Mohammed F Hamza

Hamed Mira

Yuezhou Wei
email: yzwei@usc.edu.cn

Salama M Aboelenin

Eric Guibal
email: eric.guibal@mines-ales.fr

Waheed M Salem

Sulfonation of chitosan for enhanced sorption of Li(I) from acidic solutions -Application to metal recovery from waste Li-ion mobile battery

The sulfonation of chitosan strongly enhances the recovery of lithium from aqueous solutions Uptake kinetics are fitted by the pseudo-first-order rate equation and sorption isotherms by the Langmuir dual site equation Lithium sorption remains highly selective for Li (I) against alkali-earth and heavy metals High stability in performances of sorption and desorption for eighteen cycles At pH 2.2 sulfonated chitosan selectively recovers lithium from acidic leachate of mobile phone Li-ion battery (for sustainable valoriza-tion of strategic metal)

A B S T R A C T

The functionalization of cross-linked chitosan (Chito) by an original sulfonating process allows synthesizing a highly efficient sorbent (Sulfo-C) for Li(I) recovery from acidic solution (at pH ≈2). FTIR and XPS analyses show the main contributions of amine and sulfonic groups in the binding mechanism. Maximum sorption capacity reaches 20 mmol Li g -1 , and the sorption isotherms are fitted by the Langmuir Dual Site equation; the process being spontaneous, exothermic and enthalpy-driven. The sorption remains highly efficient in 1 M NaCl solution, while the uptake remains strongly selective in the presence of equimolar concentrations of alkali-earth and heavy elements (Fe > Zn > Mg > Ca > Ni > Al cations). This selectivity (SC Li/Metal varying between 14 and 36) is controlled by the pH (optimum close to 2.2) and slightly increases (up to 16-69) with adding oxalic acid (1% w/ w). The equilibrium is achieved, under selected experimental conditions, within 30-90 min of contact (depending on temperature). Uptake kinetics are fitted by the pseudo-first order rate equation, though the contribution of resistance to intraparticle diffusion cannot be neglected. Bound metal can be completely desorbed using hydrochloric acid solution (0.2 M); the kinetics of desorption is fast (within 20 to 30 min). Both sorption and desorption performances are remarkably stable for 18 cycles (loss in sorption less than 10.4% at the 18 th cycle). The functionalized sorbent is highly efficient for the recovery of lithium from a spent Li-ion battery (mobile cell battery). The sorbent shows high selectivity for Li(I) against Co(II), Mn(II), Ni(II), Fe(III) and Al(III) (SC Li/Metal : 26-85); much lower against Cd(II) (i.e., 2.3).

Introduction

In the last decade (apart its historical use in ceramic, glass manufacture and aluminum production), the "explosive" development of rechargeable batteries for electronics, hybrid and electric vehicles has driven the demand for lithium to unexpected levels [START_REF] Gerold | Separation and efficient recovery of lithium from spent lithium-ion batteries[END_REF]. The global consumption of lithium-ion batteries for automotive applications is expected to represent a business of about 220 US billions from 2015 to 2024 [START_REF] Yang | Selective recovery of lithium from spent lithium iron phosphate batteries: a sustainable process[END_REF]. Currently, the conventional production of lithium comes from brine and mineral resources. About 80% of mining resources (production and reserves) is identified in a limited number of countries (Australia, China, Chile, Canada and Argentina) [START_REF] Gerold | Separation and efficient recovery of lithium from spent lithium-ion batteries[END_REF][START_REF]Periodic Table[END_REF]. These strategic issues as well as the increasing demand may explain the strong incentive politics of national and regional regulations for developing the recycling of waste and spent equipment, including for lithium, though no significant industrial production from secondary resource has been reported for Li. The rate of recycling of Li from spent materials is actually evaluated to less than 10% [START_REF]Periodic Table[END_REF].

The composition of Li-ion battery makes these spent materials a remarkable "alternative" as mining resource. Indeed, Gerold et al. [START_REF] Gerold | Separation and efficient recovery of lithium from spent lithium-ion batteries[END_REF] compiling the literature gave average composition of Li-ion battery as: 5-7% Li, 5-20% Co, 5-10% Ni and a certain variety of complementary heavy metals accounting for 5-10% (including Cu, Al, Fe, Cr, etc.). After the separation of plastic-based cartridges, and the pre-treatment of spent (a) precipitation [START_REF] Alsabbagh | Lithium enrichment optimization from Dead Sea end brine by chemical precipitation technique[END_REF][START_REF] Gu | Feasible route for the recovery of strategic metals from mixed lithium-ion batteries cathode materials by precipitation and carbonation[END_REF], (b) crystallization [START_REF] Ooi | Recovery of lithium from salt-brine eluates by direct crystallization as lithium sulfate[END_REF], (c) solvent extraction [START_REF] Nguyen | Separation of Co(II), Ni(II), Mn(II) and Li(I) from synthetic sulfuric acid leaching solution of spent lithium ion batteries by solvent extraction[END_REF][START_REF] Takahashi | Enhancing cobalt recovery from Li-ion batteries using grinding treatment prior to the leaching and solvent extraction process[END_REF][START_REF] Hong | A review on the separation of lithium ion from leach liquors of primary and secondary resources by solvent extraction with commercial extractants[END_REF], (d) ion-exchange [START_REF] Chiu | Recovery and separation of valuable metals from cathode materials of spent lithium-ion batteries (LIBs) by ion exchange[END_REF][START_REF] Arroyo | Lithium recovery from desalination brines using specific ion-exchange resins[END_REF][START_REF] Guneysu | Lithium sorption from aqueous solution with cationic resins, Desalin[END_REF][START_REF] Virolainen | Removal of iron, aluminium, manganese and copper from leach solutions of lithium-ion battery waste using ion exchange[END_REF][START_REF] Volkova | Using chelate cationites for the purification of a lithium chloride solution[END_REF] (including ion-exchange membranes, [START_REF] Ipekci | Application of heterogeneous ion exchange membranes for simultaneous separation and recovery of lithium and boron from aqueous solution with bipolar membrane electrodialysis (EDBM)[END_REF]), or.

(e) chelating resin [START_REF] Cicek | Removal of lithium from water by aminomethylphosphonic acid-containing resin[END_REF][START_REF] Koo | Lithiumchelating resins functionalized with oligoethylene glycols toward lithium-ion battery recycling[END_REF].

The choice of the treatment process (including combined techniques) strongly depends on the composition of the leachates, the relative concentrations of valuable metals, and the process strategy (purification of Li leachates by removal of co-metals [START_REF] Virolainen | Removal of iron, aluminium, manganese and copper from leach solutions of lithium-ion battery waste using ion exchange[END_REF][START_REF] Strauss | Separation of cobalt, nickel, and manganese in leach solutions of waste lithium-ion batteries using Dowex M4195 ion exchange resin[END_REF] or direct removal of lithium). The complexity of Li-containing effluents makes the selectivity a critical challenge in the design of sorbent. Alexandratos et al. [START_REF] Alexandratos | Immobilization of lithium-selective 14-crown-4 on crosslinked polymer supports[END_REF] immobilized crown-ethers onto polymer supports for developing Liselective sorbents. Among ion-exchange resins, Arroyo et al. [START_REF] Arroyo | Lithium recovery from desalination brines using specific ion-exchange resins[END_REF] compared different types of reactive groups for Li(I) sorption: iminodiacetic reactive groups (Lewatit TP 207 and 208 resins) and sulfonic acid groups (Lewatit K2629 resin). The sorption levels from brines (under weakly acidic solutions) range between 0.17 and 0.38 mmol Li g -1 . Güneysu [START_REF] Guneysu | Lithium sorption from aqueous solution with cationic resins, Desalin[END_REF] demonstrated that at near-neutral pH the weakly cationic Lewatit CNP 80 resin (acrylic acid-based resins) is more efficient for Li (I) sorption than strong acidic Armfield resin. Xu et al. [START_REF] Xu | Efficient adsorption performance of lithium ion onto cellulose microspheres with sulfonic acid groups[END_REF] reported the synthesis of sulfonated cellulose microspheres (activated with glycidyl methacrylate) and applied this new sorbent for Li(I) sorption at mild pH (around 6-8). Volkova et al. [START_REF] Volkova | Influence of pH on distribution coefficients of lithium and impurity elements in sorption[END_REF] discussed the order of affinity for Li(I) of a series of cation-exchange resins bearing sulfonic groups (KU 2-8, Purolite C100, and Resinex KW-8), carboxylic groups (SG-1, Purolite C104FL, and Resinex KW-H) and phosphoric groups (Purolite S957). While comparing the distribution coefficients of Li and other impurities (Na, K, Mg Ca and Fe cations) at different pH values, they conclude that carboxylic-based resins are highly efficient for separating alkali metals from lithium (not sorbed), at pH 2-3. On the other hand, the preferential sorption of alkali-earth metals (compared with lithium) by sulfonic-based resins makes these sorbents efficient for purifying mixed solutions at pH 2.

These different results oriented the current research towards the development of sulfonic-bearing sorbents for Li(I) recovery from solutions of increasing complexity (synthetic pure solutions → multicomponent synthetic solutions → acidic leachates of Li-ion batteries).

Herein, the strategy for the design of the new sorbent is based on the functionalization of chitosan (aminopolysaccharide commercially produced by deacetylation of chitin, major constituent of crustacean shells). Chitosan bears amine groups that can be used for metal sorption by chelation at near-neutral pH of metal cations through free electrondoublet on N, or by ion-exchange/electrostatic attraction of metal anions on protonated amine groups (in acidic solutions) [START_REF]Interactions of metal ions with chitosan-based sorbents: a review[END_REF]. These amine groups can be used for the functionalization of the biopolymer and for its shaping (preparation of hydrogels, foams, membranes, and so on). The high density of hydroxyl groups (responsible of its hydrophilic behavior) also offers possibilities for grafting new functional groups; this is precisely the strategy used for the sulfonation of chitosan in this work (Sulfo-C). The sorption properties of Chito (glutaraldehyde-crosslinked chitosan) and Sulfo-C micro-particles for Li(I) are widely investigated (study of pH effect, uptake kinetics, sorption isotherms at different temperatures, selectivity and competition effects). The desorption of lithium from loaded sorbents and the recycling are carried out. Finally, Sulfo-C is tested for the recovery and separation of Li(I) from a complex solution (acidic leachate of Li-ion phone cell battery). In parallel, the sorbent is characterized by a series of techniques to characterize both the structure of the sorbent and its interactions with Li(I).

The work described for the first time the sulfonation of chitosan using an original procedure involving a specific sulfating agent (synthesized by reaction of sodium nitrite and sodium hydrogen sulfate, under reflux). This method is cost-effective and does not require complex reagents, or hazardous solvent (as mentioned in most of the available literature). The study will illustrate the outstanding sorption properties (compared with conventional Li(I) sorbent), the remarkable stability of the sorption and desorption properties, the fast kinetics for metal sorption, and the effectiveness of the sorbent for the recovery of target metal from complex (synthetic or industrial) effluent containing several competitor ions.

Materials and methods

Materials

Chitosan (acetylation degree, DA, close to 25%), glutaraldehyde solution (25% w/w) as well as NaHSO 3 (>99.9%), NaOH (≥97.0%), and NaNO 2 (≥99.0%), LiNO 3 (used for most sorption tests) and LiCl (≥99.98%) (used for salinity and selectivity tests) were provided by Sigma Aldrich (Merck KGa, Darmstadt, Germany). Sodium chloride (≥99.98%), MgCl 2 ⋅6H 2 O (99%), AlCl 3 ⋅6H 2 O (95%), FeCl 3 (≥99.5%), ZnCl 2 (≥99.5%), NiSO 4 (≥99.8%) were obtained from Guangdong Guanghua, Sci-Tech Co., Ltd (Guangdong, China). Calcium chloride (≥99.1%) (used for selectivity tests) was purchased from Shanghai Makclin Biochemical Co., Ltd. (Shanghai, China).

Sorbent synthesis and characterization

Scheme S1 (Supplementary Information) shows the different steps in the synthesis of both Chito and Sulfo-C sorbents. Detailed descriptions of the synthesis procedures are reported in Supplementary Information (Section A). Briefly, chitosan sorbent (Chito) was produced as glutaraldehyde cross-linked biopolymer, as the reference material (support, SI, Section A.1). The functionalized sorbent (i.e., Sulfo-C) was obtained by a two-stage process. A sulfating agent (N(SO 3 Na) 3 ) was first synthesized (SI, Section A.2). This intermediary product was reacted with chitosan (in solution) under controlled thermal conditions (85 • C) before processing the glutaraldehyde crosslinking (under reflux) and the precipitation in alkaline conditions (SI, Section A.3). Scheme 1 shows the structures of Chito and Sulfo-C sorbents. The methods used for the characterization of the materials are reported in Section B.1 (Supplementary Information).

Sorption and desorption studies 2.3.1. Sorption tests

Sorption tests were performed in batch systems using a reciprocal shaker at fixed agitation speed (i.e., 210 rpm). A fixed volume of solution (V, L) containing a given amount of metal ion (i.e., C 0 , mmol Li L -1 ) was mixed with a fixed amount of sorbent (m, g). The pH was adjusted to target value using either HNO 3 or NaOH molar solutions. Though the pH was not controlled during the sorption step, the equilibrium value was systematically monitored at equilibrium. After fixed values of contact times (for kinetics) or after 24/48 h, samples were collected and filtrated before being analyzed for assessing residual metal concentrations (C eq , mmol L -1 ). The mass balance equation was used for calculating the sorption capacity (q eq , mmol Li g -1 ): q eq = (C 0 -C eq ) × V/m.

The precise experimental conditions are systematically reported in the caption of the Figures. Experiments were systematically duplicated and shown in Figures (for illustrating the good reproducibility of sorption tests).

Desorption studies

For the study of metal desorption, the metal-loaded sorbents (which were collected from previous kinetic or equilibrium experiments) were mixed for given contact times with an eluent solution (0.2 M HCl solution). The sorbent dose (SD) was set to 2 g L -1 for desorption. For sorbent batteries (though grinding, magnetic and densimetric separations), the solid is usually submitted to acidic leaching (as for the processing of mineral resources) [START_REF] Meshram | Extraction of lithium from primary and secondary sources by pre-treatment, leaching and separation: A comprehensive review[END_REF][START_REF] Nayl | Acid leaching of mixed spent Li-ion batteries[END_REF][START_REF] Mossali | Lithium-ion batteries towards circular economy: A literature review of opportunities and issues of recycling treatments[END_REF]. A large variety of techniques may be used for the treatment of Li-ion battery (or ore) leachates and brines, and the recovery of valuable metals including: re-use, the sorbent was rinsed several times with water between each operating step. The sorbed Li amounts at the different steps and in the eluate (for desorption step) were compared to calculate the loss in sorption efficiency and the effectiveness of metal desorption.

In the case of sorbent loaded with multi-component solutions, several types of eluents were tested (including hydrochloric, sulfuric, acetic, oxalic acids and EDTA, 0.2 M) in order to evaluate the possibility to separate these metal ions not only during the sorption step but also at the desorption stage. The sorbent dose was: 0.2 g L -1 (other experimental conditions: v: 210 rpm; time: 2 h).

Modeling of sorption

Conventional equations were used for modeling uptake kinetics and sorption isotherms. The relevant models are summarized in Table S1 (see Supplementary Information, SI). The determination of model parameters was derived from non-linear regression analysis (using the facilities of Mathematica®). The quality of the fits was analyzed by comparison of experimental and calculated data using the determination coefficient (i.e., R 2 ) and the Akaike Information Criterion (AIC; lower values researched, and differences significant when |ΔAIC|>2).

Application to the treatment of acidic leachate of waste Li-ion battery 2.4.1. Leaching

Different types of Li-ion mobile batteries were used for the application test (phone cell batteries as in Figure S1). The batteries were dismantled and the plastic parts were discarded. After grinding to 0.5 mm size, the material was quartered. The sample (0.5 g) was mineralized in a Teflon beaker at 110-160 • C using a series of acids. Concentrated HF acid was used for Si digestion (S/L ~ 1:10). In a second step, after evaporation, a mixture of HNO 3 and HCl (1:3 M ratio; volume: 20 mL) was added in the presence of hydrogen peroxide (1.5-2 mL; 32%, w/w) for the dissolution of organic electrolyte. After complete dissolution (total volume: 10 mL); the solution was diluted with demineralized water into a 100-mL volumetric flask for analyzing metal content. Table S2 reports the metal contents in the Li-ion battery (powder).

The batteries were grinded and the powder (particle size range around 1 mm) was initially mixed for 2 h with 5 M HCl solution, at T: 100 • C (in a closed reactor). The velocity of agitation was set to 150 rpm. The solid/liquid (S/L) ratio was fixed to 1:3 (50 g for 150 mL). In order to enhance the leaching of the metals from the battery, a few drops of concentrated HCl and H 2 O 2 were progressively added to the mixture (final HCl concentration ≈7 M). The leachate was collected for processing Li(I) sorption using Sulfo-C sorbent. The volume of leachate collected was 125 mL.

Processing of leachates

The effect of pH on metal sorption was tested in the range pH eq :

1-3.8. The selectivity in metal recovery was measured both for asproduced leachates and for oxalic acid-completed leachates (1% oxalic acid solution, w/w). The sorbent dose was 1 g L -1 , T: 20 ± 1 • C, and the contact time: 10 h.

Results and discussion

Summary of the characterization of sorbents

The detailed physicochemical characterization of the sorbents is discussed in section B (in SI). Herein, main characterizations are summarized. The particle size (sieving and SEM observation) allows fixing the average particle size close to 20 ± 5 µm (Figure S2). The textural analysis shows that the functionalization of crosslinked chitosan increases the specific surface area from 9 to 27 m 2 g -1 (with relevant increase of porous volume) (Section B.2.1., Figure S3). The study of thermal degradation of Chito and Sulfo-C sorbents shows 3 and 4 transitions steps (respectively) with a global weight loss that decreases from 88.5% to 80.3 % after sulfonation of Chito (Section B.2.2., Figure S4).

The effective grafting of sulfonate groups is clearly demonstrated by elemental analysis (Table S3): while N content hardly changes between Chito and Sulfo-C, the functionalized sorbent contains S element (1.27 mmol S g -1 ; with simultaneous increase in O content, as RSO 3 -moiety).

This chemical modification is also characterized by FTIR analysis (Section B.2.3., Table S4, Figures 1 andS5). The sulfonation shifts the large band at 3400 to 3421 cm -1 (corresponding to the overlapping of NH and OH stretching vibrations). The C=O ester stretching vibration (at 1729 cm -1 ) almost disappears; this is followed by the enlargement of the Amide I band (C=O stretching at 1646 cm -1 , which is shifted to 1621 cm -1 ). The reinforcement of C-N stretching appears on the 1418 cm -1 band (shifted to 1451 cm -1 ), while several bands representative of Sbased vibrations appear after modification; meaning doublet at 633 and 590 cm -1 for sulfonate groups, 1261 cm -1 (-S=O), 1161 and 607 cm -1 (sulfonamide), 788 cm -1 (C-O-S stretching), and 527 cm -1 (C-S). The Xray photoelectron spectroscopy also contributes to demonstrate the efficiency of sulfonation as appearing in Fig. 2 (survey XPS curve). (a) the broad band in the range 1350-1100 cm -1 , and. (b) the bands at 800 cm -1 (C-O-C of the polysaccharides and nitrate ions from the medium [START_REF] Coates | Interpretation of Infrared Spectra, A Practical Approach[END_REF]), 662 cm -1 (broad band that may include N-O contribution, from nitrate), 606 cm -1 (associated with C-S and C-O-S stretching vibrations), and 445 cm -1 (for polysulfide (S-S) stretching [START_REF] Coates | Interpretation of Infrared Spectra, A Practical Approach[END_REF]).

These changes result from shifts of existing bands and formation of new vibrations associated with the modification of the environment of amine and sulfonate reactive groups (due to protonation and/or interactions with Li cations). In order to verify the proper effect of sorbent protonation on its spectrum, Figure S5 compares the spectra of the Sulfo-C as-produced, after being in contact with water at pH 0 2, and after Li(I) sorption (also at pH 0 2). Table S4 also reports the assignments of the different bands. Most of the shifts observed after Li(I) sorption are also reported for the protonated sorbent (Sulfo-C(P)). The most significant differences (which could be more precisely attributed to Li-sorbent interaction) correspond to:

(a) the band at 3422 cm -1 for Sulfo-C(P) (comparable to raw sorbent) and lower than in the case of Li-loaded sorbent (at 3446 cm -1 ), (b) the band appearing at 1621 cm -1 in Sulfo-C, shifted to 1628 cm -1 for Sulfo-C(P) and to 1637 cm -1 after metal binding; this band is assigned to C=O stretching vibration (Amide I band), (c) the band at 1418 cm -1 in Sulfo-C, shifted to 1429 cm -1 for Sulfo-C(P) and up to 1438 cm -1 after Li(I) uptake; the C-N stretching vibration is affected by metal binding (modification of C-N chemical environment).

(d) the weak band at 1161-1157 cm -1 for sorbent (raw and protonated) is replaced with a strong and broad band shifted to 1188 cm -1 after Li(I) binding (which overlaps other signal present in this region on the sorbent); sulfonamide and S=O groups are involved in Li(I) binding.

(e) the new bands appearing at 446 cm -1 and 416 cm -1 (with high intensity) after metal binding can be assigned to polysulfide-type band (appearing as a small signal at 467 cm -1 on protonated sorbent). This is probably due to the tautomeric effect that displaces some bonds to form this polysulfide-based compound. The same mechanism of acidic tautomerization can explain the formation of the band at 445 cm -1 for the sorbent analyzed after fives cycles of sorption and desorption.

After metal desorption, the FTIR spectrum of the sorbent is partially restored. The main differences concern the broad band at 3444 cm -1 (not restored to 3421 cm -1 ). The desorption is operated with 0.2 M HCl solution (and water rinsing), which does not allow turning back to Sulfo-C FTIR spectrum. The other FTIR region that is substantially different from original Sulfo-C spectrum concerns the broad band initially present at 1621 cm -1 ; this band is replaced with a doublet at 1737 cm -1 and 1634 cm -1 . These bands (well-resolved) are probably associated with carbonyl groups (1737 cm -1 ) and protonated amine groups (1634 cm -1 ). The intermolecular rearrangements resulting from tautomerization and acidic conditions (associated with elution step) induce formation of carbonyl bands as shown by the appearance of the band at 1737 cm -1 , but also the broadening and increase in intensity of the band at 1164 cm -1 (which may be assigned to sulfonamide group). Therefore the changes observed in these regions are probably more associated with the protonation of the sorbent than to the direct interactions with Li(I) (which, apparently, concern more specifically the broad band 1350-1100 cm -1 and the band at 662 cm -1 ).

XPS characterization

Figures S8 and S9 show the high resolution XPS spectra of the sorbents after lithium binding (as a complement to XPS survey curve, Fig. 2). Table S5 summarizes the assignments of the binding energies (BEs) for selected signals (and their atomic percentages, At.%) for the sorbents before and after metal binding. After Li(I) sorption, the signals are also strongly affected. The weak sorption of Li(I) onto Chito leads to poorly resolved Li 1 s signal appearing as a single peak at BE: 55.5 eV (assigned to Li form). The sulfonation enhances metal sorption, and the relevant signal is considerably more intense; in addition, the deconvolution of the signal shows two components at BE: 55.9 eV and 58.1 eV, assigned to Li-N and Li-SO 3 , respectively. The signals for N 1 s are affected with the appearance of a specific band at BE 400.4 eV (N-Li) and the decrease in intensity of the other N bands. In the case of Sulfo-C, a new component for O 1 s band appears at 530.0 eV associated with O-Li bond (in sulfonate environment). The other components are slightly shifted in BE, and relative atomic percentages (At., %): substantial increase in the contribution of O-N and strong decrease in the intensity of the O-S bond. This confirms that both amine groups and sulfonic groups are involved in lithium uptake with Sulfo-C sorbent. On the opposite noteworthy that the same titration (pH-drift) method allows observing a new decrease of the pH PZC of Sulfo-C after Li(I) sorption, while after metal desorption (including after 5 cycles of sorption/desorption) the titration curve almost overlaps with the initial curve (as a proof of the macroscopic stability of the sorbent).

Chemical characterization of Li(I) interactions with sorbents 3.2.1. FTIR characterization

The sorption of Li(I) from acidic acid solution may involve substantial changes in the FTIR spectra due to the shifts of bands associated with the protonation of some reactive groups and to the direct interaction of some reactive groups with Li cations (Fig. 1). Hence, the sorption of lithium induces the shift of the 3421 cm -1 band to 3446 cm -1 probably due to the protonation of amine groups and to Li(I) binding with amine groups. The changes observed in the region 1640-1620 cm -1 confirm the contribution of amine/amide groups. A series of strong bands are observed after Li(I) sorption:

The kinetic profiles are compared in Fig. 4 for Chito and Sulfo-C. The superposition of duplicate curves confirms the reproducibility of sorption tests. The equilibrium is reached within 60-90 min at T: 20 • C. This equilibrium time is relatively long, taking into account the small size of sorbent particles (i.e., 20 ± 5 µm). Ones would expect faster sorption for micron-size sorbents where the diffusion path is limited to 10 µm-length. This means that the resistance to intraparticle diffusion, which is one of the possible limiting steps, may play a non-negligible role in the overall control of uptake kinetics. The agitation rate was set to 210 rpm, based on preliminary studies that showed fine dispersion of particles and negligible effect of resistance to bulk and film diffusion. The other mechanism that may control uptake kinetics is the proper reaction rate of sorption, which can be described by pseudo-first and pseudo-second order rate equations (Table S1a). Fig. 4 also reports the uptake kinetics for T: 50 • C in the case of Sulfo-C sorbent (under similar experimental conditions). Consistently with Fig. 3, the equilibrium concentration is higher than that at T: 20 • C, because the sorption is exothermic. However, the equilibrium is reached within a shorter contact time (i.e., ≈30 min). The thermal activation globally enhances the mass transfer characteristics (probably by improvement of diffusional movements inside the sorbent) or the proper reaction rate.

Table 1 summarizes the parameters of the three models (PFORE, PSORE and RIDE). Based on both R 2 and AIC values, the kinetic profiles are better fitted by the PFORE. In Fig. 4, the lines represent the fits of experimental profiles with the PFORE (PSORE and RIDE fits are reported in Figure S12). The calculated values for q m slightly overestimate the experimental values of the sorption capacity at equilibrium (actually by less than 10 %, meaning much less than PSORE). As expected from experimental profiles, the apparent kinetic rate (i.e., k 1 ) are of the same order of magnitude for Chito and Sulfo-C at T: 20 • C, in the range 2.58-3.47 × 10 -2 min -1 , while for T: 50 • C k 1 is almost tripled (8.94-8.75 × 10 -2 min -1 ). Simonin [START_REF] Simonin | On the comparison of pseudo-first order and pseudo-second order rate laws in the modeling of adsorption kinetics[END_REF] and Hubbe et al. [START_REF] Hubbe | Implications of apparent pseudo-second-order adsorption kinetics onto cellulosic materials: A review[END_REF] published comprehensive analyses of the literature dedicated to the discussion of kinetic modeling. They both commented that in most cases the PSORE model is abusively assigned to chemisorption because of inappropriate selection of experimental conditions or processing of data. More specifically, this model corresponds, in many cases (involving inappropriately designed experimental conditions), to systems controlled by the resistance to intraparticle diffusion.

Despite the poor fitting of experimental profiles (especially in the zone of high curvature) by the RIDE (i.e., Crank equation), the effective diffusivity coefficient is used here for comparing the systems and as a confirmation of the contribution of resistance to intraparticle diffusion in the control of uptake kinetics. Indeed, D e values are close to 7.8-6.7 × 10 -13 m 2 min -1 , 4.7-4.5 × 10 -13 m 2 min -1 , and 1.8-1.7 × 10 -12 m 2 min -1 for Chito (at T: 20 • C), Sulfo-C at T: 20 • C, and Sulfo-C at T: 50 • C, respectively. These values are several orders of magnitude lower than the molecular diffusivity of lithium in water (i.e., 6.17 × 10 -8 m 2 min -1 , [START_REF] Marcus | Ion Properties[END_REF]). This is another confirmation that the resistance to intraparticle diffusion cannot be neglected in the control of the kinetics of lithium sorption. The thermal activation favors the Brownian diffusion in the

Li(I) sorption properties 3.3.1. Effect of pH

Fig. 3 compares the pH effect on Li(I) sorption using both Chito and Sulfo-C. Duplicate experiments confirm the experimental reproducibility and the stability in sorption performances (especially for Sulfo-C, in the complete pH range). The sulfonation of the precursor (i.e., Chito) strongly improves the maximum sorption capacity from ≈1.8 mmol Li g -1 to ≈12 mmol Li g -1 . The optimum pH for the two sorbents is close to 2-2.5. It is noteworthy that for Sulfo-C the sorption capacity strongly increases with the pH between 1 and 2 and sharply decreases with the pH above pH 3. It is also remarkable that the lowest sorption capacities at pH ≈1 and pH ≈5.5 remain in the range 1-2 mmol Li g -1 (i.e., comparable to the highest q eq values for Chito). In the case of Chito, the sorption capacity weakly decreases above the optimum (i.e., pH 2-2.5): from 1.8 to 1.4 mmol Li g -1 . The weaker impact of pH for Chito sorbent is probably associated to the highest value of pH PZC that maintains the surface of the sorbent positively-charged. In strongly acidic solutions (i. e., pH 1), the competition of protons limits the sorption of Li + . Above pH 3, the protonation of the surface of Sulfo-C progressively decreases, which, in turn, may limit the ion-exchange of Li cation with protonated reactive groups of the sorbent. Sulfonic groups are more affected by deprotonation effect than amine groups; this may explain that at pH 5.5 Sulfo-C maintains sorption capacity at relatively high level (same order of magnitude than Chito). Fig. 3 also shows the effect of pH on Li(I) sorption at a higher temperature (i.e., 50 ± 1 • C). The optimum pH is not influenced by the temperature, remaining close to 2-2.5. However, the maximum sorption capacity significantly decreases at T: 50 • C compared with 20 • C: 8.2 vs. 11.7 mmol Li g -1 . This is a first indication of the exothermic nature of Li(I) sorption. Physical sorption is frequently associated with exothermic characteristics.

Figure S11a and Section C.1. (in SI) show that the pH hardly varies with metal sorption. The slope of the log 10 plot of the distribution ratio (D = q eq /C eq , L g -1 ) vs. pH eq significantly varies with the sorbent (Figure S11b). The slope for Sulfo-C is close to + 1; this is consistent with the ion-exchange of one proton with one Li + . On the opposite hand, for Chito the slope is reduced (close to + 0.38); the sorption mechanism is different (consistently with FTIR and XPS characterizations).

Uptake kinetics

sorbents, and the global mass transfer (though at the expense of lower sorption capacities).

Sorption isotherms and thermodynamic parameters

The sorption isotherms summarize the distribution of the solute between liquid and solid phases at equilibrium for increasing concentrations, under the same experimental conditions (including pH and temperature). Fig. 5 shows the comparison of sorption isotherms at pH 2 for Chito (at T: 20 • C) with the profiles for Sulfo-C at T: 20, 30, 40, and 50 • C. Again, the good superposition of experimental curves demonstrates the reproducibility of sorption tests. As expected from previous results, the sorption efficiency strongly increases with the sulfonation of the precursor, and significantly decreases with the increase of the temperature for Sulfo-C. The superiority of Sulfo-C may be explained by acid-base properties (Section 3.1), the diversity, and reactivity of functional groups (Section 3.2.). Lithium is classified among the hard acids according the HSAB principles [START_REF] Pearson | [END_REF]. Therefore, the cation has a marked preference for hard bases (i.e., higher affinity for O-bearing ligands than N-bearing ligands).

The experimental profiles have been modeled using the Langmuir, the Freundlich, the Sips, and the Temkin equations. Table 2 reports the Freundlich parameters for the sorption isotherms, while those values for the other equations are summarized in Table S6. The comparison of R 2 and AIC values allows discarding the Langmuir, Sips and Temkin equations; Figure S13 shows the Langmuir fits of experimental profiles, which fail to correlate the isotherms in the highly-curved sections. This conclusion may be partly explained by the fact that the saturation of the sorbents is not reached even with a residual concentration as high as 60-68 mmol Li L -1 . The power-type expression of the Freundlich equation accounts for the unsaturated shape of the isotherm. Another explanation may be associated to the coexistence of two types of reactive groups, as observed through the study of interaction modes (Section 3.2.). In order to address this hypothesis, the Langmuir Dual Site (LDS) equation was tested for fitting experimental data [START_REF] Astuti | Methyl violet dye removal using coal fly ash (CFA) as a dual sites adsorbent[END_REF]:

q eq = q m1 × b 1 × C eq 1 + b 1 × C eq + q m2 × b 2 × C eq 1 + b 2 × C eq (1)
where (q m1 , b 1 ) and (q m2 , b 2 ) are the Langmuir parameters for sorption sites S1 and S2, respectively (with mmol g -1 and L mmol -1 units, respectively).

The parameters and the statistical indices are reported in Table 2 (together with Freundlich modeling, Fig. 5a). Though the LDS equation fits less accurately the experimental profiles than the Freundlich equation, the quality of the correlation is globally better than with those of the other equations. Fig. 5b displays the LDS fitting of experimental profile. The coexistence of different reactive groups with different affinity for Li(I) is illustrated by the large variation in the b Li coefficients. The sites S 1 are characterized by high q m,L1 values and very low b L1 , while the sites S 2 show lower q m,L2 values (2-3 times) but much higher affinity coefficients than for S 1 sites (85-244 times). S 1 sites are probably associated with amine groups, while S 2 sites are assigned to sulfonic groups (almost irreversible/rectangular isotherm). Section C.2. (in SI) reports complementary analysis of the effect of temperature on the relative contributions of the two types of reactive group; apparently the contribution of amine groups increases with temperature (Figures S14-S15).

Tran et al. [START_REF] Tran | Thermodynamic parameters of liquid-phase adsorption process calculated from different equilibrium constants related to adsorption isotherms: A comparison study[END_REF] recently commented that the Freundlich constants cannot be used for calculating thermodynamic parameters using the van't Hoff equation due to the inappropriateness of units. Herein, the van't Hoff equation was individually applied to the two terms of the Langmuir dual site equation (i.e., b L1 and b L2 ). Lima et al. [START_REF] Lima | A critical review of the estimation of the thermodynamic parameters on adsorption equilibria. Wrong use of equilibrium constant in the Van't Hoof equation for calculation of thermodynamic parameters of adsorption[END_REF] pointed out the importance to convert the affinity coefficient of the Langmuir equation to make it dimensionless before applying the van't Hoff equation [START_REF] Tran | Thermodynamic parameters of liquid-phase adsorption process calculated from different equilibrium constants related to adsorption isotherms: A comparison study[END_REF].

ΔG • = -RTlnK 0 Eq (2a) ΔG • = ΔH • -TΔS • (2b) lnK 0 Eq = - ΔH • RT + ΔS • R (2c)
K eq 0 is the standard thermodynamic equilibrium constant (dimensionless), where in the case of Langmuir equation gives:

K 0 eq = b L × C 0 Li γ Li = b 0 L,i (2d)
With C Sorbate • (1 mol L -1 ) and γ Sorbate (dimensionless) are the unitary standard concentrations of sorbate and the activity coefficient of sorbate in solution, respectively. Frequently, in the studies on sorption processes, the concentration range is limited enough to consider that the activity of the sorbate is close to the unity. The maximum residual concentration in Fig. 5 reaches 70 mmol Li L -1 ; this makes the approximation inappropriate. The activity coefficient was evaluated close to 0.874 (using Eq. (2e)).

ln(γ

Li ) = -0.509z 2 Li ̅̅ I √ , withI = 1 2 ∑ i z 2 i c i (2e) ln ( b L × C 0 Li γ Li ) = - ΔH • RT + ΔS • R (2f)
The enthalpy changes are systematically negative as a confirmation of the exothermic nature of Li(I) sorption onto Sulfo-C (Table 3). In addition, the order of magnitude of ΔH • is relatively low (i.e., -13/-21 kJ mol -1 ), meaning that the uptake proceeds through physisorption. The entropy changes are also systematically negative: the global randomness of the system decreases with Li(I) sorption and the sorption Units: q, mmol g -1 ; k 1 , min -1 ; k 2 , g mmol -1 min -1 ; D e , m 2 min -1 .

Table 1

Modeling of uptake kinetics for Li(I) removal using Chito and Sulfo-C sorbents.

is enthalpy-driven. Table 4 compares the sorption properties of Chito and Sulfo-C sorbents with alternative sorbents recently characterized in the literature. The functionalized sorbent is ranked among the most efficient, at least in terms of maximum sorption capacity although the affinity coefficient (b L ) remains relatively low compared with, for example, Lewatit TP207 [START_REF] Arroyo | Lithium recovery from desalination brines using specific ion-exchange resins[END_REF], MnO 2 and ion sieves [START_REF] Wang | Lithium uptake in fixed-pH solution by ion sieves[END_REF][START_REF] Liu | Lithium extraction from seawater by manganese oxide ion sieve MnO 2 .0.5H 2 O, Colloids Surf[END_REF] or spinel Li 4 Mn 5 O 12 sorbent [START_REF] Xiao | Lithium ion adsorption-desorption properties on spinel Li 4 Mn 5 O 12 and pH-dependent ion-exchange model[END_REF]. It is noteworthy that most of the existing literature focuses on Li recovery from neutral or alkaline solutions; this makes the strict comparison relatively difficult since the current work considers lithium sorption from acidic solutions (i.e., pH 2). The closest experimental conditions (at pH 3 and above) were reported by Cicek et al. [START_REF] Cicek | Removal of lithium from water by aminomethylphosphonic acid-containing resin[END_REF] using an aminomethylphosphonic resin: the maximum sorption capacity reached 2 mmol Li g -1 and the affinity coefficient was reported close to 0.0003 L mmol -1 ; i.e., significantly lower than the values reported for Sulfo-C.

Sorption mechanism

The different data collected from FTIR and XPS analyses (and pH PZC determination), the study of pH effect leads to the identification of different modes of interaction with Li(I) cations. Indeed, the coexistence of free amine groups (and hydroxyl groups) and sulfonic groups (grafted onto the biopolymer) opens the possibility for lithium to interact with NH 2 groups (XPS analysis showed that amine groups are not totally engaged in crosslinking with GA), -OH, and -SO 3 H. XPS analysis showed also that the environments of O-bearing and N-bearing groups are affected by lithium binding (O-Li, and N-Li signals). The interactions of lithium with sulfonic groups may involve direct ion-exchange mechanism but also interactions associated with the tautomerization of sulfone groups. This is also shown in the FTIR analysis through (a) the decrease of the relative intensity of OH and NH groups and/or (b) shifts in the wavenumbers following lithium sorption. Since the sorption is enhanced at pH lower than the full deprotonation of the sorbent, this means that the cation exchange properties of the sorbent are engaged in Li + uptake (via Li + exchange with H + from OH, NH and SO 3 H groups)

and tautomerization of available π bonds of sulfonic moieties. Scheme 2

shows tentative modes of interaction between functional groups and lithium cation.

Competitor effects and sorption selectivity

The sorption performance may be affected by the complexity of the solution, including the effect of ionic strength. A very abundant literature deals with lithium recovery from brines [START_REF] Arroyo | Lithium recovery from desalination brines using specific ion-exchange resins[END_REF][START_REF] Luo | Extraction of lithium from salt lake brines by granulated adsorbents[END_REF]. Figure S16 focuses on the impact of NaCl on the sorption properties of Li(I) using Sulfo-C. The optimum pH remains close to 2-2.5 (Figure S16a); however, the sorption capacity progressively decreases with the concentration of the salt: from 11.7 mmol Li g -1 (as the reference) to 4.57 mmol Li g -1 in 5 M (Na,Cl) solution. The strong impact of ionic strength is frequently associated with ion-exchange mechanisms. In seawater, the Na,Cl salt concentration is around 0.5 mol kg -1 , the sorption capacity (at pH ≈2) for 1 M Na,Cl solution remains as high as 10.87 mmol Li g -1 (loss lower than 7%). This means that the sorbent remains highly selective against Na + (even with an excess as large as 63 times). In Figure S16b, the distribution ratio, D = q eq /C eq (L g -1 ), linearly varies with the concentration of Na,Cl in the solution (or ionic strength of the solution, neglecting the contribution of LiNO 3 ).

Complementary tests were performed on multi-component equimolar solutions (i.e., 1 mmol L -1 ), at different pH values for optimizing the selectivity of Li(I) sorption using Sulfo-C sorbent. The selectivity coefficient (SC Li/Me ) for Li(I) against other metal ions is defined as the quotient of distribution ratios:

SC Li/Me = D Li D Me = q Li × C Me C Li × q Me (3)
Fig. 6a reports the SC values for Li against selected competitor metal cations for different pH values. On the opposite hand, Fig. 6b reports the separation properties for the same system with the addition of oxalic Units: Units: q, mmol g -1 ; b L , (L mmol -1 ) nS ; n S , dimensionless; k F , mmol 1-1/nF L - 1/nF g -1 ; A T , L mol -1 ; b T , J kg mmol -2 .

Table 3

Thermodynamic parameters for Li(I) sorption using Sulfo-C sorbent at pH 0 : 2 -Individual contributions of Sites S 1 and S 2 (deduced from Langmuir dual site modeling). Units: acid (at 1% concentration, w/w). Indeed, the addition of oxalic acid was frequently reported for improving the separation of Li(I) from other cations [START_REF] Virolainen | Removal of iron, aluminium, manganese and copper from leach solutions of lithium-ion battery waste using ion exchange[END_REF][START_REF] Roobavannan | Enhancing the performance of membrane distillation and ion-exchange manganese oxide for recovery of water and lithium from seawater[END_REF][START_REF] Verma | Lithium and cobalt recovery for lithium-ion battery recycle using an improved oxalate process with hydrogen peroxide[END_REF]. This is explained by the preferential complexation of heavy metals by oxalate (compared with lithium).

Sites S 1 S 2 Parameter ΔH • ΔS • ΔG • ΔH • ΔS • ΔG • 20 - 13
ΔH • and ΔG • , kJ mol -1 ; ΔS • , L mol -1 K -1 .
The SC Li/Me values are strongly influenced by the equilibrium pH. Consistently with the high sensitivity of Li(I) removal to the pH (optimum sorption at pH 2-2.5 for Sulfo-C), the highest selectivity for Li against other metals is obtained at pH close to 2. The selectivity at pH 2 (in absence of oxalic acid) may be ranked according to: The presence of oxalate that complexes metal cations limits their availability for being sorbed onto Sulfo-C and their ionic affinity for binding groups (especially sulfonic groups). Therefore, the competitor effect of these cations is reduced and the selectivity coefficient for Li(I) increases with the presence of oxalic acid; this is more specifically observed for Fe(III) ≫ Mg(II) > Zn(II) > Al(III) ≈ Zn(II) ≫ Ca(II) ≫ Ni (II), consistently with the ranking of the stability constant of metaloxalate complexes [START_REF] Martell | Carboxylic Acids, in: Other Organic Ligands[END_REF].

Section C.3. (in SI) shows complementary analysis of competition effects (including the variation of distribution ratios with pH, Figure S17). This allows clearly illustrating the optimum separation of Li (I) from other cations at pH 2. Some correlations are also reported (in Figures S18-S19) between the selectivity coefficients and the intrinsic physico-chemical properties of metal ions (Table S7). These SC values cannot be directly correlated to specific criteria; however, the metals were grouped (as doublets) regarding their respective impacts on competitor effect mainly through criteria such as hydrated radius and solution-phase electronegativity.

Metal desorption and sorbent recycling

Figure S20 compares the kinetic profiles for Li(I) using 0.2 M HCl solution for Chito (loaded at T: 20 • C) and Sulfo-C (loaded at T: 20 • C and 50 • C). The eluent is highly efficient: lithium is completely desorbed within 30 min of contact (under selected experimental conditions). It is noteworthy that the desorption kinetics is significantly faster for Sulfo-C sorbent loaded at T: 20 • C than for the other systems; indeed, 15 min are sufficient for total metal desorption. Fig. 7 compares the sorption and desorption efficiencies for Chito and Sulfo-C along 15 and 18 successive re-use cycles, respectively. First, for the two sorbents, the desorption remains highly efficient (around 100%) at each step, using 0.2 M HCl solution. The comparison of the sorption efficiencies shows that for Sulfo-C the performance remains remarkably stable: the loss remains close to 10% at the 18th cycle, contrary to Chito that loses about 43% at the 15th cycle. Surprisingly, (c) other heavy metals are efficiently leached (efficiency varies between 88% and 98%).

Treatment of acid leachate

Figure S22 reports the effect of pH on the sorption capacities of Sulfo-C for selected metals in absence (Fig. S22a) and presence of oxalic acid (Fig. S22b). Consistently with their relatively high proportions in the leachates, the highest sorption capacities are reported for Li(I)≫Fe(III)≈ Al(III)≫>≫Co(II) > Mn(II) > Ni(II) > Cd(II) in absence of oxalic acid. The introduction of oxalic acid slightly changes the ranking for the trace elements according Li(I)>>Al(III)≈Fe(III)>>>Ni(II) > Co(II) > Mn(II) ≈Cd(II). It is noteworthy that consistently with the sensitivity of Li(I) sorption onto Sulfo-C to pH, the sorption capacity reaches a maximum at pH eq 2-2.5, while for the other elements the sorption capacity continuously increases with pH. Obviously, derived from these results, the highest selectivity for Li(I) recovery is obtained in this specific pH range (i.e., pH 2-2.5). This conclusion is confirmed by Fig. 8. The highest SC Li/ metal values are systematically obtained at pH eq 2.28 (in absence of oxalic acid) according: Al(III)(85.7) >> Fe(III)(60.6) > Ni(II)(56.4) > Mn(II) (53.0) >> Co(II)(26.5) >> Cd(II) (10.6). The introduction of oxalic acid changes the order of selectivity coefficients at pH eq 2.17 according: Fe (III)(74.6) ≈ Al(III)(72.8) >> Co(II)(34.7) > Mn(II)(27.0) >> Ni(II) (9.7) >> Cd(II)(2.6). It is noteworthy that the presence of oxalic acid moves the optimum pH value for the separation of Li(I) from Co(II) to pH 1.07 (SC Li/Co reaches 61.4). Table S8 summarizes the values of the main criteria (i.e., C eq , sorption efficiency, q eq and distribution ratio) for the different metal ions at pH eq ≈ 2.2. Oxalic acid enhances the sorption properties of Sulfo-C for Cd(II), Ni(II) and Mn(II) (to a lesser extent). On the other hand, the recovery of Co(II), Mn(II), and Li(I) is slightly reduced (while Al(III) is hardly affected). These trends are comforted by the plots of the concentration factor (CF = q eq /C 0 , L g -1 ) appearing in Figure S23. These data allow observing:

(a) optimum concentrating effect of Li(I) at pH 2-2.5 (greater without oxalic acid), the concentrating effect increases with the pH for other metals, (b) the second metal showing the highest concentrating effect is Cd (II), (c) the presence of oxalic acid reinforces the concentrating effect of Ni(II) (and slightly reduces the values of CF for the other metal ions).

Section C.5. provides a complementary analysis of cadmium separation from battery leachate, since cadmium shows an atypical behavior compared with other divalent cations (Figures S23-S24). The addition of oxalic acid increases the selectivity of the sorbent for Cd(II) at pH ≈ 3.4 against other divalent cations. However, these conditions do not allow efficiently separating cadmium from lithium.

Despite the complexity of leachate composition (compared with synthetic pure solution -Fig. 7), the sulfonated sorbent shows good stability for Li(I) recovery (and complete metal desorption along 15 cycles) (Fig. 9): the loss in sorption efficiency does not exceed 15% at the 15 th cycle. This is a little higher than the loss observed at the 15 th step for synthetic pure solution (i.e., 8.6%). The presence of other metal ions the functionalized material reveals remarkably more stable than the precursor, which may be either chemically modified or degraded along recycling steps. Indeed, the stability in desorption performance means that the loss in sorption efficiency is not associated with the accumulation of the metal onto the sorbent (and its progressive saturation).

Section C4 (in SI) compares different eluents for testing the possibility to separate the metals through specific desorption. None of the tested eluents succeeds in separating the metals loaded on Sulfo-C (samples collected from studies on multi-component equimolar solutions) (Figure S21). Hydrochloric acid solution is the most efficient (but non-selective). Oxalic acid is highly efficient for releasing competitor ions (due to complexing effect) but not selectively (10% of lithium is codesorbed).

Application to metal recovery from waste battery 3.4.1. Acidic leaching

Table S2 shows the composition of the acidic leachate, which is characterized by the presence of three major elements: Li (≈40.0%), Al (≈30.9%), and Fe (≈22.2%), counting for 93.1% of total metal content. Valuable traces are constituted of Co (≈2.5%), Ni (≈1.9%). The remaining (≈2.5%) is formed of other heavy metals. Compared with the contents of metals in the batteries (as derived from the analysis of mineralized batch), these data show that:

Conclusion

The characterization of sulfonated chitosan by elemental analysis, FTIR, and XPS spectroscopy clearly shows the effectiveness of the functionalization. The sulfonation drastically increases the sorption of Li (I): from 4.9 mmol Li g -1 (for Chito) to 19.9 mmol Li g -1 (for Sulfo-C). Apparently both amine groups (weak) and sulfonic groups (strong) are involved in Li(I) binding; this makes the sorption isotherms better fitted by the Freundlich and Langmuir dual site equations. The strong affinity of sulfonate groups for Li(I) is the major contributor to metal sorption, especially at low metal concentration, while amine groups have lower affinity. The sorption is fast; 60 min are sufficient for reaching equilibrium. However, the contribution of resistance to intraparticle diffusion cannot be neglected. The kinetic profiles are fitted by the pseudo-first order rate equation. Increasing the temperature speeds up the kinetics but decreases the sorption capacity. Lithium sorption is exothermic. Metal desorption is highly effective using 0.2 M HCl solutions and the sorption efficiency of Sulfo-C is maintained at high level for at least 18 cycles of sorption and desorption. In acidic solutions (i.e. pH 2-2.5), the sorbent is highly selective for Li(I) against selected alkali-earth and heavy metals; the selectivity is enhanced at this pH when oxalic acid (1%, w/w) is added to the solution. These results are confirmed in the processing of acidic leachates (produced from Li-ion mobile battery). At pH 2-2.5, the sorbent is selective to Li(I) against other heavy metals. At pH ≈ 3.4, the sorbent is selective for both Li(I) and Cd(II); however, the addition of oxalic acid causes in this case a loss in the selectivity of the sorbent for Cd(II) against Ni(II).

Sulfo-C sorbent appears very promising for the recovery of Li(I) from acidic complex solutions such as acidic leachates of batteries. The recycling of Sulfo-C for 12 successive sorption/desorption cycles shows a relatively limited decrease in performances: desorption remains close to 100%, while at the 15 th cycle the decrease in sorption efficiency does not exceed 15%. The presence of large excess of NaCl affects the sorption performance; however, the sorbent maintains relatively high capacities even in concentrations of Na,Cl as high as 1 M. This preliminary result shows that the sorbent could be also applied to metal recovery from brines. For practical applications, the small size of micron-size particles would make the processing complex (in terms of solid/liquid separation). Obviously, this limits the application of the sorbent in fixed-bed columns. It would be necessary applying this sulfonation process onto hydrogel chitosan beads. An alternatively could consist in functionalizing magnetic/chitosan composites (for batch processing).

1 .

 1 Scheme 1. Suggested structures for Chito and Sulfo-C sorbents.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Comparison of the FTIR spectra of Sulfo-C (raw), Sulfo-C (after contact with pH 0 2 solution), and Sulfo-C after Li(I) sorption at pH 0 2. Fig. 2. XPS survey of sorbent before and after Li(I) sorption.

Fig. 3 .

 3 Fig. 3. Effect of pH on Li(I) sorption using Chito and Sulfo-C sorbents (Sorbent dose, SD: 0.4 g L -1 ; Time: 48 h; v: 210 rpm; T: 20 ± 1 • C).

Fig. 4 .

 4 Fig. 4. Uptake kinetics for Li(I) removal using Chito and Sulfo-C sorbents at pH 0 2 (including temperature effect on lithium sorption onto Sulfo-C) -Modeling with the PFORE (C 0 : 15 mmol Li L -1 ; SD: 0.4 g L -1 ; v: 210 rpm).

Fig. 5 .

 5 Fig. 5. Modeling of Li(I) sorption isotherms at pH 2 using Chito and Sulfo-C sorbents using Freundlich equation (a) and Langmuir dual site equation (b) (SD: 0.4 g L -1 ; time: 48 h; v: 210 rpm; T: 20 ± 1 • C; pH eq : 2-2.13 for Chito and 2.2-2.3 for Sulfo-C; in (b) panel the Langmuir equation is applied for Chito instead of Langmuir dual site equation).

  Fe(III) (35.8) ≫ Zn(II) (22.7) > Mg(II) (20.0) > Ca(II) (16.4) > Ni(II) (15.4) > Al(III) (14.4). In the presence of oxalic acid, the selectivity slightly increases (except for Fe(III) where the SC Li/Fe is almost doubled) but the ranking is weakly affected (larger variations reported for Ca(II), Al(III) and Ni(II)): Fe(III) (69.3) ≫ Zn(II) (28.2) ≈ Mg(II) (28.1) > Al(III) (21.4) ≈ Ca(II) (20.9) > Ni(II) (16.6).

2 .

 2 Tentative mechanisms of Li(I) sorption onto Sulfo-C.

Fig. 6 .

 6 Fig. 6. Effect of pH on the selectivity coefficient (SC Li/metal ) for metal sorption from multi-component equimolar concentrations using Sulfo-C sorbentwithout (a) and with oxalic acid (1%, w/w) (b) (C 0 : 1 mmol L -1 ; SD: 1 g L -1 ; Time: 10 h; v: 210 rpm).

Fig. 7 .

 7 Fig. 7. Sorption and desorption efficiencies for the recycling of Chito and Sulfo-C sorbents for Li(I) recovery (Sorption, C 0 : 14.4 mmol Li L -1 ; SD: 0.4 g L -1 ; time: 24 h; pH 2 -Desorption, 0.2 M HCl eluent; SD: 2 g L -1 ; time: 2 h; water rinsing between each step).

  (a) Li extraction exceeds 90%, (b) Fe (the most abundant heavy metal in Li-ion batteries) is weakly leached (around 60%),

Fig. 8 .

 8 Fig. 8. Effect of pH on the selectivity coefficient (SC Li/metal ) for metal sorption from battery leachate using Sulfo-C sorbentwithout (a) and with oxalic acid (1%, w/w) (b) (SD: 1 g L -1 ; Time: 10 h; v: 210 rpm).

Fig. 9 .

 9 Fig.9. Sorption and desorption efficiencies for the recycling of Sulfo-C sorbent for Li(I) recovery using Li-ion mobile phone batterieseffect of oxalic acid addition (Sorption, C 0 : 9.33 mmol Li L -1 ; SD: 1 g L -1 ; time: 5 h; pH 2.28 -Desorption, 0.2 M HCl eluent; SD: 2.5 g L -1 ; time: 2 h; water rinsing between each step).

Table 4

 4 Comparison of Li sorption properties of Chito and Sulfo-C with literature.

	Sorbent	pH	time	q m,exp.	q m,L	b L	Ref.
	Ion-sieve Ca-alginate beads	8 7	1440 1440	-0.031	3.75 0.081	148 3 × 10 -5	[34] [41]
	K-birnessite MnO 2 ion-sieve	5-7 10	60 -a	0.332 3.05	0.399 3.13	0.0024 32.6	[42] [35]
	Spinel Li 4 Mn 5 O 12	9.74	600	0.859	0.815	139.5	[36]
	Polyacrylamide/MnO 2 ion-sieve	10.1	1800	2.64	2.95	8.60	[43]
	Spinel H 1.1 Li 0.08 Mn 1.73 O 4.05	12	720	-	7.20	0.00085	[44]
	Imprinted crown ether/macroporous membrane	9	150	3.96	4.35	0.417	[45]
	Aminomethylphosphonic resin	> 3	15	-	1.97	0.0003	[19]
	Multi-wall carbon nanotube	6	30	-	2.00	0.059	[46]
	Ion-imprinted mesoporous sorbent Lewatit K2629 (sulfonic) resin Lewatit TP207 (imino-diacetic acid) resin	7 7.84 b 7.84 b	240 30 30	1.80 --	2.32 0.265 0.366	5900 0.750 16.9	[47] [14] [14]
	Photo-responsive Li-ion imprinted polymer/C 3 N 4 Ti-intercalated LiMnO 2 composite Mo-doped/TiO 2 sieve	7 c 12 d	30 5760 1200	3.28 -11.5	7.30 3.16 12.0	0.035 0.347 0.0212	[48] [49] [50]
	H 4 Mn 5 O 12 nanotubes	12	120	5.33	5.43	0.887	[51]
	Spinel LiMn 2 O 4 nanofiber	11	1440	4.97	4.81	2.46	[52]
	K marxianus yeast	9	5	-	0.788	0.389	[53]
	Sulfonated cellulose beads	10	60	2.31	2.31	-	[54]
	Phosphorylated hazelnut shell	5.8	6	0.792	1.11	0.402	[55]
	Chito	2	90	4.87	6.58	0.037	This work
	Sulfo-C	2	120	19.9	19.9	0.284	This work

a : fixed-bed column experiment; b : sea-water reverse osmosis brine; c : Li-spiked sea-water; d : simulated brine;

Table 2

 2 Modeling of Li(I) sorption isotherms using Chito and Sulfo-C sorbents at pH 0 2 -Parameters of Freundlich and Langmuir dual site equations (only for Sulfo-C sorbent).

Acknowledgements S. M. A. acknowledges Taif University Supporting Project (TURSP-2020/105). E. G., M. F H., and H.M acknowledge the support of the French Government (Institut Français d'Egypte) and Egyptian Academy of Science & Technology (Science and Technology Development Fund) through the IMHOTEP Program (MetalValor project). W.Y acknowledges the support of National Natural Science Foundation of China [U1967218, and 11975082].

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi. org/10.1016/j.cej.2022.135941.


Vitting Jørgen

 Philippe Andersen

De Peretti

Heuristics in experiments with infinitely large strategy spaces

Keywords: complexity theory, bounded rationality, multi-period games, infinite strategy space, decoupling, agent-based modeling

We introduce a new methodology that enables detection of the onset of convergence towards Nash equilibria in simple repeated games with infinitely large strategy spaces, thereby revealing the heuristics used in decision-making. The method works by constraining on a special finite subset of strategies, called decoupled strategies. We show how the technique can be applied to understand price formation in financial market experiments by introducing a predictive measure Δ𝐷: the different between positive decoupled strategies (recommending to buy) and negative decoupled strategies (recommending to sell). Using Δ𝐷 we illustrate how the method can predict (at certain special times) participants' actions with a high success rate in a series of experiments.

Introduction

It is often difficult to obtain a proper behavioral prediction of people's actions in simple oneshot games, since they are obscured by non-equilibrium behavior which can persist even when the games are played repeatedly [START_REF] Chong | A generalized cognitive hierarchy model of games[END_REF][START_REF] Muller | Who acts more like a game theorist? Group and individual play in a sequential market game and the effect of the time horizon[END_REF]. One question then is: how can we get a better understanding of the decision-making in repeated games, and if possible the dynamics thereof, before a proper equilibrium has set in? In [START_REF] Chong | A generalized cognitive hierarchy model of games[END_REF], a generalized cognitive hierarchy was introduced to capture the fact that subjects frequently do not play a Nash equilibrium in simple one-shot games, a behavioral trait which could arise because players are heterogeneous and have different thinking abilities. As will be explained below, the fact that there are different levels of thinking ability, and the impact this could have on convergence to equilibrium, is an important issue in both the experiments and the modeling described in this article. At the same time, it illustrates a method to understand the heuristics people use in experiments where the decision-making becomes increasingly complicated.

The situation we describe here is complex since we are considering the case when the available strategy space becomes infinitely large. In situations like this it is unrealistic to think that people analyze an infinite range of strategies. Rather, it becomes more likely that people use heuristics [START_REF] Crawford | Boundedly Rational versus Optimization-Based Models of Strategic Thinking and Learning in Games[END_REF], limiting the analyses to a few simple and understandable cases. Then the question is how can we try to model such an analysis in a game theoretical framework?

Here we suggest one solution, which is to focus on a certain subset of strategies. The solution is reminiscent of the study by Ioannou and Romeo (2014), who introduced a methodology to facilitate the operability of belief-learning models with repeated-game strategies. Since the set of possible strategies in repeated games is infinite (uncountable), expecting a player to fully explore such an infinite set is unrealistic. As noted in Ioannu and Romeo (2014), and in [START_REF] Mckelvey | Playing in the dark: information, learning and coordination in repeated games[END_REF], there is also an inference problem, even for games with only two players. In repeated games there is no unique way to identify an opponent's strategy just on the basis of the history, since several different strategies can lead to the same history. This point becomes even more relevant for the type of games we present in this study, where the history is created by several (N>2) players, making it impossible for a given player to know exactly what strategies other players have adopted. Therefore, even though the history of play is publicly observable, a player will never be able to deduce an opponent's strategy in any detail.

The solution proposed by Ioannu and Romeo (2014) was to introduce repeated-game strategies implemented via a type of finite automaton, called a Moore machine, thereby limiting the infinite repeated-game strategy space to a finite subset. In our study we will focus on another special subset of repeated-game strategies, called decoupled strategies, which have the special property that their actions are independent of the action history, conditioned on observing certain histories. As will be seen, this property allows us to identify heuristic strategies automatically, since at special moments the decoupled repeated-game strategies then effectively correspond to simple one-shot strategies. We would also like to point out the similarity with collective action games; see for example [START_REF] Anauati | Collective action: experimental evidence[END_REF], who introduced a method using stability sets. They were able to capture the case where increases in the payoff of a successful collective action led to an update in prior beliefs about the expected share of cooperators. As will be seen, we encounter similar situations where increases in the payoff of a successful aggregate trading action, leads to an increase in the probability of the use of certain trading strategies.

As will be explained below, in the following we will use the repeated minority game (or more precisely: a slightly modified version called the $-game) as a case study for this paper. The definition of the minority game is presented in the section below, it was introduced by Challet and Zhang (1997) as a simple model to describe price formation in financial markets. Some of the first analytical results on the model was made by Challet et al (2000b), see also [START_REF] Challet | On the minority game analytical and numerical studies[END_REF]. [START_REF] Lamper | Predictability of large future changes in a competitive evolving population[END_REF] were the first to point out that the minority game had the particular property of showing predictability of large future price changes. The mechanism leading to such large price changes were however not identified in [START_REF] Lamper | Predictability of large future changes in a competitive evolving population[END_REF]. [START_REF] Andersen | A Mechanism for Pockets of Predictability in Complex Adaptive Systems[END_REF] introduced a mechanism that could describe such "pockets of predictability", by considering a certain class of strategies, called decoupled strategies. In the following we will use such decoupled strategies to identify the heuristics used by the participants in experiments on financial markets. For some of the few experimental efforts that has been done to understand the minority game, a nice and general multi-round strategy experiment on the minority game can be found in [START_REF] Linde | Strategies and evolution in the minority game: a multi-round strategy experiment[END_REF].

2 Simple multi-period market games: theoretical framework When the economist Brian Arthur introduced his famous El Farol bar game [START_REF] Arthur | Inductive reasoning and bounded rationality[END_REF], it set off a flurry of research on simple binary choice decision-making models, with decisions like "go/stay", "yes/no", "buy/sell", "right/left", etc. In Arthur's model, a population of N=100 people want to go to the El Farol bar (a bar that really exists in Sante Fe) to listen to folk music. However, the El Farol is quite small, and people are only satisfied if they can actually sit on one of the 60 available seats. People have to make their decision at the same time, so if everybody uses the same pure strategy, it will fail: suggesting "go" (assuming an empty bar), everybody will go and the bar will be crowded, but suggesting "stay" (assuming a crowded bar) will instead lead to an empty bar. [START_REF] Challet | Emergence of cooperation and organization in an evolutionary game[END_REF] extended the El Farol model to describe financial market behavior, where the binary choice was now a simple "buy/sell" decision. More precisely, in the minority game, there is an odd number N of players who use different strategies to try to remain on the minority side. If a strategy 𝑆 predicts that the majority will buy, then that strategy will recommend to sell, 𝑆 = -1. Otherwise, if it predicts that the majority will sell, the recommendation will be to buy, 𝑆 = 1. The payoff of strategy i is 𝜋 (𝑠 ) = -𝑠 𝐴, 𝐴 = 𝑠 where A is the order imbalance (i.e., the difference in buy and sell orders). As mentioned in [START_REF] Linde | Strategies and evolution in the minority game: a multi-round strategy experiment[END_REF], the one-shot minority game already has quite a large number of Nash equilibria. This happens since any case where exactly (N-1)/2 players choose one side (and (N+1)/2 players the other side) constitutes a Nash equilibrium. The number

! ! !
of Nash equilibria is already quite large even for moderate values of N.

In the multi-round minority game, players then use strategies that consider the outcome of not only the last, but the past M market price directions (for a positive order imbalance A, the market goes up; when A is negative, the market goes down). Each player holds the same number of strategies S, assigned randomly at the beginning of the game. A strategy in the multi-round game issues a prediction of the next market move (buy/sell) for each of the possible 2 past price histories. A strategy in the multi-round minority game is therefore a vector of size 2 instructing whether to buy (1), or sell (-1) for each possible price history. An example of a multi-round strategy could for example be (1,0,1)  1, (M=3), which means that in a market where the last price movement was up, the one before down, and the price movement three time steps ago up, it would recommend to buy. The total number of different strategies is therefore 2 . In the multi-round minority game, the players record the cumulative payoff for each of their S strategies, and at any given time, use the one which has the highest cumulative payoff.

For an extensive review of the multi-round minority game, see for example [START_REF] Cavagna | Thermal model for adaptive competition in a market[END_REF]Challet et al. 2000a;Challet, et al. 2000b;[START_REF] Challet | Stylized facts of financial markets and market crashes in minority games[END_REF][START_REF] Challet | On the minority game analytical and numerical studies[END_REF][START_REF] Johnson | Crowd effects and volatility in markets with competing agents[END_REF][START_REF] Lamper | Predictability of large future changes in a competitive evolving population[END_REF].

As mentioned in [START_REF] Andersen | The $-game[END_REF], one problem with the multi-round minority game describing simple market price dynamics is that it does not account for speculative behavior, in which investors invest to gain a return. It should be noted that the minority dynamics prevents any trend to develop, giving rise to mean-reverting price dynamics. In order to capture speculative behavior as seen in bubble/crash phases of financial markets, a modification of the payoff function was suggested in [START_REF] Andersen | The $-game[END_REF]. Denoting the game, the $-game (to describe players that speculate), the modified payoff reads:

𝜋 $ (𝑠 (𝑡)) = 𝑠 (𝑡 -1)𝑅(𝑡)
The payoff favors strategies which are able to predict the price movement over the following time step. Predicting at time t-1 a price increment over the following time step, the strategy i proposes to enter a buy position at time t-1, so 𝑠 (𝑡 -1) = 1. If the prediction was successful (a failure), the payoff gained (lost) is the return of the market over that time step.

Theorem 2.1 The Nash equilibria in the multi-period $-game consist of the two strategies

(1,1,….,1), (-1,-1,…,-1).

Proof We will prove theorem 2.1 for the simple case where each agent has only s=2 strategies.

The dynamics of the $-game is driven by a nonlinear feedback mechanism, because each agent uses his/her best strategy (fundamental/technical analysis) at each time step. In its turn, the sign of the order imbalance, ∑ 𝑎 * ℎ ⃗ (𝑡) , determines the value of the last bit 𝑏(𝑡) at time 𝑡 for the price movement history ℎ ⃗ (𝑡 + 1) = (𝑏(𝑡 -𝑚 + 1), 𝑏(𝑡 -𝑚), … , 𝑏(𝑡)). The * in 𝑎 * denotes the best strategy (out of s possible) for agent i. The dynamics of the $-game can then be expressed in terms of an equation that describes the dynamics of 𝑏(𝑡) as

𝑏(𝑡 + 1) =  ∑ 𝑎 * ℎ(𝑡) (1)
where  is a Heaviside function taking the value 1 whenever its argument is greater than 0 and the value 0 otherwise, and ℎ(𝑡) = ∑ 𝑏(𝑡 -𝑗 + 1)2 is now expressed as a scalar instead of a vector. The nonlinearity of the game can be seen formally from

𝑎 * ℎ(𝑡) = 𝑎 | ,…, ∑ ( ) ∑ * ( ) ℎ(𝑡) (2)
Equation ( 2) expresses the idea that the optimal strategy * of agent i is the strategy j which maximizes the $G payoff between times k=1 and k=t. Inserting (2) in (1), we obtain an expression that describes the $-game in terms of just a single equation for 𝑏(𝑡) depending on the values of the 5 base parameters (𝑚; 𝑠; 𝑁; ; 𝐷(𝑡)) and the random variables 𝑎 (i.e., their initial random assignments).

We would like first to point out an important difference compared to traditional game theory, since in our game the agents have no direct information concerning the action of the other players. The only (indirect) information a given agent can have of another agent's action comes through the aggregate actions of the past, i.e., the past price behavior.

In the following we will consider the simple case where each agent has only s=2 strategies.

Then considering only the relative payoff between strategy 𝑎 and 𝑎 , the equations simplify considerably. Let the action of the optimal strategy 𝑎 * be expressed in terms of the relative payoff, 𝑞 , so as to formulate ∑ 𝑎 * ℎ(𝑡) as follows:

∑ 𝑎 * ℎ(𝑡) = ∑  𝑞 ℎ(𝑡) 𝑎 ℎ(𝑡) + 1 - 𝑞 ℎ(𝑡) 𝑎 ℎ(𝑡) (3)
Inserting ( 3) into (1) and taking the derivative of 𝑏 at 𝑡 + 1,

= 𝛿 ∑ 𝑎 * ℎ(𝑡) ∑ 𝛿 𝑞 ℎ(𝑡) ( ) 𝑎 ℎ(𝑡) -𝑎 ℎ(𝑡) + 𝑞 ℎ(𝑡) ( ) + 1 - 𝑞 ℎ(𝑡) ( ) (4)
Looking at the terms inside the curly bracket in (4), it follows that a change in ∑ 𝑎 * ℎ(𝑡)

can occur in two different ways: (i) because the optimal strategy changes and the two strategies for a given ℎ(𝑡), 𝑎 ℎ(𝑡) and 𝑎 ℎ(𝑡) , differ from each other (first term in the bracket); (ii) because the optimal strategy changes its prediction for the given ℎ(𝑡) (second and third terms in the bracket).

The rate of change of the relative payoff 𝑞 is computed from

= 𝑎 ℎ(𝑡 -2) ∑ 𝑎 * ℎ(𝑡 -1) -𝑎 ℎ(𝑡 -2) ∑ 𝑎 * ℎ(𝑡 -1) (5)
Using ℎ(𝑡) = ∑ 𝑏(𝑡 -𝑗 + 1)2 and inserting (5) in (4), we obtain

= 𝛿 ∑ 𝑎 * ℎ(𝑡) ∑ 𝛿 𝑞 ℎ(𝑡) ∑ 𝑎 * ℎ(𝑡 -1) a b(t -j -1)2 -a ∑ b(t -j -1)2  𝑎 ∑ 𝑏(𝑡 -𝑗 + 1)2 -𝑎 ∑ 𝑏(𝑡 -𝑗 + 1)2   𝑞 ℎ(𝑡) ∑ ( ) + 1 - 𝑞 ℎ(𝑡) ∑ ( ) (6)
If ∑ 𝑎 * ℎ(𝑡 -1) , ∑ 𝑎 * ℎ(𝑡 -2) , … , ∑ 𝑎 * ℎ(𝑡 -𝑚) all have the same sign, the right-hand-side of (6) becomes 0, thus proving that a constant bit 𝑏(𝑡), corresponding to either an exponential increase or decrease in price, is a Nash equilibrium.

3 Introducing decoupled strategies to determine the onset of convergence towards Nashequilibria in multi-period market games: theoretical framework

As mentioned, the mere size of the strategy space for one-shot games is 2 , which considerably complicates a proper understanding of simple market models like the minority game and the $-game, despite the simplicity of their payoff functions. It is therefore out of the question to explore the full strategy space in order to gain insights into the way people would react, even in simple market games like this. We propose instead to concentrate on a certain subclass of strategies.

Let us call 𝑠 (𝑡 | ℎ ⃗ (𝑡)) the action of strategy 𝑠 at time t, conditioned on observing a given price history, ℎ ⃗ (𝑡), at time t over the last M time steps. ℎ ⃗ (𝑡) is a binary string of -1's and +1's describing the last M directions of price movements observed at time t. We now note that some strategies will be independent of ℎ ⃗ (𝑡) over the next L time steps. That is, whatever the price history, over the next t + Q time steps, the strategy 𝑠 (𝑡 + 𝑄) will always issue the same prediction independently of the price history between t and t+Q. We call such strategies Q-timesteps decoupled [START_REF] Andersen | A Mechanism for Pockets of Predictability in Complex Adaptive Systems[END_REF]. The simplest example of a strategy that is decoupled is the strategy that always issues a buy (sell) action, independently of the past price history ℎ ⃗ (𝑡). Such a strategy is trivially an infinite-number-of-time-steps decoupled.

However, the probability that any player will hold this specific strategy is very small indeed, with a probability that goes as .

As will be shown in the following, it is advantageous to split the order balance in two, so that it can be written in terms of decoupled and coupled strategies:

𝐴(𝑡) ⃗ = 𝐴(𝑡) ⃗ + 𝐴(𝑡) ⃗
We have included the superscript ℎ ⃗ to emphasize that the order imbalance is conditioned on observing the price history ℎ ⃗ at time t.

The size of 𝐴(𝑡) ⃗ /𝑁 then gives the percentage of decoupled strategies at time t, and will be used as a predictor for the actions of the participants. In the extreme case where the condition

|𝐴(𝑡) ⃗ | > 𝑁/2
is fulfilled, the prediction becomes certain, since the action at time t+2 of more than half the population takes the same sign (buy/sell) independently of what happens at time t+1. The greater the value of 𝐴(𝑡) ⃗ /𝑁, the better we should be able to predict the action at time t+2. This fact will be used in experiments where we try to predict the actions of the participants.

For a discussion of the effect of group size N, see also [START_REF] Nosenzo | Cooperation in small groups : the effect of group size[END_REF].

Laboratory experiments

We performed a series of 10 experiments at the Laboratory of Experimental Economics in Paris (LEEP). The experiments ran over 60 periods. In each period, the students received general economic news and could decide whether to buy or sell an asset, or simply do nothing.

At the end of the 60 periods, the students were paid pro rata according to their performance (for more details about the way the experiments were set up, see Appendix B).

At the beginning of the experiment, the students were told that the asset was, at this initial stage, properly priced according to rational expectations [START_REF] Fama | Efficient capital markets: a review of theory and empirical work[END_REF][START_REF] Muth | Rational expectations and the theory of price movements[END_REF]. This meant that only information regarding changes in the dividends on the asset or interest rates should have a direct influence on the price of the asset (for an interesting study with varying fundamental values, see, e.g., Stockl et al. 2015). The information flow consisted of general news from real past records of Bloomberg news items. News was selected in such a way that the general trend over the 60 consecutive periods was neutral. Then, according to rational expectations, there should be no overall price movement of the asset at the end of the 60 time periods. The price was thus expected to oscillate around the fundamental value throughout the experiment.

Results

Figures Dashed line: d -for decoupled strategies recommending sell as a function of time.

We now use the difference Δ𝐷(𝑡) ≡ |𝑑 (𝑡)-𝑑 (t)| as a predictor of the action of the participants in the next time step t+1. Table 1 shows ΔD for values greater than 0.2, the success rate in predicting the next price direction, and the number of periods used in the calculation of the success rate. The experiment E1 corresponds to the simplest case of all the 10 experiments, since in this case the price dynamics created by the participants shows the formation of a clear financial bubble. This is captured by the split in decoupled strategies measured by 𝛥D leading to a 100% success rate. However, this is less trivial than it appears since, as noted by [START_REF] Roszczynska | Short and Long Term Investor Synchronization Caused by Decoupling[END_REF], decoupling of strategies is sufficient but not necessary for financial bubble formation. In other words, the onset of speculation is possible without a split developing between 𝑑 (𝑡) and 𝑑 (t). For a more complete discussion of this point, see [START_REF] Roszczynska | Short and Long Term Investor Synchronization Caused by Decoupling[END_REF].

Insert table 1 around here

The lower two plots of Fig. 1 show the price evolution and corresponding percentage of decoupled strategies for experiment E2. Once again, we see a tendency for speculative action to develop, creating a financial bubble in prices, but a factor of 10 less than in E1. Using 𝛥D as predictor, we are again able to predict the actions of the participants with a high success rate (see Table 2).

Insert table 2 around here

Figure 2 shows the price evolution and corresponding percentage of decoupled strategies for experiments E3 and E4. In E3, a very weak overall trend develops, whereas the first part of E4 is trendless, followed by a weak upward trend, and ending with a small downward trend. Using 𝛥D as predictor, we are again able to predict the actions of the participants with a rather high success rate (see Tables 3 and4). It should be noted that the first half of E4 is trendless, but 𝛥D is still large, as can be seen from Figure 2, and, a reliable predictor even in this trendless case, as can be seen from Table 4.

Insert figure 2 around here Dashed line: d -for decoupled strategies recommending sell as a function of time.

Insert table 3 To summarize: We have made a series of 10 experiments, each ran over 60 time periods, in which students received general economic news, and could at each time period decide whether to buy or sell an asset. Some of the experiments showed the tendency to price bubble formation, whereas others fluctuated around the (constant) fundamental value of the asset. After the performance of the experiments, the price time series generated were used as input to agent based MC simulations. By studying the difference of decoupled strategies of the MC agents, Δ𝐷(𝑡) ≡ |𝑑 (𝑡)-𝑑 (t)|, we were able to issue a prediction of the action of the participants in the next time step t+1. In 5 out of the 10 experiments we were not able to use our method to issue predictions. However in 5 out of the 10 experiments we were able over several time periods to issue predictions out-of-sample of the behavior of the participants. In 5 of these 4 experiments, we obtained very high success rates, most of the events predicted with 65% succes or more. In one experiment our predictions were mostly random. Finally we used a bootstrap simulation to verify that our predictions were stable, and independent of the parameters used in the MC simulations. Overall our technique of using decoupled strategies appears to be a promissing way to probe the heuristics in participants in financial market games.

Insert figure 6 around here

Conclusion

We have introduced a new methodology to describe the onset of convergence towards Nash equilibria in simple repeated-games with infinitely large strategy spaces. This was shown to enable a better understanding of the heuristics used in decision-making. The method works by restricting to a special finite subset of strategies which we called decoupled strategies. We performed 10 multi-period simple financial market experiments, and applied our methodology within the framework of a simple financial market game called the $-game.

  Figures 1-5 show the price history versus time for the ten experiments (E1-E10), as well as the decoupling parameter versus time obtained via Monte Carlo (MC) simulations. Each figure represents the data and simulation from two experiments. The MC simulations "slaved" the price history from the experiments as input to the agents in the $G simulations. That is, instead

Fig. 2 .

 2 Fig. 2. Experiments E3 and E4. First and third plot: price evolution of experiments E3 and E4, respectively. Second (E3) and fourth (E4) plots: percentage of decoupled strategies used by the agents. Solid line: d + for decoupled strategies recommending buy as a function of time.

Fig. 6 .

 6 Fig. 6. Bootstrap simulations performed on the data for the experiments E1, E2, and E3. The figures were obtained by doing 100 Monte Carlo simulations for each experiment and calculating d + and d -. For each of the 100 Monte Carlo simulations, 100 more replica Monte Carlo simulations were then performed in order to be able to assign 90% and 10% confidence levels. Solid lines represent the result for a MC realization of d + and d -, while dashed lines give the 90% and 10% confidence levels calculated using the 100 replica MC simulations, i.e., in only 10% of the cases would the replica MC simulations find a value of d + (or d -) below what was found by the MC simulation, and in 90% of the cases the replica MC simulations would find a value of d + (or d -) not exceeding what was found by the MC simulation.

  around here Insert table 4 around here Insert table 5 around here Figure3shows the price evolution and corresponding percentage of decoupled strategies for experiments E5 and E6. Both experiments are more or less without trends, with the price fluctuating around the fundamental value. A small split in 𝑑 (𝑡) and 𝑑 (t) is only seen for initial times in E5, whereas a split exists throughout E6. For E5, 𝛥D never exceeds 0.2 and no prediction can be made. For E6, 𝛥D only exceeds the threshold value 0.2 in a few time periods, but the few predictions made appear to be slightly better than random or random (see Table6). . For each of the 100 Monte Carlo simulations, 100 more replica Monte Carlo simulations were then performed in order to be able to assign 90% and 10% confidence levels. The results are presented in Fig.6, where the solid lines represent a MC realisation of d + and d -, while the dashed lines give the 90% and 10% confidence levels calculated using the 100 replica MC simulations, i.e., in only 10% of the cases would the replica MC simulations find a value of d + (or d -) below what was found by the MC simulation, and in 90% of the cases the replica MC simulations would find a value of d + (or d -) not exceeding what was found by the MC simulation. As can be seen, the levels of d + and d -in the MC simulations are very stable, confirming the stability of our predictions obtained via the split in d + and d -.

	Insert figure 4 around here
	Fig. 4. Experiments E7 and E8. First and third plot: price evolution of experiments E7 and E8 In order to test the stability of the Monte Carlo simulations, we performed a series of
	respectively. Second (E7) and fourth (E8) plots: percentage of decoupled strategies used by bootstrap simulations. Figure 6 shows the bootstrap simulations performed on the data for
	the agents. Solid line: d + for decoupled strategies recommending buy as a function of time. the experiments E1, E2, and E3. Similar bootstrap simulations were performed for E4 and
	Dashed line: d -for decoupled strategies recommending sell as a function of time. E5, with the same conclusions as noted in the following for E1, E2, and E3. The figures
	were obtained by doing 100 Monte Carlo simulations for each experiment and calculating
	d + and d
	Insert table 7 around here
	Insert table 8 around here
	Figure 5 shows the price evolution and corresponding percentage of decoupled strategies for
	experiments E9 and E10. There is no trend present in E9, whereas a very weak trend develops
	Insert figure 3 around here in E10. A small number of splits 𝛥D develop in E9, more in E10, but in neither case can
	predictions be made.
	Fig. 3. Experiments E5 and E6. First and third plot: price evolution of experiments E5 and E6,
	respectively. Second (E5) and fourth (E6) plots: percentage of decoupled strategies used by
	the agents. Solid line: d + for decoupled strategies recommending buy as a function of time.
	Dashed line: d -for decoupled strategies recommending sell as a function of time.
	Insert figure 5 around here
	Insert table 6 around here Fig. 5. Experiments E9 and E10. First and third plot: price evolution of experiments E9 and
	E10, respectively. Second (E9) and fourth (E10) plots: percentage of decoupled strategies
	used by the agents. Solid line: d + for decoupled strategies recommending buy as a function of
	time. Dashed line: d -for decoupled strategies recommending sell as a function of time.
	Figure 4 shows the price evolution and corresponding percentage of decoupled strategies for
	experiments E7 and E8. Weak trends develop in the price formation, but not in the split of 𝑑 (𝑡)
	and 𝑑 (t). For these two experiments, the days on which we can issue a prediction are too few Insert table 9 around here
	to have any meaningful statistical significance.
	Insert table 10 around here

-

Table 1 ΔD

 1 Stockl, T., Huber, J., Kirchler, M., 2015. Multi-period experimental asset markets with distinct fundamental value regimes. Exp. Econ. 18: 314-334. , success rate, and corresponding number of events used to calculate the success rate for experiment E1.

	FIGURE 1

Table 2

 2 ΔD , success rate, and corresponding number of events used to calculate the success rate for experiment E2.

	ΔD	Success rate	Number of events
	0.2	0.70	27
	0.22	0.68	25
	0.24	0.73	22
	0.26	0.69	16
	0.28	0.62	13
	0.3	0.64	11
	0.32	0.67	6
	0.34	0.75	4
	0.36	-	0

Table 3

 3 ΔD , success rate, and corresponding number of events used to calculate the success rate for

		experiment E3
	ΔD	Success rate	Number of events
	0.2	0.57	23
	0.22	0.67	15
	0.24	0.67	3
	0.26	-	0

We have illustrated how the method was able to predict (at certain special times) participants' actions with a high success rate. We note that the decoupled strategy methodology is not limited to the $-game presented in this article, but can be applied in the context of any multiperiod game.

Appendix A: Implementation of the experiments

The number of participants in each experiment was fixed at 10. There was only one asset in our financial market that participants could either buy or sell, and short selling was allowed. The initial price of the asset was fixed at 5 euros with an expectation of a 10 cent dividend payout at the end of 60 time periods. Each of the 60 time periods lasted 15 seconds. In each time period, participants were presented with brief statements of economic news and they could either buy or sell ONE asset or do nothing. The participants were told that the asset was correctly priced according to rational expectations [START_REF] Muth | Rational expectations and the theory of price movements[END_REF], that is, the price of the asset was supposed to correctly reflect all future discounted cash flow accrued to the asset. The participants could, at zero interest rate, borrow money to buy shares, and short selling was allowed. The general financial information was taken from real financial news items obtained on Bloomberg over a two-week time period. Students were told the asset represented a portfolio of assets like an ETF or an index. They were all simultaneously presented with the same information, meant to reflect general financial news, e.g., good or bad US employment figures, commodity price changes, etc. The news items were the same in all experiments and were chosen without positive or negative bias. At the end of each time period, the participants' orders were gathered and a new market price calculated, based on the order imbalance (with sign and magnitude determining the direction and size of the price movement) [START_REF] Holthausen | The effect of large block transactions on security prices: A cross-sectional analysis[END_REF]. That is, the price at time t was given by 𝑃(𝑡) = 𝑃(𝑡 -1)𝑒 ( )/ , where A(t) is the order imbalance at time t and b the liquidity of the market (in the experiments chosen as b≡ 10 * 𝑁, representing a market crash/bubble of 10 percent when all N participants chose the same action). This was then shown to the participants graphically on their computer screen. Throughout the experiment, participants had a continuous update of the number of shares held and their gains/losses. At the end of each experiment, a pool of 200 euro was distributed pro rata among the participants who had made a gain.