Structure and Interactions of Block Copolymer Micelles of Brij 700

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Structure and Interactions of Block Copolymer Micelles of Brij 700 Studied by Combining Small-Angle X-ray and Neutron Scattering Cornelia Sommer† and Jan Skov Pedersen* Department of Chemistry, University of Aarhus, Langelandsgade 140, DK-8000 Aarhus C, Denmark

Vasil M. Garamus GKSS Research Centre, Abt. WFS, Max Planck Strasse, D-21502 Geesthacht, Germany Received October 11, 2004. In Final Form: December 22, 2004 Spherical micelles of the diblock copolymer/surfactant Brij 700 (C18EO100) in water (D2O) solution have been investigated by small-angle X-ray scattering (SAXS) and small-angle neutron scattering (SANS). SAXS and SANS experiments are combined to obtain complementary information from the two different contrast conditions of the two techniques. Solutions in a concentration range from 0.25 to 10 wt % and at temperatures from 10 to 80 °C have been investigated. The data have been analyzed on absolute scale using a model based on Monte Carlo simulations, where the micelles have a spherical homogeneous core with a graded interface surrounded by a corona of self-avoiding, semiflexible interacting chains. SANS and SAXS data were fitted simultaneously, which allows one to obtain extensive quantitative information on the structure and profile of the core and corona, the chain interactions, and the concentration effects. The model describes the scattering data very well, when part of the EO chains are taken as a “background” contribution belonging to the solvent. The effect of this becomes non-negligible at polymer concentrations as low as 2 wt %, where overlap of the micellar coronas sets in. The results from the analysis on the micellar structure, interchain interactions, and structure factor effects are all consistent with a decrease in solvent quality of water for the PEO block as the theta temperature of PEO is approached.

Introduction Block copolymers in a selective solvent spontaneously form micelles with the poorly soluble block making up a compact core and the soluble block forming a swollen corona.1 Amphiphilic block copolymers in water, with an aliphatic chain and a polar part chemically bonded together, resemble in many respects low molecular weight ordinary surfactants, which show a wide range of micellar and liquid-crystalline phases when dissolved in water. However, the larger molecular weight of the components of block copolymer systems makes it possible to structurally design the molecules by changing the molar mass of the blocks, the number of blocks, and their sequence, and in that way vary the properties of the system. Amphiphilic block copolymers of poly(ethylene oxide) (PEO) in aqueous solution have been the subject of a great number of studies. Micellization processes, phase diagrams, morphology, and structural properties as a function of the size of the blocks, temperature, solvent, and concentration are the subjects of many investigations.2-18 † Present address: Nestle ´ Research Center, P.O. Box 44, Verschez-les-Blanc, CH-1000 Lausanne 26, Switzerland.

(1) Hamley, I. W. The Physics of Block Copolymers; Oxford University Press: Oxford, 1998. (2) Park, M. J.; Char, K.; Kim H. D.; Lee, C. H.; Seong, B. S.; Han, Y. S. Macromol. Res. 2002, 10 (6), 325. (3) Deng, Y.; Young, R. N.; Ryan, A. J.; Fairclough, J. P. A.; Norman, A. I.; Tack, R. D. Polymer 2002, 43 (25), 7155. (4) Zhou, D. L.; Alexandridis, P.; Khan, A. J. Colloid Interface Sci. 1996, 183 (2), 339. (5) Svensson, B.; Olsson, U.; Alexandridis, P. Langmuir 2000, 16, 6839. (6) Linse, P. Macromolecules 1993, 26 (17), 4437. (7) Hamley, I. W.; Pedersen, J. S.; Booth, C.; Nace, V. M. Langmuir 2001, 17 (20), 6386.

Among these, a major part uses scattering techniques as the investigation tool. Reviews on PEO related block copolymers have been published by Mortensen19,20 and contain many references. Such systems have been widely used as model systems for studying interactions in tethered chain systems.21-24 PEO block copolymers with short hydrocarbon blocks, known as hydrophobically modified PEO block copolymers (HMPEOs), are also associative polymers.25-28 These molecules have also been used as adsorbing molecules at (8) Derici, L.; Ledger, S.; Mai, S. M.; Booth, C.; Hamley, I. W.; Pedersen, J. S. Phys. Chem. Chem. Phys. 1999, 1, 2773. (9) Hamley, I. W.; Castelletto, V.; Fundin, J.; Yang, Z.; Price, C.; Booth, C. Langmuir 2002, 18 (4), 1051. (10) Dean, J. M.; Lipic, P. M.; Grubbs, R. B.; Cook, R. F.; Bates, F. S. J. Polym. Sci. B 2001, 39 (23), 2996. (11) Ivanova, R.; Alexandridis, P.; Lindman, B. Colloids Surf. A 2001, 183, 41. (12) Hickl, P.; Ballauff, M.; Linder, P.; Jada, A. Colloid Polym. Sci. 1997, 275 (11), 1027. (13) Hickl, P.; Ballauff, M.; Jada, A. Macromolecules 1996, 29 (11), 4006. (14) Yang, L.; Alexandridis, P. Langmuir 2000, 16, 4819. (15) Svensson, B.; Olsson, U.; Alexandridis, P.; Mortensen, K. Macromolecules 1999, 32, 6725. (16) Yang, L.; Alexandridis, P.; Steytler, D. C.; Kositza, M. J.; Holzwarth, J. F. Langmuir 2000, 16, 8555. (17) Bryskhe, K.; Schillen, K.; Lofroth, J. E.; Olsson, U. Phys. Chem. Chem. Phys. 2001, 3 (7), 1303. (18) Castelletto, V.; Caillet, C.; Fundin, J.; Hamley, I. W.; Yang, Z.; Kelarakis, A. J. Chem. Phys. 2002, 116 (24), 10947. (19) Mortensen, K. Colloids Surf. A 2001, 183, 277. (20) Mortensen, K. Curr. Opin. Colloid Interface Sci. 1998, 3, 12. (21) Castelletto, V.; Hamley, I. W.; Pedersen, J. S. J. Chem. Phys. 2002, 117 (17), 8124. (22) Pedersen, J. S.; Svaneborg, C.; Almdal, K.; Hamley, I. W.; Young, R. N. Macromolecules 2003, 36, 416. (23) Pedersen, J. S.; Gerstenberg, M. C. Colloids Surf. A 2003, 213, 175. (24) Gast, A. P. Langmuir 1996, 12 (17), 4060.

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the interface of microemulsions29-31 in order to better understand how surfactant and block copolymer molecules arrange themselves at interfaces in colloidal systems and their influence on the phase behavior. The Brij 700 surfactant (C18EO100) used in the present study can also be regarded as a HMPEO due to its relatively large PEO block and small hydrocarbon chain. The aim of the present study is to investigate the influence of concentration and solvent quality on the coronal chains and the structure of the micelle. We have chosen to study the Brij 700 micelles because the solvent quality of the PEO blocks can be varied simply by changing the temperature. Water is a good solvent at room temperature and becomes gradually a poorer solvent as the theta temperature (at about 100 °C, see results and discussion sections) is approached. Since the C18 chains of the Brij 700 are hydrophobic at all temperatures, the change of temperature can be expected to have only a minor influence on the aggregation state. The Brij micelles are studied by scattering methods, which are powerful techniques for obtaining structural information on colloidal and other nanosized systems.32 Two recent review articles summarize the scattering and modeling of such systems.33,34 In small-angle X-ray scattering (SAXS), the contrast factor is proportional to the electronic density difference between a component and the surrounding solvent. The hydrocarbon chains of the Brij micelles, which form the micellar core, have a lower electron density than the surrounding water, whereas the PEO chains in the corona have a higher electron density than the water. This difference in signs of the contrasts allows a clear distinction between the distributions of hydrocarbon and poly(ethylene oxide). In the case of smallangle neutron scattering (SANS), the contrasts are given by the nuclear scattering length densities and are dependent on the isotope. For hydrogenated Brij molecules, the use of D2O as a solvent gives a good contrast for the particles and furthermore, the two blocks of the polymer have nearly the same contrast with respect to the solvent. The combination of SANS and SAXS therefore allows us to obtain data at two very different contrast situations and thus enhance the amount of available information without performing specific labeling of the molecules, which might be a tedious task. To obtain detailed structural information on the micellar structure, the most recent modeling approaches are used in the present work.18,35-38 The model is based on recently (25) Preu, H.; Zradba, A.; Rast, S.; Kunz, W.; Hardy, E. H.; Zeidler, M. D. Phys. Chem. Chem. Phys. 1999, 1 (14), 3321. (26) Beaudoin, E.; Borisov, O.; Lapp, A.; Billon, L.; Hiorns, R. C.; Francois, J. Macromolecules 2002, 35 (19), 7436. (27) Beaudoin, E.; Gourier, C.; Hiorns, R. C.; Francois, J. J. Colloid Interface Sci. 2002, 251 (2), 398. (28) Abrahmsen-Alami, S.; Alami, E.; Francois, J. J. Colloid Interface Sci. 1996, 179 (1), 20. (29) Washington, C.; King, S. M.; Attwood, D.; Booth, C.; Mai, S. M.; Yang, Y. W.; Cosgrove, T. Macromolecules 2000, 33, 1289. (30) King, S.; Washington, C.; Attwood, D.; Booth, C.; Mai, S.; Yang, Y. W.; Cosgrove, T. J. Appl. Crystallogr. 2000, 33, 664. (31) Sommer, C.; Pedersen, J. S.; Garamus, V. M.; Strunz, P. To be published. (32) Fairclough, J. P. A.; Hamley, I. W.; Terril, N. J. Radiat. Phys. Chem. 1999, 56, 159. (33) Pedersen, J. S.; Svaneborg, C. Curr. Opin. Colloid Interface Sci. 2002, 7, 158. (34) Castelletto, V.; Hamley, I. W. Curr. Opin. Colloid Interface Sci. 2002, 7 (3-4), 167. (35) Pedersen, J. S.; Gerstenberg, M. C. Macromolecules 1996, 29, 1363. (36) Borbely, S.; Pedersen, J. S. Physica B 2000, 276-278, 363. (37) Svaneborg, C.; Pedersen, J. S. J. Chem. Phys. 2000, 112 (21), 9661. (38) Svaneborg, C.; Pedersen, J. S. Macromolecules 2002, 35 (3), 1028.

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derived expressions from Monte Carlo simulations by Svaneborg and Pedersen where the micelles are described as consisting of a corona of semiflexible, self-avoiding and interacting chains tethered to a spherical core. Such a model is also able to describe the scattering in the full range of the modulus of the scattering vector, q, including the high-q region where the scattering originates from the dissolved chains in the corona (“blob” scattering). The advanced models based on computer simulations have previously been used only in a few studies: in a study of poly(ethylene oxide)-poly(butylene oxide) (PEOPBO) in water,21,39 in a study of poly(styrene)-poly(isoprene) micelles in decane,22 in a study of P85 Pluronic micelles in D2O,23 and in a recent SAXS investigation of Brij micelles40 related to the present paper. The present study is an extension of the SAXS work in which we also investigate the concentration dependence in a broad temperature range. The amount of structural information is increased by combining SAXS and SANS data on the same system, and by fitting simultaneously the SAXS and SANS data the model is exposed to more extensive tests than previously. The simultaneous analysis provides quantitative information on core size, corona size and shape, hard-sphere interaction parameters, interactions between the chains in the corona and the fraction of PEO chains belonging to the solvent. From the fit results, we also derive radial core and corona profiles, profiles of polymer around a single micelle and in the surrounding micelles. Our results give detailed insight into the various effects that control and influence the structural and thermodynamic behavior of the system. Experimental Section Materials and Sample Preparation. C18EO100 block copolymer, also called Brij 700, was obtained from Sigma-Aldrich and used as received. D2O with D purity higher than 99.8% was used as the solvent. Solutions at weight fractions between 0.25 and 10% were prepared gravimetrically by dissolving appropriate amounts of polymer in D2O. The samples were kept at room temperature for several days until complete dissolution, where transparent and homogeneous solutions form, before measurements were performed. Small-Angle X-ray Scattering. The measurements were performed on the SAXS instrument at the University of Aarhus.41 The instrument is a modified version of commercially available small-angle X-ray equipment (NanoSTAR), which is produced by Anton Paar, Graz, and distributed by Bruker AXS. The camera consists of a rotating anode (Cu, 0.3 × 0.3 mm2 source point, 6 kW power) and a three-pinhole collimation. The instrument is optimized with respect to flux and background, so that it is ideally suited for solution scattering. The Cu KR radiation is monochromatized and made parallel by two Go¨bel mirrors. The sample is kept in a reusable thermostated quartz capillary, which is placed in the integrated vacuum chamber of the camera. A homebuilt capillary holder with good thermal contact to the thermostated surrounding block was used. The instrument configuration gives access to a range of scattering vectors q between 0.01 and 0.33 Å-1. The scattering vector modulus is given by q ) (4π/λ) sin θ, where 2θ is the scattering angle and λ the wavelength. The two-dimensional data sets are recorded using a two-dimensional position-sensitive gas detector (HiSTAR). The spectra of all samples were isotropic and the data were azimuthally averaged, corrected for variations in detector efficiency and for spatial distortions. The background scattering from a pure D2O sample was subtracted, and the scattering intensity was converted to absolute scale using the scattering from pure water as a primary standard.42 (39) Castelletto, V.; Hamley, I. W.; Pedersen, J. S. Langmuir 2004, 20, 2992. (40) Sommer, C.; Pedersen, J. S. Macromolecules 2004, 37, 1682. (41) Pedersen, J. S. J. Appl. Crystallogr. 2004, 37, 369.

Block Copolymer Micelles of Brij 700

Figure 1. (a) SAXS experimental data for Brij solutions at varying concentrations. From top to bottom at low q: 0.25% (spheres), 0.5% (triangles, down), 1% (squares), 2% (diamonds), 5% (triangles, up), and 10 wt % (hexagons). Data are normalized for concentration, where the concentration is taken as a weight fraction. (b) SAXS experimental data and fits (lines) for Brij solutions as a function of concentration. From bottom to top at high q: 0.25%, 0.5%, 1%, 2, 5, and 10 wt %. Small-Angle Neutron Scattering. The SANS experiments were performed at the SANS1 instrument at the FRG1 research reactor at GKSS Research Centre, Geesthacht, Germany.43 The range of scattering vectors q from 0.005 to 0.26 Å-1 was covered by four sample-to-detector distances (from 0.7 to 9.7 m). The neutron wavelength was 8.1 Å, and the wavelength spread of the mechanical velocity selector was 10% (fwhm). The samples were kept in quartz cells (Hellma, Germany) with a path length of 1 or 2 mm, depending on polymer concentration. For isothermal conditions, a thermostated sample holder was used. The raw spectra were corrected for backgrounds from the solvent, sample cell, and other sources by conventional procedures. The twodimensional isotropic scattering spectra were azimuthally averaged, converted to absolute scale, and corrected for detector efficiency by dividing by the incoherent scattering spectrum of pure water, which was measured with a 1 mm path-length quartz cell. The smearing induced by the different instrumental settings was included in the data analysis.44 For both SAXS and SANS techniques, the evaluation of the scattering length densities requires that the partial specific densities of the components and the solvent are known. Precise values of the partial specific densities are particularly important for X-ray analysis, where it is the usually small electron density (42) Orthaber, D.; Bergmann, A.; Glatter, O. J. Appl. Crystallogr. 2000, 33, 218. (43) Stuhrmann, H. B.; Burkhardt, N.; Dietrich, G.; Ju¨nemann, R.; Meerwinck, W.; Schmitt, M.; Wadzack, J.; Willumeit, R.; Zhao, J.; Nierhaus, K. H. Nucl. Instrum. Methods 1995, A356, 133. (44) Pedersen, J. S.; Posselt, D.; Mortensen, K. J. Appl. Crystallogr. 1990, 23, 321.

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Figure 2. (a) SAXS (lower points) and SANS data (upper points) with fits at 20 °C and concentrations of 2, 5, and 10 wt % Brij, from bottom to top at high q. (b) SAXS and SANS data with fits at 5 wt % Brij at temperatures of 20, 40, 60, and 80 °C, from bottom to top at low q. differences that provide the contrasts. The partial specific densities depend on temperature, and it is important to consider this in the present work where the scattering experiments have been performed in a broad range of temperatures. The scattering length densities for SAXS and SANS were calculated for each temperature from the values of the specific density of the C18, EO100, and D2O components as determined in an accurate densiometry study.45

Model Typical SAXS and SANS data sets are shown in Figures 1 and 2. The data show at all conditions studied clearly the formation of micelles in agreement with the previous SAXS study.40 Beside the contrast factors, the scattering cross section depends on the number density of particles, the intraparticle form factor P(q), and the interparticle structure factor S(q), where q is the modulus of the scattering vector. To obtain quantitative information about the structure of the micelles, a model scattering curve has been fitted to the experimental data by means of leastsquares methods.46 The form factor of spherical block copolymer micelles can be written as47

Fmic(q) ) Nagg2βs2Vs2Φ2(q,R) + Naggβc2Vc2P′(q) + Nagg[Nagg - P′(0)]βc2Vc2Ac2(q) + 2Nagg2βsβcVsVcΦ(q,R)Ac(q) (1) (45) Sommer, C.; Pedersen, J. S.; Stein, P. C. J. Chem. Phys. B 2004, 108, 6242.

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where Nagg is the aggregation number of micelles and βs and βc are the excess scattering length densities of the chains in, respectively, the spherical core and the corona. Vs and Vc are the total partial volume of a chain in the core and of a chain in the corona, respectively. Note that the expression is normalized to Fmic(q ) 0) ) Nagg2[βsVs + βcVc ]2. The first term in eq 1 is the self-correlation of the spherical core of radius R, and it is expressed in terms of the form factor amplitude of a sphere48 with a smoothly decaying scattering density at the surface:

(

)

sin(qR) - qR cos(qR) q2σ2 Φ(q,R) ) 3 exp 2 (qR)3

Pexv(q) 1 + νPexv(q)

(2)

1 1+ν

(4)

where ν is a parameter related to the compressibility of the quasi two-dimensional solution of the chains in the corona and increases with concentration. It is a function of both the reduced surface coverage Σ/Σ* of the chains in the corona and the strength of the excluded volume interactions. The reduced surface coverage Σ/Σ* can be estimated by49

NaggRg2 Σ ) Σ* 4(R + R )2

(5)

(6)

where VC18 is the specific volume of one C18H37 chain, which we again note depends on the temperature. The third term in eq 1 is the cross term between different chains, whereas the last term is the cross term of the core and corona. Both terms involve the form factor amplitude of the corona, which is given by

∫Fc(r)r2 qr Ac(q) ) ∫Fc(r)r2 dr

dr

(

exp -

)

q2σ2 2

(7)

where Fc(r) is the average radial profile of the corona. The Gaussian factor gives a smooth interface at the core surface. In the present work the Fc(r) function was described as a sum of three partial cubic b splines. They consist of piecewise third-order polynomials and can be Fourier transformed analytically. Writing the (normalized) Fourier transforms as B1(q), B2(q), and B3(q), one gets

(3)

where Pexv(q) is the form factor of semiflexible self-avoiding chains with contour length L and Kuhn length b.50 From the molecular structure of Brij 700, the contour length of the PEO chains, L, was calculated to be 450 Å, obtained from L ) Nl0, N ) 100, and l0 ) 4.5 Å, where N is the number of EO repeat units and l0 the projected length of one repeat unit. The scattering of the chains at zero angle is

P′(0) )

4πR3 3VC18

sin(qr)

The second term in eq 1 is the effective form factor of a chain in the corona. It is related to the self-correlation of the chain but also contains the effects of chain-chain interactions. A Monte Carlo simulation study49 has shown that it can be described by a random phase approximation (RPA) expression:

P′(q) )

Nagg )

Ac(q) )

(

)

B1(q) + a1B2(q) + a2B3(q) q2σ2 exp (8) 1 + a1 + a2 2

In this expression a1 and a2 are fitting parameters together with a parameter scorona, which is related to the maximum width of the corona with Fc(r). When using the three b splines, the corona is going to zero at r )3scorona. In eq 1, it is only the terms related to the radial scattering length density profile which are multiplied by the term exp(-q2σ2/2) in order to have a smooth interface between core and corona. For the scattering intensity, the influence of this term is that one has a faster decay of the radial contributions so that the intensity approaches more rapidly that of the chains at high q. The intermicellar interactions have to be taken into account; since the solvated EO chains in the corona are very swollen, the interaction effects show up already at very low concentrations. In the present work, the interparticle correlation has been taken into account using the structure factor of a monodisperse hard-sphere liquid. The analytical expression for it can be found in ref 51. The final expression for the cross section of the monodisperse system is given by52

g

The reduced surface coverage is defined as the ratio between the projected area of a single chain, given by its radius of gyration Rg, and the surface area available per chain at the distance R + Rg from the center of the micelle, which is approximately in the middle of the corona. The reduced surface coverage is the two-dimensional equivalence of the reduced concentration c/c* used in semidilute polymer solutions, where c is the actual polymer concentration and c* is the overlap concentration, above which the chains start to interpenetrate. The aggregation number Nagg can be calculated from the core dimension. If we assume that the core consists only of the C18 chains, we have (46) Pedersen, J. S. Adv. Colloid Interface Sci. 1997, 70, 171. (47) Pedersen, J. S. J. Appl. Crystallogr. 2000, 33, 637. (48) Rayleigh, Lord Proc. R. Soc. London, Ser. A 1911, 84, 25. (49) Svaneborg, C.; Pedersen, J. S. Phys. Rev. E 2001, 64, 010802. (50) Pedersen, J. S.; Schurtenberger, P. Macromolecules 1996, 29, 7602.

I(q) ) n{Fmic(q) + Amic2(q)[S(q) - 1]}

(9)

where S(q) is the hard-sphere structure factor, Amic(q) is the form factor amplitude of the radial scattering length density distribution of the micelle, and n is the number density of micelles. Expression 9 takes into account the fact that the micelles are not strictly centrosymmetric, due to the random configurations of the chains in the corona. Combined with the use of a hard-sphere structure factor, the expression has been shown to give a good description of the interaction effects for block copolymer micelles.53,54 The fits were performed on absolute scale, and therefore the number density n of the micelles was calculated as n (51) Kinning, D. J.; Thomas, E. L. Macromolecules 1984, 17, 1712. (52) Pedersen, J. S. J. Chem. Phys. 2001, 114 (6), 2839. (53) Borbe´ly, S. Langmuir 2000, 16, 5540. (54) Pedersen, J. S.; Hamley, I. W.; Ryu, C. Y.; Lodge, T. P. Macromolecules 2000, 33, 542.


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) c/(MBrijNagg), where c is the weight concentration (in g/mL) and MBrij is the molecular mass of a Brij molecule (in g). A systematic fitting of the data as a function of concentration showed that the fits became increasingly poorer for higher concentrations. In a particular formulation of the model, it was possible to fit the total excess scattering length of the C18 core and the PEO chains in the corona. The fits became much better for a higher contrast of the core and a lower contrast of the corona. These changes in contrast can be obtained if part of the PEO chains is considered as a sort of homogeneous contribution to the solvent. This approach is in agreement with the “uniform sea of blobs” considered by Daoud and Cotton55 and Marques et al.56 for solutions of star polymers at high concentration. The contribution from the chains to the solvent changes the scattering length of the solvent to

βsolv )

βD2OηD2O + βPEOηPEOfPEO ηD2O + ηPEOfPEO

(10)

where βsolv, βD2O, and βPEO are the scattering length density of the solvent, D2O, and PEO, respectively, and ηD2O and ηPEO are the volume fraction of, respectively, water and PEO in the sample. The parameter fPEO is the fraction of the total PEO, which should be taken as being part of the solvent. The part of the PEO in the solvent is not completely homogeneous with respect to scattering inasmuch as it still contributes to the fluctuation term, i.e., the effective single-chain scattering. With this, the expression for the form factor is

Fmic(q) ) Nagg2βs2Vs2Φ2(q,R) + Naggβc2Vc2P′(q) + Nagg[Nagg - P′(0)]βc2Vc2(1 - fPEO)2Ac2(q) + 2Nagg2βsβcVsVc(1 - fPEO)Φ(q,R)Ac(q) (11) where the contrast factors are now calculated using the modified solvent scattering length density. The form factor amplitude of the micelle is then given by

Amic(q) ) [NaggβsVsΦ(q,R) + NaggβcVc(1 - fPEO)Ac(q)] (12) The model for fitting the scattering data does not take into account micellar polydispersity. Although the presence of such effects is expected due to the equilibrium nature of the system, we could obtain satisfactory fits without including it, and we have concluded that the polydispersity is low and that it is not necessary to include it in the model. The results obtained by simultaneously fitting the SAXS and SANS data have been used for calculating the radial distribution of the micelles. A novel method has been used, which allows the average surrounding of each micelle to be calculated. Details are given in the Appendix. Fitting Parameters. The fitting parameters consist of the following parameters for describing the radial profile of the micelles: the core radius, R, the corona width, scorona, and a1 and a2 describing the shape of the corona profile; in addition, the parameter σ, which gives the width of the core-corona interface. The “blob” scattering from the internal structure of the corona is given by P′(0), the (55) Daoud, M.; Cotton, J. P. J. Phys. (France) 1982, 43, 531. (56) Marques, C. M.; Izzo, D.; Charitat, T.; Mendes, E. Eur. Phys. J. 1998, 3, 353.

effective forward scattering of the chains, and b, the effective Kuhn length of the EO chains. The structure factor is described by the hard-sphere radius, RHS, and volume fraction, ηHS. The last parameter is fPEO, the fraction of PEO chains belonging to the solvent. P′(0) and b are determined from the high-q part of the data, whereas RHS, and ηHS are determined from the low-q part of the data. The fits were performed by conventional leastsquares methods, and the standard errors on the resulting parameters were similarly obtained by standard methods.57 The total number of parameters for describing the structure and the interactions of the micelles is 10. This can be compared to the maximum number of parameters allowed according to the sampling theorem of Fourier transformation which is Nmax ) Dmaxqmax/π,58 where Dmax is the maximum diameter of the objects under consideration and qmax is the maximum q of the experimental data. The analysis of the micelles (see Figure 9) gives Dmax ≈ 200 Å and qmax ≈ 0.2 Å-1 and therefore Nmax ≈ 13. However, this is for one data set and we can therefore expect that for two data sets obtained at different contrast, Nmax ≈ 26. The information content should therefore allow us to obtain reliable results when fitting the model to the combined SAXS and SANS data. The optimum of the leastsquares fits was well defined and this, together with the overall consistency of the fit results, confirms that the information content is high enough to allow the model to be used. Results Scattering Data and Fits. In Figure 1a, SAXS data for solutions at 25 °C in the entire range of concentrations from 0.25 to 10% are shown. The intensities have been normalized with respect to the polymer concentration. For dilute solutions the effect of concentration is small as it can be judged from the very similar behavior for concentrations between 0.25 and 1%. We note that the curves superimpose at intermediate and high q for all the concentrations, which shows that the micellar size and shape do not undergo large changes in the range of concentrations explored. The increase in concentration mainly affects the low-q part of the data, where we clearly observe a decrease of the forward intensity and the development of a peak at increasing concentrations, which is related to interparticle interactions. The correlation peak gets more pronounced at higher concentration and moves to higher values of q, as the minimum distance between particles becomes smaller. Figure 1b shows the experimental data plotted together with the fitted curves. The curves are not normalized for concentration, as this allows a better view of the quality of the fits. There is a very good agreement between the fits and the data, even at high concentration, where interaction effects are pronounced. We have performed measurements as a function of temperature for temperatures between 10 and 80 °C for every 10 °C and for concentrations of 2, 5, and 10% by both SAXS and SANS. Figure 2a shows SAXS and SANS data together with fits at 20 °C for 2, 5, and 10% polymer concentration. The shape of the SAXS and SANS curves is quite different as a result of the different contrasts of the two techniques, demonstrating that information is added by combining the two techniques. Despite the different contrast, the scattering curves display similar features, namely, that the main concentration effects occur (57) Pedersen, J. S. Adv. Colloid Interface Sci. 1997, 70, 171. (58) Glatter, O. J. Appl. Crystallogr. 1980, 13, 584.

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at low q and that the changes in the high-q domain, which reflect the internal structure of the micelles, are small. As already mentioned, the data are on absolute scale and the fitted curves are obtained by simultaneous leastsquares fitting of the SAXS and SANS experimental data. High-quality fits are obtained both for SAXS and SANS curves, which demonstrates that the model is able to describe the system in the whole range of concentration investigated. Figure 2b shows a selection of data for the 5% samples as a function of temperature. For the sake of clarity, only four temperatures of the nine investigated are plotted. For both SAXS and SANS data, the influence of concentration effects at low q decreases with increasing temperature. There are also relatively large changes for the SANS data in the intermediate range of q ) 0.04-0.07 Å-1 and for the SAXS data in the region q ) 0.07-0.15 Å-1. The main effect of changing the temperature is to change the solvent quality of water for the PEO chains in the corona, and the observed changes can be rationalized in terms of this. The theta point of PEO in water lies in the vicinity of the boiling point of water, and it is often determined by extrapolation. It is therefore difficult to determine with precision. Literature data situate it somewhere between 90 and 130 °C. This large scattering of results is due to the variety of methods in consideration, to the use of PEO of different molar masses, and to extrapolations often covering large temperature intervals, inducing systematic errors. The observation of the cloudpoint temperature leads to values of 96 °C59,60 and 117.5 °C.61 The use of intrinsic viscosity measurements leads to a value of 108.5 °C.62 Using values of the second virial coefficient A2 determined by light scattering measurements of PEO solutions leads to theta temperatures of 96 and 101 °C,63 95 and 80 °C,64 and 108, 113, and 105 °C65 when the molar masses are, respectively, 11 500, 32 800, 10 000, 20 000, 100 000, 300 000, and 600 000 g/mol. A study where the theta temperature was extrapolated from calculations of the interaction parameters obtained from critical point data and using the Flory-Huggins theory gives values of 93 and 96.5 °C.66 From differential scanning calorimetry and NMR measurements, it was found that PEO polymers have on average 2-3 structured H2O molecules per EO monomer at ambient temperature, which decreases to zero at a temperature of 100 °C for polymers with masses of the order of some thousands.67 From this survey of the literature, we expect the theta temperature to be located around 100 °C. The decrease of the solvent quality with increasing temperature reduces the strength of the intermicellar interactions and the related concentration effects. As we will see in the following, when presenting the results from the model fits, the temperature dependence also leads to a change of the micellar structure. We note that the hydrophobicity of the C18 chains is hardly influenced by the temperature and therefore it is reasonable that any change of the core size and structure of the micelles is attributed to the changes in the PEO-water interaction. (59) Boucher, E. A.; Hines, P. M. J. Polym. Sci., Polym. Sci. Part B 1978, 16, 501. (60) Ataman, M. Colloid Polym. Sci. 1987, 265, 19. (61) Napper, D. H. J. Colloid Interface Sci. 1970, 33 (3), 384. (62) Amu, T. C. Polymer 1982, 23 (12), 1775. (63) Strazielle, C. Makromol. Chem. 1968, 119, 50. (64) Venohr, H.; Fraaije, V.; Strunk, H.; Borchard, W. Eur. Polym. J. 1996, 34 (5-6), 723. (65) Venohr, H. Diplomarbeit, Universita¨t Duitsburg, Germany, 1992. (66) Fischer, V.; Borchard, W. J. Phys. Chem. B 2000, 104, 4463. (67) Graham, N. B.; Zulfiqar, M.; Mwachuku, N. E.; Rashid, A. Polymer 1989, 30, 528.

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Figure 3. Radial volume distribution profile for a single micelle at 10 wt % and 20 °C: full curve, total polymer distribution; broken curve, C18 core distribution; dotted curve, PEO corona distribut ion.

Micellar Structure. The radial volume distribution profiles of the micelles related to the fits were calculated as described in the Appendix. Figure 3 shows the volume fraction profiles of the whole micelle, of the core, and of the corona, respectively, at 10 wt % and 20 °C. The total polymer distribution displays a steep decrease close to the center of the micelle, largely due to the core contribution, whereas the behavior at large distance is mainly due to the corona chains. At about 100 Å from the center, a value equal to fPEO, the PEO volume fraction in the uniform background, is reached. The general shapes of the profile are only weakly modified by concentration and temperature, and the main effects originate from change in the size of the coronas. The core-corona interface is very broad, characterized with a value of σ ≈ 5 Å, and there is thus a broad region, where C18 and PEO chains are partly mixed. The value of σ was found to be slightly increasing with temperature and relatively independent of concentration. The core volume fraction profile is shown in Figure 4a,b as a function of temperature and concentration. The polymer concentration has only a small influence on the core size (and thus on the aggregation number) except at the highest concentration where the micelles seem to grow substantially. The evaluation of Nagg using eq 6 gives values of about 30 for all concentrations, except at 10%, where the value is 40. The increase of temperature induces a slight but systematic increase in the core size. This effect can be explained as being a consequence of the decrease of chain-chain interactions in the corona when the temperature increases. It is therefore possible to arrange a larger number of chains at the surface of the micelles, and consequently the aggregation number increases. Since Nagg varies with the third power of the core radius R, a small change in R corresponds to a relatively large change in Nagg. At 10% Brij and high temperature, the highest value close to 50 is reached. Figure 4c,d shows the corona profiles as a function of concentration and of temperature for the 10 wt % sample, respectively. A major change of the profiles occurs at the highest concentration of 10%, where a PEO volume fraction of about 35% is reached close to the surface of the micelles, compared to a value of 25% for the remaining concentrations. For 5 and 10 wt %, it is clear that the profile does not go to zero between the micelles but reaches the value corresponding to the homogeneous background (see also Figure 7). The presence of the homogeneous


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Figure 4. (a) Volume fraction profile of the C18 core chains as a function of concentration at 25 °C and (b) as a function of temperature at 10 wt %. (c) Volume fraction profile of the PEO corona chains as a function of concentration at 25 °C and (d) as a function of temperature at 10 wt %.

background contribution is due to a strong overlap of the coronas of neighboring micelles. The increase in PEO concentration at the surface corresponds very well to the volume fraction of the homogeneous contribution. It can be explained as a loss of available volume at larger distances in the overlap region, which makes it less costly, with respect to entropy, to increase the concentration close to the surface. Figure 4d shows an evolution in agreement with interchain interactions within the corona becoming weaker as a result of the decrease of the solvent quality at high temperature. The change of profile shape with temperature corresponds to a contraction of the corona to the surface, where the volume fraction of the chain segments in the vicinity of the surface increases. The profile becomes more compact at the same time as the solvent quality decreases. Scaling theory68 predicts a power-law behavior r-R with an exponent of -4/3 for the corona of the starlike micelles, and the agreement with this prediction has been checked by making log-log plots of the profiles (plots not shown). For the concentration series, the profiles follow the scaling behavior, but only from 30 to 60 Å, which is too small to establish scaling behavior. Figure 5 shows the parameter scorona describing the width of the excess component (with respect to the homogeneous background) of the corona. This parameter is strongly influenced by both concentration and temperature. Only data for 2, 5, and 10 wt % are shown, as the width at lower concentrations is in agreement with the 2 wt % results. (68) Halperin, A. Macromolecules 1987, 20, 2943.

Figure 5. Width parameter of the excess component (above homogeneous background) of the corona as a function of temperature at 2 (circles), 5 (triangles), and 10 wt % (squares).

The decrease of the corona size with increasing temperature can easily be rationalized as a consequence of the decreasing solvent quality of water for the PEO chains. As the excluded volume effects and interchain interactions are reduced, the radius of gyration of the chains becomes smaller and the width of the corona is reduced. The effect of concentration is less straightforward, and the overlap of the coronas of different micelles has to be considered. At overlap, the fraction of PEO chains belonging to the solvent fPEO (i.e., the homogeneous background contribution) becomes nonzero, and the effective corona width is now only related to the excess parts, which do not overlap.

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Figure 6. Fraction of PEO chains in the solvent as a function of concentration at 25 °C (a) and as a function of temperature at 2 (circles), 5 (triangles), and 10 wt % (squares) (b).

Figure 6a shows that fPEO becomes nonzero at concentrations as low as 2 wt % polymer at room temperature which demonstrates the high degree of swelling of the corona. The behavior of fPEO is consistent with the width parameter of the corona, scorona, being constant below 2 wt %. Figure 6a shows a very steep increase of fPEO from zero at 1 wt % to about 55% of PEO at 10 wt % concentration. As a function of temperature at 10 wt % polymer, fPEO decreases with increasing temperature, as the θ point is approached, and the expansion of the chains decreases. The average radial distribution of polymer around the center of a micelle, in which the polymer in the surrounding micelles is included, has also been calculated. The expressions used for this are given in the Appendix. The profiles give a real space view of the concentration effects. The shells of neighbors are clearly visible in Figure 7a,b, and they become more pronounced as the concentration is increased (Figure 7a). At the same time the intermicellar distance decreases, as the micelles come closer to each other due to corona overlap. The increase in temperature leads, as expected, to a weakening of the correlations and the neighboring shells. The real space picture in Figure 7a is in agreement with a corona overlap, which sets in between concentrations of 1 and 2%. The figure also gives a good impression of the development of the homogeneous PEO contribution to the solvent. Internal Structure of the Corona. The internal structure and the interactions within the corona are characterized by, respectively, the single-chain form factor P(q) and the effective forward scattering P′(q ) 0) (eq 4). The Kuhn length b is the only fitting parameter entering

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Figure 7. Radial volume distribution profile of the total polymer around a micelle as a function of concentration at 20 °C (a) and as a function of temperature at 10 wt % (b).

the form factor, since the contour length is calculated from the molecular weight and the bond lengths. Due to the concentration effects (which are described by eq 3), the overall size, i.e., the radius of gyration of the chains, is not directly reflected in the data. However, the variation of the Kuhn length can still be observed at relatively high q, where the form factor crosses over from a q-1.67 behavior, originating from the excluded volume interactions within the chain, to a 1/q behavior, which is due to local stiffness of the chain. The crossover between the two scaling regions is around q ) 3.8/b, and it is reflected at high q in the measured data. Compared to the situation of homopolymers in solution, the “attachment” of the chains to the core surface leads to crowding at the surface and an associated stretching of the chains close to the surface can be expected. The Kuhn length determined for the whole chain will reflect this. In addition, the relatively high polymer concentration in the corona can also be expected to give rise to screening of excluded volume effects as one has in a bulk semidilute solution at relatively high concentrations69 and this might also influence the observed Kuhn length. Considering these points, the determined Kuhn length has to be regarded as an effective Kuhn length of the chains. Figure 8 shows the effective Kuhn length obtained by simultaneous fits of the SAXS and SANS data. The overall behavior is a slight, but quite systematic, decrease of b when the solvent becomes poorer at increased temperatures and a relatively large decrease for increasing concentrations. At low temperature, the larger excluded (69) Flory, P. J. Chem. Phys. 1949, 17, 303.


Figure 8. Effective Kuhn length as a function of temperature for the simultaneous fit of the SAXS and SANS data at 2 (circles), 5 (triangles), and 10 wt % (squares).

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As mentioned above, the interchain interactions within the corona are characterized by the effective forward scattering P′(q ) 0), which is directly influenced by concentration effects and the solvent quality, through the parameter ν. Figure 9a shows the concentration dependence at 25 °C obtained from the SAXS data in Figure 1a. The error bars are relatively large below 1 wt %, but there is a systematic decrease with increasing concentration from 0.4 at 1 wt % to about 0.25 at 10 wt %. This can be explained by an average increase in concentration within the corona as the coronas of neighboring micelles overlap. The influence of temperature on the forward chain scattering P′(0) obtained from the combined analysis of the SANS and SAXS data is very pronounced (Figure 9b). P′(0) displays a clear increase with temperature, in agreement with a decrease of solvent quality and the related reduction of the strength of interaction between the chains in the corona. One can try to extract the theta temperature Tθ from the plot of P′(0) versus T. Since the interaction parameter ν vanishes at T ) Tθ at low concentration, the simplest expression for ν is to have it being proportional to (T - Tθ). Figure 9b includes a common fit of the data in which we have neglected the minor concentration dependence. The values of Tθ obtained is 107 ( 4 °C. This value is in surprisingly good agreement with the literature,59-67 since the “extrapolation” we have done by performing the fit is accompanied by a large uncertainty. In addition, the concentrations within the corona are relatively high so that higher order terms beyond the second virial coefficient are expected to contribute and make our simple assumption ν ∝ (T - Tθ) a poor approximation. In a previous SAXS study40 over a slightly larger temperature range (10-90 °C), a somewhat different analysis of P′(0) was performed, which included the effects of finite concentration within the corona. By this approach, agreement with the expected value of Tθ ) 100 °C was found. In the following, this approach will also be applied to the combined SAXS and SANS results for P′(0). It can be shown that the recently derived expression from the Flory-Huggins RPA theory70,71 for semidilute solutions of homopolymers predicts

VPEO 1 - 2χ VH2O 1 - η

ν)η

Figure 9. Effective forward scattering of the chains (a) as a function of concentration at 25 °C and (b) as a function of temperature at 2 (circles), 5 (triangles), and 10 wt % (squares). The broken curve is the fit for ν ∝ (T - Tθ), and the full curve is the fit with the Flory-Huggins RPA theory.

volume effects favor a more extended conformation of the chains, corresponding to larger values of the radius of gyration (Rg). At higher temperatures the solvent quality is decreased and Rg decreases. This results in lower values of the effective b. At a high concentration of micelles, the coronas of different micelles strongly overlap, and the excluded volume interactions are screened and this gives rise to a decrease of b.

[

]

(13)

where η is the polymer volume fraction, VPEO is the PEO volume fraction, VH2O is the volume of a water molecule, and χ is the Flory-Huggins interaction parameter. The equation fulfils that νf 0 for χf 1/2 and η f 0. We will apply this expression to the P′(0) data and assume that Tθ ) 100 °C and then estimate an effective average value for η in the corona. To do so, we use the simplest possible temperature dependence for χ, i.e., that it has a linear temperature dependence χ ) 1/2 + R(T - Tθ). The temperature dependence of VPEO and VH2O was derived from the density measurements.45 When fitting to the data, one gets η ) 0.03 ( 0.01 and R ) 0.0038 ( 0.0016 1/°C for Tθ ) 100 °C with a reasonable fit (Figure 9b) to the data. These values deviate somewhat from those found for the SAXS data, which were only for a 2 wt % sample. The value for η is relatively low for the present data compared to the 7% determined in the previous SAXS study. The results in the present work are somewhat scattered, and therefore the results in the present work have larger errors than those for the SAXS data.40 In (70) Graessley, W. W. Macromolecules 2002, 35, 3184. (71) De Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell University Press: London, 1979.

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addition, the present results display a stronger temperature variation and this results in a smaller value of the effective average volume fraction of the corona. The value of R corresponds to a linear variation of χ from 0.16 at 10 °C to 1/2 at 100 °C. Monte Carlo simulations on micelles under good solvent conditions for the corona chains have shown that the reduced coverage density Σ/Σ* (eq 5) is a convenient parameter for characterizing the concentration within the corona. Using the values obtained from the fits, values ranging from 3 to 4.5 for Σ/Σ* were obtained in the whole range of concentration and temperature. This corresponds to a relatively large overlap between the chains within the corona and shows that there are strong interactions between the chains. There is a tendency of increasing values of Σ/Σ* with concentration, especially at high concentrations, but no clear dependence as a function of temperature, which could be expected owing to the decrease in chain interaction as the system approaches the theta point. When discussing the chain interactions within the corona, the dimensionless curvature of the micelles, defined as κ ) Rg/R,49 should also be considered. This parameter varies from 2.5 to 1.5, and there is a clear decrease of κ at every concentration as the temperature increases (results not shown) as a result of the decreasing strength of the excluded volume interactions between the corona chains. When the concentration increases, the curvature decreases, and this effect is related to the previously discussed screening effect occurring for interacting chains at high concentrations. These effects are much larger at low temperature, where there is less space per chain and the chains are also more stretched, as shown by the larger values of b. At high concentration, where the coronas of different micelles overlap, the chains tend to adopt a more coiled conformation, and the simultaneous decrease in Rg and slight increase in R lead to decreasing values of κ for increasing concentrations. We have also compared the values of the (dimensionless) osmotic compressibility 1/P′(0) as a function of Σ/Σ* with the relationship obtained by Monte Carlo (MC) simulations60 where a universal behavior is obtained, independently of the number of chains, the chain length, and the core radius, however, with some dependence on curvature. Using the universal curve from simulations, the values of Σ/Σ* in the range of 3-4.5 as determined for the micelles in this work are found to correspond to values of 1/P′(0) between 3 and 6. The experimental values of the compressibility are of the same magnitude, namely, between 2 and 5.5. The simulations show that relatively large curvatures, like that of the present micelles, should give a reduction in the compressibility, and it is thus a quite satisfactory agreement. Finally it should be noted that we compare experimental results in a large range of concentration and temperature where the solvent quality is varying, with the MC results corresponding specifically to good solvent conditions. Intermicellar Interactions. Figure 10 shows the parameters entering the hard-sphere structure factor, which was used for describing the concentration effects. The hard-sphere volume fraction ηHS increases as expected with concentration. At 20 °C the relation between ηHS and the polymer concentration is linear for low concentrations; however, there are clear deviations at 5 and 10%. The deviations are related to the interpenetration of the micellar coronas, which reduces ηHS below the linear relation found at low concentrations. For the lowest concentrations, the hard-sphere volume fraction is about 8 times larger than the weight concentration, which reflects the highly swollen character of the coronas.

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Figure 10. (a) Hard-sphere volume fraction as a function of temperature at 1 (rhomboids), 2 (circles), 5 (triangles), and 10 wt % (squares). (b) Hard-sphere radius as a function of temperature (same signatures as in panel a).

Figure 10a demonstrates that there is a significant dependence of ηHS on temperature. The hard-sphere volume fraction ηHS of the micelles at 80 °C is approximately half of the value at 20 °C, showing clearly the reduction of solvent quality at elevated temperatures. The curvature as a function of the temperature suggests that ηHS goes to zero between 100 and 110 °C in agreement with the expected value for Tθ. Figure 10b displays the hard-sphere radius RHS for the three highest concentrations. Below 2 wt % it is constant and in agreement with the values at 2 wt %. RHS has a systematic decrease with concentration between 2 and 10% at all temperatures. This decrease is due to interpenetration of the coronas. In addition, there is a pronounced decrease of RHS with increasing temperature. This behavior is fully consistent with the effect of the reduced solvent quality already discussed and the decrease of the corona width with increasing temperature (Figure 5). The values of RHS can be compared with the size of the micelles. The width of the corona is 3scorona, and the total outer radius of a micelle is thus R + 3scorona. A comparison of this quantity with RHS shows that RHS is much smaller. One can thus conclude that the outer low-density parts of the corona are not important with respect to the intermicellar interactions. Inspection of Figure 4c shows that the polymer volume fraction is about 3% at RHS in agreement with previous findings for the Pluronic P85.23 Discussion The combined analysis of SANS and SAXS data has provided very detailed information about the structure


and interactions of the C18EO100 block copolymer micelles as a function of concentration and temperature. The SANS and SAXS data, which are obtained under very different contrast conditions, have a relatively large information content. A simultaneous analysis of the data, which are on absolute scale, and the use of an advanced model recently derived from Monte Carlo simulations make it possible to extract this information. The applied model has the novel feature that part of the PEO chains is taken as a “background” concentration belonging to the solvent. This leads to large changes of the scattering length density of the components of the micellar model at high concentrations. This feature is crucial for obtaining high-quality fits at high concentration. The background contribution to the solvent is a result of overlap of the coronas of neighboring micelles, and it becomes important as soon as the polymer concentration exceeds 1 wt %. For a concentration of 10 wt %, more than half of the PEO chains belong to the background contribution. The radial volume distribution functions of the two components of the micelles were calculated by Fourier transformation of scattering amplitudes. The values for the core radius are in the range of 15-17 Å. These values can be compared to maximum and minimum length of a C18 chain. Following Tanford,72 the maximum length for a fully stretched chain is lmax ) 1.54 + 17 × 1.265 ) 23 Å whereas the minimum length for a flexible chain is lmin ) 1.54 + 17 × 0.925 ) 17 Å. The values for the core radius thus suggest that the alkyl chains are in a flexible conformation. The interface of the core and the interface between core and corona were found to be very broad. It is rarely that such profiles have been calculated previously, and the literature contains only a few examples with which the results can be compared. Tomsic et al.73 have recently published a study of Brij 35 (C12E23) in various simple alcohols. SAXS data were analyzed by indirect Fourier tranformation and square-root deconvolution, and the radial excess electron density distributions were obtained. For this profile, the core-corona interface was very broad and had a width similar to the one determined in the present work. The radius of the core was about 10 Å, which is a little less than the value for flexible chains obtained according to Tanford72 in agreement with the situation for the present micelles. Studies of sodium alkyl sulfate micelles by SAXS and detailed modeling74 have shown that for short alkyl chains (C9), the interface is more narrow (σ ) 1.5 Å) than found for the present Brij micelles. However, as the length increases, the interface becomes broader and for C11 the interface is described by σ ) 2.8 Å, which approaches the value of σ ≈ 5 Å for the C18 chains in the present study. The previous works on the sodium alkyl sulfate micelles74,75 have shown that the radius of the micelles closely follows R [Å] ) 0.73 + 1.37nc [Å], where nc is the number of carbon atoms in the chains, in reasonable agreement with the Tanford expression for straight chains. Our data for the present system are in disagreement with this behavior of the core radius. A possible explanation could be that the large PEO chains induce a large curvature of the core-corona interface and that this leads to a relatively low aggregation number. As it is desirable to have space-filling chains, this can lead (72) Tanford, C. The hydrophobic effect, 2nd ed.; John Wiley & Sons: New York, 1980. (73) Tomsic, M.; Bester-Rogac, M.; Jamnik, A.; Kunz, W.; Touraud, D.; Bergmann, A.; Glatter, O. J. Phys. Chem. 2004, 108, 7021. (74) Vass, S. J. Phys. Chem. B 2001, 105, 455. (75) Vass, S.; Gilanyi, T.; Borbely, S. J. Phys. Chem. B 2000, 104, 2081.

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to interdigitation of the chains located on opposite sides of the core and to a broad core-corona interface. The radial profiles of the corona show that there is a very high volume fraction of PEO (30-50%) at the corecorona interface. The corona is very extended and reaches typically up to a distance of 100 Å from the micelle center. Despite this, the range of the corona is not large enough that it can be established that its decay follows the prediction from scaling theory. The results for the radial profiles can also be compared with the recent results from molecular dynamics simulations for lower molecular weight nonionic micelles of the same type, namely, C12E6.76 The mass density radial distribution profiles were determined at 10 and 40 °C, and only weak temperature dependence was observed. The profiles are qualitatively similar to those determined in the present work; however, the width of the core-corona interface is narrower, only about half of the width found for the present micelles. The radius of the core is in agreement with the length of a fully stretched C12 chain. So this does not agree with the results in the present work for C18E100 micelles and those in Tomsic et al.73 for C12E23 micelles. The main difference of the molecules is a relatively larger PEO block in these two latter studies. A possible explanation of the difference in interface width and in core size could be that the stronger interactions within the corona of these micelles are due to the relatively larger PEO block as mentioned above. In the present study, the overall conclusion with respect to the variation of the micellar structure with temperature is that it can be explained by the variation of the solvent quality of the coronal chains and the associated reduced intra- and interchain interactions. When the interactions decrease, the corona shrinks and simultaneously the aggregation number increases. In a novel approach, the structure factors were also considered in the Fourier transformation, so that the average distribution around each micelle, including the neighboring micelles, could be calculated. These profiles show clearly the overlap of the corona of one micelle with the coronas of the neighboring micelles as the polymer concentration exceeds 1 wt %. The profiles also show how the corona shrinks, when the solvent quality decreases with increasing temperature, so that the solvent background contribution is reduced. A model with a monodisperse size of spherical micelles provided excellent fits to the measured scattering data in the full range of temperatures and concentrations, and it can be concluded that the polydispersity of the micelles is quite low. The low polydispersity is in agreement with the micelles having spherical cores, so that the system follows a closed aggregation model, which for certain parameters can predict such a narrow size distribution. As the micellar aggregation number increases with concentration even at room temperature, it can be concluded that molecular exchange takes place in the full temperature range investigated despite the large hydrophobicity of the C18 chains. Even at the highest temperatures investigated, micelles persist due to the large hydrophobicity of the C18 chains. As mentioned, the scattering data are in agreement with a spherical core at all temperatures investigated. The interaction between the coronal chains becomes weaker at high temperature; however, this weakening is not so large that it leads to a change in the core shape, but it is still large enough that the aggregation number increases with increasing tem(76) Sterpone, F.; Pierleone, C.; Brigante, G.; Marchi, M. Langmuir 2004, 20, 4311.

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perature. The relatively strong interchain interaction is supported by the high values (3.5-4.5) of the reduced surface coverage, which demonstrate significant overlap between the chains in the corona. The weakening of the interchain interactions with increasing temperature is directly observed in the temperature dependence of the effective single-chain scattering. The forward value P′(0) reflects this temperature dependence and could be analyzed in terms of the FloryHuggins theory of polymer solutions. The behavior is in agreement with a theta temperature of 100 °C when it is considered that there is a finite polymer concentration within the corona. The concentration effects could be satisfactorily described in the full range of temperature and concentration by an effective hard-sphere structure factor. Some minor deviations are observed at low scattering angles; however, these are not systematic. The coronas are highly swollen, and therefore the effective hard-sphere volume fraction is at room temperature about 8 times the polymer concentration. At high concentration, the coronas of different micelles interpenetrate and this gives rise to a reduction of the effective hard-sphere interaction radius. The reduction of solvent quality with increasing temperature leads to a large reduction of effective hard-sphere volume fraction, and the contraction of the corona due to the reduced solvent quality leads to a reduction of the effective hard-sphere interaction radius. The scattering data do not show any clear signs of intermicellar attractions supporting that the solvent quality does not become poor in the investigated range of temperatures. Conclusions The present work has shown that advanced modeling and combination of SAXS and SANS can provide very detailed information on the structure of block copolymer micelles. Brij 700 (C18EO100) micelles have been studied in a wide concentration range and in a wide temperature range. A satisfactory modeling of the data required that part of PEO belongs to a solvent background contribution in agreement with previous suggestions.55,56 The micelle structure depends weakly on concentration and displays more pronounced temperature dependence. The micelles persist even at the highest temperatures investigated (90 °C). The overall conclusion is that the variation of the micellar structure with temperature can be explained by the variation of the solvent quality of the coronal chains and the associated reduced intra- and interchain interactions. When the interactions decrease as the theta temperature of PEO is approached, the corona shrinks and simultaneously the aggregation number increases. The variation of the intermicellar interactions is also in agreement with the variation of the solvent quality. The size of the micellar core is small compared to the length of the C18 alkyl chains, and the width of the corecorona interface is quite broad. This has been associated with the strong interactions between the coronal chains and the large curvature of the micellar core relative to the corona width. A more systematic investigation of the dependence on molecular weight of the blocks of the copolymer in the future could give more information to elucidate this point further.

Sommer et al.

Appendix: Calculation of Radial Profiles To visualize the structures of the micelles, various radial volume distributions have been calculated. The results and the model used for fitting the data can be used to calculate for the average radial volume distribution of polymer around a single micelle Fpol(r). It is obtained from the Fourier transform of the scattering amplitude Amic(q) (eq 12) weighted by volume instead of the scattering length. Note that (1 - fPEO)Vc is the part of the PEO belonging to the micelle. Assuming that there is no solvent or PEO in the center of the core, Fpol(r) is easily normalized to unity at r ) 0. The distribution of the C18 in the core Fs(r) can be obtained as the Fourier transform of

AVol s (q) ) NaggVsΦ(q,Rcore)

where the normalization is again Fs(r) equal to unity at r ) 0. The profile FPEO(r) of the corona is similarly given as the Fourier transform of

AVol c (q) ) NaggVc(1 - fPEO)Ac(q)

(A-2)

with the same normalization as used for Fs(r). The distribution of the PEO part in the solvent, FPEO,solv(r), assuming that there is no overlap of the coronas of different micelles, is given by

FPEO,solv(r) ) [1 - Fs(r)]ηPEOfPEO

(A-3)

The expressions and procedures given above are for the single micelle. However, it is also relevant to calculate the average polymer distribution around a micelle. In the following we describe how the average polymer distribution around the center of a micelle can be calculated. The center distribution of the micelles g(r) is given by the Fourier transform of the structure factor S(q), which in our case is that of a hard-sphere liquid. The pair correlation function is

g(r) - 1 )

1 2π2n

sin(qr) 2 q dq (A-4) qr

∫(S(q) - 1)

where n is the particle number density. The average radial distribution of polymer in the surrounding micelles Fpol,surr(r) is the convolution of the product of g(r) - 1 and the distribution around a single micelle Fpol(r). Using the convolution theorem of Fourier transforms, the distribution can be obtained as

Fpol,surr(r) - 1 ) C 2π2n


∫(S(q) - 1)Amic(q)

where C is a constant determined by normalization so that Fpol(r) + Fpol,surr(r) at large r should approach ηpol, the volume fraction of polymer in the sample. The final correction that needs to be included is due to the fact that the solvent contribution of PEO is not homogeneous in the neighborhood of a micelle as the PEO is not present inside the cores of the surrounding micelles. The PEO distribution is modulated by

1 - Fs,surr(r) ) Cs

Acknowledgment. The financial support of The Danish Natural Science Research Council is gratefully acknowledged.

(A-1)


∫(S(q) - 1)As(q) 2π n 2

where 1 - Fs,surr(r) is normalized so that it goes to unity


at r ) 0 and to zero for large r. Thus

FPEO,solv(r) ) [1 - Fs(r)]ηPEOfPEO[1 - Fs,surr(r)ηC18] (A-7) where ηC18 is the volume fraction of the C18 chains. The average polymer profile around a micelle is the sum of the three contributions:

Langmuir, Vol. 21, No. 6, 2005 2149

Fpol,tot(r) ) Fs(r) + Fpol,surr(r) + FPEO,solv(r) (A-8) For short distances, this function describes the average polymer distribution within a micelle starting out from the center. For longer distances, it describes the average polymer distribution from the surrounding micelles again starting from the center of a micelle. LA047489K

Structure and Interactions of Block Copolymer Micelles of Brij 700

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