Dr Mathilde Laird

Professor Niklas Herrmann

Naseem Ramsahye

Dr Cédric Totée

Dr Carole Carcel

Professor Masafumi Unno

Professor John R Bartlett
email: j.bartlett@westernsydney.edu.au

Michel Wong

Chi Man

Large polyhedral oligomeric silsesquioxane cagesthe isolation of functionalized POSS with an unprecedented Si 18 O 27 core

Keywords: polyhedral oligomeric silsesquioxanes, styryl-functionalized POSS, T18 POSS cage, silsesquioxane, siloxane ring strain model. . References

The synthesis of organo-functionalized polyhedral oligomeric silsesquioxanes (POSS, (R-SiO1.5)n, Tn) is currently an area of significant activity. To date, T14 is the largest such cage synthesized and isolated as a single isomer. Herein, we report the successful synthesis and isolation of an unprecedented, single-isomer styryl-functionalized T18 POSS. Unambiguously identified among nine possible isomers by multinuclear solution NMR ( 1 H, 13 C and 29 Si), MALDI-MS, FTIR and computational studies, this represents the largest single-isomer functionalized Tn compound isolated to date.

 Si resonances to silicon atoms in cage silsesquioxanes and providing a useful tool for predicting silicon-atom environments.

Introduction

The development of polyhedral oligomeric silsesquioxanes (POSS, R-(SiO1.5)n, where n=6, 8, 10, 12, etc) is currently an area of intense activity within the materials chemistry community, due in part to the wide range of fields in which they can be applied, including polymer chemistry [1][2][3][4][5][6][7][START_REF] Lickiss | Advances in Organometallic Chemistry[END_REF] , bio/nanomedicine [START_REF] Kannan | [END_REF][10][11][12][13][14] , catalysis [15][16][17][18] , optical and sensing applications [3, 19- 20] , optoelectronics [21] , flame retardancy [22] , etc. Of these, the cage silsesquioxanes, which are composed of a well-defined (SiO1.5)n polyhedral core (Tn) covalently linked to organic moieties, have been of particular interest [23] , with n=8 being the most widely reported, studied and employed in several application fields. These hybrid molecules can be readily incorporated into nanostructured materials either via hydrolysis-condensation processes involving silanol groups [24] or by polymerization on the organic moiety [4,[25][26] . Porous materials can be readily obtained in such systems, [24][25] where the porosity can either arise from the voids between the cages or the core of the cage (0.3 nm for T8) [25] . In addition, selforganized systems [27][28][29] , POSS micelles and templated POSS systems leading to mesoporous materials with T8 cages embedded with the walls [30] , and organic polymer-based nanohybrids [1,3,[31][32][33][34] can be obtained through judicious selection of functional organics.

We recently described the synthesis of new styryl-functionalized T8, T10 and T12 POSS compounds, in which the styryl group was linked to the cage via the phenylene group as spacer rather than the vinyl group [35] . Such functionalized Tn molecules bearing organic moieties that can be further modified via conventional and well-established reactions such as hydrosilylation, Heck coupling, thiolene click reactions, metathesis, etc, without modifying the structure of the Tn core, are particularly interesting. Indeed, they pave the way to an even wider range of potential applications. Accordingly, the availability of Tn precursors bearing styryl moieties grafted onto all "n" vertices usefully extends the range of versatile POSS precursors. However, most of the Tn precursors successfully isolated and unambiguously identified to date have exhibited values of n = 6, 8,10, or 12, with larger cages being comparatively rare. Indeed, to the best of our knowledge, the largest such cages isolated and unambiguously identified to date as pure compounds are the D3h and C2v isomers of (H-SiO1.5)14 [36] . The availability of such large, structurally well-defined cages would significantly extend the range of potential applications for POSS compounds as building blocks. In addition to providing the opportunity to modulate the size and shape of inorganic nodes in hybrid networks, the presence of functional organics bonded to each of the n vertices in (R-SiO1.5)n with larger values of n would confer additional flexibility and degrees-of-freedom on organic networks constructed around the functional organics. However, their synthesis is complicated by the fact that increasing values of n lead to a corresponding increase in the number of possible theoretical isomers in the system. For example, while cage silsesquioxanes with n=8 or 10 are obtained as single isomers, those with n=12, 14 or 16 could theoretically yield up to two, four or seven isomers, respectively [36] . Individual isomers have been obtained for n=14 in the (H-SiO1.5)14 system, although only mixtures have been obtained for n=16 [36] .

During subsequent work on the styryl-functionalized Tn system previously reported [35] , we identified the presence of small quantities of an additional cage silsesquioxane compound, with n = 18. In this paper, we describe the synthesis, isolation and characterization of this large styryl-functionalized cage and demonstrate that it is obtained as a single isomer. A model relating the effect of ring strain in a series of styryl-functionalized Tn cages with n=8, 10, 12 and 18 to the 29 Si chemical shifts arising from non-equivalent silicon sites in the cages is also described and used to confirm the structure of the T18 isomer obtained.

Results and Discussion

As indicated above, most studies of functionalized Tn cage silsesquioxane species to date have focused on the T8 system, together with the less common T10 and T12 compounds. However, several studies have also used various chromatographic techniques to isolate larger functionalized Tn systems. Frye and Collins first reported the preparation of a mixture of (HSiO1.5)n (n = 10, 12, 14 and 16) cage silsesquioxanes, together with a resin-like product, and their separation by gas chromatography [37] . The resulting materials were characterized by 1 H NMR and IR spectroscopy, although their structures were not determined. Agaskar et al. described the isolation and characterization by single-crystal X-ray diffraction of two isomers of (HSiO1.5)14 with D3h and C2v symmetry [36,38] . A subsequent chromatographic study by the same authors [36] revealed evidence for the presence of (HSiO1.5)n with all possible values of n from 16 to 32 (in steps of 2). In this work, the authors succeeded in isolating a mixture of isomers for n = 16, although it was not possible to isolate and identify the individual compounds within the mixture. Bürgy and Calzaferri employed size exclusion chromatography to obtain (HSiO1.5)n for values of n up to 18 [39] , although no structural data were presented. The approach used by us to synthesize and isolate styryl-functionalized Tn compounds for n = 8, 10 and 12 has been previously described . These compounds were isolated from a complex mixture, which was characterized by MALDI MS, as illustrated in Figure 1. Peaks arising from M+Na adducts of the T8, T10 and T12 compounds (each separated by the expected 310 m/z, corresponding to two R-SiO1.5 groups [35] ) are clearly seen, together with weaker peaks associated with larger cages. Thin layer chromatography was used to isolate the various fractions present in the mixture. Apart from the T8, T10, and T12 components, most of the other fractions obtained exhibited broad 1 H NMR signals, which often arise from polymeric species and were thus discarded. However, one fraction, which was obtained with a yield of 4.5 %, consistently exhibited narrow and well-defined 1 H NMR signals and was thus characterized in detail. The MALDI mass spectrum of this compound, Figure 2, revealed a strong peak at 2813.4 m/z, which is attributed to the presence of a styryl-functionalized T18 cage (expected m/z = 2813.4 for T18+Na, higher by 3 x 310.1 m/z than the mass peak for the corresponding T12 cage compound). It is intriguing that no T14 or T16 compounds could be identified by MALDI-MS, with a single, pure fraction of T18 being obtained. We note that two isomers in the H-T14 series have previously been isolated as pure products and characterized by single crystal X-ray diffraction (with D3h [38] and C2v [36] symmetry), suggesting that substituent effects, such as steric interactions, may play a role in controlling the formation and stabilization of the silsesquioxane cages in such large POSS systems. Unfortunately, attempts to crystallise the styryl-functionalized T 18 compound from a variety of solvents were unsuccessful. Apart from its attribution to styryl-functionalized T18 cage silsesquioxane, other possible attributions for the 2813.4 m/z peak were also considered but were discounted. As discussed below, a feature of the compound is that the 29 Si spectrum is consistent with all Si sites having T 3 speciation, with no T 2 or T 1 sites being evident. This is consistent with a closed-cage structure and excludes the possibility of open or partially-condensed cages terminated by hydroxyl or bridging oxo species, since all such compounds would exhibit either T 2 signals around 10 ppm downfield from the observed T 3 features (see Table S1) or T 1 signals around 20 ppm downfield. The potential formation of open or partially-condensed cages terminated by hydroxyl groups is further excluded by the IR spectrum of the compound (Figure S1), which does not exhibit any evidence of ν(O-H) modes above 3200 cm -1 . No other 29 Si signals were observed in the range from +50 to -150 ppm that might arise from other Si(IV) or pentacoordinated moieties (such as ≡Si-F). Finally, the 1 H and 13 C spectra (Figures S2 andS3, respectively, discussed in more detail below) are consistent with functionalization of all Si sites with styryl moieties, with no other organic moieties covalently bonded to Si being evident. This precludes the presence of pendant methoxy groups in the compound, which would give rise to signals near 3.5 and 50 ppm in the 1 H and 13 C solution NMR spectra, respectively.

As indicated above, Agaskar and Klemperer [36] previously reported the identification of a mixture of isomers of (H-Si18O27) via chromatographic separation of a mixture of HTn species (where n=14-32), although no pure T18 compounds were isolated or characterized in their study. They proposed nine possible isomers in the HT18 system (Figure 3), which differ mainly in the number of 6-, 5-and 4-silicon-atom rings and in the corresponding silicon environments. The isomers proposed by Agaskar and Klemperer exhibit between four and seven non-equivalent silicon atoms, and hence it would be anticipated that 29 Si NMR spectroscopy would provide useful insights into the structure of the isomer obtained in our work.

The 29 Si solution NMR spectrum of the styryl-functionalized T18 silsesquioxane, Figure 4, exhibits five distinct signals at -77.40, -78.35, -79.12, -79.18 and -79.35 ppm, suggesting the presence of five distinct T 3 Si sites in our product. Of the possible isomers suggested by Agaskar and Klemperer [36] , only two of the nine possess five non-equivalent Si sites (isomers E and F in Figure 3). The first of these contains six Si in 6 1 5 2 , four Si in 6 1 5 1 4 1 , two Si in 6 1 4 2 , two Si in 5 3 , and four Si in 5 2 4 1 sites, while the second contains four Si in 6 2 5 1 , four Si in 6 2 4 1 , four Si in 6 1 5 1 4 1 , four Si in 6 1 4 2 and two Si in 5 1 4 2 sites. Here, the notation 6 x 5 y 4 z , where x,y,z >0 and x+y+z=3, refers to the number of six-, five-and/or four-Si-atom rings that each non-equivalent Si atom is part of (see coloured Si assignments in Figure 3). In order to unambiguously identify the isomer, quantitative 29 Si NMR was performed. A zg30 pulse with a relaxation time, D1 = 100 s, was used to ensure complete relaxation of the silicons between each acquisition. The integrated intensities of the proton-decoupled signals presented in Figure 4 are consistent with the second possibility, indicating that the structure of the T18 isomer obtained in our work is that illustrated in Figure 3-F. It is interesting to note that modelling studies (see Experimental) indicated that there is only around 5 kJ/mol difference in the calculated energies of the optimized structures of isomers E and F in Figure 3, indicating that purely thermodynamic considerations alone cannot explain the formation of one isomer instead of the other. This suggests that the formation of isomer F is favored kinetically rather than thermodynamically, since only isomer F is observed in our study. 29 Si chemical shifts [36] , although these correlations could not be successfully applied to our system. Rikowski and Marsmann also presented empirical correlations between the 29 Si chemical shifts in organofunctionalized T8, T10 and T12 compounds, [40] which were attributed to differences in the strain in 4-and 5-Si-atom rings. However, again, these correlations could not be successfully applied in our case; moreover, no correlations for compounds containing 6-Si-atom rings were presented. Accordingly, we have developed an empirical "ring-strain" model to predict the chemical shifts in styryl-functionalized T8, T10, T12 and T18 cages and to assign the signals in the 29 Si T18 spectrum to specific Si atoms in each of the five non-equivalent sites. As discussed above, it has previously been established that 4-Si-atom rings in cage silsesquioxane systems are more strained than 5-Si-atom rings, although no such correlations have previously been reported in the case of 6-Si-atom rings. Indeed, the only reports of chemical shifts in 6-Si-atom ring systems appear to be in the case of less-constrained cyclic polysilanols such as [PhSi(O)(OH)] 6 [41] and silsesquioxanes such as [PhSi(O)(OSiMe2R)]6 [42] . The assignments of signals in the 29 Si NMR spectra of T8 (-78.19 ppm, 4 3 ), T10 (-79.59 ppm, 5 1 4 2 ) and T12 (-79.45 ppm, 5 1 4 2 ; -81.04 ppm, 5 2 4 1 ) have already been established using ring-strain correlations, as indicated previously and are consistent with the relative abundance observed by 29 Si NMR spectroscopy. The chemical shifts of the signals in the T18 spectrum are largely bounded by the values established in the T8, T10 and T12 systems, although one signal (-77.40 ppm) appears more downfield than that of the single resonance in the T8 spectrum (-78.35 ppm, 4 3 ). This suggests that the former is associated with a more constrained system than the T8 (which contains only 4 3 silicon sites). Although not previously observed, one explanation of this effect would be to postulate that 6-Si-atom rings are more constrained than 4-Si-atom rings in cage silsesquioxane systems. Accordingly, the T 18 signal at -77.40 ppm would logically be assigned to 6 2 4 1 sites, which would then be identified as the most constrained Si environments in the Tn systems. Based on the integrated intensities shown in Figure 4, the signal at -79.35 ppm is directly attributable to the (two) 5 1 4 2 sites in the T18 cage, in close agreement with the chemical shifts of Si atoms in 5 1 4 2 sites in the T 10 and T 12 compounds (-79.59 and -79.45 ppm, respectively). Based on these initial correlations, the proposed attributions of signals in the 29 Si NMR spectrum of the styryl-functionalized T18 cage are summarized in Table 1, together with the corresponding assignments for the T8, T10, and T12 compounds. To further investigate the role of ring-strain in modulating the positions of the 29 Si signals, we then derived a simple linear relationship between the observed chemical shifts and the number of 4-, 5-and 6-Si-atom rings associated with each silicon site: 𝛿 𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 = 𝐴. 𝑛 4-𝑆𝑖-𝑟𝑖𝑛𝑔 + 𝐵. 𝑛 5-𝑆𝑖-𝑟𝑖𝑛𝑔 + 𝐶. 𝑛 6-𝑆𝑖-𝑟𝑖𝑛𝑔 + 𝐷 where the values of A, B, C and D are obtained via non-linear regression using the data in Table 1. The results obtained are illustrated in Figure 5, which illustrates a reasonably good fit of the simple ring-strain model to the experimental data. The relative magnitudes of A, B and C suggest that the impact of 5-Si-atom rings on 29 Si chemical shifts is five-to-six-fold lower than that of 4-and 6-Si-atom rings. In addition, the impact of 6-Si-atom rings is around 15 % higher than that of 4-Si-atom rings. As far as we are aware, this is the first report of an empirical approach linking the relative strain in the various rings present in such cage silsesquioxanes to the positions of 29 Si NMR signals, with earlier studies [40] not considering 6-Si-atom ring systems. The proposed "ring-strain" approach is general, and hence the same values of A, B and C should be applicable to other functionalized Tn cage silsesquioxane families. However, the value of D would be expected to change with different organic moieties, to account for the different shielding of aliphatic and aromatic substituents, different NMR solvents, etc. To test this hypothesis, the comprehensive 29 Si NMR data for H-Tn (for n = 8, 10, 12 and 14 (both D3h and C2v isomers)) reported by Agaskar and Klemperer [36] were modelled by using the values of A, B and C derived from the styryl system and varying the value of D via a least-squares minimization. The results obtained, Figure 6, demonstrate that the ring-strain model fits the data for the H-Tn system reasonably well, with the value of D being somewhat different to that for the styryl-functionalized system as expected. As far as we are aware, there are currently no other examples of R-Tn cage silsesquioxane series for n = 8 to >14 against which to test our approach. However, it is of interest to use the correlations developed here to consider the stability of other potential isomers in the T12, T14 and T16 systems. For example, the postulated D6h T12 isomer has never been successfully isolated (despite being widely and incorrectly used as a model of the structure of the T12 cage silsesquioxanes), with the D2d isomer always being exclusively obtained. Based on "ring-strain" considerations, this could arise from the fact that the D6h isomer would consist of 12 x 6 2 4 1 Si species, and hence, would be highly strained compared to the D2d isomer (8 x 5 2 4 1 and 4 x 5 1 4 2 ).

The 1 H solution NMR spectrum of the styryl-functionalized T18 silsesquioxane, Figure S2, exhibits the series of overlapping signal clusters expected for five groups of non-equivalent styryl moieties, as discussed in detail in the Supplementary Information. Similarly, the 13 C solution NMR spectrum, shown in Figure S3, exhibits the expected chemical shifts of the styryl moieties, as previously described [35] . The effect of the five non-equivalent Si environments can also be observed (expanded view in Figure S3), with the cluster of signals arising from the quaternary carbons directly bonded to the five non-equivalent Si atoms being evident.

Finally, the FTIR spectrum of the styryl-functionalized T18 is compared to the spectra of the corresponding T8 and T10 compounds in Figure 7, with the intensities of peaks in the spectra normalized against the styryl modes. Apart from the characteristic features arising from the styryl function (e.g. ring C-C stretching at 1601 cm -1 and the C=C stretching at 1630 cm -1 ), the comparison highlights a broader full-width at half-maximum in the Si-O-Si mode region for the T18 compound compared to the other cages. The T8 and T10 cages are both highly symmetrical with equivalent Si-O-Si species in each cage, resulting in relatively narrow Si-O-Si band profiles in each case. In contrast, there are five non-equivalent Si environments in the T 18 cage, and the broader profile observed in this region is entirely consistent with the presence of such non-equivalent sites.

Conclusions

In summary, we report the synthesis and characterization of an unprecedented styryl-functionalized T18 cage silsesquioxane as a pure isomer, in which styryl groups are linked to all 18 vertices of the T18 compound via the phenylene groups. Complementary analyses via MALDI mass spectrometry and multinuclear NMR enabled the particular isomer obtained to be identified from the nine possible isomers previously postulated for the T18 system. To the best of our knowledge, this is the largest POSS compound synthesized and isolated to date as a pure, single-isomer product, thus extending the range of available organo-functionalized POSS compounds. Modelling studies are currently ongoing to better characterize the structure and properties of this novel cage silsesquioxane.

A ring strain model is presented that attributes observed 29 Si resonances in the styryl-functionalized Tn POSS family (where n=8, 10, 12, and 18) to non-equivalent Si atoms on the basis of their location in six-, five-and/or four-Si-atom rings (6 x 5 y 4 z , where x,y,z >0 and x+y+z=3). The model successfully predicted the speciation non-equivalent Si atoms in other families of organo-functionalized Tn compounds, demonstrating its general applicability for assigning 29 Si resonances to silicon atoms in cage silsesquioxanes.

Figure 1 .

 1 Figure 1. MALDI-TOF mass spectrum of the product mixture from which styryl-functionalized Tn compounds were isolated.

Figure 2 .

 2 Figure 2. MALDI-TOF mass spectrum of styryl-functionalised T18.

Figure 3 .

 3 Figure 3. Proposed isomers in the T18 system.

Figure 4 .

 4 Figure 4. Quantitative 29 Si NMR spectrum of the styryl T18 cage silsesquioxaneAn unequivocal attribution of the29 Si NMR signals in Figure4to Si atoms in the five non-equivalent sites illustrated in Figure3-F is non-trivial. Agaskar and Klemperer presented correlations between the number of 4-Si-atom rings around each silicon site in selected HTn systems and the corresponding29 Si chemical shifts[36] , although these correlations could not be successfully applied to our system. Rikowski and Marsmann also presented empirical correlations between the29 Si chemical shifts in organofunctionalized T8, T10 and T12 compounds,[40] which were attributed to differences in the strain in 4-and 5-Si-atom rings. However, again, these correlations could not be successfully applied in our case; moreover, no correlations for compounds containing 6-Si-atom rings were presented. Accordingly, we have developed an empirical "ring-strain" model to predict the chemical shifts in styryl-functionalized T8, T10, T12 and T18 cages and to assign the signals in the29 Si T18 spectrum to specific Si atoms in each of the five non-equivalent sites. As discussed above, it has previously been established that 4-Si-atom rings in cage silsesquioxane systems are more strained than 5-Si-atom rings, although no such correlations have previously been reported in the case of 6-Si-atom rings. Indeed, the only reports of chemical shifts in 6-Si-atom ring systems appear to be in the case of less-constrained cyclic polysilanols such as [PhSi(O)(OH)] 6[41] and silsesquioxanes such as [PhSi(O)(OSiMe2R)]6[42] . The assignments of signals in the29 Si NMR spectra of T8 (-78.19 ppm, 4 3 ), T10 (-79.59 ppm, 5 1 4 2 ) and T12 (-79.45 ppm, 5 1 4 2 ; -81.04 ppm, 5 2 4 1 ) have already been established using ring-strain correlations, as indicated previously and are consistent with the relative abundance observed by29 Si NMR

Figure 5 .

 5 Figure 5. Experimental vs predicted chemical shifts for 29 Si signals in the styryl-functionalized Tn system (n = 8, 10, 12, and 18).

Figure 6 .

 6 Figure 6. Experimental vs predicted chemical shifts for 29 Si signals in the H-Tn system (n = 8, 10, 12, and 14).

Figure 7 .

 7 Figure 7. FTIR spectra of the styryl-functionalized T8, T10 and T18 cage silsesquioxanes.

Table 1 .

 1 Assignments of signals in the29 Si NMR spectra of Tn compounds and predicted chemical shifts based on the "ring-strain" model.

	Cage	Number of Si in ring	Chemical	Model
	type	6-ring	5-ring	4-ring	Shift (ppm)	(ppm)
	T18	2	0	1	-77.40	-77.62
	T8	1	0	2	-78.19	-78.17
	T18	0	0	3	-78.35	-77.90
	T18	2	1	0	-79.12	-79.05
	T18	1	1	1	-79.18	-79.33
	T18	0	1	2	-79.35	-79.60
	T12	0	1	2	-79.45	-79.60
	T10	0	1	2	-79.59	-79.60
	T12	0	2	1	-81.29	-81.04

Acknowledgements

The authors gratefully acknowledge the support of the Balard Plateforme d'Analyses et de Caractérisation (PAC Balard). Funding from the French Ministère de l'Enseignement Supérieur et de la Recherche to support the PhD scholarship of ML, together with travel funding from the Japan Student Services Organization, is also gratefully acknowledged.


I R Viscardy

Justin T Thomas

S Erwin

L Robert

L Trompet

G L Neary

G L Villanueva

Liuzzi

S Aoki

F Daerden

S Viscardy

I R Thomas

J T Erwin

S Robert

L Neary

G Liuzzi

M M J Crismani

R T Clancy

J Whiteway

F Schmidt

M A Lopez-Valverde

B Ristic

M R Patel

G Bellucci

J.-J Lopez- Moreno

K S Olsen

F Lefèvre

F Montmessin

A Trokhimovskiy

A A Fedorova

O Korablev

A C Vandaele

Annual Appearance of Hydrogen Chloride on Mars and a Striking Similarity With the Water Vapor

published or not. The documents may come

Introduction

Chlorine chemistry plays an important role in the atmosphere of Earth and Venus. On Earth, small amounts of chlorine can have a dramatic impact on stratospheric ozone (e.g. [START_REF] Solomon | Stratospheric ozone depletion: A review of concepts and history[END_REF]. On Venus, chlorine is involved in the main photochemical cycles (e.g. [START_REF] Mahieux | Hydrogen halides measurements in the Venus mesosphere retrieved from SOIR onboard Venus Express[END_REF][START_REF] Sandor | First measurements of ClO in the Venus atmosphere -Altitude dependence and temporal variation[END_REF]. In contrast, on Mars, chlorine photochemistry was considered to be negligible because of the low upper limits provided by the previous measurements (Krasnopolsky et al., 1997[START_REF] Hartogh | Herschel/HIFI observations of Mars: First detection of O2 at submillimetre wavelengths and upper limits on HCl and H2O2[END_REF][START_REF] Villanueva | A sensitive search for organics (CH 4 , CH 3 OH, H 2 CO, C 2 H 6 , C 2 H 2 , C 2 H 4 ), hydroperoxyl (HO 2 ), nitrogen compounds (N 2 O, NH 3 , HCN) and chlorine species (HCl, CH 3 Cl) on Mars using ground-based high-resolution infrared spectroscopy[END_REF].

However, the first detection of hydrogen chloride (HCl) in the atmosphere of Mars was made by the Atmospheric Chemistry Suite (ACS) and also confirmed by the Nadir and Occultation for Mars Discovery (NOMAD) onboard the ESA-Roskosmos ExoMars Trace Gas Orbiter (TGO) [START_REF] Korablev | Transient HCl in the atmosphere of Mars[END_REF]. This discovery of HCl is the first detection of any chlorine species in the Martian atmosphere. It demonstrates an active presence of chlorine photochemistry and HCl is expected to be the main reservoir of chlorine species in the atmosphere of Mars [START_REF] Lefèvre | Atmospheric Photochemistry. in The Atmosphere and Climate of Mars[END_REF]. Perchlorate salts ClO 4 - [START_REF] Hecht | Detection of Perchlorate and the Soluble Chemistry of Martian Soil at the Phoenix Lander Site[END_REF][START_REF] Glavin | Evidence for perchlorates and the origin of chlorinated hydrocarbons detected by SAM at the Rocknest aeolian deposit in Gale Crater[END_REF] and chloride-bearing minerals Cl - (Osterloo et al., 2008(Osterloo et al., , 2010) ) are known to be widely present at the manuscript submitted to GRL Mars surface, thus an interaction between surface and atmosphere may support chlorine photochemistry. Model and laboratory studies investigated the potential role of atmospheric photochemical processes in the formation of perchlorates (e.g. [START_REF] Catling | Atmospheric origins of perchlorate on Mars and in the Atacama[END_REF][START_REF] Smith | The formation of sulfate, nitrate and perchlorate salts in the martian atmosphere[END_REF][START_REF] Wilson | Perchlorate formation on Mars through surface radiolysis-initiated atmospheric chemistry: A potential mechanism[END_REF]Wu et al., 2018, and references therein). However, the chlorine cycle on Mars is still not understood. The discovery made by [START_REF] Korablev | Transient HCl in the atmosphere of Mars[END_REF] hints at the transient nature of HCl: the detections are mainly confined after the global dust storm in the southern summer season. The observed fast temporal variation can be a key to identify the formation and destruction mechanisms of HCl. The discovery was based on the data taken in Martian year (MY) 34. This study presents the results of the extended NOMAD dataset that also includes almost the full MY 35. We investigate the vertical distribution of HCl volume mixing ratios (vmr) and their similarities with simultaneously obtained water vapor and aerosol profiles.

An analysis of the Cl isotopic ratio in HCl with the NOMAD data is presented in an accompanying paper (Luizzi et al., 2021).

Methods

Instrument -NOMAD onboard TGO

NOMAD, a spectrometer operating in the spectral ranges between 0.2 and 4.3 μm onboard ExoMars TGO [START_REF] Vandaele | NOMAD, an integrated suite of three spectrometers for the ExoMars Trace Gas mission: technical description, science objectives and expected performance[END_REF], has 3 spectral channels: a solar occultation provided by an echelle grating used in combination with an Acousto Optical Tunable Filter (AOTF) which selects diffraction orders [START_REF] Neefs | NOMAD spectrometer on the ExoMars trace gas orbiter mission: part 1design, manufacturing and testing of the infrared channels[END_REF]. The sampling rate for the solar occultation measurement is 1 second, which provides high vertical sampling (~1 km) with high vertical resolution (~2 km) from near the surface to 200 km altitude. Thanks to the instantaneous change of the observing diffraction orders achieved by the AOTF, the SO channel is able to measure five or six different diffraction orders per second in solar occultation mode. One of the most remarkable capabilities of NOMAD is its high spectral resolution in the infrared range. This supports investigations of: (1) vertical profiles for atmospheric constituents such as carbon dioxide, carbon monoxide, water vapor, and their isotopic ratios [START_REF] Vandaele | Martian dust storm impact on atmospheric H 2 O and D/H observed by ExoMars Trace Gas Orbiter[END_REF]Aoki et al., 2019;[START_REF] Villanueva | Water heavily fractionated with altitude on Mars as revealed by ExoMars/NOMAD[END_REF] or 130 (2920-2943 cm -1 ) have been processed. These two diffraction orders contain a strong HCl line (see Figure 1) and are normally acquired together with diffraction order 134 (3010-3034 cm - 1 ), order 136 (3056-3080 cm -1 ), or order 145 (3258-3284 cm -1 ), which include strong water lines in support of direct comparisons between HCl and water vapor vertical profiles.

Retrieval of HCl and water vapor volume mixing ratio

manuscript submitted to GRL

The reduction and analysis of the NOMAD data is basically conducted with the same methodology as the previous water vapor study [START_REF] Vandaele | Martian dust storm impact on atmospheric H 2 O and D/H observed by ExoMars Trace Gas Orbiter[END_REF][START_REF] Aoki | Water vapor vertical profiles on Mars in dust storms observed by TGO/NOMAD[END_REF]. The retrievals of HCl and H 2 O are performed with the ASIMUT-ALVL radiative transfer and inversion code [START_REF] Vandaele | Simulation and retrieval of atmospheric spectra using ASIMUT[END_REF]. H 2 O, HDO, CO 2 and HCl molecular absorptions are taken into account in the radiative transfer calculation and the absorptions are calculated using the following spectroscopic database: HITRAN 2016 database (Gordon et al. 2016) for CO 2 and HCl, and the water line list for CO 2 -rich atmospheres by [START_REF] Gamache | A spectral line list for water isotopologues in the 1100-4100 cm -1 region for application to CO 2 -rich planetary atmospheres[END_REF] for H 2 O and HDO. The temperature, pressure, and CO 2 volume mixing ratio of the simulated atmosphere is obtained from the Global Environmental Multiscale Mars model (GEM-Mars) [START_REF] Neary | The GEM-Mars general circulation model for Mars: Description and evaluation[END_REF][START_REF] Daerden | Mars atmospheric chemistry simulations with the GEM-Mars general circulation model[END_REF] which takes into account the effects of the global dust storm in MY34 [START_REF] Neary | Explanation for the increase in high altitude water on Mars observed by NOMAD during the 2018 global dust storm[END_REF]. The HCl and H 2 O retrievals are performed using the Optimal Estimation Method (OEM) [START_REF] Rodgers | Inverse Methods for Atmospheric Sounding -Theory and Practice, Inverse Methods for Atmospheric Sounding -Theory and Practice[END_REF] for each spectrum at each tangential altitude independently. The retrieved HCl and H 2 O abundances are finally averaged with an interval of 2 km to obtain the vertical profiles. The averages are weighted by their standard errors (weighted average). In addition to the HCl and H 2 O abundances, the vertical profiles of aerosols extinction are investigated. The extinction profiles are simply derived from the averaged transmittances (at diffraction order 129 or 130) based on the onion-peeling method [START_REF] Wilquet | Optical extinction due to aerosols in the upper haze of Venus: Four years of SOIR/VEX observations from 2006 to[END_REF]. Figure 1 shows an example of the NOMAD spectra at diffraction order 129 (Fig. 1a)

and order 130 (Fig. 1b). Clear signatures of HCl are identified at 2906.24 cm -1 in order 129 and 2925.89 cm -1 in order 130. HCl is usually detected in the lower atmosphere (< 20-25 km) but sometimes up to 35-40 km altitude as shown in Fig 1b. L s =190°-250° and the regional dust storm at L s =320°-340° (e.g., [START_REF] Aoki | Water vapor vertical profiles on Mars in dust storms observed by TGO/NOMAD[END_REF][START_REF] Liuzzi | Strong variability of Martian water ice clouds during manuscript submitted to GRL dust storms revealed from ExoMars Trace Gas Orbiter/NOMAD[END_REF]. Almost all of the HCl detections were made in the period between the two dust storms (L s =240°-320°) (Fig. 2a). The period of HCl detections also correspond to the southern summer season on Mars. The temporal variation of the HCl detections in MY 35 is similar to that in MY 34: most of the HCl detections are made in the southern summer period (L s =240°-320°) (Fig. 2b). We use the dust climatology of MY 31 as a proxy, as it looks similar to that of MY 35. Since there is no global dust storm in MY35, it is demonstrated that the formation of HCl is independent from a global dust storm event. In both MY 34 and 35, higher abundances of HCl are observed in the southern hemisphere (local summer, L s =270°), particularly at high latitudes, which corresponds in time and location to the water vapor enhancement caused by the south polar cap sublimation (Fig. 2c). Because the observations before L s ~240° were not possible in the southern hemisphere due to dust obscuration, it is unclear whether this represents an increase in HCl, or if high abundances were already present. In contrast, the volume mixing ratio of HCl is below the detection limit (~0.3 ppbv) at the northern hemisphere summer solstice (L s =90°), even at high latitudes where the column integrated abundances of water vapor are large.

Results

Seasonal, spatial, and interannual variations of HCl

Figure 2d shows the geographical distribution of the maximum HCl volume mixing ratio below 30 km observed by NOMAD in the southern summer period (L s =240°-320°) of MY 34 manuscript submitted to GRL and 35. The detections are dispersed over all observed longitudes, such that HCl is not confined to a specific longitude region. Most of the detections are performed at latitudes higher than 45°. This is because higher dust content in the atmosphere at lower latitudes obscures solar occultation viewing. In other words, the non-detection of HCl at low-latitudes means that the abundance of HCl cannot be measured there. There is a clear north-south asymmetrythe maximum volume mixing ratio of HCl is larger in the southern hemisphere than in the northern hemisphere. around perihelion (L s =240°-270°) (Fig. 3a), water vapor volume mixing ratios increased towards the higher altitudes with a maximum around 30 km. This decrease below 30 km reflects water vapor loss at lower altitudes due to the condensation over the seasonal polar cap (e.g. Daerden et al., 2019). In MY 34, because of increased upper level transport associated with the global dust storm, water is also enhanced at higher altitudes [START_REF] Aoki | Water vapor vertical profiles on Mars in dust storms observed by TGO/NOMAD[END_REF][START_REF] Neary | Explanation for the increase in high altitude water on Mars observed by NOMAD during the 2018 global dust storm[END_REF][START_REF] Fedorova | Stormy water on Mars: The distribution and saturation of atmospheric water during the dusty season[END_REF]. HCl also shows a decrease towards lower altitudes. At the southern mid-to-high latitudes (50°-70°S) during the same period (Fig. 3b), water vapor volume mixing ratios are constant throughout the atmosphere and HCl follows this same uniform distribution.

Vertical distribution of hydrogen chloride, water vapor, and aerosols

This constant mixing continues after perihelion (L s =270°-300°), although in MY 35, water and HCl vmr start to decrease between 20-30 km (Fig. 3c). Note that in this seasonal range at L s =270°-300°, we observe larger water vapor mixing ratios (because of the sublimation of the south polar cap) as well as the largest observed HCl vmr (4 ppbv) both in MY 34 and 35.

Moreover, HCl is detected at the highest altitudes (35-40 km). Water vapor is still abundant in the later period (L s =300°-330°) but rapidly drops above the hygropause, which is located around 15-20 km (Fig. 3d). Interestingly, HCl also dramatically drops at the same altitudes. Finally, in the northern hemisphere during the summer solstice (L s =90°-120°) (Fig. 3e), HCl is not detected anymore while a large amount of water vapor is observed at lower altitudes (below 10 km) because of the sublimation of the northern polar cap. This is a low-dust season, and the upper limit of HCl below 10 km is ~ 0.3 ppbv, which is well below the abundances detected in the southern summer season.

Such a temporal variation of the HCl vertical profiles observed by NOMAD is consistent with the simultaneous measurements by ACS reported earlier [START_REF] Korablev | Transient HCl in the atmosphere of Mars[END_REF]. In this study, we emphasize that the vertical structures of HCl and water vapor are strikingly similar. counteracting the ascend of water vapor with the normal circulation [START_REF] Neary | Explanation for the increase in high altitude water on Mars observed by NOMAD during the 2018 global dust storm[END_REF]. This causes a decrease of the water vapor vmr at the "hygropause" level (e.g., [START_REF] Daerden | Mars atmospheric chemistry simulations with the GEM-Mars general circulation model[END_REF] their Fig. 13). In polar winter conditions, the water vapor profile is also decreasing towards the surface due to strong surface deposition. We see both types of profiles in Fig. 3. The similarity of the HCl profiles to those of water vapor suggests that HCl is depositing with water vapor, both at the hygropause and at the surface in polar winter. Laboratory work for temperatures (194-228 K) and pressures (133 Pa) representative of Mars (Kippenberger et al., 2019), and for similar HCl partial pressures as found here, demonstrated an extensive, continuous uptake of HCl in water ice during growth of ice particles. This seems to indicate that HCl, once introduced in the atmosphere, tracks with water vapor. However, this behavior does not address the ultimate source or loss processes for Mars atmospheric HCl, as will be discussed below. When the ratio HCl/H 2 O is plotted for all the profiles combined, it is not decreasing over time and altitude (Fig. 4c-d). This could imply that the uptake of HCl on water ice clouds is not a permanent sink and that HCl is released again when water ice sublimates. However, it can also mean that new chlorine-bearing species are formed at the water ice cloud particles, that can become a new source of HCl (see Discussion for more details).

Discussion

Source of HCl

The first detections of HCl were obtained mainly just after the MY 34 global dust storm.

Therefore, [START_REF] Korablev | Transient HCl in the atmosphere of Mars[END_REF] proposed that dustincluding chloride-bearing materialslifted to high altitudes could release chlorine into the atmosphere by UV irradiation. This would ultimately lead to the formation of HCl through the fast reaction of chlorine with hydroperoxyl that results from photolysis of water vapor also lifted to high altitudes [START_REF] Wilson | Perchlorate formation on Mars through surface radiolysis-initiated atmospheric chemistry: A potential mechanism[END_REF][START_REF] Lefèvre | Atmospheric Photochemistry. in The Atmosphere and Climate of Mars[END_REF]:

Cl + HO 2 → HCl + O 2 (1)
However, the repeated detections of equivalent HCl abundances in the same season in MY 35 clearly indicates that a global dust storm is not required, the annually enhanced dust abundances in the perihelion season appearing to be sufficient to produce HCl.

The strong enhancement of HCl at the southern high latitudes in the southern summer season suggests that the main source of atmospheric chlorine could be there. We note that further measurement are needed to confirm the source as a large part of the NOMAD observations at low-and mid-latitudes could not detect HCl because of strong dust presence. Nevertheless, if the source is the southern high latitudes, the produced HCl can be transported to the northern hemisphere by the global circulation. Theoretical studies show that the time scale of the global mixing from 60°S latitude to north polar latitudes can be as low as 20 sols or less in this season manuscript submitted to GRL (Waugh et al., 2019, their Fig. 1d). This is fast enough to explain the observed latitudinal distribution of HCl.

The combination of enhanced dust and water abundances seems to be the key ingredient for the production of HCl. This may explain why there is no HCl detected in the northern summer season, which, despite the presence of large amounts of water vapor, is relatively dustfree. HCl may still be there, but below the detection limit. Dust optical maps from [START_REF] Montabone | Eight-year climatology of dust optical depth on Mars[END_REF][START_REF] Montabone | Martian Year 34 Column Dust Climatology from Mars Climate Sounder Observations: Reconstructed Maps and Model Simulations[END_REF] show a ratio of dust optical depths between perihelion and aphelion of ~5 (even in a low dust year), which could be sufficient to either preclude the formation of HCl in northern summer, or to keep it below detectability. However, detection limits reported in Korablev et al.

(2021) (0.1-0.2 ppbv) and here (0.3 ppbv) are much lower than simply scaling the observed HCl abundances by a factor of 5. It is possible that a certain threshold of dust abundance is required for the formation of HCl. Another possibility is that the HCl is released from the southern polar cap together with water vapor. But then the question remains why the same process is not occurring at the northern summer cap. If chlorine species are stored in the seasonal caps and released with water vapor during local spring and summer to form HCl, HCl should be strongly present during northern summer, which is not the case.

Sink of HCl

Another important issue arising from the reported observations regards the relatively rapid decay of atmospheric HCl abundances, requiring a strong HCl loss rate (Korablev et al., 2021 and Fig. 2 of this article). Indeed, the photochemical lifetime of HCl in the low atmosphere of Mars is estimated to be relatively long (see SI, Section S2). It can vary between >90 sols to >1000 sols below 15 km at the southern and northern mid-latitudes during the northern hemisphere winter solstice (see Fig. S2). On the other hand, the production of HCl through the reaction between Cl and HO 2 is much faster (typically on timescales <10 minutes in the lower atmosphere; see Fig. S2). As a result, HCl is expected to be the primary Cl reservoir species in the Martian atmosphere. In this context, the sudden loss of HCl which is observed around L s =320° (Fig. 2) cannot be explained by gas-phase photochemistry alone. A sink process, in addition to photochemical loss, has to be invoked. [START_REF] Korablev | Transient HCl in the atmosphere of Mars[END_REF] suggested that HCl could be removed from the atmosphere by uptake in growing water ice clouds particles and/or surface ice (Kippenberger et al., 2019). The striking similarity between HCl and water vapor vertical profiles shown in this study suggests that the vertical distribution of HCl is controlled by water ice clouds. However, the sublimation of those ice particles would be accompanied by the release of chlorine and thus the formation of atmospheric HCl, and thus it looks problematic to explain the permanent removal of HCl from the atmosphere by the uptake of HCl (reversible or irreversible) by water ice clouds.

Another possibility to remove HCl may be heterogeneous reactions on water ice cloud particles. This process is well known on Earth, where such reactions play a role in the formation of the polar ozone hole: ClONO 2 + HCl → Cl 2 + HNO 3 , N 2 O 5 + HCl → ClNO 2 + HNO 3 , HOCl [START_REF] Solomon | Stratospheric ozone depletion: A review of concepts and history[END_REF]. However, even if those reactions could take place efficiently on Mars, the produced chlorinated compounds would be readily converted to Cl radicals by photolysis (on timescales of a few tens of minutes), and subsequently create HCl by Reaction (1). Chlorine can also be removed by the formation of perchlorates (ClO 4 -

+ HCl → Cl 2 + H 2 O
). The uptake of ClO on ice particles was experimentally observed to lead to the release of chlorine dioxide (OClO) upon ice evaporation [START_REF] Mckeachie | Observation of a Heterogeneous Source of OClO from the Reaction of ClO Radicals on Ice[END_REF]. Although the enhanced concentrations of manuscript submitted to GRL OClO could favor the production of HClO 4 [START_REF] Catling | Atmospheric origins of perchlorate on Mars and in the Atacama[END_REF][START_REF] Wilson | Perchlorate formation on Mars through surface radiolysis-initiated atmospheric chemistry: A potential mechanism[END_REF] and the permanent removal of chlorine from the gas phase, this process should be much too slow to explain the rapid loss of HCl.

It remains difficult to explain the disappearance of HCl after L s =320°. Interestingly, the time of this sudden HCl loss corresponds to the onset of the regional storm (Fig. 2) that is annually repeated and recognized as "c" dust storm. This raises the possibility that this dust storm acts as the sink for HCl. In this respect, note that mineral dust was found to be the seat of HCl removal in the Earth's atmosphere [START_REF] Sullivan | Mineral dust is a sink for chlorine in the marine boundary layer[END_REF]. However, these arguments require robust dust-HCl interactions that are not yet established by laboratory studies, and so need to be addressed in the future.

Conclusions

In this paper, we present the full dataset of vertically resolved HCl detections derived from the TGO/NOMAD measurements taken between April 2018 and December 2020, which covers southern summer periods in MY 34 and MY 35. The main results are as follows:

1. We find enhancement of HCl at the southern summer period (L s =240-320°) both in MY 34 and MY 35. It suggests that the global dust storm (which occurred only in MY 34) is not necessary for the formation of HCl.

2. Both in MY 34 and MY 35, a larger abundance of HCl is observed at the southern high latitudes where water vapor sublimates from the southern polar cap.

3. We find that the vertical distributions of HCl and water vapor are strikingly similar in shape. It suggests that once HCl is introduced in the atmosphere, it follows water vapor including uptake in water ice clouds.

4. We show that the lifetime of HCl in the Mars atmosphere is too long to explain the observed rapid loss of HCl in the Mars atmosphere around Ls=320°, which thus requires a strong sink of HCl independent from the photochemical loss.

Acknowledgments, Samples, and Data

ExoMars is a space mission of the European Space Agency and Roscosmos.

Figures

  Figures 2a-c show the season-latitude maps of the maximum HCl volume mixing ratio below 30 km observed by NOMAD in MY 34 (Fig. 2a), MY 35 (Fig. 2b), and plotted together (Fig. 2c). As shown in Korablev et al. (2021), a few ppbv of transient HCl was detected in the Mars atmosphere. In MY 34, two strong dust storms occurred on Marsthe global dust storm at

Figure 3

 3 Figure 3 shows several representative vertical profiles of HCl, water vapor, and aerosol extinction observed during the same occultation. The seasonal variations of the HCl vertical distributions measured in MY 34 and 35 are quite similar. At northern mid-latitudes (50°-60°N)

Fig. 4a -

 4a Fig.4a-bshows the HCl vmr against scaled water vapor vmr for all the HCl detections below 25 km. The water vapor mixing ratios in this figure were scaled for each profile by the ratio of the partial column densities of HCl and water vapor below 25 km (see Supporting Information, SI, Section S1, for more details). This scaling process comes down to shifting the upper and lower xaxes in Fig.3such that the lower part of the HCl and H 2 O profiles are overlapping, and so allows us to evaluate the similarity in the shapes of the vertical distributions. The partial column densities were calculated using collocated GEM-Mars temperature and pressure profiles[START_REF] Daerden | Mars atmospheric chemistry simulations with the GEM-Mars general circulation model[END_REF][START_REF] Neary | Explanation for the increase in high altitude water on Mars observed by NOMAD during the 2018 global dust storm[END_REF] for the altitudes where HCl was detected. It was preferred to work with partial column densities rather than the ratio of the vmr at one specific

Figure 1 .

 1 Figure 1. Examples of the NOMAD-SO spectra with signatures of HCl: (a) measurements in the diffraction order 129 performed on 2nd January 2019 (L s =317.4°, Latitude=72°S, Longitude=46°W); (b) measurements in the diffraction order 130 performed on 8th November 2018 (L s =284.5°, Latitude=63°S, Longitude=149°E). The best-fit synthetic spectra are shown by

Figure 2 .Figure 3 .Figure 3 .

 233 Figure 2. (a-c) Solar longitude -Latitude map of the maximum HCl mixing ratio (ppbv) below 30 km in (a) MY 34, (b) MY 35, and (c) MY34-35 plotted together. Only 3-σ detections are shown here. The gray points show observations corresponding to dust top altitude greater than 25 km, for which it is generally difficult to perform robust retrievals. The background color maps show the column-integrated water vapor density (pr-µm) obtained by MGS/TES for Mars year 26 (Smith, 2002; 2006) and the column-integrated dust opacity at 9.3 µm normalized for 610 Pa for MY 31 (b) and MY 34 (a) (Montabone et al., 2020). (d) Geographical distribution of HCl below 30 km in the seasonal range between L s =240° and 320°.

  Participation in the TGO/NOMAD Investigation of Trace Gases on Mars". The IAA/CSIC team acknowledges financial support from the State Agency for Research of the Spanish MCIU through the "Center of Excellence Severo Ochoa" award for the Instituto de Astrofísica de Andalucía (SEV 2017 0709). This work was supported by the Belgian Fonds de la Recherche Scientifique FNRS under grant numbers 30442502 (ET_HOME) and T.0171.16 manuscript submitted to GRL (CRAMIC) and Belgian Science Policy Office BrainBe SCOOP and MICROBE Projects. S. A. is "Chargé de Recherches" at the F.R.S. FNRS. U.S. investigators were supported by the National Aeronautics and Space Administration. We acknowledge support from the Institut National des Sciences de l'Univers" (INSU), the "Centre National de la Recherche Scientifique" (CNRS) and "Centre National d'Etudes Spatiales" (CNES) through the "Programme National de

	The NOMAD experiment is led by the Royal Belgian Institute for Space Aeronomy (IASB BIRA), assisted by Co PI teams from Spain (IAA CSIC), Italy (INAF IAPS), and the United Kingdom (Open University). This project acknowledges funding by the Belgian Science Policy Office, with the financial and contractual coordination by the European Space Agency Prodex Office (PEA 4000103401 and 4000121493), by the Spanish MICINN through its Plan Nacional and by European funds under grants PGC2018 101836 B I00 and ESP2017 87143 R (MINECO/FEDER), as well as by UK Space Agency through grants ST/V002295/1, ST/V005332/1 ST/S00145X/1 ST/S00145X/1 and ST/T002069/1, and Italian Space Agency through grant 2018 2 HH.0. This work was supported by NASA's Mars Program Office under The NOMAD data used in this study are available from ESA's Planetary Science Archive WBS 604796, "Planetologie". at https://archives.esac.esa.int/psa/#!Table%20View/NOMAD=instrument.

When HCl and H 2 O profiles are similar in shape, their points will fall on or near the identity line.

The color scale represents the latitude of the observations. (c-d) HCl/H 2 O ratio for all profiles.

The color scale represents solar longitude (L s ) of the observations.


A Spica

J Roche

L Armault

M Horville

J Rolet

Evolution of model roughness on quasi-pure aluminum during plasma electrolytic polishing

The present paper studies the material removal evolution, during Plasma Electrolytic Polishing (PeP), on mode! roughnesses prepared on 1050 aluminum alloy (AA), i.e. a quasi-pure aluminum, used as a mode! substrate.

Plasma electrolytic polishing Mode! roughness

The resuils obtained in this srudy confirm the significance of the applied voltage on aluminum removal. In particular, the minimal voltage to be applied to generate the plasma is seen to be above 230 V. Meanwhile, at a higher voltage of 300 V, high quality polishing is obtained. Using this technique, rolling streaks are eliminated rapidly without the workpiece's geometry being adversely affected.

Finally, roughness evolution was accurately srudied on a series of samples with various well-defined levels of mode! roughness generated by microindentation. The srudy shows that only a few minutes of processing are needed to obtain a significant reduction in roughness, even representing up to 70%. Results show also that roughness valleys do not undergo material removal at ail as long as peak to valley height remains high enough. As the process goes on, roughness (i.e. mainly peaks) is reduced and valleys become submitted to material removal as well.

Introduction

Various polishing processes can be applied for metal parts, with chemical, electrochemical and mechanical polishing now being the most conunonly used, especially for parts with large smfaces. Sandblasting provides a rapid and easy solution but is not effective in polishing parts with complex shapes. Meanwhile, such more or Jess complex shapes can be treated by liquid phase polishing (i.e. electropolishing and chemical polishing). However, these two processes have two main drawbacks. Firstly, this method is tin1e consuming, sometimes lasting several hours. Secondly, the electrolytes used are dangerous for operators and repre sent an environmental hazard.

Polishing techniques can also be combined to treat the sanie work piece. However, the present paper focuses on an innovative smface treatment known as Plasma Electrolytic Polishing (PeP). PeP is based on the formation of a thin vapor skin resulting from the vaporization of the aqueous electrolyte at the metal surface. Applied voltages are suffi ciently high to induce the ionization of the elements present in this vapor skin and thus a plasma is fom1ed, inducing material removal and thus polishing of the pait.

PeP can be applied to produce very smooth and shiny surfaces within

• Corresponding author.

E-mail address: arurault@chinlie.ups-tlse.fr (L. Arurault).

a short treatment time (a few seconds to 10 min). In addition, the electrolytes used for PeP treatment generally comprise at least 80% by mass of water, making it an advantageous alternative to the use of conventional electropolishing electrolytes. Finally, the technique ap pears not to encounter any technical problems in uniformly treating parts with complex shapes.

The technology was already studied on various metals and alloys, e. g. stainless steels [START_REF] Nevyantseva | The influence of vapor-gaseous envelope behavior on plasma electrolytic coating removal[END_REF][START_REF] Gupta | Electrolytic plasma technol ogy: science and engineering-an overview[END_REF][START_REF] Podhor-..Ky | The poosibilities of plasma polishing of he steel dinl.0570 in electrolyte[END_REF][START_REF] Yerokhin | Pulse current plasma =i..--ted electrolytic cleaning of AIS! 4340 steel[END_REF][START_REF] Yerokhin | Charge transfer mechanisms underlying contact glow discharge electrolysis[END_REF][START_REF] Wang | Optimization of processing parameters for electrolysis and plasma po lishing[END_REF][START_REF] Ln | Investigation of the influence of plasma-dectrolytic processing on the surface of austenitic chromiwn-nickel steels[END_REF][START_REF] Vana | Swfuce properties of the stainless steel XI0 CrNi 18/10 after application of plasma polishing in electrolyte[END_REF][START_REF] Liang | Localized surface modification on 1018 low-carbon steel by electrolytic plasma process and its impact on corrosion behavior[END_REF][START_REF] Kalenchukova | About changes in the chemical composition of the electrolyte in the process of electrolytyc-plasma treatment of mate[END_REF][START_REF] Parfenov | Electric field effect on surface layer removal during electrolytic plasma polishing[END_REF][12][13][14][15][16][17][18]CoCr alloys [22,23),copper [24),brass [23),magnesium [25) and . Some papers have investigated this process with respect to aluminum substrates. Previous studies thus used unspecified aluminum substrates for Duradji et al. [24,27), AA 2017 for Kaputkin et al. [16), and other alloys: AMG (i. e. AA 5XXX), D-1 (i.e. AA 2098), D-16 (i.e. AA 2024) and D16P (i.e. AA 2094), AMTs (i.e. AA 3XXX), AD33 (i.e. AA 6061), AD35 (i.e. AA 6082) [28). Therefore, previous studies about Aluminum PeP, were performed on aluminum alloys, i.e. multiphased materials showing different initial roughness.

Moreover, according to Duradji [24,27), the applied voltage is the most significant paran1eter since it controls PeP initiation and de tem1ines the final quality of polishing. Their results show in particular that there is an optinmm voltage range (300-350 V for unspecified https:/ /doi.org/10.1 Ol 6/j.surfcoat.2021.127839

Heights peak/valley

(a)

Steps �z valleys and peaks aluminum) below which the plasma is difficult to initiate and the vapor skin is unstable around the elect: rode. Above this voltage range (> 400 V), polishing quality tends to become degraded, with an increase rather than a reduction in roughness.

(b)

Non-polished area

The present study's intention is to study evolution of mode! rough ness profiles prepared on a quasi-pure aluminum substrate (i.e. AA 1050, with 99.5 wo/o Al minimum), in order to avoid the influence of composition inhomogeneity and multiphases microstructures and to finally study precisely aluminum removal during the PeP process.

First, we study, in our expertmental conditions, the influence of the applied voltage on both plasma stability (via current response) and polishing quality (i.e. smface state and geomett" ic integrity). Secondly, we propose an original method in order to create a mode! roughness on quasi-pure aluminum substrate. To do so, a row of five consecutive pyrantidal indents of contt• olled depth and geometty are created by Vickers indentation, effectively leading to a series of ''valleys" and "peaks" with precisely contt•olled topography. Using 3D inte1ferometty, this mode! roughness allows to follow precisely where and when ma terial is ren1oved during PeP process, depending purely on the initial smface topography.

Experimental

Material and process

The substrate material was 1050 aluminum alloy (AA). Its chemical composition in weight percent is <0.40%Fe, < 0.25%Si, < 0.07%Zn, < 0.05%Cu, < 0.05%Mg, < 0.05%Ti, < 0.05%Mn while Al accounts for the remainder. Three types of substrate were used: wires (500 μm diameter; 0.20 to 0.25 cm 2 of immersed surface), thin strips (3 mm width, 1 mm thick, 4 mm of immersed depth for a treated surface of 0.25 cm 2 ) and large strips (1 cm width, 1 mm thick, 2.5 cm of immersed depth for a treated surface of 5 cm 2 ).

The experimental setup for the PeP treatment was based on those described in the bibliography [START_REF] Podhor-..Ky | The poosibilities of plasma polishing of he steel dinl.0570 in electrolyte[END_REF]23] for the treatment of other metals. This consisted of a five-liter beaker (a single-wall glass beaker) into which three to four liters of electrolyte (oxalic acid 3w%, ammonium chloride 10w%, potassium chloride 4w% [27]) introduced and heated initially to 60 • C. The workpiece (i.e. the anode) was surrounded by two counter-electrodes, made of platinized titanium grids (160*120 mm, i.e. about 115 cm 2 ), located at 4 cm on both sides. Finally, the generator used was a TDK lambda 1500 model, which can deliver a maximum of 5 A and 300 V for a power of 1500 W. Experiments were conducted in the 20-300 V voltage range, while processing times ranged from 15 s to 10 min.

Characterizations

Surface views were taken using either optical microscopy (Keyence 3D VHX-1000) or scanning electron microscopy (JEOL JSM-6510LV).

Surface roughness was studied by means of an interferometric 3D microscope in Vertical Scanning Interferometry (VSI) mode, allowing surfaces with significant height discontinuities to be measured. The equipment used was an S-Neox microscope from Sensofar with a ×10 lens (scanned surface 1754 × 1320 μm) or ×50 (scanned surface 351 × 264 μm) of the Mirau type. The norm ISO25178 determined how measurements of roughness parameters were obtained. The interferometric microscope produced digital data maps of the surface topography that usually required subsequent processing. The roughness value, Sa , was thus measured by applying two filters:

-A "Restore" filter was used to interpolate points that could not be measured. The missing "z" altitude of these points was calculated by averaging the "z" altitude of the 8 points around the vacant point. It is important to note that less than 10% of the data were thus corrected. This was the minimum requirement for the measurements to be validated. -A "Form Removal" filter allowed the tilt of the microscope stage or the curvature of the test sample to be corrected.

Having applied these two filters, the Sa roughness parameter was extracted. The Sa roughness parameter is the arithmetic mean of the absolute value for the height in a sampling area:

S a 1 A ∫ A |z(x, y) |dxdy (1)
The roughness parameters were always measured on three areas spaced by 5 mm at the center of one face of each sample.

Model roughness defects were artificially created, thus offering reproducible shapes to be studied accurately. To do so, series of peaks and valleys of different sizes were generated thanks to a Vickers microindentation machine. To create defects of different sizes, different masses (100, 400, 500 and 600 g) were applied to the Vickers type indent. The untreated part of the specimen was masked by an electrically insulating and chemically resistant polymer protection (SOCO-MASK PRT 2130). After PeP treatment, the peaks and valleys (.6.zp, .6.z,,) were measured according to the following procedure (Fig. 1 (a)): the protective mask was first removed so as to be able to measure steps between the unpo lished area and the various indentations. For each series of indentations, a first step was measured with the top of each peak .6.zp and another step was measured with the bottom of each valley . 6.zv (Fig. l(b)).

Results and discussion

Specific PeP cr eam1enc on AA 1050

A first study on AA 1050 wire was conducted to detemline the voltage range within which the plasma (and therefore the PeP treat ment) is triggered under these operating conditions. A cmve indicating the change in current density as a fonction of the applied voltage is shown in Fig. 2. The behavior is sinülar to that obtained by Kellog [29] on platinum wire and is divided into three specific zones.

In the first zone (A to C), i.e. for low applied voltages from 1 0 to 50 V, there is a peak of current that causes significant electrodissolution of the anode. Then, in the second zone (C to D) for applied voltages between 50 and 220 V, there are current fluctuations, associated with a partial for mation of the vapor skin, with the anode showing zones still in contact with the elect:rolyte. In zone (D to E), between 230 and 300 V, the plasma extends along the length of the wire and the current now stable is about 0.1 A. The voltage suited to instigate PeP thus seen� to be abov; 230 V. In addition, the vapor skin thickness is estimated to be about 250 ± 22 µm looking at a number of iniages taken during the t: reatment.

The influence of the applied voltage was then studied on rolled sheet, with the n•eated surface being 0.25 cni2. The first optical obse1vations (Fig. 3) show that polishing is of poor quality at an applied voltage of 200 V, with the geometty and sm f ace condition of the sample being totally degraded. At 250 V, the rolling stt• eaks are eliminated and enhanced brightness is found. However, the geometty is significantly in1pacted, with considerable Joss of niaterial and a rounding of the an gles. As for 300 V, the surface condition was in1proved with the disap pearance of rolling streaks and few modifications to angular geometly.

The 3D profiles for these san1e surfaces (Fig. 4) also show modifi cation of angles at 200 and 250 V. Moreover, these profiles clearly confim1 the absence of rolling stt• eaks after 3 nlÎil of polishing at 250 V and 300 V. The SEM in1ages of the surface before and after polishing at 300 V are presented in Fig. 5. These pictures also confüm the disap pearance of rolling streaks as from 30 s of polishing (Fig. Sa andb ). Some defects (i.e. micromenic holes) remain after 30 and 60 s (Fig. Sb andc), but an extended polishing time (150 s) allows a smooth surface with few defects to be obtained (Fig. Sd).

Fig. 6 shows the changes in the current density for different imposed voltages (200 V, 250 V and 300 V). The behavior observed appears similar to that observed in Wang's study [30]. In general, the intensity decreases as the applied voltage increases. At 200 V, the current remains unstable throughout the duration of treatment. This is due to the fact that the vapor skin is unstable and the electrolyte periodically comes into contact with the anode (see the C-D portion of the curve (Fig. 2)). At 250 and 300 V however, the current remains stable. It seems that for these two voltages, the vapor skin, and thus the plasma, is stable and covers the entire anode surface.

The study then went on to consider the influence of the PeP processing time at 300 V. For an initial roughness of Sa 2.61 ± 0.25 μm (obtained by prior abrasion with 180 SiC paper), the roughness rapidly decreases monotonously with the processing time (Fig. 7), to reach a final value of about 0.98 ± 0.19 μm, corresponding to a roughness reduction of 62%.

To summarize, experiments on AA 1050 wire and rolled sheet demonstrated the relevance of the applied voltage with respect to PeP quality, and a minimum plasma initiation voltage above 230 V was demonstrated. Below this voltage, no effective polishing was observed on the aluminum alloy, while with applied voltages of 250 and 300 V, the PeP process was successful.

Evolution of model roughness

The indentation profiles before PeP obtained by optical interferometry are shown in Fig. 8, with the x-axis and y-axis scales being identical for all these profiles so that these different defects can be readily compared. For an applied mass of 100 g (the lowest), the peak/valley mean height (Δh) is 9.8 μm, while for the highest applied mass (600 g) Δh is 24.1 μm.

For PeP performed at 300 V, the change over time (15, 30, 45 and 60 s) of the indentations profile for an applied mass of 100 g is shown in Fig. 9. Profiles corresponding to the other masses can be found in Figs. [START_REF] Kalenchukova | About changes in the chemical composition of the electrolyte in the process of electrolytyc-plasma treatment of mate[END_REF][START_REF] Parfenov | Electric field effect on surface layer removal during electrolytic plasma polishing[END_REF]12). On all these profiles, it emerges that the tops of the peaks become rounded after 30 s of PeP. Furthermore, after 60 s of diminishes monotonously over time for all defects. For the largest defects (m applied 400 to 600 g), the decrease is 53, 50 and 44% after 60 s of PeP. For the same treatment time, the reduction reaches up to 70% of the height of small defects (m applied 100 g). PeP thus allows "small" defects to be polished faster than "large" defects.

In addition, measurements of steps between peaks and valleys (Fig. 14) show that the peaks of the various defects undergo a monotonous material removal over time. Removal of material from the peaks (Fig. 14) is comparable for the four defect sizes, since after 60 s of PeP, step values Δz peak were very close. However, material removal behavior is not the same for the valleys (Fig. 14). For large defects (m applied 400 to 600 g), the valley steps change little over time during PeP. However, for smaller defects (m applied 100 g), the step Δz valley increases significantly with the PeP duration. It is readily observable on the profiles for m applied 600 g (Fig. 12) that the geometry of the peaks is more greatly impacted over time, as compared with the geometry of the valleys.

If it is clear that material removal is always faster on peaks than on valleys, inducing indeed a polishing effect over time, those results show that valleys remain entirely unaffected only beyond a certain depth, at least on those model pyramidal shaped defects and on quasi-pure aluminum alloy. This conclusion may arguably not be extended to multiphases alloys such as AA2024 where composition inhomogeneity can have an important impact on material removal [START_REF] Yerokhin | Charge transfer mechanisms underlying contact glow discharge electrolysis[END_REF][START_REF] Ln | Investigation of the influence of plasma-dectrolytic processing on the surface of austenitic chromiwn-nickel steels[END_REF][START_REF] Vana | Swfuce properties of the stainless steel XI0 CrNi 18/10 after application of plasma polishing in electrolyte[END_REF]16]. Furthermore, the pyramidal shape of those defects, even if providing interesting information on material removal during PeP, are quite different from the roughness shape observed on materials being obtained by either conventional or additive manufacturing. Nevertheless, those results show that, depending on experimental conditions, PeP can induce, to a lower extend, material removal in the valleys of roughness, even if the plasma discharge is triggered on peaks where the electric field is higher.

Conclusion

PeP experiments were performed on a quasi-pure aluminum (i.e. AA 1050), confirming the significance of the treatment time and of the applied voltage. Under these experimental conditions, the voltage to be applied to generate a plasma was seen to be above 230 V.

At a polishing voltage of 300 V, the polishing is of good quality, meaning that rolling streaks generated during rolling are rapidly eliminated without any modification to the workpiece's geometry. Following this, material removal was studied on model roughness of different sizes generated by microindentation. In particular, the results show that roughness peaks undergo material removal while the deepest valleys remain unaffected by the process. As the process go on though, peak to valley height decreases and valleys start to undergo material removal as well. A useful follow-up to the present study will be to investigate polishing of parts from additive manufacturing or those with more complex shapes.

CRediT authorship

Fig. 1 .

 1 Fig. 1. Diagrams of the different measurement parameters made on the indentations: (a) peak to valley heights; (b) step heights between an unpolished area and the top of the peaks or the bottom of the valleys.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Optical microscopy images of samples after 3 min of polishing at three different voltages.

FigFig. 6 .

 6 Fig. S. SEM surface views of samples (0.25 cm 2 ) (a) raw, and after (b) 30 s, (c) 60 s and (d) 150 s of polishing at 300 v.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Evolution of surface roughness parameter S a with polishing time at 300 V on samples (5 cm 2 ) with an initial roughness increased to 3 μm.

  treatment, defects of up to 10 μm in height are almost entirely removed by the PeP under such conditions. Fig.13summarizes the peak/valley height (Δh) reduction for all sizes of defects in relation to polishing time. This peak/valley height

Fig. 9 .

 9 Fig. 9. Evolution of indentation shape with polishing duration at 300 V for Vickers indentations obtained with m = 100 g.

Fig. 10 .

 10 Fig. 10. Evolution of indentation shape with polishing duration at 300 V, for Vickers indentations with m = 400 g.

  contribution statement A. Spica: Conceptualization, Methodology, Investigation, Writingoriginal draft, Visualization. J. Roche: Conceptualization, Methodology, Validation, Investigation, Writingoriginal draft, Writingreview & editing, Supervision, Visualization. L. Arurault: Conceptualization, Methodology, Validation, Writingoriginal draft, Writingreview & editing, Supervision, Visualization, Project administration, Funding acquisition. M. Horville: Investigation, Visualization. J. Rolet: Conceptualization, Project administration, Funding acquisition.

Fig. 11 . 10 • 2 0 10 • 10 • 10 • 10 • 10 •Fig. 12 .

 11102101010101012 Fig. 11. Evolution of indentation shape with polishing duration at 300 V, for Vickers indentations with m = 500 g.

ÊFig. 13 .Fig. 14 .

 1314 Fig. 13. Evolution of peak to valley height (.6.h) for different indentation sizes as a function of the polishing duration.

Acknowledgments

These results were obtained at the CIRIMAT laboratory at Toulouse III University under the research project "AFTER ALM" headed and financially supported by the "Institut de Recherche Technologique Matéiiaux, Métallurgie et Procédés" (IRT M2P).

Declaration of competing interest

The authors declare that they have no known competing financial interests or persona! relationships that could have appeai• ed to influence the work reported in this paper.


Hugo Gimbert
email: hugo.gimbert@labri.fr

Florian Horn
email: florian.horn@liafa.jussieu.fr

Solving Simple Stochastic Games ⋆

We present a new algorithm for solving Simple Stochastic Games (SSGs), which is fixed parameter tractable when parametrized with the number of random vertices. This algorithm is based on an exhaustive search of a special kind of positional optimal strategies, the f-strategies. The running time is O( |VR|! • (log(|V |)|E| + |p|) ), where |V |, |VR|, |E| and |p| are respectively the number of vertices, random vertices and edges, and the maximum bit-length of a transition probability.

Our algorithm improves existing algorithms for solving SSGs in three aspects. First, our algorithm performs well on SSGs with few random vertices, second it does not rely on linear or quadratic programming, third it applies to all SSGs, not only stopping SSGs.

Introduction

Simple Stochastic Games (SSGs for short) are played by two players Max and Min in a sequence of steps. Players move a pebble along edges of a directed graph (V, E). There are three type of vertices: V Max is the set of vertices of player Max, V Min the set of vertices of player Min and V R the set of random vertices. When the pebble is on a vertex of V Max or V Min , the corresponding player chooses an outgoing edge and moves the pebble along it. When the pebble is on a random vertex, the successor is chosen randomly according to some fixed probability distribution: from vertex v ∈ V R the pebble moves towards vertex w ∈ V with some probability p(w|v) and the probability that the game stops is 0, i.e. w∈V p(w|v) = 1. An SSG is depicted on Figure 1, with vertices of V Max represented as , vertices of V Min represented as , and vertices of V R represented as ♦.

Player Max and Min have opposite goals, indeed player Max wants the pebble to reach a special vertex t ∈ V called the target vertex, if this happens the play is won by player Max. In the opposite case, the play proceeds forever without reaching t and is won by player Min. For technical reasons, we assume that t is a vertex of player Max and is absorbing. Strategies are used by players to choose their moves, a strategy tells where to move the pebble depending on the sequence of previous vertices, i.e. the finite path followed by the pebble from the beginning of the play. The value of a vertex v is the maximal probability with which player Max can enforce the play to reach the target vertex. When player Max, respectively player Min, uses an optimal strategy he ensures reaching the target with a probability greater, respectively smaller, than the value of the initial vertex. We are interesting in solving SSGs, that is computing values and optimal strategies.

Existing algorithms for solving SSGs. The complexity of solving SSGs was first considered by Condon [START_REF] Condon | The complexity of stochastic games[END_REF], who proved that deciding whether the value of an SSG is greater than 1 2 is in NP ∩ co-NP. The algorithm provided in [START_REF] Condon | The complexity of stochastic games[END_REF] consists in first transforming the input SSG in a stopping SSG where the probability to reach a sink vertex is 1. The transformation keeps unchanged the fact that the initial vertex has value strictly greater than 1 2 but induces a quadratic blowup of the size of the SSG. The algorithm then non-deterministically guesses the values of vertices, which are rational numbers of linear size, and checks that these values are the unique solutions of some local optimality equations.

Three other kinds of algorithms for solving SSGs are presented in [START_REF] Condon | On algorithms for simple stochastic games[END_REF]. These algorithms require transformation of the initial SSG into an equivalent stopping SSG and are based on local optimality equations. First algorithm computes values of vertices using a quadratic program with linear constraints. Second algorithm computes iteratively from below the values of the SSGs, and the third is a strategy improvement algorithm à la Hoffman-Karp. These two last algorithms require solving an exponential number of linear programs, as it is the case for the algorithm recently proposed in [START_REF] Somla | New algorithms for solving simple stochastic games[END_REF].

Finally, these four algorithms suffer three main drawbacks.

First, these algorithms rely on solving either an exponential number of linear programs or a quadratic program, which may have prohibitive algorithmic cost and makes the implementation tedious.

Second, these algorithms only apply to the special class of stopping SSGs. Although it is possible to transform any SSG into a stopping SSG with arbitrarily small change of values, computing exact values this way requires to modify drastically the original SSG, introducing either |V | 2 new random vertices or new transition probabilities of bit-length quadratic in the original bit-length. This also makes the implementation tedious.

Third, the running time of these algorithms may a priori be exponential whatever be the number of random vertices of the input SSG, including the case of SSGs with no random vertices at all, also known as reachability games on graphs. However it is well-known that reachability games on graphs are solvable in linear time.

Notice that randomized algorithms do not perform much better since the best randomized algorithms [START_REF] Ludwig | A subexponential randomized algorithm for the simple stochastic game problem[END_REF][START_REF] Halman | Simple stochastic games, parity games, mean payoff games and discounted payoff games are all LP-type problems[END_REF] The key point of our algorithm is the fact that optimal strategies may be looked for in a strict subset of positional strategies, called the class of f -strategies. The f -strategies are in correspondence with permutations of random vertices. Our algorithm does an exhaustive search of optimal f -strategies among the |V R |! available ones and check their optimality. Optimality is easy to check, it consists in computing a reachability probability in a Markov Chain with V R states, which amounts to solving a linear system with at most |V R | equations.

Comparison with existing work. We improve existing results by three aspects: our algorithm performs better on SSGs with few random vertices, it is arguably much simpler, and we provide new insight about the structure of optimal strategies.

Our algorithm performs much better on SSGs with few random vertices than previously known algorithms. Indeed, its complexity is O( |V R |! • (log(|V |)|E| + |p|) ), hence when there are no random vertices at all, our algorithm matches the usual quadratic complexity for solving reachability games on graphs. When the number of random vertices is fixed, our algorithm performs in polynomial time, and on the class of SSGs such that

|V R | ≤ |V Max | + |V Min | our algorithm is sub-exponential.
Whereas the complexity is optimal when there are no random vertices, this is no more the case when there are no vertices for player Max or Min. In that case, there exists polynomial time algorithm, whereas the complexity of our algorithm remains exponential in the number of random vertices.

Our algorithm is arguably simpler than previously known algorithms. Indeed, it does not require use of linear or quadratic programming. Although linear programs can be solved in polynomial time [START_REF] Kachiyan | A polynomial time algorithm for linear programming[END_REF][START_REF] Renegar | A polynomial-time algorithm, based on newton's method, for linear programming[END_REF], this requires high-precision arithmetic. By contrast, our algorithm is very elementary: it enu-merates permutations of the random vertices and for each permutation, it solves a linear system of equations.

Our algorithm is also simpler because it applies directly to any kind of SSGs, whereas previously known algorithms require the transformation of the input SSG into a stopping SSG of quadratic size. As a consequence, we can use the same algorithm for solving other types of infinite duration game; this is ongoing work.

The full paper is available online [START_REF] Gimbert | Solving simple stochastic games with few random vertices[END_REF].

Fig. 1 .

 1 Fig. 1. A Simple Stochastic Game.

  known so far run in sub-exponential expected time e O( √ n) . Our results. In this paper we present an algorithm that computes values and optimal strategies of an SSG in time O( |V R |! • (log(|V |)|E| + |p|) ), where |V R | is the number of random vertices, |V | is the number of vertices and |p| is the maximal bit-length of transition probabilities.

⋆ This research was partially supported by french project ANR "DOTS".

Acknowledgments We thank an anonymous referee for his very insightful and helpful comments.