Vitalii Tkachenko

Camélia Matei Ghimbeu

Cyril Vaulot

Loïc Vidal

Julien Poly

Abraham Chemtob

RAFT-Photomediated PISA in Dispersion: Mechanism, Optical Properties and Application in Templated Synthesis

Diblock copolymer nanoparticles were prepared without initiator by visible-light photopolymerization-induced selfassembly ("photo-PISA") in dispersion. Both copolymer blocks were synthesized via an original photomediated reversible deactivation radical polymerization mechanism combining reversible addition-fragmentation chain transfer (RAFT) and reversible photolysis of the C-S bond of the chain-transfer agent (CTA). The photosensitive CTA was a commercially available trithiocarbonate, namely cyanomethyl dodecyl trithiocarbonate. The first hydrophilic poly(hydroxyethyl acrylate) block was synthesized in solution, then used as a macro-CTA for the dispersion polymerization of styrene in methanol/water. Spherical nanoparticles with a narrow size distribution and an average diameter of 20-40 nm were achieved. The copolymers were characterized by size-exclusion chromatography (SEC) in DMF, which confirmed the control of the reaction, with relatively narrow molecular weight distributions (Mw/Mn ≤ 1.30). Two regimes can be discerned in the RAFT-mediated dispersion photo-PISA: a slow particle nucleation (4-15 % conv.) followed by particle growth. A steady number of particles was obtained when steric stabilization was adequate. Study of the optical properties by UV-Vis spectrophotometry enabled to separate the respective contributions of absorption and scattering during the PISA process. Finally, we demonstrated that the copolymer nanoparticles could act as soft templates for the synthesis of mesoporous carbon materials.

Introduction

Over the last decade, polymerization-induced self-assembly (PISA) has emerged as a versatile method for the preparation of well-defined block copolymer nanoparticles with various morphologies. 1 In a PISA mechanism, self-assembly occurs during the synthesis itself of the amphiphilic diblock copolymer, when the first solvophilic block is chain-extended with the second solvophobic one. 2 Reaching a critical chainlength for this latter block results in the spontaneous copolymer self-assembly. The solvophilic block acting as a steric stabilizer is generally prepared via solution polymerization, while the insoluble block can be formed via two different methods: aqueous emulsion polymerization involving insoluble monomers, 3-5 or dispersion polymerization, using on the contrary, soluble monomers. 6 There are two main advantages of PISA over more conventional post-polymerization self-assembly techniques: 7-9 (i) higher solids content (10 -50 wt% instead of < 1 wt%); 10 and

(ii) reduced processing and purification steps.

Therefore, nano-objects derived from PISA have attracted much attention for a diverse range of applications, including enzyme catalysis, 12 drug delivery, 3,13 nanostructured films, 14 stimuli-responsive smart nanomaterials 15 etc. With the exception of recent works resorting to ring-opening metathesis 16 or non-controlled 17 mechanisms, the fast development of PISA has been mostly driven by reversibledeactivation radical polymerizations. Also referred to as Scheme 1 Overview of the photomediated controlled radical polymerization mechanism combining RAFT with the reversible photolysis of the chain-transfer agent. This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins controlled radical polymerizations (CRPs), these mechanisms have made possible to synthesize a broad range of diblock copolymer structures. Among the different reported CRPs, reversible addition-fragmentation chain transfer (RAFT, Scheme 1) utilizing thiocarbonylthio compounds as chaintransfer agents (CTA) is by far the most widely used in PISA, owing to its versatility in terms of compatible monomers and experimental procedures. 9 RAFT-mediated PISA has been developed mostly using thermal initiators, but the alternative use of radiation-mediated processes grouped under the term "photo-PISA", can be an even more advantageous route. 18 A major asset is the possibility to perform PISA over a wide temperature range. For example, ambient temperature contributes to reducing energy consumption or making the process safer. In addition, the use of low temperatures has proved beneficial for the synthesis of polymer nanoparticles encapsulating thermally sensitive proteins, 19 possessing CO 2 -or thermo-responsive properties, [20][21][22] and bearing reactive functional groups. 23 A second advantage of photo-PISA is to drive high-throughput synthesis. To maximize the number of tests run in parallel, Tan et al. demonstrated for example that photo-PISA could be performed in multi-well open plates through an enzyme (glucose oxidase) reacting with dissolved oxygen. 24 Exploiting the different photo-induced RAFT mechanisms, 25 three photo-PISA approaches have been already reported in the literature. The most common strategy consists simply in using a photoinitiator 26-37 rather than a thermal initiator. In order to avoid non desired absorption by the RAFT agent, near UV 35,36 or visible irradiation 37 is usually preferred. In 2014, a second approach was proposed by Boyer and co-workers, who demonstrated that visible photoredox catalysts such as Ir(ppy) 3 and [Ru(bpy) 3 ]Cl 2 can be used to activate the RAFT agent via a photo-induced electron transfer. [38][39][40][41] At the center of the current study is a third approach based on the reversiblecleavage of the excited state, formed under irradiation, of the thiocarbonylthio compound (Scheme 1), which yields an initiating alkyl radical and a thiyl radical. As result, the CTA acts also as a photoinitiator. Another distinctive feature of this mechanism over its photoinitiated homologue is the temporal control of polymerization. Because the RAFT agent photoactivation becomes reversible, it is possible to (re)initiate or to terminate reversibly the propagating chains by switching on or off the light source. The photochemical reactivity of thiocarbonylthio compounds has been known for a long time.

42,43

However, its interest for photomediated polymerization has been recently revived thanks to the discovery that photolysis can occur not only under UV irradiation but also visible light. In this latter case, the weak absorption corresponding to the n → π* transition is exploited instead of the main and dramatically more intense π → π* band. 44,[START_REF] Xu | Controlled Radical Polymerization: Mechanisms[END_REF] Using visible light, a better control of RAFT-mediated polymerizations was reported in solution, possibly as a result of minimized side/termination reactions involving the thiyl radicals. 36,43,44,[START_REF] Yeow | [END_REF][47][48] Although this route obviates the need of additional photoinitiator or photocatalyst, it is surprisingly the least developed in photo-PISA. There has been only two reports on RAFT-mediated photo-PISA relying on the reversible photolysis of a CTA, namely 4-cyano-4-((dodecylsulfanylthiocarbonyl)sulfanyl)pentanoic acid, under visible light at 460 nm. In these studies, the dispersion polymerization of benzyl methacrylate [START_REF] Yeow | [END_REF] and hydroxypropyl methacrylate 49 was performed in presence of a solvophilic macro-CTA based on poly(oligo(ethylene oxide)). Following a similar methodology, new spherical and narrowly distributed diblock copolymer nanoparticles (20 -40 nm) were synthesized by RAFT dispersion polymerization of styrene (St) at 35 °C and 20 % w/w solids using a poly(hydroxyethyl acrylate) (PHEA) macro-CTA as steric stabilizer block (Fig. 1).

Both solvophilic PHEA block and solvophobic polystyrene (PS) block were obtained via photomediated RAFT polymerization by leveraging the reversible -fragmentation of the commercial S-cyanomethyl-S-dodecyltrithiocarbonate (TTC) RAFT CTA under blue light (λ max = 472 nm). Emphasis was made on two domains poorly investigated so far in photo-PISA: the mechanism of particle formation/growth, as well as the characterization of optical properties throughout the dispersion polymerization. Finally, we showed that the resultant PHEA-b-PS nanoparticles can be utilized as template in the synthesis of mesoporous carbon materials. Interconnected and tuneable mesopores were achieved in the size range of 20-30 nm, a domain generally inaccessible with conventional liquid crystal mesophases.

Experimental section

Materials

Hydroxyethyl acrylate (HEA; Tokyo Chemical Industry (TCI); >96%) was purified by passing through a basic alumina column (Sigma-Aldrich) prior to use. Styrene (St; Sigma-Aldrich ; ≥ 99%) and 1,4-dioxane (Sigma-Aldrich; 99.5%) were distilled under reduced pressure over CaH 2 prior to use. Methanol (Fischer Chemical; ≥ 99%), S-cyanomethyl-S-dodecyltrithiocarbonate Please do not adjust margins Please do not adjust margins (TTC; Sigma-Aldrich; 98%), pholoroglucinol (Sigma-Aldrich; ≥ 99%), glyoxylic acid (Sigma-Aldrich; ≥ 99%) and diethyl ether (Sigma-Aldrich; ≥ 99%) were used as received.

Synthesis

Photomediated RAFT polymerization of HEA in solution (ex. PHEA 25 -TTC). In a typical synthesis, HEA (4.04 g, 34 mmol), TTC (0.396 g, 1.2 mmol) and dioxane (12.16 g) were charged into a Schlenk tube equipped with a stirring bar. The mixture was degassed by three freeze-pump-thaw cycles before being placed under nitrogen. The tube was then placed in a preheated oil bath at 25 °C and irradiated during 5 h in an immersion type photoreactor containing a blue LED spot light (λ = 472 nm, 547 mW cm -2 ) (Fig. S1). Aliquots were regularly withdrawn to determine the conversion by ). In a typical synthesis, PHEA 25 -TTC (0.266 g, 0.082 mmol), St (1.721 g, 16.5 mmol) and a methanol/water mixture (8.18 g/0.43 g, 95/5 w/w %) were charged into a Schlenk tube equipped with a stirring bar. The mixture was degassed by three freezepump-thaw cycles before being placed under nitrogen. The tube was then placed in a pre-heated oil bath at 35 °C and irradiated during 70 h in an immersion type photoreactor (λ = 472 nm, 547 mW cm -2

). In the kinetic study, the mean degree of polymerization of the PS block was systematically varied by adjusting the irradiation time. The syntheses involving PHEA macro-CTA of different lengths were performed at the same monomer/solvent weight ratio of 1/3. Three aliquots of 0.1 mL were regularly withdrawn during reaction. The first one was used for 1 H NMR analysis (after dissolution in 0.5 mL of CDCl 3 ) to determine St conversion. The second one was employed for SEC measurements to determine polymer molecular weights. Finally, DLS and TEM measurements were carried out using the third aliquot (see next section for details). Synthesis of mesoporous carbon materials. Phloroglucinol (0.205 g, 1.63 mmol) and glyoxylic acid monohydrate (0.152 g, 2.05 mmol) were dissolved under stirring at room temperature in methanol (2.96 g). This solution was then added to a nonpurified PHEA 25 -PS 118 copolymer dispersion (4.96 g, mass of copolymer = 0.62 g). The obtained turbid solution was transferred into smooth glass dishes and left at room temperature for about 6 h in order to evaporate methanol solution and the non-reacted St. Subsequently, a thermal consolidation of the phenolic resin was performed at 60 °C overnight to increase the cross-linking density of the phenolic resin framework. The resultant material was pyrolysed at 600 °C under inert atmosphere (Ar) for 2 h.

Characterizations

Size exclusion chromatography (SEC-THF). Accurate molecular weights of macro-CTAs were determined by SEC in THF at 35°C. Samples with precise concentrations around 1-3 mg mL -1 were prepared and filtered (PTFE membrane; 0.20 μm) before injection. The flow rate was 1.0 mL min -1

. The following Agilent 1260 Infinity series setup was used: a G1310B isocratic pump; a G1322A degasser; a G1329B auto-sampler; a G1316A thermostated column compartment equipped with a set of Polymer Laboratories ResiPore columns (nominal particle size: 3 μm; porosity: 2 μm) composed of a guard column (50 × 7.5 mm) and two columns (300 × 7.5 mm); a G1314B variable wavelength detector; a G7800A multidetector suite equipped with a MDS refractive index detector and a MDS viscometer detector. Universal calibration was performed using a set of EasiVial polystyrene (PS-M) standards. Agilent GPC/SEC software and multi-detector upgrade were used to determine molar mass values and distributions.

Size exclusion chromatography (SEC-DMF).

Molecular weights of PHEA-PS copolymers were determined by SEC in DMF with additive of 0.01 M LiBr at 50 °C. For sample preparation, an aliquot was precipitated in 2 mL of diethyl ether under intense stirring, then the copolymer collected after filtration was dried and dissolved in DMF. Solutions of samples with precise concentrations around 1-3 mg mL -1 were then prepared and filtered (PTFE membrane; 0.20 μm) before injection. The flow rate was 0.9 mL min -1 . The following Agilent 1260 Infinity series setup was used: a G1311A isocratic pump; a G1322A degasser; a G1313A auto-sampler; a G1316A thermostated column compartment equipped with a set of Polymer Laboratories ResiPore columns (nominal particle size: 3 μm; porosity: 2 μm) composed of a guard column (50 × 7.5 mm) and two columns (300 × 7.5 mm); a G1314B variable wavelength detector; a G7800A multidetector suite equipped with a MDS refractive index detector. Relative calibration was performed using a set of EasiVial poly(methyl methacrylate) (PM) standards. Agilent GPC/SEC software and multi-detector upgrade were used to determine molar mass values and distributions.

Nuclear magnetic resonance (NMR).

1 H spectra were recorded in D 2 O (homopolymers) or CDCl 3 (diblock copolymer) at 300 MHz on a Varian Mercury spectrometer. Dynamic light scattering (DLS). Hydrodynamic diameter of diblock copolymer nano-objects were determined using a VASCO nanoparticle size analyzer (Cordouan technologies) with a 15 mW laser operating at a wavelength of 658 nm. The scattered light is detected at an angle of 135°. Prior to measurements, samples were diluted 20-fold. Transmission electron microscopy (TEM). The TEM images of copolymer nanoparticles and carbon mesoporous materials were obtained with a JEOL ARM-200F instrument working at 200 kV. Latex samples were diluted 200 times, cast onto the TEM grid, and dried overnight at room temperature. UV-Spectroscopy. UV-Vis spectra were obtained using a JascoV730 UV/Vis/NIR spectrophotometer. The selected range of wavelengths was 280-800 nm. LED lamp. The emission spectrum and irradiance of the LED were determined using a Field Max II (COHERENT) optical

4 | J. Name., 2012, 00, 1-3
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Please do not adjust margins spectrum analyzer and a Power Max PM10 powermeter at a distance of 1 cm. Details on nitrogen adsorption measurements are available in the Electronic Supplementary Information (ESI).

Results and discussion

I. Synthesis of PHEA macro-CTA by photomediated RAFT polymerization in solution

In order to synthesize nano-objects based on amphiphilic diblock copolymers, a series of hydrophilic macro-CTAs was first synthesized by photomediated RAFT polymerization of hydroxyethyl acrylate (HEA) in 1,4-dioxane at ambient temperature (25°C). The initiator-free synthesis was carried out under blue light irradiation (provided by a 472 nm LED) and using the trithiocarbonate TTC as RAFT CTA (Fig. 1). TTC was selected because of its -photocleavage ability at this wavelength 47 and the fact that relatively unstable primary alkyl radicals can be produced. A short radical lifetime drives fast radical recombination allowing therefore to minimize degradation reactions. 48 Fig. 2a shows a typical kinetic profile for the synthesis of PHEA 58 -TTC. The determination of the conversion of HEA by 1 H NMR (Fig. S2) evidenced a pseudo first order kinetics, in agreement with a constant concentration of propagating species during the polymerization. A conversion of 58 % was achieved within 5 h of irradiation. The SEC traces revealed narrow dispersities (Fig. 2b) over the reaction, suggesting a controlled mechanism. In addition, the linear relationship between the measured molecular weights and conversions (Fig. 2c) fully supported a well-controlled RAFT synthesis. Following the same procedure, other PHEA macro-CTAs exhibiting different chain lengths (DP n = 25, 52 and 82) were successfully synthesized (see complete data in Table S1 of ESI).

II. Synthesis of PHEA-PS by photomediated RAFT dispersion polymerization of styrene

II.1 Kinetic study

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The PHEA macro-CTAs discussed above were then chainextended with St at 35°C in a methanol/water mixture (95/5 % w/w) using a similar photomediated RAFT approach (λ = 472 nm), and in absence of any external photoinitiator. Consistently with a PISA process in dispersion, the reaction medium progressively turned turbid as polymerization proceeded (see pictures of the reaction medium taken at different reaction times in Fig. 3). As depicted in Fig. 4a, the polymerization rate was slow and a monomer conversion of approximately 60 % was achieved after 70 h. Note that even conventional RAFT alcoholic dispersion polymerizations of St carried out at higher temperatures (T ≥ 70 °C) is known to suffer from significantly incomplete conversion; 50,51 a problem somehow mitigated by the addition of water. 52 In our case, the slow polymerization can be mostly ascribed to the dramatic effect of temperature on the propagation rate constant of St (k p = 132 L mol -1 s -1 at T = 35 °C vs k p = 482 L mol -1 s -1 at 70 °C). [START_REF] Matyjaszewski | Handbook of Radical Polymerization[END_REF] Two distinct intervals (I, II) corresponding to different rate behaviors were observed in the ln([M] 0 /[M]) versus time plot of Fig. 4a. This substantially differs from the first step of solution polymerization where a single linear regime was obtained (Fig. 2a). Irrespective of the initiation method employed, this dual rate behavior was reported by many authors in RAFT dispersion polymerization, 26,31 and was interpreted broadly as the transformation from a "homogeneous" to "heterogeneous" polymerization. In Interval I, lasting typically 30-40 h, the reaction rate was extremely slow (15 % after 30 h). As can be seen in Fig. 3 (t = 0 -30 h), the medium was macroscopically homogeneous at the beginning, then slightly scattering at the end (t = 30 h). This suggests that this first regime includes not only the solution polymerization of the PHEA 25 -PS n -TTC diblock copolymer chains but also the formation of micellar particle by selfassembly (particle nucleation).Starting after 35 h, interval II was marked by a significant increase of the polymerization rate (2 times greater). Meanwhile, the medium became increasingly scattering (Fig. 3, t = 47-70 h), most likely because of the gradual increase of particle (micelle) diameter (see TEM and DLS size data in Fig. 4a). The acceleration of St conversion translates the change of polymerization site from continuous phase to copolymer particles. The particles become favored as the main reaction site since, firstly, most copolymer chains have precipitated, and secondly, the monomer concentration in particles [M] p is higher to the one in solution [M] s . The high local monomer concentration within the particle core, similar to bulk monomer concentration, drives a higher propagation rate in this second interval. To support this assumption, a control experiment of PS extension performed in solution (Fig. S3) showed only a single and slow regime reaching only 30 % conv. after 70 h under irradiation. As polymerization proceeds, the monomer diffuses from the continuous phase into the polymer particles to replace that which has been reacted. An equilibrium (saturation) monomer concentration is thus maintained in the particles, accounting for the steady Please do not adjust margins Please do not adjust margins polymerization rate observed. Because the reaction is stopped before completion, there is no hypothetical interval III where monomer depletion within the particles could cause a decrease of reaction rate. The kinetic study was completed by SEC analysis in DMF as function of irradiation time, clearly evidencing the efficient reinitiation of the RAFT chain-ends and propagation of the chains (Fig. 4b). Narrow dispersities were measured (Đ ~ 1.1), accordingly with a controlled mechanism. The controlled character was definitely confirmed by the linear relationship between the measured molecular weights and conversions (Fig. 4c). We thus demonstrate the versatility and broad applicability of the photomediated PISA synthesis by showing that the photolysis of the RAFT chain-end can be exploited to synthesize successively the solvophilic (PHEA) and solvophobic (PS) blocks.

II.2 Colloidal study

Fig. 5 shows a series of representative TEM images obtained from samples withdrawn at different intervals during the dispersion photopolymerization of St starting from PHEA 25 -TTC macro-CTA. After 47 h of irradiation, spherical and narrowly distributed nanoparticles with increasing mean diameters were observed. Particle nucleation is assumed to take place earlier, but at shorter times, the PS block is likely too short to drive a high enough glass transition temperature necessary to preserve the integrity of the particles under the TEM beam. In agreement with this assumption, the samples taken before 30 h showed a coalesced structure. After 47 h of irradiation, narrow particle size distributions (PDI = 1.05) were obtained, irrespective of the samples, with a mean diameter increasing from 22.5 nm for PHEA 25 -PS 68 (47 h) to 31.6 nm for PHEA 25 -PS 118 (70 h) as detailed in Table 1. As a result, varying irradiation time proved to be a suitable means of controlling particle size in the size range 20-40 nm. DLS studies (Fig. 4a) confirmed the particle growth as a function of time at a rate similar to that of the TEM data. As expected, substantially larger mean particle diameters (+ 20 nm) were measured by DLS (intensity-average diameters) that takes into account the solvation of the PHEA layer on the nanoparticle surface and the swelling of the polymer particles by residual monomer. It is also worth mentioning that these diblock copolymer nanoparticles invariably possess a spherical morphology whatever the length of the PS chains. The causes do not lie in the specificities of the mechanism (RAFT agent photolysis, low temperature) but rather in the choices made as regards DP of the stabilizer and core-forming blocks. Other trials, which go beyond the framework of this study, showed that utilizing a relatively short PHEA block and targeting sufficiently high DP for the PS block enabled to access higher-order morphologies, as extensively reported in literature. 11 An example leading to vesicles is reported in Supporting Information (Fig. S4).

III Progress of RAFT-mediated dispersion photo-PISA

Although the synthesis of diblock copolymer nanoparticles during RAFT dispersion polymerization is well documented, 26 only a few studies addressed the issues of nucleation process and the way particles grow. [START_REF] Jones | [END_REF] The fact that our system displays slow kinetics of polymerization and a single uniform spherical morphology provides a simplified and appropriate framework to explore these mechanistic points.

III.1 Micellization

A conventional dispersion polymerization 6 starts from a homogeneous solution. [START_REF] Yeow | [END_REF] Given that PHEA 25 -TTC contains both a lyophilic group (PHEA block) and a lyophobic moiety (-SC 12 H 25 as -Z group of the RAFT agent), one might legitimately wonder about its capacity to form micelles initially in the alcoholic reaction medium. , proving the absence of surface activity under our reaction conditions. By contrast, surface tension reduction was demonstrated in water in which PHEA 25 -TTC has probably a lower affinity (72 mN m -1 in pure water vs 52.5 mN m -1 at 10 -3 mol L -1 ). As a conclusion, micelles

were not present at the beginning, and polymerization of St undoubtedly took place in solution initially. This accounts from the slow polymerization in Interval I since the monomer concentration was low. To detect when nucleation (micellization) starts, we used a DLS-based method originally developed to determine the critical micellar concentration of amphiphilic diblock copolymers formed in solution. 55 Our hypothesis is that the onset of micellization (beginning of particle nucleation) should be marked by a sudden increase in Please do not adjust margins Please do not adjust margins the intensity of scattered light (I s ). Fig. 6b shows the evolution of scattered

III.2 Particles growth

To elaborate the mechanism of photo-PISA, several colloidal data were determined quantitatively: particles number (N p , concentration of polymer particles in units of number of particles per liter), mean aggregation number (N agg , i.e. the number of block copolymer chains in a single nanoparticle), surface area stabilized by a copolymer chain (S agg ). Table 1 summarizes their evolution for different mean DP (m) of the PS core-forming block (details on calculation are provided in ESI). These data were obtained from a combination of TEM and 1 H NMR spectroscopy analysis for the dispersion polymerization of the PHEA 25 macro-CTA. Because TEM particle size values were accessible from Interval II, only the mechanism of particle growth could be examined in detail. Between 47 and 70 h, the particle size increase (from 22.5 nm to 31.6 nm) was accompanied by a 2-fold decrease in N p (from 1.3⋅10 19 L -1 to 6.7⋅10 18 L -1 ) and larger N agg (from 368 to 686). In this case, the steric stabilization of PHEA 25 block is likely to be insufficient to withstand the growth of the PS core and the resultant increase of particle surface area. In accordance with this hypothesis, S agg was found to be relatively constant (~4.3-4.6 nm 2

), regardless of PS chain length or particle size. Such mechanism where particle growth requires the incorporation of new copolymer chains to stabilize the newly formed surface area was already reported by the groups of Zhang and Armes. [START_REF] Jones | [END_REF]57 This trend was exacerbated when the reaction was carried at 55 °C instead of 35°C (Table 1), which resulted in larger particle diameters (42.1 nm). In this case, the drop of N p was even more pronounced because the larger DP of PS (160 instead of 118) can drive the incorporation more polymer chains. Another plausible explanation is that a higher temperature causes more particles coalescence. Table 2 provides similar colloidal characteristics for PISA syntheses performed with PHEA 52 -TTC and PHEA 85 -TTC. In both instances, N p and N agg remain unchanged during the polymerization. Consistently, the area occupied by PHEA blocks on the particle surface is larger for longer blocks and tends to increase with conversion. This result can be understood on the basis of the stronger steric stabilization of longer solvophilic blocks, which makes possible particle growth without the need for additional copolymer chains in the particle. This indicates that the observed increase in particle size in this case is simply the result of the growth of the PS core. ). With shorter PHEA stabilizers (PHEA 25 ), the particle diameter increase requires the addition of new copolymer chains through possibly two ways: particle coalescence or migration of individual chains from one particle to another. The occurrence of this second mechanism was evidenced by a simple experiment where a shot of divinylbenzene acting as a cross-linker (3 w % with respect to St) was added to the PHEA 25 -PS 118 latex. Gelation was observed within 24 h of irradiation, and clear indications of cross-linking between particles were evidenced by TEM (Fig. S5). Conversely, in the absence of exchange of copolymer chains between particles, cross-linking would be limited to PS core, thus preventing gelation.

III.3 General mechanism of particle nucleation/growth

IV Optical properties

Despite its impact on polymerization progress, very few studies on photo-PISA address the issue of optical properties. 55 In regular transmission, the apparent absorbance of a diluted diblock copolymer latex may be expressed as: (1) where : extinction coefficient [cm Please do not adjust margins Please do not adjust margins absorption and light scattering as soon as particles are formed (t > 47 h). The λ max (310 nm) of the absorption band of the π,π * -transition of TTC-based RAFT agent is clearly visible in all spectra, while scattering contribution increases during irradiation. In the selected spectral range > 280 nm, only the RAFT agent absorbs (St, PS and PHEA have a negligible absorption above 300 nm). Consequently, the eq. 1 can be simplified as:

(

) 2
The challenge is the quantitative differentiation of both absorption (cm-1) and scattering (cm-1) coefficients. Because there is no absorption between 500 nm and 800 nm ( = 0), in this range, is thus simply equal to (eq. 2). Given the small droplet diameters (less than 0.1 times the wavelength), Rayleigh theoryapplies and the experimentally determined may be proportional to the inverse of the wavelength to the power four. In agreement with this hypothesis, a range of simple fitting plots showing an excellent agreement with the experimental plots could be built (Fig. S6). The validity of our fitting was also supported by the linear correlation between and the intensityaverage diameters of particles (D p ) to the power six predicted by the Rayleigh theory (Fig. 7b). The final step consisted in using the calculated data in the 280-500 nm range to estimate separately the absorption coefficient . Fig. 7c shows the values during the dispersion polymerization. Clearly, a decrease of within the time of reaction can be observed. An analogous study performed in solution revealed a similar trend, suggesting that the change in the microenvironment of the RAFT agent, from solution to particles, was not accountable (Fig. S3). As emphasized by Qiao and co-workers, a more likely reason is the partial photolytic degradation of RAFT agent. 48,58 Up to 12% of photodegradation was determined after 15 hours of irradiation by these authors. In our case, longer irradiation (70 h) together with higher irradiance (547 mW cm -2 ) explain the higher extent of degradation.

V Template synthesis of mesoporous carbon materials

The PHEA-PS spherical nanoparticles described above were utilized as template for the synthesis of three-dimensional mesoporous carbons, i.e. with pores ranging from 2 to 50 nm.

Intensive research currently takes place on these materials due to their high technological potential in fields such as adsorption, 59 catalysis 60 or sensors.

61

Generally, only macroporous structures (> 50 nm) can be synthesized using latex templates. 60 With the exception of microemulsion polymerization, 62 most polymerization processes in dispersed system giving birth to polymer particles are not appropriate, because the accessible sizes are larger than 50 nm under standard synthesis conditions. More suitable templates to elaborate mesoporous carbons include rather colloidal (or mesoporous) silica as well as block copolymer mesophases.

45 Unlike these conventional approaches, we propose photo-PISA-derived particles as highly suitable soft template because of several unique features: access to spherical nanosized particles with diameters < 50 nm, size uniformity (PDI = 1.04) and tunability (see Table 1).

59

In a typical synthesis, the as-synthesized copolymer nanoparticles PHEA 25 -PS 118 (D p = 31.6 nm) were dispersed in a phloroglucinol/glyoxylic acid mixture without any sign of colloidal destabilization. Subsequently, a copolymer/phenolic resin hybrid including a template content of 62 wt% was created after solvent evaporation (see precise conditions in Please do not adjust margins Please do not adjust margins the experimental section). Finally, pyrolysis enabled in a single step to carbonize the phenolic resin and degrade the copolymer nanoparticles. As can be seen in Fig. 8 (image a), which shows a typical TEM image after carbonization, the structure consists of randomly distributed interconnected spherical mesopores. The size of these pores (22.0 nm) was rather uniform, which results from the narrow size distribution of the initial colloidal particles. Control experiment without nanoparticles led to a system devoid of mesopores.

Remarkably, the size of the pores could be manipulated by changing the diameter of the copolymer particles. For example, the PHEA 25 -PS 160 latex (D p = 41.6 nm, Table 1) possessing a higher DP of the core-forming block led to an average mesopore size of 29 nm (Fig. 8, image b). For these two samples, the BET (Brunauer-Emmett-Teller theory) surface area was found to be 691 and 702 m 2 g -1

, respectively (see nitrogen adsorption/desorption isotherms in Fig. S7). The BJH (Barrett-Joyner-Halenda theory) cumulative pore volumes (corresponding to pores > 2 nm) were also high: 1.01 (PHEA 25 -PS 118 ) and 1.23 cm 3 g -1 (PHEA 25 -PS 160 ). For comparison, a third mesoporous carbon (Fig. 8, image c) was prepared using as template a conventional lyotropic mesophase formed by the self-assembly of an amphiphilic triblock copolymer (poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide), Pluronic F127). 63 Due to size limitation of liquid crystal phases, much smaller mesopores (7 nm) were created. In addition, lower volumes (0.49 cm 3 g -1 ) were obtained for a BET surface area which remained similar (689 m 2 g -1 ). Despite the absence of ordered structure and their cost, PISA-derived particles appear as an excellent template for the synthesis of uniform nanoscale porous carbons with large mesopores.

Conclusions

Well-defined spherical nanoparticles were prepared by an original approach combining PISA in dispersion with a photomediated RAFT mechanism under visible-light irradiation. PHEAs were first synthesized in solution before being used as macro-CTAs for a subsequent chain-extension with St in dispersion. Besides the confirmation of the control of the polymerization, the kinetic studies evidenced two distinct regimes related to particle nucleation and particle growth. In the second interval, the high number of particles swollen by monomer resulted in much faster polymerization rate. UV-visible measurements revealed that absorption due to the TTC functions decreased during a first phase of the polymerization, before reaching a constant value, suggesting that a partial degradation of the terminal functions could first occur in the early stage of reaction. Complementary investigations are carried out for a better understanding of this observation. As expected, it was possible to modify particle size by playing on the respective lengths of the solvophilic and solvophobic blocks. Interestingly, photo-PISA could be conducted at ambient or lower temperatures than the ones commonly used in conventional PISA. This wider range of possible experimental conditions is expected to broaden the scope of PISA to new applications, involving for instance thermally sensitive compounds. The temporal control of the polymerization is undoubtedly a supplementary advantage, which could lead to more finely controlled or even adaptable morphologies. Finally, it has been demonstrated that PISAderived particles appear as excellent template for the synthesis of uniform nanoscale porous carbons with large mesopores size, difficult to achieve with block copolymer mesophases.

Fig. 1

 1 Fig. 1 Synthesis of poly(hydroxyethyl acrylate) macro-CTA via photomediated RAFT polymerization in 1,4-dioxane.Subsequent chain extension via dispersion photo-PISA of styrene in a methanol/water mixture.

Fig. 2

 2 Fig. 2 Visible light photomediated polymerization of HEA in solution during the synthesis of PHEA58-TTC. (a) Polymerization kinetics, conversion measured by 1 H NMR; (b) SEC chromatograms (RI detector) in THF; (c) Number-average molecular weight (Mn) and dispersity (Đ = Mw/Mn) with conversion; dashed line: theoretical molecular weight calculated with conversion. Experimental conditions: HEA : TTC =100 : 1; [HEA]0 = 2.97 M in dioxane, blue LED (547 mW cm -2 , 25 °C).

Fig. 3

 3 Fig.3 Series of photographs showing the evolution of the reaction medium during the photomediated RAFT dispersion polymerization of St (synthesis of PHEA25-PSm diblock copolymer particles).

Fig. 4 1 H 2 ,

 412 Fig. 4 (a) Polymerization kinetics for the synthesis of a PHEA25-PS118 copolymer. Conversion measured by 1 H NMR (Fig. S2). Evolution of particle size determined by DLS and TEM. (b) SEC chromatograms in DMF at 50 °C (RI detector) of PHEA25-TTC and PHEA25-PSm diblock copolymers synthesized via photomediated RAFT polymerization in dispersion. (c) Evolution of Mn and dispersity index with monomer conversion; dashed line: theoretical molecular weight calculated with conversion. Experimental conditions: PHEA25-TTC:St = 1:200; [PHEA25-TTC]0 = 7.68 mM in methanol/water mixture (95/5 w/w %), blue LED (547 mW cm -2 , 35 °C).

  Fig. 6a displays the evolution of surface tension σ as a function of PHEA 25 -TTC concentration for measurements performed in MeOH/H 2 O (95/5, w/w %), containing also St to reproduce reaction medium. No significant surface tension reduction was observed in the concentration range 0 -10 -2 mol L -1

Fig. 5

 5 Fig. 5 TEM images of PHEA25-PSm samples taken at different times during dispersion PISA of St.

Dp

  -Diameter of particles determined by TEM. Np -Number of particles per liter of the reaction mixture. PDI -Dispersity index. 56 Nagg -Number of RAFT chain-ends per particle. Sagg -Area occupied by 1 RAFT chain-end function on the surface of the particles.intensity as function of time within the first 20 h of irradiation. DLS measurements were performed without any dilution and at constant incident light intensity enabling comparisons between samples. In the first 8 h following light exposure, the scattering intensities had constant and low values consistently with a homogeneous medium devoid of colloidal clusters. As early as 10 h (well before interval II), scattered intensity displayed a linear and sharp increase, which marked the formation of the first micelles. St conversion was only 3-4 % at this time, leading to a critical degree of polymerization of ~6-8 units for the PS block (PHEA 25 -PS 6-8 -TTC).

Fig. 6

 6 Fig.6 (a) Surface tension of a MeOH:H2O (95/5, w/w %) mixture for different concentrations of PHEA25-TTC as determined by the Wilhelmy plate method. The typical concentration of macro-CTA used in photomediated RAFT polymerization in dispersion is 7.68 mM. (b) Scattered intensity (mcps-mega counts per second, at low concentration is proportional to particles concentration) determined by DLS technique from aliquots withdrawn during photo-PISA via RAFT polymerization in dispersion.

Dp

  -Diameter of particles determined by TEM. Np-Number of particles per liter of the reaction mixture. PDI -Dispersity index. 56 Nagg -Number of RAFT chain-ends per particle. Sagg -Area occupied by 1 RAFT chain-end function on the surface of the particles.

Scheme 2 .

 2 Fig.7ashows a series of absorption spectra of PHEA 25 -TTC macro-CTA in the range 300 -400 nm taken throughout St dispersion polymerization. They clearly show that the extinction coefficient is the combined result of both

Fig. 7 (Fig. 8 10 |

 7810 Fig. 7(a) UV-Vis absorption spectra of PHEA25-TTC macro-CTA in the range 300 -400 nm throughout St dispersion polymerization.(b) Fitted scattering coefficient( at 500 nm versus particles average diameter (Dp).(c) Estimated absorption coefficient of PHEA25-PSm during dispersion photo-PISA.

Table 1 .

 1 Colloidal data of PHEA25-PSm particles formed by dispersion photo-PISA at two different temperatures (35 and 55 °C).

Table 2 .

 2 Colloidal data of PHEA52-PSmand PHEA85-PSm particles formed by dispersion photo-PISA.

	Time	m (PS degree of	Dp	PDI	Np	Nagg	Sagg
	h	polymerization)	nm		L -1		nm 2
				PHEA52-PSm (35°C)
	47	48	20.3±0.4	1.03	1.90⋅10 19	244	5.3
	54	52	21.0±0.4	1.03	1.78⋅10 19	260	5.3
	70	110	24.0±0.4	1.03	1.80⋅10 19	257	7.0
				PHEA85-PSm (35°C)
	47	78	21.0±0.4	1.06	2.76⋅10 19	168	8.3
	54	92	20.0±0.4	1.06	3.45⋅10 19

This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 5

Acknowledgements

French ministry of higher education and research is acknowledged for providing a UHA scholarship to Vitalii Tkachenko.

Notes and references

Conflicts of interest

There are no conflicts to declare.


Raouf Boucekkine
email: raouf.boucekkine@univ-amu.fr

Carmen Camacho
email: car-men.camacho@psemail.eu.

Benteng Zou
email: ben-teng.zou@uni.lu.

Optimal switching from competition to cooperation: a preliminary exploration

Keywords: Cooperation, competition, dynamic games, multi-stage optimal control. JEL classification:

In this paper, we tackle a generic optimal regime switching problem where the decision making process is not the same from a regime to another. Precisely, we consider a simple model of optimal switching from competition to cooperation. To this end, we solve a twostage optimal control problem. In the first stage, two players engage in a dynamic game with a common state variable and one control for each player. We solve for open-loop strategies with a linear state equation and linear-quadratic payoffs. More importantly, the players may also consider the possibility to switch at finite time to a cooperative regime with the associated joint optimization of the sum of the individual payoffs. Using theoretical analysis and numerical exercises, we study the optimal switching strategy from competition to cooperation. We also discuss the reverse switching.

Introduction

The recent years have noticed the emergence of numerous debates on the opportunity and timing of very different types of transitions, each associated with a bunch of academic works in the economic and operational research literatures: environmental transitions (in particular the energy transition, see the early work of [START_REF] Tsur | Optimal transition to backstop substitutes for nonrenewable resources[END_REF], political transitions (among others, transition to democracy following [START_REF] Acemoglu | Economic Origins of Dictatorship and Democracy[END_REF], organizational transitions (either in markets or in the workplace, see [START_REF] Vallée | Optimal time switching from tayloristic to holistic workplace organization[END_REF] etc...In most of these works, the optimal timing of transitions (if any) are only implicitly tackled though the vast majority of the models developed are dynamic. 1There exist however an increasing number of papers interested in optimal regime transition and the inherent timing. As a common feature, all these papers use multistage optimal control techniques, first developed by Tomiyama (1984). This notably covers technological regimes switching [START_REF] Boucekkine | Technology adoption under embodiment: A two-stage optimal control approach[END_REF][START_REF] Saglam | Optimal pattern of technology adoptions under embodiment: A multistage optimal control approach[END_REF]) and environmental transitions (see [START_REF] Boucekkine | Optimal regime switching and threshold effects[END_REF][START_REF] Moser | History-dependence in productionpollution-trade-off models: a multi-stage approach[END_REF]). 2 Not surpisingly, given the nature of problems tackled in the latter literature, the decisions are taken by a single player, say the central planner. However, in many cases, from one stage to another, the decision makers may differ. For example, from dictatorship to democracy, we have to move from a regime in which initially almost all the decisions are taken by a dictator, to a regime in which at the very least, the decisions are taken in a more collective way (see [START_REF] Acemoglu | Economic Origins of Dictatorship and Democracy[END_REF], in a frame without multi-stage optimal control, and Boucekkine et al., 2016, with multi-stage optimal control). Similar considerations arise when analyzing international climate agreements processes where we typically switch from a regime with country-level decision making and no cooperation, for example prior to the 2015 Paris agreement, to a regime of institutional cooperation with joint decision making. On a more technical ground, the economic literature is rather very poor in papers merging multi-stage optimal control and dynamic games ingredients. The corresponding operational research literature is less poor (see for example [START_REF] Boucekkine | Environmental quality versus economic performance: a dynamic game approach[END_REF].

In this paper, we shall deal with a generic regime switching problem where the decision making process is not the same from a regime to another. Precisely, we consider a simple model of optimal switching from competition to cooperation. We believe this problem is generic enough to cover a large set of problems. In addition to the dawn of multinational agreements, it can be also applied to political parties or companies mergers within a country. 3 Needless to say, the reverse can happen, and one can notice situations in which long time cooperation comes to an end and a further competition regime sets in. Canada in 2011 pulled out of the Kyoto protocol, USA withdrew from the Paris Agreement in 2017, and it is very likely that the UK will brexit by the end of January 2020. In this kind of situation, the decision making process changes as well.

A quite rich set of questions arises from the examples given above: what are the tradeoffs involved in the decision to move from competition to cooperation (and vice versa), and what is the optimal timing for the institutional regime change if any? Since addressing these questions involves dealing with strategic tradeoffs in dynamic settings, it implies embedding dynamic games ingredients in multi-stage optimal control problems. As outlined above, the economic literature is rather thin in this respect.

There is a substantial applied game theory literature of endogenous coalition formation (see for example Di [START_REF] Bartolomeo | Staying together or breaking apart: policy-makers' endogenous coalitions formation in the European Economic and Monetary Union[END_REF], who study how coalitions among fiscal and monetary authorities are formed and what are their effects on the stabilization of output and prices). However, it generally pays no attention to the regime switching problem described above. 4In this paper, we propose a preliminary exploration of the latter switching problem. To this end, we solve a two-stage optimal control problem. In the first stage, two players "compete" on a common state variable which could be a public good or a public bad. They engage in a dynamic game in this first stage, and we solve for openloop strategies. Arguably, under the current setting, the interesting equilibrium is the path strategies, i.e., open-loop strategies, given the fact that usually the commitment extend over the entire future time and negotiation results depend on the initial condition.

Thus, the strategy may not be subgame perfect by definition, which gives possibility for changing in some future time. The decision rule strategies, i.e., the Markovian strategies, are subgame perfect Nash equilibria, but no commitment at all is possible, thus does not fit our examples above. As the state equation and the individual payoffs are linear-quadratic, the resulting dynamic game would be a conceptually trivial problem if no perspective of switching to a cooperative regime emerges. We do introduce such a possibility with the associated joint optimization of the sum of the individual payoffs.

The paper is organized as follows. Section 2 briefly presents the differential game setting. Section 3 provides the open-loop strategies during the non-cooperate and cooperative periods, thus the optimal switching conditions can be obtained. In Section 4, we turn to the numerical analysis that illustrates and deepens the theoretical findings.

Section 5 provides extra discussion of the other direction of the game, that is, from cooperate to noncooperative, and Section 6 concludes.

A simple model of optimal switching

There are two players: players 1 and 2, who share a common variable, y ∈ [0, Y ], which could be public good or public bad. Each player chooses the level of a variable +∞), which provides her with utility. At the same time, their choice increases the level of y, which induces a loss in utility. Let us assume that at time 0, players play a non-cooperative dynamic game, choosing their optimal trajectory for and Veugelers (2002). Essentially, these models are static and can not show switching conditions from noncooperative to cooperative games and vise versa.

x i ∈ [0, X] ⊂ [0,
x i . Then, since their individual choices affect equally the common variable, they could decide to switch strategies at a time T and continue playing strategically. Indeed, in this paper players can optimally choose a time T to start playing cooperatively. Let us provide an example from Economics that will accompany us throughout the paper. We can assume that there exists a unique final good, which requires only a polluting resource as input. With a quantity x i of pollution, the firm produces an amount a i x i of the final good. Consumption provides the player with utility, but at the same time it increases the level of CO 2 emissions, y. Obviously, the level of CO 2 affects both players. In the end, the player can obtain utility directly from the consumption of x i , but she also suffers from pollution, so she will receive disutility from y.

The objective of player i = 1, 2 is to maximize overall welfare, defined as

max x i W i = +∞ 0 e -rt a i x i - x 2 i 2 - b i y 2 2 dt,
which depends on her individual choice, x i and which is subject to the dynamic constraint:

ẏ(t) = x i + x j -δy(t), i, j = 1, 2; i = j, y(0) = y 0 given. (1)
Welfare is the sum of instantaneous utility from time zero. Instantaneous utility at a time t ≥ 0 is given by a i x i -

x 2 i 2 -b i y 2 2 .
As mentioned, x i is the choice of player i. The cost of choosing x i is

x 2 i 2 , thus the net gain from x i is a i x i - x 2 i
2 , with a i (≥ X) positive constant measuring the unit gain.

An increase in the state variable y has a negative effect on utility. Indeed, as in our example where y is the stock of CO 2 emissions, then any increase in y induces a damage b i y 2 2 , with scaling parameter b i , a positive constant.

Suppose that at some future date T , the two players decide to play cooperatively.

Then, the join objective is

max x i ,x j W II (T ) = +∞ T e -rt a i x i + a j x j - x 2 i + x 2 j 2 - (b i + b j )y 2 2 dt,
subject to the same state equation [START_REF] Acemoglu | Economic Origins of Dictatorship and Democracy[END_REF].

For simplicity reasons, we only consider here the symmetric case where a i = a j = a

and b i = b j = b.

Remark 1

The difficulty of considering the asymmetric case is to provide the sharing principal in the second period. Here, in the symmetric case, the two share equally the gain.

In the following, we compute the optimal switching time T . Denote by W I (T ) welfare of a player playing non-cooperative, from time 0 to the switching time T , when she starts playing cooperatively. Then, the player solves the following problem

max x i W I (T ) = T 0 e -rt a i x i - x 2 i 2 - b i y 2 2 dt,
subject to the law of motion [START_REF] Acemoglu | Economic Origins of Dictatorship and Democracy[END_REF].

The optimal choice of T is given by the solution of

max T W I (T ) + e -rT W II (T ) 2 . ( 2
)
If T = 0, then it is optimal for the two players to play cooperatively immediately;

while if T = +∞, the noncooperative game should continue forever.

The optimal switching strategy

As in [START_REF] Boucekkine | Technology adoption under embodiment: A two-stage optimal control approach[END_REF] and the references after them, we solve first the second stage problem. Optimal trajectories for x and y are obtained in case the players play cooperatively from T onwards. Second, taking T as given, we solve the first noncooperative game. We obtain optimal trajectories in both phases that depend on y 0 and T . Finally, the optimal switching time T is obtained substituting the resulting W I and W II into (2), and solving the resulting problem. As a result, the optimal solution will be made of an optimal switching time T , and the optimal trajectories for y and x before and after the switch.

The cooperative regime

The joint symmetric optimization problem can be restated as:

max x W II (T ) = +∞ T e -rt 2ax -x 2 -by 2 dt,
subject to the state equation

ẏ(t) = 2x -δy(t),
for i, j = 1, 2, i = j, and where y(0) = y 0 is given. Define the associated Hamiltonian as

H II (x, y, λ II ) = 2ax -x 2 -by 2 + λ II (2x -δy(t)), (3)
with λ II the co-state variable. The first order conditions yield the following set of

optimal conditions        x II = a + λ II , ẏII (t) = -δy + 2λ II + 2a, λII (t) = 2by + (r + δ)λ II , (4)
with transversality condition lim t→+∞ e -rt y(t)λ II (t) = 0.

Next proposition provides the analytical solution to (4). We do not reproduce here the proof since it is a standard exercise.

Proposition 1 Suppose that the two players play the cooperative game from time T onwards, then the optimal effort is given by (4). The corresponding state and costate variables are given by:

   y II (t) = 2C 1 e µt + y, λ II (t) = (µ + δ)C 1 e µt + λ, (5)
where t ≥ T and µ

= r- √ r 2 +4[δ(r+δ)+4b] 2
(< 0). Constant C 1 will be determined later by the transversality condition y(T -) = y(T + ).

Moreover, there exists a unique steady state (y, λ) given by

y = 2a(r + δ) δ(r + δ) + 4b , λ = - 4ab δ(r + δ) + 4b
.

Associated to this steady state, we can compute the steady state of x II ,

x = a + λ = a 1 - 4b δ(r + δ) + 4b
.

Corollary 1 Both ȳ and x increase with a, they decrease with b.

Obviously if T = 0, that is, if the cooperation starts from the beginning, then y(T ) = y(0) = y 0 and C 1 = y 0 -y 2 . It is easy to check that the optimal joint welfare is in this case

W * II (0) = 2a(a + λ) -(a + λ) 2 -by r + (µ + δ) 2 C 2 1 2µ -r + 2C 1 (b + (µ + δ)λ) µ -r . ( 6
)

The non-cooperative regime

Let us solve the first period non-cooperative game for a switching time T given. The first period optimization problem for player i is

max x i W I (T ) = T 0 e -rt a i x i - x 2 i 2 - b i y 2 2 dt, subject to ẏ(t) = x i + x j -δy(t),
for i, j = 1, 2, i = j, and where y(0) = y 0 is given. The associated Hamiltonian is

H I (x i , y, λ I ) = a i x i - x 2 i 2 - b i y 2 2 + λ I (x i + x j -δy(t)), (7)
with λ I co-state variable in the first period. In the symmetric setting, the first order conditions yield the following set of optimal conditions:

       x I = a + λ I , ẏI (t) = -δy I + 2λ I + 2a, λI (t) = by + (r + δ)λ I , (8)
Proposition 2 below provides the analytical solution to the first order conditions in [START_REF] Bartolomeo | Staying together or breaking apart: policy-makers' endogenous coalitions formation in the European Economic and Monetary Union[END_REF]. Its proof can be found in the appendix.

Proposition 2 Suppose that at time T , the two players start playing the cooperative game.

Then, for t ∈ [0, T ], the unique symmetric strategic Nash equilibrium is x I (t) = a + λ I , with state, y I , and costate variables, λ I , given by

     y I (t) = â e ν 1 t + b e ν 2 t + y P (t), λ I (t) = ĉ e ν 1 t + d e ν 2 t + λ P (t), (9)
in which ν 1 and ν 2 are eigenvalues given by

ν 1 = r -r 2 + 4[δ(r + δ) + 2b] 2 , ν 2 = r + r 2 + 4[δ(r + δ) + 2b] 2 .
(y P (t), λ P (t)) is a special solution:

y P (t) = ŷP (1 -e ν 1 t ) + y P (1 -e ν 2 t ), λ P (t) = λP (1 -e ν 1 t ) + λ P (1 -e ν 2 t ), with ŷP = 2a(ν 2 + δ) ν 1 (ν 1 -ν 2 ) , y P = - 2a(ν 1 + δ) ν 2 (ν 1 -ν 2 ) , λP = a(ν 1 + δ)(ν 2 + δ) ν 1 (ν 1 -ν 2 ) , λ P = - a(ν 1 + δ)(ν 2 + δ) ν 2 (ν 1 -ν 2 ) .
Constants a, b, c, d are given by

a = -(ν 2 + δ)y 0 + 2λ 0 ν 1 -ν 2 , b = (ν 1 + δ)y 0 -2λ 0 ν 1 -ν 2 and c = (ν 1 + δ)[-(ν 2 + δ)y 0 + 2λ 0 ] 2(ν 1 -ν 2 ) , d = (ν 2 + δ)[(ν 1 + δ)y 0 -2λ 0 ] 2(ν 1 -ν 2 ) ,
where y 0 is given initial condition, while λ 0 will be determined by the switching condition at time T .

A special case arises when T = +∞, that is, when no cooperation is possible. Here the solution of ( 8) with transversality condition lim t→+∞ e -rt y(t)λ

I (t) = 0, is      y I (t) = 2C 1I e ν 1 t + y I , λ I (t) = C 1I e ν 1 t + λ I , (10)
where

C 1I = y 0 -y I 2
and the steady state (y I , λ I ) = 2a(r+δ) δ(r+δ)+2b , -2ab δ(r+δ)+2b . In this special case, each player's social welfare is

W * I (+∞) = a(a + λ I ) -(a + λ I ) 2 /2 -by I /2 r + (ν 1 + δ) 2 C 2 1I 2(2ν 1 -r) + C 1I (b + (ν 1 + δ)λ I ) ν 1 -r . (11)
From ( 6) and [START_REF] Mair | The electoral payoffs of fission and fusion[END_REF], it is easy to check that

W * II 2 (+∞) -W * I (+∞) 0,
depending on the combination of parameters. In other words, it is possible that the switching happens at T = 0, or T = +∞, or T ∈ (0, +∞), depending on the situation under study, which can also be seen in the Section 4 with numerical illustration.

Optimal switching time

At the switching time, T , and by continuity, the state variable y must verify that

y I (T ) = y II (T ). (12)
Similarly, the costate variable must verify that

λ I (T ) = λ II (T ). (13)
Usually, as in [START_REF] Boucekkine | Technology adoption under embodiment: A two-stage optimal control approach[END_REF][START_REF] Boucekkine | Optimal regime switching and threshold effects[END_REF][START_REF] Boucekkine | On the timing of political regime changes in resource-dependent economies[END_REF], in the study of the optimal switching in optimal control problems with a unique decision maker, a transversality condition is imposed on the maximized Hamiltonian at the switching time T . In that case, the maximized Hamiltonian is exactly the same immediately before and immediately after the switch.

However, under the current dynamic game setting, it is different. Before and after the switch, there are different decision makers as stated in the Introduction. Before the switch, each individual player takes her own optimal decisions, while after the switch, it is a join choice. Thus, it is improper to equalize the maximized Hamiltonian before and after the switch, even with identical players.

Therefore, we impose that the shadow values are the same before and after the switching. In our example where y measures the stock of CO 2 , this transversality condition implies that the shadow value of pollution does not change immediately due to the signature of some agreement or protocol. Instead it takes some time for the shadow value to change, while immediate changes come from the choice variables.

         λ 0 (T ) = 2E 1 (T ) -(µ + δ)A 1 (T ) (µ + δ)A 2 (T ) -2E 2 (T ) , C 1 (T ) = E 1 (T ) + E 2 (T )λ 0 µ + δ e -µT , (14)
where

A 1 (t) = (ν 1 + δ)e ν 2 t -(ν 2 + δ)e ν 1 t ν 1 -ν 2 y 0 + (y P (t) -y), A 2 (t) = 2(e ν 1 t -e ν 2 t ) ν 1 -ν 2 ,
and

E 1 (t) = (ν 1 + δ)(ν 2 + δ)(e ν 2 t -e ν 1 t ) 2(ν 1 -ν 2 ) y 0 + (λ P (t) -λ), E 2 (t) = (ν 1 + δ)e ν 1 t -(ν 2 + δ)e ν 2 t ν 1 -ν 2 .
Thus, substituting ( 14) into ( 9) and ( 5), we could obtain the explicit optimal trajectories in both periods.

The optimal switching time T is given by the optimization problem (2). With the above explicit solutions, it only remains to calculate the social welfare in the two periods, i.e. W I (T ) and W II (T ). It is easy to check that

e -rT 2 W II = a 2 -λ 2 -by 2 2 e -2rT r -(µ + δ)λ + 2by C 1 e (µ-2r)T r -µ - (µ + δ) 2 + 4b 2 C 2 1 e (2µ-2r)T r -2µ , (15)
in which C 1 = C 1 (T ) is given by [START_REF] Stern | The Economics of Climate Change: The Stern Review[END_REF].

Similarly, the first period social welfare W I is given by

W I = a 2 -( λ P + λ P ) 2 -b( y P + y P ) 2 2 (e -rT -1) -r - ( c -λ P ) 2 2 + b( a -y P ) 2 2 (e (2ν 1 -r)T -1) 2ν 1 -r - ( d -λ P ) 2 2 + b ( b -y P ) 2 2 (e (2ν 2 -r)T -1) 2ν 2 -r -( c -λ P )( d -λ P ) + b ( a -y P )( b -y P ) (e (ν 1 +ν 2 -r)T -1) ν 1 + ν 2 -r -( c -λ P )( λ P + λ P ) + b ( a -y P )( y P + y P ) (e (ν 1 -r)T -1) ν 1 -r -( d -λ P )( λ P + λ P ) + b ( b -y P )( y P + y P ) (e (ν 2 -r)T -1) ν 2 -r , (16)
in which a, b, c, d depend on the switching time T as well, via λ 0 (T ).

The first order condition from (2) yields

dW I (T ) dT + e -rT 2 -rW II (T ) + dW II (T ) dT = 0. (17)
Intuitively, the first term should be nonnegative, i.e. the longer the time period [0, T ], the higher the welfare it yields (otherwise T = 0 already). The second term should be non-positive (otherwise T = +∞). Nevertheless, due to the complexity of expressions ( 15) and ( 16), the study of an explicit form becomes cumbersome, and the study of the properties and impacts become impossible. Thus, it is not wise to continue working on the search of explicit solutions. Instead, we focus in next section on numerical simulations to illustrate the impacts of important parameters, such as, the efficient parameter, a, and dis-utility parameter, b, as well as the initial condition, y 0 .

Although the model is not realistically calibrated, the qualitative pattern is illustrative.

Numerical illustration

This section illustrates numerically the theoretical results obtained in the previous sections. First, we compare the optimal switching time as a function of the initial endow-ment of the state variable, y 0 . In a second set of exercises, we compute the optimal dynamic trajectories of the state variable y and of the optimal choice x.

Optimal switching time

To seize the importance of all the elements of the model, we have computed the optimal switching time under various scenarios for both a and b. In all exercises, r = 0.0015 and δ = 0.01. The time discount is in line with the most recent literature (see the literature following [START_REF] Stern | The Economics of Climate Change: The Stern Review[END_REF].

We begin by exploring the role of a. Setting b = 0.000095 we choose an economy which loses little utility with y. Two economies are compared: a performant economy, which can extract high utility from the same amount of the choice variable x, for which a = 0.115; and a second less efficient economy, in which a = 0.109. Figure 1 shows the optimal switching time and the associated overall welfare when the initial endowment of the state variable ranges from y 0 = 50 to y 0 = 100. Interpreting y as the CO 2 stock and x as consumption, a is a measure of the effi-ciency of the economy. Indeed, with the same amount of the resource, the higher a, the more of the final consumption good we obtain. Low levels of y 0 correspond to economies that are not too polluted initially. By the same token, high y 0 correspond to economies that are already very polluted at the beginning of the game.

On the left panel of Figure 1, we see that players always decide to play cooperatively, that is 0 ≤ T < ∞. They cooperate from T = 0 when y 0 is relatively low. When the environmental quality is high, players join as soon as possible to preserve the environment and avoid the loss of utility induced by pollution. Besides, switching times are increasing functions of the initial stock of y 0 . If y is the stock of CO 2 emissions, then clean economies start cooperating earlier. Indeed, earlier cooperation reduces global emissions and ensures a higher production in the future. Conversely, when economies are relatively dirtier, y 0 is relatively high, then they start cooperating later as if the damage was already too large to struggle against. Finally, note that T is always smaller for a = 0.115. Efficient economies cooperate sooner since their advantage in production allows them to obtain more of the final good producing the same amount of pollution. Note that better performing economies can cope better with possible losses in consumption arising upon cooperation.

Let us study next the negative effect of the state variable in utility, as captured by parameter b. The right panel in Figure 1 shows optimal switching times when a = 0.09 for two values of b: a high level of b = 0.00015 and a low level b = 0.000095. Again, understanding y as CO 2 emissions, an economy with a low b corresponds to an economy less sensitive to pollution, such as a very small open economy or an economy embodied with sufficient forest or water source to absorb important amounts of CO 2 . These economies start playing cooperatively earlier in order to maintain y at a moderate level.

Optimal dynamic trajectories

Next, let us illustrate the dynamic trajectories for the economy underlining the roles of a and b in the dynamics of the optimal control and state variables. In the first exercise, a = 0.5 and we compare two economies which differ in their sensitivity to pollution.

the economy is initially endowed with y 0 = 75. Figure 2 shows our results for the state variable, y, and for the control variable per capita, x. The right panel of Figure 2 reveals the thinking of the policy maker. Compared to their steady states, both economies start with a highly polluted environment, which makes them lose a significant amount of welfare. Hence, the policy maker decides not to consume during a certain period of time. This is true when b = 0.000095 and that players play together from t = 0, as well as in the more sensitive case, when b = 0.00015, and that there are two policy makers from t = 0 till T = 18.4. Note that the less sensitive economy starts consuming before and it consumes more.

Our last exercise compares two economies which differ in their efficiency level letting b = 0.000095. In the first, a = 0.5, and in the second, the efficiency level doubles the first, a = 1. As shown in Proposition 1, the steady state of y keeps the same proportion than efficiency. Indeed, y ss = 23.23 when a = 0.5 and y ss = 46.46 when a = 1.

Our results are displayed in Figure 4. The most salient feature is that the most efficient economy does not sacrifice initial consumption in order to preserve y at a low level. Its technological advantage allows the policy maker to choose positive levels of consumption from t = 0.

From the cooperative to the non-cooperative game

In the previous sections, if players decide to join at time T , they will play cooperatively forever after. In other words, their commitment to some protocol, or union, or agreement is irreversible. However, if it was possible to exit in some future time, choices and trajectories would be more complicated. In this section, we extend the previous study to include the possibility of a future exit. The steps of the game are the following:

Step 1. The two players play the non-cooperative differential game until time T 1 .

Under the symmetric assumption, the revenue of the identical players is V I (T 1 , T 2 )

Step 2. From time T 1 until T 2 , the two players play cooperatively. Denote the join revenue as V II (T 1 , T 2 ).

Step 3. One of the two players exits the cooperative game at T 2 , thus from T 2 onwards, the two players play the non-cooperative differential games again. Denote the revenue in this period as V III (T 1 , T 2 ).

Then the optimal choice of switching times should be given by max

T 1 ,T 2 V I (T 1 , T 2 ) + e -rT 1 V II (T 1 , T 2 ) + e -rT 2 V III (T 1 , T 2 ) . (18)
Mathematically, this more complicated differential game can be solved by backward induction by reversing the calculation order of the above Section 3. Similarly, numerical investigation can be performed the same way as Section 4. Therefore, we do not need to do this exercise here in detail.

Conclusion

We investigate both theoretically and numerically some dynamic game settings where players choose to switch from noncooperative competition to cooperate (or vise versa) and which automatically change the decision makers as well. The explicit conditions of open-loop strategic switching conditions are presented, though it can not be fully analytical. We rely on numerical simulation and demonstrate the importance of parameters. Not surprising, given the intrinsic nature of open-loop strategies, both switching time and the social welfare depend essentially on the initial condition of the game, but neither were monotonic. The efficient parameter plays very important role in not only the decision of switching from noncooperative to cooperative, but also the choice of consumption. High efficient parameter may cover the side effects, such as leading to high pollution as a by product. Of course, different damaging parameter may be introduced between non-cooperative and cooperative games, which is a future work.

Furthermore, it is easy to imagine the situation where the two players are asymmetric or there are more than two players. We do believe that the current study paves the way to handle a much wider class of problems, beyond the examples we presented in the Introduction.

A Appendix. Proof of Proposition 2

Recall the dynamic system as with ν 1 < 0 and ν 2 > 0, and associated eigenvectors

- → v i =   2 ν i + δ   , i = 1, 2.
Define matrix

M (t) = ( - → v 1 - → v 2 )   e ν 1 t 0 0 e ν 2 t   =   2e ν 1 t 2e ν 2 t
(ν 1 + δ)e ν 1 t (ν 2 + δ)e ν 2 t   .

Thus the inverse matrix of M (t) is

M -1 (t) =    -ν 2 +δ 2(ν 1 -2 ) e -ν 1 t e -ν 1 t ν 1 -ν 2 ν 1 +δ 2(ν 1 -ν 2 ) e -ν 2 t -e -ν 2 t ν 1 -ν 2    .
The unique solution of the system is then given by

5

 5

Figure 1 :

 1 Figure 1: Optimal T . Left: role of a when b = 0.000095. Right: role of b when a = 0.09.

Figure 2 :

 2 Figure 2: Dynamics of y(•) and x(•). a = 0.5.

Figure 3 :

 3 Figure 3: Dynamics of y(•) and x(•). b = 0.000095.

  ) = -δy I + 2λ I + 2a, λI (t) = by + (r + δ)λ I ,It is easy to obtain the associated eigenvaluesν 1,2 = r ± r 2 + 4(δ(r + δ) + 2b) 2

  in which λ(0) is undetermined. Substituting M (t), M -1 (t) and M -1 (0) into the above matrix algebra and taking integrals, we obtain the explicit solution in Proposition 2.That completes the proof. B Appendix. Second order condition.

Figure 4

 4 Figure 4 shows the value of the second order derivative of (2) associated to the examples in Section 4. The graphs show that the second derivative is negative for all values of y 0 in the four examples. Hence, the switching times T in Figure 1 is always a maximum.

Figure 4 :

 4 Figure 4: Second order conditions.

This is specially true in the political transitions literature, with the notable exception of Boucekkine et al.,

[START_REF] Boucekkine | Technology adoption under embodiment: A two-stage optimal control approach[END_REF] Applications to macroeconomic policy switching, e.g.Zampolli et al. (2016), or to workplace organization as in the above cited paper by[START_REF] Vallée | Optimal time switching from tayloristic to holistic workplace organization[END_REF], can be also found.

For example, in December 2003, the Progressive-Conservative Party and the Reform/Canadian Alliance parties merged and created a new right-wing political formation, the Conservative Party of Canada. In October 2007, the two most important Italian left-wing parties merged into a single political entity, the Democratic Party.

Also, and even more clearly, our research questions and inherent settings are quite different from the classical literature on cooperative and noncooperative R&D (D'Aspremont and Jacquemin, 1988; Suzumura, 1992), research joint ventures[START_REF] Kamien | Research joint ventures and R&D cartels[END_REF] or the empirical work of Cassiman

The switching time trajectories displayed in Figures1 and 2are maxima of (2), since they verify the second order condition as shown in Appendix B.