Bruno Assis Pessi

Caroline Baroukh

Anais Bacquet

Olivier Bernard

A universal dynamical

Keywords: Chlorella, metabolic modelling, heterotrophy, mixotrophy, diauxic growth, dynamical modelling

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Introduction

Microalgae have been extensively studied during the past decade. Some species are capable of producing lipids or carbohydrates that can in turn be converted into biofuel [START_REF] Sajjadi | Microalgae lipid and biomass for biofuel production: A comprehensive review on lipid enhancement strategies and their effects on fatty acid composition[END_REF]. Microalgae use light energy, via photosynthesis, to fix carbon dioxide. Not only, their growth rate is much faster than that of higher plants, but they can also be cultivated in wastewater, reducing the environmental impact of bioproducts [START_REF] Morales | Chapter 20 -Life-cycle assessment of microalgal-based biofuel[END_REF][START_REF] Arashiro | Life cycle assessment of microalgae systems for wastewater treatment and bioproducts recovery: Natural pigments, biofertilizer and biogas[END_REF]. An emerging idea suggests using wastewater treatment to obtain biofuels as a co-product [START_REF] Barsanti | Is exploitation of microalgae economically and energetically sustainable?[END_REF]. Even though the production efficiency appears to be attractive, many optimization steps still need to be carried out for this process to become sufficiently cost-effective and environmentally-friendly [START_REF] Tan | Cultivation of microalgae for biodiesel production: A review on upstream and downstream processing[END_REF].

There is already an extensive list of works about wastewater treatment with microalgae. Nevertheless there is no unified framework for the modeling of this process [START_REF] Shoener | Microalgae and cyanobacteria modeling in water resource recovery facilities: A critical review[END_REF]. Most of these models are variants of active sludge models (ASM) and anaerobic digestion models (ADM), which were originally designed for bacterial systems (Casagli et al., 2021b[START_REF] Wágner | Towards a consensus-based biokinetic model for green microalgae -The ASM-A[END_REF]. In addition mixotrophy is rarely considered in these models and microalgae are assumed to grow autotrophically, neglecting the simultaneous use of organic compounds. In contrast to these models, metabolic models can acquire the reaction yields, including photosynthesis and organic carbon uptake by using the knowledge of the biochemical reactions taking place in the organism which is reconstructed from genomic data.

Despite this, less than 5% of models of water recover facilities have metabolic reconstructions and usually rely on empirical yield coefficients [START_REF] Shoener | Microalgae and cyanobacteria modeling in water resource recovery facilities: A critical review[END_REF].

Metabolic models also have the advantage of estimating internal metabolic fluxes, providing valuable information for future strain improvement via metabolic engineering. Nevertheless, the applicability of metabolic models to predict dynamical systems is constrained by the size of the metabolic network -which in genome scale models usually consists of thousands of reactions and metabolites. Consequently, to embed mixotrophic microalgal models in larger frameworks representing the mass fluxes within the wastewater process, it is necessary to use techniques to reduce the metabolic models to a reasonable size, while still keeping the phenotype of the original network.

The development of techniques to reduce the size of metabolic networks started more than 20 years ago [START_REF] Singh | Network reduction methods for genome-scale metabolic models[END_REF]. Since then new methods have been developed relying on a variety of different approaches, such as linear programming, graph-based search and elementary flux modes [START_REF] Singh | Network reduction methods for genome-scale metabolic models[END_REF].

Our increasing understanding of genomic information led to the construction of more complex genome scale networks. This has motivated the field of bioinformatics to research new ways to reduce and analyze metabolic models, consequently new methods are still being developed [START_REF] Küken | A structural property for reduction of biochemical networks[END_REF][START_REF] Hameri | The effects of model complexity and size on metabolic flux distribution and control: Case study in Escherichia coli[END_REF]. The choice of the reduction method will depend on the intentions of the modeler, the wished degree of flexibility of the phenotype prediction and the final size of the model. In this work, we will rely on the DRUM framework [START_REF] Baroukh | DRUM: A new framework for metabolic modeling under non-balanced growth. Application to the carbon metabolism of unicellular microalgae[END_REF][START_REF] Baroukh | Reply to the Comment on "Mathematical modeling of unicellular microalgae and cyanobacteria metabolism for biofuel production[END_REF], which although requires input from the modeler and prior knowledge about the network, it is able to reproduce behaviour outside the steady state, while greatly reducing the number of metabolites and reactions in the final model. On top of this, such dynamic model can support a control strategy to enhance process efficiency.

During dark fermentation of organic wastes, anaerobic bacteria and archaea convert complex and non-assimilable compounds into Volatile Fatty Acids (VFAs) us-able by microalgae. Indeed, VFAs can support heterotrophic growth of microalgae, while they use ammonium and phosphate in the wastewater as source for nitrogen and phosphorus [START_REF] Baroukh | Dynamic metabolic modeling of heterotrophic and mixotrophic microalgal growth on fermentative wastes[END_REF], Turon et al., 2015b[START_REF] Gao | Metagenomics and network analysis elucidating the coordination between fermentative bacteria and microalgae in a novel bacterial-algal coupling reactor (BACR) for mariculture wastewater treatment[END_REF]. In this perspective, Chlorella sp. was selected for its potential in associating biofuel production with effluent treatment (Casagli et al., 2021b[START_REF] Gao | Metagenomics and network analysis elucidating the coordination between fermentative bacteria and microalgae in a novel bacterial-algal coupling reactor (BACR) for mariculture wastewater treatment[END_REF][START_REF] Wágner | Towards a consensus-based biokinetic model for green microalgae -The ASM-A[END_REF]. Indeed, this species can accumulate up to 50% of its dry weight in lipids, essentially in the form of triacylglycerol (TAG). In addition, its capacity to grow in the main dark fermentation effluents has been demonstrated, under heterotrophic or mixotrophic conditions (Turon et al., 2015a[START_REF] Lacroux | Microalgae screening for heterotrophic and mixotrophic growth on butyrate[END_REF].

The VFA mixture resulting from dark fermentation is typically made of a blend composed of about 30% acetate and 70% butyrate. Other organic acids can also be found [START_REF] Rafrafi | Sub-dominant bacteria as keystone species in microbial communities producing bio-hydrogen[END_REF], Turon et al., 2015a), among which lactate, which is not consumed and does not affect the growth of Chlorella (Turon et al., 2015a). Turon et al. (2015a) first proposed a kinetic model of the consumption of butyrate and acetate by Chlorella. Later, an extended metabolic model was proposed using the DRUM framework [START_REF] Baroukh | Dynamic metabolic modeling of heterotrophic and mixotrophic microalgal growth on fermentative wastes[END_REF]. Here, we extend further this model by including the consumption of glucose and glycerol by Chlorella and propose a universal multi-substrate dynamic reduced metabolic model. This model is the cornerstone for tackling the major issue due to the high concentration of butyrate which is slowly consumed by microalgae and inhibited by acetate [START_REF] Hu | Enhanced mixotrophic growth of microalga Chlorella sp. on pretreated swine manure for simultaneous biofuel feedstock production and nutrient removal[END_REF], Turon et al., 2015b[START_REF] Lacroux | Microalgae screening for heterotrophic and mixotrophic growth on butyrate[END_REF]. To address this problem, another organic substrate can be added to lever the inhibition effect of butyrate and eventually accelerate growth in dark fermentation effluents. Microalgae will first use a more efficient carbon substrate to reach a higher biomass concentration. It follows that acetate will be faster consumed, and in the end butyrate. Glucose could ideally play this role, but its cost is not compatible with process economics. Acetate addition is more reasonable but would generate large variations in pH. Glycerol is a by-product of biodiesel synthesis by transesterification. Its low cost is likely to be compensated by the enhanced microalgal productivity. Developing a mixotrophic multi-substrate metabolic model is the main objective of this paper, and such approach can be used to identify strategies to more efficiently use dark fermentation products and optimize their conversion into algal biomass.

The model developed here represents, in detail, the growth under different autotrophic, heterotrophic and mixotrophic conditions and for four organic substrates.

To our knowledge, this is the first model including such a large range of potential substrates in mixotophic regimes. The model was validated in different cultivation conditions using the abundant literature available on autotrophic, heterotrophic or mixotrophic growth of Chlorella. To this end, data from 122 experiments was extracted from 15 publications, amounting to more than 2600 concentration data points (see Table 1), thus reaching an unprecedented level of validation. This model [START_REF] Sen | Glycerol and nitrate utilisation by marine microalgae Nannochloropsis salina and Chlorella sp. and associated bacteria during mixotrophic and heterotrophic growth[END_REF] is shown to support a strategy to enhance the bioconversion of VFA into microalgal biomass by managing the way the different substrates are supplied.

Materials and methods

General Principles of the DRUM approach

The dynamic model development follows the DRUM (Dynamic Reduction of Unbalanced Metabolism) approach. The full description and complete explanation of the approach is available in [START_REF] Baroukh | DRUM: A new framework for metabolic modeling under non-balanced growth. Application to the carbon metabolism of unicellular microalgae[END_REF].

Briefly, the metabolism of a microorganism can be described by its metabolic network composed of a set of n r biochemical reactions (here n r = 188) involving n m metabolites (here n m = 173) and represented by the stoichiometric matrix K ∈ R nm×nr (see Appendix for the full list of reactions and metabolites). The biomass B is produced from a set of substrates S and excretes a set of products P. In a perfectly mixed reactor with a constant volume, the system can be described by the following set of ordinary differential equations:

dM dt = d dt     S C P B     =     K S K C K P K B     .v(M ).B -DM +DM in = K.v(M ).B -DM +DM in (1)
where M represents the vector of the concentrations of metabolites composed of substrate (S), intracellular metabolites (C), excreted products (P) and biomass (B).

M in is the influent concentration of these quantities. The dilution rate of the reactor (ratio of influent flow rate over the reactor volume) is D (D = 0 for a batch process).

All the concentrations are expressed as total concentrations in the solution. v ∈ R nr is the reaction kinetic vector, while the matrices K S ∈ R n S ×nr , K C ∈ R n C ×nr , K P ∈ R n P ×nr and K B ∈ R 1×nr correspond, respectively, to the stoichiometric matrices of substrates S, products P, intracellular metabolites C and biomass B

(n S + n C + n P + 1 = n m ).
In metabolic models, intracellular metabolites are generally assumed to be quasistationary ( dC dt = K.v = 0), i.e. they are assumed to be consumed as soon as they have been synthesised. However, in the case of microalgae, this hypothesis has proven to be false for certain of its metabolites (denoted A) during mixotrophic or autotrophic growth [START_REF] Baroukh | DRUM: A new framework for metabolic modeling under non-balanced growth. Application to the carbon metabolism of unicellular microalgae[END_REF][START_REF] Baroukh | Dynamic metabolic modeling of heterotrophic and mixotrophic microalgal growth on fermentative wastes[END_REF]. The DRUM method [START_REF] Baroukh | DRUM: A new framework for metabolic modeling under non-balanced growth. Application to the carbon metabolism of unicellular microalgae[END_REF], consists in dividing the metabolic network into n quasi-stationary subnetworks (K = (K SN 1 , . . . , K SNn ) , K SN i .v SN i = 0 f or i ∈ 1, . . . , n). These are linked by the A metabolites that are, in contrast, non-stationary, can accumulate and be later consumed. This division into subnetworks is justified by the presence of metabolic pathways that correspond to metabolic functions, to reaction groups that are regulated simultaneously and to the presence of compartments within the cell.

Cellular mechanisms are therefore employed for assessing the subnetwork. Hence, the A metabolites can either be found at the junction of several metabolic pathways, or they can be transported from one compartment to another, or they can be final products that accumulate in the cell. The system of ordinary differential equations

(1) therefore becomes:

dM ′ dt = d dt     S A P B     =     K S K A K P K B     .α.B = K ′ .α.B -DM ′ + DM ′ in (2)
With K ′ the stoichiometric matrix of macroscopic reactions obtained through the analysis of elementary modes [START_REF] Schuster | Detection of elementary flux modes in biochemical networks: A promising tool for pathway analysis and metabolic engineering[END_REF] on the subnetworks, and α the kinetic vector associated to these macroscopic reactions. B now represents the structural biomass, i.e. the fraction of biomass that does not contain the inert compartments of reserve A. The total biomass can be deduced using a mass balance of the elemental compounds (C, N, P, O, ...).

Construction of the model

The core of the metabolic network from [START_REF] Baroukh | Dynamic metabolic modeling of heterotrophic and mixotrophic microalgal growth on fermentative wastes[END_REF] has been used and modified in order to add the glucose and glycerol consumption pathways (Figure 1). This network contains the central autotrophic, mixotrophic and heterotrophic metabolic pathways including photosynthesis, glycolysis, the pentose phosphate pathway, the Krebs cycle, oxidative phosphorylation and the synthesis of chlorophyll, carbohydrates (e.g. starch), amino-acids and nucleotides. The synthesis pathways of macromolecules such as proteins, lipids, starch, DNA, RNA as well as the functional biomass are represented through macroscopic reactions.

The DRUM method requires the partitioning of the metabolism into subnetworks as well as the identification of the metabolites, in between the subnetworks, which can accumulate. The subnetworks are defined by their metabolic function and/or their affiliation to a cellular compartment. Different partitions among the 188 reactions have been tested, with a view to select the one which minimized the number of parameters to assess while providing a correct representation of the experimental data. The best result was obtained when the network is divided into four subnetworks (Figure 1), corresponding to, 1) the glyoxysome, 2) the chloroplast, 3) the absorption of glycerol and 4) the synthesis of biomass. The glyoxysome and chloroplast subnetworks remain unchanged in comparison with the initial [START_REF] Baroukh | Dynamic metabolic modeling of heterotrophic and mixotrophic microalgal growth on fermentative wastes[END_REF] model.

The macroscopic reactions associated to each subnetwork are deduced from the elementary mode analysis [START_REF] Klamt | Two approaches for metabolic pathway analysis[END_REF]. The Matlab "efmtool" was run to calculate the Elementary Flux Modes (EFMs) [START_REF] Terzer | Large-scale computation of elementary flux modes with bit pattern trees[END_REF]. In total, 86, 142 EFMs including 3, 310 associated to futile cycles (dissipation of carbon substrate in the form of CO 2 ) have been achieved. These macroscopic reactions are further used to determine the mass fluxes in the different parts of the network by assembling reactions belonging to the same kinetics.

Analysis of the sub-networks

Motivations

In this section, we present the macroscopic reactions which result from the reduction of each subnetwork by the computation of the EFMs. As recommended by [START_REF] Baroukh | DRUM: A new framework for metabolic modeling under non-balanced growth. Application to the carbon metabolism of unicellular microalgae[END_REF], the reaction kinetics must be mathematically represented Butyrate is known for inhibiting algal growth under heterotrophic and mixotrophic conditions (Turon et al., 2015a). Furthermore, acetate inhibits the absorption of butyrate, thus leading to diauxic growth (Turon et al., 2015a). Thereby, Michaelis-Menten kinetics have been proposed to describe the absorption of acetate (α M R1 ).

Subnetwork Macroscopic reaction M R 1 Glyoxysome 2 ACE + 3.5 H + 0.5 O 2 → SU C + 0.5 H 2 O M R 2 BU T + 7H + 1.5O 2 → SU C + 5H 2 O M R 3 Chloroplast Light + 3CO 2 + 2H 2 O + P i → GAP + 3O 2 M R 4 Glycerol pathway GLY + P i → GAP + H 2 O M R 5 (N H 4 ) Biomass synthesis 4.15 GAP + 2.54O 2 + 0.99N H 4 + 0.02SO 4 + 0.01M g 2 → B + 0.99H + 2.90H 2 O + 3.92CO 2 + 4.02P i M R 5 (N O 3 ) 4.64GAP + 2.04O 2 + 0.99N O 3 + 0.98H + 0.02SO 4 + 0.01M g 2 → B + 5.39CO 2 + 2.90H 2 O + 4.51P i M R 6 (N H 4 ) Biomass synthesis 4.15 SU C + 7.30H + 4.61O 2 + 0.99N H 4 + 0.12P i + 0.02SO 4 + 0.01M g 2 → B + 7.04H 2 O + 8.06CO 2 M R 6 (N O 3 ) 4.90SU C5.28O 2 + 0.99N O 3 + 0.12P i + 10.78H + 0.02SO 4 + 0.01M g 2 → B + 11.07CO 2 + 8.31H 2 O M R 7 (N H 4 ) Biomass synthesis 2.07 GLC + 2.54O 2 + 0.99N H 4 + 0.12P i + 0.02SO 4 + 0.01M g 2 → B + 3.91CO 2 + 7.04H 2 O + 0.99H M R 7 (N O 3 ) 2.34GLC + 2.14O 2 + 0.99N O 3 + 0.12P i + 0.98H + 0.02SO 4 + 0.01M g 2 → B + 5.49CO 2 + 7.63H 2 O
α M R 1 = k M R1 .ACE KS M R1 + ACE (3)
For butyrate (α M R2 ) Haldane kinetics have been chosen with an inhibition by acetate term.

α M R2 = k M R2 .BU T BU T + k M R2 β M R2 .( BU T Sopt M R2 -1) 2 k d (ACE + k d ) (4) 2.3.3. Chloroplast subnetwork
The chloroplast subnetwork is composed of 21 reactions, from which 7 are exchange reactions. The glycerate-3-phosphate produced by photosynthesis is assumed to be transferred from the chloroplast towards the cytosol where it can be converted by glycolysis into glucose-6-phosphate or pyruvate. These metabolites are essential for the synthesis of functional biomass.

For autotrophic growth, light drives the reaction rate. When algae are growing on a turbid medium like wastewater, the average light intensity stays low and the local photoinhibition impact can be neglected [START_REF] Martínez | Theory of turbid microalgae cultures[END_REF]. Also under mixotrophic conditions with elevated concentration of carbon substrates, large biomass densities can be reached. Dense microalgal cultures strongly attenuate light. In line with [START_REF] Baroukh | Dynamic metabolic modeling of heterotrophic and mixotrophic microalgal growth on fermentative wastes[END_REF], photosynthesis rate is assumed to be linearly depending upon the average light intensity I µ in the culture (see Equation 6).

Moreover, light attenuation in the culture medium is expected to follow the Beer-Lambert law. The light intensity at depth z depends on the incident light I 0 and the extinction coefficient α due to the biomass (the turbidity of the medium without algae is negligible):

I (z) = I 0 .e -α.z.B (5)
The average light intensity of the culture medium in the reactor of depth L is given as follows (with β M R3 = α.L):

I µ = I 0 L L 0 e -α.B.z dz = I 0 (1 -e -β M R3 .B) β M R3 .B (6)
The kinetics in the chloroplast subnetwork is finally given by:

α M R3 = k M R3 .I µ (7)

Glycerol absorption subnetwork

The glycerol pathway subnetwork consists of 5 core reactions, plus the exchange reactions. Only one EFM was found for the glycerol absorption subnetwork (MR4).

The glycerol in the medium is transferred to the cytosol. Within three steps, it is transformed into glycerate-3-phosphate. During glycolysis, this glycerate-3phosphate is then used for the synthesis of precursor metabolites that are in turn required for the synthesis of functional biomass. Since inhibition has been observed

for glycerol assimilation [START_REF] Chen | Biomass and lipid production of heterotrophic microalgae Chlorella protothecoides by using biodiesel-derived crude glycerol[END_REF][START_REF] Ma | Cultivation of Chlorella vulgaris in wastewater with waste glycerol: Strategies for improving nutrients removal and enhancing lipid production[END_REF][START_REF] Liang | Biomass and lipid productivities of Chlorella vulgaris under autotrophic, heterotrophic and mixotrophic growth conditions[END_REF], a Haldane reaction kinetics with inhibition was chosen (α M R4 ):

α M R4 = k M R4 .GLY GLY + k M R4 β M R4 .( GLY Sopt M R4 -1) 2 (8)

Functional biomass synthesis subnetwork

The reactions for the synthesis of lipids, proteins, DNA, RNA, chlorophyll and carbohydrates are all lumped together in the functional biomass synthesis subnetwork. This subnetwork includes glycolysis, the Krebs cycle, oxidative phosphorylation, the pentose phosphate pathway, carbohydrate, lipid, amino-acid and nucleotide synthesis, as well as the assimilation of nitrogen, sulphur and glucose. In total there are 141 reactions in the functional biomass subnetwork.

This subnetwork generated 86, 167 EFMs, including 3310 that did not produce biomass. Nearly all of the calculated EFMs are part of the biomass synthesis network. They can be sorted by using a similar method to the FBA (Flux Balance Analysis). The standard hypothesis supporting FBA is that evolution has selected metabolisms maximising biomass growth on each substrate [START_REF] Orth | What is flux balance analysis?[END_REF], or equivalently, minimising the loss of carbon as CO 2 . Therefore, for each substrate, the EFM presenting the highest GAP/BIOM, SUC/BIOM and GLC/BIOM yields were selected. In this way, the use of GAP, SUC or GLC for the synthesis of biomass resulted from three macroscopic reactions (MR5, MR6 and MR7). The yield of biomass on the carbon substrate depends on the nitrogen source (N H 4 or N O 3 ).

Table 1 shows the resulting macroscopic reactions for both cases.

Glycerate-3-phosphate originates from the chloroplast and from the assimilation of glycerol. It is injected into the glycolysis so as to produce the necessary metabolites for growth (α M R5 ). The kinetics is supposed to be linear with respect to glycerate-3-phosphate (GAP ):

α M E5 = k M R5 .GAP (9)
Succinate originates from the glyoxysome. It enters the Krebs cycle, thus also leading to the production of metabolites required for growth (α M R6 ). Assuming a linear kinetics we get:

α M R6 = k M R6 .SU C (10)
Glucose in the medium is transferred to the cytosol where it enables the production of biomass. Glucose is inhibiting at high concentration (osmotic stress), and its consumption is assumed to follow a Haldane kinetics (α M R7 ) [START_REF] Azma | Improvement of medium composition for heterotrophic cultivation of green microalgae, Tetraselmis suecica, using response surface methodology[END_REF][START_REF] Wu | Optimization for high-density cultivation of heterotrophic Chlorella based on a hybrid neural network model[END_REF][START_REF] Liang | Biomass and lipid productivities of Chlorella vulgaris under autotrophic, heterotrophic and mixotrophic growth conditions[END_REF]. MR7:

α M R7 = k M R7 .GLC GLC + k M R7 β M R7 .( GLC Sopt M R7 -1) 2 (11)

Global dynamics of the network

Finally, the dynamical evolution of the metabolic fluxes associated to the 188 considered metabolic core reactions can be derived from a system with 17 ordinary differential equations comprising 17 metabolites and 7 macroscopic reactions:

dM " dt = d   S A B   dt = K " .α.B -DM " + DM " in ( 12
)
Where M " is the metabolite vector (17x1) comprising the substrates S, the metabolites that can accumulate A (SUC and GAP) and the functional biomass B. K' is the stoichiometric matrix (17x7) of the macroscopic reactions and α the associated kinetics vector (7x1). It is worth noting that, even if the model results from a reduction process through the DRUM approach, it can still predict the evolution of the 188 metabolic fluxes in the cell:

Moreover, the total biomass comprising the functional biomass and the metabolites A can be described as follows:

X z (t) = A Z A .A (t) + Z B .B (t) ( 13
)
Where Z is a chemical element (Z ∈ {C; N ; O; H; P ; . . .}), Z A and Z B are the number of chemical elements Z per mole of metabolites A and biomass B, A(t) and B(t) are the concentrations of A and B at time t X Z (t) is the concentration of the chemical element in the total biomass X at time t.

Finally, the metabolic fluxes within the whole network can be derived from the α kinetics and the elementary modes associated to the

E SN i , i ∈ 1, 2, 3 subnetworks: v =   v SN 1 . . . v SN k   =   E SN 1 .α SN 1 . . . E SN k .α SN k   (14)

Sensitivity analysis

We calculate the sensitivity coefficient for the model parameters as defined in [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF]:

σ ∆p y = 1 t f t f 0 y(p + ∆p, x 0 , u, τ ) -y(p, x 0 , u, τ ) y(p, x 0 , u, τ ) dτ ( 15
)
where y is the simulated output at time τ with parameter set p, initial condition

x 0 and input variables u (e.g. light intensity and dilution rate). We calculate the global sensitivity using Morris's sampling method implemented in the SALib Python toolbox (Morris, 1991, Herman and[START_REF] Herman | SALib: An open-source Python library for Sensitivity Analysis[END_REF], replacing the standard elementary effect by the sensitivity coefficient defined above. We analyse the region between ±20% of the calibrated values of the parameters. The analysis is conducted separately for each of the four carbon substrates in mixotrophic conditions at a light intensity of 500 µmol/(m 2 .s). σ ∆p y is calculated as the average of the simulations considering a cultivation of 6 hours and three different initial concentration of the considered carbon substrate (0.1, 1 and 10 g/L). Table 3 shows the result of the sensibility analysis for all the parameters.

Reduced model calibration

In order to calibrate, and then validate the model, a large set of experiments from the literature have been used. In total 122 selected experiments (see Table 1) gather data on growth i) under autotrophic conditions, without any organic carbon input and submitted to light intensities ranging from 30 to 540 µE.m -2 .s -1 ii) under heterotrophic conditions, without any light, and with varying concentrations 3.60 10

-2 M h -1 .M B -1 9.9610 -05 1.4010 -03 B Sopt M R2 1.90 10 -5 M 2.4810 -04 1.4110 -03 B β M R2 2.58 10 5 h -1 .M B -1 1.8910 -04 1.4010 -03 B k M R3 1.90 10 -3 M h -1 .M B -1 .(m 2 .s.µmol -1 ) 9.2110 -04 5.7010 -04 B β M R3 2.48 10 3 M B -1 -5.3810 -04 8.8210 -05 B k M R4 1.01 M.h -1 .M B -1 5.5410 -04 1.8610 -03 A Sopt M R4 7.00 10 -2 M 9.2710 -04 2.2510 -03 A β M R4 9.05 h -1 .M B -1 1.1010 -04 2.4310 -03 A k M R5 2.82 10 1 h -1 .M B -1 9.7310 -04 1.1610 -03 B k D 5.39 10 -10 M -1.1010 -04 1.4310 -03 B k M R6 2.37 10 5 h -1 .M B -1 3.0710 -06 3.3310 -03 B k M R7 6.81 10 -2 M.h -1 .M B -1 2.3910 -03 2.1610 -03 A Sopt M R7 5.07 10 -2 M 9.5410 -05 3.2210 -03 A β M R7 2.95 10 +1 h -1 .M B -1 4.2410 -04 3.2410 -03 A
in acetate, butyrate, glucose and glycerol, pure or combined iii) under mixotrophic conditions, with light and varying concentrations in acetate, butyrate, glucose and glycerol. Depending on the studies, different combinations of these substrates were tested.

Only the parameters for glucose and glycerol consumption were calibrated. All the other kinetic parameters for the macro reactions are taken from [START_REF] Baroukh | Dynamic metabolic modeling of heterotrophic and mixotrophic microalgal growth on fermentative wastes[END_REF]. The calibration was done following a two-step process. First a stochastic global optimizer, Differential Evolution algorithm [START_REF] Storn | Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces[END_REF], calculates the set of parameters minimizing the relative error between model and experimental data of biomass and substrate concentration over time. This parameter set is, then, used as initial point in a Markov Chain Monte Carlo sampler, which returns the parameters set inside a confidence interval [START_REF] Foreman-Mackey | Emcee : The MCMC Hammer[END_REF]. Glucose kinetics parameters were calibrated using concentration data from 5 experiments of [START_REF] Li | High productivity cultivation of a heatresistant microalga Chlorella sorokiniana for biofuel production[END_REF], while glycerol parameters were fitted using data from 6 experiments of [START_REF] Ma | Cultivation of Chlorella vulgaris in wastewater with waste glycerol: Strategies for improving nutrients removal and enhancing lipid production[END_REF]. Table 3 shows the value of calibrated parameters and Figure 2 shows the simulation of the calibrated model together with the experimental data used for calibration.

Optimization of butyrate consumption

We consider the optimal control problem [START_REF] Harmand | Optimal Control in Bioprocesses: Pontryagin's Maximum Principle in Practice[END_REF], whose objective is to minimize the time t f , where the chemical oxygen demand of the remaining waste substrates falls below the regulation threshold. The control variable is the concentration of the substrate to be added to the dark fermentation effluent (glucose or glycerol).

min

S 0 t f : S(t f ) ≤ S (16)
To solve the minimization problem we use a Nelder-Mead algorithm [START_REF] Gao | Implementing the Nelder-Mead simplex algorithm with adaptive parameters[END_REF]. The output function simulates the metabolic model for a given S 0 returning t f , the time required to reach the regulation threshold.

Results and discussion

Validation of the model

The experimental data not used during the calibration stage were used to validate the model. A coherent set of experiments, representing various experimental conditions, was kept for this validation stage (see Table 1 ). We used a Markov Chain Monte Carlo method to select parameters [START_REF] Foreman-Mackey | Emcee : The MCMC Hammer[END_REF].

We considered a ±20% uncertainty in the initial concentration of substrates. In Figure 3, the results of the model simulations are compared with the experimental data with more than one substrate.

As illustrated in Figure 2 for single substrates, the model efficiently predicts the production of biomass and the consumption of substrates, whether in autotrophic, heterotrophic or in mixotrophic conditions. Furthermore, the model is still accurate when there are two substrates (Figure 3). More generally, the predictive performance of the model is summarized in the Taylor diagram in Figure 4 for the whole data set. This diagram represents at the same time the standard deviation of the biomass prediction error and the Pearson correlation coefficient [START_REF] Taylor | Summarizing multiple aspects of model performance in a single diagram[END_REF]. It illustrates both the centered and reduced quadratic errors between the experimental data sets and the associated simulations, as well as the correlation between the model and the data. It thus summarizes the degree of resemblance between the data and the simulations for the vast range of considered data. Indeed, the closer a data point is to (0;1), the better the model reproduces the experimental data [START_REF] Taylor | Summarizing multiple aspects of model performance in a single diagram[END_REF].

Figure 5 represents all the data points versus model prediction, also demonstrating the goodness of fit of the model.

The results of the sensibility analysis is also show in Table 3. Most parameters have the same order of sensibility (10 -4 ) showing that all have an importance in the model. The high standard deviation of the sensibility coefficient demonstrates the intertwined influence of parameters, for example maximum uptake rate and the optimal concentration of the substrate, also and the dependence of the actual substrate and biomass concentration in the dynamics of the system. This demonstrates the necessity of calibrating the model in a wide range of conditions. Figure 6: Time to reach the regulation threshold of a typical dark fermentation effluent using different conditions of cultivation -Batch (typical waste effluent), Batch GLC/GLY (typical waste with addition of an optimal concentration of glucose and glycerol), Fed-batch GLC, GLY, GLC/GLY (typical waste feeding, respectively, glucose, glycerol and a mixture of both.)

Optimization of microalgae growing on a mixture of dark fermentation products

Acetate and butyrate are the main volatile fatty acids (VFA) products of dark fermentation. Butyrate will inherently lead to growth inhibition, and a strategy must be found to unblock this inhibition. Here, we propose a strategy to enhance batch and fed-batch cultivation of microalgae from dark fermentation wastes, by adding glucose and glycerol. We first study mixotrophic conditions, considering a typical effluent from dark fermentation with 3.5g/L of butyrate and 1.7g/L of acetate [START_REF] Lacroux | Mixotrophic growth of microalgae on volatile fatty acids is determined by their undissociated form[END_REF][START_REF] Ghimire | A review on dark fermentative biohydrogen production from organic biomass: Process parameters and use of by-products[END_REF] and a continuous light intensity of 500 µmol/(m 2 .s). The objective was that the chemical oxygen demand (COD) of the effluent at the end of the effluent treatment must be below S (here S = 125mg/L) so that it can satisfy the state policies for discharge in the environment. We call this objective the regulation threshold. We solve the resulting optimal control problem in minimal time, with the objective that the COD of the remaining waste substrates falls below the regulation threshold. The control variable is the concentration of the substrate to be added to the dark fermentation effluent (glucose or glycerol).

In the initial situation, without addition of organic carbon substrates, 16 days are necessary to reach the the regulation threshold.

Considering that glucose can be added, it turns out that the addition 98g/L of glucose reduces the time to reach the COD threshold to 4.0 days (see Figures 6 and7). Using only glycerol takes longer: 5.6 days with an addition of 39.5g/L glycerol.

Using a mixture of both substrates, the optimal starting concentration for the batch Optimal fed-batch with glycerol (blue), optimal fed-batch with glucose (red), batch with optimal addition of glycerol and glucose (orange), batch with only butyrate and acetate as substrates (green).

is 115g/L of glucose and 19g/L of glycerol, reducing the time to consume the VFAs to only 2.7 days.

Considering now a fed-batch cultivation systems instead of a batch one, the minimisation problem can be rewritten as following:

min D t f : S(t f ) ≤ S ( 17
)
where D is the dilution rate of the inflow containing only the additional substrates, at a high concentration, so that the volume of the reactor does not change. t f is the time to reach the regulation threshold. The optimal strategy can be approximated into a sub-optimal strategy, which would maximize the reaction rates for glucose and glycerol consumption. The strategy thus consists in computing the dilution rate such that glucose and glycerol concentrations stay constant close to the values for which the consumption of glucose and glycerol is maximum. The control problem is then reduced to finding the optimal final time of the inlet flux in the cases of a glucose, glycerol, and mixture inlet. In the case of the mixture, a fraction of 0.21 of glucose and 0.79 of glycerol is obtained, keeping glycerol at an optimal concentration.

Using this control strategy, the final times for fed-batch cultivation are: 3.2 days for glucose, 1.4 days for glycerol and 1.2 days for a mixture of glucose and glycerol (see Figure 6), to be compared to the 16 days whithout inorganic carbon addition.

Analysis of metabolic maps

One of the advantages of metabolic model is that the main metabolic fluxes can be estimated. They are represented in Figure 8 No reaction takes place in the glyoxysome, neither in the glycerol utilization pathway for growth on glucose only. Glucose carried from the medium into the cytoplasm is directly injected into the upper glycolysis for the production of precursor metabolites necessary for the synthesis of functional biomass. Glycolysis is therefore entirely in a downward direction, as the anaplerotic pathways that enable the synthesis of oxaloacetate (OXA) fueling the Krebs cycle.

For growth on glycerol only, the greatest fluxes are located in the glycerol utilization pathway subnetwork, where the uptaken glycerol is converted to glycerate-3phosphate. The glycerate-3-phosphate is then injected into the middle of glycolysis, this time in an upward direction for the high glycolysis and in a downward direction for the lower glycolysis. The remnants of the fluxes are similar to those observed for glucose.

In mixotrophic conditions, when all carbon substrates are present in the medium, all metabolic reactions are activated. First, diauxic growth occurs, and acetate is consumed instead of butyrate. Glycerol and glucose are also used, but the flux remains after the acetate is depleted. Secondly, once all acetate has disappeared, butyrate is in turn consumed.

Figure 8 shows the metabolic fluxes after 30 hours of batch culture with optimal initial conditions (110 g/l of glucose and 20 g/L of glycerol). At this point, the acetate in the culture is practically depleted, but the concentration of butyrate remains similar to the initial condition. The rate of consumption of butyrate then remains stable for 2 days, and the rate rapidly increases until the concentration threshold is reached. Since the optimal concentration is about 40 times lower than the regulation threshold, the butyrate consumption rate does not reach the maximum value. Glycerol has a much lower concentration, while the flux of glucose concentration is still high. As a result of the higher concentration of biomass and the lower availability of light per cell, the metabolic fluxes in the chloroplast are reduced.

RuBP

Model limitations and perspectives

The reduced metabolic model efficiently represents the microalgal growth under various substrates in heterotrophic or mixotrophic conditions. More accurate predictions could probably be obtained by expanding the model to include other factors such as pH, CO 2 , O 2 and temperature which were differing among the large experimental data set considered in our study. According to [START_REF] Lacroux | Mixotrophic growth of microalgae on volatile fatty acids is determined by their undissociated form[END_REF], pH fluctuations when algae consume VFA can strongly impact growth and should now be included in the model. Associating a pH model that accounts for the various chemical species and their speciation, as proposed by Casagli et al. (2021a), would allow the calculation of the pH and the concentration of the undissociated form of the acids, which is the one actually taken up by the microalgae.

The effect of mixotrophic growth, here considered as the sum of autotrophic and heterotrophic conditions, can be more subtle in some cases. There is no consensus in the literature for a general model that fits all cases. For example, according to [START_REF] Martínez | Interactions between Glucose and Inorganic Carbon Metabolism in Chlorella vulgaris Strain UAM 101[END_REF], mixotrophic growth is greater than the sum of autotrophic and heterotrophic conditions when the concentration of CO 2 is limiting, since CO 2 produced by respiration can be recycled for the photosynthesis pathway.

Recently, circular use of CO 2 and O 2 and preferential consumption of inorganic carbon has also been evidenced for Chlorella vulgaris, nonetheless heterotrophic growth was promoted at high 0 2 concentrations [START_REF] Manhaeghe | Experimental assessment and mathematical modelling of the growth of Chlorella vulgaris under photoautotrophic, heterotrophic and mixotrophic conditions[END_REF]. Furthermore, under certain conditions an inverse relationship between light intensity and glucose consumption has been recorded [START_REF] Patel | Effect of light conditions on mixotrophic cultivation of green microalgae[END_REF][START_REF] Wan | The effect of mixotrophy on microalgal growth, lipid content, and expression levels of three pathway genes in Chlorella sorokiniana[END_REF].

Temperature is one of the most important factors affecting microalgae growth, even if only a minority of the models account for it [START_REF] Shoener | Microalgae and cyanobacteria modeling in water resource recovery facilities: A critical review[END_REF]. The experimental data considered here were carried out at various temperatures and accounting for it would increase model accuracy. In the case of metabolic models, including temperature is challenging, and will involve a large number of parameters to characterize the influence of temperature on each individual reaction. The influence of short variations of temperature could be well estimated by the Arrhenius equation, but large variations of temperature typically occurring in outdoor cultivation require more advanced models (Casagli and[START_REF] Pessi | Does temperature shift justify microalgae production under greenhouse?[END_REF].

For these reasons, calibrating the temperature effect would require a large amount of dedicated experiments.

It is likely that other factors were influencing the experimental outcomes in the considered data base, especially for the experiments carried out with high concentrations of glucose and glycerol. It seems that another substrate was sometimes limiting growth, most probably nitrogen according to the mass balance from the medium initial composition.

Extending the metabolic model to account for all these mechanisms is beyond the scope of this paper. It will require a large number of experiments to further calibrate and validate the model. Our objective was primarily to validate the model across a wide range of conditions, demonstrating a strong foundation for future improvements in the model, knowing that most models of water resource recovery facilities are calibrated, but not validated [START_REF] Shoener | Microalgae and cyanobacteria modeling in water resource recovery facilities: A critical review[END_REF]. Overall, despite the simplicity of the model in its present form, it is already very efficient. Especially when accounting for the diversity of strain and experimental conditions considered through the 15 studied papers. The model can then, already in its present form, can be used as a tool for optimizing microalgae growth on a mix of substrates.

Better predictions will be achieved if biochemical and also cellular level processes (e.g. metabolic reactions) are considered in the next generation of water treatment models [START_REF] Batstone | Metabolic modelling of mixed culture anaerobic microbial processes[END_REF]. Metabolic models are able to accurately predict the specific VFAs produced by a mixed-culture depending on the components of the input medium [START_REF] Regueira | A metabolic model for targeted volatile fatty acids production by cofermentation of carbohydrates and proteins[END_REF]. We could envisage the coupling of these models -VFA production by mixed-culture and treatment by microalgae -to optimize production of hydrogen constrained by the capacities of the waste treatment system, or to predict in advance the necessity of adding another carbon substrate such as glycerol. Metabolic knowledge could also give a strong foundation for estimating possible medium conditions take could lead to undesirable emissions, such as N 2 O, and which are not well modeled through the current approaches (Casagli et al., 2021a).

Conclusion

The metabolic model developed in this work for Chlorella accurately predicts growth under autotrophic, heterotrophic and mixotrophic conditions with acetate, butyrate, glucose and glycerol for more than a hundred experiments from the literature. Covering a large range of conditions, strains and substrates, the predictive capacity of this reduced metabolic model remains remarkable. Moreover, to lift the inhibition exerted by the presence of butyrate, the optimal addition of different substrates in the medium has been predicted by the model. Optimising the cultivation conditions reduce the time to reach the regulation threshold from 16 to less than 2 days.

Thanks to this approach, it will become possible to streamline two-stage water treatment strategies, and to recycle, carbon, nitrogen and phosphorus into the microalgal biomass. In particular, this approach highlights noteworthy synergies between waste molecules, and the important role of of the metabolic network for future models. The anticipated management of these molecules could improve productivity significantly.

Figure 1 :

 1 Figure 1: Considered metabolic subnetworks to represent growth of Chlorella on a mixture of glycerol, glucose, acetate and butyrate

Figure 2 :

 2 Figure 2: Comparison of the model against experimental data used for the calibration of glucose and glycerol kinetic parameters. Lines represent the average of 100 simulations and the colored region represents ±1 standard deviation of estimated concentrations using Markov Chain Monte Carlo method.

Figure 3 :

 3 Figure 3: Model simulation and experimental data with two substrates. From left to right: glycerol and glucose; butyrate and acetate. The line represents the average of 100 simulations and the colored region represents ±1 standard deviation of estimated concentrations using Markov Chain Monte Carlo method.

Figure 4 :

 4 Figure 4: Taylor Diagram where each point represents the Pearson correlation coefficient and a normalized standard deviation of one experiment and model simulation. The semi circles centered at standard deviation 1.0 show the root-mean-square error.

Figure 5 :

 5 Figure 5: Validation of model predictions based on experimental data of butyrate, glycerol, glucose, acetate and biomass. All p-values for the regression are below 10 -3 . R 2 for the lines are, respectively, 0.97, 0.71, 0.95, 0.97 and 0.64. Darker colors represent a concentration of data points.

Figure 7 :

 7 Figure 7: Concentrations of butyrate and acetate over time in different cultivation conditions.Optimal fed-batch with glycerol (blue), optimal fed-batch with glucose (red), batch with optimal addition of glycerol and glucose (orange), batch with only butyrate and acetate as substrates (green).

  considering the exponential growth phase. During autotrophic growth, a strong activity takes place in the chloroplast subnetwork where photosynthesis occurs. The greatest fluxes are associated to the fixation of CO 2 by RUBISCO, the conversion to 3PG (3-phosphoglycerate) and the conversion of 3PG to glycerate-3-phosphate (GAP). GAP is then mainly transported toward the cytoplasm, where it is injected into the glycolysis. From then on, it enables the synthesis of precursor metabolites that are necessary for the synthesis of functional biomass composed of proteins, DNA, RNA, chlorophyll, carbohydrates, and lipids. During pure heterotrophic growth, in the dark, no reaction occurs in the chloroplast subnetwork. Considering a a mix of acetate or butyrate, the largest fluxes are concentrated in the glyoxysome subnetwork where carbon substrates are converted into succinate. The synthesised succinate is exported from the glyoxysome and injected into the Krebs cycle. The latter then enables the synthesis of precursor metabolites for the production of functional biomass. To ascend the glycolysis, the anaplerotic pathways are in an upward direction.

Figure 8 :

 8 Figure 8: Metabolic charts showing the fluxes of an optimized batch. Above, A, at initial concentration of acetate and butyrate with optimal conditions of glucose and glycerol. Below, B, 30 hours after the beginning of the batch. The widths are linearly proportional to the calculated flux.

Table 1 :

 1 Considered experiments for each substrate. In parenthesis, the number of experiments in mixotrophic conditions out of the total experiments.

	Substrate	# exp. Species	References
	Glycerol	12 (9) C. sorokiniana	León-Vaz et al. (2019)
			C. sp.	Sen and Martin (2018)
			C. protothecoides	Chen and Walker (2011), O'Grady and Morgan (2011)
			C. vulgaris	Ma et al. (2016)
	Glucose	22 (11) C. sorokiniana	León-Vaz et al. (2019), Li et al. (2013, 2014)
			C. protothecoides	Espinosa-Gonzalez et al. (2014), Chen and Walker (2011)
			C. pyrenoidosa	Ogbonna et al. (1997)
	Glucose/Glycerol 2 (2)	C. protothecoides O'Grady and Morgan (2011)
				León-Vaz et al. (2019), Turon et al.
	Acetate	40 (25) C. sorokiniana	(2015a,b), Chen et al. (2017a,b), Xie
				et al. (2020a)
			C. sacchrarophila Xie et al. (2020b)
	Butyrate	10 (4) C. sorokiniana	Turon et al. (2015b,a)
	Acetate/Butyrate 23 (6) C. sorokiniana	Turon et al. (2015b,a)
	Autotrophic	13	C. sorokiniana	Li et al. (2014), León-Vaz et al. (2019), Turon et al. (2015b,a)
			C. sp.

Table 2 :

 2 List of the macroscopic reactions in each respective subnetwork. For biomass production, the stoichiometric values differ if the nitrogen source is nitrate or ammonium.using minimal hypotheses, and when possible applying a mass action law. A list of all sub-networks and the macroscopic reactions can be found at Table2. Here carbon compounds are converted to succinate, also allowing the production of glucose from lipids. In this compartment, fatty acids can be used as a source of energy and carbon for growth is produced when no photosynthesis takes place. Two EFMs have been achieved for this subnetwork (MR1 and MR2). In the glyoxysome, butyrate and acetate are converted into acetyl-CoA, which is in turn converted, via the glyoxylate cycle, into succinate. The succinate then enters the cytosol and is injected into the Krebs cycle, thus producing the different metabolites necessary for the synthesis of biomass.

	2.3.2. Glyoxysome subnetwork
	The glyoxysome pathway consists of 26 reactions, from which 8 are exchange
	reactions. The glyoxysome is the peroxysome compartment where the glyoxylate
	cycle occurs.

Table 3 :

 3 Kinetic parameters obtained after model calibration. Results of sensibility analysis ( σ ∆p y and standard deviation (SD) of σ ∆p y . Parameters calibrated in this work A and in Baroukh et al. (2017) B . MB: mole of biomass

	Parameter Value	Unit	σ ∆p y	SD σ ∆p y	Ref.
	k M R1	3.79 10 -1 M.h -1 .M B -1	4.5910 -03	2.5210 -03 B
	KS M R1	5.50 10 -5 M	9.2010 -05	5.2410 -03 B
	k M R2

Acknowledgment

The authors acknowledge the support of the ControlAB ANR project (ANR-20-CE45-0014) and of the BIOMSA Ademe project. The UCAJEDI and EUR DS4H investments in the Future projects managed by the National Research Agency (ANR) with the reference numbers ANR-15-IDEX-0001.


Antoine Ishak

Rodolphe Sonnier

Belkacem Otazaghine

Claire Longuet
email: claire.longuet@mines-ales.fr

Silazanes, a novel flax fibers functionalization: Effect on silicone-based composites

Keywords: A, Natural fibers A, Polymer-matrix composites (PMCs) B, Interface/interphase B, Fibre/matrix bond

Introduction:

In recent years and due to the growing environmental concern, the use of natural fibers instead of synthetic fibers as reinforcement for polymer matrices have become a solution of great interest to introduce new composites with reduced environmental footprint [START_REF] Yan | Flax fibre and its composites -a review[END_REF][START_REF] Pantaloni | Interfacial and mechanical characterisation of biodegradable polymer-flax fibre composites[END_REF][START_REF] Liu | Structural biocomposites from flax -Part II: The use of PEG and PVA as interfacial compatibilising agents[END_REF][START_REF] Cantero | Effects of fibre treatment on wettability and mechanical behaviour of flax/polypropylene composites[END_REF][START_REF] Assarar | Influence of water ageing on mechanical properties and damage events of two reinforced composite materials: flax-fibres and glass-fibres[END_REF][START_REF] Summerscales | A review of bast fibres and their composites. Part 1 -Fibres as reinforcements[END_REF][START_REF] Song | Viscoelastic and thermal behavior of woven hemp fiber reinforced poly(lactic acid) composites[END_REF][START_REF] John | Biofibres and biocomposites[END_REF][START_REF] Yan | Improving the mechanical properties of natural fibre fabric reinforced epoxy composites by alkali treatment[END_REF][START_REF] Joshi | Are natural fiber composites environmentally superior to glass fiber reinforced composites? Compos Part A[END_REF][START_REF] Wambua | Natural fibres: can they replace glass in fibre reinforced plastics?[END_REF].

Flax fibers are one of the most widely used natural fibers [START_REF] Yan | Flax fibre and its composites -a review[END_REF], due to their advantages such as: low density [START_REF] Gall | Recommended flax fibre density values for composite property predictions[END_REF], good mechanical properties [13,[START_REF] Lotfi | Natural fiber-reinforced composites: a review on material, manufacturing, and machinability[END_REF], better specific tensile strength compared to glass fibers [START_REF] Peças | Natural fibre composites and their applications: a review[END_REF] and reasonable cost [START_REF] Gautreau | Fibre Individualisation and Mechanical Properties of a To cite this version[END_REF][START_REF] Thakur | Review: raw natural fiber-based polymer composites[END_REF]. Furthermore, flax fibers have a lower environmental impact than glass fibers depending on the energy source in the producing country [START_REF] Gomez-Campos | Flax fiber for technical textile: a life cycle inventory[END_REF][START_REF] Deng | Assessing the environmental impact of flax fibre reinforced polymer composite from a consequential life cycle assessment perspective[END_REF].

Flax fibers have been used to reinforce many kinds of thermoplastic and thermosetting polymer matrices such as polypropylene, epoxy, bioepoxy, bio-phenolic resin [13], poly-β-hydroxybutyrate [START_REF] Gautreau | Fibre Individualisation and Mechanical Properties of a To cite this version[END_REF], polylactic acid [START_REF] Bax | Impact and tensile properties of PLA/Cordenka and PLA/flax composites[END_REF]. Moreover, the surface treatment of flax fibers such as atmospheric pressure plasma [START_REF] Bozaci | Effects of the atmospheric plasma treatments on surface and mechanical properties of flax fiber and adhesion between fiber-matrix for composite materials[END_REF], epoxy [START_REF] Van De Weyenberg | Influence of processing and chemical treatment of flax fibres on their composites[END_REF], alkali and silane treatments [START_REF] Van De Weyenberg | Influence of processing and chemical treatment of flax fibres on their composites[END_REF][START_REF] Perremans | Investigation of the tensile behavior of treated flax fibre bio-composites at ambient humidity[END_REF] was performed in order to improve the interfacial adhesion between flax fibers and polymer matrices. This in turn enhances the mechanical properties of the composites.

To increase even more the interfacial adhesion, the fibers treatment with coupling agents bearing a functional group able to react with the polymer matrix is very effective, it leads to a further increase in the mechanical properties [START_REF] Yan | Flax fibre and its composites -a review[END_REF]. Silane coupling agents are commonly used to modify the surface of cellulose fibers due to their ability to react with hydroxyl groups of cellulose chains. This modification leads to an improvement in the mechanical properties and a greater interfacial compatibility of the cellulose fibers with the matrix by forming covalent bonds, which contributes to better adhesion [START_REF] Gadhave | Surface modification of cellulose with silanes for adhesive application: review, open[END_REF][START_REF] Abdelmouleh | Interaction of silane coupling agents with cellulose[END_REF]. For example, Abdelmouleh et al. [START_REF] Abdelmouleh | Short natural-fibre reinforced polyethylene and natural rubber composites: effect of silane coupling agents and fibres loading[END_REF][START_REF] Abdelmouleh | Modification of cellulose fibers with functionalized silanes: effect of the fiber treatment on the mechanical performances of cellulose-thermoset composites[END_REF] have modified cellulose fibers by (3-aminopropyl)triethoxysilane (APTS) and 3-(trimethoxysilyl) propyl methacrylate (MPS) to reinforce epoxy and natural rubber composites, respectively. These modified fibers bearing amine or methacrylate groups can then participate to the formation of a three-dimensional connected network between fibers and matrix by condensation or radical reactions. The formation of covalent bonds between the fibers surface and the polymer chains of the matrix allows the improvement of the fibers/matrix interface and mechanical properties of the obtained composite. The surface treatment of cellulose fibres with MPS and APTS increases the Young modulus (E) up to 10 MPa comparing with 3.6 MPa for the untreated fibers-natural rubber composite and to 3.2 GPa instead of 2.55 GPa for the untreated fibres-epoxy composite.

Since flax fibers contain about 70 wt% of cellulose [START_REF] Komuraiah | Chemical composition of natural fibers and its influence on their mechanical properties[END_REF][START_REF] Dittenber | Critical review of recent publications on use of natural composites in infrastructure[END_REF][START_REF] Bledzki | The effects of acetylation on properties of flax fibre and its polypropylene composites[END_REF], it is possible to take advantage of the presence of cellulose hydroxyl groups = 5 mm, Ø=16 ± 2 μm) were obtained from Terre de Lin Groupe France.

All the chemical products in this study were used as received without any further purification.

Preparation and pre-treatment of flax fibers

2 g of pristine flax fibers were dipped into a 200 ml of NaOH solution (0.5 M) at 70 • C for 3 h under stirring. Then, a neutralization process was performed with 200 ml of acetic acid (0.6 M) for 30 min at 70 • C followed by rinsing with deionised water until pH neutral to remove all acetic acid traces. The obtained pre-treated fibers were dried for 2 h at 60 • C.

Functionalization process of flax fibers

4 g of DVTS was dissolved in 16 g of (DMSO/Toluene) (35/65 wt%) co-solvent mixture. The prepared solution was added into a bottom flask of 50 ml fitted with a reflux condenser. Then, 0.6 g of pre-treated fibers were dipped in this solution for 3 h at 80 • C under stirring. Finally, the fibers were filtered, squeezed by paper towel and washed twice with 20 ml of toluene over 10 min to remove non-reacted and non-grafted molecules.

After reaction, the obtained flax fibers are functionalized with vinyl groups via -Si-O-C-covalent bonds due to the reaction between DVTS and hydroxyl groups of the flax fibers. Furtat et al. [START_REF] Furtat | Synthesis of fluorinemodified polysilazanes: via Si-H bond activation and their application as protective hydrophobic coatings[END_REF] have mentioned a proposed mechanism for the decomposition of Si-N backbone of polysilazanes in the presence of hydroxyl groups: the hydroxyl function reacts with the Si-N bond to forms a Si-O-C bond, releasing NH 3 gas. When DVTS is used, for each molecule of coupling agent that reacts, two vinyl groups are assumed to graft onto the flax fibers (Fig. 1).

The functionalization rate (R F ) was calculated according to the percentage ratio between the mass of grafted molecule and that of corresponding pre-treated fibers.

R F = (

Functionalized fibers weightpretreated fibers weight pretreated fibers weight

) × 100
This calculation is not accurate but gives an indication of the amount of ((CH 3 ) 2 Si-CH = CH 2 ) groups that are grafted onto the fibers during the functionalization process.

Preparation of silicone composite

The composites based on a crosslinked silicone elastomer were prepared using short flax fibers.

The silicone matrix was obtained by mixing a PDMS part bearing Si-H groups (0.39 wt%) and a PDMS part bearing vinyl groups (0.22 wt%). The system reacts by hydrosilylation and forms a cross-linked network in the presence of Karstedt's catalyst at room temperature. The vinyl groups grafted on the flax fibers can participate to the formation of this network by reaction with one part of the Si-H functions to form covalent bonds with PDMS chains of the cross-linked network (Fig. 2).

A series of silicon-based composites were prepared by mixing 3 g of each type of flax fibers (pristine /pre-treated/functionalized) with 180 g and 18 g of BLUESIL RTV 141 A and B, respectively. Then, the obtained mixtures were molded using a dog bone shape mold, after removing all gas bubbles under vacuum (Fig. 3). The mold was kept at room temperature for 24 h before samples were removed. The fibers / silicone matrix ratio for the composite products was 1.5 /100 (wt/wt).

Measurements

Fourier transform infrared spectroscopy (FTIR)

Fourier transform infrared spectra were obtained using a Bruker VERTEX 70 spectrometer used in attenuated total reflectance mode, at a resolution of 4 cm -1 , with 32 scans between 400 and 4000 cm -1 .

Scanning electron microscope coupled with energy dispersive X-ray spectrometry (SEM-EDS)

The fibers section was analysed using a scanning electron microscope (FEI Quanta 200 FEG). After being cut with a single edge blade, the samples were placed on a vertical sample holder under high vacuum at a voltage of 12.5 kV and a working distance of 10 mm. To locate the presence of vinyl groups associated with silicon element grafted on the flax fibers, SEM analysis was coupled with an energy dispersive X-ray spectroscopy, EDS, (Oxford INCA Energy system).

Scanning electron microscope was also used to perform a crosssectional survey of the composites in order to observe the interface between the fibers and the silicone matrix. The samples were prepared using cryofracture method.

Swelling of flax fibers

Swelling behaviour of elementary flax fibers in DMSO, Toluene and in order to modify the surface fibers with a coupling agent.

From this point of view, silazane functions may be useful because they have the same ability as silanes to react with hydroxyl functions due to the reactivity of Si-NH groups [START_REF] Steffen | Fire-retardant hybrid thermosetting resins from unsaturated polyesters and polysilazanes[END_REF]. Nguyen [START_REF] Nguyen | Revêtement polysilazane à activés antibactérienne -Thèse-école doctorale science chimiques (ED250) Laboratoire Matériaux Polymères Interfaces Environnement Marin[END_REF] have mentioned that silazane molecules have an excellent adhesion on a wide variety of polar substrates. He has attributed this strong adhesion to the formation of covalent bonds Si-O by reaction of the Si-NH groups with the hydroxyl groups of the substrate, releasing NH 3 as a by-product.

Nevertheless, to the best of our knowledge, silazane compounds have never been used as coupling agents for natural fibers using as reinforcement in polymer composites. Such compounds appear to be, nevertheless, very attractive for the surface modification of hydroxylrich cellulosic fibers.

Silazanes are usually used as ceramic precursors to prepare coatings [START_REF] Kim | Melt-processable aggregated boron nitride particle via polysilazane coating for thermal conductive composite[END_REF][START_REF] Tangermann-Gerk | Laser pyrolysis of an organosilazane-based glass/ZrO2 composite coating system[END_REF][START_REF] Xiao | Fabrication of ceramic coatings from polysilazane/aluminum: effect of aluminum content on chemical composition, microstructure, and mechanical properties[END_REF]. As an example, Shayed et al. [START_REF] Shayed | Carbon and glass fibers modified by polysilazane based thermal resistant coating[END_REF] have coated glass and carbon fibers with silazane molecules to protect the fibers against thermooxidation. Polysilazanes can be also involved for the development of light-emitting materials [START_REF] Wang | Synthesis, characterization, and properties of novel phenylene-silazane-acetylene polymers[END_REF], fire-retardant polymers [START_REF] Gardelle | Thermal degradation and fire performance of polysilazane-based coatings[END_REF] or transparent reinforced composites [START_REF] Seo | Synthesis of refractive index tunable silazane networks for transparent glass fiber reinforced composite[END_REF].

On the other hand, silicone rubber is highly used for various applications such as automotive, aerospace industry, electronics, injection molding [START_REF] Shit | A review on silicone rubber[END_REF] and medical industry [START_REF] Rahimi | Review on rubbers in medicine: natural, silicone and polyurethane rubbers[END_REF] due to its excellent properties including high elasticity, thermal stability [START_REF] Meng | Structure, morphology, and mechanical properties of polysiloxane elastomer composites prepared by in situ polymerization of zinc dimethacrylate[END_REF], good chemical resistance [START_REF] Morari | Electrical conductivity and electromag-netic shielding effectiveness of silicone rubber filled with ferrite and graphite powders[END_REF] and electrical insulation [START_REF] Morari | Electrical conductivity and electromag-netic shielding effectiveness of silicone rubber filled with ferrite and graphite powders[END_REF][START_REF] Amin | Hydrophobicity of silicone rubber used for outdoor insulation (an overview)[END_REF]. The reinforcement of silicone rubbers with functionalized flax fibers would be useful in order to improve the mechanical properties of these materials and thus, expand their application field.

In the present study, 1,3-divinyltetramethyldisilazane (DVTS) was used to modify flax fibers. Thanks to the reaction between hydroxyl groups of the flax and silazane groups, functionalized fibers bearing vinyl groups were prepared and used to reinforce a silicone rubber matrix (polydimethylsiloxane (PDMS)). Indeed, in a second step the vinyl groups of the functionalized flax fibers can react with Si-H functions of the silicone chains in presence of Karstedt's catalyst to form covalent bonds between fibers and matrix. The effect of conditions on the grafting reaction has been studied in detail. Moreover, the effect of the functionalization of flax fibers on the mechanical properties for the flax/silicone composite was investigated.

Materials and methods

Materials

1,3-divinyltetramethyldisilazane 97 % (DVTS), was purchased from ABCR, dimethylsulphoxide (DMSO) 99.9 % and Toluene 99.8 % were supplied by Sigma-Aldrich, Acetic acid was purchased from Fisher Scientific Co. Sodium hydroxide (NaOH) was purchased from PanReac AppliChem. A two-component silicone elastomer BLUESIL RTV 141 A&B was kindly provided by Elkem Silicone France. Short flax fibers (L co-solvent (DMSO/Toluene) (35/65 wt%) mixture was observed between two glass plates by optical microscopy in transmission mode with a Laborlux 11 Pol (Leica, Germany) equipped with a mono-CDD Sony digital camera (resolution 1600 × 1200 pixels) and an image acquisition software Archimed® (Microvision Instruments, France). The flax fiber was fixed on the bottom glass plate between two double-sided tapes with around 1 cm apart. The solvent was then injected by capillarity between the two glass plates.

A measurement of apparent diameter was carried out before and after wetting, five times along the fiber in order to calculate an average diameter. Then, the swelling rate was calculated as follow:

This measure was done for all samples with each solvent four times. The standard deviation was calculated according to the swelling values obtained for these four samples.

Thermogravimetric analysis (TGA)

A thermogravimetric analyser instrument (Setaram Setsys) was used to study the fibers thermo-oxidative decomposition. In alumina ceramic crucibles, 10 mg ± 1 mg of each kind of fibers (pristine, pre-treated and functionalized fiber) was heated from 30 to 900 • C (heating rate of 10 • C/min) under air atmosphere (40 ml/min). During the analysis, 15 min of isotherm at 100 • C was done in order to remove adsorbed water and facilitate the comparison between samples.

Rheology measurements

Rheological instrument MCR 702 (Anton Paar) equipped with parallel plates (PP25, 25 mm diameter) was used to investigate and measure the storage and loss modulus of the silicone matrix during the curing reaction of the different formulations. The oscillation experiments were conducted at constant temperature, frequency and deformation of 25 ± 1 • C, 1 Hz and 1 %, respectively for 24 h.

Tensile test

The tensile test of prepared composites was performed on a tensile machine (Zwick/Roell) equipped with a 2.5 kN load cell and a clip-on extensometer with a reference length of 30 mm at room temperature (25 • C). The equipment is calibrated according to ISO 7500-1 and ISO 9513 methods for force and displacement sensors, respectively.

A crosshead speed of 50 mm/min was applied. The Young's modulus was determined by tangent method between 0.05 and 0.25 % of elongation. Each test was carried out at least 4 times.

Results and discussion

Solvent and pre-treatment step effect on the functionalization process

Firstly, the functionalization process was applied directly on the pristine fibers without pre-treatment step using toluene as a solvent. The obtained functionalization rate was negligible (0.1 %) and the reaction almost did not occur. This result is assumed to be due to the low accessibility of the silazane agent to the hydroxyl groups onto flax fibers when toluene is used.

Swelling rate% = (

Average diameter after wetting -Average diameter before wetting Average diameter before wetting

) × 100
The hypothesis is that the swelling of flax fibers is linked to their affinity with the used solvent. In order to define the solvent influence on the fiber swelling [START_REF] Fidale | Cellulose swelling by aprotic and protic solvents: what are the similarities and differences?[END_REF], the measurement of apparent fiber diameter for different solvents was performed using microscopy analysis. Pristine fibers were impregnated with DMSO, DMSO/toluene (35/65 wt%) mixture and toluene. DMSO was chosen because of its high polarity and toluene because it is a very good solvent for DVTS. The microscopy analysis allows measuring the average diameter of fiber before and after solvent wetting (Fig. 4).

The average of flax fiber diameter increased from 16. when it is wetted with toluene (Fig. 5a).

In order to improve the RF, the toluene was replaced by the solvent mixture, DMSO/Toluene (35/65 wt%) tested previously, and RF value increased up to 3.35 % ± 0.4 in the same conditions. The diameters difference measured by microscopy can explain the increase of RF with the DMSO/toluene mixture in comparison to toluene used alone. However, DMSO alone cannot be used because of its immiscible with DVTS and the DMSO/toluene ratio (35/65 wt%) is the highest at which DVTS remains soluble.

Finally, the pristine fibers were replaced by pre-treated ones. The pre-treatment by NaOH solution removes one part of the lignin product from flax fibers [START_REF] Keshav | Bioconversion of alkali delignified cotton stalk using two-stage dilute acid hydrolysis and fermentation of detoxified hydrolysate into ethanol[END_REF]. Then, the RF increased up to 8.3 % ± 1.1 (Fig. 5 b). Generally, the lignin removal from the natural fibers promotes the availability of the cellulose-silane interaction sites. Furthermore, the pre-treated fibers surface is rougher than the surface of pristine ones which results in a larger effective contact surface. Moreover, the lignin elimination permits to facilitate the diffusion of the coupling agents inside the fibers [START_REF] Valadez-Gonzalez | Chemical modification of henequen fibers with an organosilane coupling agent[END_REF]. For example, Valadez-Gonzalez et al [START_REF] Valadez-Gonzalez | Effect of fiber surface treatment on the fiber-matrix bond strength of natural fiber reinforced composites[END_REF] have

FTIR spectroscopy

DVTS, pristine, pre-treated and functionalized fibers were analysed by FTIR spectroscopy to confirm the reaction between silazane and hydroxyl groups, and to detect the presence of vinyl groups on the functionalized fibers surface (Fig. 7). The bands corresponding to the vinyl group (C--C stretching at 1592 cm -1 , --C -H bending at 830 cm -1 ), the Si-CH 3 groups at (1250 cm -1 /780 cm -1 ) and the silazane groups (N -H at 1175 cm -1 , Si-N at 925 cm -1 ) are highlighted in the DVTS spectrum.

The pristine and pre-treated fibers have exactly the same spectrum, the bands corresponding to the hydroxyl group (-OH at 3330 cm -1 ) and the ether group (-C -O -C-at 1025 cm -1 ) of the cellulose were observed. Concerning the functionalized fibers spectrum, vinyl and Si-CH 3 bands are present while the silazane bands are not observable. The disappearance of the characteristic bands of the SiN-H and Si-NH bonds confirms the elimination of the silazane functions, while the presence of vinyl and Si-CH 3 bands confirms the grafting of Me 2 Si-CH--CH 2 on the flax fibers surface (Fig. 1).

Scanning electron microscopy (SEM) with energy dispersive X-ray spectroscopy (EDS)

A cross-section of pristine, pre-treated and functionalized fibers were analysed by SEM/EDS with silicon mapping spectroscopy to confirm the presence of the grafted silicon part after DVTS grafting (Fig. 8). As it is clearly in EDS images, silicon is naturally present in flax fibers but only in trace amounts. The percentage of Si was ≤1 wt% according to the statistical calculation of EDS device. On the other hand, for the pretreated fibers, silicon is almost non-present due to the pre-treatment with NaOH solution. The pre-treatment seems to remove the main silicon compounds from the fibers. Finally, for the functionalized fibers, silicon is highly concentrated about 4.1 wt% ± 0.7. The presence of this silicon is implicitly linked to the presence of the grafted vinyl groups.

Thermogravimetric analysis (TGA)

Fig. 9 shows the TGA/DTG curves of pristine, pre-treated and functionalized fibers. The initial decomposition temperature (T in ), the temperature of the maximum degradation rate (T max ), and the ash yield (Y ash ) are listed in (Table 1). A clear difference was observed in T in and T max between the pristine fiber on the one hand and the pre-treated and functionalized fibers on the other hand. The temperature at 5 wt% weight loss of pristine fiber is 256 • C, while for pre-treated and functionalized fibers, it is almost the same value 300 and 298 • C, respectively. Moreover, the highest rate of degradation of pristine fiber occurs at 355 • C, while for pre-treated and functionalized fibers, it occurs at 340 and 342 • C, respectively. This shift in T in and T max can be attributed to lignin removal from the flax fiber during the pre-treatment process. Indeed, it is mentioned in the literatures that alkali treatment of flax fibers leads to a slight improvement in thermal stability [START_REF] Gassan | Thermal degradation of flax and jute fibers[END_REF] which corresponds to the increase of T in for the pre-treated fibers. Furthermore, DTG curve of pristine fiber displays a second peak at 490 • C while, this peak is too weak in the curves of pre-treated and functionalized fibers. The second peak represents the degradation of lignin [START_REF] Van De Velde | Thermal degradation of flax: the determination of kinetic parameters with thermogravimetric analysis[END_REF]. The decrease of this peak confirms the lignin removal for pre-treated and functionalized fibers.

The value of Y ash gradually increases from pristine and pre-treated fibers to functionalized ones. The lignin removal is probably responsible for the increase in Y ash concerning the pre-treated fibers. While TGA was performed in air (aerobic pyrolysis), the oxygen allows the formation of silicon dioxide compounds from the pyrolysis of functionalized fibers which leads to a further Y ash increase [START_REF] Wright | Thermal oxidation of silicon in a residual oxygen atmosphere -the RESOX process -For self-limiting growth of thin silicon dioxide films[END_REF][START_REF] Lin | Facile fabrication of superhydrophobic and flameretardant coatings on cotton fabrics via layer-by-layer assembly[END_REF]. The silicon present in the functionalized fibers at a concentration of 4.1 wt% should react with oxygen to increase Y ash up to 6.4 wt% if the reaction was complete. In our results, the presence of this initial amount of silicon increases Y ash up to 5.8 wt%. This result is therefore of the same order as expected. Fig. 7. FTIR spectra of DVTS, pristine and functionalized flax fibers. used vinyltris (2-methoxy-ethoxy) silane as a coupling agent to modify henequen fibers, it was found that the amount of silane adsorbed onto the alkali treated fibers was higher than that absorbed by the pristine ones.

The effect of reaction conditions on the functionalization process

The effect of solution concentration, reaction time and temperature on the functionalization process were investigated.

Three series of pre-treated fibers were functionalized with various concentrations of DVTS, for different reaction time and at several temperatures (Table S1).

In series 1, pre-treated fibers were functionalized with different concentrations of DVTS at 80 • C for 3 h. It was found that the R F increases up to around (8.3 %) when the DVTS concentration increases up to 20 wt%. The relationship between R F and the DVTS concentration is linear in this range. After this value, no further increase in R F was observed (Fig. 6 a).

In series 2, pre-treated fibers were functionalized with different reaction times at 80 • C considering 20 wt% DVTS concentration. The R F increases during the first 3 h. The relationship between R F and reaction time is also linear in this range. Beyond this time, any prolongation of reaction time is useless. (Fig. 6 b).

In series 3 and 4, pre-treated fibers were functionalized at several temperatures for 2 and 4 h, respectively with 20 wt% DVTS concentration. The same behaviour was observed, the R F increases with the rise in the reaction temperature. For 2 h of reaction, the R F is approximately stable up to 60 • C. Thereafter, a sharp increase appears with the rise in temperature. For 4 h of reaction, the same behaviour is observed but less intense (Fig. 6c). R F of 7.5 % could be achieved at 60 • C for 4 h and at 80 • for 2 h.

According to these results, the highest R F achieved was (8.3 %) at 80 • C for 3 h in the DMSO/toluene mixture using pre-treated flax fibers with 20 wt% DVTS concentration that are considered as the optimal conditions for the functionalization process.

Characterizations of functionalized fibers

All characterizations were performed on the functionalized fibers prepared under the optimal conditions allowing to obtain the maximum R F (20 wt% of DVTS, 80 • C, 3 h, in the DMSO/Toluene co-solvent using pre-treated flax fibers) and after filtering, squeezing and washing twice with toluene during 10 min.

Effect of fibers presence on the curing reaction of PDMS formulations

Formulations of PDMS matrix, PDMS matrix with pristine fibers and PDMS matrix with functionalized fibers were prepared and then studied by rheological measurements in order to follow the evolution of the storage and loss modulus of the silicone system during the curing reaction (Fig. 10). Before the start of the crosslinking reaction, the Log G' value for the silicone matrix filled with pristine or functionalized fibers is greater than the unfilled one. This difference of rheology is attributed only to the presence of the flax fibers. After 3 h for unfilled matrix and 4 h in presence of flax fibers, Log G ′ starts to increase sharply until 12 h of reaction. Then, Log G' get close values for all formulations until the end of the reaction. Thus, the presence of the flax fibers has little influence on the crosslinking kinetic of the silicone matrix. Moreover, the grafting of vinyl groups on the flax fibers does not affect this crosslinking either. In addition, before the start of the crosslinking reaction, the Tan δ value for the unfilled matrix is greater than those of the formulations filled with pristine or functionalized fibers. At the gelation point (≈ 9 h), the Tan δ values for all silicone formulations are very close and this until the end of the crosslinking reaction. These results also confirm that the presence of flax fibers and their grafting with vinyl groups do not disturb the crosslinking reaction of the silicone system.

Characterisations of prepared composites

Pristine, pre-treated and functionalized fibers -reinforced silicone composites were characterized in order to observe the influence of flax fibers functionalization process on the adhesion with the matrix following the improvement of mechanical properties.

Tensile test

The values of maximum tensile force stress (Fmax), elongation at maximum force (dL (Fmax)), breaking point, elongation at break and Young modulus (E) obtained by tensile test for the silicone composites filled with the different flax fibers are listed in (Table S2).

Stress strain curves of samples (Fig. 11) displayed the difference between prepared composites in terms of mechanical properties compared to the fiber-free matrix.

Young modulus E (Fig. 12) increased for the composites with pristine and pre-treated fibers from 0.62 ± 0.05 MPa to 2.42 ± 0.12 MPa and 2.5 ± 0.08 MPa, respectively. Both of them have approximately the same Young modulus E value, while composite with functionalized fibers has a much higher Young modulus E, close to 3.6 ± 0.5 MPa. The flax fibers increased the stiffness of elastomer composite about 4 times but the functionalized one increased it about 5.8 times. The difference in Young modulus E between the pristine fibers-reinforced composite and the functionalized fibers-reinforced composite is about 40 %. Oppositely, elongation values at maximum force decreased for the composites filled with pristine and pre-treated fibers from 91 ± 7 % to 53 ± 16 % and 45 ± 10 %, respectively. The elongation values for both composites are very close to each other, while composite with functionalized fibers has a highly decreased elongation about 27 ± 4 %. These values are fully consistent with the evolution of the Young E modulus values. Unlike Young modulus and elongation at maximum force (dL%), breaking point values of composites with pristine and pre-treated fibers are not identical and not even close to each other. Breaking point value of the composites has a progressive increase with pristine, pre-treated and functionalized fibers, respectively. Breaking point increased from 470 kPa for fiber-free silicone matrix to 550, 750 and 910 kPa for pristine, pre-treated and functionalized fibers-reinforced composite, respectively. It means that the pre-treatment of flax fibers increases the tensile strength of the silicone matrix but the functionalized ones increase it even more. Wu and Zuo [START_REF] Wu | Silicone rubber composites modified by chopped basalt fibers treated with coupling agent[END_REF], have prepared basalt fibers-reinforced silicone composites, they found that the treatment of basalt fibers by APTS coupling agent improves the compatibility between fibers and silicone, resulting in an increase in the tensile strength of composites.

Scanning electron microscope (SEM)

SEM analysis (Fig. 13) was used to observe the fibers-matrix interface in cross-section of broken composites by cryofracture method. For composites reinforced with pristine and pre-treated flax fibers (Fig. 13 a and b, respectively), the adhesion between fibers and the matrix is very weak. The traces of fibers pull-out at the matrix surface are easily noticeable. The compatibility of pristine and pre-treated fibers with the silicone matrix is almost the same; the fibers are well dispersed in the matrix with very poor adhesion. Oppositely, there are not voids at the interfacial zone between fibers and matrix in composite reinforced with functionalized fibers. The ability of vinyl groups to react with the matrix during the curing process and to form covalent bonds between fibres and the matrix can explain this result. As it is showed in (Fig. 13c) the fibers are integrated in the matrix with a high level of adhesion. This could support the tensile test results that showed a great improvement in the mechanical properties of the functionalized fibers-reinforced composite.

Conclusion

Flax fibers have been successfully functionalized with vinyl groups using DVTS.

The used solvents nature for the grafting reaction and the use of a fibers alkaline pre-treatment play a great role for the functionalization efficiency. The solvent can promote the swelling of the flax fibers while the pre-treatment removes lignin from fibers, both of which increase the functionalization rate.

Temperature, reaction time and DVTS concentration have also an influence on the R F of flax fibers. The optimum R F is 8.3 % and it was achieved at 80 • C for 3 h with 20 wt% of DVTS.

The functionalized fibers are effective to prepare a composite material based on silicone elastomer with high fibers/matrix adhesion.

The functionalized fibers is supposed to participate to the formation of the cross-linked network due to the presence of the vinyl groups on flax that can react with PDMS chains.

The mechanical properties increase when the silicone matrix was reinforced with functionalized flax fibers in comparison with pristine ones.

This study permitted to confirm the possibility to functionalize efficiency natural fibers via covalent bonding, using a silazane molecule.

These first results open novel way possibilities in order to functionalize easily natural fibers with numerous added functional groups.

Fig. 1 .

 1 Fig. 1. Mechanism proposed for DVTS grafting on the flax fibers.

Fig. 2 .

 2 Fig. 2. Scheme representing the crosslinking step with formation of covalent bonds between silicone matrix and the functionalized flax fibers.

Fig. 3 .

 3 Fig. 3. Dog bone test specimens obtained for the different flax/silicone formulations. (a) fiber-free silicone matrix, (b) silicone matrix/pristine flax fibers, (c) silicone matrix/pre-treated flax fibers, (d) silicone matrix/functionalized flax fibers.

Fig. 4 .

 4 Fig. 4. Microscopy images of pristine flax fibers before and after wetting with DMSO.

Fig. 5 .Fig. 6 .

 56 Fig. 5. (a) swelling measurement of pristine fiber impregnated with toluene, dmso and (dmso/toluene) (35/65 wt%) mixture. (b) Functionalization rate according to the used solvent and the pre-treatment.

Fig. 8 .

 8 Fig. 8. SEM/EDS images of pristine, pre-treated and functionalized flax fibers.

Fig. 9 .

 9 Fig. 9. (a) TGA and (b) DTG curves of pristine, pre-treated and functionalized fibers.

Fig. 10 .

 10 Fig. 10. Log G' and Tan δ vs crosslinking reaction time for the PDMS matrix, PDMS matrix with pristine fibers and PDMS matrix with functionalized fibers.

Fig. 11 .

 11 Fig. 11. Stress strain curves of silicone composites filled with pristine, pretreated and functionalized fibers. Fig. 12. Young modulus E of silicone composites filled with pristine, pretreated and functionalized fibers.

Fig. 12 .

 12 Fig. 11. Stress strain curves of silicone composites filled with pristine, pretreated and functionalized fibers. Fig. 12. Young modulus E of silicone composites filled with pristine, pretreated and functionalized fibers.

Table 1

 1 TGA results of pristine, pre-treated and functionalized flax fibers.

	Samples	T in ( • C)	T max ( • C)	Y ash (%)
	Pristine fiber	256	355	1.5
	Pre-treated fiber	300	340	3.9
	Functionalized fiber	298	342	9.7

Acknowledgements:

We gratefully acknowledge Jean-Claude ROUX (C2MA, PCH, IMT Mines Alès) for SEM images, Loic DUMAZERT (C2MA, PCH, IMT Mines Alès) for TGA measurements and Alexandre CHERON (LMGC, IMT Mines Ales, Univ Montpellier) for mechanical tests. IMT Mines Alès to finance this work.

Data availability

No data was used for the research described in the article.

CRediT authorship contribution statement

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Supplementary material

Supplementary data to this article can be found online at https://doi. org/10.1016/j.compositesa.2022.107382.