; Glatter, O. J. Appl. Crystallogr. 2000, 33 (2), 218-225. (18) Thiyagarajan, P.; Epperson, J. E.; Crawford, R. K.; Carpenter, J. M.; Klippert, T. E.; Wozniak, D. G. J. Appl. Crystallogr. 1997, 30 (3), 280-293.

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Figure 1. Typical component of egg yolk PC and DSPEPEG. Egg yolk PC is a mixture of PC with oleyl and palmatoyl fatty chains. σ(〈q〉)/〈q〉, ranges from 2 to 8% depending on the q value, where 〈q〉 denotes the nominal q value. The resolution effects or instrumental smearing of the data is, therefore, small but not negligible on this instrument. It has been shown that, for the time-of-flight instrument, the resolution function resembles a Gaussian.19 The standard deviation of the resolution function has been measured and parametrized. The curve given by σ(〈q〉) ) 0.0305〈q〉 exp(-〈q〉/2.2) + 0.2〈q〉 exp(-70〈q〉) follows the experimental σ(〈q〉) within a few percent over the measured q range. The resolution effects are automatically taken into account in the programs for analyzing the scattering data by convoluting the theoretical expression for the scattering data by the resolution function.

III. General Theory: Small-Angle Scattering The small-angle scattering from a dispersion of identical, noninteracting, randomly oriented particles can be expressed as the number density of the particles, n, times the single particle scattering form factor, P(q). P(q) is the orientational averaged absolute square of the scattering amplitude from the single particles. This amplitude is calculated by Fourier transformation of the scattering length density profile, F(r b), of the single scatterers, and the scattering intensity can be expressed as

∫particle[F(rb) b ∫particle[F(r b) - F0]e-iqb.rb dr b〉Ω F0]eiqb.rb dr

I(q) ) nP(q) ) n〈

(1)

where F0 is the scattering length density of the medium in which the particles are suspended and the brackets 〈〉Ω denote the orientational average. The integrals have been solved analytically for a large number of different shapes yielding more convenient expressions. A thorough review of many of these form factors can be found in ref 20. (19) Hjelm, R. P. J. Appl. Crystallogr. 1988, 21, 618-628.

Rigorously, eq 1 only applies in the limit of infinite dilution. For samples with finite concentrations, particleparticle interactions normally have to be considered, and the scattering intensity becomes

I(q) ) nP(q) Seff(q) where Seff(q), the effective structure factor, is a function that describes the effect of the particle-particle interactions on the scattering pattern. IV. Results and Analysis A. Visual Inspection of the Scattering Data. The SAXS and SANS data obtained for the different mixing ratios are shown in Figure 2. We first notice that the SAXS scattering data, Figure 2A,C, have a minimum between 0.04 and 0.06 Å-1, depending on the system, while all the SANS scattering data, Figure 2B,D, decrease monotonically with increasing q value. This behavior is typical for a micelle in water system: with X-rays, the micelles have a negative excess scattering length density in the hydrocarbon core and a positive excess scattering length density in the surrounding hydrophilic shell.21 This gives rise to the oscillating behavior of the SAXS data. The exact position of the SAXS minimum depends on the relation between the characteristic length scales and the excess scattering length densities involved in the system. With (20) Pedersen, J. S. Adv. Colloid Interface Sci. 1997, 70, 171-210. Pedersen, J. S. In Neutrons, X-rays and light. Scattering methods applied to soft condensed matter; Lindner, P., Zemb, T., Eds.; North-Holland Delta Series; Elsevier: New York, 2002; Chapter 16, pp 391-420. (21) Using the values in Table 1 for ν and respectively bN and bX, we obtain, as an example, the following values for the excess scattering length densities for pure DSPEPEG micelles in water as seen by SAXS: ∆Fcore ) -1.4 × 1010 1/cm-2, ∆Fshell ) 5.9 × 1010 1/cm-2, and ∆FPEG ) 1.8 × 1010 1/cm-2. For the same micelles in D2O and observed by SANS we obtain ∆Fcore ) -5.7 × 1010 1/cm-2, ∆Fshell ) -7.8 × 1010 1/cm-2, and ∆FPEG ) -5.7 × 1010 1/cm-2.

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Figure 2. SAXS and SANS data from the mixed micelle systems. (A) DSPEPEG2000/PC, SAXS data. The DSPEPEG2000/PC molar mixing ratios are indicated on the graph. (B) DSPEPEG2000/PC, SANS data. (C) DSPEPEG5000/PC, SAXS data. (D) DSPEPEG5000/PC, SANS data. Points: experimental data. Lines: model fits. The data are plotted on an absolute scale. For better visibility the data sets in each plot are rescaled by 10n where n runs from -5 to 0 starting from the lower-most spectrum.

neutrons, the contrast situation is different. The micelles are suspended in D2O, and as all parts of the micelles are relatively hydrogen-rich, the excess scattering length densities of both the hydrophobic core and the surrounding hydrophilic shell are negative.21 For dilute to semidilute micelle systems this contrast situation generally gives rise to nonoscillating SANS curves as observed in Figure 2B,D. For samples with high DSPEPEG/PC ratios the scattering intensity is almost constant over a wide q range at low q. This indicates that these micelles are relatively small and nearly spherical. The scattering intensity at low q increases with increasing PC content, indicating that the micelles grow. At sufficiently high PC contents, I(q) ∝ q-1 at low q. This indicates that the system contains rodlike micelles. At high q values, a scattering intensity proportional to q-4 is expected for particles with a smooth outer surface. From the SANS data we observe that I(q) approximately decreases as q-3. This indicates that the outer surface of the micelles is rough. A more precise picture of the size and shape of the micelles is obtained by transforming the scattering data to a direct space representation. B. Size and Shape of the Micelles As Determined from their Pair-Distance Distribution Functions. By using the method of indirect Fourier transform (IFT)22 a model-independent direct space representation of the scattering data can be obtained in terms of the pairdistance distribution function, p(r). For an isotropic solution p(r) and I(q) are related via the Fourier transform

∫0∞p(r)sinqrqr dr

I(q) ) 4π

For samples that are sufficiently dilute, so that interaction effects can be neglected, the p(r) only contains information about the distribution of distances within the single aggregates. This implies that p(r) ) 0 for r > Dmax, where Dmax is the maximal distance within a single (22) Glatter, O. J. Appl. Crystallogr. 1977, 10, 415-421.

aggregate. The single particle radius of gyration, RG, can be calculated by integration over p(r):

∫0D r2p(r) dr ) D 2∫0 p(r) dr max

2

RG

max

At the sample concentration of 1 mM used in the present work the structure factor effects were visible in the scattering data, and the corresponding p(r) became negative at large r values. This showed that the interactions between the micelles were repulsive (data not shown). By omitting the lowest q values in the IFT analysis, the structure factor effects could to a good approximation be filtered out of the p(r) and the information corresponding to the pure particle form factor could be obtained.23 This approach was used to determine the p(r) functions showed in this article. Characteristic examples of the pair-distance distribution functions are shown in Figures 3 and 4. A bell-shaped p(r) curve is expected from monodisperse spherical particles with a constant excess scattering length density. For corresponding rodlike particles, the pair-distance distribution function can be separated into the small crosssectional distances and the larger longitudinal distances, and the p(r) curve becomes bell-shaped at small r and descends linearly at larger r.24 The p(r)’s for the pure DSPEPEG micelles (Figure 3) are close to being bellshaped, and we conclude that these micelles are nearly spherical. The p(r)’s for the mixed DSPEPEG/PC (50:50) micelles (Figure 4) are nearly bell-shaped at low r and descend linearly at high r, and we conclude that these micelles are rodlike. (23) Mu¨ller, K.; Glatter, O. Makromol. Chem. 1982, 183, 465-479. (24) Glatter, O. In Small-angle X-ray scattering, Glatter O., Kratky O., Eds.; Academic Press, Inc.: London, U.K., 1982; Chapter 5, pp 167196. Glatter, O. In Neutron, X-ray and Light scattering: Introduction to an investigative tool for Colloidal and Polymeric systems; Lindner, P., Zemb, T., Eds.; North-Holland Delta Series; Elsevier Science Publishers B. V.: Amsterdam, The Netherlands, 1991; pp 33-82.


Figure 3. Pair-distance distribution functions determined from the SAXS and SANS data from the pure DSPEPEG samples. (A) DSPEPEG2000/PC (100:0), SAXS data (B) DSPEPEG5000/ PC (100:0), SAXS data. (C) DSPEPEG2000/PC (100:0), SANS data. (D) DSPEPEG5000/PC (100:0), SANS data.

Figure 4. Pair-distance distribution functions determined from the SAXS and SANS data from the mixed DSPEPEG/PC samples. (A) DSPEPEG2000/PC (50:50), SAXS data. (B) DSPEPEG5000/PC (50:50), SAXS data. (C) DSPEPEG2000/ PC (50:50), SANS data. (D) DSPEPEG5000/PC (50:50), SANS data.

As mentioned earlier, the excess scattering length density profile of the micelles as seen with X-rays is nonhomogeneous: negative in the micelle hydrocarbon core and positive in the surrounding hydrophilic shell. This gives rise to the shoulder seen at low r values in the p(r)’s determined from the SAXS data. The PEG chains of the DSPEPEG molecules are expected to form a highly hydrated corona surrounding the more dense micellar cores. The slightly skewed shape of the p(r)’s determined from the pure DSPEPEG samples (Figure 3) is consistent with such a corona. For the pure DSPEPEG2000 micelles we find a Dmax of 180 Å for both SAXS and SANS data. The RG of the micelles as determined from the SAXS data is 62 Å while we obtain RG ) 54 Å from the corresponding SANS data. For the pure DSPEPEG5000 micelles we find a Dmax of 240 Å for both SAXS and SANS data and RG values of 87 and 78 Å

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for SAXS and SANS, respectively. The different sizes of the DSPEPEG2000 and DSPEPEG5000 micelles are readily explained by the larger PEG corona in DSPEPEG5000. The different RG values, as determined from SAXS and SANS data on the same micelles, are due to different contrast mechanisms for X-rays and neutrons. The maximal distance within a single micelle, the Dmax, is found to be 600 Å for the DSPEPEG2000/PC (50:50) micelles and 400 Å for the DSPEPEG5000/PC (50:50) micelles (see Figure 4). As the DSPEPEG5000/PC micelles are expected to have more extended coronas than the DSPEPEG2000/PC micelles, the slightly larger Dmax indicates a much larger aggregation number for the DSPEPEG2000/PC (50:50) micelles. C. Detailed Structure of the Mixed Micelles As Determined via Structural Modeling. From our knowledge about the molecular structures of DSPEPEG and PC and the information obtained from the scattering data presented in sections IV.A and IV.B, we propose the following model for the micelles: The hydrocarbon chains of the DSPEPEG and the PC are assembled in a hydrophobic core. This hydrophobic core is surrounded by a dense hydrophilic shell consisting of the DSPE and PC polar headgroups, a fraction of the PEG chains, and a number of water molecules. The remaining fraction of the PEG chains forms a more loosely structured and highly hydrated corona that surrounds the dense part of the micelles. 1. Scattering Form Factor. We have used the theoretical scattering form factor for particles with Gaussian random coils attached to the surface as the analytical model for the micelles.12,13,20,25 The core-shell-corona structure of the micelles is modeled for tri-axial ellipsoidal micelles and for rodlike micelles with an elliptical cross section and corona end caps. This is a generalization of models already presented in the literature.12,13,20,25 See appendix A for details. The form factor for the tri-axial ellipsoids reduces to the form factor of a sphere when the axis ratios are set to unity. Correspondingly, the form factor for the rodlike micelles with an elliptical cross section reduces to the form factor for rods with a circular cross section when the cross-sectional axis ratio is set to unity. 2. Molecular Constrained Modeling. The information on the sample concentrations, sample compositions, and molecular structures of the DSPEPEG and PC is implemented in the model calculations. We refer to this as using molecular constraints. By using molecular constraints, we can reduce both the number of fitting parameters and the number of possible solutions to the minimization problem due to the elimination of many unphysical solutions. The use of the molecular constrained model also provides the possibility of fitting SANS and SAXS data simultaneously in a self-consistent way. By insisting that the model fits both SANS and SAXS data, the minimization problem is constrained further. This simultaneous fitting approach was first successfully applied in ref 26. From previous experiments27 we know that the DSPEPEG free monomer concentration is on the order of 1 µM, while the free monomer concentration of PC is even lower.28 In the studied 1 mM samples, the monomer molar fraction is, therefore, no more than 0.001 and, thus, negligible in the measurements. For the modeling we, (25) Pedersen, J. S.; Gerstenberg, M. C. Macromolecules 1996, 29, 1363-1365. (26) Arleth, L.; Posselt, D.; Gazeau, D.; Larpent, C.; Zemb, T.; Mortensen, K.; Pedersen, J. S. Langmuir 1997, 13, 1887-1896. (27) Ashok, B.; Arleth, L.; Hjelm, R. P.; Rubinstein, I.; O ¨ nyu¨ksel, H. J. Pharm. Sci. 2004, 93, 2476-2487. (28) Tanford, C. The hydrophobic effect: Formation of micelles and biological membranes, 2nd ed.; John Wiley & Sons: New York, 1980.

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Table 1. Chemical Formulae, Molecular Volumes, and Scattering Lengths of Components of the DSPEPEG/PC Mixturesa molecule PC, polar headgroup PC, apolar, lipid chains DSPE, polar headgroup DSPE, apolar, lipid chains PEG2000 PEG5000 D2O, buffer a

chemical composition C10H21NO8P C32H62 C6H10NO8P C34H70 (C2H4O)45CH3O (C2H4O)113CH3O D2O

ν (Å3) 377 889 243 969 2750 6830 30

bN (cm) 10-12

4.88 × -1.91 × 10-12 6.33 × 10-12 -3.58 × 10-12 1.88 × 10-11 4.70 × 10-11 1.91 × 10-12

bX (cm) 4.71 × 10-11 7.16 × 10-11 3.72 × 10-11 7.73 × 10-11 3.09 × 10-10 7.70 × 10-10 2.82 × 10-12

ν is the partial specific molecular volume. bN and bX are the molecular scattering lengths for neutrons and X-rays, respectively.

therefore, neglect the monomer concentration. We also assume that the DSPEPEG and PC are fully mixed, such that the micellar molar fraction of DSPEPEG equals the macroscopic molar fraction of DSPEPEG, XDSPEPEG, and that the DSPEPEG and PC molecules are evenly distributed within the single micelles. The number of DSPEPEG molecules in a micelle with total aggregation number, Nagg, can be expressed by NDSPEPEG ) XDSPEPEGNagg, while the number of PC molecules in the micelle is NPC ) (1 XDSPEPEG)Nagg. Hydrophobic Core. The total scattering length of the micellar core is conveniently expressed in terms of NDSPEPEG and NPC: bcore ) NDSPEPEGbDSPE,apolar + NPCbPC,apolar, where bDSPE,apolar and bPC,apolar are the molecular scattering lengths of the apolar parts of DSPEPEG and PC, respectively. The b values for neutrons and X-rays are given in Table 1. The corresponding volume of the micellar core can be expressed in a similar way: Vcore ) NDSPEPEGνDSPE,apolar + NPCνPC,apolar, where νDSPE,apolar and νPC,apolar are the molecular volumes of the apolar parts of DSPEPEG and PC, respectively. The total scattering length density of the core is Fcore ) bcore/Vcore. Hydrophilic Shell. The hydrophilic shell consists of the DSPE and PC polar headgroups. We further assume that a fraction, 1 - XGauss, of the PEG chains are packed in the hydrophilic shell. Finally, a number of water molecules, Nh, will be present in the hydrophilic shell. The total scattering length is bshell ) NDSPEPEG[bDSPE,polar + (1 XGauss)bPEG] + NPCbPC,polar + NhbD20, where bDSPE,polar, bPEG, bPC,polar, and bD2O denote the scattering lengths of the DSPEPEG polar headgroup, the PEG chains, the PC polar headgroups, and a D2O molecule, respectively (see Table 1). The volume of the hydrophilic shell is Vshell ) NDSPEPEG[νDSPE,apolar + (1 - XGauss)νPEG] + NPCνPC,apolar + NhνD2O, and the scattering length density of the shell is Fshell ) bshell/ Vshell. Corona. The micelles contain NDSPEPEG PEG molecules, but only the outer part of the molecules, given by the fraction, XGauss, has the Gaussian random chain conformation and forms the corona. The total scattering length of the corona is then bcorona ) NDSPEPEGXGaussbPEG. The scattering length density of the PEG chains in the corona equals the scattering length density of the single PEG chains, so FPEG ) bPEG/νPEG. The excess scattering length, βc, of the PEG chains that is used in the calculation of eq 3 in appendix A is calculated from βc ) XGaussνPEG(bPEG/ νPEG - bD2O/νD2O). Particle Shape. There is an upper limit for how much the alkyl chains can be extended. According to Tanford29 the maximal extension, lmax, of a hydrocarbon chain is lmax ) 1.5 + 1.265nc [Å], where nc is the number of carbon chains. Because the minimal distance from the center of the hydrophobic core to the hydrophobic-hydrophilic interface cannot exceed lmax, this provides an upper constraint for the size of the hydrophobic core. Combining this information with the molecular volume of the hydrophobic groups, we obtain a constraint on the shape (29) Tanford, C. J. Phys. Chem. 1972, 76 (1), 3020-3024.

of the hydrophobic core for a given aggregation number. For DSPEPEG, nc ) 17, lmax ) 23 Å, and νc ) 969 Å3. This implies that spherical micelles with aggregation numbers above (4πlmax3/3)/νc ) 53 are not favored energetically. If the micelles have aggregation numbers above 53, they have to be ellipsoidal or rodlike (or other shapes; see, e.g., ref 11, Ch. 17). Similarly, if the mean cross-sectional radius of the hydrophobic core of the rodlike micelles exceeds 23 Å, the cross section has to be elliptical (or another shape). 3. Effective Structure Factor. The IFT analysis (section IV.B) showed that the interactions between the micelles are dominatingly repulsive, and for the modeling of the structure factor, we assumed that the interactions could be described by excluded volume effects between the micelles. Ellipsoidal Micelles. For the tri-axial ellipsoids the effective structure factor, Seff(q), is calculated by combining the monodisperse hard-spheres structure factor calculated using the Percus-Yevick formalism (see, e.g., ref 30) with the decoupling approximation.31 The decoupling approximation provides a means for taking into account polydispersity and/or anisotropy of interacting particles: It is assumed that there is no correlation between the position and the orientation of the particles.31 This is a very crude approximation; however, to our knowledge, no better alternative is yet available. For monodisperse hard spheres the structure factor, S(q), is a function of the radius of the hard-sphere interaction, RHS, and the hard-sphere volume fraction, νHS. For slightly nonspherical particles the decoupling approximation provides a method for taking into account the effect of the anisotropy of the micelles. This allows for approximating the effective structure factor Seff(q) from the particle form factor, P(q), and the PercusYevick hard-spheres structure factor, S(q). Seff(q) is calculated from the form factor parameters and have set RHS ) R + (R + 1)RG, where R is the radius of a sphere with the same volume as the micelle without the corona. RG is the radius of gyration of the Gaussian random coils that describe the PEG molecules in the corona and RRG gives the radial distance from the outer surface of the micellar hydrophilic shell to the center of mass of the Gaussian random coils. Then (R + 1)RG gives the approximate thickness of the corona. The displacement parameter, R, for the center of mass of the Gaussian random coils is a form factor fitting parameter which is expected to be close to unity (see appendix A for further details). The default value for the hard-spheres volume fraction is taken to be νHS ) n4πRHS3/3, where n is the number density of the micelles. Rodlike Micelles. The interaction effects of the rodlike micelles are taken into account using an approach for hard rigid rods similar to that explained in ref 32. An expression for the effective structure factor based on the (30) Kinning, D. J.; Thomas, E. L.; Macromolecules 1984, 17, 17121718. (31) Kotlarchyk, M.; Chen, S.-H. J. Chem. Phys. 1983, 79, 24612469. (32) Garamus, V. M.; Pedersen, J. S.; Maeda, H.; Schurtenberger, P.; Langmuir 2003, 19, 3656-3665.


Langmuir, Vol. 21, No. 8, 2005 3285

polymer reference interaction site model33 is used, and Seff(q) becomes

Seff(q) )

1 1 + νc(q) Prod(q, L)

(2)

where ν is related to the shape and effective volume fraction of the hard rigid rods. It can be calculated from ν ) [1 - S(0)]/S(0), where S(0) is the structure factor at q ) 0. For a system of hard rods, S(0) can be estimated from the expression for the osmotic compressibility as determined in the scaled particle approximation34

S(0) )

(1 - B - C)4 [1 + 2(B + C)]2 + 2D(1 + B + 5C/4)

where B ) νHR ) πRHR2LHRn, C ) (4/3)πRHR3n, and D ) (1/2)πRHRLHR2n, where RHR is the radius, LHR is the length, and n is the number density of the hard rigid rods. B ) νHR is the volume fraction of the hard rigid rods. By default we have set RHR ) R + (R + 1)RG and LHR ) L + 2(R + 1)RG, where R is the average cross-sectional radius of the hydrophilic shell, L is the length of the micelle core, and (R + 1)RG gives the approximate thickness of the corona. The function c(q) of eq 2 is related to the normalized Fourier transform of the function that describes the radial correlation hole from the rod. In the present work we have used

sin(qd) - qd cos(qd) c(q) ) 3 (qd)3 where d ) 2RHR is the diameter of the correlation hole. The same approach for c(q) has recently been (empirically) found to give a good description of the structure factor for a system of rigid rods.32 Finally, the Prod(q, L) of eq 2 is the form factor intensity of an infinitely thin rigid rod35,20 of the same length L as the micelle without a corona. 4. Fitting Parameters. Particle Shape and Size. The total aggregation number, Nagg, the number of water molecules in the hydrophilic shell, Nh, and the fraction of the PEG chains in the corona, XGauss, are taken as fitting parameters. Nagg determines the core volume and the SANS and SAXS form factor scattering intensities (assuming fixed partial density of the amphiphilic molecules), while Nh and XGauss determine the volume and scattering length density of the shell as well as the total scattering length of the corona. The core of the tri-axial ellipsoids is defined by the length of the minor axis, RC, and the axis ratios 1 and 2. Because the core volume is already determined by Nagg, only two of these three parameters are free, and we take the axis ratios 1 and 2 as fitting parameters. By assuming that the axis ratios of the polar shell are the same as those for the cores, all parameters of the shell are already fixed by Nagg, Nh, XGauss, 1, and 2. Likewise, there are five free parameters for the rods with an elliptical cross section, and for these particles the axis ratio, , the cylinder length, L, and the minor semi-axis of the hydrophobic core, RC, together with Nh and XGauss are fitted. Partial Specific Molecular Volumes. The DSPEPEG/ PC micelles in water are very close to their contrast match point for X-rays. This makes the modeling of the SAXS (33) Schweizer, K. S.; Curro, J. G. Adv. Polym. Sci. 1994, 116, 319377. (34) Cotter, M. A.; Martire, D. E.; J. Chem. Phys. 1970, 52, 19091919. (35) Neugebauer, T. Ann. Phys. (Leipzig) 1943, 42, 509-533.

data extremely sensitive to the accuracy of the values for the scattering length densities. The scattering length densities are calculated from the molecular scattering lengths which are precisely known36 and literature results for the partial specific molecular volumes of the different parts of the molecules.37 Very accurate density measurements can be performed, but the partial specific volumes of the different parts of the molecules (lipid chains, polar headgroups, and PEG chains) cannot be deduced as accurately. Furthermore, the partial specific molecular volumes of DSPEPEG and PC in the mixed micelles may vary slightly with the DSPEPEG/PC mixing ratio. To take into account small variations of the molecular volumes, three multiplication factors, Xνcore, Xνshell, and XνPEG were included as fitting parameters in the following way: νcore ) Xνcoreνcore, νshell ) Xνshellνshell, and νPEG ) XνPEGνPEG. There is a strong body of evidence in the literature supporting that the partial specific molecular volumes of the polar and apolar parts of amphiphilic molecules are nearly constant and, thus, almost independent of whether an amphiphilic molecule is in the monomer state or sits in a small spherical micelle or in a long rodlike micelle.37 However, slightly different results for the partial specific volume, for example, for the PEG chains, are reported in the literature.38 In accordance with these reports, the values of the multiplication factors were allowed to deviate up to 10% from unity. Background and Absolute Scale. Some uncertainty on the absolute scale is expected. For the SAXS data, small errors on the absolute scale can be accounted for by finetuning the molecular volumes, as explained above. For the SANS data the absolute intensity is only slightly affected by this fine-tuning, because the micelles in D2O are far from their contrast match point. However, a larger than expected amount of H2O in the D2O buffer will have a strong impact on the absolute intensity due to the change in the excess scattering length density. In the present experiments it is not possible to separate uncertainties originating from different possible sources. The combined effective uncertainty on the SANS absolute scale was, therefore, taken into account by fitting pD2O, the purity of D2O (by assuming that impurities were in the form of H2O). The values for the purity of the D2O so obtained should, thus, be regarded as an effective parameter that accounts for the sum of all factors that affect the absolute intensity. In addition, constant residual backgrounds for the SANS and SAXS data, BN and BX, not accounted for by the subtraction of measured backgrounds, were included as fitting parameters. The pD2O is expected to be close to unity; whereas, the residual backgrounds are expected to be close to zero. 5. Fitting Routine. A standard least-squares fitting routine based on the Levenberg-Marquardt algorithm was used for the molecular constrained model fitting. As a result of the very different counting statistics in the SANS and SAXS data, the experimental error bars could not be used as weights in the fits. Instead the I(q) values were weighted by c × I(q), where c is a constant.39 6. Results of Model Fits. The model fits are plotted together with the experimental data in Figure 2. The corresponding fit parameters are for the DSPEPEG2000/ PC system given in parts A (ellipsoidal shapes) and B (rodlike shapes) of Table 2. The parameters used for the (36) For details on how to calculate these values, see, for example, www.ncnr.nist.gov/resources (accessed Mar 2005). (37) Chevalier, Y.; Zemb, Th. Rep. Prog. Phys. 1990, 53 (3), 279-371. (38) Sommer, C.; Pedersen, J. S.; Stein, P. C. J. Phys. Chem. B 2004, 108 (20), 6242-6249. (39) We used c ) 0.02; however, the choice of c does not affect the quality of the fits. c only affects the size of the measure for the fit quality, for example, χ2 and standard deviation.

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Table 2. Fit Parameters for (A) the Tri-Axial Ellipsoids Formed in the DSPEPEG2000/PC System at High DSPEPEG2000 Molar Ratios and (B) the Rodlike Micelles Formed in the DSPEPEG2000/PC System at Intermediate DSPEPEG2000 Molar Ratiosa A. Tri-Axial Ellipsoids sample 100:0 85:15

Nagg

RC (Å)

RS (Å)

1

2

XGauss

Nh/Nagg

R

RG (Å)

pD2O

93c 118 ( 2

22.6 22.7

31.6 29.7

1.34c 1.5c

1.34c 1.5c

0.79 ( 0.01 0.88 ( 0.01

40. ( 3 13 ( 2

1.21 ( 0.03 1.26 ( 0.02

17.6 ( 0.3 18.0 ( 0.3

0.905 ( 0.005 0.9c

b

b

B. Rodlike Micelles sample 75:25 60:40 50:50 30:70

Naggb

L (Å)

RC (Å)

175 374 773 2680

100 ( 2 250c 500c 1400c

19.8 ( 0.2 15.8 ( 0.2 16.4 ( 0.2 16.7 ( 0.5

RSb (Å)

XGauss

Nh/Nagg

R

RG (Å)

28.8 27.2 26.6 28.4

1.31 ( 0.02 1.75 ( 0.02 1.65 ( 0.03 1.92 ( 0.06

0.84 ( 0.01 0.55 ( 0.02 0.71 ( 0.02 0.34 ( 0.07

13 ( 2 24 ( 2 24 ( 2 25 ( 5

1.28 ( 0.03 1.23 ( 0.06 0.99 ( 0.05 0.78 ( 0.46

14.4 ( 0.2 14.5 ( 0.5 14.9 ( 0.5 20.1 ( 8.5

pD2O 0.9c

0.918 ( 0.007 0.879 ( 0.006 0.915 ( 0.014

a N agg is the aggregation number; RC and RS are the minor semi-axes for the core and shell, respectively; 1 and 2 are the axis ratios for the tri-axial micelles; XGauss is the fraction of the PEG that has a Gaussian random coil conformation, the rest is located in the hydrophilic shell; Nh the number of water molecules per lipid molecule in the hydrophilic shell; RRG gives the distance between the outer surface of the hydrophilic shell and the center of mass of the Gaussian random chains; RG is the radius of gyration of the Gaussian random chains; pD2O is the apparent purity of the D2O; L is the length of the hydrophobic core, and is the axis ratios of the elliptical cross section. b Parameter calculated from the explicit fit parameters. c Parameter fitted manually.

Table 3. Structure Factor Parameters and Corona Structure Parameters for (A) the Tri-Axial DSPEPEG2000/PC Micelles Formed at Low DSPEPEG2000 Molar Ratios and (B) the Rodlike DSPEPEG2000/PC Micelles Formed at Intermediate DSPEPEG2000 Molar Ratiosa A. Tri-Axial DSPEPEG2000/PC Micelles sample

RHS (Å)

νHS

S(0)

D (Å)

C∞

s/Σ

100:0 85:15

79.7 78.0

0.014 0.011

0.921 0.934

50.3 ( 1.2 48.1 ( 1.0

3.8 3.8

2.2 2.0

B. Rodlike DSPEPEG2000/PC Micelles sample RHR (Å) LHR (Å) 75:25 60:40 50:50 30:70

65.8 68.1 64.0 76.2

166 314 560 1474

νHR

S(0)

0.0077 0.0073 0.0056 0.0060

0.893 0.897 0.906 0.852

D (Å)

C∞

41.7 ( 0.9 2.4 43.5 ( 1.9 3.7 40.0 ( 1.7 3.1 48.5 ( 20.7 12.3

s/Σ 1.4 1.3 1.5 2.1

a See text for the calculation of the parameters. Structure factor parameters: RHS, hard-sphere radius of interaction; νHS, effective hard-sphere volume fraction; S(0), structure factor at zero angle; RHR and LHR, hard-rod radius and hard-rod length, respectively; and νHR, effective volume fraction of the hard rods. Corona structure parameters: D, approximate thickness of hydrophilic shell plus corona; c∞, Flory’s scale invariant expansion factor; and s/Σ, coilcoil overlap ratio.

corresponding structure factor calculations are given in parts A and B of Table 3 for ellipsoidal and rodlike shapes, respectively. These parameters are calculated according to section IV.C.3 and are, thus, not fitting parameters. For the DSPEPEG5000/PC system, the fit parameters are given in parts A (ellipsoidal shapes) and B (rodlike shapes) of Table 4, and the corresponding structure factor parameters are given in Table 5A,B. The micellar shapes derived from the scattering data are schematically shown in Figure 5. General Observations. The approach of simultaneous fitting of the SAXS and SANS data significantly constrained the model fitting. Better fits to respectively SANS and SAXS data could be obtained when these were fitted separately. However, the fitted values for certain parameters were less significant and had larger error bars and in some cases the values turned out to be unphysical. For example, the SAXS data were only slightly sensitive to the aggregation number of the micelles; whereas the SANS data were only slightly sensitive to the internal structure of the micelles in terms of the radii of the hydrophobic core and hydrophilic shell and the axis ratios.

The overall quality and consistency of the fits are generally good enough to conclude that the hairy mixed micelle model gives a suitable description of the shape and the structure of the micelles. An exception is seen in Figure 2B where the SANS data for the DSPEPEG2000/PC (30:70) system show an unusual hump between the q value of 0.03 and the q value of 0.04 Å-1 as well as an upturn at low q values. The model accounts for neither feature. This sample may be very close to the transition region between the micelle phase and a phase with planar geometry, that is, the lamellar/ vesicle phase (or the disk phase as suggested by ref 8). Indeed, if the hump is interpreted as a (smeared) Bragg peak, the corresponding lamellar repeat distance would be ∼180 Å (d ) 2π/q), in very good agreement with the 170 Å reported for the comparable mixed DSPEPEG1900SOPC (stearoyl-oleoyl-phosphatidyl-choline) system.9 However, similar features are significantly not apparent in the corresponding SAXS data. The SANS data were measured a few days after the SAXS data, and the different behavior of SANS and SAXS data may indicate that the actual sample was close to the transition region and not fully equilibrated when the SAXS measurements were done. For a few of the DSPEPEG5000/PC samples with high DSPEPEG5000 molar contents, the model fits deviate from the data at high q values (see Figure 2C,D). A possible explanation is that in these cases the PEG surface coverage is very high; thus, the PEG corona will be densely packed, and the assumption of a Gaussian random coil conformation of the PEG chains may no longer hold. For spherical particles it has been demonstrated that chain-chain excluded volume effects affect the corona density profile and, thus, affect the scattering pattern significantly at high polymer densities.40,41 However, it is not trivial to take into account these excluded volume effects for the nonspherical particle shapes of relevance in the present work. For example, the corona density profile will depend on the local curvature of the surface where the PEG chain is attached and, thus, be different for the end caps and along the long axis of the rodlike micelles. In the absence of a more complete theoretical model and without the guidance that could be provided by computer simulations of the corona density profiles in the relevant nonspherical (40) Svaneborg, C.; Pedersen, J. S. J. Chem. Phys. 2000, 112 (21), 9661-9670. (41) Pedersen, J. S.; Svaneborg, C.; Almdal, K.; Hamley, I. W.; Young, R. N. Macromolecules 2003, 36, 416-433.


Langmuir, Vol. 21, No. 8, 2005 3287

Table 4. (A) Structure Factor Parameters and Corona Structure Parameters for the Tri-Axial DSPEPEG5000/PC Micelles Formed at Low DSPEPEG5000 Molar Ratios and (B) Fit Parameters for the Rodlike Micelles Formed in the DSPEPEG5000/PC System at Intermediate DSPEPEG5000 Molar Ratiosa A. Structure Factor Parameters and Corona Structure Parameters sample

Nagg

RC (Å)

RS (Å)

1

2

XGauss

Nh/Nagg

R

RG (Å)

pD2O

100:0 85:15 75:25

95c 110c 120c

22.4 22.5 22.8

35.9 29.6 33.0

1.33 ( 0.14 1.44 ( 0.14 1.48 ( 0.20

1.41 ( 0.19 1.5 ( 0.16 1.53 ( 0.17

0.93 ( 0.01 0.98 ( 0.006 0.93 ( 0.01

74 ( 7 30 ( 4 44 ( 7

0.85 ( 0.03 0.74 ( 0.02 0.85 ( 0.04

41.2 ( 0.9 46.8 ( 0.9 42.8 ( 1.2

0.98c 0.98c 0.98c

sample

Naggb

L (Å)

RC (Å)

RS (Å)

XGauss

60:40 50:50 30:70

285 369 2720

97 ( 4 137 ( 3 1200c

25.1 ( 0.6 22.1 ( 0.2 18.0 ( 0.3

32.7 28.8 26.0

1.34 ( 0.07 1.58 ( 0.03 1.96 ( 0.04

0.92 ( 0.01 0.91 ( 0.004 0.81 ( 0.01

b

b

B. Fit Parameters b

Nh/Nagg 0c 0c 6(1

R

RG (Å)

pD2O

1c 1.04 ( 0.02 1.32 ( 0.03

40.0 ( 1.2 34.9 ( 0.4 27.9 ( 0.6

0.80c 0.788 ( 0.004 0.795 ( 0.006

a See footnote a of Table 2 for an explanation of the nomenclature. b Parameter calculated from the explicit fit parameters. c Parameter fitted manually.

Figure 5. Schematic model of the relation between micelle composition and micelle structure for the DSPEPEG2000/PC and the DSPEPEG5000 micelles. The hydrophobic cores are constituted by the acyl chains of DSPE and PC. The surrounding hydrophilic shells are constituted by the DSPE and PC polar headgroups, the inner part of the PEG chains, and a small amount of water molecules. The outer part of the PEG chains have a Gaussian random walk conformation and form a highly hydrated, loosestructured corona around the micellar core.

geometries, it would be questionable to introduce a model that is more detailed than the one used in the present work. The fitted values for Xνcore, Xνshell, and XνPEG are not given in the tables. With only small variations with sample composition, these values were found to be Xνcore ) 0.96, Xνshell ) 1.00, and XνPEG ) 1.06. According to these results, the partial specific molecular volumes of the hydrophobic chains are, thus, slightly smaller and those of the PEG chains are slightly larger than the values given in Table 1. This also affected the calculations of the excess scattering length densities slightly. However, because the deviations were small and fell within the expected38 accuracy of the nominal values (Table 1) we will not address this issue further. As seen from Tables 2 and 4, the value for pD2O lies near 0.9 in the PEG2000 samples, while ranging from unity to a value as low as 0.8 at high PC contents in the PEG5000 samples. As mentioned in section IV.C.4 pD2O should be regarded as an effective parameter which takes into account both errors on the absolute scale and impurities (in the form of H2O) in the D2O as well as other effects. We notice that for most of the fits the apparent pD2O is smaller than expected from the uncertainty of the absolute calibration of the instruments and the nominal purity of the D2O. This observation reveals that we have slightly less scattering intensity that would be expected from our structural models of the micelles. We will comment on this point later.

Micelle Shape and Polydispersity. From the fit parameters in Tables 2 and 4 we notice that the micelles are never perfectly spherical. Perfect spheres are unphysical for the observed aggregation numbers; in the present case the micelles have a slightly oblate ellipsoidal shape with axis ratios 1 and 2 ranging from 1.2 to 1.7 (including error bars). As the PC content is increased, the micelles become elongated along one of the axes and form rodlike micelles. The cross sections of the rodlike micelles are elliptical. The axis ratio of the cross section of the micelles increases with increasing PC content. In both systems the radius of the minor axis of the micelles is largest for the ellipsoidal and DSPEPEG-rich micelles and decreases slightly with increasing PC ratio. The polydispersity of the micelles has so far not been addressed. Generally, it is expected that surfactant micelle size will be polydisperse with a distribution having a significant standard deviation (see, e.g., Ch. 16 of ref 11). Multiple chemical-equilibrium theories predict quite significant length polydispersity for rodlike micelles.42 However, previous studies show that the effects of such a polydispersity on the structure factor tend to vanish,43 and, hence, we did not include a length polydispersity. For spherical (or nearly spherical) micelles, on the other hand, polydispersity will generally be very visible in the (42) Cates, M. E.; Candau, S. J. J. Phys.: Condens. Matter 1990, 2, 6869-6892. (43) Arleth, L.; Bauer, R.; O ¨ gendal, L. H.; Egelhaaf, S. U.; Schurtenberger, P.; Pedersen, J. S. Langmuir 2003, 19 (10), 4096-4104.

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Table 5. (A) Parameters for the Structure Factor Calculations for the Tri-Axial DSPEPEG5000/PC Micelles Formed at Low DSPEPEG5000 Molar Ratios and (B) Structure Factor Parameters and Corona Structure Parameters for the Rodlike DSPEPEG5000/PC Micelles Formed at Intermediate DSPEPEG5000 Molar Ratiosa A. Parameters for the Structure Factor Calculations sample

RHS (Å)

νHS

S(0)

D (Å)

C∞

s/Σ

100:0 85:15 75:25

121 120 122

0.047 0.040 0.039

0.721 0.778 0.781

89.7 ( 2.9 88.7 ( 2.5 89.3 ( 3.9

7.1 8.7 7.6

6.3 9.5 6.5

B. Structure Factor Parameters and Corona Structure Parameters sample RHR (Å) LHR (Å) 60:40 50:50 30:70 a

118 108 101

257 280 1330

νHR

S(0)

D (Å)

C∞

be forced to stretch away from the surface at which they are bound and are, therefore, losing conformational entropy. As more space for the single chains becomes available with decreasing DSPEPEG content the chains will take on a structure which is closer to the Gaussian random coil, thus, gaining conformational entropy. Structure Factor. The parameters used for the structure factor calculations are given in Tables 3 and 5. Within the approximations, S(0) varies only slightly in each system. The structure factor is more dominant in the DSPEPEG5000 system than in the DSPEPEG2000 system, due to the larger effective hard-sphere and hard-rod volume fractions resulting from the larger corona in this system.

s/Σ

0.023 0.707 87.5 ( 2.4 6.7 6.9 0.017 0.787 78.1 ( 1.5 5.2 5.5 0.0095 0.820 72.9 ( 2.2 3.8 3.2

See footnote a of Table 3 for an explanation of the parameters.

high q range and show up as a smearing of the scattering data. In our initial fits a polydispersity was, therefore, taken into account by assuming a Gaussian distribution for the micelle sizes and fitting the standard deviation. In the fits, the standard deviation values were nearly zero. For this reason we omitted the polydispersity effects in the fits presented in this article. Hydrophilic Shell and Structure of the Corona Region. We have assumed a Gaussian random coil structure for the PEG chains where the center of mass of the coil is placed at a distance of RRG from the outer surface of the hydrophilic shell. To take into account the effect of the high PEG density near the surface of the hydrophilic shell, we have allowed that a significant fraction (1 - XGauss) is not in the Gaussian random coil form, but rather is incorporated in the hydrophilic shell, along with hydration water. The values for 1 - XGauss, multiplied by the number of EG monomers (respectively, PEG2000 ∼45 EG units and PEG5000 ∼113 EG units), give the number of EG units that are incorporated in the hydrophilic shell. Our results suggest that, for each DSPEPEG molecule, approximately 10 of the EG monomers are incorporated in the hydrophilic shell. The thickness of the thus obtained dense hydrophilic layer of the micelles ranges from 7 to 12 Å depending on the results obtained for the hydration numbers and 1 - XGauss. From a purely small-angle scattering point of view, the physical interpretation of this layer is that it is a region where the scattering length density is homogeneous within the resolution of the technique. In the micelles, the transition from this region to the corona region is not well-defined, and a more detailed interpretation of the values for XGauss and Nh is not likely to be fruitful. According to ref 25, R is expected to be close to or larger than unity to avoid penetration of the Gaussian random coils into the particle cores. The observed values of R are sufficiently close to unity that such a penetration is avoided to a good approximation. The small variations of the obtained R values is probably mainly a result of the approximations made for the corona structure. An approximate total thickness, D, of the hydrophilic and corona layer can be calculated from D ) DS + (R + 1)RG, where DS denotes the average thickness of the dense hydrophilic layer and (R + 1)RG gives the approximate thickness of the corona layer. The obtained values for D are given in Tables 3 and 5. As is seen, D decreases slightly with decreasing DSPEPEG content. This can be understood in terms of a polymer “brush” picture of the PEG chains on the surface of the micelles.14,15 The densely packed PEG corona in the DSPEPEG-rich samples will

V. Discussion PEG Conformation. The end-to-end length, r, and the radius of gyration, RG, of the Gaussian random coils are related by 〈r2〉 ) 6〈RG2〉 where the brackets denote the average over all configurations.44 It is convenient to discuss the conformation of the polymer chains in terms of Flory’s scale invariant characteristic ratio44 which is defined by C∞ ) 〈r2〉/na2, where a is the PEG monomer unit length of 3.7 Å45 and n is the number of monomers in the chain (see Table 1). Taking into account that only a fraction of each chain, XGauss, is in the Gaussian random coil conformation, we obtain the values for the expansion factors listed in Tables 3 and 5. The values should be compared with Mark and Florys experimental result for a free PEG chain under θ conditions (in aqueous salt solutions) where C∞ ) 4.1 ( 0.4 was obtained,45 as well as the results of Needham et al.,9 where the extension of PEG1900 chains attached to a lipid bilayer was found to be 50 Å, which corresponds to a characteristic ratio of 4.3. For the PEG2000 system our results for C∞ are comparable to, though generally slightly smaller than, the result for the free PEG chain (except for the above-mentioned 30:70 deviant sample). The coil-coil interactions on the surface of the micelles can be investigated by comparing the number of coils on each micelle, NDSPEPEG, times the surface area covered by each coil (estimated by πRG2) divided by the surface area of the micelle taken at the interface described by the center of mass of the attached Gaussian random coils.12 The obtained coil-coil overlap ratios, s/Σ, are listed in Tables 3 and 5. As seen, the values are all significantly above unity, showing that the coils are overlapping. As expected, the largest values for the surface coverage are obtained with DSPEPEG5000 and high DSPEPEG molar ratios. The results for C∞ and s/Σ indicate that despite the fact that the PEG coils overlap there is apparently enough room for a relatively unperturbed Gaussian random coil configuration of the PEG2000 chains as assumed in our structural model for the micelles. For the PEG5000 system, the overlap ratios are high and we generally obtain larger values for C∞ than for the free PEG chains. This indicates a more expanded polymer conformation, and, thus, the chains are not in an unperturbed Gaussian random coil conformation. The largest values for the characteristic ratios are obtained for the highest PEG5000 ratios, whereas the C∞ approaches the result for the free PEG chain for the 30:70 sample. This interpretation is in good qualitative agreement with the above-mentioned “mushroom/brush” picture of the system14,15 and suggests, more generally, that for a PEGylated system, the surface (44) Flory, P. J. Statistical mechanics of chain molecules; John Wiley & Sons: New York, 1969. (45) Mark, J. E.; Flory, P. J. J. Am. Chem. Soc. 1965, 87 (7), 14151423.


coverage has to be as large as 4-5 before the PEG chains are significantly forced into another structure than the mushroom/Gaussian random coil conformation. Micelle Shape. The cryo-TEM work of Edwards et al.3 considered a number of factors in determining particle morphology in DSPEPEG2000/PC extruded mixtures, including the presence of cholesterol (in both extruded and nonextruded samples), substitution of the PEGylated 16-carbon saturated lipid, dipalmatoylphosphatidylethanolamine-PEG for DSPEPEG and distearoylphosphatidylcholine and dipalmitoylphosphatidylcholine for egg yolk PC. The same sequence of phases was observed in each case. For the extruded DSPEPEG/PC mixtures, long, wormlike micelles and disklike bilayers were found to coexist with about 100-nm bilayer vesicles with as little as 11 mol % DSPEPEG2000 mixed with PC. The proportion of wormlike micelles to vesicles was observed to be greater in DSPEPEG2000/PC at a molar ratio of 23:77, a ratio just slightly less than the lowest DSPEPEG2000/ PC ratio studied here of 30:70. As described above, the SANS scattering pattern from the 30:70 sample might suggest that a fraction of this sample forms a lamellarlike structure. However, no signs of lamellar or vesicle structures are observed in any of the remaining spectra. At a molar ratio of 40:60 the cryo-TEM images show largely long, wormlike micelles coexisting with small, globular micelles. Neither the presence of other molecules in the mixture nor the absence of extrusion in sample preparation changes the general characteristics of this picture, although extrusion reduces the size of the vesicles and the presence of cholesterol at a 0.40 mole fraction appears to prevent the formation of the wormlike micelles. In our small-angle scattering data the stronger signal from the rods might mask the scattering from a small population of small, globular micelles. Thus, although our data are consistent with our interpretation that as the PC content is increased from 0 to 0.7 mole fraction, globular micelles generally transform to rodlike micelles, we cannot rule out the coexistence of small amounts of other structures observed by cryo-TEM. Actually, the fact that our measured scattering intensity is generally slightly smaller than the model intensity would be consistent with a minor population of small globular micelles. Hristova and Needham1 previously predicted that the DSPEPEG5000/PC system would form spherical micelles in the [DSPEPEG/PC] molar range from 100:0 to 18:82. Between 18:82 and 11:89 cylindrical micelles would form, while between 11:89 and 0:100 vesicles, lamella, or other planar structures would form. For the DSPEPEG2000/ PC system spherical micelles would form between 100:0 and 32:68, cylindrical between 32:68 and 14:86, and planar structures between 14:86 and 0:100. The predictions were based on a description of the free energy of the system as controlled by a balance between the free (conformational) energy stored in the attached polymer chain and the elastic stretching and bending energies of the lipid film as obtained by minimization of the free energy, where the “lipid film” in our case refers to the DSPE and PC mixture. The free energy of the attached polymer chain was estimated on the basis of the “mushroom/brush” model for the system.14,15 According to the model, packing constrains the bound polymer to an entropically less favored extended brush conformation above coil-coil overlap ratios of unity. This creates a lateral pressure in the polymer corona region that, when countered by the cohesiveness of the lipid membrane, will lead to a bending of the lipid corona interface. This, again, creates more space for the polymer coil and allows it to approach the

Langmuir, Vol. 21, No. 8, 2005 3289

energetically more favored Gaussian random coil conformation, however, at the cost of the bending of the lipid film. Experimentally, we never observe perfect spherical micelles; as it turns out, ellipsoidal micelle shapes are favored at the highest DSPEPEG/PC molar fractions. Therefore, we do not also observe a sphere-to-rod transition as predicted by Hristova and Needham.1 According to their predictions, rodlike micelles will not form until at very low DSPEPEG contents, and, furthermore, the range where rodlike micelles grow is very narrow so that the length of the micelles varies strongly with the DSPEPEG/ PC molar ratio. In other words, a very fast growth is predicted. Theoretically, the fastest growth is obtained in the DSPEPEG5000/PC system. In contrast to this we observe that the growth of the rodlike micelles takes off at a much higher DSPEPEG molar fractions (see also ref 27) and that the range of molar ratios where the growth takes place is quite extended, so that a much weaker dependence of the DSPEPEG/PC molar fraction and, thus, a much slower growth than predicted by Hristova and Needham are observed. More generally, they predict that the DSPEPEG2000 system will become rodlike at higher DSPEPEG molar ratios than the DSPEPEG5000 system. This prediction is in good agreement with our observation that the growth of the DSPEPEG2000 micelles takes off at lower PC ratios than it does in the DSPEPEG5000 system. When comparing their theory with our findings, it should be kept in mind that the detailed theoretical predictions will depend on the values chosen for the surface energy and so forth used for the calculations. Other values for these parameters might lead to theoretical predictions in better agreement with our experimental findings. However, our data suggest that the use of the assumption that the polymer coils are, significantly, forced into a brushlike structure above coil-coil overlap ratios of around unity14,15 may not be adequate for a PEGylated system such as, for example, the DSPEPEG/PC system. Our data suggest that the PEG chains maintain a structure that resembles the Gaussian random coil structure well above coil-coil overlap ratios of unity. This may explain our experimental observation of cylindrical geometries at much higher DSPEPEG fractions than those predicted theoretically.1 In the region where the growth of the cylindrical micelles takes place, we observe, as mentioned above, a much weaker dependence on changes of the DSPEPEG/PC molar fraction than that predicted theoretically. In their theoretical predictions, Hristova and Needham take the micelle conformation that minimizes the free energy (calculated as the sum of the contributions from the curvature and the polymer packing) as the preferred micelle conformation. A multiple equilibrium approach for the micelle size distribution and the mean micelle size (see, e.g., refs 42 and 46) in combination with the Hristova and Needhams expression for the free energy might yield a more realistic prediction of the dependence between the DSPEPEG/PC molar ratio and the length of the cylindrical micelles. VI. Conclusion The DSPEPEG5000/PC and DSPEPEG2000/PC systems were investigated in a part of the phase diagram where ellipsoidal and rodlike micelles are formed. The structures of the micelles as a function of the mixing ratio were determined by fitting simultaneously a molecular (46) Bergstro¨m, M. Thermodynamics of surfactant micelles and vesicles. In Handbook of Surfaces and Interfaces of Materials; Nalwa, H. S., Ed.; Academic: San Diego, 2001; Vol. 5, pp 233-264.

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constrained model for the micelles to SANS and SAXS data. A model for ellipsoidal and rodlike micelles with a hydrophobic core, surrounded by a hydrophilic shell that is again surrounded by a loose structured corona of PEG chains in a Gaussian random coil conformation, gives a satisfactory interpretation of the scattering data, and our work is a critical test of the analytical scattering form factors for the nonspherical versions of such micelles.12 The micelles are never perfectly spherical; oblate ellipsoidal micelles are formed even at 100% DSPEPEG. The micelles, which have elliptical cross sections, become elongated and then rodlike with increasing PC content. In the domain of PC content where micelle elongation takes place, the DSPEPEG2000/PC micelles have larger aggregation numbers than the DSPEPEG5000 micelles for identical mixing ratios. Although these results are in qualitative agreement with previous theoretical predictions1 as well as the general understanding of the relation between curvature and micelle shape,10,11 the detailed picture derived here questions the direct coupling between the “mushroom-brush” transition and the shape of the micelles assumed in theoretical predictions of the phase behavior of the mixed DSPEPEG/PC system.1 Acknowledgment. This work benefited from the use of BESSRC-CAT at the Advanced Photon Source and the Small-Angle Neutron Diffractometer (SAND) at the Intense Pulsed Neutron Source, which are funded by the U.S. Department of Energy BES (Basic Energy Sciences) under Contract No. W-31-109-ENG-38 to the University of Chicago as well as from the use of LQD at the Manuel Lujan, Jr., Neutron Scattering Center supported by the U.S. Department of Energy at Los Alamos National Laboratory operated by the University of California under Contract No. W-7405-ENG-36. Appendix A The theoretical scattering form factor for spherical particles with Gaussian random coils attached to the surface was calculated by Pedersen and Gerstenberg in 1996.25 This form factor was later generalized to other shapes.12 Below, this model is generalized to nonspherical core-shell particles with Gaussian random coils attached to the surface, according to the principles explained in ref 20:

P(q) ) Ps(q) + Ncβc2FD(q) + 2NcβcSsc(q) + Nc(Nc - 1)βc2Scc(q) (3) The first term, Ps(q), is the self-correlation term or scattering form factor for the micellar core, that is, the hydrophobic core surrounded by a dense hydrophilic layer. Ps(q) can be expressed generally by

Ps(q) )

∫Ω{VC(FC - FS) f[qRC(x, y, z)] + (VS + VC)(FS - F0)f[qRS(x, y, z)]}2 dΩ (4)

The subscripts C and S refer to core and shell, respectively. For spherical particles, RC(x, y, z) ) RC, where RC is the radius of the core, and for other shapes, RC(x, y, z) is the generalized core radius that describes the distance from the center to the surface of the particle. RS(x, y, z) is the generalized radius of the shell. VC and VS are the volumes of the core and shell, respectively. f[qRC(x, y, z)] and f[qRS(x, y, z)] denote the scattering form factor amplitudes corresponding to a particle shape and size defined by R(x, y, z). The f (q)’s are normalized so that f (0) equals unity. The integration over Ω signifies that an orientational

Arleth et al.

average over the particles has to be carried out for nonspherical shapes. For the present work we have used the particle form factors for tri-axial ellipsoids, short rods with an elliptical cross section, and long rods with an elliptical cross section. The expressions for these particle form factors and many others can be found in ref 20. The second term of eq 3 is the self-correlation term for the fraction, XGauss, of the PEG chains that has a Gaussian random coil structure. Nc is the number of chains, in our case Nc ) NaggXDSPEPEG, where Nagg is the total aggregation number of the micelle and XDSPEPEG is the molar fraction of DSPEPEG. βc is the excess scattering length of a single chain. FD(q) is the Debye function:

FD(q) )

2(exp(-x) - 1 + x) x2

(5)

where x ) q2RG2 and RG is the radius of gyration of the chains. The third term of eq 3 is the chain-core correlation term. The q-dependent factor, Ssc(q), is calculated from the product of the form factor amplitude corresponding to the particle core and the form factor amplitude corresponding to the corona formed by the surrounding Gaussian random chains. For a general shaped coreshell-corona particle, this becomes

Ssc(q) )

∫ΩAs(q) ψ(qRG) Ξ[q, RS(x, y, z) + RRG] dΩ (6)

Again an orientational average has to be performed. As(q) is the form factor amplitude of the solid cores:

As(q) ) VC(FC - FS) f[qRC(x, y, z)] + (VS + VC)(FS - F0)f[qRS(x, y, z)] (7) The term ψ(qRG) Ξ(q, RRG) describes the form factor amplitude for a corona formed by Gaussian random coils. The centers of these coils are evenly distributed on an infinitely thin fictive shell surrounding the particle cores. ψ(x) is the form factor amplitude of the Gaussian random chain:47

ψ(x) ) [1 - exp(-x)]/x.

(8)

Ξ[q, RS(x, y, z) + RRG] is the form factor amplitude of the infinitely thin shell of the same center as the cores. The generalized radius of the shell is RS(x, y, z) + RRG, where RRG gives the distance between the center of mass of the Gaussian random coil and the hydrophilic shell. The value of R is expected to be close to unity to avoid penetration of the Gaussian random chains into the particle cores.25 The expressions for Ξ[q, RS(x, y, z) + RRG] for the relevant shapes (tri-axial ellipsoids, short and long rods with an elliptical cross section) are listed in refs 12 and 20. Finally, the fourth term of eq 3 is the chain-chain correlation term or the scattering intensity corresponding to the corona formed by the surrounding Gaussian random chains. For a general shaped corona, the q-dependent part, Scc(q), is calculated by

Scc(q) )

∫Ω{ψ(qRG) Ξ[q, RS(x, y, z) + RRG]}2 dΩ

(9)

where ψ and Ξ are defined above. LA047588Y (47) Hammouda, B. J. Polym. Sci., Part B: Polym. Phys. 1992, 30, 1387-1390.

Detailed Structure of Hairy Mixed Micelles Formed by

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