Julien Christmann

Christian Ley
email: christian.ley@uha.fr

Xavier Allonas

Céline Ahmad Ibrahim

Céline Croutxé-Barghorn

Experimental and theoretical investigations of free radical photopolymerization: Inhibition and termination reactions

Keywords:

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

Introduction

Photopolymerization is nowadays a high-performance technology for the synthesis of polymer materials in various industrial fields such as coatings (paints, inks, varnishes…), dental resins, automotive, holographic data storage, stereolithography, etc. Several advantages over the thermally-induced polymerization process make photopolymerization an eco-friendly technology, with time, money and energy savings, waste reduction, absence of solvent, spatio-temporal control of the reaction, etc. [START_REF]Radiation Curing in Polymer Science and Technology Volume I: Fundamentals and Methods[END_REF][START_REF] Fouassier | Photoinitiation, Photopolymerization, and Photocuring: Fundamentals and Applications[END_REF][START_REF] Decker | Photoinitiated crosslinking polymerization[END_REF][START_REF] Yagci | Photoinitiated polymerization: advances, challenges, and opportunities[END_REF]. The photopolymerization process basically consists in the conversion of a liquid monomer mixture into a solid three-dimensional polymer network under light irradiation. Depending on the photogenerated initiating species, different types of photopolymerization processes can be distinguished: radical, cationic and anionic. Among them, free radical photopolymerization (FRP) remains the main process currently implemented [START_REF] Nesvadba | Radical polymerization in industry[END_REF].

The FRP mechanism consists in a complex combination of reactions which can be divided into three main steps: photoinitiation, propagation and termination [START_REF] Fouassier | Photoinitiation, Photopolymerization, and Photocuring: Fundamentals and Applications[END_REF][START_REF] Decker | Photoinitiated crosslinking polymerization[END_REF][START_REF] Odian | Principles of Polymerization[END_REF][START_REF] Gnanou | Organic and Physical Chemistry of Polymers[END_REF][START_REF] Andrzejewska | Photopolymerization kinetics of multifunctional monomers[END_REF][START_REF] Ibrahim | Investigation of termination reactions in free radical photopolymerization of UV powder formulations[END_REF]. During the photoinitiation, initiating radicals R • are produced under light exposure. A photoinitiating system (PIS) is thus generally required to convert photons into such a chemical potential [START_REF] Fouassier | Photoinitiation, Photopolymerization, and Photocuring: Fundamentals and Applications[END_REF][START_REF] Fouassier | Photoinitiators for Polymer Synthesis: Scope, Reactivity, and Efficiency[END_REF][START_REF] Green | Industrial Photoinitiators: A Technical Guide[END_REF][START_REF] Fouassier | Photoinitiators for free radical polymerization reactions[END_REF][START_REF] Ibrahim | High-performance photoinitiating systems for free radical photopolymerization. Application to holographic recording[END_REF].

Type-I PIS rely on the bond dissociation of a photoinitiator molecule to yield one (or two) initiating radical(s), while type-II systems associate a photoinitiator with a coinitiator to insure the production of active species. Photoproduced radicals are then able to react onto monomer C=C double bonds and generate chain-initiating radicals R(C=C)1 • . Macroradicals R(C=C)n • grow during the propagation step, through successive additions of monomer units associated to the regeneration of the active radical center after each addition. Termination reactions finally stop the growth of the macromolecules, either by bimolecular (combination, disproportion [START_REF] Odian | Principles of Polymerization[END_REF][START_REF] Gnanou | Organic and Physical Chemistry of Polymers[END_REF][START_REF] Andrzejewska | Photopolymerization kinetics of multifunctional monomers[END_REF][START_REF] Ibrahim | Investigation of termination reactions in free radical photopolymerization of UV powder formulations[END_REF] or primary radical termination PRT [START_REF] Ibrahim | Investigation of termination reactions in free radical photopolymerization of UV powder formulations[END_REF][START_REF] Goodner | Modeling primary radical termination and its effects on autoacceleration in photopolymerization kinetics[END_REF][START_REF] David | Synthesis of carboxy-telechelic oligostyrene by deadend polymerization: evaluation of primary radical termination by kinetic study and kinetic simulation model[END_REF][START_REF] Kelly | Temporal and non-ideal behavior in photopolymers[END_REF][START_REF] Kelly | Holographic photopolymer materials: nonlocal polymerization-driven diffusion under nonideal kinetic conditions[END_REF]) or monomolecular (radical trapping by occlusion [START_REF] Decker | Photoinitiated crosslinking polymerization[END_REF][START_REF] Ibrahim | Investigation of termination reactions in free radical photopolymerization of UV powder formulations[END_REF][START_REF] Fraenkel | Detection of polymerization radicals by paramagnetic resonance[END_REF][START_REF] Kloosterboer | Structure and stability of polyacrylate radicals trapped in a network[END_REF][START_REF] Decker | Radical trapping in photopolymerized acrylic networks[END_REF][START_REF] Bellobono | Kinetic effect of radiation intensity and thermal after-treatment in the photopolymerization of diallyl oxydiethylene dicarbonate[END_REF][START_REF] Best | Electron spin resonance study of radicals in photopolymerized di(meth)acrylate network[END_REF][START_REF] Zhu | Termination of trapped radicals at elevated temperatures during copolymerization of MMA/EGDMA[END_REF][START_REF] Zhu | Radical trapping and termination in freeradical polymerization of methyl methacrylate[END_REF][START_REF]Radiation Curing in Polymer Science and Technology Volume III: Polymerization Mechanism[END_REF][START_REF] Anseth | Radical concentrations, environments, and reactivities during crosslinking polymerizations[END_REF][START_REF] Berchtold | Propagation and termination kinetics of cross-linking photopolymerizations studied using electron paramagnetic resonance spectroscopy in conjunction with near IR spectroscopy[END_REF][START_REF] Wen | A kinetic model for radical trapping in photopolymerization of multifunctional monomers[END_REF][START_REF] Zhang | Study of UV cure kinetics resulting from a changing concentration of mobile and trapped radicals[END_REF][START_REF] Zhang | Modeling and measuring UV cure kinetics of thick dimethacrylate samples[END_REF]) mechanisms. All termination modes should occur simultaneously but the proportion of each of them is expected to evolve during the polymerization, because of the progressive increase of the medium viscosity which limits the species motion as the tridimensional network develops. At low and medium conversion, bimolecular termination is supposed to prevail, while radical trapping by occlusion should be the major termination way when the reaction medium becomes vitrified. Nevertheless, there are only a few studies in the literature dealing with the evolution of the termination modes during the FRP process [START_REF] Ibrahim | Investigation of termination reactions in free radical photopolymerization of UV powder formulations[END_REF].

Radical photopolymerization is also known to be strongly inhibited by molecular oxygen, either by its reaction with the excited state of the photoinitiator or the (macro)radicals to yield inactive peroxy radicals ROO • and R(C=C)nOO • [START_REF] Fouassier | Photoinitiation, Photopolymerization, and Photocuring: Fundamentals and Applications[END_REF][START_REF] Decker | Photoinitiated crosslinking polymerization[END_REF][START_REF] Yagci | Photoinitiated polymerization: advances, challenges, and opportunities[END_REF][START_REF] Nesvadba | Radical polymerization in industry[END_REF][START_REF] Odian | Principles of Polymerization[END_REF][START_REF] Gnanou | Organic and Physical Chemistry of Polymers[END_REF][START_REF] Andrzejewska | Photopolymerization kinetics of multifunctional monomers[END_REF][START_REF] Ibrahim | Investigation of termination reactions in free radical photopolymerization of UV powder formulations[END_REF][START_REF] Fouassier | Photoinitiators for Polymer Synthesis: Scope, Reactivity, and Efficiency[END_REF][START_REF]Radiation Curing in Polymer Science and Technology Volume III: Polymerization Mechanism[END_REF][START_REF]Radiation Curing in Polymer Science and Technology Volume II: Photoinitiating Systems[END_REF][START_REF] Pynaert | Effect of reactive oxygen species on the kinetics of free radical photopolymerization[END_REF][START_REF] Ligon | Strategies to reduce oxygen inhibition in photoinduced polymerization[END_REF]. However, the distinction between the inhibition reactions has only been scarcely addressed in a few papers [START_REF] Gleeson | Effects of absorption and inhibition during grating formation in photopolymer materials[END_REF][START_REF] Gleeson | Modeling the photochemical effects present during holographic grating formation in photopolymer materials[END_REF].

The progressive increase of the medium viscosity and the diffusional control of the elementary reactions mainly contribute to the complexity of the (photo)polymerization process, with effects such as the autoacceleration (increase of the polymerization rate Rp when bimolecular termination becomes restricted) and the autodeceleration (decrease of Rp when propagation also becomes diffusion-controlled) [START_REF] Decker | Photoinitiated crosslinking polymerization[END_REF][START_REF] Odian | Principles of Polymerization[END_REF][START_REF] Gnanou | Organic and Physical Chemistry of Polymers[END_REF][START_REF] Andrzejewska | Photopolymerization kinetics of multifunctional monomers[END_REF][START_REF] Von Trommsdorff | Zur Polymerisation des Methacrylsäuremethylesters[END_REF]. Reaction diffusion termination is also specific to this strong increase of viscosity. Indeed, immobilized macroradicals are able to pursue their growth through the addition of small monomer units by propagation, until their chain-ends come together and react bimolecularly (by combination or disproportion) [START_REF] Decker | Photoinitiated crosslinking polymerization[END_REF][START_REF] Schulz | Über die Polymerisationskinetik in Hochkonzentrierten Systemen[END_REF][START_REF] Soh | Diffusion-controlled vinyl polymerization. II. Limitations on the gel effect[END_REF][START_REF] Stickler | Free-radical polymerization kinetics of methyl methacrylate at very high conversions[END_REF][START_REF] Ballard | Termination-rate coefficients in methyl methacrylate polymerizations[END_REF][START_REF] Buback | Conversion dependence of free-radical polymerization rate coefficients from laser-induced experiments, 1. Butyl acrylate[END_REF][START_REF] Buback | Free-radical polymerization kinetics in extended ranges of temperature, pressure and monomer conversion[END_REF][START_REF] Buback | Free-radical polymerization up to high conversion. A general kinetic treatment[END_REF][START_REF] Buback | Modeling of termination in intermediate and high conversion free radical polymerizations[END_REF][START_REF] Cook | Photopolymerization kinetics of oligo(ethylene oxide) and oligo(methylene) oxide dimethacrylates[END_REF][START_REF] Anseth | Reaction diffusion enhanced termination in polymerizations of multifunctional monomers[END_REF][START_REF] Berchtold | Coupling chain length dependent and reaction diffusion controlled termination in the free radical polymerization of multivinyl (meth)acrylates[END_REF][START_REF] Anseth | Reaction behaviour and kinetic constants for photopolymerizations of multi(meth)acrylate monomers[END_REF][START_REF] Anseth | Polymerization kinetics and volume relaxation behavior of photopolymerized multifunctional monomers producing highly crosslinked networks[END_REF][START_REF] Anseth | Kinetic evidence of reaction diffusion during the polymerization of multi(meth)acrylate monomers[END_REF][START_REF] Anseth | Reaction kinetics and volume relaxation during polymerizations of multiethylene glycol dimethacrylates[END_REF][START_REF] Young | Effect of polymerization temperature and cross-linker concentration on reaction diffusion controlled termination[END_REF][START_REF] White | The influence of N-vinyl pyrrolidone on polymerization kinetics and thermo-mechanical properties of crosslinked acrylate polymers[END_REF].

The corresponding bimolecular termination rate constant kt,b is then proportional to the propagation rate constant kp with a reaction diffusion factor Rrd (Eq. 1). Reaction diffusion termination starts as soon as diffusional bimolecular termination is hindered by the increase of the medium viscosity.

kt,b = Rrdkp [C=C] (1)

Better understanding of the complex FRP mechanism and its related effects is a recurrent topic in polymer science. For that purpose, several modeling methods have been developed, starting with the statistical approach developed by Flory in the 1940's [START_REF] Flory | Molecular size distribution in three dimensional polymers. I. Gelation[END_REF][START_REF] Flory | Molecular size distribution in three dimensional polymers. II. Trifunctional branching units[END_REF][START_REF] Flory | Molecular size distribution in three dimensional polymers. III. Tetrafunctional branching units[END_REF][START_REF] Flory | Statistical Mechanics of Chain Molecules[END_REF] and followed by spatial [START_REF] Manneville | Percolation and gelation by additive polymerization[END_REF][START_REF] Schröder | Computer simulation of network formation via crosslinking copolymerization[END_REF][START_REF] Ghiass | Simulation of non-linear free-radical polymerization using a percolation kinetic gelation model[END_REF][START_REF] Wen | Kinetic gelation modeling: structural inhomogeneity during cross-linking polymerization[END_REF][START_REF] Wen | Kinetic gelation modeling: kinetics of crosslinking polymerization[END_REF] and kinetic modeling [START_REF] Goodner | Modeling primary radical termination and its effects on autoacceleration in photopolymerization kinetics[END_REF][START_REF] David | Synthesis of carboxy-telechelic oligostyrene by deadend polymerization: evaluation of primary radical termination by kinetic study and kinetic simulation model[END_REF][START_REF] Kelly | Temporal and non-ideal behavior in photopolymers[END_REF][START_REF] Kelly | Holographic photopolymer materials: nonlocal polymerization-driven diffusion under nonideal kinetic conditions[END_REF][START_REF] Wen | A kinetic model for radical trapping in photopolymerization of multifunctional monomers[END_REF][START_REF] Zhang | Study of UV cure kinetics resulting from a changing concentration of mobile and trapped radicals[END_REF][START_REF] Zhang | Modeling and measuring UV cure kinetics of thick dimethacrylate samples[END_REF][START_REF] Soh | Diffusion-controlled vinyl polymerization. II. Limitations on the gel effect[END_REF][START_REF] Buback | Free-radical polymerization up to high conversion. A general kinetic treatment[END_REF][START_REF] Buback | Modeling of termination in intermediate and high conversion free radical polymerizations[END_REF][START_REF] Anseth | Reaction diffusion enhanced termination in polymerizations of multifunctional monomers[END_REF][START_REF] Berchtold | Coupling chain length dependent and reaction diffusion controlled termination in the free radical polymerization of multivinyl (meth)acrylates[END_REF]. This last method appears especially interesting as it: i. enables direct comparison between experimental and simulated polymerization kinetics, ii. is relatively simple to implement, iii. gives access to the concentration of species which are difficult or even impossible to obtain from experiment (macroradicals, initiating radicals…). The one developed by Bowman's group is currently the most complete, considering the whole photopolymerization process from light absorption by a type-I PIS to termination reactions [START_REF] Goodner | Modeling primary radical termination and its effects on autoacceleration in photopolymerization kinetics[END_REF][START_REF] Anseth | Reaction diffusion enhanced termination in polymerizations of multifunctional monomers[END_REF][START_REF] Berchtold | Coupling chain length dependent and reaction diffusion controlled termination in the free radical polymerization of multivinyl (meth)acrylates[END_REF][START_REF] Bowman | Coupling of kinetics and volume relaxation during polymerizations of multiacrylates and multimethacrylates[END_REF][START_REF] Goodner | Method for determining the kinetic parameters in diffusion-controlled free-radical homopolymerizations[END_REF][START_REF] Goodner | Development of a comprehensive free radical photopolymerization model incorporating heat and mass transfer effects in thick films[END_REF][START_REF] Lovestead | Models of multivinyl free radical photopolymerization kinetics[END_REF]. It has been notably used to determine the effect of oxygen inhibition on the propagation and termination rate constants [START_REF] Taki | Effect of oxygen inhibition on the kinetic constants of the UV-radical photopolymerization of diurethane dimethacrylate/photoinitiator systems[END_REF], as well as model layer-by-layer photopolymerization [START_REF] Taki | Oxygen concentration and conversion distributions in a layer-by-layer UV-cured film used as a simplified model of a 3D UV inkjet printing system[END_REF] by Taki et al.. Such a kinetic model has been recently adapted for the photoinitiation involving a type-II PIS combining isopropylthioxanthone and triazine derivatives, evidencing the crucial role of back electron transfer in the kinetics of three-dimensional polymer network formation [START_REF] Christmann | Triazine-based type-II photoinitiating system for free radical photopolymerization: mechanism, efficiency, and modeling[END_REF]. However, there is still a lack of quantitative data concerning the amount and the competition between the termination reactions for such a simple system as a type-I photoinitator. Indeed, among the photopolymerization kinetic models available in the literature, most of them does not take into account simultaneously all the termination pathways [START_REF] Kelly | Temporal and non-ideal behavior in photopolymers[END_REF][START_REF] Kelly | Holographic photopolymer materials: nonlocal polymerization-driven diffusion under nonideal kinetic conditions[END_REF][START_REF] Zhang | Study of UV cure kinetics resulting from a changing concentration of mobile and trapped radicals[END_REF][START_REF] Zhang | Modeling and measuring UV cure kinetics of thick dimethacrylate samples[END_REF][START_REF] Soh | Diffusion-controlled vinyl polymerization. II. Limitations on the gel effect[END_REF][START_REF] Buback | Free-radical polymerization up to high conversion. A general kinetic treatment[END_REF][START_REF] Buback | Modeling of termination in intermediate and high conversion free radical polymerizations[END_REF][START_REF] Anseth | Reaction diffusion enhanced termination in polymerizations of multifunctional monomers[END_REF][START_REF] Berchtold | Coupling chain length dependent and reaction diffusion controlled termination in the free radical polymerization of multivinyl (meth)acrylates[END_REF][START_REF] Lecamp | Simulation of temperature distributions within monomer film during photopolymerization[END_REF][START_REF] Lecamp | Simulation of conversion profiles and temperature distributions within dimethacrylate thick material during photopolymerization[END_REF][START_REF] Lecamp | Influence of UV radiation wavelength on conversion and temperature distribution profiles within dimethacrylate thick material during photopolymerization[END_REF][START_REF] Kerbouc'h | Numerical simulation to correlate photopolymerization kinetics monitoring by RT-NIR spectroscopy and photocalorimetry[END_REF][START_REF] Désilles | Simulation of conversion profiles within dimethacrylate thick material during photopolymerization. Validation of the simulation by thermal analysis data. Application to the synthesis of a gradient structure material[END_REF][START_REF] Azan | Simulation of the photopolymerization gradient inside a pigmented coating[END_REF][START_REF] Bayou | Simulation of conversion profiles inside a thick dental material photopolymerized in the presence of nanofillers[END_REF][START_REF] Terrones | Effects of optical attenuation and consumption of a photobleaching initiator on local initiation rates in photopolymerizations[END_REF][START_REF] Terrones | Effects of kinetics and optical attenuation on the completeness, uniformity, and dynamics of monomer conversion in free-radical photopolymerizations[END_REF][START_REF] Terrones | Nonuniformity of chain-length distributions in photopolymerized layers[END_REF][START_REF] Terrones | Diffusion-induced nonuniformity of photoinitiation in a photobleaching medium[END_REF][START_REF] Gleeson | A review of the modelling of free-radical photopolymerization in the formation of holographic gratings[END_REF][START_REF] Li | Analysis of the absorptive behavior of photopolymer materials. Part I. Theoretical modeling[END_REF][START_REF] Li | Analysis of the absorptive behavior of photopolymer materials. Part II. Experimental validation[END_REF][START_REF] Bowman | Coupling of kinetics and volume relaxation during polymerizations of multiacrylates and multimethacrylates[END_REF][START_REF] Goodner | Method for determining the kinetic parameters in diffusion-controlled free-radical homopolymerizations[END_REF][START_REF] Lovestead | Models of multivinyl free radical photopolymerization kinetics[END_REF][START_REF] Taki | Effect of oxygen inhibition on the kinetic constants of the UV-radical photopolymerization of diurethane dimethacrylate/photoinitiator systems[END_REF][START_REF] Taki | Oxygen concentration and conversion distributions in a layer-by-layer UV-cured film used as a simplified model of a 3D UV inkjet printing system[END_REF] and/or does not address their evolution all along the photopolymerization process [START_REF] Kelly | Temporal and non-ideal behavior in photopolymers[END_REF][START_REF] Kelly | Holographic photopolymer materials: nonlocal polymerization-driven diffusion under nonideal kinetic conditions[END_REF][START_REF] Soh | Diffusion-controlled vinyl polymerization. II. Limitations on the gel effect[END_REF][START_REF] Achilias | A review of modeling of diffusion controlled polymerization reactions[END_REF][START_REF] Lecamp | Simulation of temperature distributions within monomer film during photopolymerization[END_REF][START_REF] Lecamp | Simulation of conversion profiles and temperature distributions within dimethacrylate thick material during photopolymerization[END_REF][START_REF] Lecamp | Influence of UV radiation wavelength on conversion and temperature distribution profiles within dimethacrylate thick material during photopolymerization[END_REF][START_REF] Kerbouc'h | Numerical simulation to correlate photopolymerization kinetics monitoring by RT-NIR spectroscopy and photocalorimetry[END_REF][START_REF] Désilles | Simulation of conversion profiles within dimethacrylate thick material during photopolymerization. Validation of the simulation by thermal analysis data. Application to the synthesis of a gradient structure material[END_REF][START_REF] Azan | Simulation of the photopolymerization gradient inside a pigmented coating[END_REF][START_REF] Bayou | Simulation of conversion profiles inside a thick dental material photopolymerized in the presence of nanofillers[END_REF][START_REF] Terrones | Effects of optical attenuation and consumption of a photobleaching initiator on local initiation rates in photopolymerizations[END_REF][START_REF] Terrones | Effects of kinetics and optical attenuation on the completeness, uniformity, and dynamics of monomer conversion in free-radical photopolymerizations[END_REF][START_REF] Terrones | Nonuniformity of chain-length distributions in photopolymerized layers[END_REF][START_REF] Terrones | Diffusion-induced nonuniformity of photoinitiation in a photobleaching medium[END_REF][START_REF] Goodner | Development of a comprehensive free radical photopolymerization model incorporating heat and mass transfer effects in thick films[END_REF][START_REF] Lovestead | Models of multivinyl free radical photopolymerization kinetics[END_REF][START_REF] Taki | Effect of oxygen inhibition on the kinetic constants of the UV-radical photopolymerization of diurethane dimethacrylate/photoinitiator systems[END_REF][START_REF] Taki | Oxygen concentration and conversion distributions in a layer-by-layer UV-cured film used as a simplified model of a 3D UV inkjet printing system[END_REF].

Experimental section

Materials

Type-I PIS diphenyl(2,4,6-trimethylbenzoyl)phosphine oxide (TPO) was purchased from BASF. Diacrylic monomer ethoxylated bisphenol A diacrylate (SR349) was supplied by Sartomer and dimethylsulfoxide (DMSO, ≥ 99.8 %) by Sigma-Aldrich. Their structures are given in the ESI (Figure S1). All compounds were used as received.

Experimental methods

The formulations for the experimental kinetic analyses were based on a mixture of SR349 (90 wt%) and DMSO (10 wt%), in which 0.5 to 1 wt% of TPO has been dissolved. They were stirred overnight and kept in the dark. The photopolymerization kinetics were experimentally followed with a real-time Fourier transform infrared (RT-FTIR) spectrometer Vertex 70 from Brucker Optics, equipped with a nitrogen liquid-cooled mercury cadmium telluride (MCT)

detector [START_REF] Decker | Real-time kinetic study of laser-induced polymerization[END_REF]. It operated in rapid scan mode with a sampling interval of 0.12 s and 4 cm -1 resolution. The light irradiation was provided by a 395 nm LED (Roithner, LaserTechnik) whose incident intensity was checked with a calibrated fiber optic spectrometer (Ocean Optics, USB4000), on which a polypropylene (PP) film and a CaF2 pellet were placed to take into account light loss on the interfaces. Oxygen diffusion from the atmosphere was avoided by performing laminated experiments (the photosensitive resin was placed between two PP films and two CaF2 pellets). A 25 µm Teflon spacer was also used in order to adjust the thickness of the formulation.

The FRP kinetics were measured by following in real time the disappearance of the C=C bond stretching vibration band of the diacryclic monomer at 1637 cm -1 . The degree of conversion was calculated as:

Conv (%) = (A 1637 ) 0 -(A 1637 ) t (A 1637 ) 0 × 100 (2)
where (A1637)0 and (A1637)t are, respectively, the areas of the 1637 cm -1 vibration band before light irradiation and at an irradiation time t. The rate of polymerization Rp is derived from the degree of conversion by:

R p = k p [R(C=C) n • ][C=C] = dConv(%) dt × [C=C] 0 100 (3)
with [C=C]0 the initial concentration of acrylic doubles bonds in the formulation. Each experiment was repeated three times to ensure a good reproducibility and the data presented here result from an average.

Computational methods

Kinetic modeling was implemented in Wolfram Mathematica 9 software. The rate constants were defined as functions of the fractional free volume f, and the reactions of the photopolymerization mechanism were expressed as a set of differential equations, which was numerically solved with the NDSolve function [START_REF] Christmann | Triazine-based type-II photoinitiating system for free radical photopolymerization: mechanism, efficiency, and modeling[END_REF]. Conversion was directly calculated with the values of the acrylic double bond concentration [C=C] at zero and t time (Eq. 4) and Rp was calculated after numerical resolution with Eq. 3.

Conv (%) = [C=C] 0 -[C=C] t [C=C] 0 × 100 (4)

Results and discussion

Kinetic model conception

The main kinetic rate constants are defined as functions of the fractional free volume f in order to take into account their progressive diffusional control throughout the photopolymerization process [START_REF] Goodner | Modeling primary radical termination and its effects on autoacceleration in photopolymerization kinetics[END_REF][START_REF] Anseth | Reaction diffusion enhanced termination in polymerizations of multifunctional monomers[END_REF][START_REF] Berchtold | Coupling chain length dependent and reaction diffusion controlled termination in the free radical polymerization of multivinyl (meth)acrylates[END_REF][START_REF] Bowman | Coupling of kinetics and volume relaxation during polymerizations of multiacrylates and multimethacrylates[END_REF][START_REF] Goodner | Method for determining the kinetic parameters in diffusion-controlled free-radical homopolymerizations[END_REF][START_REF] Goodner | Development of a comprehensive free radical photopolymerization model incorporating heat and mass transfer effects in thick films[END_REF][START_REF] Lovestead | Models of multivinyl free radical photopolymerization kinetics[END_REF]. f represents the fraction of unoccupied volume in the reaction medium. It decreases as the polymerization proceeds according to:

ν f = 0.025 + α M (T -T g,M )Φ M + α P (T -T g,P )(1 -Φ M ) (5)
Φ M = 1 -Conv 1 -Conv + Conv × ( ρ M ρ P ⁄ ) (6)
k p = k p0 (1 + exp(A p ( 1 ν f - 1 ν f,cp ))) (7) k t,b = k t,b0 (1 + 1 R rd k p [C=C] k t,b0 + exp(-A t,b ( 1 ν f - 1 ν f,ct,b )) ) -1 (8)
The initiation rate constant ki and the PRT rate constant kt,PRT have the same expression as the propagation rate kp (Eq. 7). The exponential factors A and the critical fractional free volume f,c coefficients for initiation and PRT are taken equal to Ap and f,cp, meaning that ki and kt,PRT start decreasing at the same moment and at the same rate as kp does [START_REF] Goodner | Modeling primary radical termination and its effects on autoacceleration in photopolymerization kinetics[END_REF]. The values of the initiation (ki0) and PRT (kt,PRT0) intrinsic rate constants depend on the nature of the primary initiating radical. As introduced in a previous paper [START_REF] Christmann | Triazine-based type-II photoinitiating system for free radical photopolymerization: mechanism, efficiency, and modeling[END_REF], the oxygen inhibition and primary radicals (re)combination rate constants can be assimilated to the diffusion rate constant kdiff given by the Stokes-Einstein equation (Eq. 9), as it involves the mass transport of oxygen molecules or primary radicals. Following a proposition by Buback [START_REF] Buback | Free-radical polymerization up to high conversion. A general kinetic treatment[END_REF], the viscosity  exponentially increases with conversion from its initial value 0, with an exponential factor B optimized to 2 (Eq. 10) [START_REF] Christmann | Triazine-based type-II photoinitiating system for free radical photopolymerization: mechanism, efficiency, and modeling[END_REF]. The determination of the kinetic parameters for TPO-initiated photopolymerization has been fully described in [START_REF] Christmann | Triazine-based type-II photoinitiating system for free radical photopolymerization: mechanism, efficiency, and modeling[END_REF] and their values are listed in the ESI. Among them, the initial dissolved oxygen concentration has been optimized to 3.210 -4 M by a trial-and-error fit to experimental data, in agreement with the experimental value obtained in [START_REF] Scherzer | The effect of temperature on the induction period in the photoinitiated polymerization of tripropylene glycol diacrylate[END_REF] for a pure acrylic monomer.

k diff = 8RT 3η (9) η = η 0 exp (B η × Conv) (10)

Comparison to experiment

Characteristic values for experimental FRP kinetic curves are the final conversion Conv f (%)

and the maximal rate of polymerization Rp max . Calculated Conv(%) and Rp vs. irradiation time curves are compared with experimental data obtained for 0.5 and 1 wt% of TPO under 395 nm LED irradiation (10 mW.cm -2 ). A good agreement between experimental and calculated curves can be seen in Figure 2 a) and b), respectively on the final conversion and maximal rate of polymerization values. The corresponding results are listed in Table 1. Figure 3 plots the FTIR spectra before and after light irradiation for 1 wt% of TPO (see Figure S3 in the ESI for 0.5 wt%).

Oxygen inhibition in laminated samples

As stated before, experiments and simulations were performed in the case of laminated samples. However, oxygen inhibition actually occurs until the O2 molecules initially dissolved in the resin are totally consumed. The production of inactive peroxy radicals from the reaction of oxygen on primary radicals (RA • and RB • ) and macroradicals is described by the following set of differential equations:

d[R A OO • ] dt = k O 2 R A • [R A • ][O 2 ] (11)
d[R B OO • ] dt = k O 2 R B • [R B • ][O 2 ] (12)
d[R(C=C) n OO • ] dt = k O 2 R(C=C) n • [R(C=C) n • ][O 2 ] (13)
RAOO • (Eq. 11) and RBOO • (Eq. 12) are the inhibited primary initiating radicals and R(C=C)nOO • (Eq. 13) the inhibited macroradicals. The temporal evolution of

[RAOO • ], [RBOO • ],
and [R(C=C)nOO • ] (on a logarithmic scale) is compared to that of oxygen for 0.5 wt% TPO in (macro)radicals. More interestingly, it is also shown that the concentration of oxygen-inhibited macroradicals [R(C=C)nOO • ] is orders of magnitude higher than that of primary peroxy radicals

[RAOO • ] and [RBOO • ].

Termination reactions

Three different bimolecular termination reactions can occur during FRP: i. formation of a chemical bond between two macroradicals (combination), ii. hydrogen abstraction from a macroradical to a second one with formation of a double bond on the former (disproportion),

iii. reaction between a primary radical and a macroradical (PRT). As they both involve a reaction between two macroradicals, combination and disproportion are lumped into a single mechanism called bimolecular termination (kt,b), while PRT by RA • and RB • are distinguished (kt,PRT RA• and kt,PRT RB• ). It is possible to model these termination reactions by the following set of differential equations:

d[bimol] dt = 2k t,b [R(C=C) n • ]² (14)
d[R(C=C) n R A ] dt = k t,PRT R A • [R(C=C) n • ][R A • ] (15)
d[R(C=C) n R B ] dt = k t,PRT R B • [R(C=C) n • ][R B • ] (16)
[bimol] represents the concentration of macroradicals terminated by bimolecular termination (combination or disproportion), either diffusional or through reaction diffusion (Eq. 14).

[ From Figure 5 and Table 2, it can be seen that bimolecular termination is actually the main termination mode at the end of the reaction, followed by occlusion and PRT by RB Indeed, intrinsic initiation rate constant ki0 is 100 times higher for the phosphonyl radical RA •

(1.8-3.510 7 [START_REF] Sumiyoshi | On the reactivity of phosphonyl radicals towards olefinic compounds[END_REF][START_REF] Sumiyoshi | On the photolysis of acylphosphine oxides: 1. Laser flash photolysis studies with 2,4,6-trimethylbenzoyldiphenylphosphine oxide[END_REF][START_REF] Sluggett | Laser flash photolysis and timeresolved ESR study of phosphinoyl radical structure and reactivity[END_REF][START_REF] Jockusch | Phosphinoyl radicals: structure and reactivity. A laser flash photolysis and time-resolved ESR investigation[END_REF][START_REF] Gatlik | Structure-reactivity relationships in radical reactions: a novel method for the simultaneous determination of absolute rate constants and structural features[END_REF][START_REF] Weber | Electron spin resonance and laser flash photolysis study of radical addition to vinyl acrylate and related alkenes[END_REF][START_REF] Weber | A novel approach for measuring absolute rate constants by pulsed electron spin resonance: addition of phosphinoyl and 2-hydroxy-2-propyl radicals to several alkenes[END_REF][START_REF] Takahashi | Addition rate constants of phosphorus-and carbon-centered radicals to double bond of monomers as studied by a pulsed electron paramagnetic resonance method[END_REF]) than for its benzoyl equivalent RB • (1.8-3.610 5 M -1 .s -1 [START_REF] Gatlik | Structure-reactivity relationships in radical reactions: a novel method for the simultaneous determination of absolute rate constants and structural features[END_REF][START_REF] Colley | Probing the reactivity of photoinitiators for free radical polymerization: time-resolved infrared spectroscopic study of benzoyl radicals[END_REF]). The latter thus becomes more available for PRT reactions.

A deeper insight into the evolution of termination modes all along the photopolymerization is given in Figure 6, where the fractions of macroradicals and terminated species are It can be seen that the ratio of these different pathways actually evolves throughout the polymerization process. Indeed, the bimolecular fraction increases up to 92 % at 40 % of conversion, then decreases before a final increase at around 80 % of conversion. In parallel, the fraction of macroradicals decreases from the start of the polymerization process until 40 % of conversion, increases up to a plateau value around 70 % of conversion and finally decreases after 80 % of conversion. Termination by RB • radicals is only significant for a conversion higher than 50 %. It first increases, before a slight decrease at the end of the process. The noticeable role of RA • radicals in termination is lower and delayed to higher conversions (> 70 %), with a similar evolution than that of RB • . Finally, Figure 6 also shows that TPO photolysis occurs at high conversion (when the process is slowed down), its concentration being almost constant up to 60 % of conversion.

Bimolecular termination is clearly the main termination process throughout the polymerization, even at high conversion. This result is consistent with the introduction of 10 wt% of DMSO in the model resin used here, these small molecules playing the role of plasticizers at high conversion. The strong growth of the bimolecular termination during the first 20 % of conversion can be explained by the continuous production of macroradicals as initiation occurs, associated to the high value of kt,b in this range (represented on a logarithmic scale in Figure 7). Despite its decrease (due to the diffusion control of pure bimolecular termination), the effective kt,b stays higher than 210 5 M -1 .s -1 and thus overcomes the other termination modes. As PRT is not yet significant, the fraction of macroradicals is consequently reduced. kt,b stays almost constant from 20 to 40 % of conversion because termination by reaction diffusion takes over. Indeed, kt,b becomes then proportional to kp (see Eq. 1). As can be seen in Figure 7, kp stays constant up to 40 % of conversion because propagation becomes diffusion-controlled at a higher conversion than bimolecular termination (small monomer molecules are more free to diffuse than macroradicals). As a consequence, the increase of bimolecular termination (mainly by reaction diffusion) carries on between 20 and 40 % of conversion but is slower (Figure 6), and the fraction of macroradicals decreases (PRT is still not yet efficient).

The decrease of bimolecular termination after 40 % of conversion is also associated to the proportionality between kt,b and kp. Indeed, propagation becomes diffusion-controlled at high conversion (here after 40 % of conversion) and effective kp falls (see Figure 7). The reaction diffusion rate constant being proportional to kp (Eq. 1), kt,b also diminishes and bimolecular termination then decreases. As PRT is also not significant in this range, the amount of macroradicals consequently raises. The predominance of bimolecular termination over PRT throughout the polymerization can be explained by its higher rate compared to that of PRT (see Figure S6 in the ESI). The evolution of kp and kt,b all along the photopolymerization is actually in agreement with what have been experimentally observed [START_REF] Buback | Free-radical polymerization up to high conversion. A general kinetic treatment[END_REF][START_REF] Anseth | Reaction behaviour and kinetic constants for photopolymerizations of multi(meth)acrylate monomers[END_REF][START_REF] Anseth | Polymerization kinetics and volume relaxation behavior of photopolymerized multifunctional monomers producing highly crosslinked networks[END_REF][START_REF] Goodner | Method for determining the kinetic parameters in diffusion-controlled free-radical homopolymerizations[END_REF] and simulated [START_REF] Buback | Free-radical polymerization up to high conversion. A general kinetic treatment[END_REF][START_REF] Anseth | Reaction diffusion enhanced termination in polymerizations of multifunctional monomers[END_REF][START_REF] Goodner | Method for determining the kinetic parameters in diffusion-controlled free-radical homopolymerizations[END_REF]. As previously mentioned, PRT becomes efficient only in the last stages of the process. This delayed effectiveness of PRT has to be related to the competition between PRT and initiation reactions for the primary radicals. This can be expressed by the ratio between the PRT rate Rt,PRT = kt,PRT[R(C=C)n • ][R • ] and the initiation rate Ri = ki[C=C][R • ] (with R • = RA • or RB • , Eq. 17).

The evolution of this ratio as a function of polymer conversion is given in Figure 8. It can be seen that the Rt,PRT/Ri ratio is close to zero for a large part of the process, only increasing after 50 % of conversion for both RB • and RA • . After a maximum at 82 % of conversion, the ratio finally falls for both primary radicals. However, it has to be noticed that the ratio for phosphonyl RA • radicals is largely lower than that for benzoyl RB • ones throughout the process. An initial low value of Rt,PRT/Ri means that initiation is largely more efficient than PRT for a large part of the FRP reaction. The following raise shows that competition of PRT with initiation becomes more efficient when the conversion increases. Indeed, the effective kt,PRT and ki rate constants decrease by means of the progressive diffusional control of the associated reactions during the process. In the model, the exponential factor A and critical free volume f,c parameters are identical for both kt,PRT and ki (equal to Ap and f,cp, vide supra), thus the reaction rate constants start decreasing at the same time and at the same rate. In parallel, the monomer concentration [C=C] also decreases all along the polymerization while [R(C=C)n • ] increases due to the decrease of kt,b (vide supra), with a maximum around 82 % of conversion (Figure 8). As a consequence, the Rt,PRT/Ri ratio increases as long as [R(C=C)n • ] raises. PRT becomes hence more efficient, and its proportion in the termination modes increases (see Figure 6). The lower Rt,PRT/Ri ratio for RA • could explain the delay in the significant role of PRT by RA • in Figure 6. This is caused by the higher intrinsic initiation rate constant ki0 for phosphonyl radicals RA • than for benzoyl RB • ones. Initiation by RA • thus longer prevails over PRT and the concentration of RA • stays lower than that of RB • . This emphasizes that the primary radicals may play a dual role during the photopolymerization, either initiator of the reaction or terminating agent. The choice of the photoinitiator structure would also have to take into account this effect, in addition to the classical requirements (absorption region, dissociation efficiency…) to optimize the polymerization performances. The increase of PRT by RB • after 60 % of conversion seems to impact only the bimolecular termination fraction, as a plateau is observed in the fraction of macroradicals (Figure 6). The final decrease of Rt,PRT/Ri ratio after 82 % of conversion is associated to the decrease of [R(C=C)n • ] (Figure 8), [RA • ] and [RB • ], which could be due to the almost complete photolysis of TPO and the cessation of (macro)radical formation.

R t,PRT /R i = k t,PRT [R(C=C) n • ][R • ] k i [C=C][R • ] (17)
Figures 6 and8 both present a strong change at around 80 % of conversion. This has to be related to the evolution of [TPO] plotted in Figure 6. Indeed, most part of the initial TPO is consumed after 80 % of conversion, so the production of primary initiating radicals falls.

Initiation and PRT are then strongly reduced. As a consequence of the reduction of initiation, less macroradicals are produced (up to a complete cessation when TPO totally disappears).

However, bimolecular termination still occurs between the remaining macroradicals, so its fraction increases while that of macroradical decreases. This final enhancement of bimolecular termination explains the final decrease of [R(C=C)n • ] observed in Figure 8, causing that of the Rt,PRT/Ri ratio and thus the final decline of PRT fraction for both initiating radicals.

Effect of experimental parameters I0 s and [TPO]0

The incident light intensity on the surface sample I0 s and/or the initial photoinitiator concentration are generally tuned in order to optimize the photopolymerization process.

However, a modification of these experimental parameters will affect the initiation and termination reactions. Kinetic modeling represents an efficient way to quantify this influence with numerical values, especially that on the termination modes.

Influence of incident light intensity I0 s on termination reactions

The ability of the kinetic model to reproduce changes in the incident light intensity was confirmed with photopolymerization experiments made at 2 and 5 mW.cm -2 (see Figures S7 to S10 in the ESI). The effect of incident light intensity (from 2 to 50 mW.cm -2 ) on conversion and Rp curves is shown in Figure 9. Final conversion is almost not affected: only a slight lowering can be observed at higher I0 s . However, it is clear that the rate of free radical photopolymerization is enhanced and the inhibition time reduced. As expected, the shape of Rp vs. irradiation time curves is strongly modified when I0 s is raised: i. the inhibition time is reduced, ii. the Rp max position is switched to shorter times, iii. the Rp max values are greater, iv.

the curves are sharper, revealing both faster autoacceleration and autodeceleration processes. Increasing the incident intensity enhances the rate of primary radical production at early times, hence that of macroradical formation (see Figure S11 in the ESI). As the macroradicals are the principal scavengers of dissolved O2 molecules in laminated systems (vide supra), this faster production of macroradicals explains the reduced inhibition time with increasing incident intensity. Moreover, the vitrification of the reaction medium occurs faster, consequently leading to an earlier and faster autoacceleration process (i.e. diffusion control of bimolecular termination). For the same reason, autodeceleration (i.e. diffusion control of propagation) occurs faster. The higher amount of photogenerated initiating radicals and macroradicals per second at higher intensities should explain the higher maximum rates of polymerization observed.

Increasing the incident intensity I0 s has a strong impact on termination proportions at final conversion, as shown in Figure 10. Indeed, both bimolecular termination and PRT fractions increase with I0 s , while occlusion falls. They all reach a plateau at higher incident intensities. It clearly shows that, contrary to what is generally suggested [START_REF] Goodner | Modeling primary radical termination and its effects on autoacceleration in photopolymerization kinetics[END_REF], PRT never becomes the major way of termination, even at high light intensities. The increase of bimolecular termination and the reduction of radical trapping by occlusion are surprising. Indeed, it is generally stated in the literature that increasing incident intensity causes an increase of both PRT, by means of a higher concentration of primary radicals [START_REF] Goodner | Modeling and experimental investigation of light intensity and initiator effects on solvent-free photopolymerizations[END_REF][START_REF] Goodner | Modeling primary radical termination and its effects on autoacceleration in photopolymerization kinetics[END_REF], and occlusion at final conversion, due to a faster vitrification of the reaction medium [START_REF] Anseth | Radical concentrations, environments, and reactivities during crosslinking polymerizations[END_REF][START_REF] Berchtold | Propagation and termination kinetics of cross-linking photopolymerizations studied using electron paramagnetic resonance spectroscopy in conjunction with near IR spectroscopy[END_REF][START_REF] Wen | A kinetic model for radical trapping in photopolymerization of multifunctional monomers[END_REF]. In order to understand the influence of I0 s on the final termination ratio, the change in the fractions of terminated species and macroradicals for different light intensities is compared to that of TPO concentration (Figure 11). For a given light intensity, the successive steps in the change of bimolecular termination, PRT and macroradical fractions are identical to that discussed in the previous section. When I0 s is modified, change in these fractions is also identical before 60 % of conversion, but reveals interesting differences after. Indeed, the PRT fraction increases earlier and becomes higher when I0 s is enhanced for both RA • and RB • . It also decreases sooner at the end of the process. The final increase of bimolecular termination, as well as the corresponding decrease of macroradical fraction, is also shifted to shorter times when the incident intensity is increased. Finally, the TPO photolysis is faster when I0 s is enhanced, by means of a higher amount of photons per second being absorbed by the photoinitiator molecules. The earlier increase of PRT could be explained by a higher rate of macroradical production when I0 s is increased. This shifts the efficient competition between initiation and PRT reactions for primary radicals to a lower conversion, a fact supported by an earlier enhancement of the Rt,PRT/Ri ratio when I0 s is increased (see Figure S12 in the ESI). PRT has then more time to occur until the end of the process, and finally represents a higher amount of terminated species. The faster photolysis of TPO with increasing I0 s causes an earlier increase of bimolecular termination and a decrease of the macroradical and PRT fractions at the end of the polymerization. Indeed, as initiation ceases earlier during the process, the formation of This increase could be explained by the diffusional control of both the propagation and bimolecular termination steps (vide supra). Indeed, the macroradical concentration strongly increases when bimolecular termination is reduced. The final decrease is associated to the progressive photolysis of TPO and the associated cessation of initiation. As a consequence, macroradicals are not produced but termination reactions take place. Figure 12 then reveals that the steady-state assumption should be valid between 5 and 40 % of conversion, but not for higher values. Indeed, the macroradical concentration strongly evolves in this range. It confirms previous observations made in the literature with a simpler kinetic model [START_REF] Bowman | Coupling of kinetics and volume relaxation during polymerizations of multiacrylates and multimethacrylates[END_REF] and then reconsiders the validity of the steady-state assumption still generally applied on [R(C=C)n • ] in photopolymerization kinetics.

Influence of initial photoinitiator concentration on termination reactions

The comparison of experimental and simulated data for 0.5 and 1 wt% TPO in Figure 2 proves the ability of the model to account for the effect of the initial TPO concentration on the kinetics. Figure 13 It can be seen that the successive steps for the change in the termination and macroradical fractions are not identical to what was observed when the incident light intensity is increased, meaning that the [TPO]0 value has different effects on the final increase of bimolecular termination and the decrease of macroradical fractions, as well as the onset of PRT. Indeed, as the TPO initial concentration is increased, the proportion of bimolecular termination decreases faster and to a lower value. Moreover, its final increase is delayed to higher conversions. In parallel, the macroradical fraction increases while PRT becomes significant earlier and represents a higher proportion of termination reactions when [TPO]0 is increased.

Finally, TPO is consumed slowly when its initial concentration is raised.

The previous sections of this work revealed that the final increase of bimolecular termination and the decrease of macroradical fraction are directly linked to the disappearance of TPO. The delayed final increase of the bimolecular termination proportion and the decrease of the macroradical fraction observed in Figure 15 are then consistent with the slower photolysis of the photoinitiator when [TPO]0 is raised. Indeed, the macroradical production ceases at a higher conversion and bimolecular termination has less time to occur between the remaining [TPO] (M)

Conv (%)

 5 macroradicals, which explains the higher final fraction of macroradicals. The earlier efficiency of PRT could be explained by the higher concentration of primary radicals and their increased formation rate when [TPO]0 is raised. Indeed, the Rt,PRT/Ri ratio is higher for a same conversion when [TPO]0 increases, promoting PRT over initiation at a lower conversion (see Figure S14 in the ESI). As a consequence, this termination way finally represents a higher proportion of the macroradicals produced.

Conclusion

A kinetic model taking into account simultaneously all the possible termination pathways (bimolecular termination, primary radical termination and radical trapping by occlusion) was successfully applied to the photopolymerization kinetics initiated by a type-I photoinitiator.

Comparison with experiments reveals a good agreement both on conversion and rate of polymerization curves. The inhibition by the initially dissolved oxygen molecules in laminated systems was considered in order to take into account the experimental inhibition time. It points out for the first time that the macroradicals act as the principal O2 scavengers at the early stages of the process, whereas the primary initiating radicals are almost totally involved in initiation reactions with the monomer.

The model was then adapted to identify the relative contribution of the different termination ways throughout the photopolymerization process with numerical values. It was shown that bimolecular termination remains the major termination reaction during the whole process.

However, its ratio, as well as that of primary radical termination and macroradicals, actually evolves because of: i. the progressive diffusion control of the polymerization reactions as the polymer network grows, ii. the cessation of initiation when the photoinitiator is totally consumed. The impact of the incident light intensity and the initial photoinitiator concentration on the termination modes was also investigated. As expected, the photopolymerization process is faster and the inhibition time lower when both parameters are increased, but the final conversion only increases when [TPO]0 is raised. Increasing the incident light intensity enhances the fraction of bimolecular termination and that of PRT. However, it is interesting to notice that the former remains the major termination pathway even at high incident intensity.

When the initial photoinitiator concentration is raised, PRT becomes more efficient, but radical trapping by occlusion in the glassy 3D polymer network also surprisingly increases.

It will be very interesting to extend this approach to more complex photoinitiating systems (type-II or three-component photocyclic ones), as the underlying photochemistry will be impacted by the formation of the polymer network during the irradiation but will also influence back the polymerization process. This will be the subject of a forthcoming paper.

  photoinitiator. A kinetic model considering simultaneously all the termination pathways is thus optimized by comparison with experimental photopolymerization kinetics. It is subsequently modified to identify the main oxygen trapping agent in the case of a laminated photopolymerization reaction, and discuss the dual role of the initiating radicals between initiation and termination in function of their reactivity towards the monomer and the macroradicals as the diffusion control settles. For the first time, the evolution of each of the termination modes is discussed with numerical values all along the photopolymerization reaction in conjunction with the evolution of the polymerization rate constants and the concentration of the reactive species. This study is finally extended to different incident light intensities and photoinitiator initial concentrations, and a new view on the validity of the pseudo-steady-state assumption is provided in relation with the advancement of the polymerization reaction.

Figure 1 :

 1 Figure 1: General mechanism of FRP initiated by TPO.

Figure 2 :Figure 3 :

 23 Figure 2: Comparison between experimental (dashed) and simulated (plain) a) acrylate double bond conversion b) rate of polymerization for 0.5 (black) and 1 wt% (red) of TPO (irr = 395 nm, 10 mW.cm -2 ).

Figure 4 (

 4 Figure4(for 1 wt%, see FigureS4in the ESI). As known, the initially dissolved oxygen is rapidly consumed (0.45 and 0.22 s for 0.5 and 1 wt% TPO, respectively). This consumption is faster for the most concentrated formulation by means of a faster generation of

Figure 4 :

 4 Figure 4: Evolution of the oxygen and inactive peroxy radical (primary and macroradicals) concentrations (logarithmic scale) as a function of the irradiation time (0.5 wt% TPO, irr = 395 nm, 10 mW.cm -2 ).

  R(C=C)nRA] and [R(C=C)nRB] stand for the concentrations of macroradicals terminated through PRT by RA • and RB • respectively (Eqs.15 and 16). At final conversion, the polymer network is vitrified and it can be assumed that all remaining macroradicals are trapped by occlusion in the tridimensional network.Evolution of [bimol], [R(C=C)nRA], [R(C=C)nRB] and [R(C=C)n • ] as a function of irradiationtime is plotted in Figure5for 1 wt% TPO (see FigureS5in the ESI for 0.5 wt%). It can be seen that the concentration of macroradicals terminated by bimolecular reactions increases from the onset to the end of the polymerization. It is largely higher than [R(C=C)nRA] and [R(C=C)nRB] which also continuously increase, or [R(C=C)n • ] which increases up to 100 s and finally decreases until the polymerization ceases. Moreover, the growth of [R(C=C)nRA] and [R(C=C)nRB] starts at longer times than that of macroradicals terminated by bimolecular termination. Considering the final values of these concentrations, the fractions of macroradicals terminated by each termination mode at the end of the photopolymerization can be calculated and the corresponding results are given in Table2.

Figure 5 :

 5 Figure 5: Evolution of the terminated species and macroradical concentrations (logarithmic scale) as a function of the irradiation time (1 wt% TPO, irr = 395 nm, 10 mW.cm -2 ).

  • . In comparison, PRT by RA • is only a minor way of termination. Relative values of [R(C=C)nRA] and [R(C=C)nRB] also show that PRT by benzoyl RB • radicals is more efficient than that by phosphonyl RA • ones, a fact which can be explained by their different initiation efficiencies.

  represented as a function of the acrylate conversion (for 1 wt% TPO) all along the polymerization. The corresponding change in TPO concentration is also plotted for discussion.

Figure 6 :

 6 Figure 6: Evolution of the fractions of terminated species and propagating or trapped macroradicals (left) and of the TPO concentration (right) as a function of the acrylate conversion (1 wt% TPO, irr = 395 nm, 10 mW.cm -2 ). The horizontal arrows indicate the corresponding vertical axisblue curve is magnified by a factor 5 for clarity.

Figure 7 :

 7 Figure 7: Effective propagation (kp, black) and bimolecular termination (kt,b, red) rate constants (logarithmic scale) as a function of the acrylate conversion (1 wt% TPO, irr = 395 nm, 10 mW.cm -2 ).

Figure 8 :

 8 Figure 8: Evolution of the Rt,PRT/Ri ratio for RA • (green) and RB • (blue) radicals (left) and of [R(C=C)n • ] (dashed, right) as a function of the acrylate conversion (1 wt% TPO, irr = 395 nm, 10 mW.cm -2 ). The horizontal arrows indicate the corresponding vertical axisblue curve is magnified by a factor 10 for clarity.

Figure 9 :

 9 Figure 9: Effect of the incident light intensity I0 s on the simulated a) conversion b) rate of polymerization (1 wt% TPO, irr = 395 nm).

Figure 10 :

 10 Figure 10: Influence of the incident light intensity I0 s on the termination modes at final conversion (1 wt% TPO, irr = 395 nm).

Figure 11 :

 11 Figure 11: Change in the fractions of terminated species and macroradicals (left) and of the TPO concentration (right) for 2 (solid), 10 (dashed) and 50 (dotted) mW.cm -2 as a function of the acrylate conversion (1 wt% TPO, irr = 395 nm). The horizontal arrows indicate the corresponding axisblue curve is magnified by a factor 10 for clarity.

  sooner. Bimolecular termination has then more time to occur between the remaining macroradicals. The final fraction of bimolecular termination is then enhanced, while that of radical trapping by occlusion is consequently reduced. As TPO is consumed, PRT efficiency is also reduced because of the decrease of the primary radical formation. Its fraction then decreases earlier when I0 s is enhanced.The influence of the incident light intensity on the temporal evolution of the macroradical concentration is especially interesting because the steady-state assumption is generally applied to [R(C=C)n • ] in order to derive relations between the rate of polymerization Rp and the incident light intensity[START_REF] Odian | Principles of Polymerization[END_REF][START_REF] Andrzejewska | Photopolymerization kinetics of multifunctional monomers[END_REF][START_REF] Ibrahim | Investigation of termination reactions in free radical photopolymerization of UV powder formulations[END_REF][START_REF]Radiation Curing in Polymer Science and Technology Volume III: Polymerization Mechanism[END_REF]. This assumption notably implies that [R(C=C)n • ] is constant throughout the polymerization, hence d[R(C=C)n • ]/dt = 0. Figure12shows the evolution of the rate of polymerization Rp and that of d[R(C=C)n • ]/dt as a function of the conversion for different light intensities.

Figure 12 :

 12 Figure 12: Evolution of the rate of polymerization Rp (circles, left) and the rate of macroradical formation d[R(C=C)n • ]/dt (plain, right) as a function of the acrylate conversion for different incident light intensities (1 wt% TPO, irr = 395 nm). The horizontal arrows indicate the corresponding axis.

  represents the results obtained for [TPO]0 values comprised between 0.510 -2 and 1010 -2 M (0.15 to 3 wt%). Contrary to the effect of I0 s , the final conversion increases when [TPO]0 is enhanced. However, the rate of the FRP process is once again increased and the inhibition time reduced, as expected from the literature. The effects of increasing [TPO]0 on Rp curves are similar to the ones obtained for the increase of I0 s : i. the inhibition time is reduced, ii. the Rp max position is moved to shorter times, iii. the Rp max values are greater, iv. the curves are sharper (both faster autoacceleration and autodeceleration).

Figure 13 :

 13 Figure 13: Effect of the TPO initial concentration on the simulated a) conversion b) rate of polymerization (irr = 395 nm, 10 mW.cm -2 ).

Figure 14

 14 Figure 14 represents the evolution of the termination fractions at final conversion as a function of the initial TPO concentration. It can be seen that macroradical bimolecular termination decreases with [TPO]0, while radical trapping by occlusion and PRT are enhanced. These results can be explained with Figure 15, which represents the evolution of the termination and macroradical fractions and that of the TPO concentration as a function of the polymer conversion for three initial [TPO]0 values.

Figure 14 :Figure 15 :

 1415 Figure 14: Influence of the initial TPO concentration on the termination modes at final conversion (irr = 395 nm, 10 mW.cm -2 ).

  Subscripts M and P refer to monomer and polymer respectively, with  the thermal expansion coefficient (difference between liquid and glassy state expansion coefficients), Tg the glass transition temperature,  the volume fraction and  the volumetric mass density.

	The expressions for the propagation (kp) and bimolecular termination (kt,b) rate constants
	are given in Eqs. 7 and 8 [14,46,47,78-81]. The expression of kt,b includes both diffusional
	bimolecular termination and subsequent reaction diffusion processes. kp0 and kt,b0 correspond
	to the propagation and bimolecular termination intrinsic rate constants respectively, i.e. without
	any diffusional control. Ap and At,b are the parameters which govern the rate at which,
	respectively, kp and kt,b decrease with viscosity. Finally, f,cp and f,ct,b represent the critical
	fractional free volumes at which propagation and bimolecular termination become diffusion-
	limited respectively.

Table 1 :

 1 Comparison between experimental and simulated data for final conversion Conv f (%) and maximal rate of polymerization Rp max .

	TPO	0.5 wt%		1 wt%
		Experiment	Model	Experiment	Model
	Conv f (%)	80	82.5	84.5	83.8
	Rp max (M.s -1 )	2.7	2.5	3.3	3.6

Table 2 :

 2 Relative ratio of macroradical termination reactions at final conversion (tirr = 300 s).

			Termination mode (%)
	TPO	Bimolecular	PRT by RA •	PRT by RB •	Occlusion*
	0.5 wt%	79.9	0.2	6.8	13.1
	1 wt%	77.7	0.4	7.6	14.3

* Macroradicals at final conversion are all considered as trapped by occlusion.

Funding

ANR and Mäder are fully acknowledged for financial support of the DeepCure project #ANR-13-CHIN-0004-01.

Declaration of interest

The authors declare that they have no competing financial interests.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, in the online version, at XX.


S Thébaud

Ch Adessi

G Bouzerar

Large enhancement of the thermoelectric power factor in disordered materials through resonant scattering

In the search for more efficient thermoelectric materials, scientists have placed high hopes in the possibility of enhancing the power factor using resonant states. In this study, we investigate theoretically the effects of randomly distributed resonant impurities on the power factor. Using the Chebyshev Polynomial Green's Function method, we compute the electron transport properties for very large systems (∼10 7 atoms) with an exact treatment of disorder. The introduction of resonant defects can lead to a large enhancement of the power factor together with a sign inversion in the Seebeck coefficient. This boost depends crucially on the position of the resonant peak, and on the interplay between elastic impurity scattering and inelastic processes. Strong electron-phonon or electron-electron scattering are found detrimental. Finally, the robustness of our results is examined in the case of anisotropic orbitals and two-dimensional confinement. Our findings are promising for the prospect of thermoelectric power generation.

I. RESONANT STATES IN THERMOELECTRIC MATERIALS

Over the past decades, the increasingly pressing need for clean energy sources and the realization that a huge proportion of the energy used worldwide is wasted in heat 1 have prompted great interest in the perspective of developping efficient thermoelectric generation modules. [2][3][4][5] The currently available devices, mainly based on (Bi,Sb) 2 Te 3 , PbTe or Si-Ge alloys, are not efficient enough to be used industrially on a large scale. 6 The efficiency of a thermoelectric module is limited by the temperature-averaged figure of merit zT of both the n and p legs of the module, [START_REF] Zlatic | Modern theory of thermoelectricity[END_REF][START_REF] Goldsmid | Introduction to Thermoelectricity[END_REF][START_REF] Snyder | [END_REF] with zT = σS 2 κ T, in which σ is the electrical conductivity, S is the Seebeck coefficient and κ is the thermal conductivity, often dominated by phonons in doped semiconductors. The use of thermoelectric devices on a large scale would require a figure of merit of at least 2 to 3, depending on the application area. 10,[START_REF] Thekdi | E3M[END_REF] The thermal averaging must be done between the temperatures of the heat source and that of the heat sink, meaning that the materials composing the legs should have a good figure of merit on a wide range of temperatures. Additionally, the efficiency and reliability of the device is partly determined by the compatibility between the two legs, so ideally they should be made of similar compounds, such as a single semiconductor host with a different doping. 10,[START_REF] He | [END_REF] So far, most progress in boosting the thermoelectric figure of merit has been accomplished by lowering the thermal conductivity. 13,14 Enhancing the power factor (PF) σS 2 is a much more formidable challenge due to the interplay between the electrical conductivity and the Seebeck coefficient. To go beyond the rigid band doping optimum, various band engineering strategies have been proposed, such as band convergence [15][16][17] and dimensional confinement of the electron gas. [18][START_REF] Heremans | Proceedings of the XXXIV International School of Semiconducting Compounds[END_REF][START_REF] Dresselhaus | [END_REF][21] Another promising method to boost the PF consists in using resonant states. [22][23][24] The concept is illustrated in Fig. 1. Dopants introduce impurity states inside the conduction band that hybridize with the extended states. This way, a sharp peak in the density of states (DOS) of the host compound is created, which alters the electronic transport properties when the Fermi level lies in its vicinity. An enhancement of the thermoelectric properties through resonant impurity states has been claimed in various compounds, such as Tl-doped PbTe, 25 Sn-doped Bi 2 Te 3 , 26 In-doped SnTe, 27 Al-doped PbSe, 28 or Sndoped β-As 2 Te 3 . 29 The case of Tl-doped PbTe is controversial since first-principles calculations reproduced the experimental values for the Seebeck coefficient with a simple rigid-band shift of the pristine material. 30 Subsequently, numerical studies using the Coherent Potential Approximation (CPA) coupled with first-principles calculations 31 have been conducted to investigate the effects of random Tl doping in PbTe. [32][33][34] However, transport properties were calculated in the absence of electronphonon scattering, and the treatment of disorder by CPA methods is known to ignore the vertex corrections and thus localization effects. [START_REF] Economou | Green's Functions in Quantum Physics[END_REF] More importantly, no clear improvement in the PF over a rigid-band shift of the Fermi level was shown for Tl-doped PbTe. As it stands, there is still no consensus in the literature whether actual resonant enhancement of the thermoelectric properties have been observed experimentally.

In this study, we aim to clarify the general conditions required for a boost of the PF using resonant substitution impurities, all with a full treatment of the disorder and resonant scattering. In particular, we will investigate the influence of the impurity concentration x, the effects of inelastic scattering, and finally we will examine the case of anisotropic orbitals.

II. DISORDERED MODEL HAMILTONIAN AND METHODOLOGY

To keep the conclusions as general as possible, we consider a single-orbital tight-binding Hamiltonian featuring hopping terms t between nearest neighbors on a cubic lattice. Here the charge carriers are electrons (n-type), but because of electron-hole symmetry, our results are valid for p-type materials as well. The resonant impurities are modelled by an on-site potential ǫ on the defect sites and an hybridization V between the host and impurity sites. The position of the defects are chosen randomly. Hence, the Hamiltonian, as illustrated in the inset of Fig. 1, reads:

Ĥ = -t i,j ,σ c † i,σ c j,σ + c † j,σ c i,σ (1)
+ ǫ m,σ c † m,σ c m,σ -V i,m ,σ c † i,σ c m,σ + c † m,σ c i,σ ,
i runs over host sites, m over impurity sites, σ is the electron spin and the brackets denote nearest neighbors only.

Since the transport properties do not depend explicitely on t, we express the other parameters in units of t (see below).

The electrical conductivity and the Seebeck coefficient at electron density n and temperature T write [START_REF] Ashcroft | Solid State Physics[END_REF][START_REF] Scheidemantel | [END_REF]

σ(n, T ) = dE - ∂f ∂E Σ(E), (2)
and

S(n, T ) = - 1 eT σ dE - ∂f ∂E (E -µ) Σ(E), (3)
with f (E, µ, T ) the Fermi distribution. The chemical potental µ is set to give the correct electron density when the DOS ρ(E) is integrated. Σ(E) is the so-called transport distribution function (TDF), which gives access to all the electronic transport properties and is therefore the key quantity to be calculated. σ is the thermal average of Σ around the Fermi level µ, while S is basically the logarithmic derivative of Σ around µ. 38 Therefore, high Seebeck coefficients arise from strong, sharp variations in the TDF (i.e. large values of | dΣ dE |). Most theoretical studies of doped thermoelectric materials compute the TDF within the framework of the Boltzmann transport equation with the relaxation time approximation. [START_REF] Ashcroft | Solid State Physics[END_REF] They either consider impurity scattering to be negligible compared to electron-phonon scattering, [39][40][41][42][43] or estimate the electron-impurity scattering rate by secondorder perturbation theory (i.e. Fermi's Golden Rule) using a model description for the impurity scattering. [44][45][46] This is reasonable when the doping does not significantly alter the electronic structure and causes only weak electron-impurity scattering, as is the case of La or Nb doped SrTiO 3 for instance. 47 But the whole point of resonant states is that they distort the band structure of the host material and introduce strong scattering. Therefore, in this study, we go beyond the semi-classical Boltzmann formalism to incorporate the full effects of disorder and multiple resonant scattering. We use the Kubo formula expressed in terms of the Green's function Ĝ of the system [48][49][50][51]

Σ(E) = e 2 πΩ Tr(Im Ĝ(E) vx Im Ĝ(E) vx ) , (4)
brackets denote disorder averaging, Ω is the total volume, vx = it <i,j>,σ

(x i -x j )(c † i,σ c j,σ -c † j,σ c i,σ
) is the velocity operator along the transport direction x, and the Green's function writes

Ĝ(E) = 1 E -Ĥ + i γin 2 . (5)
A constant imaginary part has been introduced in the denominator of Ĝ(E) to account for the presence of inelastic scattering mechanisms in the system, such as electronphonon (e-ph) or electron-electron (e-e) collisions. It will be further discussed thereafter. Σ can be expressed in terms of the adimensioned TDF Σ, Σ = e 2 a Σ, and likewise for the power factor, PF = k 2 B a PF. For definiteness, we set a = 4 Å, which gives the same volume per atom as in Si or PbTe, leading to e 2 a = 6.08 × 10 3 S/cm and

k 2 B a = 45.18 µW • cm -1 • K -2 .
The exact diagonalization of the Hamiltonian (1) would drastically limit the system sizes that could be studied with a reasonable amount of memory and computational time. Therefore, a good alternative to compute Σ(E) exactly, fully including vertex corrections, is the Chebyshev-Polynomial Green's Function method (CPGF). 52,53 It is a real-space approach particularly suitable for addressing the physics in disordered systems. A brief overview of the method follows. The Green's function Ĝ(E) is expanded on the Chebyshev polynomials basis:

Ĝ( Ẽ) = ∞ n=0 g n (z) T n ( H), (6)
where z = Ẽ + iγ in /2 with Ẽ and γin /2 rescaled in the energy interval [-1, 1], T n ( H) are the Chebyshev polynomials calculated by the recursion relation

T n+1 ( H) = 2 HT n ( H) -T n-1 ( H)
, and g n (z) are known complex functions given by

g n (z) = -i (2 -δ n,0 ) z -i √ 1 -z 2 n √ 1 -z 2 . ( 7
)
Inserting equation ( 6) into equation ( 4) yields

Σ( Ẽ) = e 2 πΩ n,n ′ µ n,n ′ Img n (z) Img n ′ (z), (8)
with

µ n,n ′ = Tr(T n ( H) ṽx T n ′ ( H) ṽx ) . (9)
The quantities µ n,n ′ are called the moments and are calculated by iterative multiplications of the Hamiltonian matrix operator. Due to the superior convergence properties of the Chebyshev polynomials [START_REF] Boyd | Chebyshev and Fourier Spectral Methods[END_REF] and to the presence of the inelastic self-energy, only a finite number of terms in equation ( 6) need to be computed in order to obtain the exact Green's function of the system. Around 1800 × 1800 moments are sufficient for the TDF to fully converge. Periodic boundary conditions are used to reach the thermodynamic limit more easily. Here, we compute the TDF on systems of size N = 1200 × 200 × 200 (48 × 10 6 sites), this slab geometry allows faster convergence. The trace in equation ( 9) is evaluated efficiently by a stochastic method involving random vectors as described in Ref. 53 and Ref. 52. When calculating the TDF for such a large system size, only a few random vectors and disorder configurations are necessary. We have checked that the clean limit is perfectly recovered for both open and periodic boundary conditions. Regarding γ in in equation ( 5), it can be interpreted as the inelastic contribution to the electron relaxation time. [START_REF] Mahan | Many Particle Physics[END_REF] In thermoelectric materials, scattering rates at room temperature typically range from 1 meV to 100 meV. In certain Half-Heuslers such as ZrNiSn, for instance, particularly weak e-ph couplings lead to inelastic scattering rates varying between 1 meV and 20 meV [START_REF] Zhou | [END_REF] while strong e-e scattering in SrTiO 3 leads to γ in ranging from 50 meV to 200 meV and even higher. 47 In between, scattering rates vary from 10 meV to 100 meV in Si 46,57,58 or from 20 meV to 60 meV in pristine and doped SnSe. 59 From the bandwidth in these compounds, we estimate t ranging from 0.3 eV to 1 eV, therefore we will consider 0.02 t ≤ γ in ≤ 0.2 t. This corresponds to γ in ranging from 10 meV to 200 meV and mean free paths between 50 Å and 500 Å in the pristine case, which is well in line with calculated values in PbTe, 40 for instance. As we will see shortly, a small inelastic scattering is most favorable for resonant enhancements of the PF, therefore we set γ in = 0.02 t unless specified otherwise. The choice of a constant γ in preserves the generality of this investigation, since incorporating energy and temperature dependences requires a material-specific study. Note also that computations for smaller γ in would imply a strong increase in the number of calculated moments before convergence is reached.

III. NUMERICAL RESULTS AND DISCUSSION

In what follows, we consider the set of values ǫ = -4 t and V = 0.3 t that were found optimal in a previous study that completely ignored disorder. 60 Unless specified otherwise, the electronic properties will be calculated using these values. Fig. 2(a) shows the TDF and the DOS (inset) for different impurity concentrations x = Nimp N , with N imp the number of randomly distributed defects. Five concentrations are considered, from x = 0% (the pristine reference case) to x = 5%. The defects introduce a local peak in the DOS, which is considered the main signature of resonant states in the literature, and the mechanism by which the transport properties are enhanced. The higher the impurity concentration, the bigger and sharper the peak, so we would expect the best thermoelectric performances from the highest concentrations. As we will see shortly, this is not the case. The resonant peak gives rise in the TDF to a sharp, asymmetrical dip, as the extended states acquire a more localized character by hybridizing with impurity states. At high defect concentrations, electron transport is more suppressed across the whole energy range so the variations of the TDF are gentler (| dΣ dE | is reduced as x increases). We compare in Fig. 2(b) the TDF and dΣ dE calculated by CPGF and by the often used Fermi's Golden Rule (FGR). Matthiessen's rule states that the total scattering rate is γ tot = γ imp + γ in with γ imp the impurity scattering rate. FGR leads to γ imp (E) = 2πxǫ 2 ρ 0 (E), ρ 0 (E) being the DOS of the clean system. The FGR transport distribution function is given by Σ FGR (E) = γin γtot Σ (0) (E) where Σ (0) (E) is the pristine TDF. Notice that the exact Σ(E) cannot be cast into such an analytical form. Clearly, the FGR approach completely fails to give the correct dependance of the TDF. In particular, the dip is entirely absent. The discrepancy is even worse for the derivatives, which are directly linked to the Seebeck coefficients. Therefore, when resonant states are involved, second-order perturbation theory breaks down.

From the results of Fig. 2(a) and equation ( 2) and (3), we compute the room-temperature electrical conductivity σ and Seebeck coefficient S, as plotted in Fig. 3 as a function of the electron density n. T is set to 0.025 t which corresponds to room temperature if t ≈ 1 eV. σ is reduced by the disorder and still exhibits the same features as Σ(E) (T being relatively small). This reduction would be detrimental to the PF, but the sharp variations in the TDF lead to a boost of the Seebeck coefficient that overcompensates the suppression of σ. This is accompanied by a sign inversion of S around n = 0.1 electrons/cell. Thus, S can change sign in the disordered systems, while it remains n-type in the absence of resonant states. This interesting feature opens the possibility of changing the thermoelectric material from n-type to p-type just by introducing the appropriate impurity or dopant. Therefore, one could build a device with both n and p legs from the same semiconductor host. This would be advantageous for device performance and reliability, provided that the PF is sufficiently large.

The PF is plotted in Fig. 4 as a function of n. The pristine system exhibits a maximum of 6.3 µW • cm -1 • K -2 around n = 10 -3 electrons/cell, corresponding to a conductivity of 400 S/cm and a Seebeck coefficient of -130 µV/K. Note that this relatively low value of the power factor is partly due to the absence of band degeneracy and anisotropy in our single-band model. The effects of resonant impurities relative to the pristine case would not be affected by band degeneracy, while the case of anisotropic orbitals will be examined thereafter. When resonant defects are introduced, the PF is suppressed at low densities (inset), because multiple impurity scatterings have a stronger effect on the long wavelength carriers. By contrast, around n = 0.1 electrons/cell, the PF now exhibits a large enhancement due to the boost of the Seebeck coefficient that overcompensates the drop in conductivity. The largest increase corresponds to x = 1%, for which the PF reaches its maximum 35.9 µW • cm -1 • K -2 , a sixfold enhancement compared to that of the clean system. For x = 5%, the boost is still present but less spectacular (a ratio less than 2) due to the gentler variations in the TDF. This is an important and surprising finding: to achieve an efficient enhancement of the thermoelectric properties with resonant states, the defect concentration should be kept relatively low, typically around 1%. From an experimental point of view, that is favorable, because such concentrations usually lie below the solubility limit. 61 Co-doping with a donor atom acting as an electron reservoir is necessary to shift the Fermi level inside the resonant peak, where the PF is enhanced and the Seebeck inversion occurs. This x = 0% 0.5% 1% 2% 5% carrier density optimization still requires at most 10% co-doping, which is reasonable. We define PF max as the optimum PF with respect to the carrier concentration. PF max extracted from Fig. 4 are shown in the first row of table I.

ǫ = -4 t V = 0.
We now address the influence of both ǫ and V on the transport properties. In Fig. 5(a), the DOS and TDF are plotted for x = 1% resonant impurities with ǫ = -5.5 t and V = 0.3 t. The increase of the on-site potential shifts the position of the resonant peak much closer to the bottom of the conduction band. The resulting TDF also features a much smaller dip in Σ(E) at the position of the peak in ρ(E). Consequently, | dΣ dE | remains quite weak, and so does the Seebeck coefficient. There is still a sign inversion, but no boost in the PF. Indeed it is even suppressed by a factor 3-4 with respect to the reference value of 6.3 µW • cm -1 • K -2 (see the second row of table I). Thus the resonant peak should not be too close to the band edge, but deep inside the conduction band. We now focus on the effect of a larger hybridization, which implies a stronger coupling between conduction and defect states. Results are depicted in Fig. 5(b). The increase in hybridization also pushes the resonant states at the very edge of the band, and severely suppresses the TDF below -5 t (notice the scale in the inset). There is still a small dip in Σ(E) and a sign inversion of S, but because the carriers are now so localized in this energy range, the PF shrinks by at least two orders of magnitude compared to that of the reference. For x = 1%, PF max is now 0.05 µW • cm -1 • K -2 (see the third row of table I for the other concentrations). This suppression of the PF is entirely due to a huge reduction in the conductivity caused by multiple scattering events that become important at low energy in the presence of stronger disorder. These findings are consistent with the results obtained in Ref. 34 for Tl-doped PbTe, in which the Tl doping creates a resonant bump at the edge of the valence band associated with a much higher resistivity compared to that of Na doping which behaves as a reservoir. It should also be mentioned that resonant states formed by antisites in Fe 2 VAl have been found to suppress the PF by more than an order of magnitude while changing the sign of S. 62 The takeaway to obtain a boost of the PF is that the substituting element should be suitably selected in order to create a resonant peak far from the band edge. This could also explain why many claims of experimental enhancement of the PF by resonant states remain controversial, and why no sign inversion of the Seebeck coefficient has been observed so far. These effects are indeed sensitive to the hybridization, on-site potential and position of the Fermi level. Additionally, it is difficult to rule out other enhancement mechanisms, such as energy filtering effects resulting from ionized impurity scattering, for instance.

Since thermoelectric materials are meant to be used in a wide range of temperature, we now discuss the Tdependence of PF max . Fig. 6 shows PF max as a function of temperature for the same impurity concentrations as in Fig. 4. It increases when the temperature rises, reaching a broad maximum, and then decreases slowly in the disordered systems. This high temperature behaviour results from the sharp variations in the TDF being smoothed out by the thermal average. Accordingly, at high values of x, the variations in the TDF are broader and are less sensitive to the thermal average, so the maximum region is shifted to higher temperatures. An important finding is that PF max itself is robust, suggesting that resonant states could be efficient for both low and high temperature power generation.

We now propose to investigate the role of inelastic scattering. In Fig. 7(a), we plot PF max as a function of γ in . Clearly, the PF is completely suppressed if the inelastic scattering is too strong. This results from the competition between resonant impurity scatterings (elastic processes) and inelastic scatterings. In Fig. 7(b) is presented the TDF for γ in = 0.02 t, 0.08 t and 0.2 t with x = 1%. The inset shows the impurity scattering rate γ imp , extracted from an analysis of the single-particle spectral function A(q, E) = -1 π ImG(q, E) . G(q, E) is the spatial Fourier transform of the disordered Green's function G ij,σ (E) = iσ| Ĝ(E)|jσ , where |iσ is the real-space basis (i runs over the lattice sites). γ imp exhibits a non- monotonic behavior and large variations across the resonant peak, from 4 × 10 -3 t at E = -4.5 t to 2 × 10 -1 t at -4 t. At the position of the dip in the TDF, where the electronic states have a stronger localized character, transport is not very sensitive to the strength of γ in because γ imp ≈ 2 × 10 -1 t dominates. In contrast, if γ imp is smaller than γ in , which is the case for states associated with large values of the TDF (γ imp ≈ 4 × 10 -3 t at E = -4.5 t), then increasing γ in strongly suppresses Σ(E). Thus, large inelastic scattering rates have the overall effect of reducing the disparities in the TDF, leading to poor values of the Seebeck coefficient. If we now consider small values of γ in (below 0.03 t) for x ≤ 2%, we observe a huge increase of PF max as γ in is reduced. If we extrapolate to γ in ≤ 0.02 t for x = 1%, an enhancement factor of more than an order of magnitude could even be reached. Hence, due to the competition between elastic and inelastic scattering, the impurity concentration should be tuned with respect to the inelastic scattering rate in the host material to reach an optimal boost of the PF. Compounds exhibiting strong e-ph or e-e scattering should not be the best candidates for resonant substitu- tion doping. Till now, we have been considering an isotropic electronic structure (s-type orbitals), but it is worth considering the influence of orbital anisotropy. 63,64 Lowdimensional confinement is expected to introduce sharp structures in the DOS and thus sharp variations in the TDF, thereby boosting the Seebeck coefficient. To evaluate the gain in the PF that could be obtained from resonant states in anistropic systems, we now introduce a different hopping t ⊥ in a direction perpendicular to transport. The optimum PF for the reference x = 0% and for x = 1% is presented in Fig. 8 as a function of t ⊥ /t. First notice that the PF of the pristine system strongly increases with the anisotropy, from 6.3 µW • cm -1 • K -2 (3D) to 73.3 µW • cm -1 • K -2 (2D). This confirms that two-dimensional confinement in itself does favor good performances. The maximum PF (p-type) for x = 1% also increases with the anisotropy, from 35.9 µW • cm -1 • K -2 (3D) to 74.4 µW • cm -1 • K -2 (2D). Surprisingly, for t ⊥ = 0 we find no boost in the PF, suggesting that the presence of resonant states in fully confined systems might not further enhance the thermoelectric properties. However, one should emphasize that even for finite but low ratios t ⊥ /t (down to 0.05), the PF can be significantly increased by resonant states. This is promising for bulk systems in which charge carriers populate highly anisotropic orbitals. This is, for instance, the case of n-doped SrTiO 3 , in which the Titanium 3d orbitals exhibit a t ⊥ /t ≈ 0.1. 47,65 Moreover, using resonant states in fully confined materials could be interesting for the sign inversion of S alone.

To conclude, we have used the Chebyshev-Polynomial Green's Function method to address the effects of resonant impurities on electron transport. Although resonant states suppress the electrical conductivity, they may also lead to a boost and a sign inversion of the Seebeck coefficient. Consequently, the power factor can increase by one order of magnitude. However, the resonant peak should be located far from the band edge, otherwise the thermoelectric performances are destroyed. Additionally, the optimal boost of the power factor depends crucially on the interplay between elastic and inelastic scattering. Strong electron-phonon or electron-electron scattering are found to preclude the possibility of enhancing thermoelectric transport. Therefore, materials featuring long electron mean free paths and weak inelastic scattering, such as PbTe, 40 certain Half-Heuslers compounds [START_REF] Zhou | [END_REF] or even graphene 66,67 should be promising candidates. Finally, the resonant boost of the power factor is found robust in the case of anisotropic orbitals. This study will hopefully contribute to a better understanding of resonant states in the context of thermoelectric power generation. Our methodology is very general and can be combined with realistic first-principles calculations to guide experimentalists and pave the way in the search for compounds exhibiting resonant effects.

Figure 1 :

 1 Figure 1: Sketch of the density of states as a function of energy corresponding to a conduction band in the presence of resonant states. Inset: illustration of the Hamiltonian terms.

2 ΣFigure 2 :

 22 Figure 2: (a) Transport distribution function Σ(E) and density of states ρ(E) (inset) for five impurity concentrations, from x = 0% (reference, black dashed line) to x = 5%. A vertical dashed line marks the position of the resonant peak. (b)Σ(E) and its derivative (inset) for x = 1% calculated exactly (CPGF) and by second-order perturbation theory (FGR).

Figure 3 :

 3 Figure 3: (a) Electrical conductivity σ and (b) Seebeck coefficient S as a function of the electron density n for five impurity concentrations, from x = 0% to x = 5%.

Figure 4 :

 4 Figure 4: Power factor σS 2 as a function of the electron density n for five impurity concentrations, from x = 0% to x = 5%. Inset: PF for lower electron densities.

  8 -6.4 -6 -5.6 -5.2

Figure 5 :

 5 Figure 5: Density of states ρ(E) (dotted line, right axis) and transport distribution function Σ(E) (left axis) for x = 1% with (a) ǫ = -5.5 t, V = 0.3 t and (b) ǫ = -4 t, V = t. Inset: zoom on the bottom of the conduction band, notice the different scales for the TDF.

2 )Figure 6 :

 26 Figure 6: Optimum power factor as a function of temperature for five impurity concentrations, from x = 0% to x = 5%.

Figure 7 : 2 )Figure 8 :

 728 Figure 7: (a) Optimum power factor as a function of the inelastic scattering rate γin for five impurity concentrations, from x = 0% to x = 5%. Inset: Ratio of the optimum power factor with respect to that of the clean system. (b) Σ(E) for x = 1% with γin = 0.02 t, 0.08 t and 0.2 t (left axis). ρ(E) is also shown (dotted line, right axis). A vertical dashed line marks the position of the resonant peak. Inset: the calculated impurity scattering rate along the Γ-X direction as a function of energy. The dashed curve is a guide to the eye.

* E-mail: simon.thebaud@univ-lyon1.fr 1 C. Forman, I. K. Muritala, R. Pardemann, and B. Meyer, Renewable and Sustainable Energy Reviews 57, 1568