B Lachet

Rousseau

E Roguet

K Akhan

N Brusselle-Dupend

V Le Corre

M Sidhom

L Cangemi

M Moreaud

G Clavier

V Lachet

B Rousseau

Investigation of the 3D crystalline network impact on the elastic properties of semi-crystalline polymers from a multi-scale modelling approach

Investigation of the 3D crystalline network impact on the elastic properties of semi-crystalline polymers from a multi-scale modelling approach

Introduction

This work addresses the general issue of the mechanical behaviour of Semi-Crystalline Polymers (SCP). These materials are widely used in engineering applications and the improvement of their mechanical properties is continuously researched due to the variety of application domains. To help for materials design, predicting the impact of the materials microstructure on their macroscopic mechanical properties is a challenge.

At the microscopic scale, various methods have been developed to investigate the structure of SCP, for instance scanning electron microscopy or X-ray scattering [START_REF] Rh | An improved permanganic etchant for polyolefines[END_REF][START_REF] Farge | Anisotropy development during HPDE necking studied at the microscale with in situ continuous 1D SAXS scans[END_REF][START_REF] Humbert | Small strain behavior of polyethylene: in situ SAXS measurements[END_REF][START_REF] Chmelai | Free and constrained amorphous phases in polyethylene: Interpretation of 1H NMR and SAXS data over a broad range of crystallinity[END_REF]. Some studies have been focused on understanding the involved crystallization mechanisms for a better knowledge of their microstructural layout [START_REF] Yamada | Isotacticity dependence of spherulitic morphology of isotactic polypropylene[END_REF][START_REF] Zhu | Morphological distribution of injection-moulded isotactic polypropylene: a study of synchrotron small angle X-ray scattering[END_REF][START_REF] Nishida | Novel morphology of isotactic polypropylene crystal generated by a rapid temperature jump method[END_REF]. At the macroscopic scale, many works have been done to model the mechanical response under several conditions such as pressure, temperature or strain rate. Based on this description, many mechanical constitutive model equations have been developed. Boyce et al. has proposed a model based on the polymer chains kinetics for glassy polymers [START_REF] Boyce | Large inelastic deformation of glassy polymers. Part I: rate dependent constitutive model[END_REF]. These authors introduce the decomposition of the plastic flow into two kinds of resistances: an intermolecular resistance to stand for a segment rotation and an entropic resistance to characterize the molecular alignment. Variations of the original model have then been proposed to consider different experimental conditions [START_REF] Boyce | Plastic flow in oriented glassy polymers[END_REF][START_REF] Tomita | Prediction of deformation behavior of glassy polymers based on molecular chain network model[END_REF] and to modify the account of entanglements in the structure [START_REF] Arruda | A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials[END_REF]. Microstructure considerations have also been considered to model Semi-Crystalline Polymers (SCP) behaviour. In literature, SCP are considered heterogeneous media with molecular chains that can be part of a free amorphous phase, a crystalline network and/or an intermediate tied amorphous phase [START_REF] Flory | The interphase in lamellar semicrystalline polymers[END_REF]. Some approaches are based on a two phase representation of Semi-Crystalline Polymers at the mesoscopic level which represent the material with a "solid" network (including the crystal and the tied amorphous zone) interacting with a saturating "soft" or fluid-like phase (essentially free amorphous) [START_REF] Cangémi | A Two-Phase Model for the Mechanical Behavior of Semicrystalline Polymers Oil & Gas Science and Technology[END_REF]. Micromechanical modelling based on Eshelby theory [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion, and related problems[END_REF] has also been developed within microstructures context. Applied to polymer materials, this modelling allowed conclusive results on prediction of elastic properties for Polypropylene (PP) or Polyethylene (PE) [START_REF] Bédoui | Micromechanical modeling of elastic properties in polyolefins[END_REF][START_REF] Bédoui | Micromechanical modeling of isotropic elastic behavior of semicrystalline polymers[END_REF]. In this approach, the crystalline lamellae are ellipsoidal inclusions whereas the amorphous phase is the matrix. The same authors have also developed two-phase models where the inclusions are dispersed randomly in the amorphous phase [START_REF] Bédoui | Micromechanical modeling of isotropic elastic behavior of semicrystalline polymers[END_REF]. A three-phase model has also been developed by Gueguen et al. [START_REF] Gueguen | A new three-phase model to estimate the effective elastic properties of semi-crystalline polymers: application to PET[END_REF] through a generalization of the double-inclusion model. All these above mentioned works do not take into consideration the specific 3D architecture of semi-crystalline materials at the mesoscopic scale, and in particular the presence of spherulitic structures where lamellae grow radially from the nucleus of the spherulite in all spatial directions, with some possible branches that grow from the principal lamellae. The molecular origin of this branching and the consequent cross-hatched structure were clarified by Lotz et al. [START_REF] Lotz | Structure and morphology of poly(propylenes): A molecular analysis[END_REF] and such growth can be observed by optical microscopy or SEM.

Promising results obtained with previously described micromechanical models may suggest that this particular structure of a semi-crystalline sample does not significantly affect the macroscopic properties. However, other works suggest that the spherulites play an important role in the deformation field [START_REF] Sell | Introduction à la mécanique des polymères INPL[END_REF][START_REF] Teixeira-Pinto | Towards the size estimation of a Representative Elementary Domain in semi-crystalline polymers[END_REF][START_REF] Oktay | Modeling of spherulite microstructures in semicrystalline Polymers[END_REF]. For instance, Uchida et al. [START_REF] Uchida | Finite element simulation of deformation behavior of semi-crystalline polymers with multi-spherulitic mesostructure[END_REF] proposed a Finite Element homogenization method on multi-spherulitic structures; they concluded that the distribution of lamellar orientation in the spherulite has an impact on the material behaviour. Laschet et al. [START_REF] Laschet | Multiscale simulation to predict microstructure dependent effective elastic properties of an injection molded polypropylene component[END_REF] have also proposed a two-level homogenization scheme to determine the influence of the lamella design and of the crystalline properties on the effective properties of spherulitic microstructures. An isotropic interface between spherulites has been introduced to manage the spherulitic layout.

At first approximation SCP are two-phase materials, consisting of crystalline lamellae embedded in amorphous phase. Only few papers tackle the role of the amorphous phase in the mechanical macroscopic behaviour. The pioneering work of Struik [START_REF] Struik | The mechanical and physical ageing in semicrystalline polymers: 1[END_REF][START_REF] Struik | The mechanical and physical ageing in semicrystalline polymers: 2[END_REF], followed by Chmelai et al. [START_REF] Chmelai | Free and constrained amorphous phases in polyethylene: Interpretation of 1H NMR and SAXS data over a broad range of crystallinity[END_REF] evidences the heterogeneous nature of the amorphous phase, depending on its confinement inside the rigid network of the crystal phase. This feature suggests considering a free amorphous region with low mechanical properties and a confined one that has a higher impact on the global rigidity. However, a recent work indicates that the amorphous part does not have a major impact on the macroscopic elastic properties [START_REF] Xiong | Amorphous phase modulus and micro-macro scale relationship in polyethylene via in situ SAXS and WAXS[END_REF]. To date, it is difficult to state the contribution of each phase on the overall elastic behaviour. The aim of this paper is to analyse the qualitative role of the crystalline mesostructure, especially the impact of the spherulite architecture on the macroscopic behaviour. Thus no amorphous phase has been considered here. To achieve this objective, different mesostructures have been designed from an in-house code and thanks to numerical calculations on them the corresponding macroscopic elastic properties are determined and compared against each other.

Method based on Finite Element (FE) or on Fast Fourier Transform (FFT) could be used to carry out these calculations. This last approach has been originally introduced by Moulinec et al. [START_REF] Moulinec | A fast numerical method for computing the linear and non linear echanical propertie of the co po ite . o pte rendu[END_REF] and has been successfully used to compute numerically the properties of composite materials [START_REF] Michel | A computational scheme for linear and nonlinear composites with arbitrary phase contrast[END_REF][START_REF] Vinogradov | An accelerated FFT algorithm for thermoelastic and non-linear composites[END_REF]. However, in our work, the mesostructures are composed of lamella in which the length, is significantly higher than the other dimensions. Besides, even if only the crystalline phase is studied in this work, and contrary to the FE method, the FFT one imposes to construct the all volume (even the void). It seems that, in our situation (slender structure in a porous medium), the use of shell elements in a FE model is more suitable than a FFT analysis. To our knowledge, the originality of this work is the direct supply of input data obtained at atomic scale from Molecular Dynamics (MD) simulation to the upper spherulitic scale, where full field mechanical calculation results can give information about the role of the mesoscopic architecture on the macroscopic properties. The method is based on a qualitative description of the spherulitic structure and consists in a sensitivity analysis of the crystalline layout on the macroscopic mechanical properties. Only the elastic properties are investigated in this first work.

In the following, the inputs for the FE calculations are presented. The mesoscopic spherulitic geometry of the models is deduced from observations of real microstructures from the literature. The orthotropic elastic properties of the crystalline lamellae are determined through MD simulations at the atomic scale. Then, the workflow employed to determine the macroscopic properties is described and a Representative Volume Element (RVE) is investigated in order to validate the size of the microstructure samples. Finally, macroscopic elastic results are presented and compared for different crystalline arrangements. In order to validate the proposed workflow with the FE method, some comparisons with a micromechanical approach and with experimental results are also discussed.

Input data for Finite Element calculations: mesostructures and material properties 2.1 Mesostructures

Semi-Crystalline Polymers, like PE, exhibit a hierarchical structure from nanoscopic to microscopic scales as illustrated in Figure 1. At the nanometric scale, SCPs are considered in a first approximation as two-phase materials, consisting of crystalline lamellae embedded in amorphous phases. At the micrometric scale, an aggregation of spherulites, in which crystalline lamellae grow radially from the nucleus of the spherulite, can be observed. These mesoscopic structures are most often in contact with each other and represent a quite cohesive network. According to Atomic Force Microscopy (AFM) analyses, a cross-hatched structure can be observed within the spherulites (Figure 2). The spherulite sizes are dependent on the crystallization process. According to [START_REF] Humbert | A re-examination of the elastic modulus dependence on crystallinity in semi-crystalline polymers[END_REF], lamellae are plate shape and following dimensions are considered to be reasonable values for PE: 1m length, 15 nm width and 5 nm thick.

Nanoscopic scale ~nm

Mesoscopic scale ~m

Macroscopic scale ~mm Spherulites network of a semi-crystalline (PE) [31]

Crystal-amorphous stacking in a model PE sample at the molecular scale [START_REF] Ier | Etude a l'echelle oleculaire de propriete ecaniques de polymeres semi-cristallins[END_REF] Tensile test specimen

In this work, no amorphous phase is considered and only the crystalline network is constructed and analysed. Different mesostructures are generated in order to understand and analyse the impact of this meso-architecture on the material macroscopic behaviour.

Representative mesostructures are obtained thanks to mathematical morphology tools, embedded in plug Im! software [START_REF]plug I !" an open acce and cu to iza le oftware for ignal and image processing[END_REF].

The impact of spherulite and cross-hatched arrangements is analysed. Three main types of mesostructures are compared (Figure 3):

 " e o tructure t pe A" with a totall rando di tri ution of la ellae  " e o tructure t pe B" with a pherulite tructure (onl principal la ellae)

 " e o tructure t pe " with a pherulite structure with branches (principal and secondary lamellae resulting in a "cro -hatched" like tructure)

To model mesostructures with a random lamellar network (Figure 3, type A), a Boolean model with randomly oriented lamellae is considered first [START_REF] Wang | Modelling mesoporous alumina microstructure with 3D random models of platelets[END_REF]. A Poisson Point process is generated with a density imposed to follow the mesostructure overall degree of crystallinity.

At each point, a randomly oriented parallelepiped lamellae is implanted. This model needs only five parameters :

-The generated volume size -The size of the lamellae: length, width, thickness -The degree of crystallinity A second approach is used to model mesostructures with a lamellar spherulitic network (Figure 3, types B). A Poisson Point process is first generated to model the germination points. To complete the crystalline part generation, lamellae are then radially disposed, with random orientation, fro the cell' centre defining germination point. The numbers of germination points and of lamellae have to be adjusted to the desired degree of crystallinity. This model needs several parameters to be adjusted :

-The generated volume size -The number of germination points following degree of crystallinity -The number of lamellae for each germination point, following also degree of crystallinity -The size of the lamellae: length, width, thickness A third model is designed using the second one and by changing simple lamellae to lamellae with branches (Figure 3, Type C). Additional parameters are the following :

-The number of branches for each principal lamella -The geometry of the branches: length, width, thickness -The branching angle: angle between the principal lamella and the secondary ones A perfect load transfer between each connected entity has been considered. It is numerically introduced by merging the connected lamellae with a perfect tie. The crystalline network is cohesive through these connections despite the absence of the amorphous phase.

Illustrations of the three types of constructed mesostructures are proposed in Figure 3.

Figure 3: Three types of mesostructures

We define various quantities to characterize the mesostructures, and to allow comparisons between them. These indicators are: the degree of crystallinity (%), the size and number of lamellae, the number of germs and bifurcations in case of spherulitic structures, the specific surface area (nm 2 /nm 3 ) and the features of the connection zones. The specific surface area is defined as the total surface of crystalline lamellae per volume unit of the mesostructure. The connection zones represent the regions where merging has been carried out to obtain a cohesive network. The number of connections/tied zones has been evaluated for each mesostructure. It defines the number of links between two entities of the structure, i.e. the number of links between two crystalline lamellae. Overlap zones can be either links between two lamellae belonging to two different spherulites or between two lamellae belonging to the same spherulite. This last type of connection cannot be found in the mesostructure B (see Figure 4). Note that the overlaps between a lamella of a spherulite and the corresponding germ are not considered in the count of connection zone, neither are the connections between a principal and a secondary lamellae in the case of mesostructure C.

Material inputs

The elastic properties of the PE crystalline phase have been determined at the atomic scale using MD calculations performed with the LAMMPS simulation package [START_REF] Plimpton | Fast Parallel Algorithms for Short-Range Molecular Dynamics[END_REF]. The PE crystal has been modelled using a simulation cell composed of 70 chains with 50 carbon atoms each Mesostructure B: no overlap between two lamellae of a same spherulite (= 5 x 7 x 25 PE unit cells). Periodic boundary conditions have been applied in the three space directions in order to mimic an infinite periodic crystal with infinite chain lengths. The Compass force field [START_REF] Sun | COMPASS: An ab Initio Force-Field Optimized for Condensed-Phase Applications -Overview with Details on Alkane and Benzene Compounds[END_REF] has been used to account for bonded and non-bonded interactions.

As discussed below, no Coulombic interactions have been considered.

A preliminary simulation has been conducted on a reference orthorhombic box in the isothermal-isobaric ensemble at 293 K and 1 bar in order to check the ability of the chosen force field to reproduce the lattice parameters of the PE crystal. Two calculations have been performed: one using the exact Compass force field including all partial charges and another one that neglects electrostatic contribution. Obtained parameters are respectively a = 7.73 Å, b = 4.95 Å, c = 2.56 Å and a = 7.72 Å, b = 4.96 Å, c = 2.57 Å for the two calculations. No significant impact of the partial charges is observed (error less than 0.5%). Obtained values in both cases are in good agreement with crystallographic data reported in reference [START_REF] Peacock | Handbook of Polyethylene: Structures: Properties, and Applications[END_REF] for PE at ambient conditions: a = 7.417 Å, b = 4.945 Å, c = 2.547 Å, with an overestimate of the lattice volume of ~5%. The Compass force field without electrostatic charges has thus been used thereafter for the study of the elastic properties of the crystalline PE.

The stiffness tensor elements (C ij ) of the crystalline PE have been calculated using an explicit deformation method that ake u e of the generali ed Hooke' law. Basically, different types of small elementary deformations ( j ) are applied to the reference orthorhombic box, the stress tensor elements (σ i ) under each small deformation are then computed during a canonical simulation of 5 ns and the C ij elements are extracted from the slope of the corresponding stress (σ i )strain ( j ) plot. The following deformations were done:

-Three elementary uniaxial strains (one for each spatial direction) with 5 positive  j values, in the range (0 -0.5) %, corresponding to a small extension of the material. -Three elementary uniaxial strains (one for each spatial direction) with 5 negative  j values in the range (0 -0.5) %, corresponding to a small compression of the material. -Three elementary shearing strains (one for each box angle) with 5 positive  j values in the range (0 -1) %.

More details about this procedure can be found in reference [START_REF] Clavier | Computation of elastic constants of solids using molecular simulation: comparison of constant volume and constant pressure ensemble methods[END_REF].

To illustrate these MD simulations, Figure 5 presents the stress-strain curves obtained in the case of the 10 uniaxial strain values performed in the direction of the crystallographic axis a. As required, a linear regime is observed in the range of deformations investigated in this work. The obtained C ij tensor, in Voigt notation, is:

All alue are expre ed in GPa. Error are confidence inter als on the slope of σ i versus  j plot. The most important stiffness corresponds to the extension of the chains in the direction of their main axis, with C 33 = 213.11 ± 0.39 GPa. This result is quite close to experimental values founded in the literature (221± 8 GPa measured by x-ray diffractions [START_REF] Matsuo | Elastic modulus of polyethylene in the crystal chain direction as measured by X-ray diffraction[END_REF]). Quite different results can be noted between the elements C 34 and C 43 or C 35 and C 53 . However, error bars associated with the C 34 and C 35 are large. With the exception of these two terms, we find that the lower-left and upper-right blocks of the tensor are zero within uncertainties. The same goes for the off-diagonal elements of the lower-right block. Orthotropic symmetry is thus assumed in the following and the corresponding nine C ij coefficients are directly used in FE calculations to account for the elastic behaviour of the crystalline regions of the 3D mesostructures. Special attention has been paid to the orientation of the Cij tensor in order to make the chain axis at the molecular level coincides with the chain axis at the mesoscopic level. At the mesoscopic level the chain axis is considered to be perpendicular to the surface of the crystalline lamellae (see Figure 6). Experimentally, the chains are found to be tilted by up to 45° from the lamellar normal, with 34° being the most common angle [START_REF] Point | General discussion[END_REF][START_REF] Oleinik | Plasticity of Semicrystalline Flexible-Chain Polymers at the Microscopic and Mesoscopic Levels[END_REF]. This tilt angle has not been considered in our simulations.

Workflow for FE calculations

Mesostructure architectures and material properties are the inputs for the FE calculations. The general workflow to carry out this up-scaling issue is proposed in the following scheme (Figure 7). The aim is to obtain elastic properties at the macroscopic scale and to determine the impact of the different mesostructures on these properties.

Geometry and mesh generation

Once the mesostructure is created with plug Im!, the FE model is obtained by means of a python script. It allows defining the geometry (conform to plug Im!) and the mesh. Abaqus® Finite Element package is used to generate meshing and to perform simulations [START_REF]Abaqus Unified FEA[END_REF]. Three-node triangular general-purpose shell elements are used. The characteristic element size is chosen to have five elements in the la ella' width. With uch a alue, 300 000 elements were typically generated for a 400 × 400 × 400 nm 3 volume. It should be noted that the use of a FFT method would led to 64 million voxels (with also 5 pixels in the lamellae thickness, equivalent to 1 pixel by nm). Calculations have also been performed with three elements in the width to analyse the mesh size impact on mesostructure C (it is assumed that if it is sufficient for the most intricate geometry, it is also the case for the others). The elastic properties (calculated as detailed in section 3.3) are proposed as supplementary data. Similar results are obtained for the two sizes of meshing. It seems reasonable to consider that the used mesh (typical size of five elements in the width) is accurate enough for this study.

Boundary conditions

This study aims at calculating the elastic properties of the mesostructures, i.e. at defining a relation between stress and strain tensors of the considered volume. By definition, the fourthorder elastic stiffness tensor is obtained by the following equation:

Equation 1
The average values can be controlled by imposing values on the contour of the domain only, i.e. by imposing specific Boundary Conditions (BC).

Different kinds of BC have been evaluated for homogenization issues. Kinematic uniform boundary conditions (KUBC), obtained by constraining node displacements, tend to overestimate the macroscopic stiffness. On the contrary, static uniform boundary conditions (SUBC) tend to under-estimate the stiffness. It is well known that periodic boundary conditions provide a better convergence than KUBC or SUBC [START_REF] Kanit | Determination of the size of the representative volume element for random composites: statistical and numerical approach[END_REF]: the homogenized elastic properties will be obtained for a smaller RVE size with periodic boundary conditions. However, our mesostructures are not periodic and this type of BC cannot be applied easily. We thus decided to apply KUBC since it is known that, as far as the considered volume is important enough or the number of numerical simulations is important enough, the KUBC conditions converge towards the same result as the one obtained with periodic boundary conditions [START_REF] Kanit | Determination of the size of the representative volume element for random composites: statistical and numerical approach[END_REF]. The KUBC conditions are applied through reference points: all the nodes of the same face are kinematically linked to a reference point. For an elongation along direction i, the displacement u i (x), with x being any node belonging to the face of normal direction i, is equal to the displacement u i of the corresponding reference point.

At the macroscopic scale, the elastic behaviour is known to be isotropic. However, the local properties at the scale of a set of lamellae is necessarily anisotropic. A sufficient volume of material has to be considered in order to respect the global isotropic behaviour. To check if this assumption is verified, it has been decided to determine nine elastic properties (as an orthotropic behaviour) and to check a posteriori if the three directions are equivalent. As an orthotropic behaviour is supposed, six elementary loadings determine the complete stiffness tensor:

-Three elementary uniaxial displacements (one for each spatial direction) with positive displacement in the range (0-10) nm. -Three elementary shearing strains (one for each shear angle) with positive shear angle in the range (0-0.025) rad.

In each computation, the stress tensor elements are determined by the FE calculations and the C ijkl elements are then extracted according to the generalized Hooke' law. Young modulii and Poisson' ratios are calculated from the uniaxial tests and shear moduli are determined from the shear test simulations.

Representative Volume Element: validity of the mesostructures

Ideally, the volume chosen for the mesostructures should be important enough to describe the statistics of the arrangement of the material leading to a so-called Representative Volume Element (RVE). In practice and due to calculation limitations, smaller volumes may be used and a statistical approach can be employed to converge towards the same results providing that a sufficient number of samples are considered.

The tested mesostructures are assumed to be ergodic and first of all, calculations are carried out to determine if the considered volumes are sufficient to lead to confident results. For each type of mesostructure (types A, B and C), several samples are constructed (with the same geometry parameters for a given mesostructure type) and for each sample, the macroscopic stiffness tensor is determined by means of six elementary loadings. The Voigt-Reuss-Hill averaging procedure [START_REF] Den Toonder | The relation between single crystal elasticity and the effective elastic behaviour of polycrystalline materials: theory, measurement and computation[END_REF] is then applied to provide average elastic constants (Young Modulus E VRH , Shear Modulus G VRH and Poi on' ratio  VRH ). This averaging method combines the upper and lower bounds (respectively Voigt and Reuss averaging method : Young Modulus E V /E R , Shear Modulus G V /G R and Poi on' ratio  V / R ) by assuming that the average of the Voigt and the Reuss elastic constants is a good approximation for the macroscopic elastic constants :

Equation 2
with Equation 3 Equation 4A, B, , A', B' and ' depend on the fourth-order elasticity and compliance tensors (C ijkl , S ijkl ) with the following equations in Voigt notation:

Equation 5 Equation 6The Reuss-Hill elastic properties are then calculated for each microstructure. In this RVE analyse, the main objective is to determine the confidence interval for the elastic properties. The relative error err z for a property Z, depending on the volume V, can be estimated by the following equation [START_REF] Kanit | Determination of the size of the representative volume element for random composites: statistical and numerical approach[END_REF]:

Equation 7
where n is the number of numerical samples, D Z is the variance of the property Z for a given volume V. The errors calculated for the elastic properties of each mesostructure configuration are given in Table 1.

Mesostructure

Table 1: Error estimations on macroscopic elastic properties for each mesostructure type

The different stiffness matrices are proposed as supplementary data. Undoubtedly, more samples or bigger volumes should be considered to have more precise values. However, the obtained estimations are considered to be sufficient to validate the workflow and to give insights for mesostructure-property relationships. Considering an uncertainty around 10%, the results are isotropic. According to this hypothesis, the elastic properties can be defined through onl three para eter : Young oduli are gi en within 7%, Poi on' ratio within 12% and shear modulus within 9% error.

Results

Mesostructure comparisons: impact of the 3D architecture

To understand the impact of the mesoscopic architecture, some structures of the three studied types are built with some identical features: a degree of crystallinity of 35%, a crystalline lamella width of 15 nm and a crystalline lamella thickness of 5 nm. All microstructural features of the different samples are gathered in Table 2. The volume of each mesostructure has been chosen to correspond to the values used in the previous section: 500x500x500 nm 3 for mesostructure A and 400x400x400 nm 3 for mesostructures B and C. The specific surface is approximately the same for the three architectures. Note that more lamellae are needed in mesostructure A to reach the same degree of crystallinity due to the bigger volume. The structural features of Table 2 are also given in terms of density: the tied zone density being the number of connection zones per volume unit. The structural quantity that allows distinguishing different structures seems to be the number of tied zones. There are very few tied zones in mesostructure A and this tendency is even accentuated when this value is normalized by the volume of the structure (500x500x500 nm 3 for the mesostructure A compared to 400x400x400 nm 3 for the others). The tied zones density in mesostructure A is about three times less than in mesostructure B and six times less than in mesostructure C. Macroscopic elastic properties obtained following the workflow proposed in Figure 7 are also given in Table 2 for the different mesostructures (the complete stiffness matrices are provided as supplementary data). As mentioned above, an isotropic behaviour is observed within statistical uncertainty (~10%), thus, only average macroscopic Young modulus, Poisson' ratio and shear modulus are reported in Table 2. Despite this 10% uncertainty, tendencies can be discussed.

First, the crystallite distribution has an important influence on the material rigidity: the spherulitic mesostructure tends to strengthen the material (the Young Modulus of mesostructure B is 92% higher than the one of mesostructure A). At the same degree of crystallinity, a random disposition tends to be less rigid than a spherulitic one. This can be explained by the tied zone density: the random architecture is less cohesive and thus the elastic properties are weaker.

Then, comparing mesostructure C to mesostructure B, the impact of a cross-hatched like structure (approximated here with branches) can be analysed. Mesostructure C is more rigid than mesostructure B (the Young Modulus of mesostructure C is 44% higher than the one of mesostructure B). Once again, the number of tied zones density can explain such results as it is significantly higher in mesostructure C. It means that the connection links are more important when branches are taken into consideration and the cohesion of the material is improved. On the one hand, in the mesostructure B, there are only inter-spherulites connections: due to its construction (Figure 4), a lamella may only be connected to a lamella belonging to another spherulite. On the other hand, connection may be inter-and intraspherulites in the mesostructure C. Due to the bifurcations in this mesostructure, tied zones can be either between lamellae belonging to two different spherulites or between two lamellae belonging to the same spherulite.

According to the obtained results, it appears that the architecture at the mesoscospic scale has an influence on the elastic properties at the macroscopic scale. The spherulitic morphology tends to strengthen the material and it seems that this stiffness increase may be linked to the existing connections between spherulites (comparisons between mesostructures A and B). Furthermore, the intra-spherulites connections have also an impact (comparisons between mesostructures B and C). These two types of tied zones bring cohesion to the material and thus influence its stiffness.

Comparison with experimental data

The numerical approach is not supposed to be predictive as the semi-crystalline polymer is not completely modelled : only the crystalline phase is considered in this numerical analysis. As a consequence, the direct comparison of numerical moduli with experimental results must be analysed with care. The objective is to compare trends, order of magnitudes, and to give a picture of the importance of the mesostructure on the elastic properties.

For a crystalline degree of 35%, available experimental data on PE provide a Young Modulus of 83 (+/-6) MPa [START_REF] Humbert | A re-examination of the elastic modulus dependence on crystallinity in semi-crystalline polymers[END_REF]. As a reminder, the results obtained with the proposed workflow are :

 56 (+/-3) MPa for mesostructure A  99 (+/-7) MPa for mesostructure B  142 (+/-10) MPa for mesostructure C

All three mesostructures provide an elastic stiffness with a good order of magnitude. As only crystalline network is here considered, this results question the amorphous phase role on the elastic properties. This topic will also be discussed in the next section. Besides, one can consider that mesostructure B provides the most predictive stiffness. At first sight, the mesostructure C could have been considered as more realistic. However, mesostructure C appears to be too stiff, but it can be argued that some geometric parameters may be modified. Actually, whereas no geometrical parameter (as the lamellae size has been obtained through Small-angle X-ray scattering [START_REF] Humbert | A re-examination of the elastic modulus dependence on crystallinity in semi-crystalline polymers[END_REF]) may be modified in mesostructure B, some others may be adjusted for mesostructure C: such as the size, the number and the orientation of the secondary branches that have been arbitrarily chosen in this study. Experimental observations like AFM analyses for instance [START_REF] Voitot | Atomic Force Microscope study of plastic mechanisms in isotactic polypropylène beta[END_REF] could be conducted on PE to have some experimental inputs on secondary branches.

Elastic stiffness of additional structures of type B and C are assessed with different degrees of crystallinity. The obtained mechanical properties are compared with experimental results [START_REF] Bédoui | Micromechanical modeling of isotropic elastic behavior of semicrystalline polymers[END_REF][START_REF] Humbert | A re-examination of the elastic modulus dependence on crystallinity in semi-crystalline polymers[END_REF] and also to a micromechanical approach proposed by Bédoui [START_REF] Bédoui | Micromechanical modeling of isotropic elastic behavior of semicrystalline polymers[END_REF] as shown in Figure 8. Up to a degree of crystallinity of 50%, our results show good agreement with experimental results. Besides, the meso-architecture appears to have the same influence as previously described for the samples with a degree of crystallinity of 35% : the branching (mesostructure C) tends to strengthen the material (compared to architecture B) with an enhancement of the structural network both inside and outside the spherulites.

For higher degree of crystallinity, it is quite intricate to conclude: we did not manage to construct mesostructures with such amount of crystal due to numerical cost of the FE method. However, it seems reasonable to assume that our methodology will underestimate the Young modulus. It can be noticed that the inclusion-matrix model proposed by Bedoui et al. [START_REF] Bédoui | Micromechanical modeling of isotropic elastic behavior of semicrystalline polymers[END_REF] allows interesting results: a very good estimation of the Young' odulu . cr tallinit ha been obtained (solid line in Figure 8). In this work, the authors considered the crystalline phase as a reinforcing phase embedded in an amorphous matrix. Crystal lamellae are approximated by ellipsoidal inclusions. To approximate the macroscopic tensor of the elastic behaviour of the equivalent homogeneous medium, the authors have used a differential scheme. In such approach, the reinforcing phase is introduced by infinitesimal increments using the dilute-distribution assumption [START_REF] Bédoui | Micromechanical modeling of elastic properties in polyolefins[END_REF]. Shape factors (width/thickness and length/thickness of lamellae) have been fitted to obtain very good agreement between model and experimental results. The chosen values (width/thickness=5 and length/thickness = 23) do not seem realistic compared to values observed in another work [START_REF] Humbert | A re-examination of the elastic modulus dependence on crystallinity in semi-crystalline polymers[END_REF]. Besides, through this inclusion-matrix approach, the mesostructure architecture has no impact on the elastic properties, contrary to the results proposed in this work. However, what remains clear is that the Bédoui et al. model [START_REF] Bédoui | Micromechanical modeling of isotropic elastic behavior of semicrystalline polymers[END_REF] provides better results and especially for high crystallinity rates. Amorphous phase is not considered in the present work and the results seem better when the degree of crystallinity is low, which is surprising at first glance. Some works suggest that percolation in the crystalline network has an increasing role in the elastic macroscopic properties with increasing the degree of crystallinity, which counterbalances more and more the role of the amorphous phase [START_REF] Xiong | Amorphous phase modulus and micro-macro scale relationship in polyethylene via in situ SAXS and WAXS[END_REF]. For degree of crystallinity higher than 50%, the amorphous phase role should be less and less important as the crystallinity increases. Two approaches may be explored to explain our results :  The under-estimation of elastic properties at high degree of crystallinity can be explained by the modification of geometric parameters: semi-crystalline polymers with such a degree of crystallinity may have been obtained with larger lamellae (which may be linked to a shape factor modification). The investigated mesostructures may not be representative of the material with such a degree of crystallinity and new sets of geometric parameters should be considered in future works. Such an idea is supported by other works. Humbert et al. suggest that a degree of crystallinity of 40% is a boundary for two different structural morphologies : under 40%, spherulites contain both bundle-like and lamellar crystals, above 40% spherulites are more well-defined [START_REF] Humbert | A re-examination of the elastic modulus dependence on crystallinity in semi-crystalline polymers[END_REF].  It can also be argued that amorphous phase can be in different state depending on the crystalline network [START_REF] Chmelai | Free and constrained amorphous phases in polyethylene: Interpretation of 1H NMR and SAXS data over a broad range of crystallinity[END_REF]. At low degree of crystallinity, the amorphous phase is supposed to be less constrained and has a weaker behaviour which does not influence sensibly the elastic properties. On the contrary, at higher degree of crystallinity, the amorphous phase is more confined by the crystal lamellae: tied amorphous phase is proportionally more important and may have more impact on the elastic properties.

Conclusion

In this paper, a workflow has been proposed to determine elastic mechanical properties of Semi-Crystalline Polymers. FE calculations have been carried out on representative mesostructures. Some required inputs namely the elastic properties of the crystalline phase have been calculated at the molecular scale thanks to MD simulations. According to the results of the FE calculations, it is shown that the crystalline network geometry at the mesoscopic scale has a major impact on the macroscopic mechanical behaviour. More precisely, a spherulitic mesostructure tends to strengthen the material as the number of cohesion points increases in comparison to a random architecture. The branching architecture seems also to have an impact. Due to branches, connections between lamellae belonging to the same spherulite is also possible: cohesion is given by inter and intra spherulite interactions. Consequently, the branched structures have a higher Young modulus than structures with no branches. Furthermore, in the studied cases and up to a degree of crystallinity of 50%, the elastic properties of the crystalline network with no amorphous phase show good agreement with macroscopic experimental properties. For higher degree of crystallinity, the shift of the numerical values from the experimental data suggests that the influence of the lamellae dimensions and/or of the percolation degree of the crystalline network have to be deepened. The mechanical contribution of the constrained amorphous phase should be investigated at this stage too. Additional numerical studies should be carried out to analyse more precisely the role of the amorphous phase. A perfect load transfer has here been considered between neighbouring spherulites, and the modification of this behaviour may be tested to analyse the impact of this cohesion. Another way to consider the amorphous phase should be the use of method with no need of discretization. For instance, a method based on the Fast Fourier Transform can be tested.

In future work, the mesostructure construction could also be improved: it should be interesting to use experimental observation to create more realistic mesostructures. For example, the use of kinematic law to reproduce the spherulite growth could be introduced.

Supplementary data

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.

RVE size validation

Mesostructures A, B and C should not be compared: used material data are not the same, these results have only been used to check the RVE size.

Stiffness matrices (GPa) for several samples of mesostructure type A.

Stiffness matrices (GPa) for several samples of mesostructure type C.

Mesh refinement

Stiffness matrices (GPa) for two identical samples with two different meshes

Figure 1 :

 1 Figure 1: Schematic illustration of the structure of SCP at different length scales

Figure 2 :

 2 Figure 2: Cross-hatched structure of polypropylene's spherulites[START_REF] Voitot | Atomic Force Microscope study of plastic mechanisms in isotactic polypropylène beta[END_REF]

Figure 4 :

 4 Figure 4: Connection zone inside a spherulite for mesotructures B and C, extracted from the geometry of figure 3.

  overlap between two lamellae of a same spherulite (red circle)

Figure 5 :

 5 Figure 5: Stress-strain curves obtained from MD simulations of crystalline PE at 293 K in the case of an uniaxial strain  1 (direction of vector a of the crystalline unit).

Figure 6 :

 6 Figure 6: Schematic representation of the axis systems at the molecular and the mesoscopic scale with axis c being parallel to the chain axis i.e. perpendicular to the surface of the crystalline lamellae.

Figure 7 :

 7 Figure 7: Schematic workflow of the study

Figure 8 :

 8 Figure 8: Comparison of experimental Young moduli from Solvay Database [16] and Humbert work [32] with numerical results and with a micro-mechanical model from Bédoui et al. [16]

  Stiffness matrices (GPa) for mesostructure A, B and C corresponding to the same material data. Data used for the analyze of the mesostructure impact.

Table 2 : Features of the different studied mesostructures and corresponding elastic properties at room temperature and atmospheric pressure

 2

Stiffness matrices (GPa) for several samples of mesostructure type B.


B L A Costa

M Sauceau

S Del Confetto

R Sescousse

M I Ré

Determination of drug-polymer solubility from supersaturated spray-dried amorphous solid dispersions: A case study with Efavirenz and Soluplus®

Keywords:

Amorphous solid dispersions (ASDs) are found to be a well-established strategy for overcoming limited aqueous solubility and poor oral bioavailability of active pharmaceutical ingredients (APIs). One of the main parameters affecting ASDs physical stability is the API solubility in the carrier, because this value determines the maximal API load without a risk of phase separation and recrystallization. Phase-diagrams can be experimentally obtained by following the recrystallization of the API from a supersaturated homogeneous API-polymer solid solution, commonly produced by processes as solvent casting or comilling, which are very time-consuming (hours). The work deals with the construction of a temperature-composition EFV-Soluplus® phase diagram, from a thermal study of recrystallization of a supersaturated solid solution (85 wt% in EFV) generated by spray drying. This supersaturated solution is kept at a given annealing temperature to reach the equilibrium state and the amount that still remains dispersed in the polymer carrier at this equilibrium temperature is determined by means of the new glass transition temperature of the binary mixture. From our knowledge, this is the first study employing a fast process (spray drying) to prepare a supersaturated solid solution of an API in a polymer aiming to determine a temperature-composition phase diagram. The EFV solubility in Soluplus ranges from 20 wt% at 25 °C to 30 wt % at 40 °C. It can be a very useful preformulation tool for researchers studying amorphous solid dispersions of Efavirenz in Soluplus, to assist for predicting the stability of EFV-Soluplus ASDs at different EFV loadings and under different thermal conditions.

Introduction

One of the biggest challenges facing the pharmaceutical world today is the poor aqueous solubility of many drugs. When a drug is administered orally, if the clinical dose of the drug cannot be dissolved in the available volume of gastrointestinal fluids, the drug will show a poor oral absorption, which leads to a weak bioavailability [START_REF] Arca | Multidrug, Anti-HIV amorphous solid dispersions: nature and mechanisms of impacts of drugs on each other's solution concentrations[END_REF].

The crystalline form of a drug offers the advantage of high purity and physical or chemical stability. However, a large amount of external energy is required to break the drug crystal lattice in dissolution. The amorphous state, on the other hand, exhibits a disordered structure in comparison to the crystalline form and possesses higher free energy (thermodynamic driving force), which consequently is reflected by higher apparent water solubility, dissolution rate, and oral absorption. However, due to a lower thermodynamic stability, the main challenge of amorphous form is evidenced: stabilization against drug recrystallization tendency both on in vivo dissolution and storage [START_REF] Singh | Spray drying formulation of amorphous solid dispersions[END_REF][START_REF] Baghel | Polymeric amorphous solid dispersions: a review of amorphization, crystallization, stabilization, solid-state characterization, and aqueous solubilization of biopharmaceutical classification system class II drugs[END_REF]. To retain the solubility advantage of the amorphous state, amorphous solid dispersions (ASDs) have been proposed as an effective strategy to "kinetically stabilize" amorphous drugs [START_REF] Patel | Revealing facts behind spray dried solid dispersion technology used for solubility enhancement[END_REF]. ASDs refer to drug -carrier systems, where the drug is dispersed at a molecular level in the carrier, usually an amorphous polymer [START_REF] Sun | Solubilities of crystalline drugs in polymers: an improved analytical method and comparison of solubilities of indomethacin and nifedipine in PVP, PVP/VA, and PVAc[END_REF][START_REF] Wlodarski | Physical stability of solid dispersions with respect to thermodynamic solubility of tadalafil in PVP-VA[END_REF]. ASDs can be produced by several processes such as freeze-drying [START_REF] Ihli | Freeze-drying yields stable and pure amorphous calcium carbonate (ACC)[END_REF][START_REF] Kojo | Improved oral absorption profile of itraconazole in hypochlorhydria by self-micellizing solid dispersion approach[END_REF], hot-melt extrusion [START_REF] Kyeremateng | A fast and reliable empirical approach for estimating solubility of crystalline drugs in polymers for hot melt extrusion formulations[END_REF][START_REF] Thiry | Continuous production of itraconazole-based solid dispersions by hot melt extrusion: Preformulation, optimization and design space determination[END_REF], rotary solvent evaporation [START_REF] Xie | Dissolution performance of high drug loading celecoxib amorphous solid dispersions formulated with polymer combinations[END_REF]; ball milling [START_REF] Caron | Amorphous solid dispersions of sulfonamide/soluplus® and sulfonamide/PVP prepared by ball milling[END_REF]; cryo-milling [START_REF] Mahieu | A new protocol to determine the solubility of drugs into polymer matrixes[END_REF]; supercooling [START_REF] Semjonov | The formation and physical stability of two-phase solid dispersion systems of indomethacin in supercooled molten mixtures with different matrix formers[END_REF] and spray drying [START_REF] Wlodarski | Physical stability of solid dispersions with respect to thermodynamic solubility of tadalafil in PVP-VA[END_REF][START_REF] Pawar | Development of amorphous dispersions of artemether with hydrophilic polymers via spray drying: physicochemical and in silico studies[END_REF][START_REF] Ghanavati | Anomalous dissolution behavior of celecoxib in PVP/Isomalt solid dispersions prepared using spray drier[END_REF].

The choice of the carrier is a critical parameter for ASDs because it impacts several factors such as molecular mobility, preparation methods and the physical/chemical stability of the amorphous form. To stabilize an ASD thermodynamically, it is of great importance to disperse molecularly the drug in the carrier because, if the drug is molecularly dispersed in the carrier at concentrations below the drug solubility in the carrier at a fixed temperature, this system is, in theory, thermodynamically stable. Consequently, for the rational development of ASDs, it is essential to know the solubility of the drug in the carrier because this value determines the maximal drug load without a risk of phase separation and recrystallization [START_REF] Murdande | Aqueous solubility of crystalline and amorphous drugs: Challenges in measurement[END_REF][START_REF] Shah | Amorphous Solid Dispersions: Theory and Practice[END_REF].

However, measuring the API solubility in polymers is still a challenge due to the high polymers viscosity that hampers the solubility equilibrium achievement (equilibrium between a crystalline solute and its solution), and makes difficult the experimental construction of solubility curves in a low time consuming [START_REF] Kyeremateng | A fast and reliable empirical approach for estimating solubility of crystalline drugs in polymers for hot melt extrusion formulations[END_REF]. For this reason, different methods have been developed to determine the solubility of APIs in polymeric carriers: estimation of the solubility of a API in a polymer from the solubility of the API in a low molecular weight liquid, chemically analogue to the polymer [START_REF] Knopp | Comparative study of different methods for the prediction of drug-polymer solubility[END_REF]; methods based on thermal analysis such as the melting point depression method [START_REF] Marsac | Theoretical and practical approaches for prediction of drug-polymer miscibility and solubility[END_REF] and the dissolution end point method [START_REF] Tao | Solubility of small-molecule crystals in polymers: d-Mannitol in PVP, Indomethacin in PVP/VA, and Nifedipine in PVP/VA[END_REF].

Theoretically, melting of a crystalline API occurs at the temperature (T m ) when the chemical potential of the crystalline API equals to the chemical potential of the molten API. The melting point depression method explores the depressed melting point of the API in the presence of a miscible polymer [START_REF] Knopp | Comparative study of different methods for the prediction of drug-polymer solubility[END_REF]. Assuming that the amorphous API behaves as a solvent and thus by extending the equations presented by Flory-Huggins systems, it is possible to relate solubility of a API in an amorphous polymer to melting point depression of the API [START_REF] Lehmkemper | Long-term physical stability of PVP-and PVPVA-amorphous solid dispersions[END_REF]. The onset of the melting endotherm is considered as the equilibrium solubility temperature of the given composition. The choice of the melting onset eliminates the impact on T m of particle size and preparation method of the physical mixtures. However, the onset only marks the beginning of the API dissolution in the polymer but does not provide information concerning the amount of API in the binary mixture that can be dissolved at that respective temperature [START_REF] Knopp | Comparative study of different methods for the prediction of drug-polymer solubility[END_REF]. Progressing the protocol for thermal analysis, the cryomilling of the physical mixtures was proposed in order to compensate the slow dissolution kinetics by reducing the size crystals prior to DSC experiments [START_REF] Sun | Solubilities of crystalline drugs in polymers: an improved analytical method and comparison of solubilities of indomethacin and nifedipine in PVP, PVP/VA, and PVAc[END_REF]. However, cryomilling is a longstanding process and have some limitations to control and to standardize particle characteristics such as size, morphology, shape and surface properties [START_REF] Kumar | Cryomilling: an environment friendly approach of preparation large quantity ultra refined pure aluminium nanoparticles[END_REF]. In contrast, the dissolution endpoint method enables the measurement of the endpoint of the dissolution step and thus might be more accurate [START_REF] Yang | Molecular implications of drug-polymer solubility in understanding the destabilization of solid dispersions by milling[END_REF]. The end point value represents the melting point of the final composition, assuming complete mixing has occurred. The disadvantage of this method is its strong dependence on the particle size. If the particle size is not sufficiently small, the melt is not able to reach its equilibrium state during heating. This may result in a higher dissolution end point, leading to underestimation of the API-polymer solubility.

The mentioned methods follow the dissolution kinetics of APIs into polymers at under saturated concentration solutions. Another way to determine the API-polymer solubility is to reach faster the equilibrium solubility by following the recrystallization (recrystallization method) of the API from a supersaturated homogeneous API-polymer solid solution [START_REF] Mahieu | A new protocol to determine the solubility of drugs into polymer matrixes[END_REF][START_REF] Knopp | Comparative study of different methods for the prediction of drug-polymer solubility[END_REF]. The demixing kinetics of APIs from supersaturated systems is expected to be much faster than the dissolution kinetics of APIs into under saturated systems. This difference arises directly from the molecular mobility in the polymer matrix, which is higher in supersaturated API-polymer solutions than in under saturated ones.

The supersaturation of the API into the polymer can be obtained from a physical mixture of crystalline API and polymer carrier by different techniques such as film casting method [START_REF] Knopp | Comparative study of different methods for the prediction of drug-polymer solubility[END_REF] and comilling [START_REF] Mahieu | A new protocol to determine the solubility of drugs into polymer matrixes[END_REF][START_REF] Yang | Molecular implications of drug-polymer solubility in understanding the destabilization of solid dispersions by milling[END_REF] that forces the miscibility of the two compounds to reach supersaturation. The supersaturated glass solution is then loaded in a DSC analyzer and annealed above its T g during a sufficient time to complete the demixing process: release the excess of API in the mixture and reach the equilibrium solution concentration at the annealing temperature.

We propose here to prepare supersaturated API-polymer solid solutions by spray drying as an alternative to determine API-carrier solubility from demixing, because it is a fast process (some seconds) compared to other ones commonly used as solvent casting or comilling, which are very time-consuming (hours) [START_REF] Sosnik | Advantages and challenges of the spray-drying technology for the production of pure drug particles and drug-loaded polymeric carriers[END_REF]. From our knowledge, this is the first study employing spray drying to prepare a supersaturated solid solution of an API in a polymer aiming to determine a temperature-composition phase diagram by following the recrystallization of the API from a supersaturated homogeneous API-polymer solid solution. The binary mixture case-control study is Efavirenz-Soluplus®. The chosen API, Efavirenz (EFV), is classified as class II API according to the biopharmaceutical classification system (BCS) [START_REF] Kamble | Efavirenz self-nano-emulsifying drug delivery system. in vitro and in vivo evaluation[END_REF] with high permeability and poor-water solubility water (below 10 µg/mL) in its crystalline form [START_REF] Chiappetta | Efavirenz-loaded polymeric micelles for pediatric anti-HIV pharmacotherapy with significantly higher oral bioavailability[END_REF]. EFV is recommended by the World Health Organization [START_REF]HIV/AIDS[END_REF] as a first-line anti-HIV API, largely used as a non-nucleoside reverse transcriptase inhibitor as part of antiretroviral therapies. However, it exhibits low oral bioavailability (40-45 %) [START_REF] Hoffmeister | Efavirenz dissolution enhancement III: Colloid milling, pharmacokinetics and electronic tongue evaluation[END_REF][START_REF] Sathigari | Amorphous-state characterization of efavirenz-polymer hot-melt extrusion systems for dissolution enhancement[END_REF]. Amphiphilic polymers have gained much attention in recent studies to inhibit recrystallization in ASDs. Composed by a hydrophilic unit (contact with aqueous media) and a hydrophobic unit (contact with the API), Soluplus® (SOL) is an example of amphiphilic polymer [START_REF] Kojo | Improved oral absorption profile of itraconazole in hypochlorhydria by self-micellizing solid dispersion approach[END_REF].

Experimental section

Materials

Efavirenz, EFV, ((S)-6-chloro-4-(cyclopropylethynyl)-1,4-dihydro-4-(trifluoromethyl)-2H-3,1-benzoxazin-2-one, Fig. 4) was kindly provided by Cristalia Ltd (Itapira, Brazil). Soluplus®, SOL, (polyvinyl caprolactam-polyvynil acetate-polyethylene glycol graft copolymer/PCL-PVAc-PEG, Fig. 4) was provided by BASF corporation (Ludwigshafen, Germany). Ethanol used as organic solvent in the preparation of solid dispersions was purchased from Carlo ERBA Reagent (Italy). For the purpose of analysis, distilled and purified water (conductivity of 18.2 MΩ.cm at 23 °C) was obtained by the purification system Milli-Q (classic Purelab DI, MK2, E3lga, UK). Physicochemical characteristics of API and polymer are summarized in Table 1.

Supersaturated EFV ASD

Preparation

A supersaturated amorphous solid dispersion (SD) EFV: SOL was prepared by spray drying with a drug load proportion of 85:15 wt%.

A Buchi B-290 mini spray dryer (Buchi Labortechnik AG, Flawil, Switzerland) equipped with Inert Loop B-295 and an integrated twofluid 0.7 mm nozzle was used to produce the ASD samples.

Solvent choice is an essential step in the spray dryer process, and it was based on high solubility of API and polymer, generation of a feed solution with acceptable viscosity, low toxicity and high volatility to be easily evaporated during the drying of the liquid droplets. Ethanol respects these specifications and was thus used as the common solvent. The feeding solution was prepared by dissolving EFV in a 10 % (w/w) solution of SOL in absolute ethanol 99 %.

Compressed nitrogen was used as the drying/carrying gas with a flow rate of 600 L/h. The solution feed rate was ranged from 50 to 117 kg/h, the inlet air temperature was set to 80 ± 2 °C and the resultant outlet temperature (59 ± 2 °C) was determined by the inlet temperature and relative factors such as air and liquid flow rates. Ethanol was continuously condensed in the process for further disposal. The dry powder was recovered in a cyclone.

Solid state characterization 2.2.2.1. Thermal analysis (DSC).

Thermal analyses were performed by using a DSC Q200 with the base module and mDSC (TA instruments, USA). On this, it was used a RCS90 cooling system which explains that we can precisely control the cooling rate. Nitrogen was used as the purging gas at 50 mL/min, the analysis was made in non-hermetic aluminum pans, indium standards were used for enthalpy and temperature calibration and an empty aluminum pan was used as a blank. Concerning mDSC, sapphire was used to calibrate in C P . Samples were heated with a rate of 2 °C/min. To obtain endothermic peaks and analyze the apparent API-polymer miscibility, samples were heated from 10 to 150 °C. Moreover, experiments were also done on modulated-DSC to determine glass transition temperatures of SOL, EFV and their mixture. In these analyzes, the samples were heated at 2 °C/min from 10 to 150 °C, with a modulation period of 40 s and an amplitude of 0.2 °C.

Cristallinity studies (PXRD).

Powder X-ray diffraction patterns were obtained by X'Pert Panalytical X-ray diffractometer (Philips, USA) using CuKα radiation, putting 40 mA of current and 45 kV of voltage. The recording spectral range was set at 7-50°with a measuring step (angular deviation between 2 consecutives points) of 0.0167°and an acquisition time of 100 s per point. In addition, the powder was rotated in its sample holder (1 s/revolution) during the acquisition of the results.

Fourier

transform infrared spectroscopy (FTIR). FTIR investigations were realized on a Thermo Scientific Nicolet 5700 spectrometer. The samples were premixed with KBr using mortar and pestle at 10 % of the samples in KBr and the discs were produced by compressing the powders at a working pressure of 1 t. The scanning range was 4000-400 cm -1 with a 4 cm -1 resolution.

Surface morphology studies (SEM)

. SEM images of pure SOL, pure EFV and solid dispersions were obtained using a scanning electron microscope Philips XL30 ESEM-FEG (Philips, USA) with acceleration voltage of 20 kV. Samples were fixed on a support using a double-sided adhesive and covered with platinum using a high-resolution SEM coated spray Polaron SC7640 (Quorom Technologies, England).

2.2.2.5. Raman mapping. Raman mapping was performed using a Raman confocal microscope 300AR Alpha-AFM (Witec, Germany), equipped with a laser at a wavelength of 532 nm and the analyses were conducted at room temperature. Samples were analyzed by a 50× objective with a lateral resolution of 432 nm and an axial resolution of 1320 nm. A surface (10 × 10 μm) and deep mapping (12 × 10 μm) were used to predict drug and polymer distribution. Images obtained in 3D representing special distribution of the chemical species present in the sample.

API-polymer solubility determination by recrystallization

The supersaturated SD was loaded into the DSC analyzer and annealed at different temperatures below the T m (melting point) of the API under investigation (EFV) for 2-28 h to recrystallize the excess of drug in the mixture and to reach equilibrium solubility. After annealing, the sample was cooled to 10 °C and ramped at a rate of 2 °C/min to determine the T g (glass transition temperature) of the annealed mate- rial.

The recrystallization method is illustrated in Fig. 1 that presents a schema of DSC running. Run 1 (first heat cycle) represents a change in the solid state evidenced by an unique glass transition temperature of the binary mixture, followed by an exothermic phenomenon (recrystallization). Run 2 (isotherm) aims to reach the equilibrium state for the sample at a given annealing temperature. In turn, Run 3 is the cooling step. Finishing the cycle, Run 4 (second heat cycle) gives the new glass transition temperature of the binary mixture, corresponding to the API amount that still remains dispersed in the polymer carrier at the equilibrium temperature, which corresponds to the melting temperature of this mixture T MIX m .

Annealing temperature and annealing time are essentials parameters and play an important role in the recrystallization method. At lower annealing temperatures, a longer time is required to reach the equilibrium state. The reason is probably a slower demixing kinetics due to an insufficient molecular mobility, leading consequently to slower API diffusion in the polymer matrix [START_REF] Mahieu | A new protocol to determine the solubility of drugs into polymer matrixes[END_REF]. Completing the DSC analysis, the new equilibrium dissolution concentration of the API remaining dispersed in the polymer was calculated using the Gordon-Taylor equation [START_REF] Baird | Evaluation of amorphous solid dispersion properties using thermal analysis techniques[END_REF][START_REF] Fule | Development and evaluation of lafutidine solid dispersion via hot melt extrusion: investigating drug-polymer miscibility with advanced characterisation[END_REF]. ) . . ( 1)

= + - + - T W T K W T W K W . . ( 1
MIX g PA PA g PA PO g
PA PA [START_REF] Arca | Multidrug, Anti-HIV amorphous solid dispersions: nature and mechanisms of impacts of drugs on each other's solution concentrations[END_REF] T PA g and T PO g are the API and polymer glass transition temperatures, respectively. W PA is the API mass fraction in the mixture. K can be defined as a parameter characterizing the intermolecular interactions between both components resulting in the curvature of the evolution and is defined by Eq. ( 2) [START_REF] Baird | Evaluation of amorphous solid dispersion properties using thermal analysis techniques[END_REF]. A value of 1 for K means that the interactions between both components are similar to the ones of the pure components.

= K T ρ T ρ . . PA g PA PO g PO (2)
Densities of the drug and polymer, ρ PA and, ρ PO respectively, are defined in Table 1.

Solubility curve fitting

Flory-Huggins modeling was used to construct the solubility curve, which can estimate API solubility at a range of annealing temperatures (see Table 2). The calculated data (Eq. ( 3)) were compared to the experimental results obtained with the DSC protocol. Binary interactions between polymer and API are commonly described by Flory-Huggins parameter (χ) obtained from the solubility curve. Immiscibility between the components is represented by a positive χ, while miscibility is indicated by a negative χ [START_REF] Mahieu | A new protocol to determine the solubility of drugs into polymer matrixes[END_REF][START_REF] Meng | Qualitative and quantitative methods to determine miscibility in amorphous drug-polymer systems[END_REF][START_REF] Donnelly | Probing the effects of experimental conditions on the character of drug-polymer phase diagrams constructed using Flory-Huggins theory[END_REF].

⎜ - = - ⎛ ⎝ ⎜ + ⎛ ⎝ - ⎞ ⎠ ⎟ + ⎞ ⎠ ⎟ T T R H ν λ ν χ ν 1 1 ∆ . ln 1 1 . MIX m PA m PA m PA PO PO 2
(3)

T MIX m
is the mixture melting temperature (in our case annealing temperature at which equilibrium is measured), T PA m is the melting point of the pure API, H ∆ PA m is the melting heat of the pure API, R is the ideal gas constant, λ is the molar volume ratio of the polymer and the API, ν PA and ν Po are the volume fractions of the API and the polymer, re- spectively. The volume fractions ν i can be calculated according Eq. ( 4):

= + - ν W W W i ρ ρ ρ 1 PA PA i PA PO i (4)
3. Results and discussions

Supersaturated ASD characteristics

Solid states of samples were characterized by XRD analysis. The XRD patterns obtained for pure SOL, unprocessed EFV crystals and SD sample were depicted in Fig. 2. As expected, SOL is completely amorphous in nature (absence of any peaks on its diffractogram). Unprocessed EFV crystals exhibit principal sharp diffraction peaks at 2θ of 12.20°, 17.50°, 21.35°and 30.00°which represents characteristic peaks for crystalline nature of EFV in accordance to the literature [START_REF] Lavra | Solubility and dissolution performances of spray-dried solid dispersion of Efavirenz in Soluplus[END_REF]. SD sample did not show peaks of Bragg's, which confirmed that it became amorphous during spray-drying process.

The Raman spectra of the individual components (SOL and EFV) and SD EFV were carried out to understand the drug distribution inside polymer matrix (Fig. 3). SOL spectrum showed characteristic peaks around 2987 cm -1 and 2900 cm -1 due to CeH stretching bands. For EFV spectrum, characteristic peaks at a wavelength around 3100 cm -1 and 2250 cm -1 due to amine group NeH and CeC triple bond respectively, were identified. SD Raman spectrum revealed all EFV and SOL vibrations bands in the binary mixture. Still in Fig. 3, 3D images represent a scan of the characteristic peak of each component in a SD sample region (10 µm × 10 μm). It confirms the presence of SOL and EFV and their good distribution with however a better uniformity for SOL. In fact, the drug concentration in SD corresponds to a supersaturated state of EFV in SOL and this EFV excess might be distributed in amorphous clusters in the sample as suggested by EFV spot (yellow area observed in 3D Raman imaging of EFV peak in Fig. 3).

FTIR spectra for EFV, SOL and SD are exhibited in Fig. 4. Pure SOL spectra shows distinctive peaks in accordance to the literature [START_REF] Lan | Characterization of Soluplus® by FTIR and Raman Spectroscopy[END_REF]: around 3675 cm -1 due to the hydrogen bonds O-H group, 2987 cm -1 and 2900 cm -1 due to CeH stretching bands, 1735 cm -1 and 1635 cm -1 corresponding to carbonyl bonds C]O group and 1406 cm -1 to CeOeC stretching vibrations. In turn, the characteristic peaks previously reported for EFV [START_REF] Lavra | Solubility and dissolution performances of spray-dried solid dispersion of Efavirenz in Soluplus[END_REF] were also identified here: around 3320 cm -1 , 1600 cm -1 , 2250 cm -1 , 1750 cm -1 and 1250 cm -1 due to the amine group NeH, CeC triple bond, C]C bond and CeF stretching vibrations, respectively. The spectral analysis of SD confirmed that the characteristic peaks for the pure components were present in the binary mixture, without modifications like shifts or disappearance of peaks that could suggest specific interactions between EFV and SOL components.

Microstructure morphologies of the spray-dried SOL and SD are represented in Fig. 5. Spray-dried amorphous EFV were not observed by SEM because, during production, the dry powder remained adhered to the column wall due to the low T g (T PA g = 36.1 °C) of amorphous pure EFV. SOL spray-dried particles took a nearly spherical shape and had a wrinkled appearance (Fig. 5A). In comparison, the "wrinkled collapsed morphology" of SOL spray-dried particles was completely affected by the presence of EFV (Fig. 5B). Indeed, SD particles are mostly spherical with seemingly smooth surfaces without surface discontinuities, which suggests a good mixture between the two amorphous components. SD particles are constituted by some fused interconnected particles, which could be explained by a still low T g since this sample contains a ma- jority of the API that has a very low T g . Besides composition, the feed concentration plays an important role in defining spray-dried morphology [START_REF] Lavra | Solubility and dissolution performances of spray-dried solid dispersion of Efavirenz in Soluplus[END_REF]. Spray-dried SOL particles were generated from a feed concentration of about 10 % (w/w). In this case, the solid concentration is low and then, through the evaporation process, the droplet surface remained wet for a longer time creating hollow particles with a thin solid skin that could deflate resulting in a wrinkled morphology. With the API incorporation in the SOL solution, SD were generated from a feed concentration of about 43 % (w/w), what leads the droplets to dry faster, developing a dried hard surface and preventing subsequent deflation and shrinkage.

API-polymer solubility determination by recrystallization

As proposed here, a supersaturated EFV-SOL solution with 85 wt% EFV (SD) was successfully prepared by spray drying. The glass transition temperatures of SD sample and pure components were determined by DSC (Table 3). SD revealed a single and distinctive mixture glass transition temperature (T MIX g ) in the reversing heat flow signals, which indicates a single relaxation process, homogeneity and mixing at molecular level between the components in the supersaturated EFV-SOL solution produced by spray-drying. DSC results are in good agreement with XRD, Raman microscopy and FTIR results already presented [START_REF] Qian | Is a distinctive single Tg a reliable indicator for the homogeneity of amorphous solid dispersion?[END_REF].

The experimental T MIX g is compared (Table 2) with the theoretical value calculated by the Gordon-Taylor equation previously detailed. The parameter K is determined from Eq. ( 2) with the data from Table 2 and an apparent linear variation of T g is achieved with a K close to 1 (1.03). The gap between experimental and theoretical (Gordon-Taylor) T g values increased with the increase of the EFV loading in the solid dispersions. For the sample with 85 wt% of EFV, this difference is even more significant. This could be explained by the large amount of EFV in the sample, which is possibly responsible for disturbing the measure of T MIX g due to the low value of ΔC P (0.16 J/g/K). The high API load SD produced by spray drying was annealed from 90 °C until 130 °C to promote the recrystallization of excess EFV amount. The choice of this temperature range was based on the molecular mobility in the mixture, which became slower for the lowest temperature and for the melting point of pure API. For instance, the mDSC corresponding to 120 °C showed a mixture glass transition (T MIX g )

in Run 1 at 36.7 ± 3 °C followed by a recrystallization which starts at about 90 °C and continues until 120 °C. In Run 2 (isotherm), the sample is heated during 2 h at 120 °C and then, it is cooled (Run 3). The step 2 is essential, because it favors the recrystallization of the remaining excess EFV in order to reach the equilibrium state.

Once reached the equilibrium state, the sample was rescanned (Run 4) in the same conditions as the first cycle. It is detected a higher second mixture glass transition (T MIX g = 51.2 ± 3 °C) by a decrease in C ∆ p (0.32 J/g/K for Run 1 and 0.16 J/g/K for Run 4). This increase in the T MIX g value and the smaller amplitude of C ∆ p is expected to be due to the reduction of amorphous API amount in the SD that recrystallized in comparison with the Run 1.

The new EFV composition can be determined from Gordon-Taylor equation. Table 3 regroups the saturated EFV concentrations at equilibrium obtained for all annealing temperatures.

In the case of lower temperature than 110 °C, 2 h wasn't enough to reach the saturation equilibrium because of the slower kinetic of demixing. In that case, the time required to reach the saturation equilibrium had to be increased to 28 h.

Flory-Huggins model (Eq. ( 3)) was used to estimate the χ parameter by lowering of the T g mixture temperature. The Eq. ( 3) is expressed in a linear form for which χ is the slope of the obtained line (Fig. 6A). The full line represents the linear regression of the data obtained with the least squares method and the dashed lines, in the same figure, represents the limit of the 95 % confidence interval (CI 95% ) for the slope. The number of data used for the regression was decreased until all used data are in this confidence interval. Finally, the χ parameter of Flory-Huggins obtained is -1.165 with CI 95% = [-1.70;-0.63]. A negative χ parameter represents attractive API-polymer interaction [START_REF] Kyeremateng | A fast and reliable empirical approach for estimating solubility of crystalline drugs in polymers for hot melt extrusion formulations[END_REF]. Fig. 6B presents the best fit of Flory-Huggins model with the corresponding limits of χ confidence interval, as represented in Fig. 6A. To refine these results, a polynomial fitting is also represented. The two fitting models are in good agreement between them for low temperatures and give the solubility curve of EFV in SOL. Thanks to these plots, it is possible to identify regions of kinetic and thermodynamic stability. For example, approximately 20 wt% (Flory-Huggins model) of EFV is soluble in SOL at room temperature (25 °C). It means that a thermodynamic stability should be achieved at room temperature for API loads lower than 20 wt %. In contrast, a high API load SD with 85 wt% of EFV might be phase separate and eventually recrystallize at 25 °C.

The lower temperature, the more difficult the determination of the solubility curve, since the time required to reach the saturation equilibrium is relevant due to slow kinectic of demixing. Although there is no thermodynamic stability, the sample was still stable from the kinetic point of view. As showed in Table 3, below 110 °C, the slow molecular mobility makes the demixing process slower and then, a longer time was required to reach the saturation equilibrium (28 h in our experiments).

This method can not be applied for the lowest temperature given by the crossing point between the solubility and Gordon-Taylor curves. In our case, the crossing point for the Flory-Huggins model is 63. [START_REF] Patel | Revealing facts behind spray dried solid dispersion technology used for solubility enhancement[END_REF] °C, what means that, if the annealing is done under this temperature, it would result in an abrupt reduction of molecular mobility and consequently of the demixing process avoiding the reach of saturation equilibrium [START_REF] Mahieu | A new protocol to determine the solubility of drugs into polymer matrixes[END_REF].

Conclusion

Spray drying is one of the main processes capable of being scaled-up into large manufacturing scale to produce ASDs but it could be demonstrated in this work that this process also represent an interesting alternative to produce supersaturated solid solutions, from which APIpolymer solubility is determined from demixing, because it is a fast process compared to other ones commonly used to produce supersaturated solid solutions as solvent casting or comilling.

A supersaturated solid solution containing high API load (85 wt% Efavirenz) in Soluplus® was generated by spray drying, in only few seconds, and then exposed to a recrystallization protocol. The phase diagram of Efavirenz (EFV)-Soluplus (SOL) obtained from this work can be used to predict the stability of EFV-SOL ASDs at different EFV loadings and under different thermal conditions. For example, samples are stressed under different temperature conditions from room to 40 °C in order to investigate physical stability. The EFV solubility in Soluplus ranges from 20 wt% (25 °C) to 30 wt% (40 °C). Amorphous solid dispersions of EFV in Soluplus containing more than 30 wt% of EFV in mass are supersaturated and their stability should be monitored over storage from room to accelerated (40 °C) temperature conditions.

2. 3 .

 3 Theoretical considerations 2.3.1. Prediction of the amorphous solid dispersion glass transition temperature (T MIX g ) Gordon-Taylor equation is used to estimate the theoretical glass transition temperature of the mixture T MIX g . It is calculated based on the individual contributions of weight fraction and glass transition temperature of each component in the binary mixture, without considering the intermolecular interactions between them. Applying Gordon-Taylor equation, the T MIX g is described by Eq. (1).

Fig. 1 .

 1 Fig. 1. Principle of DSC analysis for recrystallization method.

Fig. 2 .

 2 Fig. 2. XRD diffractograms of pure compounds and spray-dried dispersion.

Fig. 3 .

 3 Fig. 3. 3D images and spectra from Raman spectroscopy of EFV, SOL and SD.

Fig. 4 .

 4 Fig. 4. FTIR spectra chemical formulas of pure components and solid dispersion.

Fig. 5 .

 5 Fig. 5. SEM images of SOL (A) and SD (B).

Fig. 6 .

 6 Fig. 6. (A) Estimation of Flory-Huggins parameter χ by linear regression (B) Phase diagram of EFV-SOL.

Table 1

 1 Physico-chemical properties of products[START_REF]14H9ClF3NO2 -PubChem[END_REF][START_REF]Soluplus -For better solubility & bioavailability[END_REF].

		Molecular weight (g/mol)	Melting point (°C)	Density (g/cm 3 ) *
	Efavirenz	315.68	136-141	1.39
	Soluplus®	90,000-140,000	-	1.18

* Determined experimentally by gas pycnometry.

Table 2

 2 Glass transition temperatures of solid dispersion and pure components determined by DSC and Gordon-Taylor equation.

	W PA	T MIX g	(°C) ± 2	°C
		Gordon Taylor equation		DSC
	0.00	-		79.9
	0.40	62.8		62.2
	0.70	49.6		47.1
	0.85	47.2		36.7
	1.00	-		36.2

Table 3

 3 Equilibrium saturated concentration of EFV in SOL at different temperatures(W )

	PA
	Annealing temperature (°C)	(°C) Annealing time (h) T MIX g	during Run 4	W PA
	90	28	55.0		0.58
	100	28	54.9		0.58
	110	2	52.2		0.64
	120	2	51.2		0.66
	125	2	47.1		0.75
	130	2	45.9		0.78
	137	-	-		1.00

Acknowledgements

The authors are grateful to Cristalia (Brazil) for providing Efavirenz and to V. Nallet, L. Haurie and C. Rolland from Rapsodee Center for respectively DRX, Raman and SEM analyses. Gala® platform/Rapsodee Center also provided technical facilities.