Christophe Travelet
email: christophe.travelet@cermav.cnrs.fr

Mylene Stemmelen

Vincent Lapinte

Frédéric Dubreuil

Jean- Jacques Jacques Robin

Redouane Borsali
email: borsali@cermav.cnrs.fr

Mylène Stemmelen

Amphiphilic copolymers based on

Keywords: Amphiphilic copolymer, polyoxazoline, grape seed vegetable oil, self-assembly, selforganization, nanoparticle, dynamic light scattering

The behavior in solution of original structures of amphiphilic partially natural copolymers based on polyoxazoline (POx) (poly(2-methyl-2-oxazoline)) and grape seed vegetable oil derivatives (linear, T-and trident-structure) are investigated. The results show that such systems are found,using dynamic light scattering (DLS), to spontaneously self-organize into monomodal, narrow-size and stable nanoparticles in aqueous medium. The obtained hydrodynamic diameters (D h ) range from 8.6 up to 32.5 nm. Specifically, such size increases strongly with increasing natural block (i.e. lipophilic species) length due to higher hydrophobic interactions (from 10.1 nm for C 19 to 19.2 nm for C 57 ). Furthermore, increasing the polyoxazoline (i.e. hydrophilic block) length leads to a moderate linear increase of the D hvalues. Therefore, the first order size effect comes from the natural lipophilic block whereas the characteristic size can be tuned more finely (i.e. in a second order) by choosing appropriately the polyoxazoline length. The DLS results in terms of characteristic size are corroborated using nanoparticle tracking analysis (NTA), and also by atomic force microscopy (AFM) and transmission electron microscopy (TEM) imaging where welldefined, spherical and individual nanoparticles exhibit a very good mechanical resistance upon drying. Moreover, changing the lipophilic block architecture from linear to T-shape and keeping the same molar mass generates a branching and thus a shrinking by a factor of 2 of the nanoparticle volume, as shown by DLS. In this paper, we clearly show that the selfassemblies of amphiphilic block copolymer obtained from grape seed vegetable oil derivatives (sustainable renewable resources) as well as their tunability are of great interest for biomass valorization at the nanoscale level.

Introduction:

Combining polymers having different properties allows to obtain interesting architectures [1] such as linear block copolymers, star-shape copolymers, graft copolymers or dendrimers. [2,3] Amphiphilic diblock copolymers are especially interesting for their self-assembling properties in liquid medium and their ability to produce soft objects [4,5] such as micelles, vesicles [6] or polymersomes. [7] The size of these soft objects ranges from few nanometers to hundred micrometers [8] and the resulting self-organization properties are used in a wide range of applications such as emulsifiers [9] and stabilizers for food or cosmetics. Other important application fields are nanotechnologies [10] and drug delivery since the self-assembling of copolymers allows the formation of nanoreactors and the encapsulation of active molecules such as drugs. [11,12] So far, diblock copolymers were mainly obtained from petroleumorigin but the rarefaction of fossil resources is now favoring research focusing on the development and the characterization of fully or partially bio-based copolymers. Many investigations concern now the synthesis of non-ionic copolymers and the study of their physico-chemical properties.

More recently, fully bio-sourced copolymers have been reported such as alkyl polyglycosides, [13] glycolipids, [14] lipids linked to polyglycerol, [START_REF] Johansson | Surfactants from Renewable Resources[END_REF] glycoproteins forming vesicles, [START_REF] Schatz | [END_REF] oligosaccharides linked to hydrophobized oligosaccharides forming micelles [17] or nucleolipids leading to various supramolecular assemblies. [18] Partially bio-sourced copolymers can be divided into two main categories: copolymers based on sugar derivatives or based on lipids. Within the first family, various glycopolymers associating responsive poly(N-isopropylacrylamide) and oligosaccharides such as linear maltoheptaose, [19] branched xyloglucooligosaccharide [20] or cyclic β-cyclodextrin [21] were synthesized and their ability to form nanoparticles or vesicles was also investigated. Glycopolymers linked to a polystyrene block were reported to self-organize into films with inter-domain spacing of about 10 nm which underwent transition phase in presence of bipyridine. [22] Specific glycopolymers containing a silicon-based block [23] and fluorescent π-conjugated poly(3-hexylthiophene) [24] showed interesting organization behaviors in periodic films made by approximately 5 nm diameter cylinders (in dry state) and vesicles (in liquid state), respectively. The second category gathers lipopolymers based on vegetable oils or their derivatives and a hydrophilic block such as polyacrylamide, [25] poly(ethylene oxide) (PEO) or peculiar polyoxazolines such as poly(2-methyl-2-oxazoline), poly(2-ethyl-2-oxazoline) or poly(2-(2-N-pyrrolidonylethyl)-2-oxazoline). [26] The most reported lipopolymer is based on PEO since it is easily available and its physical properties in solution are well-known. For instance, lipid mono-and di-end capped PEO molecules were found to self-assemble in micelle and flower-like micelle, respectively. An analysis in the dilute regime showed that the effective interaction radius was close to the hydrodynamic radius. Moreover, a study inthe semi-dilute regime indicated that micelles were organized in a liquid-like ordered structure, [27] and recent investigations proved that they were good emulsifiers especially for high internal phase emulsions. [9,28] More complex structures such as PEO-phosphatidyl ethanolamine were investigated for drug delivery systems because they had a low critical micelle concentration (CMC) (about 10 -6 mol L -1 ), a good drug loading with an efficiency varying between 1.5 and 50% of introduced molecules and a high retention (over 75%). [3,29] One may also add that lipopolymers based on polyoxazoline, especially on poly(2-methyl-2-oxazoline) (POx), were found to have similar behavior as PEO-based ones in terms of self-organization and biomedical requirements. [26,30] Furthermore, poly(2-methyl-2-oxazoline) has been proved to be non-toxic whereas recent studies highlighted the bioaccumulation of PEO chains into human organisms. [31,32] Only few lipopolymers based on lipidic chain and poly(2-methyl-2oxazoline), so-called LipoPOx, have been synthesized and reported in the literature (Figure 1).On one hand, dialkyl-lipopolymers based on 1,2-O-dioctadecyl-sn-glycerol (Figure 1b), [33] disteaorylphosphatidylethanolamine (Figure 1c) [34] and 1,2-didodecanoylpropyl (Figure 1d) [31] were developed. So far, they have been studied in air-water interface as mono-layer [35][36][37][38][39] or supported membranes [40,41] but never in solution. On other hand, lipid mono-and di-end capped POx molecules, leading thus to linear AB (Figure 1a) [42] and BAB [43] structures, were studied in solution as reported by Volet et al. [42,43] Nano-hairy micelles were observed and the influence of the lipophilic and hydrophilic parts on theradius of gyrationwas studied by smallangle neutron scattering in deuterated dichloromethane and water. [44] Fluorimetry and viscosimetry analyses were also performed and the effect of adding cyclodextrin in order to reduce hydrogen bonding was detailed. [42] Recently, we have developed LipoPOx based on castor oil by modifying natural hydroxylated groups with tosylate groups to obtain macroinitiator for the polymerization of 2-methyl-2-oxazoline (Figure 1e). [45] Such lipopolymers were found to form aggregates in solution and star morphologies in solid state. [45] To the best of our knowledge, the self-assembly in water and in organic solvents of complex structures of LipoPOx has never been detailed in the literature.

In the present paper, the behavior in solution of original structures of LipoPOx is investigated.

These lipopolymers are based on either fatty esters or triglycerides bearing one poly(2methyl-2-oxazoline) chain grafted on one fatty chain, in the case of the triacylglycerol (Figure 1f and 1g, respectively). The synthesis of such structures was described in our previous paper [46] and the present contribution reports the study of the self-assembly of LipoPOx in aqueous medium by dynamic light scattering (DLS) associated with various complementary techniques such as nanoparticle tracking analysis (NTA), atomic force microscopy (AFM) and transmission electron microscopy (TEM). The influence of both lipophilic and hydrophilic parts on the particle size is investigated as well as the behavior of self-assembled mixtures of two different lipopolymers. Finally, the effect of adding an immiscible solvent (water / toluene) on the particle size is highlighted.

Experimental section:

Synthesis: The synthesis of the copolymers based on polyoxazoline and grape seed vegetable oil (GSO) or fatty ester (methyloleate (MO))wasextensively described in our previous joint publication mainly focused on chemistry. [46] Dynamic light scattering (DLS): DLS measurements were performed at CERMAV [47,48] using an apparatus of type ALV/CGS-8F S/N 069 (ALV, Langen, Germany) [START_REF]ALV-Laser Vertriebsgesellschaft mit beschränkter Haftung[END_REF] equipped with a 35 mW red helium-neon linearly polarized laser operating at a wavelength of 632.8 nm (JDSU, Milpitas, USA) and an ALV/LSE-5004 multiple τ digital correlator with a 125 ns initial sampling time. The aqueous copolymer suspensions were successively filtered through 0.45 and then 0.1 µm hydrophilic PTFE syringe filters (Whatman, Maidstone, UK), loaded in 10 mm diameter cylindrical cells and thermalized at a constant temperature of 25.0±0.1 °C prior to measurement. The wave vector modulus (q) is equal to (4πn/ )sin(θ/2) where n represents the refractive index of the pure solvent (or pure solvent mixture), is the laser wavelength and θ designates the scattering angle. Data were collected typically for a counting time of 120 s at different scattering angles ranging from 20 to 155°(i.e.4.59×10 -3 ≤ q ≤ 2.58×10 -2 nm -1 in pure water)by step of 5° using the digital ALV correlator control software.

The reproducibility of the carried out measurements was checked at least 3 times. The relaxation time distribution was obtained using the CONTIN analysis [START_REF] Provencher | [END_REF]51] of the autocorrelation function (g (2) -1).

Ultraviolet-visible spectrophotometry:

The aqueous copolymer suspensions were checked at CERMAV with the help of a Cary 50 Bio ultraviolet-visible spectrophotometer (Varian/Agilent Technologies, Palo Alto, USA) not to absorb light at the helium-neon laser wavelength used in DLS experiments (i.e. 632.8 nm), thus avoiding local convection currents. Indeed, at 2 mg mL -1 and 632.8 nm, no local maximum or shoulder is present and the absorbances are typically 5.93×10 -3 after removal of the solvent signal (data not shown).

Atomic force microscopy (AFM):

Sample preparation was done as follows: Silica wafers were thoroughly washed with Milli-Q water characterized by a resistivity of 18.2 MΩ cm, with acetone and then for 2 h with fuming nitric acid. They were subsequently rinsed with Milli-Q water and acetone. Then, the aqueous copolymer suspensions at 0.04 mg mL -1 were successively filtered through 0.45 and then 0.1 µm hydrophilic PTFE syringe filters (Whatman, Maidstone, UK) and 50 L were dropped onto the silica wafers which were allowed to dry for 4 days in an exicator under static vacuum in presence of silica gel. AFM imaging was carried out at ICMG at room temperature using a Pico Plus commercial instrument (Molecular Imaging, Phoenix, USA). A 100×100 µm 2 piezoelectric scanner was used and 512×512 data points were acquired. Images were obtained in tapping mode using silicon tips with a spring constant of 48 N m -1 and a resonance frequency of approximately 190 kHz (Vista Probes, Phoenix, USA). Data treatment (i.e. height measurement after baseline correction only) and presentation were realized with the help of the Gwyddion software.

Transmission electron microscopy (TEM): Sample preparation was done as follows:

The aqueous copolymer suspensions at 0.04 mg mL -1 were successively filtered through 0.45 and then 0.1 µm hydrophilic PTFE syringe filters (Whatman, Maidstone, UK) and 4 L weredropped onto glow discharged carbon coated copper grids. Then, 4 L of a 2 w/v% aqueous uranyl acetate negative stain solution was added prior to complete drying. After a few minutes, the excess liquid was blotted with filter paper and the grids were allowed to dry for one day in presence of silica gel. TEM imaging was performed at CERMAV using a CM200 microscope operating at 80 kV (Philips/FEI, Hillsboro, USA). Images were recorded on SO163 films (Kodak, Rochester, USA) and the nanoparticle mean diameters were determined with the Scandium software.

Nanoparticle tracking analysis (NTA): NTA was carried out at CERMAV using a LM10HSoptical microscope setup equipped with a camera and a chamber mounted on the modified microscope stage (Nanosight, Amesbury, UK). The original aqueous copolymer suspension at 2 mg mL -1 was diluted with Milli-Q water down to 0.08 mg mL -1 and introduced into the chamber by a syringe. A video clip of the nanoparticles submitted to their natural Brownian motion was captured over 30 s at 25.0 °C and analyzed by the analytical software version 2.1.

Results and Discussion:

1. LipoPOx copolymer self-assembly in water [START_REF]The use of other less hydrophilic solvents such as acetonitrile or dimetylformamide is not successful[END_REF] / Effect of the lipophilic block length:

The synthesis of the copolymers based on polyoxazoline and grape seed vegetable oil or fatty ester (LipoPOx) was reported elsewhere. [46] L-C 18 -POx n designates a linear structure made with an alkyl chain (Figure 1a) where n is the oxazoline monomer unit number (n), T-MO-POx n represents a T-structure made with a fatty ester especially methyloleate (MO) (Figure 1f) and Tri-GSO-POx n designates a trident-structure made with a triglyceride of grape seedvegetable oil(GSO) (Figure 1g). When LipoPOx are mixed with water, [START_REF]The use of other less hydrophilic solvents such as acetonitrile or dimetylformamide is not successful[END_REF] they spontaneously self-assemble as deduced from the dynamic light scattering (DLS) experiments (Figure 2) carried out in a wide range of scattering angles. Indeed, the relaxation time distributions obtained from the autocorrelation functions (g (2) -1) and showed at a scattering angle of 40, 90 and 140° for T-MO-POx 25 and Tri-GSO-POx 23 exhibit a monomodal and narrow exponential decay (Figure 2a and2b). From the x-positions of the relaxation time distribution maxima (1/Γ), the proportional dependence of the Γ-values (i.e.the relaxation frequencies) on the square of the wave vector modulus (q 2 )(Figure 2c and2d) is attributed to the Brownian diffusive motion of particles. [START_REF] Berne | Dynamic Light Scattering With Applications to Chemistry[END_REF][START_REF] Daniel | Les Latex Synthétiques: Elaboration, Propriétés, Applications[END_REF] Indeed, internal modes such as the Rouse or Zimm modes are preferentially probed in conditions such as qD G >4 [START_REF] Berne | Dynamic Light Scattering With Applications to Chemistry[END_REF][START_REF] Galinsky | [END_REF]56] what is presently not at all fulfilled(D G designates the mean-square diameter of gyration and is in the same order of magnitude as the hereafter given D h -values). Moreover, transient uncontrolled aggregates are not observed. The slope of Γ = f(q 2 ) dependence is equal to the diffusion coefficient (D) of the particles from which the hydrodynamic diameter (D h ) [START_REF]The hydrodynamic diameter extrapolated from the slope of the angular dependency[END_REF] of the particles is calculated for each sample using the well-known Stokes-Einstein equation. [START_REF] Einstein | [END_REF]59] The so-calculated D hvalues are 10. polyoxazoline), driving the self-organization in aqueous medium towards a much bigger characteristic size.Similar experimentaland modeling tendency as a function of the lipophilic block length was observed for other amphiphilic block copolymers such as poly(styrene-b-4vinylpyridine). [60] Furthermore, Volet et al. [43,44] reported that, for a moderate change in the lipophilic block length (i.e. by 50 %, from C 12 to C 18 ), the characteristic size in dichloromethane remains constant whatever for similar linear diblock [44] or similar linear triblock lipophilic/hydrophilic/lipophilic systems. [43] The evolution of the lipophilic block length is, for the present systems, much more drastic (i.e. by 200 %, from C 19 to C 57 ).

Experimentally, the critical micelle concentration (CMC) is found to be equal to 0.01 -0.02 mg mL -1 as determined by fluorescence and DLS experiments for Tri-GSO-POx 23 in water (data not shown). The CMC-values published by Volet et al. [42] and determined by fluorescence measurements in water were in the same order of magnitude. Thus, the copolymer concentrations used in the present contribution (from 0.04 to 4 mg mL -1 ) are higher than our CMC-value, guaranteeing the self-assembling.

Effect of the lipophilic block structure on the LipoPOx copolymer self-assembly in water:

So as to better apprehend the influence of the lipophilic block structure on the self-assembly, the D h -values are measured by DLS for L-C 18 -POx 26 (having a linear shape) and T-MO-POx 25 (having a T-shape) copolymer suspensions (Figure 3). The lipophilic block molecular weight is almost the same for both compounds (254 versus 297 g mol -1 ), their architectures only differ from one to another. The D h -value decreases by 16 % from the linear to the T-shape.

This result confirms that a branched polymer has a smaller characteristic size than a linear polymer (all other things being held constant and particularly the molecular weight). [START_REF] Arndt | Polymercharakterisierung[END_REF][START_REF] Kurata | [END_REF] This behavior was already reported for polystyrene [START_REF] Podzimek | Light Scattering, Size Exclusion Chromatography and Asymmetric Flow Field Flow Fractionation: Powerful Tools for the Characterization of Polymers, Proteins and Nanoparticles[END_REF] and even for polyelectrolytes. [START_REF] Burchard | [END_REF] The contraction is characterized by the so-called shrinking factor defined as the ratio at the power 3 of the D h -values of the T-structure to the linear structure. [START_REF] Arndt | Polymercharakterisierung[END_REF][START_REF] Kurata | [END_REF] It is equal to 0.596, meaning that the spherical nanoparticle volume is roughly divided by a factor of 2 when modifying the lipophilic block structure.

T-MO-POx n (having a T-shape) copolymer self-assembly in water / Effect of the hydrophilic block length:

In order to probe the influence of the hydrophilic block length on the self-assembly, the D hvalues are measured by DLS for T-MO-POx n copolymer suspensions from n = 0 to 55 (Figure 4). The error bars relatively to the LipoPOx copolymers are small (typically 4 % in relative value), indicating a satisfying reproducibility of the measurements. For n = 0, the compound corresponds simply to hydroxylated methyloleate (i.e. methyloleate functionalized with 2mercaptoethanol). [46] Its lipophilic character is not counter-balanced by numerous hydrophilic functional groups (only one alcohol function and one ester function per 374 g mol -1 molecule), thus leading to a large characteristic size of 314 nm and a high relative standard deviation of 8 % (versus typically 30 times less and 4 % for the three compounds whose n-value differs from 0). With increasing n-values, the D h -values increase linearly but moderately (Figure 4). The characteristic size increases by 73 % when n increases by 323 %. It means that the first order size effect comes from the lipophilic block (as discussed in paragraph 1) whereas the characteristic size can be tuned more finely (i.e. in a second order) by choosing appropriately the hydrophilic block length. Such a linear behavior was reported on similar linear triblock alkane/POx/alkane systems. [43] Furthermore, the stability of the copolymer suspensions, particularly for T-MO-POx 13 at 2 mg mL -1 in water, is successfully checked using DLS over an aging time of 30 days in terms of D h -value (8.7±0.2versus 8.6±0.2 nm at short times, Atomic force microscopy (AFM) and transmission electron microscopy (TEM) imaging are performed on T-MO-POx 25 (Figure 5a and5c).Well-defined spherical nanoparticles are observed on the substrates. They are individual and no sign of coalescence upon drying is observed. Statistic treatment of the images carried out on few hundred nanoparticles gives by AFM measurements the average height H AFM, MAX = 7.7 nm and by TEM measurements the average diameter D TEM, MAX = 8.9 nm (Figure 5b and5d). Both values are within experimental error in very good agreement, and as expected slightly smaller than the value D h = 10.2 nm obtained in the swollen state by DLS (Figure 4).

Tri-GSO-POx n (having a trident-shape) copolymer self-assembly in water / Effect of the hydrophilic block length:

The D h -values are measured by DLS for Tri-GSO-POx n copolymer suspensions from n = 0 up to 56 (Figure 6a). The obtained profile is quite comparable with the one measured for the Tshape compounds (Figure 4). However, for any given n-value, the D h -values are higher (typically 2 times bigger, except for n = 0) for Tri-GSO-POx n due to their longer lipophilic blocks and thus stronger lipophilic character in aqueous medium.

In order to complete the DLS results indicating the presence of nano-objects, nanoparticle tracking analysis (NTA)is carried out onTri-GSO-POx 56 at 0.08 mg mL -1 in water (Figure 6b).

The values n = 11 and 23 cannot be probed since the expected characteristic sizes are at the low resolution limit (i.e. about 20 -30 nm for organic systems due to their moderate nanoparticle refractive indexes compared to inorganic systems). [65][66][START_REF]Nanosight Operating Manual[END_REF] The nanoparticles are illuminated by a laser and tracked individually in a video clip filmed with a charge-coupled device (CCD) camera. Those that just undergo natural Brownian motion [START_REF] Einstein | [END_REF] in their liquid medium are taken in account by the software, their diffusion coefficients are individually estimated from which their hydrodynamic diameters (D h, NTA ) are calculated using the Stokes-Einstein equation. [START_REF] Einstein | [END_REF]59] The hydrodynamic diameter distribution is narrow and gives the average diameter D h, NTA, MAX = 24.0 nm (Figure 6c). This value is notably smaller than D h = 30.3 nm obtained by DLS (Figure 6a). This shift is reasonably attributed to the fact that NTA gives by nature access to number weighted hydrodynamic diameter distributions whereas DLS consists of mass weighted distributions. [69,70] No shift is thus observed for strictly speaking monodisperse samples(what is presently not the case). Both characterization methods are complementary and in very good agreement.

AFM and TEM imaging are performed on Tri-GSO-POx 11 , showing dry spherical nanoparticles (Figure 7a and7c). As for T-MO-POx 25 , no drastic coalescence or aggregation of nanoparticles is observed. The average height H AFM, MAX = 5.5 nm and the average diameter D TEM, MAX = 11.1nm are obtained by the statistic treatment of the AFM and TEM images, respectively (Figure 7b and7d). This difference lies on different affinities of the nanoparticles with the substrates (silica wafer and glow discharged carbon coated copper grid, respectively) combined with weaker mechanical resistance in the unswollen state of the nanoparticles based on Tri-GSO-POx 11 compared to the ones based on T-MO-POx 25 (they spread a bit, mainly on the silica wafer). Indeed, these two values differ by a factor of 4 and 2, respectively, from the value D h = 20.4 nm obtained by DLS (Figure 6a). Thus, the drying step affects more strongly the nanoparticles based on Tri-GSO-POx 11 in terms ofcharacteristic size, but not in terms of morphology.

Effect of the concentration on the LipoPOx copolymer self-assembly in water:

Todetermine the possible influence of the copolymer concentration on the self-assembly, the D h -values are measured by DLS for T-MO-POx 55 (having a T-shape) and Tri-GSO-POx 56 (having a trident-shape) copolymer suspensions (Figure 8). In the studied concentration range going from 1 to 4 mg mL -1 (i.e. for concentrations above the typical CMC-value discussed in paragraph 1), the D h -values remain constant at about 15.0 and 32.0 nm, respectively.

In order to go further on the interpretation of the results, and knowing that polyoxazoline is a semi-rigid molecule with a persistence length l P = 2.0 nm, [44] we propose to have a close look at the morphology of the self-assembled LipoPOx copolymer structure: micelle or other structure? The notion of persistence length is strongly correlated to worm-like chain model [71,72] valid for semi-flexible polymers in dilute solution and could be used to appreciate, in a first approximation, the mean-square diameter of gyration (D G ):

2 = 4 3 -4 2 + 8 3 - 8
(Equation 1) [71,72] Where L C designates the contour length. The trident-shape compound having quite a complex structure which cannot be rendered by the simplicity of the chosen equation, we should at that point focus on the T-shape compound (T-MO-POx 55 ). For n = 55, making the hypothesis that the lipophilic block behaves as the semi-flexible hydrophilic block (therefore n FICTITIOUS = 55+(18/3) = 61), we get L C = 27.3 nm and D G = 7.7 nm. [START_REF] Reassuringly | the Gaussian chain model, involving much more flexible chains in Θsolvent conditions (i.e. the polymer / polymer and polymer / solvent interactions compensate each other and the excluded volume effect is canceled out), leads to the lower mean-square diameter of gyration D G[END_REF] The result concerning the nanoparticles with a D h -value at about 15.0 nm, as experimentally measured by DLS (Figure 8):

 is not at all compatible with the simple star-like micelle model [START_REF] Ball | Designing the Molecular World: Chemistry at the Frontier[END_REF] for which the characteristic expected diameter would be roughly 2L C = 54.7 nm.

 ishowever compatible with the worm-like chain model [71,72] valid for semi-flexible polymers. More precisely, instead of having a unique copolymer chain leading to a characteristic diameter of 7.7 nm, few of them (less than 10) are self-organized and constitute the nanoparticle so as to get 15.0 nm. This experimental higher latter value may also be due to the hygroscopic character of polyoxazoline [START_REF] Rettler | [END_REF]76] which contains numerous bound water molecules, increasing its excluded volume and thus the nanoparticle characteristic size.

For these reasons, in the present case, the terminology "micelle" may be ambiguous. That is why "self-assembly" or simply "nanoparticle" should be preferred. From experimental results obtained from different amphiphilic block copolymers with molecular weights between 2,700 and 20,000 g mol -1 , Discher et al. [4,6] gave a simple unifying rule for the formation of selfassemblies in water. For this purpose, they introduced the hydrophilic ratio (f HYDRO ), defined as the ratio of the hydrophilic part to the total mass, which is equal in our case to 94 and 85 % for T-MO-POx 55 (with a molecular weight of about 5,190 g mol -1 ) [START_REF]The molecular weights are determined by size exclusion chromatography calibrated with poly(methyl methacrylate) standards[END_REF] and Tri-GSO-POx 56

(with a molecular weight of about 5,840 g mol -1 ), [START_REF]The molecular weights are determined by size exclusion chromatography calibrated with poly(methyl methacrylate) standards[END_REF] respectively. Both values are notably higher than the minimum required value f HYDRO = 45 % [4,6] from which direct structures (which cannot be vesicles) are expected to form in water.

Mixtures of LipoPOx copolymers / Mixtures of solvents:

When mixing together the T-MO-POx 25 and Tri-GSO-POx 56 powders in the weight ratios 90/10 and 98/2 w/w% before adding water, the relaxation time distributions obtained from the autocorrelation functions measured by DLS at a scattering angle of 90° show for both systems a bimodal exponential decay (Figure 9a). The calculated D h -values reach roughly 10.0 and 50.0 nm, almost independently on the feed ratio (Figure 9b). These two values differ by a factor of 5, much more than the factorof 3 mentioned by Trushkevychet al. [65] andconsidered as the low resolution limit from which two size distributions of nanoparticles can be unambiguously distinguished by DLS.The fact that the two peaks appearing in the relaxation time distributions have more or less the same surfacearea (Figure 9a, in the case of the feed ratio 90/10) does not mean that an equal number of small and big nanoparticles are present.

Indeed, DLS experiment data processing with the CONTIN analysis leads to mass weighted hydrodynamic diameter distributions: Bigger nanoparticles contribute much stronger. [69,70] Thus, to convert simply such mass weighted distributions into number weighted distributions [59] what may be more pragmatic to the physico-chemistthe hypotheses are that hard sphere-like nanoparticles are present and that the small (characterized by an hydrodynamic diameter D h, SMALL ) and big (characterized by an hydrodynamic diameter D h, BIG ) nanoparticles have the same density. In that case, we have:

= ℎ, ℎ , 3 1 -
(Equation 2) [59] Where N SMALL designates the number of small nanoparticles, N BIG is the number of big nanoparticles and A SMALL represents in the relaxation time distribution the percentage of the total surface area corresponding to the surface area under the peak related to the small nanoparticles. In the present example (Figure 9a), we have D h, SMALL = 10.8 nm, D h, BIG = 54.0 nm and A SMALL = 34%. Thus, we get N SMALL = 64.4N BIG . Therefore, small nanoparticles are the majority, making up 98 % of the nanoparticles in number. Regarding the assignment of the two here above discussed peaks, when comparing with Figures4 and 6a, the peak related to the small nanoparticles can be reasonably attributed to the pure T-MO-POx 25 copolymer whereas the peak related to the big nanoparticles may consist of the Tri-GSO-POx 56 copolymer whose outer part is filled and rendered more compact and larger by the much mobile and a bit much hydrophilic T-MO-POx 25 copolymer.

Is it relevant to think that small lipophilicmolecules can easily enter in the lipophilicinner part of the nanoparticles in aqueous medium (or the opposite using small hydrophilic molecules in organic medium)? For this purpose, Tri-GSO-POx 56 at 1 and 2 mg mL -1 in toluene (Figure 10a) and water (Figure 10b), respectively, are subsequently mixed and sonicated with their associated immiscible solvent (water and toluene, respectively). We get stable suspensions whose relaxation time distributions obtained by DLS at a scattering angle of 90° show systematically a bimodal exponential decay. The D h -values are roughly 36.0 and220nm (Figure 10). When comparing with Figure 6a, the small-scale objects are assigned to nanoparticles based on Tri-GSO-POx 56 slightly swollen by water (or toluene). The shift towards bigger sizes can be reasonably attributed to little variations in the solute viscosity and refractive index when adding the non-miscible compound. These variations are not easy to estimate due to the fact that part of the non-miscible compound may be effectively encapsulated. The large-scale objects can result in droplets of the immiscible solvent stabilized by the Tri-GSO-POx 56 copolymer. The error bars are large, particularly in the case of toluene into water (leading to direct oil-into-water suspensions) (typically 46 % in relative value), suggesting that the stabilization is not so effective. Before the addition of the immiscible solvent, the D h -values are higher in water compared to toluene, they differ by 16

%. This behavior is due to the hydrophilic and even hygroscopic character of polyoxazoline [START_REF] Rettler | [END_REF]76] and also to the fact that the Tri-GSO-POx 56 compound is highly unsymmetric in terms of steric hindrance of the lipophilic and hydrophilic blocks (Figure 1).

Thus, the spatial organizations of the direct and inverted nanoparticles are not equivalent.

Conclusion:

Original and amphiphilic partially natural copolymers based on polyoxazoline and grape seed vegetable oil derivatives were synthesized with linear, T-and trident-structure. All these structures are found to spontaneously self-organize in water in monomodal, narrow and stable colloidal suspensions as shown using DLS experiments. Globaly, the obtained D h -values are at the nanometer-scale and range from 8.6 to 32.5 nm. Specifically, the nanoparticle characteristic size increases strongly with increasing natural block (i.e. lipophilic block) length, from 10.1 nm (for C 19 ) to 19.2 nm (for C 57 ) due to higher hydrophobic interactions (all other things being held constant). Furthermore, D h -values slightly increase with the polyoxazoline (i.e. hydrophilic block) length, for both T-and trident-shape. Therefore, the first order size effect comes from the natural block whereas the characteristic size can be tuned more finely (i.e. in a second order) by choosing appropriately the polyoxazoline length.

The results of the characteristic sizes obtained by DLS on the copolymer suspensions are in very good agreement withNTAexperiments as well as AFM and TEM imaging techniques where well-definedspherical and individual nanoparticles exhibit a remarkable mechanical resistance upon drying. Moreover, changing the lipophilic block architecture from a linear to a T-shape generates a branching and thus a shrinking by a factor of 2 of the nanoparticle volume, as shown by DLS on the copolymer suspensions. The presently described use of grape seed vegetable oil derivativesaslipophilic block(or natural sugar derivatives as hydrophilic block as found elsewhere), [20,24,[START_REF] Otsuka | [END_REF] sustainable renewable resources, and the demonstrated tunability of their self-assembling physico-chemicalproperties are of great interest for biomass valorization at the nanoscale level. chemical structures taken from the literature. [31,33,34,45]

  1 and 19.2 nm for T-MO-POx 25 and Tri-GSO-POx 23 , respectively (Figure 2c and 2d). These two LipoPOx copolymersystems differ only by their lipophilic blocks, and particularly their lengths (C 19 for the fatty ester-based T-MO-POx 25 and C 57 for the triglyceride-based Tri-GSO-POx 23 ). Thus, a higher aggregation number needs to be reached in the case of Tri-GSO-POx 23 to have an efficient stabilization by the hydrophilic block (i.e.

Figure 4 )

 4 Figure 4), relaxation time distribution and scattering intensity (data not shown): No sedimentation, no coalescence or uncontrolled aggregation are observed. The experimentally obtained D h -values (typically 10 nm) are too small to envisage static light scattering measurements aiming at getting information on the molar mass or the nanoparticle morphology such as the form factor. For this reason, microscopic techniques are used hereafter.

Figure 1 .

 1 Figure 1.Chemical structures of the amphiphilic copolymers based on polyoxazoline and vegetable oil derivatives (LipoPOx): a) L-C 18 -POx n (having a linear shape), f) T-MO-POx n (having a T-shape), g) Tri-GSO-POx n (having a trident-shape), and b), c), d), e) other

Figure 2 .

 2 Figure 2. DLS: Autocorrelation function (g (2) -1) and relaxation time distribution at a scattering angle of 40, 90 and 140° for: a) T-MO-POx 25 , and b) Tri-GSO-POx 23 at 2 mg mL -1 in water. Dependence of the relaxation frequency (Γ) on the square of the wave vector modulus (q 2 )between 20 and 155° by step of 5° for: c) T-MO-POx 25 , and d) Tri-GSO-POx 23 at 2 mg mL -1 in water (diffusion coefficient (D), hydrodynamic diameter (D h ) and correlation coefficient (R 2 )).

2 ]Figure 3 .

 23 Figure 3. DLS: Hydrodynamic diameter (D h ) for a) L-C 18 -POx 26 , and b) T-MO-POx 25 at 2mg mL -1 in water.[79]

Figure 4 .Figure 5 .

 45 Figure 4. DLS: Dependence of the hydrodynamic diameter (D h ) on the oxazoline monomerunit number (n) for T-MO-POx n at 2 mg mL -1 in water.[79]

Figure 6 .Figure 7 .

 67 Figure 6. DLS: a) Dependence of the hydrodynamic diameter (D h ) on the oxazoline monomer unit number (n) for Tri-GSO-POx n at 2 mg mL -1 in water. [79] NTA: b) Image taken from the video clip, and c) corresponding hydrodynamic diameter (D h, NTA ) distribution for Tri-GSO-POx 56 at 0.08 mg mL -1 in water.

Figure 8 .

 8 Figure 8. DLS: Dependence of the hydrodynamic diameter (D h ) on the copolymer concentration (C) for T-MO-POx 55 and Tri-GSO-POx 56 at 2 mg mL -1 in water.[79]

55 Figure 9 .

 559 Figure 9. DLS: a) Autocorrelation function (g(2) -1) and relaxation time distribution at a scattering angle of 90° for a T-MO-POx 25 and Tri-GSO-POx 56 mixture (90/10 w/w%) at a total concentration of 2 mg mL -1 in water. The deconvolution of the g(2) -1-function is successfully carried out using a double exponential function. b) Hydrodynamic diameter (D h ) at a scattering angle of 90° for T-MO-POx 25 and Tri-GSO-POx 56 mixtures at a total concentration of 2 mg mL -1 in water.[79]

Figure 10 .

 10 Figure10. DLS: Hydrodynamic diameter (D h ) at a scattering angle of 90° for 1 mL of Tri-GSO-POx 56 : a) at 1 mg mL -1 in toluene, when a controlled volume of water is subsequently added, and b) at 2 mg mL -1 in water, when a controlled volume of toluene is subsequently added.[79]

1 --

Acknowledgements:

This research project has been supported by the French centre national de la recherche scientifique (CNRS). The authors are grateful to Ms. Amandine Durand-Terrasson for transmission electron microscopy (TEM) imaging at CERMAV and fruitful discussions.


Anne Condamines
email: anne.condamines@univ-tlse2.fr

Amélie Josselin-Leray

Cécile Fabre

Luce Lefeuvre

Aurélie Picton

Josette Rebeyrolle

Using Comparable Corpora to Characterize Knowledge-Rich Contexts for Various Kinds of Users: Preliminary Steps

Keywords: CAT, comparable corpora, knowledge-rich context, translation, terminology, knowledge engineering

. Using comparable corpora, it analyzes how the notion of KRC can vary according to text genre and/or type of users.

Theoretical background : the notion of KRCs and its applications

What are KRCs?

The idea that some contexts can prove useful in defining words is far from being new -Aristotle already asserted that there was such a thing as defining contexts. The growing importance of corpus linguistics and information extraction has resulted in a need for a more systematic description of those contexts in order to be able to retrieve them automatically. More recently, the needs have focused on specialized fields, for text-based terminology-or ontology-building. It is within that framework that Ingrid Meyer defined Knowledge Rich Contexts (KRCs) as follows: "By knowledge-rich context, we designate a context indicating at least one item of domain knowledge that could be useful for conceptual analysis " (2001: 281).

KRCs & Ontologies

Until now, the notion of KRC has been mainly studied for the building of ontologies [START_REF] Auger | Pattern based approaches to semantic relation extraction: a state-of-the-art[END_REF]. It is therefore very often linked to the notion of markers of conceptual relations, as illustrated by the following quotation from (Marshman, 2008: 125): "KRCs may be defined as segments of text that convey information that can be useful for concept analysis (e.g. indications of conceptual relations or attributes)". However, our hypothesis in the project is that KRCs can vary [START_REF] Aussenac-Gilles | Variation and semantic relation interpretation: Linguistic and processing issues[END_REF] depending on the domain, depending on the text genre [START_REF] Condamines | Corpus Analysis and Conceptual Relation Patterns[END_REF], depending on the language and, maybe most importantly, depending on the use (translation vs. knowledge engineering for instance).

In knowledge engineering, users are mostly interested in building networks of terms (termino-ontologies) and thus tend to focus on markers of conceptual relations such as meronymy and hyperonymy. Here are two examples of KRCs taken from the comparable corpora we are using for the project and which are described below in section 2.3.1.

(1) Un volcan se compose de trois parties : un réservoir, une cheminée et un édifice visible en surface.

(2) There are three types of lava and lava flows: pillow, pahoehoe, and aa.

In (1) there is a clear relation of meronymy between the French term volcan and the French terms réservoir, cheminée and édifice which is signalled by the use of the conceptual markers which are underlined. In (2), the conceptual marker "types of" clearly indicates a relation of hyperonymy between the hyperonyms lava and lava flows, and the hyponyms pillow, pahoehoe and aa.

From a linguistic point a view, what is at stake is to assess how Term-Relation-Term triplets can be built based on specialized texts, the whole of the triplets making up a network.

KRCS & Translation

It is a well-known fact that context is a key element in the translation process. What do translators need contextual information for? In translation, users are naturally interested in conceptual relations [START_REF] Marshman | Helping translators manage terminological relations: Storing and using occurrences of terminological relations and lexical relation markers[END_REF] but they also use the contexts for "subject-field understanding, correct term choice and idiomatic expression" (Bowker, 1998: 18). This implies they are not only interested in conceptual markers (or in contexts containing Term-Relation-Term triplets), but also in contexts containing defining elements such as (3), synonyms such as (4) or collocations such as the N+V collocation in (5) [START_REF] Josselin-Leray | La définition des termes dans les dictionnaires généraux unilingues : analyse de quelques exemples du domaine de la volcanologie à la lumière d'un corpus de vulgarisation[END_REF].

(3) Tephra is any material ejected explosively from a volcano (ash, lapilli, cinder, and spatter).

(4) Composite cones, or stratovolcanoes, are large, nearly symmetrical structures built of interbedded lavas and pyroclastic deposits.

(5) Lorsque le magma s'épanche en surface il forme une roche volcanique. Bowker (2011: 214) underlines that "translators really want usage informationsuch as contexts or collocations -which will help them to produce a target text that reads well". The problem is that the various resources at their disposal do not sufficiently meet their needs. One of the main sources translators rely on is term records found in term banks such as Termium or Le Grand Dictionnaire Terminologique. Even though translators make up the largest user of term banks, those term records also serve a wide range of other users. For various reasons that are carefully listed by Bowker (2011: 214-215), the information found on those records is rather limited and usually consists in definitions and terms presented out of context, or in only a single context. Bowker (2011: 215) thus suggests: "it would be more helpful for translators to have access not simply to term records that provide a single "best" term with a solitary context, but rather to information that would allow them to see all possible terms in a range of contexts and thus find the solution that works best in the target text at hand". She insists on the fact that looking at a wide range of contexts should not be considered as a waste of time, and that this has been made easier thanks to corpus-analysis tools that present information in an easy-to-read format.

Another tool most translators use nowadays is CAT (Computer-Aided Translation) tool suites which usually include translation memory systems and automatic term extractors, such as SDL Trados Studio 2011 or else DéjàVu. Translation memory systems are a database of aligned source and target texts, that can be considered as a type of parallel corpus (Bowker, 2011: 218). Even though these have proved very useful to translators since they allow them to work faster by reusing segments of texts that have been previously translated, using translated material as a resource may have drawbacks since this type of text is more likely to contain "conceptual or terminological errors, awkward syntactic constructions, or non-idiomatic expressions" (Bowker, 2011: 221). While comparable corpora have been considered very useful in the translation process [START_REF] Zanettin | Bilingual Comparable Corpora and Training of Translators[END_REF][START_REF] Josselin-Leray | Place et rôle des terminologies dans les dictionnaires généraux unilingues et bilingues. Etude d"un domaine de spécialité : volcanologie[END_REF], to our knowledge to this day no CAT tool relies on such corpora.

Moreover, Bowker (2011: 215) states that while the corpus-based approach has had an impact on the terminological process in the field of terminology, and while corpora and tools for corpus processing have been in widespread use in the translation profession for approximately fifteen years, "the impact on the product (i.e. term records in a term bank) has been negligible". Even if there have been a few attempts at improving the contextrelated contents of term banks (e.g. the model template devised by [START_REF] Pearson | Terms in Context[END_REF]Pearson ( : 2000) ) or the DicoInfo compiled at the University of Montreal), the notion of KRC and the potential usefulness of comparable corpora have not been seriously taken into account so far. The CRISTAL project attempts to bridge the gap.

The CRISTAL Project

General background

The CRISTAL project is a three-year project funded by the French National Agency for Research (ANR; ANR-12-CORD-0020). CRISTAL is an acronym that stands for "Knowledge-rich contexts for terminological translation" (Contextes Riches en Connaissances pour la Traduction Terminologique in French). Four different partners are involved: a Computing research team at the University of Nantes, France (LINA), a linguistics research team at the University of Toulouse-Le-Mirail, France (CLLE-ERSS), the Translation Technologies team from the Faculty of Interpreting and Translation at the University of Geneva in Switzerland, and a firm specializing in multilingual text management (Lingua et Machina). The project started in November 2012 and is thus only at its very beginning.

The aim is to retrieve the contexts from bilingual corpora that are the most relevant for translation and to provide them to users through the CAT tool developed by Lingua & Machina, the Libellex Platform. In the current state of the platform, the term to be translated and its candidate translations are enriched with basic contextual information, i.e. part-of-speech, a list of the first three cooccurrents ranked by an association score, and a passageor contextselected according to a metric based on the association scores between the cooccurrents and the length of the sentence. In other words, a "rich context" so far is a context in which significant cooccurrents of the term appear.

The linguistic issues

From a linguistic point of view, the interest of this project is threefold. First, while KRCs have been the focus of various studies which take into account variation depending on text genre [START_REF] Condamines | Corpus Analysis and Conceptual Relation Patterns[END_REF], [START_REF] Marshman | Expressions of uncertainty in candidate knowledge-rich contexts: A comparison in English and French specialized texts[END_REF], the idea of how relevant KRCs might be depending on use has never been investigated as such so far.

The second major element in this project is the use of comparable corpora as a resource for translation and for building and enhancing termino-ontologies. As far as translation is concerned, we are trying to show how using a comparable corpus differs from using an aligned corpus and whether it proves more efficient. For the building of termino-ontologies, we want to find out whether a comparable corpus simply doubles the available textual evidence or if there might be some other benefits to it.

Third, we wish to extend and compare the testing methods used in translation studies by combining feedbackbased and eye-tracking-based experimentations. Eye-tracking has not been often used with specialized texts and the few existing studies rely mostly on experimentations led by computer scientists (Jensen, 1998).

Finally, the project should allow us to assess the interpretabiblity and the relevance of the results thus obtained.

In summary, we are trying to establish how what KRCs are used forfor building termino-ontologies or as a help in translation -can help define what they really are. Besides the "variety of use" aspect, we are also trying to assess how helpful a comparable corpus can be for such a task.

Methodology

The methodology we are trying to devise is based on two essential elements: the comparable corpora and the implementation of approaches used in other studies in order to see how relevant they might be to reach our goal.

The use of comparable corpora

Let us remind here that by comparable corpora, we mean the same as (Altenberg & Granger 2002: 8): "Comparable corpora consist of original texts in each language, matched as far as possible in terms of text type, subject matter and communicative function". The two languages involved in our corpora are French and English.

Two different approaches

Two different approaches will be implemented: the first one will allow us to identify new relation markers in order to build termino-ontologies; the second one will enable us to spot and analyse which KRCs prove to be of greatest use to translators.

The recursive method for identifying new relation markers

The so-called recursive method, is well-known method in text-based Knowledge Engineering [START_REF] Hearst | Automatic Acquisition of Hyponyms From Large Text Corpora[END_REF]) (Morin & Jacquemin, 1999) and which was already used by [START_REF] Condamines | Searching for and Identifying Conceptual Relationships via a corpusbased approach to a Terminological Knowledge Base (CTKB) : method and results[END_REF]. In this method, pairs of terms linked by a clearly identified conceptual relation are first searched in texts in order to identify new patterns; then these patterns are researched in order to spot new pairs of terms and so on.

This method relies on the idea that some relation markers are not necessarily known beforehand, either because they have not been described yet in the literature, or because they are specific to the field under study. But the very fact that there is a relation between two terms can be used as a starting point for identifying new markers.

For example, in the medical field, a "generic" marker such as [N1 entraîner N2] in French can allow us to immediately sport a cause relation between anti-hypertenseur and vertiges in (6) (6) La prise d'un anti-hypertenseur a entraîné des vertiges chez Monsieur Y If we then search the same pair antihypertenseur/vertiges in a medical corpus, we can identify another marker which is specific to the medical field: ([sous N1, N3 (développer, avoir) N2], as in (7)

(7) Sous anti-hypertenseur, Monsieur X s'est mis à avoir des vertiges The comparable corpus will reveal whether the markers are equivalent from one language to another or if they often differ.

Experiment(s) with translators

Our second approach consists in carrying out one or several experiments with translators, in order to identify which contexts (containing conceptual relation markers, or not) translators find most useful when faced with several contexts aiming at helping them with their translation.

Several testing methods have been so far for empirical research in translation studies: think-aloud protocols, direct observation of translators, post-translation feedback…(see for instance [START_REF] Bowker | Using Specialized Monolingual Native-Language Corpora as a Translation Resource: A Pilot Study[END_REF][START_REF] Künzli | Experts versus novices : l"utilisation de sources d"information pendant le processus de traduction[END_REF]). More recently, experiments based on eye-tracking systems named UAD (User Activity Data) have been led. They "consist of the translator"s recorded keystroke and eye-movement behavior, which makes it possible to replay a translation session and to register the subjects" comments on their own behavior during a retrospective interview" (Carl & al., 2008, 21); see also (Gopferich, Jakobsen, & Mees, 2008).

We are planning to rely partly on the methodology devised by [START_REF] Bowker | Using Specialized Monolingual Native-Language Corpora as a Translation Resource: A Pilot Study[END_REF] who asked translator trainees to translate a semi-specialized text into their native language. The translators were divided into two groups, some using monolingual corpora and some using only conventional translation resources (i.e. no corpus). The quality of the translations thus obtained was then compared, and the translators were asked to provide comments on the process. In our case, several groups will probably be tested: one group with only conventional resources, one group which will be provided with several KRCs extracted beforehand from our comparable corpora and which will have to rate their usefulness, one with free access to our comparable corpora whose behaviour will be monitored thanks to UAD and screen-recording. We are thus considering combining several methods in our own experiment(s), which will all be based on comparable corpora and might involve expert translators as well.

Conclusion

There are many challenges to address in the near future: (i) adapt the recursive method to comparable corpora and, at the same time, refine the list of markers, (ii) complete our methodology for the experiment with translators, (iii) carry out the experiment, (iv) analyze and compare the results of the two methodological approaches.

The CRISTAL project is clearly still in its infancy but seems very promising. Linguistically speaking, the most innovative aspect lies in the refined characterization of KRCs. First, it will take into account how the needs for KRCs might vary depending on the type of user (translators vs. knowledge engineers). Second, it will study how using comparable corpora might improve the quality of KRCs proposed to users. Finally, it will assess how useful eye-tracking might be in terminology studies.

Table 1 .

 1 The following table sums up the main features of the corpora: Presentation of the corpora

		VOLCANOLOGY	BREAST CANCER
	POPULAR SCIENCE TEXTS	400,000 words/language	200,000/language
		1980-2002	2002-2008
	SPECIALIZED TEXTS	Under construction	200,000/language
		Under construction	2001-2008


Julie Diani
email: julie.diani@ensam.eu

Pierre Gilormini
email: pierre.gilormini@ensam.eu

Using a pattern-based homogenization scheme for modeling the linear viscoelasticity of nano-reinforced polymers with an interphase

Keywords: Particulate reinforced material (B), Micromechanics, Viscoelasticity, Morphological pattern

The self-consistent model based on morphological representative patterns is applied to the realistic case of the linear viscoelasticity of polymers reinforced by elastic nano-particles coated with a viscoelastic interphase. This approach allows to study the effect of such microstructure parameters as particle dispersion, particle size distribution and interparticle distance distribution. Under the assumption that the interphase has the same thickness around all reinforcing particles, it is shown that the particle size distribution has little effect on the effective properties of the heterogeneous material, whereas the particle dispersion and the interparticle distance distribution have stronger impacts.

Introduction

Few studies have investigated the interest of the morphological representative pattern (MRP) approach introduced by [START_REF] Stolz | Analyse morphologique et approches variationnelles du comportement d'un milieu élastique hétérogène[END_REF] and which defines a micromechanics framework that accounts for the microstructure characteristics of materials. [START_REF] Bornert | A generalized pattern-based self-consistent scheme[END_REF] has included the MRP approach in a self-consistent scheme which has been applied mainly by this author and his co-workers. For instance, Bilger et al. (2007) used this self-consistent scheme to study the effect of a non-uniform void distribution in porous materials and [START_REF] Chabert | Filler-filler interactions and viscoelastic behavior of polymer nanocomposites[END_REF] applied it to viscoelastic polymers reinforced by silica, but without accounting for a possible material interphase or for particle size distributions.

Taking an interphase into account in various materials has been performed by using the 4-phase self-consistent model in many papers. This model is based on a 3-phase spherical inclusion embedded in the homogeneous equivalent medium and has been given with full details by [START_REF] Maurer | An interlayer model to describe the physical properties of particulate composites[END_REF], with applications to interphases in viscoelastic materials by [START_REF] Maurer | Interphase effects on viscoelastic properties of polymer composites[END_REF], [START_REF] Schaeffer | Properties of the interphase in ternary polymer composites[END_REF], [START_REF] Eklind | Micromechanical transitions in compatibilized polymer blends[END_REF], [START_REF] Colombini | Evidence of an interphase relaxation in a ternary polymer blend through a reverse mechanical modeling[END_REF], [START_REF] Reynaud | Nanofillers in polymeric matrix: a study on silica reinforced PA6[END_REF], [START_REF] Colombini | Use of mechanical modeling to study multiphase polymeric materials[END_REF], [START_REF] Colombini | Origin of an additional mechanical transition in multicomponent polymeric materials[END_REF], among others. A derivation of the same model leading to different equations, has also been proposed by [START_REF] Hashin | An inverse method to determine the elastic properties of the interphase between the aggregate and the cement paste[END_REF], for an interphase problem in elastic materials.

In a recent work [START_REF] Diani | Micromechanical modeling of the linear viscoelasticity of carbonblack filled styrene butadiene rubbers: the role of the rubber-filler interphase[END_REF], the present authors studied the viscoelastic behavior of several carbon-black filled styrene butadiene rubbers (SBRs). The experimental data showed evidences in favor of the existence of an interphase at the rubber-filler interface, with enhanced viscoelastic properties compared to the bulk matrix viscoelasticity. The behavior and the thickness of the interphase were estimated by using the 4-phase selfconsistent model, which allowed a very good prediction of the viscoelastic behavior of several filled SBRs, but the interphase thickness was estimated to an arguably large value of 5 nm for spherical particles with a radius of 30 nm. Therefore, it seemed interesting to apply a more elaborate model that allows for the introduction of more microstructure parameters, and to see how the evaluation of the interphase thickness evolves. With the MRP approach, one may study the effect of parameters such as particle dispersion, particle size distribution and interparticle distance distribution. The method is applied here to one of the above-mentioned carbon-black filled rubbers, where the carbon-black agglomerates are approximated by elastic spherical particles, and where both the polymer interphase and the polymer matrix are viscoelastic. The mechanical behaviors considered for the constitutive phases are realistic, and therefore the homogeneous equivalent medium can be compared to the behavior of an actual carbon-black filled SBR. Considering actual materials gives a sound framework to study the impact of some of the parameters used in the MRP approach. The paper is organized as follows. In the next section, the general equations of the MRP self-consistent model are detailed in a comprehensive and an easy way, in order to favor its use among the scientific community. Additional equations that are useful for the specific case of 3-phase spherical patterns are also detailed, and the model parameters are discussed. Then, the effects of particle dispersion, particle size distribution (based on realistic carbon-black filler distributions), and interparticle distance distribution on the predicted viscoelastic behavior of the heterogeneous material are examined by accounting for a large number of patterns. The comparison between the model predictions and the behavior of an actual filled rubber provides estimates for the interphase thickness according to the microstructures considered.

Morphologically representative pattern-based self-consistent model

General theory

The main motivation of the MRP self-consistent model introduced by [START_REF] Bornert | A generalized pattern-based self-consistent scheme[END_REF] is to account for some dispersion and size effects that cannot be included in classical homogenization schemes. Given a schematic representation of a material reinforced by randomly distributed particles of various sizes and geometries (Fig. 1), the idea is to recognize the various patterns that are found within the material (Fig. 2) and to build a homogenization scheme that takes them into account. Let us assume that the heterogeneous material contains p constitutive phases with volume fractions f j and elastic stiffness tensors C j , j ∈ {1, 2, ...p}. The objective of any homogenization scheme is to compute the behavior C h of the homogeneous equivalent medium (HEM) defined by

C h = ∑ j f j C j : A j (1)
where the average strain localization tensor A j in phase j is given by

εj = A j : E (2)
where εj is the average strain in phase j and E is the overall strain applied to the heterogeneous material that may be written as

E = ∑ j f j εj . (3)
Let us account now for the description of the heterogeneous material as a perfectly disordered distribution of patterns of n different types, plus a complement of matrix material left between the patterns, as shown in Fig. 2. The latter residual matrix volume has no specific shape, but it is treated as a single spherical homogeneous pattern Ω 0 because of its statistically isotropic distribution over the heterogeneous material, as done in the [START_REF] Hashin | A variational approach to the theory of the elastic behaviour of multiphase materials[END_REF] approach for bounds. This differs from the spherical shape used for the other patterns, which originates from the actual shape of the particles. In general, several copies of each Ω λ pattern (λ > 0) are present in the heterogeneous medium, but considering a single copy is sufficient in a self-consistent model, where any pattern is assumed to behave as if surrounded by the unbounded HEM only (see Fig. 2). Various types of volume fractions come into play in the model, at either the local scale of a pattern or at the global scale of the heterogeneous material, and they must be defined carefully. Let V λ denote the total volume of the patterns of type Ω λ found in a volume V of heterogeneous material (with V = ∑ λ V λ ), and let V λ j denote the total volume of phase j over all these patterns (with

V λ = ∑ j V λ j , ∀λ, and f j = ∑ λ V λ j /V , ∀j).
Therefore, ϕ λ j = V λ j /(f j V ) denote the fraction of the total volume of phase j that belongs to the set of λ-type patterns, with λ ∈ {0, 1, ..., n}, and it must not be mixed up with the volume fraction of phase j in a Ω λ pattern, which is equal to V λ j /V λ . One has ∑ λ ϕ λ j = 1, ∀j, obviously, and the average strain εj writes also as

εj = ∑ λ ϕ λ j ϵ λ j (4)
where ϵ λ j , the average strain over phase j in pattern λ, is defined by the localization tensor A λ j associated to pattern λ,

ϵ λ j = A λ j : ϵ (5)
where ϵ denotes the auxiliary far strain field applied to the unbounded HEM surrounding pattern Ω λ (Fig. 2), which may differ from E. Combining (4) and ( 5)

gives εj = ∑ λ ϕ λ j A λ j : ϵ (6)
and substituting (6) into (3) gives the relation between the far fields ϵ and E :

ϵ = ( ∑ j f j ∑ λ ϕ λ j A λ j ) -1 : E . (7)
Introducing ( 6) and ( 7) into (2) provides the expression of the localization tensor over phase j in the heterogeneous material :

A j = ∑ λ ϕ λ j A λ j : ( ∑ j f j ∑ λ ϕ λ j A λ j ) -1 (8)
which, once introduced in (1), gives C h finally as

C h = ∑ j f j C j : ∑ λ ϕ λ j A λ j : ( ∑ j f j ∑ λ ϕ λ j A λ j ) -1 . (9)
Note that (9) simplifies into the generalized self-consistent model of [START_REF] Hervé | N-layered inclusion-based micromechanical modeling[END_REF] if a single type of heterogeneous pattern is considered, without any homogeneous pattern (Ω 0 ). Considering (9), the evaluation of the effective behavior of the heterogeneous material amounts to computing the strain localization tensors A λ j over all phases in all different patterns. These tensors depend on C h , since each pattern is surrounded by the unbounded HEM, and consequently ( 9) is an implicit equation in C h , as expected in a self-consistent scheme. Eq. ( 8) looks very similar to Eq. ( 8) of [START_REF] Marcadon | Micromechanical modeling of packing and size effects in particulate composites[END_REF] but with a significant difference, since ϕ λ j has been erroneously replaced by c λ , which denotes the volume fraction of type-λ patterns over the heterogeneous material (c λ = V λ /V ). This mistake may mislead the reader of [START_REF] Marcadon | Micromechanical modeling of packing and size effects in particulate composites[END_REF] but does not seem to have spread in the calculations.

In the following, the study will focus on an isotropic case where an isotropic polymer matrix is reinforced by a statistically isotropic distribution of spherical isotropic particles with account of an isotropic interphase at the matrix-particle interface. Therefore, Eq. ( 9) is simplified by projecting the strain localization tensors onto the spherical and deviatoric isotropic fourth-order tensors J = 1 3 i ⊗ i and K = I -J , where i and I denote the identity tensors of second and fourth order, respectively, which gives A j = A dj K + A sj J and A λ j = A λ dj K + A λ sj J . This allows working with scalar quantities, and Eq. ( 8) transforms into:

A dj = ∑ λ ϕ λ j A λ dj ∑ j f j ∑ λ ϕ λ j A λ dj and A sj = ∑ λ ϕ λ j A λ sj ∑ j f j ∑ λ ϕ λ j A λ sj . ( 10
)
The behavior of phase j being characterized by its shear modulus G j and its bulk modulus K j , the moduli G h and K h of the isotropic homogeneous equivalent medium are given, from ( 9), by

G h = ∑ j f j G j ∑ λ ϕ λ j A λ dj ∑ j f j ∑ λ ϕ λ j A λ dj and K h = ∑ j f j K j ∑ λ ϕ λ j A λ sj ∑ j f j ∑ λ ϕ λ j A λ sj . ( 11
)
This actually is a system of two coupled equations, since the strain localization tensors A λ dj and A λ sj depend on G h and K h . These tensors are detailed in the next section for two specific patterns that will be of special interest in the model application section, namely a pure matrix pattern and a 3-phase spherical pattern.

2.2. Localization tensors for a pure matrix pattern and a 3-phase spherical pattern Pattern Ω 0 (Fig. 2) is a spherical matrix domain. When embedded in the unbounded HEM, the strain localization tensor for such a pattern yields from the classical [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion and related problems[END_REF] solution:

A 0 d3 = ( 1 + 6 5 G 3 -G h 3K h + 4G h K h + 2G h G h ) -1 and A 0 s3 = ( 1 + 3 K 3 -K h 3K h + 4G h ) -1
(12) where G 3 and K 3 denote the shear and bulk moduli of the isotropic matrix.

The other patterns sketched in Fig. 2 are made of a spherical particle coated with an interphase and surrounded by a matrix shell. When this pattern is embedded in the unbounded HEM, the localization problem reminds the 4-phase micromechanical model introduced by [START_REF] Maurer | An interlayer model to describe the physical properties of particulate composites[END_REF] and generalized by [START_REF] Hervé | N-layered inclusion-based micromechanical modeling[END_REF]. We recall the expressions of the corresponding localization tensors that may be found in [START_REF] Hervé | N-layered inclusion-based micromechanical modeling[END_REF]. In an effort of consistency, we will use their notations and the filler particles, the interphase, the matrix, and the HEM are referred to as phases 1, 2, 3, and 4, respectively. According to Eq. ( 36) of [START_REF] Hervé | N-layered inclusion-based micromechanical modeling[END_REF], the deviatoric part of the strain localization tensor in a λ-type pattern is

A λ dj = A j - 7 5 ( 1 + 3K j G j ) (R λ j ) 5 -(R λ j-1 ) 5 (R λ j ) 3 -(R λ j-1 ) 3 B j ∀j ∈ {1, 2, 3} (13)
with λ > 0 and R λ j the outer radius of phase j (taking R λ 0 = 0), where1

A 1 = P (3) 22 χ , A 2 = P (3) 22 P (1)
11 -P

(3) 21 P

(1) 12 χ , A 3 = P

(3) 22 P

(2)

11 -P (3) 21 P (2) 12 χ , (14)
and

B 1 = -P (3) 21 χ , B 2 = P (3) 22 P (1)
21 -P

(3) 21 P

(1) 22 χ , B 3 = P

(3) 22 P

(2)

21 -P (3) 21 P (2) 22 χ , (15)
with χ = P

(3) 11 P

(3) 22 -P

(3) 21 P

(3)

12 and P (1) = M (1) , P (2) = M (2) .M (1) , P (3) = M (3) .M (2) .M (1) . The 4 × 4 M (j) matrices are explicitly given by Eq. ( 27) of [START_REF] Hervé | N-layered inclusion-based micromechanical modeling[END_REF] and are not repeated here for brevity.

For the spherical part of the strain localization tensor, Eqs (14) of [START_REF] Hervé | N-layered inclusion-based micromechanical modeling[END_REF] gives

A λ s1 = 1 Q (3) 11 , A λ s2 = Q (1) 11 Q (3) 11 , A λ s3 = Q (2) 11 Q (3) 11 (16)
with λ > 0 and 3) .N (2) .N (1) . Matrix N (j) is defined as

Q (1) = N (1) , Q (2) = N (2) .N (1) , Q (3) = N (
N (j) = 1 3K j+1 + 4G j+1 [ 3K j + 4G j+1 4(G j+1 -G j )/(R λ j ) 3 3(K j+1 -K j )(R λ j ) 3 3K j+1 + 4G j ] . (17)
It may be noted that matrices M (j) and N (j) depend on the outer radius R λ j of phase j and on the elastic constants of phases j and j + 1 only. Consequently, M (3) and N (3) involve the shear and bulk moduli of the HEM.

Model parameters

When dealing with a distribution of particle sizes, one has to define the volume fractions and geometric parameters involved in the MRP approach. First, when all particles are of the same chemical nature, it seems reasonable from a physical point of view to assume that the interphase thickness e is independent of the particle size, and is therefore constant for all patterns Ω λ with λ > 0. Second, let us assume that the particle sizes and the fraction number of particles of each size are known. This is provided by the particle size distribution when it is available. For each particle size, a pattern Ω λ is defined, where the particle radius is R λ 1 and the outer radius of the interphase layer is R λ 2 = R λ 1 + e (Fig. 2). The probability of finding a particle of radius R λ 1 over all particles within the reinforced material is denoted α λ (with ∑ λ>0 α λ = 1). With these notations, the fraction of the total volume of phase j that is found in type-λ patterns, ϕ λ j , is obtained from simple geometric considerations and one has

ϕ λ 1 = α λ (R λ 1 ) 3 ∑ λ>0 α λ (R λ 1 ) 3 , ϕ λ 2 = α λ [ (R λ 1 +e) 3 -(R λ 1 ) 3 ] ∑ λ>0 α λ [ (R λ 1 +e) 3 -(R λ 1 ) 3 ] , ϕ λ 3 = f 1 α λ [ (R λ 3 ) 3 -(R λ 1 +e) 3 ] ∑ λ>0 α λ [ (R λ 1 ) 3 -f 1( R λ 1 +e) 3 ] (18)
for λ > 0. From these equations, one notes that the relations ∑ λ>0 ϕ λ 1 = 1 and ∑ λ>0 ϕ λ 2 = 1 do hold and that the choice of the outer radii R λ 3 , λ ∈ {1, ..., n}, decides of the volume fractions ϕ λ 3 , λ ∈ {1, ..., n}. Equivalently, giving a set of ϕ λ 3 values decides of the outer radii R λ 3 . Pattern Ω 0 is made of matrix material only (ϕ 0 1 = ϕ 0 2 = 0), and it contains a fraction ϕ 0 3 of the total matrix volume available in the heterogeneous material, with

ϕ 0 3 = 1 - ∑ λ>0 ϕ λ 3 . ( 19
)
To sum up, in order to run an MRP approach as sketched in Fig. 2 with a given particle size distribution, one must provide the interphase thickness e and the set of matrix volume fractions ϕ λ 3 , λ ∈ {1, ..., n}, or equivalently the set of outer radii R λ 3 , λ ∈ {1, ..., n}. The coupled implicit equations ( 11) must be solved finally to get G h and K h , and a mere fixed-point method is very efficient for that purpose.

Viscoelasticity

The equations involved by the MRP self-consistent model for the patterns sketched in Fig. 2, where all phases are isotropic, have been written above within an elasticity framework, and analytical expressions have been obtained. Therefore, by applying the elastic-viscoelastic correspondence principle stated by [START_REF] Hashin | Complex moduli of viscoelastic composites. I. General theory and application to particulate composites[END_REF], these expressions extend readily to linear viscoelasticity by replacing the elastic moduli (G, K) by their respective complex viscoelastic counterparts (G * , K * ). This principle provides a way to apply micromechanics models within a linear viscoelasticity framework that is much easier than the classical Laplace-Carson transform. The complex moduli relate a harmonic strain loading to the steady harmonic stress response of a linear viscoelastic material, and they depend of the applied frequency. Actually, the real and imaginary parts of these complex moduli, defining the storage and loss moduli of the material, respectively, provide the same information as the viscoelastic relaxation or creep functions. One reason for using complex moduli is that the storage and loss shear moduli, for instance, can be measured easily over a wide range of frequencies with dynamic mechanical analysis machines. Since linear viscoelasticity provides much more information than elasticity for polymers, it helps assessing the relevance of micromechanics models more strictly. The MRP self-consistent model is therefore applied in such a viscoelastic context in the following.

Application to a carbon-black filled styrene butadiene gum

In order to apply the MRP self-consistent model to a realistic material, the linear viscoelasticity of carbon-black filled SBRs with a filler-rubber interphase is considered. When filled rubbers are prepared, a weight fraction of carbon black is added to the gum, but the former is an extremely thin powder that tends to agglomerate into what is considered in this paper as particles idealized as spheres. Therefore, the latter contain some gum trapped between filler subparticles, and an effective volume content of particles can be defined that we denote f 1 . As explained by [START_REF] Diani | Micromechanical modeling of the linear viscoelasticity of carbonblack filled styrene butadiene rubbers: the role of the rubber-filler interphase[END_REF], the values that are used here are consistent with experimental measures based on the experimental procedure of [START_REF] Medalia | Effective degree of immobilization of rubber occluded within carbon black aggregates[END_REF]. In addition to a volume fraction and an average size, these particles may also have a size distribution, and the distance between nearest neighbors may also be distributed around an average value. These are typical microstructure parameters (among many others, see Torquato, 2002, for instance) that the present study tries to account for with the MRP approach in order to evaluate their effects on the effective properties of filled rubbers.

Materials and behavior

The viscoelastic tensile behaviors of an unfilled SBR and of several carbonblack filled SBR materials (manufactured with the same SBR gum) were measured by dynamic mechanical analysis. Assuming that the bulk modulus of the unfilled rubber does not depend on frequency (K 3 = 3.5 GPa), the dynamic shear modulus of the unfilled rubber is deduced; it is shown in Fig. 3 with its approximation by a generalized Maxwell model. Note that albeit the bulk modulus of the SBR does change through the glass transition, this approximation is fairly reasonable when focusing on the viscoelastic shear modulus. Actually, this assumption is supported by the fact that in polymer networks, the bulk modulus results mainly from van der Waals interactions and not from entropy change [START_REF] Diani | Study on the temperature dependence of the bulk modulus of polyisoprene by molecular dynamics simulations[END_REF]. Then, assuming that the viscoelastic shear modulus of the interphase is enhanced compared to the viscoelastic shear modulus of the bulk gum shown in Fig. 3, because of a reduced molecular mobility, and assuming the carbon black is elastic and incompressible, with G 1 = 70 GPa, [START_REF] Diani | Micromechanical modeling of the linear viscoelasticity of carbonblack filled styrene butadiene rubbers: the role of the rubber-filler interphase[END_REF] obtained good estimates of the viscoelasticity of the filled SBRs by applying the 4-phase model with a constant interphase thickness of about 5 nm for filler aggregates with an average radius of 30 nm. An example of the comparison between model and experiment is shown in Fig. 4 for the SBR filled with 40 per hundred rubber (phr) carbon black, where a 30% effective volume fraction of filler could be evaluated. The results for other carbon-black contents can also be found in Fig. 6 of [START_REF] Diani | Micromechanical modeling of the linear viscoelasticity of carbonblack filled styrene butadiene rubbers: the role of the rubber-filler interphase[END_REF].

An interphase thickness of 5 nm in carbon-black filled SBRs may be considered as too large, though it has never been measured in systems like ours. In order to investigate the impacts of a possible effect of dispersing the particles more or less uniformly and of a probable particle size distribution on the estimate of the viscoelastic behavior of filled rubbers, the MRP selfconsistent model is applied assuming that the behavior of the interphase estimated by [START_REF] Diani | Micromechanical modeling of the linear viscoelasticity of carbonblack filled styrene butadiene rubbers: the role of the rubber-filler interphase[END_REF] is reasonable. Therefore, in the following, phase 1 consists of incompressible elastic spherical particles with G 1 = 70 GPa, with a radius of 30 nm (except when a radius distribution is defined) and with a volume fraction of f 1 = 0.3. Phase 2 is a 5-nm thick interphase with a viscoelastic shear modulus as shown in Fig. 3 and with a constant bulk modulus of K 2 = 3.5 GPa. Phase 3 is the viscoelastic SBR gum, with a viscoelastic shear modulus as shown in Fig. 3 and a constant bulk modulus of K 3 = 3.5 GPa. Moreover, the viscoelastic shear moduli provided by the MRP self-consistent model with various pattern choices are compared to the 4-phase model prediction that was found to fit the behavior of the 40 phrfilled rubber quite well (Fig. 4). This allows a more precise comparison, since experimental noise is removed.

Particle dispersion effect

By varying the fraction ϕ 0 3 of the total matrix volume that is allocated to pattern Ω 0 , the MRP approach makes possible to account for filler clusters within the material. For instance, a MRP model defined by patterns Ω 0 and Ω 1 only (Fig. 2), with a given interphase thickness in pattern Ω 1 , defines the case of particles which are monodisperse in size but with a possible dispersion effect. For instance, the matrix shell around the coated particles is thin if ϕ 0 3 is large, and the particles are close to each other. It may be pointed out that if the constraint of the same interphase thickness for all particles were relaxed, then pattern Ω 1 would describe all cases of fillers with such polydisperse sizes that all coated particles are homothetic to each other, but this is not considered here.

Considering patterns Ω 0 and Ω 1 only, the fraction of the total volume of heterogeneous material which is contained in type-1 patterns,

c 1 = V 1 /(V 0 + V 1 )
, can be used to characterize the dispersion of particles. For instance, if this parameter is large, the volume of the Ω 0 pattern is small, i.e. ϕ 0 3 is small, and the particles are far from each other. According to the Kepler conjecture proved by [START_REF] Hales | A proof of the conjecture[END_REF], no packing of identical spheres is denser than the FCC (face-centered cubic) lattice, and therefore the largest possible value for c 1 is π/ √ 18 ≃ 0.74 (and ϕ 0 3 is minimal). Since FCC lattice periodic and anisotropic, random isotropic dense packings of identical spheres have also been studied by computer simulation and a limit value of 0.64 seems reasonably acceptable (Torquato et al., 2000, Farr andGroot, 2009, for instance), which is very close to the experimental value of [START_REF] Scott | The density of random close packing of spheres[END_REF], although the problem is not well-defined mathematically [START_REF] Torquato | Is random close packing of spheres well defined?[END_REF]. By contrast, the minimum c 1 value is obtained when pattern Ω 1 is made of particle and interphase only (coated particles agglomerate and ϕ 0 3 = 1), which leads to c 1 = 1 -f 3 = (1 + e/R 1 1 ) 3 f 1 . Consequently, c 1 ranges from 0.48 to 0.74 when f 1 = 0.3, R 1 = 30 nm, and e = 5 nm. Fig. 5 shows the effect of particle dispersion on the viscoelastic shear modulus that is given by the MRP self-consistent model applied to monodisperse (in size) inclusions. It can be observed that decreasing c 1 enhances the shear viscoelastic modulus, and even though this increase is more noticeable in the rubbery state (at low frequencies), it does exist also in the glassy state (at high frequencies). Therefore, the MRP self consistent model predicts that filler clusters have a reinforcing effect with respect to well-dispersed fillers. Moreover, if pattern Ω 1 does not contain the matrix material (clustered particles), the experimental data for the 40 phr SBR is best fitted by the model for an interphase thickness of ≃ 3 nm with all other parameters unchanged. By contrast, if pattern Ω 1 contains the largest volume fraction compatible with c 1 = 0.74 (well-dispersed particles), the experimental data is well reproduced for an interphase thickness of 4.5 nm. As a consequence, the interphase thickness of 5 nm deduced by fitting the 4-phase model probably overestimated the actual value, since some filler clustering is very likely in filled rubbers.

Particle size distribution effect

The MRP approach allows taking a particle size distribution into account very directly, by varying the number and the content of the heterogeneous patterns, as illustrated in Fig. 2. In order to study the resulting effect that is predicted by the model, the particle size distribution only is varied, keeping all other parameters fixed. For instance, a volume content of f 1 = 0.3 and an average particle radius of R 1 = 30 nm are used. Moreover, the fraction of the total matrix volume that belongs to pattern Ω 0 keeps the value of ϕ 0 3 = 0.5 that was found in the previous section to fit well the case of monodisperse particles coated by a 5-nm interphase with c 0 = 1 -c 1 = 0.26. It is quite natural to use patterns of various sizes in the case of polydisperse particles, although this is not necessary, and the question arises of the maximum possible packing of random spheres of various diameters. Of course, c 0 values lower than 0.26 can easily be reached, as shown by [START_REF] Farr | Close packing density of polydisperse hard spheres[END_REF], for instance, where up to c 0 ≈ 0.17 was reached with suitable fractions of small, medium, and large spheres with size ratios 1:3:9, but the value of 0.26 will nevertheless be used below for easier comparisons with the results of the previous section. Two particle size distributions are considered here. They are presented in Fig. 6 in terms of the occurrence frequency α λ (with ∑ λ>0 α λ = 1) of particles with radius R λ 1 . They are adapted from the distributions measured in two filled SBRs by [START_REF] Klüppel | The role of disorder in filler reinforcement of elastomers on various length scales[END_REF], by applying a shift so that (∑ λ>0 (R λ 1 ) 3 α λ ) 1/3 = 30 nm. In other words, the average volume of the particles corresponds to a sphere with 30-nm radius. As mentioned in section 2.3, once the distribution (R λ 1 , α λ ) is defined for λ ∈ {1, ..., n}, the last input required is the set of ϕ λ 3 values, or equivalently R λ 3 for λ ∈ {1, ..., n}. This defines how the matrix volume that is not used in pattern Ω 0 is distributed among the other patterns, and it is easier to define a set of ϕ λ 3 (positive and such that ∑ λ>0 ϕ λ 3 = 1 -ϕ 0 3 ) than a set of admissible R λ 3 because of the constraint R λ 3 ≥ R λ 1 + e for λ ∈ {1, ..., n}. As a first step, we consider the case where the matrix volume is the same for all sets of heterogeneous patterns, i.e. ϕ λ 3 = (1 -ϕ 0 3 )/n for λ ∈ {1, ..., n}. For the two size distributions shown in Fig. 6a, this simple assumption leads to the distributions of phase 3 thickness R λ 3 -R λ 2 around the coated particles shown in Fig. 6b. All parameters being defined, the MRP self-consistent model can be applied with either 15 (first distribution) or 17 (second distribution) patterns. The viscoelastic shear moduli calculated with Eq. ( 11) are compared in Fig. 7 to the results obtained in the monodisperse case, and the effect of the size distribution appears negligible. Note that different results, with a significant effect of size distribution, would have been obtained if the distributions were shifted such that the average radius

∑ λ>0 R λ 1 α λ (instead of (∑ λ>0 (R λ 1 ) 3 α λ ) 1/3
) had been taken equal to 30 nm. The above noticeable result would simplify the use of the model significantly, since any particle size distribution could be reduced to two patterns only, without affecting the homogenization result significantly, provided the appropriate definition of the equivalent particle radius is used. Nonetheless, the crude assumption of a uniform distribution of matrix volume over all sets of heterogeneous patterns remains questionable, and variants are explored below.

Interparticle distance effect

The distribution of the volume fraction of phase 3 in the patterns, ϕ λ 3 for λ ∈ {0, ..., n}, is strongly related to the distribution of interparticle distance. Keeping distribution 1 for the particle size (Fig. 6), the above assumption of equal values of ϕ λ 3 for λ ∈ {1, ..., n} is now relaxed in order to test various schematizations of the material. One simple distribution of ϕ λ 3 considers that the thickness of phase 3 layer increases with the particle radius, which corresponds to a material where small particles are close to each other and far from large ones. By contrast, the thickness of the matrix layer may decrease when the particle radius increases in another distribution, and thus large particles would be closely packed, with small particles repelled farther. Fig. 8 shows examples of such cases (distributions A, B, and C), with the same particle size distribution (R λ 1 , α λ ) but with various sets of outer radius R λ 3 , to which the model has been applied. Fig. 9 displays the viscoelastic shear modulus predicted by the model for each material representation. By comparing to Fig. 5, it can be observed that the distribution of the matrix thickness around the particles has a stronger impact on the behavior of the homogeneous equivalent material than the distribution of the particle size, with an evident reinforcement when the matrix phase is mostly concentrated around the smaller (distribution C in Fig. 8). For such a microstructure, the viscoelastic shear modulus differs from what was obtained with the (monodisperse) two-pattern representation, which has been shown above to approximate a uniform distribution of ϕ λ 3 values very well (distribution A in Fig. 8). Therefore, specific distributions of interparticle distances may enhance the role of the interphase within particle reinforced polymers. On the contrary, although it differs significantly from distribution A, distribution B does not lead to a noticeably different effective viscoelastic behavior. Using distribution C, the model can also be used to estimate the interphase thickness by fitting to the reference curve of Fig. 4, which leads to e = 4.3 nm. Consequently, the distribution of interparticle distances, which cannot be accounted for with a simplified two-pattern model, has a non-negligible effect, but which is second-order compared to the particle dispersion effect when evaluating the interphase thickness.

Bulk modulus

Up to this point, the predicted viscoelastic shear modulus only has been presented, but the model also predicts the viscoelastic bulk modulus with Eq. ( 11). Due to expression of A λ sj (j ∈ {1, 2, 3}), K h depends on both shear and bulk moduli of all phases. In polymers, a drop of the bulk modulus by a factor between 1.5 and 3 is expected when sweeping from high frequencies to low frequencies. The assumption of an elastic response of the matrix material to a volume change, i.e. a constant bulk modulus within the matrix (and in the interphase, consequently, where the same value is used), is reasonable when the shear modulus of the homogeneous equivalent material is studied, but is questionable when its bulk modulus is analyzed. Actually, with the chosen inputs, the variations of the HEM storage bulk modulus with loading frequency have been found unrealistically small (less than 10%) in all our simulation cases. Therefore, we may focus on the results obtained at the two ends of the frequency range only, and this amounts to considering two elastic problems.

At low frequency, in the rubbery state, the matrix is very soft and the interphase is both harder than the matrix and softer than the particles. At high frequency, in the glassy state, the interphase behaves like the matrix, which is much stiffer than in the rubbery state but still softer than the reinforcing particles. The model predicts that the particle dispersion, the particle size distribution, and the distribution of the matrix phase in the patterns all have a negligible effect on the bulk modulus K h when the matrix is in the rubbery state. When the is in the glassy state, the trends found for the bulk modulus are similar to the trends already reported for the shear modulus: increasing the volume fraction of pattern Ω 0 has a stiffening effect, the particle size distribution has a very slight effect, and assigning more matrix around the small particles leads to higher K h values.

Conclusion

A previous study on the viscoelasticity of carbon-black filled SBRs had shown experimental evidences of the existence of a viscoelastic interphase at the filler-rubber interface. In order to estimate the behavior and the thickness of this interphase, which are difficult to reach experimentally, a mere 4-phase self-consistent model was applied and lead to an arguably large interphase thickness of 5 nm. In an effort to understand the effect of microstructure parameters on the material behavior, a morphological representative pattern-based self-consistent model has been applied to estimate the overall viscoelastic response of a viscoelastic matrix reinforced by elastic spherical particles coated by a thin viscoelastic interphase. The general equations of the MRP self-consistent model have been detailed to promote a wider diffusion of this potentially rich approach, which so far had been used with two patterns only. For the first time, the use of a large number of patterns has allowed studying the effects of microstructure parameters such as particle dispersion, particle size distribution, and interparticle distance distribution on the material behavior. The results show a significant impact of the particle dispersion, a weak effect of the particle distribution for a fixed average particle volume, and a moderate effect of the distribution of interparticle distance. Finally, the results also showed that the interphase thickness would be evaluated to smaller values if the particle dispersion and interparticle distance could be measured in the filled rubbers considered.

Figure 1 :

 1 Figure 1: Schematic representation of a material reinforced by randomly distributed particles of various sizes and coated with an interphase.

Figure 2 :

 2 Figure 2: Morphological representative patterns (embedded in the unbounded homogeneous equivalent medium) that may be taken into account in the MRP self-consistent model applied to the material shown in Fig. 1.

Figure 3 :

 3 Figure 3: (a) Storage and loss shear moduli of the unfilled SBR and approximation by a generalized Maxwell model. (b) Comparison between the unfilled SBR viscoelasticity and the assumed viscoelasticity of the interphase at the rubber-filler interface.

Figure 4 :

 4 Figure 4: Comparison between the experimental viscoelastic shear modulus of the SBR filled with 40 phr of carbon black and the predictions of the 4-phase model using the two behaviors shown in Fig. 3b.

Figure 5 :

 5 Figure5: Storage and loss shear moduli of the reinforced rubber predicted by the MRP self-consistent model, using patterns Ω 0 and Ω 1 only, and predicted by the 4-phase model. Same interphase thickness (5 nm), filler content (0.3) and filler radius (30 nm) in all calculations.

Figure 6 :

 6 Figure 6: (a) The distributions of particle radius R λ 1 used for carbon-black clusters, and (b) the resulting distribution of R λ3 -R λ 2 when the same amount of matrix phase is assumed in all sets of heterogeneous patterns.

Figure 7 :

 7 Figure 7: Effect of the particle size distribution on the prediction of the viscoelastic shear modulus of the filled rubber by the MRP self-consistent model.

Figure 8 :

 8 Figure 8: (a) Definition of three sets of morphological patterns using the same distribution of particle size (distribution 1 in Fig. 6) but different distributions of the volume fractions ϕ λ 3 of the matrix phase. (b) Resulting distributions of the thickness R λ 3 -R λ 2 of the matrix shell.

Figure 9 :

 9 Figure 9: Effect of the distribution of the matrix volume over the patterns on the estimate of the viscoelastic shear modulus of the filled rubber using the MRP self-consistent model. The three distributions used are defined in Fig. 8.

Up to the end of this paragraph, upperscripts between brackets refer to phase numbers and pairs of subscripts refer to matrix components.

Acknowledgments

The authors would like to acknowledge fruitful discussions with R. Sachs from Michelin company that stimulated this work.

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