Polymer-Grafted Magnetic Nanoparticles in Nanocomposites

Article pubs.acs.org/Macromolecules

Polymer-Grafted Magnetic Nanoparticles in Nanocomposites: Curvature Effects, Conformation of Grafted Chain, and Bimodal Nanotriggering of Filler Organization by Combination of Chain Grafting and Magnetic Field Anne-Sophie Robbes,†,‡ Fabrice Cousin,*,† Florian Meneau,‡ Florent Dalmas,§ Ralf Schweins,∥ Didier Gigmes,⊥ and Jacques Jestin*,† †

Laboratoire Léon Brillouin, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France Synchrotron SOLEIL, L’Orme des Merisiers, PO Box 48, Saint-Aubin, 91192 Gif sur Yvette, France § Institut de Chimie et des Matériaux Paris-Est, UMR 7182 CNRS/Université Paris-Est, 2-8 rue Henri Dunant 94320 Thiais, France ∥ Institut Laue Langevin (ILL), DS/LSS, 6 rue Jules Horowitz, 38042 Grenoble Cedex 9, France ⊥ Institut de Chimie Radicalaire, UMR 7273 Aix-Marseille Université-CNRS, Av. Esc. Normandie Niemen Service 542, 13397 Marseille Cedex 20, France ‡

ABSTRACT: We present the synthesis and structural characterization of new nanocomposites made of linear tetramers of spherical magnetic nanoparticles of maghemite (γ-Fe2O3), grafted by tethered polystyrene (PS) chains with an intermediate grafting density (∼0.15 chains/nm2), dispersed in a PS matrix. First, we studied by combination of SAXS and TEM the dispersion state of the grafted objects within the matrix for various ratio R of the mass of the grafted chains N to the mass of free chains P ranging from R = N/P = 0.09 to R = 2.83. For R < 0.4, we obtained ramified aggregates of a few tens of grafted objects whose compactness slightly depends on R. For R = 1, the objects are well dispersed, a case that we have studied by recovering the free untethered residual chains resulting from the synthesis of the PS-grafted nanoparticles. For R > 1, the objects are also well dispersed although there are some remaining aggregates, arising probably from the fact that we have used matrix’ chains below the entanglement mass to reach such a high R values. This enabled to determine the threshold of the “wet-to-dry” transition, between the wetting of the brush by the free chain and the expulsion of them from the brush due to entropic effects, between 0.4 and 1, a value higher than the one (0.24) obtained on a very similar system on silica spherical particles by Chevigny et al. [Macromolecules 2011, 44 (1), 122−133]. This shift highlights the influence of the curvature of the grafted objects on the threshold’s transition. Second, we have made a direct measurement of the conformation of the grafted chains in the case of aggregates, benefiting from the fact that the scattering length density (SLD) of maghemite has almost the same value as the one of deuterated PS chains. They are strongly collapsed in comparison with their conformation in bulk, as shown by a significant reduction of their radius of gyration accompanied by the deviation from a Gaussian behavior. Finally, by applying a magnetic field during casting on samples for R < 0.4, we showed that the aggregates can be aligned in the direction of the magnetic field. We demonstrate thus the possibility of tuning the structure of the fillers by coupling a double control: chemical and magnetic.

I. INTRODUCTION The mechanical properties of polymers or elastomers can be considerably improved by addition of hard particles inside the matrix. Although these improvements were evidenced in many experiments,1 with sometimes very spectacular effects in the case of nanosized particles, their fundamental mechanisms remain still to be described. It is however now commonly admitted that there is an intimate link between the local structure of the particles inside the matrix and the macroscopic response of the materials. One possible way toward a better understanding of the mechanisms is thus to rule such local © 2012 American Chemical Society

structure, starting from the specificities of the two basic components of the nanocomposites, i.e., the particle and the polymer, in order to foresee the macroscopic properties of the materials. This exciting but challenging way that we can call “nanotriggering” aims at triggering the aggregation of particles at the nanometer scale to produce reproducibly various controlled nanostructures within the polymeric or elastomeric Received: September 13, 2012 Revised: October 29, 2012 Published: November 7, 2012 9220

dx.doi.org/10.1021/ma301927e | Macromolecules 2012, 45, 9220−9231

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Table 1. Number-Averaged Molecular Masses and Polydispersity Index for the Grafted Chains and for the Matrix Chains Made from Hydrogenated and/or Deuterated Monomers and the Corresponding Grafted to Free Mass Chain Ratio Used in the Present Study Mn (g/mol)/PDI grafted chains (H/D) Mn (g/mol)/PDI matrix chains (H/D) ratio R grafted/matrix

26000/1.7 285000/1.1 0.09

26000/1.7 65000/1.9 0.40

17000/1.8 65000/1.9 0.26

17000/1.8 17000/1.8 1.00

26000/1.7 26000/1.7 1.00

17000/1.8 14000/2.8 1.21

26000/1.7 9200/1.05 2.83

magnetic field during the solvent casting, as it opens the way of designing anisotropic aligned reproducible nanostructures. The controlled aggregation of magnetic nanoparticles inside a polymer melt combined with orientation of particles at the nanometer scale under magnetic field is since recently of an increasing interest due to its promising potentialities,17−21 although it has still only been tested on bare magnetic particles without surface modification. Thanks to dipolar interactions, which are strongly dependent on the particle size, the nanoparticles can form chains of particles inside the matrix, and possibly a 1D filler network at high filler content, which enables thus to create materials with anisotropic macroscopic mechanical properties.17,20 Here, we propose to combine these two ways of nanotriggeringthe polymer grafting on particle surface and the magnetic alignmentin order to dispose of a bimodal level of control on the dispersion and on the organization of the particles inside the polymer matrix. A recent similar approach has been done by Jiao et al.22 at low grafting density to evaluate the competition between interparticles dipolar interactions and brush−brush entanglement effects on the final dispersion. We have worked here on magnetic nanoparticles of maghemite of ∼8 nm diameter that we have recently successfully grafted by PS chains with an intermediate density (∼0.15 chains/nm2)23 by adapting a grafting procedure previously developed for silica nanoparticles.24,25 This has enabled us to (i) probe the particles’ dispersion at intermediate grafting density for a large range of R ranging from 0.09 to 2.83 by a combination of SAXS and TEM, (ii) to make a direct measurement of the conformation of the grafted chains by SANS since we have chosen a system for which the maghemite core can be exactly matched by the chains of matrix, and (iii) to test the potentiality of the grafted objects to be aligned inside the polymer melt with an external magnetic field.

matrix. Ranging from perfectly dispersed nanoparticles to large aggregates up to the formation of a percolated network of particles, it would allow extracting fundamental physical laws and ultimately improving the design of material for dedicated applications. Among the different routes of nanotriggering, functionalizing the particles by grafting of polymeric chains before their mixing with the chains of the matrix is very promising. A recent attention has focused on nanocomposites for which the grafted chains are similar to the free chains, the χ parameter being then equal to zero. Thanks to a large recent experimental effort,2−6 a general state diagram is now emerging7 that depends both on the grafting density of the chains on the nanoparticles σ and on the ratio R of the mass of the grafted chains N to the mass of free chains P ratio (R = N/P). It can be summarized as follows: for either very low σ or very high σ, the particles are arranged as aggregates within the polymer matrix. The formation of aggregates is due to attractive interactions, either between particles at very low σ corresponding to allophobic wetting transitions7 or between grafted brushes at very high σ corresponding to autophobic wetting transitions.7 For intermediate σ, the dispersion state of the particles depends mainly on the interactions between the grafted and the free chains. It becomes then only tuned by R: when the free chains are smaller than the grafted ones, they can wet the grafted brushes owing to a favorable mixing entropy. This induces repulsive interactions between the grafted objects inside the matrix which leads to their perfect individual dispersion. On the contrary, when the free chains are larger than the grafted ones, the free chains are expulsed from the grafted coronas because of the unfavorable mixing entropy, which results in attractive interactions between objects and ultimately to the formation of dense aggregates of particles inside the matrix. This mechanism is accompanied by an elastic compression of the grafted chains6,8 for P > N and with an extension for P < N to minimize the free energy of the system. Beyond this clear view of the general behavior of these systems, several refining points remain still to be solved, as highlighted by recent simulations.9−12 It mainly concerns the threshold of the transition between the individual dispersion of particles and aggregates’ formation, usually called wet-to-dry transition,8,11 and the identification of the whole parameters that influence it. According to the pioneer work of De Gennes,13 this transition is supposed to occur at P = N in the case of planar surfaces but must be influenced by surface curvature effects when considering spherical particles,14 by the brush−brush and free chains−brush entanglement effects,15 and by the sample processing conditions. Recently, the experimental threshold of the transition has been experimentally determined at R = 0.24 on polystyrene (PS) grafted silica particles dispersed in a PS matrix, evidencing a significant shift in comparison with the theoretical expected value whose origin is still under discussion.16 Another attractive route of nanotriggering is the use of magnetic particles as fillers that can be aligned with an external

II. MATERIALS AND METHODS 1. Synthesis of the Grafted Nanoparticles. We have used PSgrafted maghemite nanoparticles whose complete description of synthesis can be found in our previous work.23 Briefly, after the synthesis of the magnetic core according to the Massart method,26 followed by their transfer in dimethylacetamide (DMAc), an organic polar solvent which is a good solvent for PS, the grafting of the chains was achieved by coupling a “grafting from” approach with a controlled radical polymerization process based on nitroxide-mediated polymerization (NMP). This “multistep” method allowed us to graft either hydrogenated or deuterated PS chains with grafting density close of 0.15 chains/nm2 while controlling the colloidal stability during the synthesis, avoiding strong aggregation of the particles. The final grafted objects, whose structure was fully characterized by a refined combination of SAXS/SANS measurements,23 were linear aggregates of 3−4 native maghemite spherical particles of diameter 8.1 nm, with a log-normal distribution, grafted by PS tethered chains, whose length can be adjusted via synthesis. In DMAc, the grafted chains behave as brushes in a theta solvent. Two batches of grafted particles have been synthesized for the present study (see Table 1): a first one using hydrogenated styrene monomers giving grafted PS chains of mass equal to Mn = 26 000 g/mol and a second one using deuterated 9221


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styrene monomers giving grafted deuterated PS chains of mass equal to Mn = 17 000 g/mol. The mass of the grafted chains was determined by GPC after cleaving from the maghemite core.23 2. Preparation of the Matrix Chains. The different masses of PS used for the matrix of the nanocomposites are reported in Table 1. The PS chains of Mn = 9200 (PDI = 1.05) and Mn = 285 000 g/mol are coming from Polymer Sources. For Mn = 17 000 and 26 000 g/ mol, they were obtained during synthesis of the grafted nanoparticles. Indeed, owing to the presence of free initiator in NMP, the free chains have been synthesized at the same time as the grafted chains. They have thus exactly the same mass as the grafted ones. At the end of the polymerization, the free chains have been separated from the grafted ones by ultrafiltration. The chains of Mn = 14 000 g/mol have been synthesized by NMP (120 °C for 2 h) using PBO as initiator. The chains of Mn = 65 000 g/mol have been synthesized by classical radical polymerization (90 °C for 4 h) using AIBN as initiator. The mass of all chains was measured by GPC using a standard method. As described in the following, the use of deuterated chains for the matrix permits to match the neutron scattering contribution coming from the maghemite particles and thus to analyze only the scattering from the grafted corona (section III.2). We also used a deuterated grafted corona around the particles in hydrogenated matrixes as a tool of choice for probing the conformation of the free chains of the matrix, a study that we will present in a forthcoming paper. 3. Preparation of the Nanocomposites. The preparation of nanocomposites follows the process developed in the laboratory as described by Jouault et al.27 The grafted nanoparticles, dispersed in DMAc, are mixed with a concentrated solution of free PS chains, typically 10% v/v also in DMAc at various fractions of particles. The mixtures are stirred (using a magnetic rod) for 2 h. They are then poured into Teflon molds (5 cm ×5 cm ×2.5 cm) and let cast in an oven at constant temperature Tcast = 130 °C for 1 week, largely below the boiling point of DMAc (T = 167°), which ensures a gentle evaporation of the solvent. This yields dry films of dimension of 5 cm ×5 cm ×0.1 cm (i.e., a volume of 2.5 cm3). For the casting under magnetic field, a constant magnetic field was imposed by two plates of iron magnetized by a series of permanents NeFeB magnets inserted in the mold. The number of permanents magnets inserted in the mold enables to vary the magnetic field up to a maximum value of 600 G. The cartography of the applied field within the mold was done with a Hall effect probe to test its homogeneity. For a desired nominal value of applied field of 600 G, the homogeneity was very satisfactory: the lowest and upper magnetic field measured were respectively 585 and 615 G. 3. SAXS Experiments. The SAXS experiments were carried out at the SWING beamline of the French Synchrotron SOLEIL. Measurements were performed at an energy of 7 keV (λ = 1.77 Å), with a twodimensional AVIEX-CCD detector placed at a distance of 6.5 and 1.4 m from the sample. A 3 mm beam stop with a photodiode inserts in its center enables to measure the transmission. The resulting q-range (the scattering vector q, defined by q = 4π sin(θ/2)/λ, where θ is the angle between direct beam and scattered beam), spanned from 1.8 × 10−3 to 0.15 Å−1. The scattering of a pure PS matrix, without magnetic nanoparticles, Ipure_PS, was first measured. In order to remove the PS chains’ contribution in the films, (1 − Φpart)Ipure_PS was subtracted from all the nanocomposite samples. The data reduction has been done by the software Foxtrot. 4. SANS Experiments. SANS measurements were performed at the Institut Laue Langevin on the instrument D11. Three configurations were used with a wavelength of 8 Å and sample-todetector distances of 34, 8, and 1.5 m, corresponding to a total q-range of 0.001−0.2 Å−1. All measurements were done under atmospheric pressure and room temperature. Standard corrections for sample volume, neutron beam transmission, empty cell signal subtraction, detector efficiency, subtraction of incoherent scattering, and solvent were applied to get the scattered intensities on absolute scale. The data reduction has been done with the software Pasinet. 5. Transmission Electronic Microscopy. In order to complete at larger scale the SAXS analysis of the nanocomposite structure, conventional TEM observations were also performed on composite

materials. The samples were cut at room temperature by ultramicrotomy using a Leica Ultracut microtome with a diamond knife. The cutting speed was set to 0.2 mm s−1. The thin sections of about 40 nm thick were floated on deionized water and collected on a 400-mesh copper grid. Transmission electron microscopy was performed on a FEI Tecnai F20 ST microscope (field-emission gun operated at 3.8 kV extraction voltage) operating at 200 kV. Precise scans of various regions of the sample were systematically done first at small magnification and then at increasing magnification. The slabs observed were stable under the electron beam. The sample aspect was the same in every spot of every piece, and typically 10 different slabs were observed. The pictures presented in the following are completely representative of the single aspect of the sample, which appears in average homogeneous.

III. RESULTS 1. Dispersion of the Grafted Nanoparticles in the Nanocomposite. We first investigate the evolution of the particles’ dispersion inside the film without magnetic field as a function of the particle concentration, when varying the mass ratio between the grafted and the matrix chains. For the intermediate grafting density investigated here (around 0.15 chains/nm2), such a dispersion is expected to depend mainly on the interactions between the grafted and the free chains. Formerly, in the case where the PS grafted chains are hydrogenated and the free chains of the matrix are deuterated chains, or in the reverse case of grafted deuterated chains and hydrogenated free chains, the Flory parameter is not null but has a very weak value (χ ∼ 4 × 10−3). However, such a value remains low enough to not affect the mechanisms of particle dispersion inside the film which is expected to depend mainly on the interactions between the grafted and the free chains. 1.1. Grafted to Free Chains Length Ratio R < 1. We have checked three values of R < 1: R = 0.09, 0.26, and 0.4. The sample corresponding to R = 0.09, made of grafted hydrogenated PS chains of Mn = 26 000 g/mol dispersed in a deuterated PS matrix of Mn = 285 000 g/mol, has been analyzed by SAXS for two particle concentrations of maghemite: 0.5% v/v and 3% v/v (Figure 1). Because of the specific interaction between the photons and the electronic cloud of the elements, X-rays permit to get a very good contrast between γ-Fe2O3 maghemite and polystyrene, whether it is hydrogenated or deuterated and/or whether it belongs to the grafted corona or the matrix, i.e only the scattering of the magnetic core is probed here. For both concentrations, the scattering curves normalized by the particle volume fraction superimpose nicely over the whole Q range. The structure is thus perfectly similar at the local scales probed by SAXS in both samples. At high Q, the curves decay like Q−4, indicating a smooth interface between the particles’ core and the polymer. When going toward low Q, there is a change of slope located around Q ∼ 0.05 Å−1 followed by a power law regime of the scattered intensity which finally ends as a plateau for the lowest Q values. Thanks to our previous characterization of the grafted particles in solution,23 the form factor of the core of the grafted objects made of maghemite particles is known and can be modeled with linear aggregates of 3−4 native particles, as shown by the full red line in Figure 1. As the PS brush was covalently grafted on these primary aggregates, such form factor cannot be affected by the mixing with free polymer chain and will remain constant in all nanocomposites. It can thus be used to extract the structure factor of the particles inside the film. It is obtained by division of the total scattered intensity by the form factor, as presented 9222


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Figure 2. TEM images of nanocomposites polymer matrix (M = 285 000 g/mol) filled with grafted particles (Mn = 26 000 g/mol) corresponding to a grafted to free chains length ration R = 0.09 at Φmag = 0.5% v/v of particle volume fractions (a) and (b) and at Φmag = 3% v/v of particle volume fractions (c) and (d) for low (a, c) and higher (b, d) magnification.

Figure 1. SAXS scattering curves for nanocomposites at grafted to free chains length ratio R = 0.09 filled with particle volume fraction Φmag = 0.5% v/v (gray open circles) and Φmag = 3% v/v (black open circles). The full red line is the form factor of the core of the grafted objects, a linear aggregate composed of 3−4 native spherical nanoparticles, as determined in solution.23 The structure factor S(Q) deduced from the division of the intensity by the form factor P(Q) is presented in the inset.

main features of the curves are similar, showing that the obtained structures are not concentration dependent. At large Q, the curves superimpose to the form factor of the grafted objects. At Q1* = 0.025 Å−1, there is a correlation peak, which is clearly visible in the structure factor in Figure 3b, that corresponds to a distance of 250 Å in direct space (D = 2π/ Q1*). In the sample, the smallest possible distance between two centers of mass would correspond to the contact, side by side, of two linear aggregates, i.e., to twice the sum of the maghemite nanoparticules’ radius (40 Å) and the thickness of the brush εbrush. In a θ solvent, the extension of the grafted brush εbrush is 92 Å, as deduced from the analysis in solution.23 The minimal distance between two centers of mass in the sample is thus of the order of 2 × (40 + 92) = 264 Å in case of noncollapsed brushes, or lower if the brushes are partially collapsed, close to the experimental obtained distance. The correlation peak corresponds thus to the contact between two primary aggregates, which are densely packed in larger aggregates. In the intermediate Q range, the scattered intensity exhibits a power law behavior. The 2.6 exponent is however much larger than for the previous case, indicating the formation of denser aggregates for this grafted to free chains length ratio, in accordance with the appearance of the correlation peak at 0.025 Å−1. For the lowest Q, we can observe a nice plateau, which is the signature of the finite size of the aggregates, and even a maximum at Q2* which is highlighted in the structure factor S(Q) (see Figure 3c). It corresponds in real space to the typical distances between the aggregates (D = 2π/Q*) and is equal respectively to 1.1 and 0.6 μm at 0.5% and 3% v/v. If the sizes of the aggregates are similar in both samples and if the dispersion is homogeneous, the shift of this distance would come from a simple dilution law, i.e., (DΦ1/DΦ2)3 = Φ2/Φ1. The two experimental values strongly support this hypothesis as one obtains (1.1/0.6)3 = 6.15. The presence of the correlation peak does not allow a proper determination of Nagg, even it can be roughly estimated at around ∼50, a value higher than for R = 0.09. We completed the study of the regime R < 1 with a third grafted to free chain mass ratio (R = 0.4) made with grafted

in inset for the two particles concentrations. The structure factors show a nice Q−1.7 dependence indicating the formation of ramified aggregates of grafted objects inside the polymer matrix. The plateau at low Q shows that the aggregates have a finite size. In this case, the value of the scattered intensity when Q tends toward 0 is a direct measurement of the aggregates’ mass: I(q)q→0 ∼ ΦΔρ2NaggV0 ∼ Magg, where V0 is the volume of the core of the grafted object. Thus, the ratio of the scattered intensity of the plateau Isample(q)q→0 (∼ΦΔρ2NaggV0) to the one of the form factor Iform_factor(q)q→0 (∼ΦΔρ2V0) is a direct measurement of the aggregation number Nagg of primary aggregates within the ramified aggregate, which can be directly obtained by the lowest value of the structure factor. Nagg is here ∼25. The same samples have been also analyzed with TEM whose pictures are presented in Figure 2. The images confirm the presence of individual ramified aggregates of finite size made of grafted entities in the real space. The number of particles per aggregate is consistent with the one determined by SAXS (∼80−100 nanoparticles, as Nagg ∼ 25 for primary aggregates of 3−4 particles). The aggregates do not appear connected one to each other, even if the thickness of the cut (40 nm) does not allow concluding unambiguously as it is thinner than the typical size of an aggregate. The dispersion is homogeneous up to the largest scale probed by TEM, and the typical size of the aggregates is consistent with the one deduced from SAXS: Rg ∼ 1/0.002 ∼ 500 Å. We have analyzed a second series of nanocomposites made of deuterated PS grafted particles (Mn = 17 000 g/mol) dispersed in a deuterated PS matrix (Mn = 65 000 g/mol) for two particle contents (0.5% and 3% v/v) corresponding to a grafted to free chain length ratio equal to R = 0.26. The SAXS curves are presented in Figure 3. The sample containing 0.5% v/v of particles has been measured with an additional Bonse− Hart setup which permits to cover another smaller decade in the Q range down to 2 × 10−4 Å−1. As for the previous case, the 9223


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Figure 3. (a) SAXS scattering curves for nanocomposites at grafted to free chains length ratio R = 0.26 filled with particle volume fraction Φmag = 0.5% v/v (pink open circles) and Φmag = 3% v/v (violet open circles). The full red line is the form factor of the core of the grafted objects, a linear aggregate composed of 3−4 native nanoparticles, as determined in solution.23 Structure factors S(Q) deduced from the division of the intensity by the form factor P(Q) are presented in the inset in log−log (b) and linear (c) representations.

Figure 4. SAXS scattering curves for nanocomposites at grafted to free chains length ratio R = 0.4 filled with particle volume fraction Φmag = 0.5% v/ v (violet open circles). The full red line is the form factor of the core of the grafted objects, a linear aggregate composed of 3−4 native nanoparticles, as determined in solution.23 The structure factor S(Q) deduced from the division of the intensity by the form factor P(Q) is presented in the inset. On the right part, the corresponding TEM images in the real space at two magnifications are shown.

values for R = 0.09 and R = 0.4: 30(3/1.6) = 600 Å and 100(3/2.1) = 700 Å, respectively . Furthermore, the correlation peak at 0.025 Å−1 is hardly visible on insets of Figures 1 and 4, confirming the opening of the aggregates in both cases. A significant discrepancy is observed for R = 0.26, which shows much denser aggregates (Df = 2.6, which is close to a fully compact structure given than one starts from linear ramified objects of 3−4 native nanoparticles) illustrated by the large intensity of the correlation peak at 0.025 Å−1. It is possible that this difference may result from the lower mass of the grafted chains used in this case (17 000 g/mol) than for the two other cases (26 000 g/mol). Indeed, once aggregates are formed due to dewetting processes, it is possible that the mass of the grafted chains influences the compactness of aggregates. It has been postulated8 that the outer part of the collapsed brush remains partially wet in an interdiffusion region within aggregates. Such

hydrogenated chains of Mn = 26 000 g/mol dispersed in a deuterated matrix of Mn = 65 000 g/mol. Figure 4a presents the SAXS scattering curves obtained for the sample containing 0.5% v/v of particles and the corresponding TEM images in Figures 4b and 4c. Again, the scattering curve exhibits a typical aspect of the formation of ramified aggregates made of selfassembled primary grafted objects whose form factor is recalled by the full red line. Although there is no clear plateau at low Q here, which makes difficult to extract a value of the aggregation number Nagg, the low Q part of the S(q) presented in inset of Figure 4 shows that the magnitude of Nagg is ∼100. The TEM picture in real space confirms this structural organization. When considering both the fractal dimension and the aggregation number, one can estimate the size Ragg of the ramified aggregates according to Ragg/Robject ∼ Nagg(3/Df), where Robject is the typical size of the grafted objects. We obtain close 9224


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Figure 5. SAXS scattering curves for nanocomposites at grafted to free chains length ratio R = 1 filled with particle volume fraction Φmag = 0.5% v/v (left, Mn,grafted = Mn,matrix = 17 000 g/mol) and Φmag = 7% v/v (right, Mn,grafted = Mn,matrix = 26 000 g/mol). The full red line is the form factor of the core of the grafted objects, a linear aggregate composed of 3−4 native nanoparticles, as determined in solution.23 Structure factors S(Q) deduced from the division of the intensity by the form factor P(Q) are presented in the insets. For the concentrated case, the structure factor S(Q) is compared to a model of Percus−Yevick (see text for details).

reveals a very slight aggregation of primary aggregates. The aggregation number can be estimated by extrapolation at Q = 0 and is ∼3 primary aggregates. When looking at the concentrated case, we can observe a nice correlation peak in the intermediate Q range, characteristic of a well-defined dispersion of the grafted objects inside the polymer matrix. The structure factor of the grafted objects S(q), obtained after division by the form factor of the primary aggregates (inset on the right part of Figure 5), recalls the one of a repulsive system. Apart from a small upturn at very low Q, which probably comes from the scattering of voids coming from the solvent casting sample processing,31 S(q) tends to a very small value when going toward low Q; i.e., it has a very weak compressibility. The position of the peak (0.018 Å−1) corresponds in the real space to a typical distance between objects equal to 350 Å. This distance, equal to 4 diameters of the native particles, suggests an individual dispersion of the primary grafted aggregates (made of 3−4 natives particles) which are very close one to each other. The main peak of the structure factor, as well as its behavior in the low Q region, can be modeled with a Percus−Yevick function, which is the exact structure factor of interacting hard-sphere systems,32 with only two free parameters: the size and the effective volume fraction of the objects. The result of the modeling is reported in the inset of Figure 5, taking an effective size of 346 Å and an effective volume fraction of particles equal to 20% v/v. The size corresponds to the one of the primary aggregates made of 3−4 native maghemite particles (4 diameters). The 20% v/v effective volume fraction is thus also consistent with the experimental value of 7% v/v of native maghemite particles because it includes both the grafted corona volume contribution and the fact that our objects are not spherical but linear aggregates. Even if the Percus−Yevick function is a very raw approximation for our system of linear primary aggregates, it permits to model correctly both the weak compressibility and the shape of the main correlation peak of the experimental structure factor which illustrates the welldefined dispersion of the grafted primary objects. The “quasi” individual dispersion of the primary grafted aggregates is also

partial wetting counteracts the full collapse of the grafted objects and leads to more open structures. The size of this interdiffusion region will increase with the increase of the mass of the grafted chain. Thus, for a given R, the compactness of the aggregates will decrease with an increase of the mass of the grafted chains. 1.2. Grafted to Free Chains Length Ratio R = 1. A second series of samples have been investigated using the same chains than the grafted ones for the matrix to probe the case R = 1. They were obtained at the end of the synthesis of the PSgrafted nanoparticles by NMP polymerization. Indeed, in NMP techniques, the control of the polymerization reaction requires a minimal concentration of controller (nitroxide radical) which is achieved by the addition in the solution of free “sacrificial” initiator.28 This results in the formation of free untethered chains in solution, of masses identical to the grafted ones as previously demonstrated,29,30 in addition to the tethered ones grafted at the particles surface. These free chains are usually removed from the solution of grafted objects by an additional purification step. If such purification may often be considered as a drawback, we have turned it to an advantage as it offers the possibility to make films in which the grafted and the free chains have been synthesized together, in order to unambiguously check with the case R = 1. After the polymerization process, the free PS chains have been removed by ultrafiltration and then mixed again with the grafted object to keep the film processing conditions identical to the ones used for R ≠ 1. We have realized two films: a dilute one at 0.5% v/v of particles made of grafted and free deuterated chains of Mn = 17 000 g/ mol and a concentrated one at 7% v/v made with hydrogenated chains of Mn = 26 000 g/mol. The limited amount of free chains extracted from the purification did not allow us to process samples with several concentrations for a given Mn. The scattering curves obtained by SAXS are presented in Figure 5. For the dilute case, the scattering almost follows the form factor of the primary grafted objects at large and intermediate Q, indicating that the system is close to the individual dispersion of the primary aggregates. We can see an upturn with a Q−1.6 decay that arises from the nanoparticle’s core and 9225


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the polymer matrix (equal to 18 000 g/mol for PS). The SAXS scattering curve obtained on the film filled with 3% v/v of particles is presented in Figure 7. It has roughly the same behavior as the one already observed for R < 1 (R = 0.09 and R = 0.26), with a power law decay like Q−1.7 at intermediate Q and a plateau at the lowest Q, which suggests the formation of ramified aggregates. There is however a large difference with respect to these previous cases, as the aggregation number Nagg obtained from the plateau is slightly lower than 10. The aggregation is thus much more limited as for R < 1, as confirmed by the corresponding TEM images in the real space on the right part of Figure 7. It is worth noting that the dispersion remains homogeneous up to a large space range on the TEM pictures. It also appears that there are both isolated primary aggregates and a limited amount of rather large aggregates, the Nagg of ∼10 being an average between two coexisting populations of isolated objects and aggregates. The final dispersion appears to be an intermediate situation between the individual dispersion obtained at R = 1 and the large ramified aggregates for R < 1. Such an intermediate situation can be explained by the fact that we are here below the entanglement mass of the polymer matrix, a situation that can strongly influence the film processing conditions and especially the kinetics of the solvent evaporation that could be faster for small chains forming the matrix than for chains upon entanglement mass. It is then possible that there are gradients of solvent content within the film during casting and that the consequent local hydrodynamic instabilities induce the formation of some large aggregates before the freezing of the system due to the increased viscosity at the end of the evaporation process. Moreover, the lack of entanglements will not slow down the diffusion of grafted objects within the sample before this freezing, enhancing the possibility to form aggregates. These aggregates which largely contribute to the scattering would then coexist with isolated objects in the sample. This behavior is confirmed when increasing the grafted to free chain ratio R to R = 2.8, decreasing more the mass of the

nicely confirmed with the TEM images in real space for both particle concentrations presented in Figure 6.

Figure 6. TEM images of nanocomposites polymer matrix (Mn = 17 000 g/mol) filled with grafted particles (Mn = 17 000 g/mol) corresponding to a grafted to free chains length ratio R = 1 at Φmag = 0.5% v/v of particles volume fractions at high (a) and low (b) magnification. TEM images of nanocomposites polymer matrix (Mn = 26 000 g/mol) filled with grafted particles (Mn = 26 000 g/mol) corresponding to a grafted to free chains length ratio R = 1 at Φmag = 7% v/v of particles volume fractions at high (c) and low (d) magnification.

1.3. Grafted to Free Chains Length Ratio R > 1. We finally investigate the last regime (R > 1), a specific case when the length of the polymer chain matrix is lower than the grafted one. The first value of R = 1.2 has been made using PS chains of Mn = 14 000 g/mol as matrix in which PS grafted particles (Mn = 17 000 g/mol) have been dispersed. Please note that probing the regime R > 1 forces us to use very short chains for the matrix, and we worked here below the entanglement mass of

Figure 7. SAXS scattering curves for nanocomposites at grafted to free chains length ratio R = 1.2 filled with particles volume fraction Φmag = 3% v/v (blue open circles). The full red line is the form factor of the core of the grafted objects, a linear aggregate composed of 3−4 native nanoparticles, as determined in solution.23 The structure factor S(Q) deduced from the division of the intensity by the form factor P(Q) is presented in inset. On the right part, the corresponding TEM images in the real space at two magnifications are shown. 9226


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2. Conformation of Grafted Chains in the Case of Dispersion of Aggregates. In this section, we perform the determination of the conformation of the grafted chains when the grafted objects are forming aggregates within the matrix. As recalled in the Introduction, the correlation between this grafted brushes conformation in grafted nanocomposite and the final dispersion state of the particles inside the film remains a key point to understand the mechanisms of particles dispersion. For such a purpose, SANS is a tool of choice. Indeed, if the neutron scattering length density of the core is at the same time similar to the one of the chains of the matrix but different from the one of the grafted chains, only the corona becomes visible under neutron irradiation. That approach has been used by Chevigny et al.;6,8 by means of it they have achieved the direct determination of the grafted brushes conformation in the case of individual dispersion of grafted particles in the matrix, for a system composed of PS chains grafted on silica particles. To this aim, they have synthesized statistical hydrogenated/ deuterated PS copolymer chains whose averaged neutron scattering length matches the one of the particles used for the formation of the matrix. However, they could only indirectly deduce the brush conformation in the case of aggregates from interparticle distances extrapolation.6,8 Our experimental system has here a unique specificity with respect to SANS: the maghemite nanoparticles have almost exactly the same neutron scattering length density ρ (6.86 × 1010 cm−2) as the deuterated PS polymer chains (6.56 × 1010 cm−2), while the hydrogenated PS chains have a very different one (1.47 × 1010 cm−2). In a composite made of hydrogenated grafted brushes dispersed inside deuterated chains, the matching of the core will thus be achieved whatever the aggregation state of the objects. We thus choose to focus on two cases of particle dispersion where we obtained aggregates: ramified (R = 0.09) and much denser (R = 0.4). For both cases, the grafted chains are hydrogenated (Mn = 26 000 g/mol) and dispersed in deuterated chains for the matrix (Mn = 65 000 g/mol for R = 0.4 and Mn = 285 000 g/mol for R = 0.09). The SANS

chain matrix down to Mn = 9200 g/mol while using grafted chains of Mn = 26 000 g/mol (Figure 8). As for the case of R =

Figure 8. SAXS scattering curves for nanocomposites at grafted to free chains length ratio R = 2.8 filled with particles volume fraction Φmag = 3% v/v (blue open circles). The full red line is the form factor of the core of the grafted objects, a linear aggregate composed of 3−4 native nanoparticles, as determined in solution.23 The structure factor S(Q) deduced from the division of the intensity by the form factor P(Q) is presented in the inset.

1.2, the scattering follows the form factor of the primary grafted objects at high Q, showing a power law at intermediate Q that ends to a plateau when going toward low Q. The mean Nagg is here very limited (∼6), suggesting a rather good dispersion of individual grafted objects coexisting with a few large aggregates whose origin of formation comes from nonentanglement of the polymer chain matrix.

Figure 9. (a) SANS scattered intensity (black open circles) by a nanocomposite filled with Φmag = 0.5% v/v of hydrogenated PS grafted particles (Mn = 26 000 g/mol) dispersed in deuterated PS matrix (Mn = 285 000 g/mol) corresponding to R = 0.09. (b) SANS scattered intensity (black open circles) by a nanocomposite filled with Φmag = 0.5% v/v of hydrogenated PS grafted particles (Mn = 26 000 g/mol) dispersed in deuterated PS matrix (Mn = 65 000 g/mol) corresponding to R = 0.4. In the insets, the structure factors between the grafted objects deduced from SAXS measurements Figures 1 and 4. The form factors P(Q) (crosses) of the grafted brushes are calculated from the division of the total SANS intensities by the SAXS structure factors. The full red lines correspond to the model of the form factor of an ellipsoid core grafted with Gaussian polymer chains, derived from a model initially developed by Pedersen33 (see text for details). 9227


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Table 2. Results of the Modeling of the Form Factor of the Grafted Ps Corona in Solution and inside the Nanocomposite Films for R = 0.09 and R = 0.4 Using the Gaussian Chain Model Developed by Pedersen

modeling in solutiona modeling in melt for R = 0.09 modeling in melt for R = 0.4 a

particle volume fraction (% v/v)

ellipticity of core ε

polydispersity of core σ

grafting density (chains/nm2)

volume of the grafted chain (Å3)

radius of gyration of the grafted chains (Å)

deviation from the Gaussian chain behavior at high Q

0.50

0.22

0.23

0.13

54.7 × 103

92

Q−(2+0)

0.55

0.22

0.23

0.13

54.7 × 103

71

Q−(2+1.3)

0.54

0.22

0.23

0.13

54.7 × 103

76

Q−(2+0.8)

From ref 23.

Figure 10. (a) SAXS scattering curves for nanocomposites at grafted to free chains length ratio R = 0.4 filled with particles volume fraction Φmag = 0.5% v/v processed with a magnetic field of 600 G, either averaged along the direction parallel to the applied magnetic field (blue open circles) or to the direction perpendicular to the applied field (red open circles) and compared with the isotropic case (black open circles). The dotted black line is the form factor of the core of the grafted objects, a linear aggregate composed of 3−4 native nanoparticles, as determined in solution.23 The inset shows the 2D SAXS spectra and the angular sectors used for averaging. (b, c) Corresponding TEM images in real space at two magnifications. The red arrow shows the direction of the applied magnetic field.

classical Q−2 power law, but with a much more pronounced slope. The chains of the grafted brushes are thus partially collapsed in the samples. In order to model the scattered intensity, we have used an analytical form factor Pgaussian_corona(q), computed by Pedersen,33 composed of an ellipsoid core of maghemite grafted with Gaussian polymer chains. A complete description of Pgaussian_corona(q) can be found in ref 23. It involves several parameters, namely the particles’ concentration, the geometric parameters of the ellipsoid core, the scattering length densities, the grafting density, and the radius of gyration of the grafted chains. We have kept constant all the parameters deduced from the modeling of the grafted objects in solution except the radius of gyration of the grafted chains, which is the lone physical value that may vary upon a change of conformation. The results are presented in Table 2, where we have also reported the deviation from the Gaussian behavior in the high Q range (Table 2). Two clear modifications of the conformation of the grafted brushes from solution to the films are highlighted by this modeling: a decrease of the mean radius of gyration of the grafted chains of around 20% (from 92 Å down to 71 Å for R = 0.09 and down to 76 Å for R = 0.4) accompanied by a significant deviation from the classical Q−2 power law (Q−3.3 for

scattering curves are presented in Figure 9. For both samples, a strong increase of the intensity can be observed at low Q, coming from the interactions between the grafted coronas. The scattering curve is indeed the product of the form factor of the grafted chains with their structure factor. Since our grafted objects are centrosymetrical, the structure factor depends only on the respective positions of the mass center of the objects. It is similar whether one probes the core of the objects or their corona and has thus been already determined with SAXS in the previous section on the maghemite core. The exact form factor of the grafted brushes can thus be extracted after division of the total SANS intensity by the SAXS structure factor and is presented in Figure 8. Such a determination is then a very nice example of the complementary combination of SANS and SAXS that enables to probe selectively the different parts of a ternary system. After normalization by the structure factor, the strong increase of the scattered intensity is nicely suppressed. The form factor shows a nice plateau at low Q and an oscillation in the intermediate Q range. Such oscillation arises here from the typical form factor of a corona and proves that one probes objects with a very well-defined size and geometry. At high Q, in the regime where the behavior of the chains is directly probed, there is a striking feature. The Gaussian behavior is lost as the scattered intensity does not decay by the 9228


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Figure 11. Left panel: (a) SAXS scattering curves for nanocomposites at grafted to free chains length ratio R = 0.09 filled with particle volume fraction Φmag = 0.5% v/v processed with a magnetic field of 600 G, either averaged along the direction parallel to the applied magnetic field (blue open circles) or to the direction perpendicular to the applied field (red open circles) and compared with the isotropic case (black open circles). The dotted black line is the form factor of the core of the grafted objects, a linear aggregate composed of 3−4 native nanoparticles, as determined in solution.23 The inset shows the 2D SAXS spectra and the angular sectors used for averaging. (b, c) Corresponding TEM images in the real space at two magnifications. The red arrow shows the direction of the applied magnetic field. Right panel presents exactly the same information (d) SAXS and (e, f) TEM images, but for a particle volume fraction of Φmag = 3% v/v. No significant effect resulting from the orientation of the aggregates by the magnetic field has been observed on the grafted chains conformation.

R = 0.09 and Q−2.8 for R = 0.4). The more the deviation from the Q−2, the more collapsed the brush is, as one would end with a Q−4 decay at high Q in the case of a fully collapsed brush because it would form a net interface with the matrix. The partial dewetting of the brush is thus very important in both samples. 3. Influence of an External Magnetic Field. We have previously demonstrated the possibility to align bare naked maghemite particles inside a PS matrix when applying a magnetic field during the sample processing.20 Because of dipolar interactions between maghemite particles, it is possible in given cases to form chains of particles, and a 1D filler network at high filler content, which results in anisotropic mechanical properties of the material. Here, we propose to combine this possibility of alignment of aggregates with the possibility to finely tune the aggregation of particles with grafting. We believe this would result in an unprecedented control of aggregation of hard particles in nanocomposites. We thus have focused on the case of R < 1 for which ramified aggregates of variable density are formed. It is indeed likely that the magnetic field would have no effect for dispersed particles at R = 1. We have synthesized three different samples with an applied magnetic field of 600 G during casting for grafted to free chain length ratio of R = 0.09, respectively filled with 0.5 and 3% v/v of grafted particles, and R = 0.4 at 0.5% v/v of particles. The concentrations have been chosen to be exactly similar to the ones of the samples casted without magnetic field presented in section III.1. Again the samples have been characterized with a combination of SAXS and TEM. During SAXS experiments, the samples were placed within the beam so that the direction of the applied magnetic field was vertical. On the TEM pictures, the direction of the field is recalled with a red arrow. The results are presented in Figure 10 for R = 0.4 and in Figure 11 for R = 0.09. The 2D SAXS patterns of the detector exhibited in all cases anisotropic features, an elliptic figure with an excess of scattering in the horizontal direction which means an alignment of the scattered objects along the direction of the magnetic field. A specific radial averaging of the data along the direction perpendicular to the magnetic field (in

red) or parallel (in blue) permits to highlight more quantitatively the anisotropy. The scattering of the isotropic sample is recalled in black for comparison (Figure 10a and Figure 11a,b). For all samples, the scattering curves of the magnetically aligned samples superimpose with the one of the isotropic reference down to ∼0.015 Å−1, showing that there is no alignment of linear primary aggregates at local scale and, more generally, no internal reorganization of the objects. The deviation of the scattering curve from the isotropic case in both parallel and perpendicular direction occurs only at low Q, i.e., at large scale, meaning that these are the aggregates themselves who have been aligned along the direction of the magnetic field. It suggests thus that the anisotropic structures result from a two-step process during casting: (i) the initial formation of aggregates of finite size followed by (ii) their alignment with the magnetic field. The alignment of the ramified aggregates along the magnetic field direction is also nicely confirmed in real space with TEM pictures at various magnifications (Figure 10b,c and Figure 11b,c,e,f). We can observe a larger anisotropy for R = 0.4 than for R = 0.09, suggesting that the efficiency of the alignment under magnetic field is correlated with the density of the aggregates. In the isotropic state, the aggregates were indeed more dense for R = 0.4 (Figure 3) than for R = 0.09 (Figure 1). Such a correlation of the efficiency of alignment with the density is consistent with the two-step process of formation of aggregates. Once the aggregates are formed, they form aligned chains thanks to dipolar magnetic interactions, as for bare maghemite nanoparticles.20 The relevant parameter becomes then the magnetic momentum of the aggregates that depends both on their size and on their magnetic content, i.e., the inner volume fraction of maghemite nanoparticles. Since the aggregates have roughly the same size in all cases, these are the denser ones, with the higher Nagg, which are better aligned.

IV. DISCUSSION Following our results, we can propose a clear view of the structural organization of the PS grafted maghemite primary cluster (tetramers of native maghemite as previously described 9229


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in solution23) dispersed inside a PS matrix as a function of the grafted to free chain length ratio R. Let us discuss first step by step the isotropic case obtained without applying an external magnetic field. For the specific value of grafting density studied here, 0.15 chain/nm2, we observed the formation of aggregates of several tens of grafted primary clusters inside the polymer matrix for all the probed values of R below 0.4. On the reverse, when decreasing the size of the free chains, the individual dispersion of the primary grafted cluster inside the matrix is achieved for R ≥ 1, when the free chains are of the same size or smaller than the grafted ones. In the case of R > 1, there were some remaining aggregates, coexisting with isolated objects within the matrix. The situation was however particular because the mass of the chains chosen for the matrix were below the entanglement mass of PS, which may modify the process of film formation, in particular with respect to kinetics of solvent evaporation. The situation for higher values, R = 1 and R < 1, was however truly unambiguous: the structural threshold transition between individual dispersion and aggregates’ formation is located between R = 0.4 and R = 1. This behavior is in good agreement with the general state diagrams recently observed for nonmagnetic particles in the literature.2,5,6,8,16 When particles are magnetic, the aggregation during casting may be affected by the competition between dipolar magnetic interactions, which strongly increase with the size of the filler particle, and isotropic forces, which is mainly the grafted to free chains interaction in grafted systems. For very low grafting density (∼2 chains/particle), Jiao et al.22 observed the formation of chains on magnetite grafted nanoparticles. When increasing the grafting density, as we did here, the isotropic grafted to free chains interactions appear to be completely dominant to the dipolar ones, as we obtained perfect isotropic dispersion out of magnetic f ield. Since dipolar interactions are negligible out of magnetic field, our study enabled us to probe the effects of the surface curvature of the grafted objects on the threshold of the “wet-todry” transition by comparison with our previous study performed on silica particles grafted by PS chains with the same chemical procedure and dispersed in a matrix made of PS chains. In this study, the threshold of the transition was clearly located at R = 0.24,6 while grafted particles are individual silica spheres.24 For the present study, the grafted magnetic fillers are not individual spheres but linear tetramers23 whose typical size, grafting densities, and length of grafted brushes are similar to the silica case. With these linear tetramers, the threshold value is superior to 0.24 and shifts toward a value closer to R = 1. This evidence experimentally, for the first time to our best knowledge, shows the direct influence of the surface curvature on the “wet-to-dry” transition. Such effect, up to date discussed only with simulations,9−12 shows that the modification of the surface curvature from sphere to linear aggregates is shifting the transition threshold toward the expected value of R = 1 for planar surfaces.13 We do not believe that the position of the transition can be influenced by the magnetic dipolar interactions considering that these interactions are limited in comparison with the van der Waals attractive ones in the case of identical nongrafted particles.18 Another key point of our study is the direct experimental proof of the correlation existing between the partial collapse of the grafted chains and the formation of aggregates for R < 0.4. Thanks to an unique property of our system with respect to neutron scattering (the SLD of the maghemite and deuterated PS are almost similar), we have been able to make a direct

measurement of the grafted chain conformation in the case of formation of aggregates. The result is clear-cut as the grafted chains are strongly collapsed in comparison with their conformation in bulk, as shown by a significant reduction of their radius of gyration accompanied by deviation from the Gaussian behavior at high Q illustrating a densification of the interface. This completes our previous results obtained on PSgrafted silica, where the direct measurement of the grafted chains’ conformation in samples presenting a perfect dispersion of grafted objects showed that the chains kept their Gaussian behavior. Taken together, these two studies enable a direct consistent experimental verification of De Gennes’ pioneering theory:13 The extension of the grafted chain is associated with the individual dispersion while the formation of the aggregates is related to a compression of the grafted chain to minimize the free energy of the system. Finally, the last but not least advance of our study is the experimental demonstration of controlling the final structure of aggregates by coupling the application of an external magnetic field during the solvent casting to the control through R. Since dipolar magnetic interactions are proportional to μagg,6 where μagg is the magnetic momentum of the aggregates that can be tuned by several parameters (R, grafting densities, choice of magnetic material, etc.), this bimodal nanotriggering route opens the way to the formation of controlled anisotropic aggregate, with an almost infinite possibility to reach a desired anisotropy.

V. SUMMARY AND CONCLUSION Thanks to a specific grafting process of PS chains at the surface of tetramers of maghemite nanoparticles, we present here a complete study of the particle dispersion in nanocomposites as a function of the grafted to free chain ratio R. Below R = 0.4, we observe the formation of aggregates of nanoparticles in the film for which the density depends of the matrix chain length, while above R = 0.4, the tetramers organize themselves inside the matrix as individual entities. These results are consistent with a general behavior of state diagram of grafted nanoparticles dispersed in homologous free chains deduced from recent experimental studies mainly performed on silica nanoparticles. In addition, we reported here an incremental experimental evidence of the influence of the surface curvature of the grafted objects on the threshold of the transition between dispersed and aggregated state: the modification of the particle surface curvature from sphere to linear aggregates is shifting the transition threshold toward the expected value of R = 1 for planar surface. Using specific deuteration and neutron scattering, we have also demonstrated that the aggregate state of the particle is associated with a significant collapse of the grafted brushes according to De Gennes pioneering theory. Finally, by combining the control of the particle dispersion with grafting and with the application of an external magnetic field during the solvent casting, we demonstrate the possibility of orientation of the aggregates along the field direction that open the way of a new bimodal route of nanotriggering to create controlled anisotropic filled materials.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (J.J.); [email protected] (F.C.). 9230


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Notes

(30) Flood, C.; Cosgrove, T.; Howell, I.; Revell, P. Langmuir 2006, 22, 6923−6930. (31) Rottler, J.; Robbins, M. O. Phys. Rev. E 2003, 68 (1), 011801. (32) Percus, J. K.; Yevick, G. J. Phys. Rev., Second Ser. 1958, 110 (1), 1−13. (33) Pedersen, J. S.; Gerstenberg, M. C. Macromolecules 1996, 29, 1363.

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank CEA and Synchrotron SOLEIL for the PhD Grant of A.-S. Robbes and for beam-time allocation and ILL for beamtime allocation. The Laboratoire Léon Brillouin is a joined facility of CEA and CNRS. We thank Jérémie Gummel for the measurement with the Bonse-Hart setup on ID2 at ESRF presented in Figure 5a.

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REFERENCES

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