Bao Yu

Mohamed Selkti

Janick Ardisson

Marie-Isabelle Lannou

Geoffroy Sorin

Access to conjugated enynes via allenyl silver formation/cyclization/decarboxylation reaction catalyzed by silver carbonate(I)

Herein, we develop a novel and straightforward access to cyclic conjugated enynes catalyzed by silver carbonate/DBU from readily available substrates with good yields. The reaction is easy to set-up, broad in scope and can also be conducted in a onepot fashion from easily accessible substrates through a sequence Michael addition reaction/cyclization. Based on our previous works, the mechanism proposed would involve an allenyl silver intermediate and decarboxylation reaction.

Since the last two decades, homogeneous silver salts catalysis has renewed a great interest in synthetic chemistry highlighted by a broad variety of reactions reported in the literature. [START_REF]For reviews and book, see for example: (a) Muñoz, M. P. Silver and platinum-catalysed addition of O-H and N-H bonds to allenes[END_REF] Silver salts possess versatile activities and were, for example, commonly used either for their ability to activate heteroatoms thanks to their Lewis acidity, 2 their oxidizing properties or for their halogenophilicity in metal catalyzed C-C bond formation. 3,4 In addition, due to its electronic configuration [Kr] 4d 10 5s 1 in group 11, silver belongs to "coinage metals" and therefore, possess similar activity as gold towards the π-activation of carbon-carbon multiple bonds. However, even if silver is cheaper than gold, it is almost as scarce and therefore, performing catalytic reactions with such metals is a crucial point regarding the depletion of the natural resources on earth. 5 Among the different types of activation by silver salts, we can underline those targeting alkynes, which have been reviewed or published in the literature these last years. [START_REF] Fang | A. Silver-mediated oxidative C-H/C-H functionalization: A strategy to construct polysubstituted furans[END_REF] Remarkably, silver π-acidity allowed very interesting and promising transformations from alkynes through different C-C triple bond activation (Scheme 1). Scheme 1. Different pathways possible from alkyne complexed by silver salts Hence, from alkynes, the formation of silver acetylides, 7 vinyl silver or silver carbenes can be envisioned. [START_REF] Bourderioux | Phosphine-Triggered Selectivity Switch in Silver-Catalyzed o-Alkynylbenzohydroxamic Acid Cycloisomerizations[END_REF]9 In addition, we recently highlighted the propensity of silver oxide(I) to isomerize terminal or silylated alkynes into allenyls silver in the synthesis of 2,3-dihydrofurans derivatives. [START_REF] Yu | I) Oxide-/DBU-Promoted Synthesis of Dihydrofuran Units through Allenyl Silver Formation[END_REF] Previous works reported by Zhang et al. also mentioned this type of isomerization catalyzed by silver salts from propargylamides. 11 Nevertheless, the formation of allenyl silver intermediates are really scarce and therefore very little attention was paid towards their reactivities. To the best of our knowledge, they have mainly been explored by the Vermeer group through the intermolecular addition of allenyl silver species onto few electrophiles. [START_REF] Westmijze | Application of allenylsilver(I) compounds in organic synthesis. A simple route to substituted allenes[END_REF] However, no intramolecular reaction involving allenyl silver species has been reported prior to our work on dihydrofuran derivatives (Scheme 1). [START_REF] Yu | I) Oxide-/DBU-Promoted Synthesis of Dihydrofuran Units through Allenyl Silver Formation[END_REF] In the course of our previous works on polysubstituted furan synthesis, 13 we screened substrates with different chain length bearing an alkyne moiety in order to reach 5-or 6-membered rings (Scheme 2). Astonishingly, depending on the substitution pattern of the ketone, different behaviors were observed concerning the cyclization. Indeed, with aromatic ketones, the expected bicyclic furan was obtained as the major product whereas aliphatic ketones mainly led to conjugated enynes. The formation of such cyclic cores appeared particularly interesting since only one step was required to reach complex enyne structures from a readily available malonate using smooth conditions. Moreover, the formation of an allenyl silver intermediate was clearly considered as key point of this unprecedented transformation. Intrigued by this promising result, we decided to pursue further our investigations in order to explore the scope and mechanism of this new transformation.

Scheme 2. Preliminary works

In a preliminary survey, behaviour of the malonate type compound 1a was studied in acetonitrile in the presence of a combination of Ag2O/DBU. After increasing the temperature to 90 °C, we were pleased to find a complete conversion of 1a into the conjugated enyne 2a over 12 h. The structure was confirmed by a single-crystal X-ray diffraction analysis of p-nitrobenzoyl derivative 3a, 14 obtained after reduction of ester and subsequent esterification of the resulting alcohol with p-nitrobenzoyl chloride (93% yield over two steps, Scheme 3).

Scheme 3. Structure determination

Then, we started the optimization of the reaction conditions, again by choosing compound 1a as the model substrate (Table 1). * 1 H-NMR yield was calculated by using 1,3,5-trimethoxybenzene as internal standard.

A rapid survey of silver catalysts demonstrated that silver carbonate(I) was the best candidate for this transformation (87% yield, Entry 1). Silver oxide(I) provided the compound 2a in 67% yield (Entry 2) whereas silver phosphate led to 60% yield (Entry 3). With other silver salts, conversions into product 2a ranged from insignificant to low with respectively AgOTf (8%, Entry 4) and AgNTf2 (23%, Entry 5). The role of the base appeared crucial since K2CO3 (Entry 6), pyrrolidine (Entry 7), NEt3 (Entry 8) and i-Pr2NEt (Entry 9) afforded poor conversions (respectively 36%, 14%, 31% and 30%). On the contrary, various solvents were well tolerated including THF (85%, Entry 10), DMF (67%, Entry 11) and toluene (68%, Entry 12). However, we noticed a slight decrease of the yield by using protic solvent like MeOH (44%, Entry 13). Finally, control reactions highlighted the mandatory combination of both reagents since no reaction occurred without silver carbonate, (0%, Entry 14) and a dramatic drop of yield was observed when the reaction was conducted without DBU (9%, Entry 15). To our delight, similar good yields were obtained by reducing the loading of Ag2CO3 to 0.25 equivalent (Entry 16, 86%) or to 0.125 equivalent (Entry 17, 86%). However, a collapse of yield was observed when 0.0625 equivalent of silver carbonate(I) was employed (Entry 18, 47%). At this stage, we tried to add oxidative agent in order to restore the catalytic species. Unfortunately, whatever the oxidant tested (O2, PhI(OAc)2, CeO2 or K2S2O8), we never succeed in improving the yield. 15 As a conclusion, the best combination for this reaction turned out to be: Ag2CO3 (0.125 eq), DBU (1 eq), CH3CN, under reflux over 12h.

Scheme 4. Scope of the reaction

With the optimized reaction conditions in hand, we subsequently explored the substrate scope of our method by modifying the substitution pattern on the ketone as well as on the malonate-type part (Scheme 4). For substrates bearing a -diester moiety, n-pentyl (1b) or cyclopentyl (1c) or benzyl (1d) derivatives afforded the corresponding enynes (2b, 2c and 2d) with satisfactory yields (82 to 87%), whereas aromatic ketones (R = Ph, 1e) resulted in moderate yield (2e, 45%). This drop of yield is explained by the competitive formation of the bicyclic furan (29%, isolated yield). This result is consistent with our preliminary work (scheme 2). 13 Hence, the main difference between aryl and alkyl ketones arises from the pKa values of the hydrogen in α position to the ketone. Competitive formation of the furan is more prone to occur as the pKa value of the -hydrogen diminishes; the balance between the two competitive reactions being sometimes very tight.

A similar behavior was also observed with -cyanoesters, since various alkyl ketones, comporting either methyl (1f), n-pentyl (1g) or cyclohexyl (1h) group successfully furnished the desired products (2f, 2g and 2h) (Scheme 4) in good yields (80%, 76% and 76%, respectively), while cyclization of the precursor bearing a phenyl group (1i) only resulted in moderate yield (2i, 40%). To our delight, this transformation also delivered phenylsulfonylesters derivatives (2j and 2k) with good yields (79% and 74%, respectively from precursors 1j and 1k). Interestingly, these reaction conditions were also found to be suitable for more hindered cyclopropyl ketone (1l) even if the expected product 2l was only obtained in moderate yield (44%).

Regarding the low difference of the pKa of the hydrogen in α position to the ketone in 1l and 2l, only steric factor at that position could help to discriminate 1l and 2l, which explains the lower yields obtained while carrying the cyclization on 1l. Finally, the case of longer chain systems was investigated: while submitting malonate 1m having a 6-C instead of 5-C spacer between the ketone and the alkyne moiety, only traces of the expected product 2m were detected by 1 H NMR even by increasing the loading of Ag2CO3 to 2 equivalents or heating in a sealed tube at 110 °C.

In order to set up a flexible, straightforward and more environmentally friendly process, we focused on a one-pot procedure Michael addition/cyclization. To this aim, we decided to take advantage of the use of DBU, which was previously identified by Kim and co-workers as an efficient base for promoting Michael addition reactions. 16 In our case, DBU could sequentially promote the 1,4-addition onto α,β-unsaturated ketone before being engaged in the cyclization process. To this aim, we applied our optimized conditions onto various Michael acceptors in the presence of malonates 5a-e (Scheme 5).

Scheme 5. One-pot process

To our delight, the cyclization conditions were found suitable for a one-pot process. First trial involving malonate 5a and 1phenylbut-3-en-2-one 4a afforded 2d with a similar yield (80%) as in the two-step process (83%); malonate 5a also reacted smoothly with 1-cyclhexylprop-2-en-1-one 4b to afford 2n in good yield (79%). Likewise, alkyne bearing a -cyanoester moiety 5b was able to react with methylvinyl ketone 4c and 1phenylbut-3-en-2-one 4a to afford expected compounds in good yields (2f: 74% vs. 80% for the two-steps process and 2o: 70%). Interestingly, one-pot process was also found to be appropriate in the case of -ketoester 5c comporting an hindered isopropyl ketone; hence while reacting with 4a and 4d, 2p and 2q were provided with satisfying yields (respectively 49 % and 65 %).

In the same manner, the reaction proceeded with -ketoester 5d comporting a benzylic ketone to afford 2r in 54% yield, this moderate yield can be explained by difficulties encountered during the purification step. Finally, the reaction was successfully carried out with -phenylsulfonylester 5e delivering 2s in 77% yield. As a conclusion, this one-pot process Michael addition/cyclization revealed especially efficient since on two selected examples, yields proved to be similar to those obtained in the two-step process. Moreover, this transformation evidenced a good functional group tolerance whether on the Michael acceptor (4a-e) or on the alkyne bearing a malonate-type moiety (5a-e).

On the basis of our previous work on 2,3-dihydrofuran and furans [START_REF] Yu | I) Oxide-/DBU-Promoted Synthesis of Dihydrofuran Units through Allenyl Silver Formation[END_REF]13 a plausible reaction mechanism for the cyclic enyne core synthesis is suggested in Scheme 6. 17 Scheme 6. Proposed mechanism

The first step would be the formation of the alkynyl silver 6a from alkyne 1a. Thanks to the support of DFT calculation performed in the course of the synthesis of dihydrofurans, [START_REF] Yu | I) Oxide-/DBU-Promoted Synthesis of Dihydrofuran Units through Allenyl Silver Formation[END_REF] we could assume that species 6a is in equilibrium with the allenyl silver 8a through the transitory formation of the silver alkyne complex 7a. Due to the very weak nucleophilicity of intermediate 6a, alkynyl silver could neither react in intra-nor in intermolecular fashion. In contrast, the allenyl silver 8a would preferentially attack the ketone leading to the silver alkoxide 9a. A retro-addition can reasonably be envisioned rendering this cyclization step reversible too. The hydroxylate thus generated 9a would attack one of the ester to form the highly strained lactone 10a. Release of CO2 from the tricyclic adduct 10a would be achieved by deprotonation in α-position of the alkyne and subsequent fragmentation/decarboxylation reaction. This irreversible transformation would push the equilibrium towards the formation of the six-membered ring 11a. This fragmentation step would constitute the driving force of the reaction. The newly formed silver enolate could be reprotonated by the conjugated acid of the DBU yielding the final compound 2a. Interestingly, during our scope investigation, we isolated compound 12t in 27% from isopropyl ketone 1t (Scheme 7).

Scheme 7. Proof of mechanism

At this stage, we presumed that tertiary silver alkoxide was partially synperiplanar to the ester, yielding 2t and partially antiperiplanar. In this latter case, it was unable to react with the ester moiety, preventing the second cyclization and thus interrupting the mechanism pathway to provide 12t. This observation reinforced the mechanism proposed in Scheme 6.

In summary, we have developed a new method for the synthesis of conjugated enynes by a silver carbonate-catalyzed cyclization reaction, conducted for the first time in the presence of DBU. The conditions developed herein are mild, open-air, easy to set-up and tolerate a wide variety of substituted malonatetype systems (nitrile, aliphatic or aromatic ketone, phenylsulfone). Interestingly, we succeeded to extend the conditions to a tandem one-pot process, which is more environmentally friendly and requires even less elaborated substrates, some of them being commercially available. Noteworthy, we assumed that the reaction could proceed with other functional groups such as polar functional groups, substituted alkyls or benzyls, (hetero)aryls since the only crucial point is the pKa value of the H in α position to the ketone. However, functional groups lowering this value would promote the unwanted formation of furan derivatives. Based on our previous works and results, a plausible mechanism involving an allenyl silver intermediate and a decarboxylation reaction was proposed. However, switching the reactivity towards ene-yne cycloadducts from aromatic ketones appeared more challenging since the competitive formation of furane derivatives spontaneously occurs as described in our previous work. Investigation of new conditions are currently underway in our laboratory to fix this issue.

Table 1 . Reaction optimization. Full data optimization is available in the ESI.

 1

	Entry	Silver (eq).	Base	Solvent Yield*
	1	Ag2CO3 (0.5)	DBU	CH3CN	87%
	2	Ag2O (0.5)	DBU	CH3CN	67%
	3	Ag3PO4 (0.33)	DBU	CH3CN	60%
	4	AgOTf (1)	DBU	CH3CN	8%
	5	AgNTf2 (1)	DBU	CH3CN	23%
	6	Ag2CO3 (0.5)	K2CO3	CH3CN	36%
	7	Ag2CO3 (0.5)	Pyrrolidine CH3CN	14%
	8	Ag2CO3 (0.5)	NEt3	CH3CN	31%
	9	Ag2CO3 (0.5)	i-Pr2NEt	CH3CN	30%
	10	Ag2CO3 (0.5)	DBU	THF	85%
	11	Ag2CO3 (0.5)	DBU	DMF	67%

ACKNOWLEDGMENT

B. Y. thanks the China Scholarship Council for PhD grant. We would like to thank single-cristal X-ray diffraction service from ICMMO-UMR8182 Orsay.

ASSOCIATED CONTENT Supporting Information

The Supporting Information is available free of charge on the ACS Publications website. General reaction optimization, catalytic attempts, preparation of the cyclization precursors, cyclization reactions, details on mechanism, 1 H, 13 C and 19 F NMR spectra (PDF)

AUTHOR INFORMATION Corresponding Author

Geoffroy Sorin -UMR 8030 CiTCoM, Faculté de Pharmacie, Université de Paris Cité; https://orcid.org/0000-0001-9656-0771 E-mail: geoffroy.sorin@univ-paris.fr Marie Isabelle Lannou -UMR 8030 CiTCoM, Faculté de Pharmacie, Université de Paris Cité; https://orcid.org/0000-0002-3042-2058 E-mail : marie-isabelle.lannou@univ-paris.fr

Present Addresses

Bao Yu -College of Traditional Chinese Medicine, Chongqing Medical University, Chongqing 400016, P.R.China.

Author Contributions

The manuscript was written through contributions of all authors. / All authors have given approval to the final version of the manuscript. /


S Berger

F Iglesias

P Bonnet

C Voisin

G Cassabois

J-S Lauret

C Delalande

P Roussignol

Optical Properties of Carbon Nanotubes in a Composite Material : the Role of Dielectric Screening and Thermal Expansion

Keywords:

We report on environmental effects on the optical properties of single-wall carbon nanotubes in a gelatin-based composite material designed to foster their photoluminescence. We show that the dielectric screening of excitons due to the surrounding medium is responsible for a sizeable shift of the luminescence lines which hardly depends on the tube geometry. In contrast, the temperature dependence (from 4 to 300 K) of the luminescence is clearly chirality dependent : the first and second excitonic lines shift in opposite directions with a magnitude that can be related quantitatively to a strain-induced modification of the electronic structure due to an expansivity mismatch between the nanotube and the matrix.

I. INTRODUCTION

Single-Wall Carbon Nanotubes (SWCNTs) which consist of a single enrolled graphene sheet are natural prototypes of quasi-unidimensional objects and therefore have drawn considerable attention in fields ranging from very fundamental physics to industrial applications. As for optical properties SWCNTs have demonstrated unique features such as unprecedented exciton binding energies [1,2], large nonlinear susceptibilities [3] and emission/absorption spectra that allow the assignment of each optical feature to a specific chirality [4]. SWCNTs have also been successfully integrated in electro-optical devices such as field-effect transistors for light emission or detection [5]. However, since they are intrinsically made of surface atoms, SWCNTs strongly and inevitably interact with their environment. A dramatic illustration of this singular property is the total quenching of luminescence that occurs when nanotubes aggregate into bundles [6] or lie on a substrate [7]. This has hampered the development of optical studies of nanotubes until an efficient way of producing isolated nanotubes had been proposed [8]. The most popular of these methods for bulk samples is based on a chemico-physical insulation of nanotubes encased in micelles of surfactant with possible further processing and integration in various matrices such as polymers or gelatins [9,10]. These composite materials are of technological interest since they are much easier to handle than solutions and allow to vary external parameters such as temperature, strain, electrical gating...

In this paper, we study the photoluminescence properties of SWCNTs in a composite material optimized for optical purposes. We show that even for such isolated nanotubes the environment plays a key role in their optical properties. The zero order effect is a sizeable shift of the resonances due to environment-induced dielectric screening of the Coulomb interactions together with a broadening of the lines. When cooling the sample down to cryogenic temperatures, we show that the expected intrinsic change of the electronic structure is completely masked by subtle environment-induced effects : the thermal expansivity mismatch between the nanotube and the surrounding matrix induces considerable stress which in turn modifies the electronic structure of the tube in a way that is remarkably accounted for by calculations of uniaxial strain-induced changes of the band-structure [11,12].

II. SAMPLES

The samples consist of a gelatin-based composite material doped with HiPCO SWCNTs (purchased from Carbon Nanotechnologies Inc.). Isolated nanotubes encased in micelles of surfactant are obtained following standard methods based on ultrasonication in an aqueous solution of Sodium Dodecyl Sulfate (SDS) (1 wt %) [8]. After ultracentrifugation at 200000g for 4h, the supernatant is collected. As pointed by several studies, this method gives suspensions strongly enriched in isolated single nanotubes, but the suspension still contains a non negligible amount of small bundles [10,13]. For similar samples we had estimated the relative amount of single tubes to be of the order of 10% [10]. We further incorporate commercial dehydrated gelatin in the solution at 70 o C (10 mg/ml). A little amount of this gelatin enriched solution is dropped on a quartz substrate and dried at room temperature.

III. RESULTS AND DISCUSSION

A. Photoluminescence Properties

Fig. 1 shows that gelatin-based samples are an excellent choice for optical studies of carbon nanotubes : the photoluminescence (PL) efficiency of such a composite material remains comparable to the one of the initial suspension over time whereas a simple deposition and drying of a drop of the initial suspension leads to a drastic quenching of the PL (more than one order of magnitude) when the solvent is evaporated. This is most likely due to a collapse of the micelles at high concentration which leads to reaggregation of the tubes into bundles. In contrast the gelatin network is known to remain highly hydrated [14] and therefore preserves the micelle structure and the insulation of the nanotubes. Adding some gelatin into a suspension of isolated nanotubes turns out to be one of the most efficient methods published so far to obtain a solid-state composite preserving the luminescence properties of nanotubes. Furthermore, these samples can endure several cryogenic cooling cycles without any deterioration of their optical properties. FIG. 1: Temporal evolution of the photoluminescence intensity (logarithmic scale) after deposition on a glass substrate of a drop of a suspension of SWCNTs in water (black dots) and a drop of gelatin-enriched material (red stars) at room temperature. The luminescence intensity of the gelatin-based sample remains constant over time whereas the luminescence of the suspension is reduced by one order of magnitude when the solvent is evaporated. The transient increase of the signal is likely due to a lens effect in the drying drop changing the collection efficiency of the setup.

Fig. 2 shows a typical PL excitation (PLE) map of a gelatin-based sample doped with HiPCO nanotubes. Each bright spot corresponds to the luminescence from the so-called S 11 excitonic level resonantly excited on the S 22 excitonic level of a given semi-conducting nanotube family (i.e. defined by a pair of chiral indices (n,m)). Following the assignment scheme proposed by Bachilo et al. [4], we assigne each spot to the chiral indices reported on the map. Additional chiral families are accessible in the same detection range by tuning the excitation energy to lower values (around 1.5 eV, not shown in Fig. 2).

FIG. 2: Photoluminescence intensity at 10 K of isolated HiPCO SWCNTs embedded in a gelatin matrix plotted as a function of emission and excitation energies. By comparison to the semi-empirical Kataura plot [4], the PL spots are assigned to a given SWCNT chirality.

B. Dielectric Screening

A comparison of the PL spectra of suspensions with those of gelatin-based samples brings a striking evidence of the high sensitivity of nanotubes to their environment. The data displayed in Fig. 3 show a global red shift of the PL lines of the nanotubes embedded in gelatin as compared to the ones of the nanotubes in aqueous suspension. This corresponds to a red shift of the order of 15±4 meV of the S 11 exciton level. Data on the S 22 levels were obtained by means of PLE measurements on the same species and reveal a red-shift of the same magnitude. Similar effects have been reported upon changing the solvant of NT suspensions [15,16]. We interpret this effect as a consequence of the dielectric screening of the Coulomb interaction in the nanotube through its environment due to the spill-out of the wave-function of the exciton. Two main effects drive the exciton energy i.e. the direct Coulomb interaction and the self-energy. The former tends to lower the exciton energy (binding energy) whereas the latter leads to an enhancement of the bandgap. Theoretical and experimental studies have shown that the latter dominates for SWCNTs [17][18][19]. Therefore an increase of the dielectric constant results in a red shift of the line.

Perebeinos et al. have proposed a scaling law for the binding energy of the exciton (E b ∝ -α with α = 1.4) where is an effective dielectric constant [20]. In the case of small diameter nanotubes such as the present ones (d t ∼ 0.9 nm), the main contribution to the dielectric constant is the one of the environment [20]. Lefebvre et al. have proposed to generalize this approach for the self-energy (E s ∝ -β with β = 1) [21]. Therefore the change of the exciton energy upon a relative change of the dielectric constant reads : ∆E = ∆ (αE b -βE s ).

The binding energy for nanotubes in micelles has been previously measured by two-photon PLE measurements and is of the order of 380 meV for 0.9 nm diameter nanotubes [1,2]. The self-energy correction has been estimated by fine spectroscopy analysis of a temperatureinduced bandgap shift transition and turns out to be of the order of E s ∼ 650 meV in solution [21]. From the experimental line-shift observed in Fig. 3, we deduce that the relative change in the dielectric environment (gelatin vs suspension) of the tube is of the order of +15%.

Let us examine the possible origin of this modification of the dielectric screening. The dielectric constant of interest in this problem is the one probed by the exciton wave-function, in the very local environment of the tube. In view of the large exciton binding energy in carbon nanotubes, one has to consider an optical dielectric constant rather than a static one [START_REF] Klingshirn | Semiconductor optics[END_REF]. This quantity is difficult to estimate in a complex environment involving long-chain-molecules in a disordered medium. However we can estimate upper and lower bounds in each environment. Molecular dynamics simulations by O'Connel et al. [8] show that for nanotubes in a micelle the local environment of the tube within a distance of 1 nm (typical exciton size [1]) is made of the close-packed alkyl chains of SDS molecules with a carbon density of the order of 30 nm -2 . The corresponding dielectric constant is of the order of 1.8 [START_REF] Aminabhavi | [END_REF]. Note that due to the sharp density profile, the value is presumably significantly dependent on the molecular arrangement. In the gelatin environment the molecular pressure from the matrix can induce a densification of the micelle and consequently an increase of the local dielectric constant. The molecular dynamics simulations in [8] show that a moderate rearrangement of the micelle structure can easily lead to a 10% increase in the local density. The extreme case would correspond to the dissociation of the micelles and direct enclosure of the nanotubes in the hydrated gelatin matrix. The dielectric constant of the latter is of the order of 2.3 (refractometric measurements [24]). Compared to the suspension environment this corresponds to a relative increase of the dielectric constant of the order of 25-30% in qualitative agreement with the spectroscopic estimate. From this analysis we can deduce a quantity of practical interest : the sensitivity of the exciton energy upon a change of surrounding dielectric constant : dS11 d =2 ∼ 45 meV (obtained for a typical dielectric constant of 2).

C. Line Broadening

An other consequence of the insertion of nanotubes in a gelatin matrix is the broadening of the emission lines (cf. Fig. 3). The effect is especially clear for the (10,2) and (9,4) lines, although the former is partially quenched by off resonance excitation conditions due to the shift of S 22 in gelatin. In average the line broadening is of the order of 5 to 10 meV that is up to 50% of the initial linewidth. As for S 22 transitions, our measurement is hampered by the large background and an initial linewidth of the order 100 meV. We see no significant extra-broadening within a 10 meV experimental resolution. The broadening of S 11 transitions may be related to several mechanisms. Bearing in mind that the PL lines stem from an ensemble of nanotubes, the extrabroadening can be of inhomogeneous nature : the global red shift mentioned previously corresponds to an increase of the average dielectric function of the environment of the tube but local variations can lead individual lines to deviate from the mean position and result in an extrainhomogeneous broadening. This would explain why the global red-shift and the extra-broadening are of the same order of magnitude. Another type of broadening mechanism especially relevant for nanostructures is related to environment-induced dephasing processes. These latter may involve a more efficient coupling to the phonon modes of the solid matrix (with respect to a liquid surrounding) or the presence of fluctuating photo-induced charges in the local environment of the nanotube. This latter effect is well established for nanostructures embedded in matrices [25] and has been previously evoked to account for spectral diffusion and blinking phenomema at the single nanotube level [26].

D. Thermal Expansion

In contrast to suspensions, solid state samples such as nanotube-doped gelatin samples allow low temperature measurements. As reported elsewhere, in the very low temperature range (4 to 40 K), these studies have permitted to gain valuable information about the exciton structure and the existence of a low-lying dark state [10].

In this study, we focus on the intermediate temperature range and the shift of the S 11 and S 22 lines. Fig. 4 shows a set of PL spectra for temperatures ranging from 10 to 300 K. A clear spectral shift of the lines (associated to a specific chiral family) is observed. However, in contrast to the previous graph (Fig. 3), this is not a global shift of the whole spectrum but a highly chiral dependent shift : the amplitude of the shift strongly varies from one line to another and even the sign of the shift changes. Table I gives a summary of the line-shifts observed when cooling the sample down to 150 K, for both the S 11 and S 22 (when observable) transitions. Although a limited range of chiral families is available, some clear trends show up : first S 11 and S 22 systematically shift in opposite directions. Second, the sign of the shift is related to the family type of the tubes : nanotubes with q = n-m = 1 (mod 3) show a positive shift for S 11 (and negative for S 22 ) whereas tubes with q = nm = -1 (mod 3) show a negative shift for S 11 (and positive for S 22 ). Finally, the magnitude of the spectral shift clearly depends on the chiral angle and is larger for smaller chiral angles (close to zig-zag geometry). The magnitude of the spectral shift of the S 11 line as a function of the chiral angle is displayed in Fig. 5. It turns out to be proportional to cos 3θ. Similar effects have been previously reported for polymer wrapped nanotubes or frozen suspensions of nanotubes [9,27,30].

Chirality q chiral angle ∆S 11 (meV) dS11 dσ (eV) ∆S 22 (meV) ( I: Spectral shifts of the S 11 line for several nanotubes under thermally induced strain. q stands for nm (mod 3). ∆S 11 stands for S11(300 K) -S11(150 K). A negative strain is induced at low temperature.

L. Yang et al. [11,12] have studied theoretically the effect of uniaxial and torsional strain on the band gap of SWCNTs. This study has been recently adjusted by Huang et al. [28] to account for the internal sublattice relaxation of the graphitic structure in the case of uniaxial strain (σ) and leads to :

dS ii dσ = 0.57q(-1) i (1 + ν)3t 0 cos(3θ) (1)
where ν is the Poisson ratio and t 0 the electron hopping parameter. This study turns out to be an excellent framework to interpret our experimental results. Especially, the sign of the spectral shifts reported here (including its dependence on the tube family (q) and the transition number i (S 11 vs S 22 )) as well as the magnitude of the effect as a function of the chiral angle (θ) can 0.9 1.0 1.1

PL Intensity (Arb. Units) Energy (eV) FIG. 4: Photoluminescence spectrum of isolated HiPCO SWCNTs embedded in a gelatin matrix as a function of temperature. The excitation energy is 1.7 eV. The lower the temperature, the larger the PL intensity.

be understood as a consequence of a moderate uniaxial strain (σ) applied on the tubes at low temperatures.

Due to the large aspect ratio of the tube and the large Young modulus mismatch between the matrix and the tube, the thermally induced deformation of the tube is expected to boil down to an almost uniaxial strain [29]. In fact, our experimental data confirm that the spectral shift observed for close to armchair nanotubes is much weaker than for smaller chiral angles, ruling out any significant torsional or radial strain contribution. This effect is related to a thermal expansion coefficient mismatch between the nanotube and the surrounding matrix [30,31]. The linear thermal expansivity is of the order of α t = -5.10 -6 K -1 for nanotubes [32] whereas for water ice it is slightly negative below 60 K and varies almost linearly from 0 at 60 K to α w = +4.10 -5 K -1 at 200 K [33]. Interestingly the observed spectral shift versus temperature (cf. Fig. 6) shows a plateau between 4 and 50 K, which corresponds to the temperature range where the thermal expansivities of water and nanotubes almost balance each other. Conversely the thermallyinduced spectral shift increases with temperature above 50 K as expected from the increasing value of the matrix expansivity. For further quantitative analysis we focus on the range 150-300 K where we fit the thermally-induced spectral shift to a linear law. The strain-induced spectral shift is deduced from that fit assuming a full load transfer from the matrix and an average value of the matrix expansivity of 4.5x10 -5 K -1 [33].

We obtain : dS11 dσ = dS11 ∆αdT ∼ -3.6 eV for the (10,2) nanotube and -0.8 eV for the (8,6) (cf. Table I). We estimate that the strain applied to the tube when the sample is cooled down to 150 K is of the order of -7.5 10 -3 , which corresponds to an applied pressure of 7.5 GPa if one assumes a Young modulus of 1 TPa for nanotubes [34]. Although the absence of jumps or plateaux in the FIG. 5: Energy shift of the emission line of several nanotubes when the sample is heated from 150 K through 300 K plotted against the geometric parameter q cos(3θ). This temperature change corresponds to a positive strain of the order of 0.7%. range 150-300 K supports the assumption that the load transfer is high, one cannot exclude that little slips of the nanotubes relative to the matrix occur at the single tube level, which are smoothed out by the ensemble measurement. Therefore the strain applied to the tube is likely to be overestimated and conversely for the strain-induced spectral shift. These results can be compared with recently published data obtained in experimental conditions where the strain is applied in a different manner : either through the deformation of a polymer matrix host [35] or through direct mechanical stretching of the nanotube [28]. Since all those publications agree on the linearity of dS 11 dσ with the geometrical factor q cos(3θ), we can compare the magnitude of the effect by measuring the slope (cf. Fig. 5). The slope reported by Leeuw et al. [35] is of the order of 9 eV, whereas the one reported by Huang et al. [28] is of the order of 4.5 eV. Our own value is 4.2 eV±0.5. According to Eq.( 1) and using a commonly accepted value of 0.15 for ν [34] we deduce an estimate of the C-C hopping integral of 2.2±0.3 eV. As discussed above, due to possible imperfect load transfer this value is likely to be underestimated ; however it is still in good agreement with usual values found in the literature (ranging from 2.4 to 3 eV) [START_REF] Saito | Physical Properties of Carbon Nanotubes[END_REF][START_REF] Reich | Carbon Nanotubes[END_REF].

We can further note that the absence of chiral dependent spectral shift for gelatin samples (as compared to suspensions) at room temperature contrasts with what is reported for other polymers [9]. This means that no sizeable stress is applied to the tube from the matrix at room temperature, which is consistent with the picture of a highly hydrated material in which the micelle structure is preserved.

IV. CONCLUSION

To summarize, we have investigated the optical properties of carbon nanotubes embedded in a functional composite material. This study brings evidence for the extreme sensitivity of nanotubes to their environment. In particular, we have shown that any possible intrinsic effect is overwhelmed by extrinsic effects such as excitonic dielectric screening and strain induced spectral shifts. The former leads to a red shift of the excitonic lines upon an increase of the local dielectric constant. The strain induced spectral shifts and their chiral dependence reveal important information about the mechanical properties of such composite material and their relationship with the band structure of the nanotubes. Furthermore it provides evidence for an efficient load transfer from the matrix to the tube, which is of outmost importance for mechanical reinforcement applications.

FIG. 6 :

 6 FIG.6: Emission energy (S 11 ) of (10,2) nanotubes as a function of temperature. The excitation energy is 1.7 eV. The full red line is a parabolic fit whereas the black dashed line is a linear fit for the range [150-300 K] used to estimatedS 11 dσ (see text).

TABLE

V. ACKNOWLEDGEMENTS

This work has been done in the framework of the GDRE n o 2756 'Science and applications of the nanotubes -NANO-E'.


A Rani

A Basu Sarbadhikari
email: a.basusarbadhikari@prl.res.in

D R Hood

O Gasnault

S Nambiar

S Karunatillake

Consolidated Chemical Provinces on Mars: Implications for Geologic Interpretations

 We have consolidated three key multivariate methods, yielding the first unified set of chemical provinces of Mars  Igneous geochemistry pervades volatile-rich regions, consistent with water-limited isochemical weathering throughout Mars's geologic history  Secular chemical trends suggest complex processes of mantle melting for highlands, which differs from a possibly garnet-rich lowland source 2

Introduction

A trilogy of multivariate cluster analysis methods applied to the chemical maps derived from Mars Odyssey gamma-ray and neutron spectroscopy (GRS) initially identified chemical provinces and supported interpretations of regional-scale geology [START_REF] Karunatillake | Chemically striking regions on Mars and Stealth revisited[END_REF][START_REF] Gasnault | Quantitative geochemical mapping of martian elemental provinces[END_REF][START_REF] Gasnault | Quantitative geochemical mapping of martian elemental provinces[END_REF]. However, the lack of an intermethodological synthesis to characterize geological processes has hindered subsequent works and stymied advances with other datasets (cf., [START_REF] Rogers | Compositional provinces of Mars from statistical analyses of TES, GRS, OMEGA and CRISM data[END_REF]. Furthermore, key chemical maps used in subsequent papers [START_REF] Baratoux | Petrological constraints on the density of the Martian crust[END_REF][START_REF] Ojha | The Medusae Fossae Formation as the single largest source of dust on Mars[END_REF][START_REF] Plesa | The thermal state and interior structure of Mars[END_REF] were unavailable or only in a preliminary state for the original trilogy: Aluminium (Al), sensitive to both igneous evolution and secondary mineralogy (cf., [START_REF] Nimmo | Early crustal evolution of Mars[END_REF][START_REF] Ehlmann | Subsurface water and clay mineral formation during the early history of Mars[END_REF][START_REF] Mclennan | Geochemistry of Sedimentary Processes on Mars, Sedimentary Geology of Mars[END_REF]; and Sulfur (S), sensitive to cycling processes within the martian critical zone across the atmosphere and crust [START_REF] Halevy | A Sulfur Dioxide Climate Feedback on Early Mars[END_REF][START_REF] King | Sulfur on Mars[END_REF].

Here we use nine elemental mass fraction maps from the GRS: key rock-forming elements (Fe, Si, Ca, and Al) with large ion lithophiles (K and Th) and volatiles (stoichiometrically equivalent H 2 O, Cl, and S) to resolve the aforementioned challenges. We synthesize and advance prior multivariate methods [START_REF] Karunatillake | Chemically striking regions on Mars and Stealth revisited[END_REF][START_REF] Gasnault | Quantitative geochemical mapping of martian elemental provinces[END_REF][START_REF] Gasnault | Quantitative geochemical mapping of martian elemental provinces[END_REF] to derive the first consolidated chemical provinces of Mars. We consider three of the provinces as case studies on geologic insight, revealing diverse mantle sources and magmatic processes in volcanic areas, characterizing possible global ashfall units, and evaluating the relative roles of igneous versus aqueous environments.

Data and Methods

We use chemical maps derived from GRS spectra to delineate the chemically distinct provinces. The average decimetre sampling depth and coarse spatial resolution of GRS overcome the effects of fine dust mantles making GRS elemental analysis ideal for primary and secondary geological process studies on broad scales. GRS measures the energy spectrum of gamma photons emitted from the martian surface and characteristic spectral peaks allow us to measure the abundance and distribution of most major rock-forming elements, along with some minor elements present on the martian surface. There are two main processes on the martian surface that give rise to gamma-photon emission. Gamma photons can be emitted from the decay of natural radioactive elements (K and Th) or emissions induced by cosmic particle interactions with the martian atmosphere and nonradioactive elements of the martian surface. However, due to increasing H (reported as stoichiometric H 2 O mass fractions) towards the poles causing mass dilution for radioactive elements and compositional modelling issues for others due to neutron moderation, derived chemical maps are restricted to the mid-to-low latitudes, roughly within ±50 o . Furthermore, chemical maps are generated at a resolution of 5 o × 5 o due to broad footprint of the GRS instrument [START_REF] Boynton | Concentration of H, Si, Cl, K, Fe, and Th in the low-and mid-latitude regions of Mars[END_REF]. We use the latest refined datasets of GRS, including Al and S used in the literature (cf., [START_REF] Hood | Assessing the geologic evolution of Greater Thaumasia, Mars[END_REF][START_REF] Hood | Contrasting regional soil alteration across the topographic dichotomy of Mars[END_REF][START_REF] Ojha | The Medusae Fossae Formation as the single largest source of dust on Mars[END_REF][START_REF] Ojha | Amagmatic hydrothermal systems on Mars from radiogenic heat[END_REF], and compare them with GRS data used a decade ago [START_REF] Boynton | Concentration of H, Si, Cl, K, Fe, and Th in the low-and mid-latitude regions of Mars[END_REF][START_REF] Karunatillake | Chemically striking regions on Mars and Stealth revisited[END_REF][START_REF] Gasnault | Quantitative geochemical mapping of martian elemental provinces[END_REF][START_REF] Gasnault | Quantitative geochemical mapping of martian elemental provinces[END_REF], including the archived data at the NASA Planetary Database System (https://pds-geosciences.wustl.edu/ody/ody-m-grs-5-elements v1/odgm_xxxx/data/smoothed/). The delineation of provinces also mitigates the substantial spatial autocorrelation inherent in the GRS data by associating large chemical deviations with multi-pixel regions rather which are larger than the GRS instrument footprint [START_REF] Karunatillake | Chemically striking regions on Mars and Stealth revisited[END_REF]. The detailed description and comparison of old versus latest refined datasets of GRS are described in Supplementary Information S1.

Multivariate Cluster Analysis

We apply three multivariate cluster analyses to discern a set of internally uniform chemical provinces that reflect the compositional variability of the martian crust. The three methods (Supporting Information S1.2 provides more details) consist of non-hierarchical clustering with principal component analysis (NHC-PCA: [START_REF] Gasnault | Quantitative geochemical mapping of martian elemental provinces[END_REF], hierarchical clustering with principal component analysis (HC-PCA: [START_REF] Gasnault | Quantitative geochemical mapping of martian elemental provinces[END_REF], and modified student's t-test to distinguish spatially overlapping Gaussian tail clusters (t-GTC: [START_REF] Karunatillake | Chemically striking regions on Mars and Stealth revisited[END_REF].

The three methods optimize clustering differently but still present spatially overlapping trends suggestive of distinct provinces. NHC-PCA optimizes Euclidian distances between cluster centers, where the number of clusters is specified a priori. In comparison, HC-PCA builds a hierarchy of clusters using a top-down clustering approach based on a divisive algorithm [START_REF] Kaufman | Finding Groups in Data: An Introduction to Cluster Analysis[END_REF], where the complete martian landscape mapped by GRS belongs to one cluster and is split iteratively, moving down the hierarchy to highlight the differences between the sub-clusters. The modified t-GTC method identifies spatially contiguous areas where two or more elements deviate from their respective average mid-to-low latitude crustal abundances by one standard deviation or greater. The t-GTCs and HC-PCA-derived provinces reduce spatial autocorrelation biases by using unclassified (boundary) pixels to separate provinces from each other. Complementarily, the NHC-PCA method delineates border-sharing provinces.

We combine the intermediate provinces from the NHC-PCA, HC-PCA, and t-GTC methods into a consolidated map using a spatial contouring approach. A contour map of provinces is made by categorizing the regions into three levels: provinces derived from all three-methods overlapping, provinces with any two-methods overlapping, and provinces with no overlap (Figure S1). We demarcate the boundaries of final provinces resulting from at least two-methods overlapping, reducing the spatial uncertainty in any method (Supporting Information S2 provides more details). Provinces that deviate minimally from the average martian crust (AMC) composition (Supporting Information S2.1 and S2.2) are omitted so that the consolidated chemical province map highlights crustally distinct regions. In addition, we use mapped geology [START_REF] Tanaka | Description of Map Units[END_REF] for context on chemical trends.

Results

All three methods lead to large chemically uniform provinces, with minimal intraprovince compositional variance, reinforcing the subdued variability in chemical maps of Mars. The intermediate chemical province sets resulting from the three underlying methods (Supporting Information S2 provides more details) are labeled numerically (Province-1 to Province-6; Figures 1a-c and Tables S1 andS2). Province-2 from NHC-PCA and HC-PCA methods lacks the minimum criterion of at least two methods overlapping (Figure S1; Supporting Information S2 provides more details), and Province-4 is compositionally identical to the AMC within statistical uncertainty. Therefore, Province-2 and 4 are deprioritized in the consolidated chemical province map (Figure 1d). Moreover, provinces that overlap spatially across the methods are compositionally similar even if driven by different multivariate methods (Tables S2,S3). This further justifies our contour-consolidation method yielding the merged chemical province map (Figure 1 and Tables 1 andS3). Overall, overlapping areas labeled as 'Province-1' in Figures 1a-c have been combined into 'Province-A', 'Province-3' into 'Province-B', 'Province-5' into 'Province-C', and 'Province-6' into 'Province-D', resulting in four consolidated provinces (Figure 1d and Tables 1 andS1). Compositional consistency across the numeric intermediate provinces enables this synthesis (Supporting Information S2). While geographically widely separated areas may constitute a single province, such regions show some compositional heterogeneity.

Distinct Geochemical Provinces

Four geographically distributed regions in the southern highlands constitute Province-A (Figure 1d). This province broadly represents mid and late Noachian highlands (> 30%) to early Hesperian aged volcanic units (> 20%) (Figures 2a andS2), containing many of the large Hesperian volcanic provinces. This province is enriched in Al and depleted in K, Th, and volatiles (H 2 O, S, Cl) relative to the AMC (Figure 2b and Tables 1,S3, and S4; Supporting Information S2.2). The K/Th and S/Cl ratios are above average, while the Fe/Si ratio is consistent with the AMC.

Province-B comprises Amazonian and Amazonian-Hesperian aged volcanic units (> 30%) with Amazonian to Hesperian transitional geological units (~ 18%) (Figures 2a andS2). The distribution of K and Th is consistent with the AMC [START_REF] Taylor | Variations in K/Th on Mars[END_REF]; however, the K/Th ratio is slightly low compared to the AMC (Figure 2b and Tables 1, S3 andS4 1d, 2a, andS2 andTable S1). These regions are enriched in volatiles (H 2 O, Cl, and S) (cf. [START_REF] Diez | H layering in the top meter of Mars[END_REF][START_REF] Feldman | Mars Odyssey neutron data: 2. Search for buried excess water ice deposits at nonpolar latitudes on Mars[END_REF][START_REF] Maurice | Mars Odyssey neutron data: 1. Data processing and models of waterequivalent-hydrogen distribution[END_REF], while Fe, Ca are slightly enriched relative to the AMC (Figure 2b and Tables 1,S3 and S4; Supporting Information S2.2). The ratio of S/Cl and K/Th is consistent with the AMC. However, Fe enrichments make the Fe/Si ratio higher relative to the AMC.

Two regions of broadly late Hesperian lowlands form Province-D (> 60%) (Figures 2a andS2). It is confined to Southern Acidalia Planitia and Utopia Planitia of the Northern lowland near the dichotomy boundary (Figure 1d and Table S1). Since both these regions are found within the northern lowlands and the northern extent of each is obscured by the chemical mapping boundary (Supporting Information S1), they may constitute a larger, contiguous region. The broader extent of Province-D, which includes the western and eastern part of this province, spatially overlaps with the TES-derived Surface Type 2 mineralogy in the martian northern lowlands [START_REF] Bandfield | A global view of Martian surface compositions from MGS-TES[END_REF][START_REF] Rogers | Global spectral classification of Martian low-albedo regions with Mars Global Surveyor Thermal Emission Spectrometer[END_REF][START_REF] Rogers | Compositional provinces of Mars from statistical analyses of TES, GRS, OMEGA and CRISM data[END_REF]. This province is highly depleted in Al, enriched in Si, Fe, K, and Th relative to the AMC (Figure 2b and Tables 1,S3 and S4), contrasting with other provinces. The K/Th and S/Cl ratio is consistent with the AMC, whereas Fe/Si is higher.

Discussion

Extent of aqueous alteration and K/Th trends

We first evaluate the extent of aqueous alteration evident at regional scales to decimeter depths. The Chemical Index of Alteration (CIA=Al 2 O 3 /(Al 2 O 3 +CaO+Na 2 O+K 2 O)×100) in standard molar abundances of oxides corresponding to the provinces A-D is < 50, which indicates no signature of weathering (Figure 2c) at these broad scales. The K/Th ratio is similar across all the provinces, and with the AMC within statistical uncertainty (Figure 2b). Specifically, K/Th ratio is effectively invariant with the abundance of the volatiles (H 2 O, SO 3 , and Cl) even across the maximum in Province-C to a minimum in Province-A (Figure 3a). The most incompatible mapped elements, K and Th (partition coefficient <<1; [START_REF] Borg | A petrogenetic model for the origin and compositional variation of the Martian basaltic meteorites[END_REF] do not fractionate in the solid phases during magmatic processes. Collectively, our study shows that even the volatilerich regions are compositionally consistent with water-limited isochemical weathering throughout Mars's geologic history [START_REF] Hurowitz | A ~3.5 Ga record of water-limited, acidic weathering conditions on Mars[END_REF][START_REF] Tosca | Fe oxidation processes at Meridiani Planum and implications for secondary Fe mineralogy on Mars[END_REF].

Consequently, the provinces are broadly representative of well-preserved igneous processes at least to decimeter depth scales and averaged over a few hundred kilometres horizontally.

Secular chemical trends on Mars

Earlier work by [START_REF] Gasnault | Quantitative geochemical mapping of martian elemental provinces[END_REF] grouped all volcanic landscapes into a single province irrespective of age. However, our study supports the findings [START_REF] Karunatillake | Chemically striking regions on Mars and Stealth revisited[END_REF][START_REF] Gasnault | Quantitative geochemical mapping of martian elemental provinces[END_REF] that volcanic regions form separate provinces by age (Figure 2a), as shown by [START_REF] Baratoux | Thermal history of Mars inferred from orbital geochemistry of volcanic provinces[END_REF]. Moreover, the minimally weathered secular volcanic Earth, considering a stagnant-lid regime of Mars [START_REF] Golombek | Mars tectonics, Planetary tectonics[END_REF][START_REF] Grott | Long-term evolution of the Martian crust-mantle system[END_REF], the decrease of SiO 2 and concomitant increase in olivine component in the parent magma from the oldest to youngest volcanic terrains of southern highlands could be explained by the growth of lithospheric thickness, which would increase the formation pressure, supporting the results of [START_REF] Baratoux | The petrological expression of early Mars volcanism[END_REF]. This contrasts with the topmost micron-layer observation by Thermal Emission Spectrometer (TES), suggestive of weathering reflected by the non-detection of olivine in the Hesperian highland terrains, at the spatial resolution of 1 pixel per degree [START_REF] Rogers | Compositional provinces of Mars from statistical analyses of TES, GRS, OMEGA and CRISM data[END_REF]. GRS-scale footprints in this study consistently reveal the igneous crustal composition albeit weathering processes was prevalent during martian surface evolution. In addition, high Ca/Al for Province-B and -D could indicate a path for sampling a high-pressure, garnet-rich source [START_REF] Treiman | The nakhlite meteorites: Augite-rich igneous rocks from Mars[END_REF][START_REF] Dasgupta | A modified iterative sandwich method for determination of near-solidus partial melt compositions. II. Application to determination of near-solidus melt compositions of carbonated peridotite[END_REF][START_REF] Balta | Water and the composition of Martian magmas[END_REF]. The abundances of K and Th are relatively low in late Noachian -early Hesperian Province-A as compared to Hesperian-Amazonian Province-B and mid-Noachian Province-C in the highland volcanic terrains (Figure 3e). Earlier studies advocated the early Noachian crust building phenomenon for the enrichment of these large ion lithophile elements (LILEs: K and Th) in Noachian than Hesperian terrains [START_REF] Rogers | Compositional provinces of Mars from statistical analyses of TES, GRS, OMEGA and CRISM data[END_REF]. However, our study shows that the youngest volcanic province also contains a high relative abundance of those LILEs, which contradicts the hypothesis of LILE fractionation during the early Noachian crust formation. Moreover, the non-systematic trend of the abundances of LILEs and other secular chemical trends may necessitate complex melting processes in a highly differentiated mantle source on a regional to global scale (cf. Basu Sarbadhikari et al., 2016;2017;[START_REF] Udry | What martian meteorites reveal about the interior and surface of Mars[END_REF] that existed from the early-stage evolution of Mars.

Compared to prior works, the emergence of distinct regions as well as the exclusion of previously suggested provinces (Figure 1d) reflects our methodological advances along with the use of Al and S. Given the collective distinctness among the consolidated chemical provinces, we consider three regions for detailed case studies: Lunae Planum, Medusae Fossae Formation, and Southern Acidalia Planitia (Figure 1d).

Lunae Planum (LP) in Province-A

LP region is a geologically distinct plateau of early Hesperian-aged highland material [START_REF] Tanaka | Description of Map Units[END_REF], which lies between Chryse Planitia and Olympus Mons in Province-A (Figure 1d). Geochemically, the LP is depleted in K, Th, and H 2 O relative to the AMC. LP follows the general K and Th crustal correlation on Mars and substantially overlaps with the K and Th distribution in Hesperia and Thaumasia Planum (Figures 3e andS3).

Consequently, LP's K and Th depletion likely reflects volcanic resurfacing, not aqueous weathering [START_REF] Taylor | Variations in K/Th on Mars[END_REF]. H 2 O depletion in the LP is also comparable in magnitude to that in the vicinity of Greater Thaumasia [START_REF] Hood | Assessing the geologic evolution of Greater Thaumasia, Mars[END_REF]. Given minimal chemical weathering across the provinces (Section 4.1), the H 2 O depletion is likely volcanic and could suggest a dry mantle source as predicted for Hesperian volcanism (cf., [START_REF] Balta | Water and the composition of Martian magmas[END_REF]. However, that mantle evolution model also predicts high Th abundances, not seen in our study. Overall, the LP likely follows the same volcanic evolution trends as identified in the Greater Thaumasia area, i.e., low H 2 O abundance and temporally decreasing K and Th, maybe a part of the same evolving volcanic province [START_REF] Hood | Assessing the geologic evolution of Greater Thaumasia, Mars[END_REF]. Furthermore, higher K and Th abundance within LP relative to Greater Thaumasia could place it as the oldest member of an underlying volcanic series.

Medusae Fossae Formation (MFF) in Province-C

The discrete wind-sculpted outcrops of the mid-Noachian MFF extend along the martian equator (Figure 1d) and their origin remains uncertain (c.f., [START_REF] Kerber | The age of the Medusae Fossae Formation: Evidence of Hesperian emplacement from crater morphology, stratigraphy, and ancient lava contacts[END_REF][START_REF] Zimbelman | Hesperian Age for Western Medusae, Mars[END_REF][START_REF] Wilson | Equatorial locations of water on Mars: Improved resolution maps based on Mars Odyssey Neutron Spectrometer data[END_REF]. Geochemically, the MFF region is highly enriched in Fe, Ca, and volatiles (SO 3 , Cl, and H 2 O) relative to the AMC (Figure S4) [START_REF] Keller | Equatorial and midlatitude distribution of chlorine measured by Mars Odyssey GRS[END_REF]. However, the abundance of K, Th, Al, Si, and the value of K/Th is consistent with the martian crust. As highly incompatible elements in igneous processes, the ratio of Cl/K would inherit the initial mantle abundance [START_REF] Gasnault | Quantitative geochemical mapping of martian elemental provinces[END_REF]. Although the ratio is globally maintained, the MFF region shows significantly different Cl/K than the rest of Mars.

Chlorine is highly abundant, and K is generally lower than that in AMC (Figures 3e andS4), consistent with Cl enrichment in airborne dust and volcanic aerosol in the MFF [START_REF] Newsom | Chemical components of the Martian soil: Melt degassing, hydrothermal alteration, and chondritic debris[END_REF][START_REF] Goetz | Indication of drier periods on Mars from the chemistry and mineralogy of atmospheric dust[END_REF]. Meanwhile, high S and Cl content, moderately high H 2 O (Figure S5), fine-grained nature, preserved primary mineralogy, and extensive erosional features suggest that MFF is the source of the global dust reservoirs of Mars [START_REF] Ojha | The Medusae Fossae Formation as the single largest source of dust on Mars[END_REF]. Thus, our work supports the provenance of martian dust from mostly mechanical weathering and transport of a pyroclastic ashfall unit, which possibly arose from SW-oriented paleo-winds during Tharsis volcanic eruptions [START_REF] Keller | Equatorial and midlatitude distribution of chlorine measured by Mars Odyssey GRS[END_REF][START_REF] Diez | Contribution of mars odyssey GRS at central elysium planitia[END_REF]. Our results don't support the hypothesis of water-ice existence in polar layer-like deposits to explain the volatile enrichment in the equatorial MFF [START_REF] Wilson | Equatorial locations of water on Mars: Improved resolution maps based on Mars Odyssey Neutron Spectrometer data[END_REF]. Summarily, the voluminous pyroclasts of MFF are most likely the product of low-moderate degree partial melting of a distinct volatile-rich mantle source during mid-Noachian age in comparison to the late Noachian -Amazonian volcanic provinces of the martian highlands.

Southern Acidalia Planitia (SAP) in Province-D

SAP is a late Hesperian plain area [START_REF] Tanaka | Description of Map Units[END_REF] in the northern lowlands that geographically lies between Tharsis and Arabia Terra and is part of Province-D (Figure 1d).

Geochemically, SAP is enriched in K, Th, Fe, Si and depleted in Al relative to the rest of the AMC (Figure S6). Despite K and Th enrichment, the K/Th ratio of SAP is consistent with the martian crust (Figures 3e andS6), which suggests primary rock composition [START_REF] Karunatillake | Composition of northern low-albedo regions of Mars: Insights from the Mars Odyssey Gamma Ray Spectrometer[END_REF][START_REF] Taylor | Variations in K/Th on Mars[END_REF], diverging from the extensive alteration posited by [START_REF] Bouley | A thick crustal block revealed by reconstructions of early Mars highlands[END_REF]. Al/Si versus Mg/Si plots of SAP (Figure S7a) show a slope different from the martian crust but resembling that of younger martian meteorites (after, McSween et al., 2009). The Al depletion in SAP best correlates with that of the mean basaltic shergottite composition [START_REF] Nimmo | Early crustal evolution of Mars[END_REF], and given the K, Th, and K/Th ratio signatures, it is more likely to be related to mantle composition than chemical weathering [START_REF] Taylor | Variations in K/Th on Mars[END_REF]) at least at the scales of our observations. Further, mineralogical observations by Compact Reconnaissance Imaging Spectrometer for Mars (CRISM) reveal mafic mineral signatures in outcrops, crater ejecta, and central peaks of the craters in SAP [START_REF] Pan | The stratigraphy and history of Mars' northern lowlands through mineralogy of impact craters. A comprehensive survey[END_REF] resembling Hesperian-aged volcanism [START_REF] Salvatore | Definitive evidence of Hesperian basalt in Acidalia and Chryse Planitiae[END_REF]. Also, the non-detection of Al-bearing sulfates (Figure S7b) that would result from low-pH aqueous alteration [START_REF] Mclennan | Geochemistry of Sedimentary Processes on Mars, Sedimentary Geology of Mars[END_REF] supports an igneous origin for the apparent depletion of Al. The collective K and Th enrichment and Al-depletion can best be explained by a low degree of partial melting at high pressure derived from a garnet-rich mantle source for SAP.

Summary and Conclusion

The Camera (MOLA-HRSC) data map [START_REF] Smith | Mars Orbiter Laser Altimeter: Experiment summary after the first year of global mapping of Mars[END_REF] to delineate the geochemical provinces. Provinces-2 and -4 do not appear in the t-GTC method (c) as they are compositionally similar to the average crustal composition (Tables S2 andS3). (d) The consolidated geochemical provinces on Mars are derived by integrating provinces from three different multivariate techniques and overlaid onto mapped geology [START_REF] Tanaka | Description of Map Units[END_REF] different colors indicate different geologic units with major units in legend). The white dashed outline areas delineate our regional (LP, SAP, and MFF) case studies. For CIA, we are using standard molar abundances of oxides. Non-GRS Na₂O is calculated using [START_REF] Baratoux | Petrological constraints on the density of the Martian crust[END_REF].

;

  Supporting Information S2.2).This province is moderately depleted in Si, Ca, and Fe and highly depleted in Al, while the volatiles (especially Cl and S) are highly enriched relative to the AMC. The S/Cl ratio is depleted from preferential Cl enrichment. Compared to the AMC, the Fe/Si ratio is below average. Province-C consists of two geographic regions: Sinus Meridiani (SM) and the Medusae Fossae Formation (MFF), dominated by mid-Noachian highlands (~ 40 %) (Figures

  composition of provinces A-C in the southern highlands of Mars shows distinct geochemical trends. Unlike southern highlands, Province-D, the late Hesperian lowland unit, deviates from the secular chemical trend (Figure 3b-d), indicating distinct lithology. The provinces A-C from mid-Noachian to Amazonian show Al-, Fe-, Si-and Ca-depletion (Figure 3b-d). As hypothesized by Balta & McSween (2013), higher SiO 2 in older regions of Mars may indicate dehydration melting of the mantle with hundreds of mg/kg water, similar to that of Province-A and -C of our study, in contrast to the relatively silica-depleted younger Province-B. Unlike

  first consolidated chemical provinces of Mars reveal several geochemical signatures of broad geologic import. Our findings demonstrate secular chemical trends among the southern highland provinces A-C may indicate complex processes of an evolving degree of partial melting within a highly differentiated mantle source. Distinct lithology of the late Hesperian lowland unit, Province-D, does not follow the secular chemical trends like highland provinces. This could provide a mechanism for sampling a high-pressure, garnetrich source for Province-D. We also find that even older volatile-rich regions preserve igneous geochemistry, indicating how throughout Mars's geologic history, water-limited isochemical weathering dominated regionally to decimeter depths scales. Specifically, as a distinct plateau of early Hesperian-aged highlands, the Lunae Planum region shows compositional links -and possibly shared provenance -with Province-A, much like Hesperia Planum, Syrtis Major, and part of Greater Thaumasia. The similarity of Province-C's K/Th trend with the average crust and enrichment of volatiles (especially S and Cl) supports mostly regional volcanic materials (e.g., ashfall pyroclastic deposits), especially for the Medusae Fossae Formation, despite volatiles' enrichment. In contrast to the provinces A-C, Southern Acidalia Planitia of Province-D in the northern lowlands is lithologically distinct with the highest abundance of K and Th of the provinces. However, the similarity of K/Th and depletion of Al relative to the average crust argues against sedimentary signatures of putative paleo-ocean(s). The northern lowlands may mostly represent the mafic crust of Mars, which might have originated by a low degree of partial melting at high pressure as suggested by high abundances of incompatible elements (K, Th) and Al-depletion rather than the conventional thought that the northern lowlands are highly fractionated. These findings also demonstrate the overall utility of the integrative multivariate methodology we have .edu/missions/odyssey/grs_cgs.html). Mars geology map taken from USGS-310 archive (https://pubs.usgs.gov/sim/3292/), and topographic data from MOLA-HRSC blended DEM data by[START_REF] Fergason | HRSC and MOLA blended digital elevation Model at 200m v2. Astrogeology PDS annex[END_REF].

Figure Caption Figure 1 :

 Caption1 Figure Caption

Figure 2 :

 2 Figure 2: (a) The relative areal fractions of secular, transitional, and volcanic units within each chemical province corresponding to the consolidated geochemical map. The prefix "N" indicates Noachian, "H" indicates Hesperian, and "A" indicates Amazonian, and suffix "v" indicates the associated volcanic units (after, Tanaka et al., 2014). A detailed version of Figure 2a is given in Figure S2, both as per Tanaka et al. (2014) geology map. (b) Modified box-and-whiskers diagram to compare the distribution of elements for all provinces to the global geochemistry of the AMC. The top of the upper box represents 75 th percentile/25 th percentile ratio, the bottom of the lower box represents the 25 th /75 th percentile ratio, and the ratio of the medians by the boundary between them. The whiskers here represent the uncertainty on the ratio of medians using the Median Absolute Deviation (MAD) (Karunatillake et al., 2011). (c) The Chemical Index of Alteration (CIA=100*[Al₂O₃/(Al₂O₃+CaO+Na₂O+K₂O)]) diagram is used to assess the extent of chemical weathering in the provinces A-D. For CIA, we are using standard molar abundances

Figure 3 :

 3 Figure 3: Diagrams of K/Th normalized to the bulk silicate Mars (~ 5300) versus GRS derived total volatile concentration (a), Fe versus Si (b), Al versus Fe (c), and Ca versus Si (d) of the provinces A-D are represented by colored boxes (southern highland provinces A-C) and ellipse (northern lowland Province-D). The modest variation in K/Th is consistent with the bulk Mars, implies to low or no alteration at decimeter depths in the consolidated provinces. The regional uncertainty is shown at the right corner of each plot, represented by

Figure 1

 1 Figure 1.

  Figure 2.

Table 1 :

 1 Arithmetic mean stoichiometric oxide mass fraction as a percentage (wt%, if not mentioned otherwise), elemental ratio, and its corresponding root means square uncertainty (σ) are calculated from mean measurement uncertainties for the grid points composing a geochemical province. Whereas, K/Th ratio uncertainty is calculated from [(K/Th)*((σ K /K) 2 +((σ Th /Th) 2 ] 1/2 (GRS derived) of the consolidated chemical provinces.

		Province-A	Province-B	Province-C	Province-D	Mid-latitudinal Average Martian Crust (AMC)
	K 2 O	0.35±0.01	0.42±0.01	0.40±0.01	0.56±0.02	0.43±0.01
	FeO	16.1±0.9	14.0±0.8	18.2±1.1	18.4±1.2	16.3±1.0
	SiO 2	42.7±2.3	39.1±2.1	42.4±2.6	44.9±3.0	42.5±2.5
	CaO	6.6±1.2	5.9±1.1	7.9±1.4	6.9±1.7	7.0±1.3
	Th mg/kg	0.48±0.04	0.62±0.05	0.59±0.05	0.80±0.06	0.61±0.05
	Al 2 O 3	10.6±2.1	5.5±1.8	8.6±2.2	6.2±2.6	8.5±2.2
	H 2 O	2.9±0.4	4.2±0.4	5.5±0.5	3.5±0.6	3.9±0.5
	Cl	0.41±0.03	0.61±0.03	0.57±0.04	0.46±0.04	0.47±0.03
	SO 3	4.9±0.5	6.0±0.5	6.5±0.6	5.2±0.7	5.4±0.6
	K/Th	6060±580	5600±430	5680±520	5800±440	5760±470

Acknowledgment AR, ABS, and SN were supported by the Indian Space Research Organization, Director PRL, Head of Planetary Sciences Division, PRL, and Director IIT Gandhinagar. DRH is supported by the Baylor Office of the Vice Provost of Research Postdoctoral Program. OG's contribution was supported by the CNES, based on observations with GRS embarked on Odyssey. SK's participation was supported by NASA MDAP 80NSSC18K1375. The authors acknowledge anonymous reviewers for providing constructive and thorough reviews of the manuscript and Andrew Dombard for editorial handling.

Data Availability Statement: Data in support of this work can be found through Mendeley data (doi: 10.17632/3jd9whd78m.1). Archival GRS-derived chemical maps are from NASA


Yoann Dieudonné

Franck Petit

Squaring the Circle with Weak Mobile Robots

Keywords: Distributed Coordination, (Uniform) Circle Formation, Mobile Robot Networks, Self-Deployment

We present two non-trivial deterministic protocols that solve the circle formation problem (CFP) with 4 and 3 robots, respectively. Both solutions do not require that each robot reaches its destination in one atomic step. This paper closes CFP for any number n (> 0) of robots in the semi-synchronous model.

Introduction

Consider a distributed system where the computing units are mobile weak robots (sensors or agents), i.e., devices equipped with sensors and designed to move in a two-dimensional plane. By weak, we mean that the robots are anonymous, autonomous, disoriented, and oblivious, i.e., devoid of (1) any local parameter (such that an identity) allowing to differentiate any of them, (2) any central coordination mechanism or scheduler, (3) any common coordinate mechanism or common sense of direction, and (4) any way to remember any previous observation nor computation performed in any previous step. Furthermore, all the robots follow the same program (uniform or homogeneous), and there is no kind of explicit communication medium. The robots implicitly "communicate" by observing the position of the others robots in the plane, and by executing a part of their program accordingly.

In such a weak model, there has been considerable interest in the design of deterministic coordination protocols. One of the common features of these works is the study of the minimal level of ability the robots are required to have to achieve the desired task. The Circle Formation Problem (CFP) consists in the design of a protocol insuring that starting from an initial arbitrary configuration (where no two robots is at the same position), n robots eventually form a circle with equal spacing between any two adjacent robots. In other words, the robots are required to form a regular n-gon in finite time.

The first attempt for formally and deterministically solving the CFP were presented in [START_REF] Defago | Circle formation for oblivious anonymous mobile robots with no common sense of orientation[END_REF]. It works in the semi-synchronous model (SSM) in which the cycles of all the robots are synchronized and their actions are atomic. They ensure only asymptotical convergence toward a configuration in which the robots are uniformly distributed on the boundary of a circle. In other words, the robots move infinitely often and never reach the desired final configuration. The first solution leading n robots in a regular n-gon in finite time is proposed in [START_REF] Katreniak | Biangular circle formation by asynchronous mobile robots[END_REF]. Designed for the fully asynchronous model (CORDA), it is also valid in SSM. It works if n ≥ 5 only. Moreover, if n is even, the robots may form a biangular circle in the final configuration, i.e., the distance between two adjacent robots is alternatively either α or β. A general solution is given in [START_REF] Dieudonné | Circle formation of weak mobile robots[END_REF]. It works in SSM, for any number n of robots, except 3 and 4. The approach in [START_REF] Dieudonné | Circle formation of weak mobile robots[END_REF] is based on a technique using tools from combinatorics on words and geometric properties of the convex hull formed by the robots. Following this work, both cases n = 4 and n = 3 remain open problems. Indeed, it is very difficult to maintain a geometric invariant with such a few number of robots, e.g., the smallest enclosing circle, concentric cycles, properties of the convex hull, or a leader. As a matter of fact, due to the high rate of symetric configurations, right now, the problem was suspected to be unsolvable with 4 robots.

In this paper, we first disprove this conjecture by presenting a non-trivial deterministic protocol that solves CFP for the case n = 4 (Section 3). Next (Section 4), we present a solution for the case n = 3. None of the two solutions requires that each robot reaches its destination in one atomic step. Since a cohort of n robots trivially always form a regular n-gon if n ∈ {1, 2}, this paper closes the circle formation problem for any number n (> 0) of robots in SSM.

Preliminaries

Model. We adopt the semi-synchronous model, below referred to as SSM. The distributed system considered in this paper consists of n mobile robots. Each robot, viewed as a point in the Euclidean plane, move on this two-dimensional space unbounded and devoid of any landmark. Any robot can observe, compute and move with infinite decimal precision. The robots are equipped with sensors enabling to detect the instantaneous position of the other robots in the plane. Each robot has its own local coordinate system and unit measure. The robots do not agree on the orientation of the axes of their local coordinate system, nor on the unit measure. They are uniform and anonymous, i.e, they all have the same program using no local parameter (such that an observable identity) allowing to differentiate any of them. They communicate only by observing the position of the others and they are oblivious, i.e., none of them can remember any previous observation nor computation performed in any previous step. At each time instant t j ( j ≥ 0), each robot r is either active or inactive. The former means that, during the computation step (t j ,t j+1 ), using a given algorithm, r computes in its local coordinate system a position p(t j+1 ) depending only on the system configuration at t j , and moves towards p(t j+1 ). In the latter case, r does not perform any local computation and remains at the same position. In every single activation, the maximum distance traveled by any robot r is bounded by σ r .

Basic Definitions and Properties. Given a set P of n ≥ 2 points p 1 , p 2 , • • • , p n on the plane, the convex hull of P, denoted H(P) (H for short), is the smallest polygon such that every point in P is either on an edges of H(P) or inside it. Informally, it is the shape of a rubber-band stretched around p 1 , p 2 , • • • , p n . The convex hull is unique and can be computed with time complexity O(n log n) [START_REF] Graham | An efficient algorithm for determining the convex hull of a finite planar set[END_REF].

A convex hull H is called a (convex) quadrilateral (respectively, triangle) if H forms a polygon with four (resp. three) sides (or edges) and vertices (or, corners). In the sequel, we consider convex quadrilaterals (resp. triangle) only. A quadrilateral is said to be perpendicular if and only if its diagonals are perpendicular. Otherwise, it is called a non-perpendicular quadrilateral.

A triangle is said to be equilateral if all its sides are of equal length. An isosceles triangle has two sides of equal length. A triangle having all sides of different lengths is said to be scalene. A trapezoid is a quadrilateral with at least one pair of opposite sides parallel. An isosceles trapezoid is a trapezoid whose the diagonals are of equal length. A parallelogram is a quadrilateral with both pairs of opposite sides parallel. A rectangle is defined as a parallelogram where all four of its angles are right angles. A square is a rectangle perpendicular quadrilateral.

Four Robots

In this section, we present our algorithm that leads 4 mobile robots to eventually form a square. We refer to Figure 1 to explain our scheme.

Consider the convex hull H formed by the robots on the plane. If the 4 robots belong to the same Line L, then H is reduced to the segment of line linking the 4 points (Figure 1, Case sL). Otherwise (the 4 robots are not aligned), there are only two possible forms for H : H forms either a quadrilateral or a triangle. If H forms a triangle, then there is a robot r being located either inside H (Case nD-T ) or between two of the three corners of the triangle (Cases nP-D and P-D). In the latter case, three out of the four robots q, r, and s are aligned on a line L (r belonging to the segment [q, s]), whereas the fourth robot t does not. Such a configuration is called a (arbitrary) delta. If the line L ′ passing through r and t is perpendicular to L, then the delta is said to be perpendicular (Case P-D).

Our propotol is made of two steps : (1) Starting from an arbitrary configuration, move the robots to eventually form an arbitrary perpendicular quadrilaterals ; (2) Starting from an arbitrary perpendicular quadrilateral, the robots eventually form a square.

Indeed, when the convex hull H forms an arbitrary perpendicular quadrilateral (Figure 1 robots away from O along their diagonal until they reach the positions being at the same distance from O than the farthest ones (d max in Figure 1). To reach such an arbitrary perpendicular quadrilateral, we aim to bring the system into an arbitrary delta. Starting from a perpendicular delta, Case P-D in Figure 1 (q, r, and s are aligned one a line L, the line L ′ passing through r and t is perpendicular to L), in one step, the system becomes an arbitrary perpendicular quadrilateral by sliding r on L ′ in the opposite direction of t. Starting from a non-perpendicular delta (Case nP-D), the system eventually becomes a perpendicular delta by moving t along L ′′ , the line passing through t that is parallel to L, until L ′ and L become perpendicular.

Clearly, the above scheme does not cover all the possible cases. In particular, it gives no details about the "arbitrary" configurations considered in the above first item. In fact, we can detail the different classes of such "arbitrary" configurations and the corresponding moves as follows : 1. The convex hull H forms an arbitrary quadrilateral that is not perpendicular, a rectangle, an isosceles trapezoid, nor a parallelogram. In the sequel, such a configuration is called an asymmetric quadrilateral (Case AQ, in Figure 1). In that case, we show that there always exists a robot r being either the unique closest or the unique farthest robot from the center O of the quadrilateral. By moving either r or the opposite robot (w.r.t. to O) along its diagonal toward O, the moving robot eventually reaches O. By the way, it crosses one side of the triangle formed by the 3 other robots. The system then becomes a non-perpendicular delta, and from this point on, adopts the above behavior. 2. The convex hull H of the 4 robots forms a symmetric non-perpendicular quadrilateral that is not reduced to a line segment. In that case, H forms either an isosceles trapezoid (Case nR-IT ) or a parallelogram (Case nR-P) -note that H can be a non-perpendicular rectangle (Case nP -R) if it is both an isosceles trapezoid and a parallelogram. In these cases, the robots move trying to form a square in one step. Clearly, if they move synchronously and reach their respective positions to form a square, then they succeed. Therefore, in every executing starting from either a non-rectangle isosceles trapezoid or a non-rectangle parallelogram, the four robots eventually form either an asymmetric quadrilateral, or a square. Starting from a non-square rectangle, the four robots eventually form either an asymmetric quadrilateral, a non-rectangle isosceles trapezoid, a non-rectangle parallelogram, or a square-refer to Figure 1. 3. The convex hull H forms a triangle that is not an arbitrary delta. So, one of the four robots is located inside the triangle (Case nD-T in Figure 1). In that case, the robot r inside the triangle moves toward the closest side of the triangle -if r is at the center of the triangle, then it arbitrarily chooses one side to move on. Again, the system reaches a configuration where the cohort of robots forms a delta. 4. The 4 robots are aligned on the same Line L (Case sL). In that case, both robots r 1 and r 2 located between the two extremities of the segment formed by the 4 robots are able to move perpendicularly to L. With respect to the asynchrony, there are 5 possible resulting configurations : either a non-perpendicular quadrilateral (possibly, an isosceles trapezoid or a parallelogram) or a triangle (possibly, a non-perpendicular delta).

Three Robots

In this section, we show that starting from an arbitrary configuration, 3 robots can form an equilateral triangle in finite time. As for the case n = 4, consider the convex hull H formed by the robots on the plane. If the 3 robots belong to the same Line L, then H is reduced to the segment of line linking the 3 points. Otherwise, H forms a non-aligned triangle. (In the following, when we consider a non-aligned triangle, we will omit the term "non-aligned".)

Let us consider the three following cases : 1. The three robots form an isosceles triangle. In that case, if the triangle is also equilateral, then the problem is solved. If the triangle is not equilateral, then let r be the unique robot being placed at the unique angle different from the two others robots s and t. Let p be the position of L, the perpendicular bisector of [s,t], such that p, s, and t form an equilateral triangle. Since H forms an isosceles triangle, r belongs to L. So, it can move along L toward p. Clearly, while the triangle is not equilateral -i.e., r does not reach p -, r remains the single robot allowed to move. By fairness, the equilateral triangle is formed in finite time. 2. The three robots are on the same line L. Let s and t be the two robots located at the extremities of the segment formed by the three robots. Let r be the median robot and d(s, r) (respectively d(t, r)) denotes the distance between s and r (resp., r and t). There are two cases to consider : a. d(s, r) = d(t, r) -r is located at the middle of [s,t]. In that case, r can move on any position on the perpendicular bisector of [s,t]. After one step, the robots form an isosceles triangle, and the system behaves as in the previous case. b. d(s, r) = d(t, r). Then, r can move toward the position p such that d(s, p) = d(t, p). Clearly, r reaches p finite time, and the three robots behaves as above thereafter. 3. The three robots form a scalene triangle. Since the three robots form a scalene triangle, the three internal angles are all different. Let r be the robots corresponding to the greatest internal angle. Then, r can move toward the intersection between the opposite side formed by the two others robots and the line passing through r that is perpendicular to the opposite side of the triangle. While the robots are not on the same line, r remains the only robots allowed to move because its internal angle increases whereas the two others internal angles decrease. By fairness, the three robots are eventually on the same line. Then, they behave as above.

Conclusion

We closed the circle formation problem for any number n(n > 0) of robots in SSM. We proposed two non-trivial deterministic protocols solving CFP for 4 and 3 robots, respectively. The proposed solutions do not require that each robot reaches its destination in one atomic step. In a future work, we would like to address and solve the problem for any number of robots in CORDA.

R éf érences

FIG. 1 :

 1 FIG. 1:General Scheme with 4 robots.