Cécile Fossé

Jesus Manuel

Alexandre Castro-Román

Yannick Freulon

Jacques Thebault

Luc Lacaze

Robbiola

C Fosse

M Castro-Roman

A Freulon

Y Thébault

J Lacaze

L Robbiola

Microstructure and thermal analysis of Cu-Cu2O eutectic -Can we mimic archeological remains?

Keywords: Cu-Cu 2 O eutectic, eutectic grain, rod-like structure

OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible Any correspondence concerning this service should be sent to the repository administrator: tech-oatao@listes-diff.inp-toulouse.fr Microstructure and thermal analysis of Cu-Cu 2 O eutectic -Can we mimic archeological remains?

Introduction

Numerous archeological artefacts and remains containing metallic copper have been discovered in various sites all around the world revealing a Cu-Cu 2 O eutectic. Some of these artefacts have been dated back to several thousands of years [1]. They all show a specific microstructure with a fibrous Cu-Cu 2 O eutectic (Figure 1) surrounding dendrites of a primary phase that can be either (Cu) -the copperrich fcc phase -or Cu 2 O. This eutectic has been rarely studied. Eastwood [2] showed that the eutectic grains consist in densely packed fibers in the center, which bend and coarsen at the periphery. The similarity of these observations with the microstructure seen in the ancient artefacts suggested that laboratory experiments could help understanding the original metal process. This report presents preliminary results obtained within this line of thinking.

Experimental details

Pure copper (better than 99.9 wt%) was used to prepare two cylindrical samples 16 mm in diameter and 13 mm in height. These samples were placed in an alumina crucible designed to fit on a thermocouple and then introduced in a thermal analysis apparatus for remelting and solidification under ambient atmosphere. They were heated up to 1190-1200°C at 4.5 °C.min -1 , hold at that temperature for two minutes (sample 1) or one hour (sample 2), and finally cooled down to room temperature at the same rate of 4.5 °C.min -1 . Then, they were cut in halves, mounted for classical metallographic preparation, mirror polished and observed with optical microscopy.

Results

Figure 2-a shows the whole thermal records of the two samples where are also indicated the Cu-Cu 2 O eutectic (1066°C) and the Cu melting (1085°C) temperatures according to Neumann et al. [3]. Examination of the heating records shows that the signals bent before a sharp decrease in temperature that must be associated to bulk melting. The first slope change occurred at 1066°C, i.e. the eutectic temperature. This suggests that some oxidation took place during the heating stage that led to incipient melting when the eutectic temperature was reached. It was verified with a third sample (not presented here) heated to 1066°C and immediately cooled to room temperature that its surface was effectively oxidized and this oxide may have led to an incipient melting at 1066°C. As the samples were further heated above 1066°C, oxidation proceeded until bulk melting occurred with the temperature falling back to nearly the eutectic temperature. This is certainly related to the fact the material was then saturated in oxygen. After a short while, the samples were fully liquid and their temperature started increasing again. During holding of sample 2 at the upper temperature, it is seen that the temperature increases from 1190°C to 1200°C when it should have been constant. This is supposedly related to heat release by further oxidation of copper.

Upon cooling, both curves exhibit a strong thermal arrest which must correspond to the eutectic reaction. It is marked by recalescence and a time plateau of about 5 min at a eutectic undercooling of 8 and 11°C for samples 1 and 2 respectively. The time-derivative of the temperature record did not show any thermal effect for sample 1 but did for sample 2 as seen in figure 2-b where the start of the primary arrest is indicated with the downwards arrow. Figure 3 shows bright field light macrographs of the samples. The hole in the middle of the bottom was the location of the thermocouple while the many other cavities formed during the melting/solidification process. This latter feature seems to correlate with the presence of a huge amount of voids in ancient copper ingots [4]. Both Figure 4 shows also that Cu 2 O dendrites are enveloped with a thick halo of fcc (Cu) solid solution, i.e. that the coupled eutectic is not initiated at the Cu 2 O/liquid interface but at the (Cu)/liquid interface. This is in line with the fact that (Cu) seems needing a significant undercooling for nucleation as already reported by Eastwood [2]. It may be stressed that this microstructure is highly similar to the one seen in archaeological artefacts. When the eutectic develops, the composition of the liquid evolves because of rejection of oxygen from (Cu). During recalescence, the oxygen content in the liquid in equilibrium with Cu 2 O increases while that of the liquid in equilibrium with (Cu) decreases. Due to the shape of the Cu-O phase diagram, this latter quantity is much higher than the former one. Further, because recalescence is very rapid, it may be suggested that this oxygen is prone to transform to gas. Though the oxide shell around the metal is porous, it certainly limits its exhaust so that there can be a pressure build-up in between the metal and the oxide shell. According to the assessed Cu-O phase diagram [3], a recalescence of 2°C as seen in figure 2 corresponds to a difference in composition along the (Cu) liquidus of 0.2 at.%. For a sample of 2600 mm 3 , this corresponds to 3.3•10 -4 moles of oxygen gas. At 1330 K, this would correspond to a pressure of 2 MPa, which is certainly sufficient to lead to some creep of the Cu 2 O crust and deformation of the sample [4]. Hence the cavities seen systematically on the samples as illustrated in figure 3, which give the material an aspect very much alike that noticed on Cu ingots [5].

The other important feature shown by the macrograph of sample 1 in figure 3 is the grain structure emanating from the bottom of the sample. These are eutectic grains which nucleated at the bottom surface and invaded the bulk sample from bottom to top. Figure 5 shows the eutectic structure at the bottom of sample 1 where it can be seen that it is much finer than at the top of the sample (compare with the micrograph in figure 4, top row). This is in line with a development of the eutectic from the bottom to the top of sample 1 with a decreasing growth rate leading to increased coarseness of the microstructure. Such a grain structure cannot be evidenced in the macrograph of sample 2, figure 3. However, figure 6 shows that the eutectic structure is again much finer at the bottom than in the middle of this sample, suggesting the eutectic developed also from bottom to top.

A last feature of the Cu-Cu 2 O eutectic that has already been stressed by Eastwood [2] is that the rods seen in the middle of the eutectic grains often bend and appear elongated on the outer grain boundaries. This is illustrated in figure 7. This suggest that Cu 2 O rods changed their b a crystallographic orientation during eutectic grain growth so as to remain perpendicular to the eutectic front at grain boundaries [6,[START_REF] Hogan | Eutectic grains[END_REF].

Conclusion

Solidification of oxidized copper samples showed the development of Cu 2 O dendrites followed by (Cu)-Cu 2 O eutectic. Thermal analysis demonstrated this latter appears with significant undercooling and some recalescence. It is proposed that this recalescence leads to gas exhaust from the sample in a quantity that is sufficient for explaining the observation that samples show large voids.

These results correspond well to archaeological observations revealed on copper-based remains as detailed for example by Thoury et al. [1] and Haupmann et al. [5]. It is interesting to notice that the Cu-Cu 2 O eutectic can be considered as a process marker of copper melting in aerated environment in relation with a precise thermal indicator. This approach opens a way to new considerations about smelting and melting copper since the beginning of the metallurgy.

  This is an author's version published in: http://oatao.univ-toulouse.fr/21513 To cite this version: Fossé, Cécile and Castro-Román, Manuel Jesus and Freulon, Alexandre and Thebault, Yannick and Lacaze, Jacques and Robbiola, Luc Microstructure and thermal analysis of Cu-Cu2O eutectic -Can we mimic archeological remains? (2017) In: SP17 6th Decennial International Conference on Solidification Processing, Old Windsor, July 2017, 25 July 2017 -28 July 2017 (Old Windsor, Beaumont Estate, United Kingdom). (Unpublished)

Figure 1 :

 1 Figure 1: Micrograph of sample 360-4_6 from Bibracte.

Figure 2 :

 2 Figure 2: a) Heating-holding-cooling schedule of samples 1 and 2; b) enlargement of the thermal record on cooling of sample 2 and time derivative of the record showing the primary arrest starting far above the eutectic arrest as indicated with the downwards (red) arrow.

Figure 3 :

 3 Figure 3: Bright field light microscopy macrographs of the vertical section of samples 1 (a) and 2 (b). The samples were 16 mm in diameter.

  hyper-eutectic structure with cuprous oxide dendrites emanating from the top surface and filling half the samples. Furthermore, equiaxed Cu 2 O dendrites are seen in the bottom half of sample 2 while none appears in sample 1. Instead, one can see in sample 1 a structure of grains emanating from the thermocouple hole in the central area.

Figure 4

 4 compares the microstructure of the upper surface of both samples. This clearly shows that sample 2 (Figure 4-b) contained much more oxygen than sample 1 (figure 4-a) when cooling was initiated and Cu 2 O dendrites started growing inwards. Accordingly, there is much more Cu 2 O-Cu eutectic in sample 1 than in sample 2. Also, the formation of equiaxed Cu 2 O dendrites at the bottom of sample 2 (figure 2-b) indicates that the remaining liquid became supersaturated in oxygen with respect to the Cu 2 O liquidus at some stage of the cooling process.

Figure 4 :

 4 Figure 4: Low enlargement optical micrographs showing the top-surface microstructure of samples 1 (a) and 2 (b). The same scale applies to both micrographs.

Figure 5 :

 5 Figure 5: Micrograph of the bottom part of sample 1.

Figure 6 :

 6 Figure 6: Micrographs showing the eutectic in the middle (a) and the bottom (b) parts of sample 2. The same scale applies to both micrographs.

Figure 7 :

 7 Figure 7: optical micrograph illustrating the rotation of Cu 2 O rods during eutectic grain growth (sample 1).


Giulia Masi
email: giulia.masi5@unibo.it

Claudie Josse
email: claudie.josse@ums-castaing.fr

Jérôme Esvan

Elena Bernardi
email: elena.bernardi@unibo.it

Carla Martini
email: carla.martini@unibo.it

Maria Chiara Bignozzi

Luka Skrelp
email: luka.skrlep@zag.si

Erika Svara Fabjan

Tadeja Kosec
email: tadeja.kosec@zag.si

Jerome Esvan
email: jerome.esvan@ensiacet.fr

Maria Chiara Bignozzi
email: maria.bignozzi@unibo.it

Erika Švara Fabjan

Maelenn Aufray

Luc Robbiola
email: robbiola@univ-tlse2.fr

Cristina Chiavari
email: cristina.chiavari@unibo.it

Micro-characterisation of innovative organic coatings applied for the protection of outdoor bronze monuments

Keywords: bronze, atmospheric corrosion, fluoropolymer, artificial ageing, XPS

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

Introduction

Fluoropolymer coatings, in particular polyvinylidene fluoride (PVDF), showed outstanding performance in outdoor applications, as exterior durability, excellent chemical resistance, good flexibility, low dirt retention, photo-stability and good moisture and fungus resistance, as reported in [START_REF] Iezzi | Acrylic-fluoropolymer mixtures and their use in coatings[END_REF]. Unfortunately, this coating exhibited poor adhesion properties on bronze substrates, due to the fact that such coating is bonded to substrate surface by very weak Van der Waals forces [START_REF] Malshe | Fluorinated acrylic copolymers: Part I: Study of clear coatings[END_REF]. So, to increase the adhesion on bronze substrates, this polymer was usually blended with an acrylic-based or silicon-based polymer [START_REF] Malshe | Fluorinated acrylic copolymers: Part I: Study of clear coatings[END_REF][START_REF] Satoh | Preparation of super-water-repellent fluorinated inorganic-organic coating films on nylon 66 by the sol-gel method using microphase separation[END_REF][START_REF] Baradie | Novel fluoro-terpolymers for coatings applications[END_REF]. Bierwagen et al. found that this type of fluoropolymer blended with acrylic coating acted as a strong barrier under electrochemical tests and so it showed very high corrosion resistance properties [START_REF] Bierwagen | Developing and testing a new generation of protective coatings for outdoor bronze sculpture[END_REF]. In addition, as the ideal coating for conservation applications should be removable, this polymer could be also easily removed using a polar solvent, such as acetone [START_REF] Sadat-Shojai | Polymeric Coatings for Protection of Historic Monuments: Opportunities and Challenges[END_REF]. For all these reasons, a fluoro-acrylate (fluorocarbon functionality together with adhesion provided by other groups, FA) was selected as a promising protective treatment for corrosion protection of outdoor bronzes in the field of Cultural Heritage. An adhesion promoter, a silane modified poly-methyl-methacrylate (MS) was also mixed together with the fluoropolymer and the blended coating (FA-MS) was applied by spraying on a representative bronze substrate. In particular, this coating was tested on pre-patinated quaternary bronze (Cu-Sn-Zn-Pb) aged by accelerated dropping method, simulating the unsheltered exposure of outdoor bronzes, already verified in previous studies [START_REF] Chiavari | Predicting the corrosion behaviour of outdoor bronzes: assessment of artificially exposed and real outdoor samples[END_REF][START_REF] Bernardi | The atmospheric corrosion of quaternary bronzes: The leaching action of acid rain[END_REF]. The aim of this study is to assess the protectiveness of fluoro-acrylate polymer for the conservation of outdoor bronzes for Cultural Heritage applications. The coated surfaces were characterised by X-ray Photoelectron Spectroscopy (XPS). The corrosion behaviour was studied by the analysis of the weathering solution used for the accelerated ageing, as well by surface and cross-section characterisation of the FA-MS coating after dropping. Finally, the performance of the coating was compared with the surface evolution of the uncoated prepatinated bronze substrates under the same experimental conditions.

Materials and methods

Materials

Quaternary bronze (Cu-Sn-Zn-Pb), supplied by Livartis d.o.o. Foundry (Ljubljana, Slovenia) in as-cast condition, showed a chemical composition (wt%) of: 6.9±0.6 Sn, 3.1±0.4 Zn, 2.0±0.9 Pb, Al, P, Mn, Si (traces), Cu to balance. This bronze alloy had the typical microstructure of as-cast bronze, with a dendritic network including shrinkage cavities due to cooling during casting. Quaternary bronze substrates were pre-patinated by dropping test for a Total Time of Wetness of 87 days [START_REF] Masi | Corrosion protection of outdoor bronze : historical quaternary alloy and modern alloy with artificial patination[END_REF]. These patinated substrates were used as representative samples for coating application, but also as reference and named "uncoated sample". A fluoro-acrylate polymer (FA) was selected as candidate coating for the protection of historical bronze (Cu-Sn-Zn-Pb). In order to improve the adhesion on the bronze substrate, a silane-modified poly-methyl-methacrylate (MS) was mixed with the fluoropolymer (FA). In particular, 5 wt% FA and 10 wt% MS were mixed together in solvent mixture solution made of 50 wt% of n-butyl acetate and 50 wt% diethyl-succinate. This coating (FA-MS) was then applied by brushing on the bronze samples.

Artificial ageing by dropping All the uncoated and coated samples were exposed to artificial acid rain solution by dropping test for a total time of Wetness of 10 days. This type of accelerated ageing method simulates unsheltered exposure of bronze objects in atmospheric conditions. In order to ensure that only the coated surface (25×25 mm 2 ) of each coupon was directly exposed to acid rain, the other surfaces were protected by epoxy resin. The artificial rain solution was prepared according to the composition of natural acid rains collected during the winter months in Bologna. The chemical composition of the artificial solution is reported in previous study [START_REF] Chiavari | Atmospheric corrosion of fire-gilded bronze: corrosion and corrosion protection during accelerated ageing tests[END_REF]. Dropping test consisted of two weekly cycles of 3 days of dropping / 1 days of dry and 2 days of dropping / 1 day of wet were performed up to a total Time of Wetness (ToW) of 10 days per sample. The samples were placed at a 45° angle on plastic sample holders. At least two samples for each coating were simultaneously tested. A peristaltic pump dropped artificial acid rain solution on the samples by means of capillary tubes with an almost constant dropped volume of 60 mL/h on each exposed sample, as reported in [START_REF] Bernardi | The atmospheric corrosion of quaternary bronzes: The leaching action of acid rain[END_REF]. The runoff solution dropped on each sample was separately collected in canisters. Mass measurements were carried out using a digital balance KERN AGB210_4 with repeatability of ± 0.1 mg and mass variation was calculated by the difference between the initial weight of the sample and the weight of the corroded samples at different ageing times during weathering. Each time the weathering solution was renewed during dropping, an amount of each solution was stored in HDPE bottles, in order to analyse the metallic ions release. These solutions were acidified with HNO 3 65% suprapur, in order to prevent absorption of metals on the interior of the bottles and dissolve any metal complexes, and stored at 4 °C. The concentration of Cu was analysed through Atomic Absorption Spectrometer (AAS) Perkin-Elmer AAnalyst 400 (with both Flame and electrothermical atomization) and Perkin-Elmer PinAAcle 900Z, with Transversely Heated and longitudinal Zeeman background correction. The protective efficiency of the tested coatings was quantified through the metal release in the ageing solutions. In particular, the Inhibiting Efficiency in terms of metal release (η M ) was calculated using Eq. ( 1):

! ! = ! !"#,!" -! !!",! ! !"#,!" ×100 (1)
where M sol is the amount of metal in the ageing solution (Cu in µg cm -2 ); NC is the uncoated sample and C is the coated sample as reported in [11].

Surface and cross-section characterization XPS analysis was carried out on coated pre-patinated bronze by FA-MS before ageing. The photoelectron emission spectra were recorded using a monochromatised Al Kα (hν = 1486.6 eV) source on a ThermoScientific K-Alpha system. The X-ray spot size was about 400 μm.

The pass energy was fixed at 130 eV with a step of 1 eV for surveys and 40 eV with a step of 0.1 eV for core levels. Ionic sputtering of the surfaces was made by Ar + ion beam accelerated under 200 eV low current for 30 s. The Shirley's method for the background subtraction was applied for all the peaks. In-situ FIB cross-sections were carried out on the uncoated and coated samples after dropping test (ToW=10 days). Before FEG-SEM observation, the samples were coated by an amorphous graphite layer with a film thickness of around 10 nm (applied by sputtering). For FIB cross-section preparation, each sample was then tilted at 52° and a platinum (Pt-C) layer of about 1 μm was first laid down for surface protection on the selected area. Then, FIB milling by Ga + ions was applied. For all the samples, representative areas were chosen, producing cross-sections from 100 to 130 µm in length. SEM observation and EDS analysis were carried out using an acceleration voltage of 10 kV.

Results and discussion

Pre-patinated Cu-Sn-Zn-Pb coated by FA-MS The surface characterization of Cu-Sn-Zn-Pb coated by FA-MS was mainly performed by XPS. The coating surface was smooth and uniform, showing that the coating is able to cover well the pre-patinated substrates. In addition, on the top surface, the FA-MS coating showed the presence of intense C 1s, O 1s and F1s peaks with minor Si 2p, S 2p and N 1s features, as depicted in the XPS survey reported in Figure 1. Very small amounts of alloying elements (mainly Cu 2p and Pb 4f) were detected in traces and considered as pollution from surface preparation.

The effect of the Ar + sputtering is evident on the top surface of FA-MS, as O 1s, F 1s and S 2p showed a strong decrease in peak intensity, while Si 2p was only detected after etching. In particular, C 1s showed higher intense peak at lower BE values (C-C, C-H) and the features at higher BE centred at 291.4 eV and 293.8 eV linked to F 1s were not detected after etching. Moreover, O 1s and F 1s showed a less intense peak, as well as S 2p that showed just a small peak centred at 163.4 eV and sulphate features were not observed anymore, as it was for the core level before etching. The behaviour described for FA-MS before and after Ar + ion sputtering was also confirmed by atomic quantification reported in Table 1. This is probably due to the inhomogeneity of the coating made of the two different components (the fluoro-acrylate and the adhesion promoter). So, on the top surface, the coating showed enrichment in F 1s (intense peak at BE=688.6±0.1 eV related to the bond with C), which mostly disappeared after etching, due to an increase of C 1s in correspondence with C-C and C-H compounds (BE=284.2±0.1 eV). In addition, on the top surface, S 2p showed a peak centred at BE=168.7±0.1 eV, characteristic Counts (a.u.)

Binding energy (eV)

of sulphates, that was not present in the theoretical chemical formula of the coating. This peak strongly decreased after etching, while small Si 2p at BE=102.2±0.1 eV to be ascribed mainly to silane was found, indicating the presence of methacryloxy-propyl-trimethoxysilane (MS). Accelerated ageing of coated and uncoated Cu-Sn-Zn-Pb substrates Cumulative gravimetric measurements of the samples at the end of dry periods of dropping test are reported in Table 2. Significant differences in the mass variations were observed for uncoated and FA-MS coated samples. In particular, uncoated bronze showed a linear trend of mass decrease, while the coated samples showed a very low mass variation.

Regarding FA-MS, the mass increase was observed since the beginning of the ageing, just after 3 days of ToW and then remained constant. This could be due to some water trapped within the polymer, as the same weight increase was measured also after storing the samples in desiccator for a week.

In parallel, the analyses on the ageing solution were also performed and Cu ions releases were calculated at increasing ToW (Table 2). The achieved results are in accordance with the gravimetric measurements. The uncoated sample showed a negative linear trend with a total Cu release up to 1600 µg/cm 2 at 10 days of wetness, indicating a strong corrosion/dissolution impact highlighted by the weight loss evolution. The samples coated by FA-MS showed a slight positive linear trend and with a marked lower slope by comparison to the uncoated sample and the total Cu release was lower than 15 µg/cm 2 . Inhibiting efficiency of FA-MS related to Cu release was calculated in accordance with Eq. ( 1) and a value of η Cu =99±22was measured, indicating an excellent performance of the coating during dropping test. In order to explain the results of mass variations and Cu ions release, surface and crosssection observations of uncoated and coated samples were carried out. They are reported in Figures 2 and3. SEM observation of uncoated Cu-Sn-Zn-Pb, aged by dropping, confirmed that the continuous runoff during the ageing induced a strong surface corrosion, compared to pre-patinated surface characterised in [START_REF] Masi | Corrosion protection of outdoor bronze : historical quaternary alloy and modern alloy with artificial patination[END_REF]. Corrosion of the uncoated sample is nonhomogenous, highlighting the bronze microstructure. In particular, due to dropping also dendrite borders showed strong corrosion and several cracks on corrosion products. The corrosion attack is preferentially developed between the Cu-rich core and the Sn-rich areas of eutectoid. Interdendritic areas with Cu-Sn eutectoid (white areas) seem not corroded at all. In addition, the presence of the polishing grooves revealed on the micrograph reported in Figure 2. It is confirmed is representative of the natural phenomena: the corrosion attack is linked to an internal phenomenon inducing a corrosion patina of Sn-rich products (decuprification) and the dropping facilitates the leaching of Cu metal ions rather than the deposit of corrosion products, as already observed in [START_REF] Bernardi | The atmospheric corrosion of quaternary bronzes: The leaching action of acid rain[END_REF]. X-ray maps of the uncoated corroded surface (Figure 2) showed that O element is the main aggressive environmental element, while S and Cl (not reported here because Cl signal was very low and the map not satisfactorily resolved) were only locally detected in the most corroded areas. Three different areas are characterised by different elemental distributions. In particular, dendrite cores are containing mainly Cu-O with ratio Cu/Sn values by weight of 12, while dendrite borders are characterised by tin-rich areas (Cu/Sn=0.3) due to strong decuprification process, where the eutectoid is apparently unaffected by corrosion as oxygen element was detected in very low amount about 1.5 wt%.

In interdendritic areas, only Pb globules, not miscible in the α-Cu solid solution matrix, were remarkably involved in the dissolution phenomena. Finally, the observation of in-situ FIB cross-section of uncoated and coated samples by FA-MS after dropping is reported in Figure 3. For accelerated aged uncoated Cu-Sn-Zn-Pb samples (Figures 3 a-b), internal cracks within the corrosion (patina) were detected and some of the corrosion products forming surface patches were lost after the ageing process. The corrosion structure is not uniform within the whole cross-section, and it is more developed in

100 µm O-K Cu-L Sn-L Pb-M eutectoid dendrites
correspondence with the interdendritic borders. In particular, a thick corrosion layer of about 3 µm is observed in the interdendritic areas, exhibiting a two-layer structure: a thin internal corrosion layer (< 300 nm) and an external homogenous and dense outermost laye, but with some nano-metric porosities close to the external/internal layers interface. However in the dendrite core, corrosion produces a thinner layer of about 1.5-2 µm, also characterised by a two layer structure but with an external one revealing an homogenous distribution of nanometric interconnected pores, as observed for the pre-patinated quaternary bronze [START_REF] Masi | Corrosion protection of outdoor bronze : historical quaternary alloy and modern alloy with artificial patination[END_REF]. The observed corrosion behaviour is in good accordance with previous results achieved in quaternary bronze with different compositions, as reported in [START_REF] Bernardi | The atmospheric corrosion of quaternary bronzes: The leaching action of acid rain[END_REF]. The α+δ eutectoid only reveals a thin corrosion film on the Cu-rich α-phase close to the surface. Investigation of the cross-section of the FA-MS coated sample is illustrated in Figures 3 c-d, showing FEG-SEM images of the FIB cross-section. FA-MS is highly sensitive to the electron beam. Thus during the FIB session, the coating also showed some peeling and degradation due to electron and ion beam. The FIB cross-section showed the corrosion layer relative to the pre-patination process and revealed also some coating accumulation characterised by higher coating thickness (here not reported) relative to some shrinkage porosity in the bronze substrate. Generally the FA-MS layer is thin, characterised by a mean thickness of 2 µm, except in correspondence of the coating accumulation areas, where the thickness ranges from 5.5 to 7 µm. In addition, the coating layer showed few porosities and the coating/substrate interface followed the substrate morphology, but it also revealed some nano-porosities of the coating.

Conclusions

The following conclusions can be drawn: • The FA-MS coating showed good performance for the protection of pre-patinated Cu-Sn-Zn-Pb bronze, shown by gravimetrical and leaching ion analysis measured by Atomic Absorption Spectrometer.

• From the observation of in-situ FIB cross-section, this coating formed a thin continuous and uniform layer with very few µm-sized pores.

• The inhibiting efficiency was measured up to 99 %, on the basis of the evolution of the Cu ions released in the weathering solutions during the accelerated ageing process.

Figure 1 :

 1 Figure 1: XPS survey of the surface coated by FA-MS before ageing.

Figure 2 :

 2 Figure 2: EDS X-ray maps of uncoated Cu-Sn-Zn-Pb surface after dropping (ToW=10 days).

Figure 3 :

 3 Figure 3: FEG-SEM images of in-situ FIB cross-sections of uncoated (a and b) and FA-MS coated Cu-Sn-Zn-Pb (c and d) after accelerated ageing.

Table 1 :

 1 XPS atomic concentrations of the main elements of FA-MS top surface before and after Ar + sputtering.

				C 1s			N 1s	O 1s	F 1s	Si 2p	S 2p
		CC, CH	C-O, C-N	O=C-O	CF 2	CF 3	N-C, N-O		CF 2	CF 3	Si-O-Si	S-C	SO 4
	Before
	etching
	Peak BE(eV)	284.7	286.4	288.8	291.4	293.8	399.5	532.8	687.3	688.6	-	163.2	168.7
	at%	39.5	13.5	4.5	5.8	2.0	3.4	15.7	1.0	11.9	-	0.1	2.5
	After
	etching
	Peak BE(eV)	284.2	286.2	288.3	-	-	400.0	532.1	687.3	688.5	102.2	163.3	168.2
	at%	63.7	12.4	3.3	-	-	3.1	10.8	2.9	2.6	0.4	0.4	0.4

Table 2 :

 2 Cumulative mass variation per unit area (mg/cm 2 ) as a function of Time of Wetness (ToW); cumulative concentration of Cu ions released in the artificial rain, normalized to the surface, as a function of ToW.

			Time of Wetness (days)
		3	5	7	10
		Cumulative mass variation (mg/cm 2 )
	Patinated,	-0.3±0.2	-0.5±0.2	-0.9±0.3	-1.4±0.4
	uncoated sample
	Patinated, FA-MS	0.17±0.04	0.13±0.05	0.11±0.04	0.091±0.04
		Cumulative Cu ions release (µg/cm 2 )
	Patinated,	429±66	718±134	1167±222	1617±255

Acknowledgments

The study was financed through M-ERA.NET research project (Bronze-IMproved nonhazardous PAtina CoaTings, B-IMPACT, which lasted from 2015-2017. Special thanks to Nina Gartner for effectively coordinating the tasks within the project.


Rida Laraki
email: rida.laraki@polytechnique.edu

Michel Balinski

Eilon Solan

Sylvain Sorin

Explicit Formulas for Repeated Games with Absorbing States *

Explicit formulas for the asymptotic value lim λ→0 v(λ) and the asymptotic minmax lim λ→0 w(λ) of finite λ-discounted absorbing games are provided. New simple proofs for the existence of the limits as λ goes zero are given. Similar characterizations for stationary Nash equilibrium payoffs are obtained. The results may be extended to absorbing games with compact action sets and jointly continuous payoff functions.

Introduction

Aumann and Maschler [START_REF] Aumann | Repeated Games with Incomplete Information[END_REF] introduced and studied long interaction two player zero-sum repeated games with incomplete information. They introduced the asymptotic and uniform approaches and obtained an explicit formulas for the asymptotic value when a player is fully informed (the famous Cav(u) theorem). Mertens and Zamir [START_REF] Mertens | The Value of Two-Person Zero-Sum Repeated Games with Lack of Information on Both Sides[END_REF] completed this work and obtained their elegant system of functional equations [START_REF] Mertens | The Value of Two-Person Zero-Sum Repeated Games with Lack of Information on Both Sides[END_REF] that characterizes the asymptotic value of a repeated game with incomplete information on both sides. Unfortunately, very few repeated games have an explicit characterization of the asymptotic value .

Stochastic games are repeated games in which a state variable follows a markov chain controlled by the actions of the players. Shapley [START_REF] Shapley | Stochastic Games[END_REF] introduced the two player zero-sum model with finitely many states and actions (i.e. the finite model). He proved the existence of the value of the λ-discounted game v (λ) by introducing a dynamic programming principle (the Shapley operator).

Kohlberg [START_REF] Kohlberg | Repeated Games with Absorbing States[END_REF] proved the existence of the asymptotic value v = lim λ→0 v (λ) in the subclass of finite absorbing games (i.e., stochastic games in which only one state is non-absorbing). His operator appraoch uses the additional information obtained from the derivative of the Shapley operator at λ = 0 to deduce the existence of lim λ→0 v (λ) and its characterization via variational inequalities.

Laraki ([6], [START_REF] Laraki | Variational Inequalities, System of Functional Equations, and Incomplete Information Repeated Games[END_REF]) used a variational approach to study games in which each player controls a martingale (including Aumann and Maschler repeated games). As in differential games with fixed duration, the approach starts with the dynamic programming principle, follows with an optimal strategy for some player for a fixed discounted factor λ, relax the constraint of the other player and lets the discount factor λ carefully tend to zero. This allows to show the existence of lim λ→0 v (λ) and its variational characterization.

The same approach gives, for absorbing games, a new proof for the existence of lim λ→0 v (λ) and its characterization as the value of a one-shot game. When the probability of absorption is controlled by only one player (as in the big match), the formula can be simplified to the value of 1 an underlying finite game. Coulomb [START_REF] Coulomb | Repeated Games with Absorbing States and Signaling Structure[END_REF] provided another explicit formula for lim λ→0 v (λ) using an algebraic approach.

Rosenberg and Sorin [START_REF] Rosenberg | An Operator Approach to Zero-sum Repeated Games[END_REF] extended the Kohlberg result to absorbing games when action sets are infinite but compact and payoffs and transitions are separately continuous. However, no explicit formula is known for v.

The minmax w (λ) of a multi-player λ-discounted absorbing game is the level at which a team of players could punish another player. From Bewley and Kohlberg [START_REF] Bewley | The Asymptotic Theory of Stochastic Games[END_REF] one deduces the existence of lim w (λ) for any finite absorbing game. However, no explicit formula exists and it is not known if the limit exists in infinite absorbing games.

The variational approach allows (1) to prove the existence of the asymptotic minmax w = lim λ→0 w (λ) of any multi-player jointly continuous compact absorbing game and (2) provides an explicit formula for w. Similar formulas could be obtained for stationary Nash equilibrium payoffs.

The value

Consider two finite sets I and J, two (payoff) functions f , g from I ×J to [-1, 1] and a (probability transition) function

p from I × J to [0, 1] .
The game is played in discrete time. At stage t = 1, 2, ... player I chooses at random i t ∈ I (according to some mixed action x t ∈ ∆ (I)1 ) and, simultaneously, player J chooses at random j t ∈ J (according to some mixed action y t ∈ ∆ (J)2 ):

(i) the payoff is f (i t , j t ) at stage t;

(ii) with probability 1 -p (i t , j t ) the game is absorbed and the payoff is g (i t , j t ) in all future stages; and (iii) with probability p (i t , j t ) the interaction continues (the situation is repeated at step t + 1).

If the stream of payoffs is r(t), t = 1, 2, ..., the λ-discounted-payoff of the game is ∞ t=1 λ(1λ) t-1 r(t). Player I maximizes the expected discounted-payoff and player J minimizes that payoff.

M + (I) = {α = (α i ) i∈I : α i ∈ [0, +∞)} is the set of positive measures on I (the I-dimensional positive orthant). For any i and j, let p * (i, j) = 1 -p(i, j) and f * (i, j) = [1 -p(i, j)] × g(i, j). For any (α, j) ∈ M + (I) × J and ϕ : I × J → [-1, 1] , ϕ is extended linearly as follows ϕ(α, j) = i∈I α i ϕ(i, j). Note that ∆(I) ⊂ M + (I).

Proposition 1 [START_REF] Shapley | Stochastic Games[END_REF]) G λ has a value, v (λ) . It is the unique real in [0, 1] satisfying,

v (λ) = max x∈∆(I) min j∈J [λf (x, j) + (1 -λ) p(x, j)v (λ) + (1 -λ) f * (x, j)] . (1)
Equation ( 1) implies that player I has an optimal stationary strategy (that plays the same mixed action x at each period). This implies in particular that the proposition holds even if the players have no memory or do not observe past actions.

Example: a quitting-game

C Q C 0 1 * Q 1 * 0 *
The game stops with probability 1 if one of the players plays Q. There are two absorbing payoffs 1 and 0 (they are marked, as usual, with a * ). The absorbing payoff 1 is achieved at some period if (C,Q) or (Q,C) is played. The absorbing payoff 0 * is achieved if (Q,Q) is played. The game is non-absorbed if both players decide to continue and play (C,C).

Consider the following strategy stationary profile in which player I plays at each period (xC, (1 -x)Q) and player 2 (yC, (1 -y)Q). The corresponding discounted payoff r λ (x, y) satisfies

r λ (x, y) = xy (λ × 0 + (1 -λ) r λ (x, y)) + ((1 -x)y + (1 -y)x) , so that r λ (x, y) = x + y -2xy 1 -xy(1 -λ) .
The value v λ ∈ [0, 1] satisfies:

v λ = value   C Q C (1 -λ)v λ 1 Q 1 0   = max x∈[0,1] min y∈[0,1] [xy(1 -λ)v λ + x(1 -y) + y(1 -x)] = min y∈[0,1] max x∈[0,1] [xy(1 -λ)v λ + x(1 -y) + y(1 -x)] .
It may be checked that

v λ = x λ = y λ = 1 - √ λ 1 -λ .
The value and the optimal strategies are not rational fractions of λ (but admit a puiseux series in power of λ). Bewley and Kohlberg [START_REF] Bewley | The Asymptotic Theory of Stochastic Games[END_REF] show this to hold for all finite stochastic games and deduce from it the existence of lim v (λ).

Lemma 2 v (λ) satisfies v (λ) = max x∈∆(I) min j∈J λf (x, j) + (1 -λ) f * (x, j) λp(x, j) + p * (x, j) .
Proof. If in the λ-discounted game, player I plays the stationary strategy x and player J plays a pure stationary strategy j ∈ J, the λ-discounted reward r (λ, x, j) satisfies:

r (λ, x, j) = λf (x, j) + (1 -λ) p(x, j)r (λ, x, j) + (1 -λ) f * (x, j). Since p * = (1 -p), r (λ, x, j) = λf (x, j) + (1 -λ) f * (x, j) λp(x, j) + p * (x, j) .
The maximizer has a stationary optimal strategy and the minimizer has a pure stationary best reply: this proves the lemma.

In the following, α ⊥ x means that for every i ∈ I,

x i > 0 ⇒ α i = 0. Theorem 3 As λ goes to zero v (λ) converges to v = sup x∈∆(I) sup α⊥x∈M + (I) min j∈J f * (x, j) p * (x, j) 1 {p * (x,j)>0} + f (x, j) + f * (α, j) p(x, j) + p * (α, j) 1 {p * (x,j)=0} .
Proof. Let w = lim n→∞ v (λ n ) be an accumulation point of v (λ).

Step 1: Consider an optimal stationary strategy x (λ n ) for player I and go to the limit using Shapley's dynamic programming principle. From the formula of v(λ n ), there exists x (λ n ) ∈ ∆(I) such that for every j ∈ J,

v (λ n ) ≤ λ n f (x(λ n ), j) + (1 -λ n ) f * (x(λ n ), j) λ n p(x(λ n ), j) + p * (x(λ n ), j) . (2)
By the compactness of ∆(I) it may be supposed that x (λ n ) → x. Case 1: p * (x, j) > 0. Letting λ n go to zero implies w ≤ f * (x,j) p * (x,j) . Case 2: p * (x, j) = i∈I x i p * (i, j) = 0. Thus, i∈S(x) p * (i, j) = 0 where S(x) = {i ∈ I :

x i > 0} is the support of x. Let α(λ n ) = x i (λn) λn 1 {x i =0} i∈I ∈ M + (I) so that α(λ n ) ⊥ x. Consequently, i∈I x i (λ n ) λ n p * (i, j) = i / ∈S(x) x i (λ n ) λ n p * (i, j) = i∈I α i (λ n ) p * (i, j) = p * (α (λ n ) , j),
and

i∈I x i (λ n ) λ n f * (i, j) = i∈I α i (λ n ) f * (i, j) = f * (α (λ n ) , j),
so, from equation ( 2), and because p(x, j) = 1,

w ≤ lim n→∞ inf f (x, j) + (1 -λ n ) f * (α(λ n ), j) p(x, j) + p * (α(λ n ), j) .
Since J is finite, for any ε > 0, there is N (ε) such that, for every j ∈ J, w ≤

f (x,j)+f * (α(λ N(ε) ),j) p(x,j)+p * (α(λ N(ε) ),j) +ε. Consequently, w ≤ v.
Step 2: Construct a strategy for player I in the λ n -discounted game that guarantees v as λ n → 0. Let (α ε , x ε ) ∈ M + (I) × ∆(I) be ε-optimal for the maximizer in the formula of v. For λ n small enough, let x ε (λ n ) be proportional to

x ε + λ n α ε (x ε (λ n ) = µ n (x ε + λ n α ε ) for some µ n > 0). Let r(λ n ) be the unique real in the interval [0, 1] that satisfies, r(λ n ) = min j∈J λ n [f (x ε (λ n ), j)] + (1 -λ n ) (p(x ε (λ n ), j)) r (λ n ) + (1 -λ n ) f * (x ε (λ n ), j) . (4)
By the linearity of f , p, f * and p * on x,

r (λ n ) = min j λ n f (x ε + λ n α ε , j) + (1 -λ n ) f * (x ε + λ n α ε , j) λ n p(x ε + λ n α ε , j) + p * (x ε + λ n α ε , j) = min j λ n f (x ε , j) + λ 2 n f (α ε , j) + (1 -λ n ) f * (x ε , j) + (1 -λ n ) λ n f * (α ε , j) λ n p(x ε , j) + λ 2 n p(α ε , j) + p * (x ε , j) + λ n p * (α ε , j) . Also, v(λ n ) ≥ r(λ n ) since r(λ n )
is the payoff of player I if he plays the stationary strategy x ε (λ n ). Let j λn ∈ J be an optimal stationary pure best response for player J against x ε (λ n ) (an element of the arg min in (4)). Since J is finite and r(λ n ) bounded, one can switch to a subsequence and suppose that j λn is constant (= j) and that r(

λ n ) → r. If p * (x ε , j) > 0 then r = f * (xε,j) p * (xε,j) . If p * (x ε , j) = 0, clearly r = f (xε,j)+f * (αε,j)
p(xε,j)+p * (αε,j) . Consequently, w ≥ v. This proof shows that for each ε > 0, a player always admits an ε-optimal strategy in the λ-discounted game proportional to x ε + λα ε for all λ small enough. The quitting game example shows that a 0-optimal strategy of the λ-discounted game is not always of that form. This identifies the asymptotic value as the value of what may be called the asymptotic game.

For any (α, β) ∈ M + (I) × M + (J) and ϕ :

I × J → [-1, 1]
, ϕ is extended linearly as follows ϕ(α, β) = i∈I,j∈J α i β j ϕ(i, j). .

Proof. Consider an ε-optimal strategy x ε (λ) proportional to x ε + λα ε in the λ-discounted game. Taking any strategy of Player J proportional to y (λ) = y + λβ yields

v (λ) -ε ≤ λf (x ε + λα ε , y + λβ) + (1 -λ) f * (x ε + λα ε , y + λβ) λp(x ε + λα ε , y + λβ) + p * (x ε + λα ε , y + λβ) . p * (x ε , y) > 0 implies v = lim v (λ) ≤ f * (xε,y) p * (xε,y) . If p * (x ε , y) = 0 then f * (x ε , y) = 0.
Using the multi-linearity of f , f * , p and p * and dividing by λ imply:

v (λ) -ε ≤ f (x ε + λα ε , y + λβ) + (1 -λ) f * (α ε , y) + (1 -λ) f * (x ε , β) + (1 -λ) λf * (α ε , β) p(x ε + λα ε , y + λβ) + p * (α ε , y) + f * (x ε , β) + λp * (α ε , β) .
Going to the limit,

v ≤ f (x ε , y) + f * (α ε , y) + f * (x ε , β) p(x ε , y) + p * (α ε , y) + f * (x ε , β) ,
which holds for all (y, β). Thus,

v ≤ sup (x,α)∈Λ(I) inf (y,β)∈Λ(J) f * (x,y) p * (x,y) 1 {p * (x,y)>0} + f (x,y)+f * (α,y)+f * (x,β)
p(x,y)+p * (α,y)+p * (x,β) 1 {p * (x,y)=0}

.

And similarly for the other inequality. Since the inf sup is always higher than the sup inf the first two equalities follow.

Taking β = 0 in the last inequality implies:

v ≤ sup (x,α)∈Λ(I) inf y∈∆(J) f * (x,y) p * (x,y) 1 {p * (x,y)>0} + f (x,y)+f * (α,y) p(x,y)+p * (α,y) 1 {p * (x,y)=0}
, and from the formula of v in theorem 3, one obtains the last equality of the corollary.

Absorption controlled by one player

Consider the following zero-sum absorbing game (the big-match).

L R T 1 * 0 * B 0 1
It is easy to show that v(λ) = 1 2 and that the unique optimal strategy for player I is to play T with probability λ 1+λ . Consequently, v = 1 2 which also happens to be the value of the underlying one-shot game 1 0 0 1 . On the other hand, the asymptotic value of the quitting-game is 1, which is not the value of the underlying one-shot game 0 1 1 0 . A natural question arises:

what are the absorbing games when v is the value of an underlying one-shot game? A game is partially-controlled by player I if the the transition function p(i, j) depends only on i (but not the associated payoffs).

Proposition 5 If a zero-sum absorbing game is partially-controlled by player I, the asymptotic value equals the value of the underlying game, namely:

v = max x∈∆(I) min j∈J i / ∈I * x i f (i, j) + i∈I *
x i g(i, j)

where I * = {i : p * (i) > 0} is the set of absorbing actions of player I.

Proof. Step 1 v ≤ u. Let x ε ∈ ∆(I) and α ε ⊥ x ε ∈ M + (I) be ε-optimal in the formula of v. If p * (x ε ) > 0 then, v-ε ≤ min j∈J f * (x ε , j) p * (x ε ) = min j∈J i∈I x i p * (i) p * (x ε ) g(i, j) = min j∈J i∈I * x i p * (i) p * (x ε ) g(i, j) ≤ max z∈∆(I * ) min j∈J i∈I * z i g(i, j) ≤ u. If p * (x ε ) = 0 then x i ε = 0 for i ∈ I * and p * (i) = 0 when i / ∈ I * so that v -ε ≤ min j∈J f (x ε , j) + f * (α ε , j) p(x ε ) + p * (α ε ) = min j∈J i / ∈I * x i ε p(x ε ) + p * (α ε ) f (i, j) + i∈I * α i ε p * (i) p(x ε ) + p * (α ε ) g(i, j) .
But when i / ∈ I * , p(i) = 1, thus:

v -ε ≤ min j∈J i / ∈I * x i ε p(i) p(x ε ) + p * (α ε ) f (i, j) + i∈I * α i ε p * (i) p(x ε ) + p * (α ε ) g(i, j) ≤ u.
Step 2 v ≥ u. Let x 0 be optimal for player I in the one shot matrix game associated with u. Define (x 1 , α 1 ) as follows. If p * (x 0 ) = 0, let (x 1 , α 1 ) = (x 0 , 0). This clearly implies that v ≥ u. If p * (x 0 ) > 0 then for all i ∈ I * , let x i 1 = 0 (so that x 1 is non-absorbing) and for i / ∈ I * let α i 1 = 0. This will imply that,

v ≥ min j∈J f (x 1 , j) + f * (α 1 , j) p(x 1 ) + p * (α 1 ) = min j∈J i / ∈I * x i 1 p(x 1 ) + p * (α 1 ) f (i, j) + i∈I * α i 1 p * (i) p(x 1 ) + p * (α 1 ) g(i, j) = min j∈J i / ∈I * x i 1 p(i) p(x 1 ) + p * (α 1 ) f (i, j) + i∈I * α i 1 p * (i) p(x 1 ) + p * (α 1 )
g(i, j) .

Complete the definition of (x 1 , α 1 ) as follows. For i / ∈ I * , let x i 0 =

x i 1 p(i) p(x 1 )+p * (α 1 ) (x 1 is proportional to x 0 on the I\I * ) and for i ∈

I * let x i 0 = α i 1 p * (i) p(x 1 )+p * (α 1 ) (α 1 is proportional to x 0 on I * ). Consequently, v ≥ min j∈J   i / ∈I x i 0 g(i, j) + i∈I x i 0 g(i, j)   = u.

The minmax

A team of N players (named I) play against player (J). Assume the finiteness of all the strategy sets. Each player k in team I has a finite set of actions I k . Player J has a finite set of actions J. Let I = I 1 × ... × I N and f , g from I × J → [-1, 1] and p : I × J → [0, 1] . The game is played as above, except that (1) at each period, players in team I randomize independently (they are not allowed to correlate their random moves); and (2) team I minimizes the expected λ-discounted-payoff and player J maximizes the payoff (players in I try to punish player J).

Let ∆ = ∆(I 1 ) × ... × ∆(I N ), p * (•) = 1 -p (•) , f * (•) = p * (•) × g (•) and M + = M + (I 1 ) × ... × M + (I N ). For x ∈ X, j ∈ J, k ∈ N and α ∈ M + , a function ϕ : I × J → [-1, 1] is extended multi-linearly as follows: ϕ(x, j) = i=(i 1 ,...,i N )∈I x i 1 1 × ... × x i N N ϕ(i, j) ϕ(α k , x -k , j) = i=(i 1 ,...,i N )∈I x i 1 1 × ... × x i k-1 k-1 × α i k k × x i k+1 k+1 ... × x i N n ϕ(i, j).
Let w (λ) denote the minimum payoff that team I can guarantee against player J. From Bewley and Kohlberg [START_REF] Bewley | The Asymptotic Theory of Stochastic Games[END_REF] one can deduce the existence of w (λ) . However, no explicit formula exists.

Theorem 6 w (λ) = min x∈∆ max j∈J λf (x,j)+(1-λ)f * (x,j) λp(x,j)+p * (x,j)
and, as λ → 0, converges to

w = inf (x,α)∈∆×M + :∀k,α k ⊥x k max j∈J   f * (x,j) p * (x,j) 1 {p * (x,j)>0} + f (x,j)+ N k=1 f * (α k ,x -k ,j) p(x,j)+ N k=1 p * (α k ,x -k ,j) 1 {p * (x,j)=0}   = inf (x,α)∈∆×M + :∀k,α k ⊥x k max y∈∆(J)   f * (x,y) p * (x,y) 1 {p * (x,y)>0} + f (x,y)+ N k=1 f * (α k ,x -k ,y) p(x,y)+ N k=1 p * (α k ,x -k ,y) 1 {p * (x,y)=0}   .
Proof. For the first formula, follow the ideas in the proof of theorem 3 and corollary 4. Let v = lim n→∞ w (λ n ) where λ n → 0.

Modifications in step 1 in theorem 3: let x (λ n ) → x be such that for every j ∈ J,

w (λ n ) ≥ λ n f (x(λ n ), j) + (1 -λ n ) f * (x(λ n ), j) λ n p(x(λ n ), j) + p * (x(λ n ), j) . Let y(λ n ) = x (λ n ) -x → 0 so that: p * (x(λ n ), j) = i=(i 1 ,...,i N )∈I x i 1 1 (λ n ) × ... × x i N N (λ n )p(i, j) = i=(i 1 ,...,i N )∈I (y i 1 1 (λ n ) + x i 1 1 ) × ... × (y i N N (λ n ) + x i N N )p(i, j) = p * (x, j) + N k=1 p * (y k (λ n ), x -k , j) + o( N k=1 p * (y k (λ n ), x -k , j)) If p * (x, j) > 0 then w ≥ f * (x,j) p * (x,j) . If p * (x, j) = 0 and if α k (λ n ) = x i k k (λn) λn 1 x i k k =0 i k ∈I k ∈ M + (I k ) then α k (λ n ) ⊥ x k and p * ( x(λ n ) λ n , j) = N k=1 p * (α k (λ n ), x -k , j) + o( N k=1 p * (α k (λ n ), x -k , j))
and the same is true for f * so that

w ≥ lim sup n→∞ f (x, j) + N k=1 f * (α k (λ n ), x -k , j) p(x, j) + N k=1 f * (α k (λ n ), x -k , j)
which implies that w ≥ v. Modifications in step 2 in theorem 3: take (α ε , x ε ) to be ε-optimal for the minimizer in the formula of w and define x ε k (λ n ) to be proportional to x ε k + λ n α ε k and deduce that w ≤ v. For the second formula, follow corollary 4. For each ε > 0, the proof above implies that players in I have an ε-optimal strategy (x ε k (λ)) k∈I where x ε k (λ) is proportional to x ε k + λα ε k in the λ-discounted game for all λ small enough. This implies that for any y ∈ ∆(J),

w (λ) + ε ≥ λf (x ε k (λ), y) + (1 -λ) f * (x ε k (λ), y) λp(x ε k (λ), y) + p * (x ε k (λ), y)
.

where the right hand is a fractional function of λ. Consequently, it admits a limit which may be computed as in step 1 (using the multi-linearity of payoffs and transitions). This will imply that

w ≥ inf x∈∆ inf α∈M + :∀k,α k ⊥x k max y∈∆(J)   f * (x,y) p * (x,y) 1 {p * (x,y)>0} + f (x,y)+ N k=1 f * (α k ,x -k ,y) p(x,y)+ N k=1 p * (α k ,x -k ,y) 1 {p * (x,y)=0}   .
The first formula of w and the fact that J ⊂ ∆(J) imply the other inequality. The game is played as above except that if at stage t player k = 1, ..., N chooses the action i k t ∈ I k then player k receives f k i 1 t , ..., i N t and if the game is absorbed he receives g k i 1 t , ..., i N t . From Fink [START_REF] Fink | Equilibrium in a Stochastic N -Person Game[END_REF], the λ-discounted game admits a stationary Nash equilibrium. Notations and calculus as above allow to establish that x λ ∈ ∆ with the corresponding payoff u λ = u 1 λ , ..., u N λ ∈ R N is a stationary equilibrium iff:

Stationary Nash Equilibria

x k λ ∈ arg max x k ∈∆(I k ) λf k (x k , x -k λ ) + (1 -λ)f * k (x k , x -k λ ) λp(x k , x -k λ ) + p * (x k , x -k λ ) u λ = max x k ∈∆(I k ) λf k (x k , x -k λ ) + (1 -λ)f * k (x k , x -k λ ) λp(x k , x -k λ ) + p * (x k , x -k λ )
, Arguments as above prove the following.

Theorem 7 Let u = (u 1 , ..., u N ) ∈ [-1, 1]
N be an accumulation point of u λ . Then u should be a limit equilibrium payoff of the game where the set of strategies of player k is:

Λ(I k ) = (x k , α k ) ∈ ∆(I k ) × M + (I k ) : α k ⊥ x k
and where the payoff function of player k is

f * k (x) p * (x) 1 {p * (x)>0} + f k (x) + N j=1 f * k (α j , x -j ) p(x) + N j=1 p * (α j , x -j ) 1 {p * (x)=0}
More precisely, for any ε > 0, there exists (x ε , α ε ) such that:

u k = lim ε→0 f * k (x ε ) p * (x ε ) 1 {p * (xε)>0} + f k (x ε ) + N j=1 f * k (α j ε , x -j ε ) p(x ε ) + N j=1 p * (α j ε , x -j ε ) 1 {p * (xε)=0} ≥ sup (x k ,α k )∈Λ(I k )    f * k (x k ,x -k ε ) p * (x k ,x -k ε ) 1 {p * (x k ,x -k ε )>0} + f k (x k ,x -k ε )+f * k (α k ,x -k ε )+ N j =k f * j (α j ε ,x k ,x -{k,j} ε ) p(x k ,x -k ε )+p * k (α k ,x -k ε )+ N j =k p * (α j ε ,x k ,x -{k,j} ε ) 1 {p * (x k ,x -k ε )=0}    -ε 6

Compact continuous games

Let us extend the model of zero-sum game. I and J are now assumed to be compact and metric sets. The game is separately (resp. jointly) continuous if f , g and p are separately (resp. jointly) continuous functions on I × J. ∆(K), K = I, J, is the set of Borel probability measures on K and M + (K) is the set of Borel positive measure on K. They are endowed with the weak* topology. For (α, β) ∈ M + (I) × M + (J) and ϕ :

I × J → [-1, 1] measurable, ϕ(α, β) = I×J ϕ(i, j)dα(i)dβ(j).
This framework was introduced in Rosenberg and Sorin [START_REF] Rosenberg | An Operator Approach to Zero-sum Repeated Games[END_REF]. Following the approach of Kohlberg [START_REF] Kohlberg | Repeated Games with Absorbing States[END_REF], Rosenberg and Sorin considered the Shapley operator r → Φ(λ, r) where

Φ(λ, r) = max x∈∆(I) min y∈∆(J) [λf (x, y) + (1 -λ) p(x, y)r + (1 -λ) f * (x, y)] = min y∈∆(J) max x∈∆(I) [λf (x, y) + (1 -λ) p(x, y)r + (1 -λ) f * (x, y)] .
The operator is well defined and the existence of the value is guaranteed via Sion's minmax theorem. As Shapley did, the operator is (1 -λ)-contracting so that the value of the λ-discounted game v(λ) is the unique fixed point. Kohlberg [START_REF] Kohlberg | Repeated Games with Absorbing States[END_REF], in finite absorbing games and Rosenberg and Sorin [START_REF] Rosenberg | An Operator Approach to Zero-sum Repeated Games[END_REF] in separately continuous absorbing games proved the existence of v = lim v(λ) and provided a variational characterization of v using the information obtained from the derivative of Φ(λ, r) around λ ≈ 0. Notations for a multi-player absorbing game are introduced similarly.

Theorem 8 If the game is jointly continuous, all the results proved above for finite games still hold (for lim v(λ), lim w(λ) and Nash equilibria).

Proof. Let us show how the first part of theorem 3 is modified. Let w = lim n→∞ v (λ n ) where λ n → 0. Take an optimal strategy x(λ n ) of player I in the λ n -discounted game and suppose w.l.o.g. that it converges to some x. Consider any strategy j of Player J so that:

v (λ n ) ≤ λ n f (x(λ n ), j) + (1 -λ n ) f * (x(λ n ), j)
λ n p(x(λ n ), j) + p * (x(λ n ), j)

If p * (x, j) > 0 then v ≤ f * (x,j) p * (x,j) . If p * (x, j) = 0 then p * (i, j) = 0 on i ∈ S(x) the support of x. Define α(λ n ) ∈ M + (I) to be dα(λ n )(i) = dx(λn)(i) λn 1 {i / ∈S(x)} . Let s n ≥ 0 to be such that α(λ n ) = s n σ(λ n ) and σ(λ n ) ∈ ∆(I) and assume w.l.o.g. that σ(λ n ) → σ and s n → t ∈ [0, +∞] (by compactness of ∆(I)). Using joint continuity, the fact that p(x, j) = 1 and that payoffs are uniformly bounded by 1 imply that for any ε > 0, there is N (ε) such that for all n ≥ N (ε) and all j ∈ J f (x(λ n ), j) + (1 -λ n ) f * (α(λ n ), j) p(x(λ n ), j) + p * (α(λ n ), j) ≤ f (x, j) + ε + f * (α(λ n ), j) -λ n f * (α(λ n ), j) p(x, j) -ε + p * (α(λ n ), j) ≤ f (x, j) + f * (α(λ n ), j) p(x, j) + p * (α(λ n ), j) min j∈J f * (x, j) p * (x, j) 1 {p * (x,j)>0} + f (x, j) + f * (α, j) p(x, j) + p * (α, j) 1 {p * (x,j)=0} .

+ 2ε 1 -ε + λ n
Step 2 of theorem 1 needs no modification. The other proofs are adapted in a similar way.

  For player I let Λ(I) = {(x, α) ∈ ∆(I) × M + (I) : α ⊥ x} and similarly for player J. Corollary 4 v satisfies the following equations:v = sup (x,α)∈Λ(I) inf (y,β)∈Λ(J) f * (x,y) p * (x,y) 1 {p * (x,y)>0} + f (x,y)+f * (α,y)+f * (x,β) p(x,y)+p * (α,y)+p * (x,β) 1 {p * (x,y)=0} = inf (y,β)∈Λ(I) sup (x,α)∈Λ(I) f * (x,y) p * (x,y) 1 {p * (x,y)>0} + f (x,y)+f * (α,y)+f * (x,β) p(x,y)+p * (α,y)+p * (x,β) 1 {p * (x,y)=0} = sup (x,α)∈Λ(I) inf y∈∆(J) f * (x,y)p * (x,y) 1 {p * (x,y)>0} + f (x,y)+f * (α,y) p(x,y)+p * (α,y) 1 {p * (x,y)=0}

  Consider a N player absorbing game where each player k ∈ N has a finite set of actions I k . Define the payoff functions f k : I → [-1, 1] and g k : I → [-1, 1], k ∈ {1, ..., N } and a probability transition p : I → [0, 1] where I = I 1 × ... × I N .

∆(I) = (x i )i∈I : x i ∈ [0, 1], i∈I x i = 1 is the set probabilities over I.

∆(J) = (y j )j∈J : y j ∈ [0, 1], j∈J y j = 1 is the set of probabilities over J.