Elnaz Gholizadeh

Rima Belibel

Laurence Mora

Didier Letourneur

Christel Barbaud
email: barbaud@univ-paris13.fr

Small changes in PDMMLA structure influence the adsorption behavior of ECM proteins and syndecan-4 on PDMMLA derivative surfaces: Experimental validation by tensiometric surface force measurements

Keywords: Protein adhesion, Biopolyesters, In-stent restenosis, Dynamic tensiometry, Syndecan-4, ECM proteins

scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Cardiovascular diseases are the main cause of mortality and morbidity in the world [START_REF] Strohbach | Polymers for Cardiovascular Stent Coatings[END_REF][START_REF] Versari | The importance of reendothelialization after arterial injury[END_REF]. In stenosis cases, where the arterial lumen narrows because of an atheromatous plaque formation, percutaneous devises (such as cardiovascular stents) and bypass surgery are utilized in order to reestablish a normal blood flow. These interventions cause a disruption of the vessel wall. After a stent implantation, in approximately 20 to 30% of the cases (where a bare metal stent (BMS) is used), in-stent restenosis (IR) is observed due to an inflammatory response of the immune system [START_REF] Kim | In-Stent Restenosis[END_REF][START_REF] Byrne | Stent thrombosis and restenosis: what have we learned and where are we going? The Andreas Gruntzig Lecture ESC 2014[END_REF]. Platelet aggregation, smooth muscle cell (SMC) proliferation and migration, and growth factor release are responsible for the neointimal hyperplasia causing a re-narrowing of the arterial lumen [START_REF] Marx | Vascular smooth muscle cell proliferation in restenosis[END_REF][START_REF] Hoffmann | Patterns and mechanisms of in-stent restenosis. A serial intravascular ultrasound study[END_REF].

Biocompatible and bioresorbable polymers are used in order to limit the direct contact of the BMS with the blood [START_REF] Babapulle | Coated stents for the prevention of restenosis: Part II[END_REF]. A favorable polymer to be used in this case would be a polymer able to inhibit the proliferation and the migration of SMCs, and promote the vessel re-endothelialization in order to re-vascularize the damaged area. The most used bioresorbable polymers for stent coating are poly(lactic acid) (PLA) and poly(lactic-co-glycolic acid) (PLGA), polyesters with suitable thermal properties and degradation time [START_REF] Kulkarni | Biodegradable poly(lactic acid) polymers[END_REF][START_REF] Westedt | Paclitaxel releasing films consisting of poly(vinyl alcohol)-graftpoly(lactide-co-glycolide) and their potential as biodegradable stent coatings[END_REF]. We present a new bioactive polyester, which would provide all the needed properties of an adequate stent coating; Poly([R,S]-3,3-dimethylmalic acid) (PDMMLA) (Scheme 1-a) is a biocompatible and biodegradable polyester with promising properties. The synthesis of amphiphilic PDMMLA derivatives takes place by incorporating on the polymer side chain, carboxyl groups (-COOH) and hexylic ester groups (-COOHex), giving the polymers their hydrophilic and hydrophobic characters respectively (Scheme 1-b) [START_REF] Ouhib | Synthesis of new statistical and block co-polyesters by ROP of α,α,β-trisubstituted β-lactones and their characterizations[END_REF]. This property is a huge advantage compared to PLA and PLGA. Indeed, the degradation kinetics of PDMMLA can be controlled by regulating its hydrophilic/hydrophobic balance [START_REF] Belibel | Poly((R,S)-3,3-dimethylmalic acid) derivatives as a promising cardiovascular metallic stent coating: Biodegradation and biocompatibility of the hydrolysis products in human endothelial cells[END_REF]. All PDMMLA derivatives studied here have a glass transition temperature (Tg) under 37°C and show interesting thermomechanical properties [START_REF] Belibel | Synthesis and characterizations of new isotactic homopolyesters, statistical and block copolyesters derived of poly((S)-3,3-dimethylmalic acid) via the lactone route[END_REF]. Also, the presence of chemically alterable functions on PDMMLA's side chain (-COOH) represents a great potential for a covalent grafting of active drugs on the polymer, which cannot be realized on the currently used biopolymers. This allows a wellmanaged controlled release of drug during the arterial healing process. When a biomedical material is implanted in the body, its surface gets covered by a layer of serum or plasma proteins. The adsorbed proteins can then freely reorient and change conformation, affecting the wettability of the material surface, and its surface force and its environment interaction [START_REF] Ruardy | Growth of fibroblasts and endothelial cells on wettability gradient surfaces[END_REF].

Understanding surface-protein interactions is important for the prediction of biological answers (surface/cell); these interactions play a central role in cell mechanism regulation (adhesion, spreading and proliferation…) [START_REF] Alves | The dynamics, kinetics and reversibility of protein adsorption onto the surface of biodegradable materials[END_REF][START_REF] Rupp | Adsorption/desorption phenomena on pure and Teflon® AF-coated titania surfaces studied by dynamic contact angle analysis[END_REF][START_REF] Müller | Protein adhesion on dental surfaces-a combined surface analytical approach[END_REF]. This surface/cell interaction depends on the physicochemical properties of the polymeric biomaterial surface (such as roughness, wettability and surface charge). Cell attachment is mediated by the adhesion proteins contained in the physiological medium (proteins/cell) [START_REF] Aiyelabegan | Fundamentals of protein and cell interactions in biomaterials[END_REF]. Thus, the ability of materials to adsorb adhesive proteins in a favorable conformation (surface/protein) will determine their ability to induce cell adhesion and spreading (surface/protein/cell), a fundamental aspect in the biocompatibility of surfaces (Fig 1).

The initial adsorption of proteins on a surface occurs very rapidly and avoids direct interactions between cells and the surface [START_REF] Sheikh | Macrophages, foreign body giant cells and their response to implantable biomaterials[END_REF]. This protein adsorption can be promoted or, on the contrary, inhibited by surface modifications involving the hydration of proteins and surfaces, the redistribution of charges at the proteins interface/conformational changes [START_REF] Hartvig | Protein adsorption at charged surfaces: The role of electrostatic interactions and interfacial charge regulation[END_REF]. The characterization of the adsorption of proteins on different surfaces makes it possible to better understand the physiological phenomena and to guide the cellular response.

DCA is a technique used for the measurement of dynamic wettability changes of a surface allowing to detect the dynamic interfacial and surface configuration changes of a biomaterial [START_REF] Belibel | Effect of chemical heterogeneity of biodegradable polymers on surface energy: A static contact angle analysis of polyester model films[END_REF][START_REF] Benni | Dynamic Contact Angle Analysis of Protein Adsorption on Polysaccharide Multilayer's Films for Biomaterial Reendothelialization[END_REF]. DCA method characterizes the surface/interface changes during the protein adsorption by measuring the surface wetting tension. It also gives information about the proteins conformational changes and the reversibility phenomenon. [START_REF] Yuan | Contact angle and wetting properties[END_REF]. Palacio and Bhushan present a schematic demonstration of protein and cell adsorption on surfaces with different characteristics [START_REF] Palacio | Research article: Bioadhesion: A review of concepts and applications[END_REF]. proteoglycan, syndecan-4 (Syn-4) is studied because of its overexpression during the healing period of a damaged arterial tissue [START_REF] Kojima | Plasma levels of syndecan-4 (ryudocan) are elevated in patients with acute myocardial infarction[END_REF]. A mixture of these proteins is also studied in order to compare each protein's individual profile to that of the case where thay are all present. This allows to identify the dominating adsorbed protein and its affinity with each polymer surface. For the preparation of polymer films, glass slides (Objektträger, 10 × 13.3mm, 0.99mm thick, MENZEL-GLASSER, Braunschweig, Germany) were cleaned in water, chloroform, and acetone, then, dried at 37°C in an oven. Polymers were dissolved in a small quantity of acetone in order to obtain a viscous solution. The polymers solution was loaded on both sides of the glass slides with a micropipette (n = 3 for each experiment). The covered slides were dried at room temperature, then at 37°C overnight in order to obtain visible, smooth and homogeneous films.

Protein solution preparation

All proteins (BSA, Fn, Fb, Vn, and Syn-4) were purchased from Sigma Aldrich and employed as received. All protein solutions were prepared in Phosphate Buffered Saline (PBS), pH = 7.4, at 0.2% of total plasma proteins' concentration, representing the physiological concentration for each protein (BSA at 86 µg/mL, Fb at 6µg/mL, Fn and Vn at 0.6µg/mL, and Syn-4 at 1µg/mL) [START_REF] Benni | Dynamic Contact Angle Analysis of Protein Adsorption on Polysaccharide Multilayer's Films for Biomaterial Reendothelialization[END_REF]. The "Mix" solution was prepared in accordance with the physiological proportions, by mixing equal volumes of each previously prepared protein solution.

Bovine Serum Albumin (BSA) is used as a model protein for albumin. It is a globular protein, with a plasma concentration range of 35 to 50 mg/mL; it is thus the most abundant protein in the plasma.

Regulating the oncotic pressure of blood is the main function of this protein [START_REF] Quinlan | Albumin: Biochemical properties and therapeutic potential[END_REF].

Fibronectin (Fn) is a multi-domain glycoprotein. Fn regulates cell adhesion, migration and differentiation [START_REF] Schwarzbauer | Fibronectins, their fibrillogenesis, and in vivo functions[END_REF] and provides structural support for cell survival, gene expression, growth factor signaling, etc [START_REF] Faralli | Functional properties of fibronectin in the trabecular meshwork[END_REF]. This protein is soluble in a globular form in the plasma (concentration range of 0,2 to 0,4 mg/mL), and insoluble in a fibrillary pattern in the ECM [START_REF] Guan | Fibronectin / integrin interaction induces tyrosine phosphorylation of a 120-kDa protein[END_REF].

Vitronectin (Vn) is a cell adhesion promoting glycoprotein present in the ECM with a concentration range of 0.25 to 0.45 mg/mL. Although it has a similar activity to Fn in terms of cell attachment and spreading it remains biochemically and immunologically different from the latter [START_REF] Li | Impact of vitronectin concentration and surface properties on the stable propagation of human embryonic stem cells[END_REF][START_REF] Preissner | Structure and biological role of vitronectin[END_REF].

Fibrinogen (Fb) is a glycoprotein that cleaved with fibrin, plays an important role in the process of hemostasis and coagulation, which is essential in limiting blood loss and wound healing. It is present in the plasma with a concentration range of 1.5 to 4.5 mg/mL [START_REF] Herrick | Fibrinogen[END_REF].

It is also known that Fb and its breakdown products promote processes like angiogenesis, migration and proliferation of various cells such as fibroblasts, smooth muscle cells (SMCs) and lymphocytes, which are important events in tissue repairing [START_REF] Koo | Control of anti-thrombogenic properties: Surface-induced self-assembly of fibrinogen fibers[END_REF].

Syndecan-4 (Syn-4): Syndecans are transmembrane heparan sulfate proteoglycans (HSPGs) synthesized by many cells. Syndecans bind extracellular ligands via their covalently attached heparan sulfate chains and influence intracellular events through transmembrane signaling pathways [START_REF] Carey | Syndecans: multifunctional cell-surface co-receptors[END_REF]. There are four syndecan family members: syn-1, syn -2, syn -3, and syn -4 [START_REF] Kim | Members of the syndecan family of heparan sulfate proteoglycans are expressed in distinct cell-, tissue-, and development-specific patterns[END_REF]. Here, we study Syn-4 thanks to its major role in biological functions such as regulation of blood coagulation, cell adhesion, and cell growth [START_REF] Couchman | Syndecan-4 and integrins: combinatorial signaling in cell adhesion[END_REF]. Studies show that Syn-4 activates flow-dependent atheroprotective pathways by sensing flow direction and promoting endothelial cell alignment [START_REF] Baeyens | Syndecan 4 is required for endothelial alignment in flow and atheroprotective signaling[END_REF].

Its presence is also essential in neointimal formation and preventing cardiac rupture after vascular injuries [START_REF] Matsui | Syndecan-4 prevents cardiac rupture and dysfunction after myocardial infarction[END_REF][START_REF] Ikesue | Syndecan-4 deficiency limits neointimal formation after vascular injury by regulating vascular smooth muscle cell proliferation and vascular progenitor cell mobilization[END_REF]. Studies show that in patients with myocardial infarction (MI), human Syn-4 expression is elevated following MI in repair region of the damaged cardiac tissue [START_REF] Echtermeyer | Syndecan-4 signalling inhibits apoptosis and controls NFAT activity during myocardial damage and remodelling[END_REF]. Our biology research team demonstrated that Syn-4 is positively involved in this process of arterial healing [START_REF] Maillard | RANTES/CCL5 mediated-biological effects depend on the syndecan-4/PKCa signaling pathway[END_REF].

Dynamic protein adsorption by DCA

Surface wetting forces of PDMMLA derivatives (10/90, 20/80, and 30/70) were registered in the presence of different proteins in order to study their interaction with different polymer surfaces. The experiments were carried out using a Wilhelmy balance tensiometer (K100MK2 from Krüss Gmbh), connected to a computer for sorption and contact angle measurement and analysis. The relation between surface tension and force is established by the Wilhelmy method using the following equation:

F = γlv.L.cosθ
where F is the force in mN, γlv the surface tension of the liquid, L the wetted length, and θ either the advancing or the receding contact angle [START_REF] Yuan | Contact angle and wetting properties[END_REF]. The contact angle hysteresis is determined by calculating the difference between the advancing and receding contact angles. Surface chemistry, morphology and roughness are the main factors affecting thermodynamic hysteresis [START_REF] Alves | The dynamics, kinetics and reversibility of protein adsorption onto the surface of biodegradable materials[END_REF].

To perform the blank experiment, 10 wetting/dewetting cycles of the polymer films were realized in a PBS solution (pH = 7.4). To perform the protein experiments 5 cycles were realized for adsorption (5 wetting/dewetting of the polymer film in the protein solution), followed by 5 cycles for desorption/rinsing (5 wetting/dewetting of the same polymer film in PBS). For all samples, immersion depth and the immersion/emersion rate were set to 10mm and 6mm/min respectively.

All experiments were carried out at 37°C.

Results and discussions

Three statistical PDMMLA surfaces (10/90, 20/80 and 30/70) were studied in contact with five proteins (BSA, Fn, Fb, Vn and Syn-4). Three kinds of interaction forces were considered in this study: ionic interactions, hydrogen bond and hydrophobic interactions (Fig1). Ionic and hydrogen bond interactions are dominant in carboxyl groups, while hydrophobic interactions are dominant in the hexylic ester groups of the polymers side chains.

In a parallel work, in vitro cell adhesion, spreading and proliferation are performed in order to evaluate the human umbilical vein endothelial cells (HUVEC) behavior on different polymer-based films (10/90, 20/80 and 30/70). First results are very encouraging (results will shortly be published).

In order to complete this work and explain the obtained results, we study in this work, the protein adhesion on these PDMMLA surfaces using DCA technique with the intention of designating the future bioactive stent coating.

In this work, we are interested in the advancing wetting force, "F" of PDMMLA derivatives in the presence of different proteins, as well as a mixture of all these studied proteins. Blank experiments show that F increases with the increase of carboxyl groups on the polymer chain.

Copolymers 10/90 and 20/80 reveal a baseline between 3 and 4mN (3.18 ± 0.07mN and 3.38 ± 0.04mN respectively). F values are slightly higher for copolymer 20/80 due to the existence of 20% of hydrophilic groups versus 10% in 10/90 (p > 0.05). This non-significant increase of F confirms the increase of the hydrophilicity of the surface but is not sufficient to make a considerable difference between these two surfaces.

It can be explained that in the case of copolymer 10/90, due to the presence of 90% of hexylic ester groups, the carboxyl functions are hidden and less exposed by self-organization of these hydrophobic side chains.

In contrast, a significant gap is observed between copolymers 20/80 and 30/70 (4.62 ± 0.07 mN) (p < 0.05). This indicates that with 30% of carboxyl functions, the polymer has more opportunities to expose these groups to the interface between the polymer surface and aqueous medium. These chains are further exposed to the surface compared to previous cases. Indeed, the percentage of hydrophilic chains has a significant impact on the chain mobility effect.

Protein adsorption/desorption

The first 5 cycles of wetting/dewetting are realized in a protein solution (cycles 1 to 5 for adsorption experiment) and the last 5 cycles are realized in a PBS solution (cycles 6 to 10 for rinsing/desorption experiment). Each profile is compared to that of PBS alone obtained with corresponding polymer film. If the adsorption curve is superimposed with the baseline (PBS curve), there is no adhesion. Above the baseline, the protein is adsorbed via hydrophobic interactions, exposing its hydrophilic parts to the surface. If the curve is below the baseline, adsorption takes place via hydrophilic interactions, exposing hydrophobic parts of the protein to the surface. As for the rinsing stage, three cases can be observed: if the rinsing curve is on the baseline, it means that either the protein is desorbed (ΔF = 0), or reoriented (ΔF ≠ 0). If the force remains stable with no change, it means that the protein is adsorbed without reorientation or desorption. If the force Figure 3 shows the BSA adsorption/desorption profiles on three polymer surfaces. BSA is strongly adhered to the 10/90 surface via hydrophobic interactions at the first contact with the polymer. The surface becomes slightly more hydrophilic during rinsing (∆F = 0.46 ± 0.005mN). Adsorbed BSA changes conformation during rinsing, it decreases the hydrophobic interaction but leaves the electrostatic interaction. The surface becomes less hydrophilic compared to adsorption case (∆F = 0.15± 0.01mN). This is confirmed by the case of 20/80 copolymer containing more hydrophilic groups. The adhesion of BSA takes place by more hydrophilic interactions compared to 10/90, but its adhesion does not change the surface hydrophilicity. The adsorption/desorption interfacial force is very close to that of PBS alone (∆F = 0.03± 0.07 and -0.05± 0.08mN, respectively). Regarding the 30/70, the surface is suitable for making hydrophilic interactions as well as hydrogen bonds and ionic interactions. All adhered BSA remains attached to the surface, and does not change conformation during the rinsing step (∆F = -3.67± 0.02 (adsorption) and -3.81± 0.01mN (rinsing)).

In conclusion, BSA adsorption power increases as hydrophilic character increases, which means that the BSA affinity is reduced by hydrophobic interactions. The results of the tests on the other studied proteins are represented in the Figure 4. According to the experimental results, it is clearly visible that from the copolymer 10/90 (the most hydrophobic surface) to the copolymer 30/70 (the most hydrophilic surface), the surfaces become more and more hydrophobic after the adhesion of different proteins (table 1 andFig [START_REF] Marx | Vascular smooth muscle cell proliferation in restenosis[END_REF]. Only the copolymer 30/70 has the surface on which all proteins expose their hydrophobic side after adhesion, meaning that increasing the hydrophilicity of the polymer surface promotes hydrophilic, ionic interactions and hydrogen bond type adhesions. Concerning Vn, which presents a hydrophobic curve for all three surfaces 10/90, 20/80 and 30/70 (∆F = -2.40 ± 0.02, -2.69 ± 0.02 and -4.17 ± 0.01mN, respectively), we can say that the latter makes only one type of interaction on all surfaces.

Syn-4, which is not a plasma protein, shows a similar behavior on all three copolymers with small difference of F values (∆F = -2.87 ± 0.04, -3.09 ± 0.04 and -3.76 ± 0.002mN, respectively).

In addition, using the whole set of studied polymers and proteins, we can see the overall trend between interfacial force and acidity percentage (%A) of polymers. Figure 5 histograms show the variations of proteins profiles on different polymer surfaces during adsorption/desorption cycles and linear correlation of F with %A. The slope corresponds to the sensitivity of the correlation and R 2 is the correlation coefficient. R 2 and slope are calculated for the interfacial force F (the straight line is drawn only from the three points corresponding to the three studied copolymers.

With the exception of membrane protein Syn-4 which gives the same profile as blank (increase of F with the increase of acidity), the hydrophobicity of surface increases linearly with %A (decrease of F) for all studied plasma proteins. Indeed, Vn exhibits the highest sensitivity associated to an excellent R 2 correlation coefficient which is very close to 1 for adsorption (R 2 = 0.97) and equal to 1 for desorption. BSA and Fn are also interesting (R 2 = 0.81 and 0.87, respectively) whereas Mix is not well fitting to %A (R 2 far from 1). R 2 of Fb and Syn-4 are close to 0.80.

Then, the "Mix" curve represents the overall behavior of the mixture of all studied proteins, giving information about affinity to the surface and competitiveness between the proteins. As observed with plasma proteins, from the most hydrophobic polymer (10/90) to the most hydrophilic one (30/70), surfaces become more hydrophobic after the adhesion of different proteins (Mix) (Table 1 and Fig 5).

In the case of the copolymers 10/90 and 20/80, the closest curve to that of the "Mix", is the Vn's, meaning that this protein has an important affinity to these surfaces. It adheres the first when all the studied proteins are present in the medium, making the surface more hydrophobic. Regarding the copolymer 30/70 though, we can see that all of the curves are gathered around the "Mix" curve.

This means that in the presence of all of the studied proteins, the surface becomes hydrophobic and all of the proteins show a similar affinity to the surface with poor competitiveness even though the domination of Vn is visible.

The mobility of the carboxyl groups on the 30/70 copolymer surface results in exposing these groups further to the surface, leading to further possible hydrophilic interactions between the surface and the proteins. Furthermore, at neutral pH these groups exist in both carboxyl (-COOH)

and carboxylate anion (-COO -) forms, which favor also the formation of ionic and hydrogen bonds.

The adhered proteins thus expose their hydrophobic parts to the surface and reduce the hydrophilicity.

Table 1

Hydrophilicity of copolymer surfaces compared to the baseline (PBS alone) after protein adhesion. ΔF is the difference between F of adsorption/desorption and F of blank.

Conclusion

DCA technique was used to measure surface forces between three PDMMLA derivatives (10/90, 20/80, and 30/70) in contact with different protein solutions, allowing to determine different protein adhesion profiles on each of these surfaces. Studying protein adhesion on biomaterial surfaces is important because of its considerable role in inducing cell adhesion on these surfaces. This work shows that there is a strong correlation between the surface hydrophilicity and protein adhesion on the studied copolymers.

The "Mix" experiment is realized in order to obtain data about the affinity and competitiveness between the different proteins. In the case of copolymer 30/70, all proteins show similar adhesion profiles with low competitiveness. All studied proteins are adhered to the surface via hydrogen bonds, hydrophilic and ionic interactions making the polymer surface more hydrophobic than the starting point.

Vn gives a hydrophobic sorption curve for all three surfaces, meaning that the latter makes only one type of interaction with all surfaces.

Syn-4, a transmembrane plasma protein, shows also similar behavior on all studied surfaces. First results of parallel work show that the HUVECs are more likely to adhere on more hydrophobic surfaces (i.e. copolymer 10/90) than hydrophilic ones. However, the spreading and growth of these cells might demand a moderately hydrophilic surface (i.e. copolymer 30/70) (results will shortly be published). On the other hand, study shows that even on hydrophobic surfaces, there is a chance for the endothelial cells to grow constantly in a time-dependent manner [START_REF] Ruardy | Growth of fibroblasts and endothelial cells on wettability gradient surfaces[END_REF]. Therefore, to validate the choice of the customized polymer 30/70, further study is needed to determine cell attachment, spreading, proliferation and growth on the surfaces covered by the protein layer, in the presence and absence of growth factors and other parameters affecting cell growth. These experiments will shortly be carried out and the results will be communicated in a next publication.

Scheme 1 ,

 1 Scheme 1, a) Poly([R,S]-3,3 dimethylmalic acid) (PDMMLA) b) Statistical and amphiphilic PDMMLA copolyester with -COOH and -COOHex groups (PDMMLAHn-co-Hexm)

Fig 1 :

 1 Fig 1: Schematic presentation of protein layer formation on a biomaterial and cell adhesion on the protein layer

2. Experimental section 2 . 1 .

 21 Polymer synthesis and polymer film preparation Three statistical PDMMLAs were synthesized through an anionic ring opening polymerization of βlactone monomers. Two different β-lactones were used in order to obtain the wanted amphiphilic copolymers. (10/90: Mn(10/90) = 148060 g/mol, Mw (10/90) = 153800 g/mol, D(10/90) = 1.038; 20/80: Mn(20/80) = 214420 g/mol, Mw (20/80) = 231800 g/mol, D(20/80) = 1.081; 30/70: Mn(30/70) = 354460 g/mol, Mw (30/70) = 364800 g/mol, D(30/70) = 1.029) The protocols for the synthesis of the copolymers, βlactones as well as different characterization techniques are described in the literature[START_REF] Belibel | Synthesis of New Optically Active α , α' , β -Trisubstituted β-Lactones as Monomers for Stereoregular Biopolyesters[END_REF][START_REF] Barbaud | Synthesis of New Homopolyester and Copolyesters by Anionic Ring-opening Polymerization of α,α′,β-Trisubstituted β-Lactones[END_REF]. The mean surface roughness values (Ra) for each copolymer are: Ra (10/90) = 0.106 nm; Ra (20/80) = 0.184 snm; Ra (20/80) = 0.216 nm[START_REF] Belibel | Effect of chemical heterogeneity of biodegradable polymers on surface energy: A static contact angle analysis of polyester model films[END_REF].

3. 1 .

 1 Blank experiments 10 cycles of wetting/dewetting in PBS (no proteins) were realized on copolymers 10/90, 20/80, and 30/70. The measured advancing wetting force values for each polymer are represented as a function of cycle numbers (Fig 2). The increase of a wetting force indicates the increase of hydrophilic character of the surface.

Fig 2 :

 2 Fig 2: blank experiments of copolymers 10/90, 20/80, and 30/70 in PBS at 37°C (n = 3)

Fig 3 :

 3 Fig 3: adsorption and desorption of BSA on a) 10/90, b) 20/80, c) 30/70 copolymer surfaces (n = 3)

Fig 4 :

 4 Fig 4: comparative behavior of different proteins on a) 10/90, b) 20/80, c) 30/70 copolymer surfaces (n = 3)

  .65 ± 0.07) ----(-4.04 ± 0.01) ----(-4.15 ± 0.01)The surface: (+) becomes hydrophilic, (-) becomes hydrophobic, and (0) does not change due to the protein adhesion.

Fig 5 :

 5 Fig 5: protein profile variations on different polymer surfaces during a) adsorption, b) desorption (rinsing), and linear correlation of F with %A


Ángel Hernández-Gómez

Victor Ramirez

Damien Guilbert
email: damien.guilbert@univ-lorraine.fr

Investigation of PEM electrolyzer modeling: Electrical domain, efficiency, and specific energy consumption

Keywords: PEM electrolyzer, Modeling, Faraday's efficiency, Voltage efficiency, Specific energy consumption, Dynamic operation

Proton exchange membrane (PEM) electrolyzer is an advanced technology considered a viable alternative for the generation of hydrogen-based on renewable energy sources (RES). Its modeling is essential to study its interaction with RES and power electronics. In the current literature, the models for the electrical domain are mainly based on semi-empirical and empirical equations. However, dynamic operations are generally neglected.

Besides, a few works about electrolyzer efficiency have been reported, especially Faraday's efficiency, which is a key parameter to express the losses due to gas diffusion. The main purpose of this review is to summarize and analyze the reported models to describe the electrical domain. Furthermore, dynamic operation issues are highlighted and recent works about modeling the dynamics are introduced. Finally, a discussion is provided about the different efficiency (Faraday, voltage, energy) and the specific energy consumption, which are important indicators linked with the performance.

Introduction

Over the last decade, the global demand for energy has been growing at a very high rate due to the increase of population and the depletion of fossil fuels [START_REF] Nie | Numerical modeling of two-phase flow in a bipolar plate of a PEM electrolyzer cell[END_REF]. These factors have prompted interest in developing at large scale renewable energy sources (RES) for electricity, clean fuel, and heat production [START_REF] Fernandes | Renewable energy scenarios in the Portuguese electricity system[END_REF].

The RES are the resources that have "the future inside" by definition because they can be reformed in the environment by natural processes. RES include solar energy, water energy, biomass energy, wind energy, earth energy, sea energy, biological, and bacterial energy [START_REF] Orecchini | The era of energy vectors[END_REF]. It is stipulated for the year 2050 several countries (Australia, Denmark, Macedonia, Portugal, etc.) will depend only on RES [START_REF] Fernandes | Renewable energy scenarios in the Portuguese electricity system[END_REF][START_REF] Lund | Energy system analysis of 100% renewable energy systems-the case of Denmark in years 2030 and[END_REF][START_REF] Mathiesen | 100% renewable energy systems, climate mitigation and economic growth[END_REF][START_REF] Elliston | Simulations of scenarios with 100% renewable electricity in the Australian National Electricity Market[END_REF][START_REF] Ćosić | A 100% renewable energy system in the year 2050: The case of Macedonia[END_REF].

Hydrogen is an energy carrier since it is one of the most efficient, cleanest and lightest fuels. There is a vast experience of handling hydrogen in the industry, which reinforces the belief that it is a safe resource. It is as versatile as electricity, it can be stored and it can be produced by RES, which is important from the sustainability point of view [START_REF] Singh | Hydrogen: A sustainable fuel for future of the transport sector[END_REF][START_REF] Yilanci | A review on solar-hydrogen/fuel cell hybrid energy systems for stationary applications[END_REF][START_REF] Barthels | Phoebus-Jülich: An autonomous energy supply system comprising photovoltaics, electrolytic hydrogen, fuel cell[END_REF][START_REF] Ulleberg | The wind/hydrogen demonstration system at Utsira in Norway: Evaluation of system performance using operational data and updated hydrogen energy system modeling tools[END_REF][START_REF] Vanhanen | Simulation of solar hydrogen energy systems[END_REF][START_REF] Kélouwani | Model for energy conversion in renewable energy system with hydrogen storage[END_REF][START_REF] Zhang | A wind-hydrogen energy storage system model for massive wind energy curtailment[END_REF][START_REF] Gahleitner | Hydrogen from renewable electricity: An international review of power-to-gas pilot plants for stationary applications[END_REF][START_REF] Albadi | Overview of wind power intermittency impacts on power systems[END_REF]. Different processes for hydrogen production can be used: photodissociation of water, thermochemical cycles, microscopic organisms as algae, etc. These processes are still far from practical is coupled directly with RES since it ensures efficient and reliable operation of them [START_REF] García-Valverde | Simple PEM water electrolyser model and experimental validation[END_REF][START_REF] Abdin | Modelling and simulation of a proton exchange membrane (PEM) electrolyser cell[END_REF][START_REF] Görgün | Dynamic modelling of a proton exchange membrane (PEM) electrolyzer[END_REF][START_REF] Rahim | An overview of polymer electrolyte membrane electrolyzer for hydrogen production: Modeling and mass transport[END_REF]. Besides, compared to alkaline technology, PEM technology is particularly suitable to be coupled with RES since it can absorb the energy from RES during transients. It is one of the most important features requested when coupling with dynamic sources such as RES [START_REF] Mohammadi | A comprehensive review on coupling different types of electrolyzer to renewable energy sources[END_REF].

In recent years, several reviews about modeling of the electrolyzers have been reported. In [START_REF] Zeng | Recent progress in alkaline water electrolysis for hydrogen production and applications[END_REF], the authors have presented a work where the fundamentals of the electrolysis are examined and the performance of various designs of alkaline electrolyzers is compared based on the thermodynamic and kinetic analysis. In [START_REF] Ursúa | Hydrogen production from water electrolysis: Current status and future trends[END_REF], a review of electrolyzer technologies for hydrogen production and the state of the art of their integration with RES has been reported. In [START_REF] Carmo | A comprehensive review on PEM water electrolysis[END_REF], a review of the areas of electrocatalysis, essential components, and modeling activities for PEM electrolyzers has been introduced. In [START_REF] Stempien | Solid oxide electrolyzer cell modeling: A review[END_REF], the authors have presented a work focused on modeling SO electrolyzers and remaining key issues. In [START_REF] Rahim | An overview of polymer electrolyte membrane electrolyzer for hydrogen production: Modeling and mass transport[END_REF], a state-of-the-art of the mass transport model for PEM electrolyzers based on analytical, semi-empirical and mechanistic methods has been presented. The work [START_REF] Olivier | Low-temperature electrolysis system modelling: A review[END_REF] has presented a review of the models of the low temperatures electrolysis system for alkaline and PEM electrolyzers. In [START_REF] Bideau | Review of necessary thermophysical properties and their sensivities with temperature and electrolyte mass fractions for alkaline water electrolysis multiphysics modelling[END_REF], a review of multiphysics models provided experimental data needed to calculate thermos-physical properties and the correlations for the simulations of alkaline electrolyzers has been carried out. In [START_REF] David | Advances in alkaline water electrolyzers: A review[END_REF], a review of different techniques to produce hydrogen in a focus on alkaline electrolysis has been reported. Finally, the work [START_REF] Kumar | Hydrogen production by pem water electrolysis -a review[END_REF] has presented a review mainly focused on the recent development of PEM electrolyzer including the evolution reaction electrocatalysts of the hydrogen and the oxygen, and by highlighting the most efficient electrocatalysts among similar materials.

From this current state-of-the-art, it can be noted that five reviews have been reported for alkaline and PEM electrolyzers; while only one review for SO has been carried out since this technology is under research and development. Among PEM electrolyzer reviews, only in [START_REF] Rahim | An overview of polymer electrolyte membrane electrolyzer for hydrogen production: Modeling and mass transport[END_REF][START_REF] Olivier | Low-temperature electrolysis system modelling: A review[END_REF], models of PEM electrolyzers are summarized and discussed from the mass transport and low-temperature electrolysis modeling point of view. Besides, static models of the electrical domain have been presented but the dynamics have not been discussed. Recently, a first work [START_REF] Guilbert | Experimental validation of an equivalent dynamic electrical model for a proton exchange membrane electrolyzer[END_REF] has been reported about the dynamic operations of PEM electrolyzers. The authors have demonstrated that the electrolyzer takes some time to reach its steady-state operation as a result of a dynamic current profile. This is mainly due to the move of charge inside the electrolyzer. The response time strongly depends on the input energy, and consequently, the response may be more or less fast. To model the dynamics of the electrolyzer, an equivalent electrical circuit has been developed and validated experimentally. The results demonstrate the effectiveness of the model in modeling accurately the dynamic behavior of PEM electrolyzers compared to classical static models.

It has to be noted also that a few works have been reported about Faraday's efficiency, which is the key parameter modeling the losses due to the gas diffusion. Based on the previously reported reviews, the objective of this work is to summarize and analyze the static models based on semi-empirical and empirical equations for the electrical domain. Besides, dynamic issues of PEM electrolyzers are discussed and recent models are presented.

Finally, a discussion is provided about the different efficiency (Faraday, voltage, energy), and the specific energy consumption which are linked with the performance of PEM electrolyzers. This work is composed of five sections. After introducing the current state-of-the-art about the previously reported reviews focused on electrolyzers and the remaining key issues, Section 2 introduces the basic principles, advantages, and disadvantages of the PEM electrolyzer. Besides, dynamic issues are discussed to emphasize their importance in modeling accurately dynamic operations of PEM electrolyzers. Then, in Section 3, a summary and an analysis of the electrical domain modeling is reported from the static and dynamic point of view. After that, in Section 4, an investigation of the efficiency, the specific energy consumption, and its relation with the performance of PEM electrolyzers is made. Besides, a discussion is provided to emphasize the two important issues which require further investigation in the future. Finally, in Section 5, the conclusion is drawn out.

PEM electrolyzer technology

Operation and characteristics

As mentioned in the introduction, three main technologies of electrolyzers have been reported such as alkaline, PEM, and SO. However, only alkaline and PEM technologies are currently available in the market; while SO technology is still under research and development [START_REF] Stempien | Solid oxide electrolyzer cell modeling: A review[END_REF]. Among the two available technologies, alkaline technology is the most mature and widespread technology since its operation principles were introduced more than 100 years ago. Therefore, so many alkaline water electrolysis plants have been developing so far. On one hand, the main benefits of this technology are low-cost catalysts (i.e. cobalt, nickel, iron), high lifespan and gas purities [START_REF] David | Advances in alkaline water electrolyzers: A review[END_REF]. On the other hand, this technology suffers from having a low current density (between 0.2-0.4 A • cm -2 ) which slows down its development in emerging markets of great potential. Furthermore, in terms of flexibility, alkaline technology is not fit in load-following operation, which is requested by coupling it with RES. To overcome the above-mentioned issues for alkaline technology (i.e. current density, flexibility), PEM technology has been developing since the 1960s. Indeed, this technology can operate at high current densities (around 2 A • cm -2 ) and is particularly suitable to be coupled with RES due to its fast response time. As a result, since RES are considered very dynamic sources, the energy flowing during transients can be absorbed by the PEM electrolyzer, enhancing the efficiency of the system. However, compared to alkaline electrolyzers, PEM electrolyzers use expensive catalysts such as platinum (at the cathode side) and iridium (at the anode side), increasing consequently the global price of the electrolyzer.

In [START_REF] Millet | Electrochemical performances of PEM water electrolysis cells and perspectives[END_REF], a survey of the most significant scientific and technological materials to build a PEM electrolyzer has been reported. The PEM electrolyzer cell is essentially a PEM as an electrolytic conductor between an anode and a cathode. The water molecules and ionic particles are transferred across the membrane from the anode to the cathode, where it is decomposed into oxygen, protons, and electrons. In the reaction process, electrical energy is supplied to the system and converted into chemical energy. The electrons exit the cell through an external circuit. After, the electrons and protons recombine at the cathode to produce hydrogen [START_REF] Rahim | An overview of polymer electrolyte membrane electrolyzer for hydrogen production: Modeling and mass transport[END_REF], see Figure 1. The PEM is responsible for providing high proton conductivity, low gas crossover, compact system design, and high-pressure operation. The low membrane thickness is in part the reason for many of the advantages. The low gas crossover rate of the polymer electrolyte membrane allows for the PEM electrolyzer to work under a wide range of power input. The high-pressure operation of an electrolyzer brings the advantage of delivering hydrogen at high pressure for the end-user, thus requiring less energy to further compress and store the hydrogen [START_REF] Carmo | A comprehensive review on PEM water electrolysis[END_REF].

Problems related to higher operational pressures in PEM electrolysis are also present, such as cross-permeation phenomenon which increases with pressure. The corrosive acidic regime provided by the PEM requires the use of distinct materials. These materials must not only resist the hard corrosive low pH (around 2) condition but also sustain the high applied over-potential (around 2 V), especially at high current densities. Corrosion resistance applies not only for the catalysts used but also for current collectors and separator plates. Only a few materials can be selected that would perform in this hard environment [START_REF] Carmo | A comprehensive review on PEM water electrolysis[END_REF].

The reactions in the PEM electrolyzer are:

Anode: H 2 O l → 1 2 O 2g + 2H + g + 2e - Cathode: 2H + + 2e -→ H 2g Overall reaction: H 2 O l → H 2g + 1 2 O 2g

Dynamic issues

Over the last decades, so many contributions have been reported in the literature about the development of static models based on semi-empirical and empirical equations [START_REF] Rahim | An overview of polymer electrolyte membrane electrolyzer for hydrogen production: Modeling and mass transport[END_REF][START_REF] Olivier | Low-temperature electrolysis system modelling: A review[END_REF]. However, the modeling of the dynamics is generally neglected due to a few studies developed on it. Like fuel cells, electrolyzers are electrochemical devices and their response time to reach a steady-state operation may be slow or fast depending on the operating conditions. Recently, a first investigation [START_REF] Guilbert | Experimental validation of an equivalent dynamic electrical model for a proton exchange membrane electrolyzer[END_REF] has been provided on PEM electrolyzer responses based on dynamic current profiles. It has been demonstrated that the responses are linked with the operating conditions, and consequently, the modeling of these dynamics is crucial to make reliable and accurate the model.

In Figures 2 and3, two dynamic operations are reported and have been carried out by using a commercial-400W

PEM electrolyzer. The specifications of the studied PEM electrolyzer can be found in [START_REF] Guilbert | Experimental validation of an equivalent dynamic electrical model for a proton exchange membrane electrolyzer[END_REF]. The first test has been performed with a step current from 0 to 10 A; while the second with a step current from 10 to 0 A. In both tests, it can be noted that the PEM electrolyzer responds immediately as a result of the step current. It is one of the most important features required for electrolyzers when coupling with RES since they are dynamic sources which their operation strongly depends on the weather conditions. The immediate response is followed by a slow voltage rise or fall before the electrolyzer reaches its steady-state operation. This slow response is linked with the dynamics of the reactions both at the anode and the cathode as reported in [START_REF] Guilbert | Experimental validation of an equivalent dynamic electrical model for a proton exchange membrane electrolyzer[END_REF].

This emphasizes that the dynamics are currently important issues and must be taken into consideration in modeling the PEM electrolyzer by offering more accuracy. Besides, since the dynamics can be slower or faster according to the input energy, it is crucial to develop accurate and reliable models based on adaptive parameters.

Electrical domain modeling

Several PEM fuel cell models have been reported, but so few PEM electrolyzers models have been [START_REF] Rahim | An overview of polymer electrolyte membrane electrolyzer for hydrogen production: Modeling and mass transport[END_REF][START_REF] Nie | Numerical and experimental study of threedimensional fluid flow in the bipolar plate of a PEM electrolysis cell[END_REF]. In recent years, the modeling of PEM electrolyzers has gained a growing interest from industrials and researchers to optimize, design and use the PEM electrolyzer in a good way, see Figure 4. In this section, static models based on semi-empirical and empirical equations are presented and analyzed. Besides, since the dynamics modeling has been neglected in previous reviews [START_REF] Rahim | An overview of polymer electrolyte membrane electrolyzer for hydrogen production: Modeling and mass transport[END_REF][START_REF] Olivier | Low-temperature electrolysis system modelling: A review[END_REF], it is reported in this section based on recently published works.

This enables to highlighting the remaining issues for dynamics modeling. A polarization curve is an important tool not only for the study of PEM electrolyzers but, for all kinds of electrolyzers, describes the relationship between the cell voltage V cell and the cell current I cell , for alkaline electrolyzer see the works [START_REF] Ulleberg | Modeling of advanced alkaline electrolyzers: A system simulation approach[END_REF][START_REF] Ursúa | Influence of the power supply on the energy efficiency of an alkaline water electrolyser[END_REF][START_REF] Smolinka | Water electrolysis[END_REF][START_REF] Tijani | Mathematical modelling and simulation analysis of advanced alkaline electrolyzer system for hydrogen production[END_REF][START_REF] Ursúa | Static-dynamic modelling of the electrical behaviour of a commercial advanced alkaline water electrolyser[END_REF][START_REF] Zouhri | Evaluation and optimization of the alkaline water electrolysis ohmic polarization: Exergy study[END_REF][START_REF] Sánchez | Semi-empirical model and experimental validation for the performance evaluation of a 15 kW alkaline water electrolyzer[END_REF][START_REF] Diéguez | Thermal performance of a commercial alkaline water electrolyzer: Experimental study and mathematical modeling[END_REF][START_REF] Sandeep | Experimental studies and modeling of advanced alkaline water electrolyser with porous nickel electrodes for hydrogen production[END_REF][START_REF] Milewski | Modeling an alkaline electrolysis cell through reduced-order and loss-estimate approaches[END_REF][START_REF] Gonzatti | Mathematical and experimental basis to model energy storage systems composed of electrolyzer, metal hydrides and fuel cells[END_REF][START_REF] Khan | Analysis of a small wind-hydrogen stand-alone hybrid energy system[END_REF][START_REF] Hammoudi | New multi-physics approach for modelling and design of alkaline electrolyzers[END_REF][START_REF] Henao | Simulation tool based on a physics model and an electrical analogy for an alkaline electrolyser[END_REF][START_REF] Busquet | A new approach to empirical electrical modelling of a fuel cell, an electrolyser or a regenerative fuel cell[END_REF][START_REF] Speckmann | Influence of rectifiers on the energy demand and gas quality of alkaline electrolysis systems in dynamic operation[END_REF][START_REF] Koponen | Effect of converter topology on the specific energy consumption of alkaline water electrolyzers[END_REF][START_REF] Artuso | Alkaline electrolysers: Model and real data analysis[END_REF][START_REF] Belmokhtar | Dynamic model of an alkaline electrolyzer based an artificial neural networks[END_REF][START_REF] Hug | Intermittent operation and operation modeling of an alkaline electrolyzer[END_REF] and for SO electrolyzer . Since the current directly corresponds to the production rate by Faraday's law, the polarization curve gives a correlation between electric power and the production rate [START_REF] Ursúa | Influence of the power supply on the energy efficiency of an alkaline water electrolyser[END_REF][START_REF] Tjarks | Energetically-optimal PEM electrolyzer pressure in power-to-gas plants[END_REF]. The polarization curve in one cell can be expressed like the product of V cell with and I cell .

P cell = V cell • I cell (1)
and the polarization curve in the stack can be expressed similarly like the product of the stack voltage V S with the stack current I S .

P S = V S • I S (2)
As the current in the electrolyzer is direct from one source the study is focused on the voltage of the electrolyzer.

The voltage at the stack terminals is the sum of the voltages of the number of cells N connected in serie, and the stack current is the sum of currents flowing through the number of cells N branches connected in parallel [START_REF] Ursúa | Hydrogen production from water electrolysis: Current status and future trends[END_REF], see Table 1.

Table 1: Kind of V S and I S depending on electrolyzer configuration.

Parameter

Parallel Serie

V S V cell N j=1 V cell,j I S N j=1 I cell,j I cell
However, it is generally assumed that all the cells of a stack have the same voltage and current. As it is widely known the real cell voltage in a PEM electrolyzer cell can be expressed as the sum of the reversible potential and its over-potentials:

V cell = V rev + η act + η ohm + η con (3)
where V rev is the reversible potential, η act , η ohm , and η con are the activation, ohmic and concentration overpotentials. In Table 2 is summarized the papers about the reversible voltage and over-potentials and in Figure 5 is shown the number of works about these over-potentials for PEM electrolyzers.

Semi-empirical modeling

First of all, a simple way to model the current-voltage curve is to use a semi-empirical equation. The most known semi-empirical model has been developed by Ulleberg in 1998 [START_REF] Ulleberg | Stand-alone power systems for the future: optimal design, operation and control of solarhydrogen energy systems[END_REF] for alkaline electrolyzers but can be applied to PEM electrolyzers. It is based on thermodynamics basics and empirical electrochemical expressions.

The semi-empirical model is described by the following expression:

V cell = V rev + r • I cell + s • log[k • I cell + 1] (4)
The equation ( 4) can be determined by a static characterization (i.e. the parameter is considered as constant)

or by using the empirical equation depending on the change in Gibbs energy, pressure, and temperature (see 3.1.3). In comparison, the second term is linked with the ohmic over-potential η ohm by its parameter r. Finally, the last term is related to the activation over-potential η act based on two parameters s and k.

As reported in [START_REF] Ulleberg | Modeling of advanced alkaline electrolyzers: A system simulation approach[END_REF], the temperature has a strong influence on the performance of the electrolyzer described by its current-voltage curve. For this reason, the effect of temperature has to be taken into account in the previous semi-empirical equation ( 4). Among the parameters, only the parameters r and k depend on the temperature T and the electrode area A; while the parameter s is considered constant. Hence, the semi-empirical model based on the temperature T and the electrode area A is given as follows [START_REF] Ulleberg | Modeling of advanced alkaline electrolyzers: A system simulation approach[END_REF]:

V cell = V rev + r 1 + r 2 • T A • I cell + s • log k 1 • T 2 + k 2 • T + k 3 A • T 2 • I cell + 1 (5)
In spite of Ulleberg's model is accurate to describe the current-voltage curve according to the temperature, it does not include other parameters such as the pressure of the gas which has a great influence on the performance of the electrolyzer. Hence, some authors have modified Ulleberg's model by adding gas pressure to obtain a more reliable model. The modified Ulleberg's model taking into account the gas pressure is expressed by the following equation [START_REF] Sánchez | Semi-empirical model and experimental validation for the performance evaluation of a 15 kW alkaline water electrolyzer[END_REF]:

V cell = V rev + [(r 1 + d 1 ) + r 2 • T + d 2 • P ] • I cell + s • log k 1 + k 2 T + k 3 T 2 • I cell + 1 (6)
To take into consideration the gas pressure P , additional parameters d 1 and d 2 can be found in equation [START_REF] Elliston | Simulations of scenarios with 100% renewable electricity in the Australian National Electricity Market[END_REF].

These parameters are related to the linear change in the ohmic over-potential. The different parameters of the model can be determined by using the experimental data (according to the pressure and temperature) and the least-squares regression algorithm. Finally, the stack voltage of the electrolyzer can be deduced from Table 1.

Empirical modeling

To improve the accuracy of the current-voltage curve a lot of authors have studied and described each term that makes up the equation (3) (i.e. the reversible potential and its over-potentials) of an empirical way.

• Reversible potential

The reversible potential or open circuit voltage is the minimum required voltage for the electrolysis can be carried out in the electrolyzer [START_REF] Ursúa | Hydrogen production from water electrolysis: Current status and future trends[END_REF][START_REF] Rahim | An overview of polymer electrolyte membrane electrolyzer for hydrogen production: Modeling and mass transport[END_REF]. The reversible potential at the cell has a relation with the from Gibbs free energy:

∆G = ∆H -T • ∆S (7)
and the reversible potential is expressed:

V rev = ∆G n • F (8)
where, at standard condition, the temperature is T = 298.15 K, the pressure in the electrolyzer is 1 atm, ∆S = 0.1631 kJ mol -1 K -1 is the entropy chance, ∆H =285.84 kJ mol -1 is the enthalpy change, n = 2 is the number of the electron moles transferred per hydrogen mole, and F = 96485 C mol -1 is the Faraday's constant.

By these conditions and equations, ( 7) and (8) many authors work with V rev = 1.23 V [START_REF] Rahim | An overview of polymer electrolyte membrane electrolyzer for hydrogen production: Modeling and mass transport[END_REF].

Another important concept related whit the reversible potential and voltage efficiency is the thermo-neutral potential V th , which is obtained employing enthalpy ∆H in its place of ∆G for the potential calculation [START_REF] Kumar | Hydrogen production by pem water electrolysis -a review[END_REF].

V th = ∆H n • F = ∆G n • F + T • ∆S n • F (9)
where at standard condition, V th = 1.48 V .

• Activation over-potential

The activation over-potential represents the over-potential to initiate the proton transfer and the electrochemical kinetic behavior in the PEM electrolyzer. Some portion of the applied voltage is lost as a result of transferring the electrons to or from the electrodes during chemical reactions at the electrodes [START_REF] Ursúa | Hydrogen production from water electrolysis: Current status and future trends[END_REF][START_REF] Rahim | An overview of polymer electrolyte membrane electrolyzer for hydrogen production: Modeling and mass transport[END_REF]. The activation energy required at both the anode and the cathode due to the activation over-potential can be modeled by relating the Butler-Volmer expression as many authors have done:

η act = η act,a + η act,c (10)
where

η act,a = R • T α a • z • F • log i a i 0,a , (11)
η act,c = R • T α c • z • F • log i c i 0,c (12)
where R = 8.314 J K -1 mol -1 is the universal gas constant, z is the stoichiometric coefficient refers to the number of electrons transferred in the global semi reactions defined by Faraday's law. The value of the stoichiometric coefficient in water electrolysis is 2. α a and α c are the charge transfer coefficients. The values of α a and α c are 0.5 on the symmetry reactions. However, in [START_REF] Harrison | Semiempirical model for determining PEM electrolyzer stack characteristics[END_REF] has been demonstrated that α a and α c are variables, for example, it can be seen in [START_REF] Marangio | Theoretical model and experimental analysis of a high pressure PEM water electrolyser for hydrogen production[END_REF] values for α a = 2 and α c = 0.5.

• Ohmic over-potential

The ohmic over-potential is the resistance caused by the flow of electrons and the electronic resistance of the PEM electrolyzer. This ohmic over-potential depends on the type of PEM, and electrode material. The ohmic over-potential due to membrane resistance, or ionic resistance, is the resistance to the proton transport through the PEM. Meanwhile, electronic resistance is caused by electronic materials such as bipolar plates, electrodes current collectors, etc. The ohmic over-potential is linearly proportional to the current [START_REF] Ursúa | Hydrogen production from water electrolysis: Current status and future trends[END_REF][START_REF] Rahim | An overview of polymer electrolyte membrane electrolyzer for hydrogen production: Modeling and mass transport[END_REF]. The membrane over-potential can be expressed as a function of the membrane thickness φ (cm), the conductivity of the membrane σ mem and i 0 .

η ohm,mem = R ion • i 0 ( 13
)
where R ion = φ/σ mem is the ionic resistance. The local ionic conductivity with water content and temperature function can be written as:

σ mem = (0.005139λ -0.00326) • exp 1268 • 1 303 - 1 T ( 14
)
where λ is the degree of membrane humidification and it can be defined as:

λ = 0.08533 • T -6.77632 (15)
The electronic over-potential can be expressed as:

η ohm,ele = R ele • i 0 (16)
The ohmic resistance of the electronic materials can be expressed as a function of the material resistivity ρ in (Ωm), the length of the path of the electrons l, and the conductor cross-sectional area A c :

R ele = ρ • l A c (17)
Therefore, as a result of the ionic resistance and electronic resistance, the ohmic over-potential can be expressed as:

η ohm = η ohm,mem + η ohm,ele (18)
• Concentration over-potential

The concentration over-potential or diffusion over-potential is caused by mass transport processes. In the case of water electrolysis, the biggest influence is likely due to the product transport limitations, if H 2 and O 2 are not removed as fast as they are produced their concentration in the reaction site increases slowing the reaction kinetics [START_REF] Ursúa | Hydrogen production from water electrolysis: Current status and future trends[END_REF][START_REF] Marangio | Theoretical model and experimental analysis of a high pressure PEM water electrolyser for hydrogen production[END_REF]. It has been reported that V con is much lower than V ohm and V act , for this reason, many authors take V con = 0 in their studies. However, the concentration over-potential is widely estimated with the Nernst's equation.

η con = η con,a + η con,c (19)
where

η con,a = R • T z • F • In C O2,mem C O2,mem,0 (20)
η con,c = R • T z • F • In C H2,mem C H2,mem,0 (21)
where C O2,mem and C H2,mem are the oxygen and hydrogen concentration at the membrane-electrode interface, respectively.

Dynamic modeling

In recent years, some authors have found a way to assimilate the electrolyzer like an electronic circuit as is showed in Figure 6, intending to study the present dynamics in the current-voltage curve. This circuit is composed of the following components [START_REF] Guilbert | Experimental validation of an equivalent dynamic electrical model for a proton exchange membrane electrolyzer[END_REF]:

-Two resistance-capacitor branches to model the dynamics both at the anode and cathode (activation overpotential).

-One resistance to model the membrane (ohmic over-potential).

-A DC voltage source (reversible potential). From Figure 6, the stack voltage of the electrolyzer is expressed as follows:

V cell = E rev + η act,a + η act,c + η ohm (22)
Furthermore, the dynamic equations for η act,a and η act,c are:

dη act,a dt = 1 C a • I cell - 1 τ a • η act,a (23)
dη act,c

dt = 1 C c • I cell - 1 τ c • η act,c (24)
where C a and C c are the capacitors for anode and cathode, R a and R c are the resistances for anode and cathode.

The time constants τ a and τ c governing the dynamics are variables according to the operating conditions at the input of the electrolyzer. The equivalent resistances R a and R c are determined based on the activation voltage and input current of the electrolyzer:

τ a = C a • R a = C a • η act,a I cell , (25)
τ c = C c • R c = C c • η act,c I cell (26)
In [START_REF] Guilbert | Experimental validation of an equivalent dynamic electrical model for a proton exchange membrane electrolyzer[END_REF], the authors have demonstrated that the dynamics in the PEM electrolyzer are slower compared to a PEM fuel cell [START_REF] Pukrushpan | Chapter three -fuel cell system model: Fuel cell stack[END_REF]. Indeed, the equivalent capacitor is equal to 37 F compared to 3 F obtained in PEM fuel cell.

Finally, the dynamics of the PEM electrolyzer are mainly governed by the anode reaction since its dynamic is slower compared to the cathode reaction [START_REF] Guilbert | Experimental validation of an equivalent dynamic electrical model for a proton exchange membrane electrolyzer[END_REF]. In the same way, it is possible to model the ohmic over-potential like an electronic circuit, see Figure 6, using Ohm's law.

η ohm = R mem • I cell (27)
where R mem represents the membrane resistance.

Based on the work reported in [START_REF] Guilbert | Experimental validation of an equivalent dynamic electrical model for a proton exchange membrane electrolyzer[END_REF], a comparison between a static and dynamic modeling (taking into consideration the electrical circuit in Figure 6) is provided in Figure 7. It can be noted that the dynamic model offers more accuracy and reliability in modeling the real behavior of a PEM electrolyzer supplied by dynamic current profiles. Indeed, the maximum error obtained with the dynamic model is around 4%; while the static model around 15%. The errors for the static model are particularly noticeable during the transients.

Finally, the parameters of the electrical circuit can be determined by using static and dynamic model identification as reported in [START_REF] Guilbert | Experimental validation of an equivalent dynamic electrical model for a proton exchange membrane electrolyzer[END_REF]. However, the determined parameters are valid for a specific current range since the dynamics of the electrolyzer may change according to the operating conditions. For this reason, the development of models based on adaptive parameters is a challenging issue to make the model more reliable and precise.

Analysis of the efficiency and specific energy consumption

The efficiency in an electrolyzer indicates the performance of the device if it has adequate energy and voltage consumption to the hydrogen production of the device. Few authors take into considerations this study for PEM electrolyzer, see Table 3 and Figure 8. The main efficiencies are the Faraday (η f ), voltage (η V ), and electrolyzer (η ele ). The three efficiency are linked by the following expression:

η ele = η f • η V (28)
Among the three efficiency, the Faraday's efficiency is a key parameter since it is related to the losses due to the gas diffusion [START_REF] Baik | Characterization of nitrogen gas crossover through the membrane in protonexchange membrane fuel cells[END_REF][START_REF] Yan | Investigation of water transport through membrane in a PEM fuel cell by water balance experiments[END_REF][START_REF] Springer | Polymer electrolyte fuel cell model[END_REF][START_REF] Zawodzinski | Water uptake by and transport through Nafion R 117 membranes[END_REF]. This parameter which is linked with the operating conditions (i.e. current, temperature, pressure) has a great influence on the energy efficiency η ele and the gas flow rates (i.e. oxygen and hydrogen)

as well [START_REF] Görgün | Dynamic modelling of a proton exchange membrane (PEM) electrolyzer[END_REF][START_REF] Marangio | Theoretical model and experimental analysis of a high pressure PEM water electrolyser for hydrogen production[END_REF]. Besides, the oxygen and hydrogen flow rates are given by the following equations:

f O2 = N • I cell 4F • η f ( 29
)
f H2 = N • I cell 2F • η f (30)
The Faraday's efficiency is investigated in the next subsection to highlight its role in the electrolysis process.

Faraday's efficiency

The Faraday's efficiency or current efficiency is relevant to model the production of hydrogen and oxygen in the electrolyzer. Usually is expressed as the ratio of the ideal electric charge Q id over the real electric charge Q re required for the production of a given amount of hydrogen or as the ratio of the hydrogen calculated over the real hydrogen produced [START_REF] Ursúa | Hydrogen production from water electrolysis: Current status and future trends[END_REF][START_REF] Ulleberg | Modeling of advanced alkaline electrolyzers: A system simulation approach[END_REF][START_REF] Tijani | Mathematical modelling and simulation analysis of advanced alkaline electrolyzer system for hydrogen production[END_REF][START_REF] Schalenbach | Pressurized PEM water electrolysis: Efficiency and gas crossover[END_REF]. A Faraday's efficiency lower than 1 is caused by current losses and by the hydrogen that is lost in the membrane [START_REF] Ursúa | Hydrogen production from water electrolysis: Current status and future trends[END_REF][START_REF] Ursúa | Influence of the power supply on the energy efficiency of an alkaline water electrolyser[END_REF]. The Faraday's efficiency decreases significantly at low current densities [START_REF] Grigoriev | High-pressure PEM water electrolysis and corresponding safety issues[END_REF].

η f = Q id Q re ( 31
)
First of all, based on second Faraday's law of electrolysis [START_REF] Millet | Electrochemical performances of PEM water electrolysis cells and perspectives[END_REF] and on the general gas equation [START_REF] Nie | Numerical and experimental study of threedimensional fluid flow in the bipolar plate of a PEM electrolysis cell[END_REF], the expression of the theoretical hydrogen quantity can be determined [START_REF] Ulleberg | Modeling of advanced alkaline electrolyzers: A system simulation approach[END_REF]:

Q rev = I cell • t = n • F • z (32)
P • V H2 = n • R • T (33)
V H2 (theoretical) = R • I cell • t • T F • P • z (34)
Q rev = P • V H2 • F • z R • T (35)
where Q rev is the total electric charge passed through the gas [C]; t is the total time the constant PEM electrolyzer current was applied [s], V H2 is the gas volume [m 3 ].

Over the last decades, a few works have been reported in the literature regarding the investigation of Faraday's efficiency [START_REF] Ulleberg | Modeling of advanced alkaline electrolyzers: A system simulation approach[END_REF][START_REF] Tijani | Mathematical modelling and simulation analysis of advanced alkaline electrolyzer system for hydrogen production[END_REF][START_REF] Hug | Intermittent operation and operation modeling of an alkaline electrolyzer[END_REF][START_REF] Tjarks | Energetically-optimal PEM electrolyzer pressure in power-to-gas plants[END_REF][START_REF] Schalenbach | Pressurized PEM water electrolysis: Efficiency and gas crossover[END_REF][START_REF] Tijani | Numerical modeling the effect of operating variables on Faraday efficiency in PEM electrolyzer[END_REF][START_REF] Ulleberg | Stand-alone power systems for the future: optimal design, operation and control of solarhydrogen energy systems[END_REF][START_REF] Khalilnejad | Optimal design of hybrid wind/photovoltaic electrolyzer for maximum hydrogen production using imperialist competitive algorithm[END_REF]. The major part of the reported works is focused on the alkaline electrolyzer. In the literature, two empirical models have been developed to express the Faraday's efficiency according to the current for alkaline electrolyzers. The first model has been proposed by Hug [START_REF] Hug | Intermittent operation and operation modeling of an alkaline electrolyzer[END_REF] in 1993; while the second one has been developed by Ulleberg [START_REF] Ulleberg | Stand-alone power systems for the future: optimal design, operation and control of solarhydrogen energy systems[END_REF]. The first model [START_REF] Hug | Intermittent operation and operation modeling of an alkaline electrolyzer[END_REF] based on five-parameters provides the Faraday's efficiency at any temperature according to the current supplying the electrolyzer:

η f = B 1 + B 2 • exp B 3 + B 4 • T + B 5 • T 2 I cell (36)
This semi-empirical model can give a good agreement with the experimental data at any temperature. Since the model is based on five parameters, they can be determined by performing tests at different temperatures and using a least-square regression algorithm. However, the model does not take into account the effect of other variables, such as the pressure of the gas.

In comparison, Ulleberg [START_REF] Ulleberg | Stand-alone power systems for the future: optimal design, operation and control of solarhydrogen energy systems[END_REF] has developed an empirical model to estimate the Faraday's efficiency according to the temperature and current. It is based on four parameters described by:

η f = I 2 cell f 11 + f 12 • T + I 2 cell • (f 21 + f 22 • T ) (37)
Compared to Hug's model, Ulleberg's model includes fewer parameters, only four against five. The four parameters can be determined in the same way. However, the effect of pressure is neglected in the model; while the pressure has an impact on Faraday's efficiency.

The equation ( 37) is reported in several works focused on electrolyzer modeling [START_REF] Sánchez | Semi-empirical model and experimental validation for the performance evaluation of a 15 kW alkaline water electrolyzer[END_REF][START_REF] Khalilnejad | Optimal design of hybrid wind/photovoltaic electrolyzer for maximum hydrogen production using imperialist competitive algorithm[END_REF][START_REF] Onar | Dynamic modeling, design and simulation of a wind/fuel cell/ultracapacitor-based hybrid power generation system[END_REF][START_REF] Sarrias-Mena | Electrolyzer models for hydrogen production from wind energy systems[END_REF]. However, this semi-empirical equation is only valid for the alkaline electrolyzer and cannot be applied to PEM electrolyzer due to the different materials used for their manufacturing. In [START_REF] Tijani | Numerical modeling the effect of operating variables on Faraday efficiency in PEM electrolyzer[END_REF], Tijani and Abdol Rahim have investigated the influence of operating temperature and membrane thickness on the Faraday's efficiency for a PEM electrolyzer.

However, the authors have not proposed a model to express the Faraday's efficiency.

Voltage efficiency

The voltage efficiency is the efficiency regarding the voltage used for the electrolyzer. Generally, the Faraday's efficiency is higher than the voltage efficiency, since it only takes into consideration the losses due to the gas diffusion [START_REF] Schalenbach | Pressurized PEM water electrolysis: Efficiency and gas crossover[END_REF][START_REF] Schalenbach | Corrigendum to 'pressurized PEM water electrolysis: Efficiency and gas crossover[END_REF]. By comparison, voltage efficiency takes into account additional losses such as membrane losses and heat losses both in the anode and cathode. Voltage efficiency is defined as the ratio of the thermo-neutral potential over the cell voltage [START_REF] Ulleberg | Modeling of advanced alkaline electrolyzers: A system simulation approach[END_REF][START_REF] Tijani | Mathematical modelling and simulation analysis of advanced alkaline electrolyzer system for hydrogen production[END_REF][START_REF] Schalenbach | Pressurized PEM water electrolysis: Efficiency and gas crossover[END_REF].

η V = V th V cell (38)

Electrolyzer efficiency and specific energy consumption

Finally, the system efficiency is obtained by the ratio between the energy contained in the hydrogen produced and the energy needed to electrolyze the water consumed during the process [START_REF] Ursúa | Hydrogen production from water electrolysis: Current status and future trends[END_REF]. Thus, the electrolyzer efficiency can be obtained by the equation [START_REF] Bideau | Review of necessary thermophysical properties and their sensivities with temperature and electrolyte mass fractions for alkaline water electrolysis multiphysics modelling[END_REF] or by the ratio between the Higher Heating Value of hydrogen HHV H2 , which has a value of 3.54 kW h/N m 3 , and the energy consumption C E [START_REF] Ursúa | Hydrogen production from water electrolysis: Current status and future trends[END_REF][START_REF] Ursúa | Influence of the power supply on the energy efficiency of an alkaline water electrolyser[END_REF].

η ele = HHV H2 C E • 100 (39)
where C E relates the energy consumed to the hydrogen produced and can be calculated, for a given time interval ∆t.

C E = ∆t 0 N • I cell • V cell • dt ∆t 0 f H2 • dt (40)

Discussion

First of all, Figure 9 allows summarizing the electrical domain modeling (semi-empirical and empirical) and its link with Faraday's efficiency, voltage efficiency, energy efficiency, and specific energy consumption. It can be noted that the voltage efficiency is linked with the electrical domain modeling (i.e. V cell ) but the Faraday's efficiency remains a key parameter for hydrogen production and the specific energy consumption, providing valuable information about the performance of the electrolyzer. As highlighted in this section, the investigation of the Faraday's efficiency has received less attention compared to the electrical domain modeling; whereas it has a great influence on the performance of the electrolyzer. Indeed, Faraday's efficiency related to the gas diffusion losses strongly depends on the operating conditions (i.e. pressure, temperature) and also the features of the electrolyzer (i.e. membrane thickness). Currently, Faraday's efficiency is based on two main empirical equations for alkaline electrolyzers developed in the 1990s. Since that period, no additional work has been reported in the literature on Faraday's efficiency modeling, especially for PEM electrolyzers. For this reason, since the membrane thickness of PEM electrolyzers is smaller compared to alkaline electrolyzers, further investigations are required to understand better the effects of operating conditions and thickness of the membrane on Faraday's efficiency.

Besides, the dynamics of PEM electrolyzers are very important issues and have to be taken into consideration in modeling accurately the behavior of electrolyzers when coupling with RES. Furthermore, since the behavior of the electrolyzer is linked with the operating conditions, further analysis of the different behaviors are requested to develop accurate models with adaptive parameters.

Conclusion

The PEM electrolyzers, independently for industrial, household, etc. needs, can produce hydrogen highly pure.

Besides, they can be coupled with RES, carried out a 100% renewable system due to their fast response time.

It is one of the most important requirements for electrolyzer when coupling with dynamic sources.

Over the last decade, there has been an increase in works focused on electrical domain modeling. Generally, it is based on semi-empirical and empirical equations. However, the investigation on the dynamics and the Faraday's efficiency have received less attention while they are important issues to take into consideration when modeling the electrolyzer. Indeed, on one hand, the study of the dynamics allows developing models more accurate and reliable compared to static models. On the other hand, since the dynamics are strongly linked with the operating conditions, the development of accurate models based on adaptive parameters is required.

On the other side, further investigations are required regarding the effects of operating conditions on Faraday's efficiency. Hence, new empirical models could be developed to estimate the performance of the electrolyzer according to the operating conditions.

Figure 1 :

 1 Figure 1: Schematic illustration of a basic water PEM electrolysis system.

Figure 2 :

 2 Figure 2: Response of the electrolyzer as a result of a rising electrical current from 0 to 10 A.

Figure 3 :

 3 Figure 3: Response of the electrolyzer as a result of a falling electrical current from 10 to 0 A.

Figure 4 :

 4 Figure 4: Evolution of the number of works about voltage to the year for the PEM electrolyzer.

  Electrical domain V cell = V rev + η act + η ohm V cell = V rev + η act + η ohm + η con PEM[START_REF] García-Valverde | Simple PEM water electrolyser model and experimental validation[END_REF][START_REF] Görgün | Dynamic modelling of a proton exchange membrane (PEM) electrolyzer[END_REF][START_REF] Rahim | An overview of polymer electrolyte membrane electrolyzer for hydrogen production: Modeling and mass transport[END_REF][START_REF] Olivier | Low-temperature electrolysis system modelling: A review[END_REF][START_REF] Kumar | Hydrogen production by pem water electrolysis -a review[END_REF][START_REF] Guilbert | Experimental validation of an equivalent dynamic electrical model for a proton exchange membrane electrolyzer[END_REF],[START_REF] Smolinka | Water electrolysis[END_REF] [[START_REF] Ursúa | Hydrogen production from water electrolysis: Current status and future trends[END_REF][START_REF] Abdin | Modelling and simulation of a proton exchange membrane (PEM) electrolyser cell[END_REF][START_REF] Carmo | A comprehensive review on PEM water electrolysis[END_REF][START_REF] Tjarks | Energetically-optimal PEM electrolyzer pressure in power-to-gas plants[END_REF][START_REF] Ruuskanen | Design and implementation of a power-hardware-in-loop simulator for water electrolysis emulation[END_REF][START_REF] Tijani | Numerical modeling the effect of operating variables on Faraday efficiency in PEM electrolyzer[END_REF][START_REF] Ruuskanen | Power quality estimation of water electrolyzers based on current and voltage measurements[END_REF][START_REF] Toghyani | Optimization of operating parameters of a polymer exchange membrane electrolyzer[END_REF][START_REF] Koponen | Control and energy efficiency of pem water electrolyzers in renewable energy systems[END_REF][START_REF] Espinosa-López | Modelling and experimental validation of a 46 kw pem high pressure water electrolyzer[END_REF][START_REF] Marangio | Theoretical model and experimental analysis of a high pressure PEM water electrolyser for hydrogen production[END_REF][START_REF] Lebbal | Identification and monitoring of a PEM electrolyser based on dynamical modelling[END_REF][START_REF] Tijani | A study of the loss characteristic of a high pressure electrolyzer system for hydrogen production[END_REF][START_REF] Ruuskanen | PEM water electrolyzer model for a power-hardware-in-loop simulator[END_REF][START_REF] Liso | Modelling and experimental analysis of a polymer electrolyte membrane water electrolysis cell at different operating temperatures[END_REF]

Figure 5 :

 5 Figure 5: Number of works about the different over-potentials for the PEM electrolyzer.

Figure 6 :

 6 Figure 6: Equivalent circuit for a PEM electrolyzer.

Figure 7 :

 7 Figure 7: Comparison between static and dynamic modeling.

Figure 8 :

 8 Figure 8: Number of works about efficiency for the PEM electrolyzer.

Figure 9 :

 9 Figure 9: Diagram of connection between the models to give the electrolyzer efficiency.

Table 2 :

 2 Works about the reversible potential and the over-potentials.

Table 3 :

 3 Works about the different efficiencies in the PEM electrolyzer.

	Electrolyzer	η f	η V	η ele
	PEM	[18, 21, 27, 30, 36],	[30, 36, 81-83, 108]	[18, 31, 36, 81-83, 90, 98, 102]
		[81-83, 108, 117, 120]		[105, 106, 109-111, 117, 120]