Mohammed-Ridha Mahi

Ilham Mokbel
email: mokbel@univ-lyon1.fr

Latifa Negadi

Fatiha Dergal

Jacques Jose

Latifanegadi

Experimental solubility of carbon dioxide in monoethanolamine, or diethanolamine or N-methyldiethanolamine (30 wt%) dissolved in deep eutectic solvent (choline chloride and ethylene glycol solution)

come     Experimental solubility of carbon dioxide in monoethanolamine, or diethanolamine or N-methyldiethanolamine (30 wt%) dissolved in deep eutectic solvent (choline chloride and ethylene glycol solution)

Introduction

The current discussion about global warming and climate change is centered on the anthropogenic greenhouse effect. It is caused by the emission and accumulation of gases such as water vapor, carbon dioxide, methane… Carbon dioxide (CO2) is the most important anthropogenic greenhouse gas because of its comparatively high concentration in the atmosphere. The combustion of fossil fuels has led mainly to an increase in the CO2 concentration in the atmosphere. CO2 contributes to more than 60% of the global warming effect [START_REF] Yamasaki | An overview of CO2 mitigation options for global warming-emphasizing CO2sequestration options[END_REF]. It is therefore essential to develop new technologies to reduce CO2 emissions from industrial fossil-fueled energy production units (ex: cement plant, refinery, etc…).

Post-combustion CO2 capture technologies are considered to be the most mature technology.

Alkanolaminessuch

as monoethanolamine (MEA), diethanolamine (DEA), triethanolamine(TEA), 2-amino-2-methyl-1-propanol (AMP), and 2-methylaminoethanol (MAE) in aqueous solutions are the most used as chemical absorbent for CO2 capture due to their high absorption ability and fast absorption rate [START_REF] Cohen | Comparing post-combustion CO2 capture operation at retrofitted coal-fired power plants in the Texas and Great Britain electric grids[END_REF][START_REF] Tarun | Technoeconomic study of CO2capture from natural gas based hydrogen plants[END_REF][START_REF] Rodríguez | Optimization of post-combustion CO2 process using DEA-MDEA mixtures[END_REF][START_REF] Dave | CO2 capture by aqueous amines and aqueous ammonia-A Comparison[END_REF][START_REF] Mofarahi | Design of CO2 absorption plant for recovery of CO2 from flue gases of gas turbine[END_REF][START_REF] Abu-Zahra | CO2 capture from power plants. Part I. A parametric study of the technical performance based on monoethanolamine[END_REF]. However, the problems associated with the aqueous-based absorbents, such as equipment corrosion, high energy consumption, currently make the process not viable [START_REF] Yu | A review of CO2 capture by absorption and adsorption[END_REF][START_REF] Camper | Room-temperature ionic liquid-amine solutions: Tunable solvents for efficient and reversible capture of CO2[END_REF][START_REF] Gray | Improved immobilized carbon dioxide capture sorbents[END_REF][START_REF] Taib | Solubilities of CO2 in aqueous solutions of ionic liquids (ILs) and monoethanolamine (MEA) at pressures from 100 to 1600kPa[END_REF][START_REF] Figueroa | Advances in CO2 capture technology-The U.S. Department of Energy's Carbon Sequestration Program[END_REF][START_REF] Rochelle | Amine scrubbing for CO2 capture[END_REF].Various researches are being carried out in order to define alternative solvents which exhibit high affinity for CO2 with easier solvent regeneration and reuse, low corrosion of equipment.

In recent years, ionic liquids (ILs) have quickly emerged as an alternative of choice. An IL is a molten salt, composed of a cation and an anion which interact via electrostatic forces. They are liquid at room temperature; their melting temperatures are often below 100°C. ILs showed several unusual characteristics such as low volatility, high thermal and chemical stability, strong solvation ability and tunability of chemical and physical properties by choice of the cation/anion combination [START_REF] Hallett | Room-temperature ionic liquids: Solvents for synthesis and catalysis[END_REF][START_REF] Brennecke | Ionic liquids: Innovative fluids for chemical processing[END_REF]. However, due to expensive raw material chemicals, poor biodegradability, an uncertain toxicity and complicated synthetic processes of ILs, it is still a challenge for their large-scale applications in industry [START_REF] Hou | Novel binary eutectic mixtures based on imidazole[END_REF][START_REF] Francisco | A new low transition temperature mixture (LTTM) formed by choline chloride + lactic acid: Characterization as solvent for CO2 capture[END_REF]. Recently, a new generation of solvent, named Deep Eutectic Solvent (DES), has emerged as a low cost alternative of ILs. A DES is a fluid generally obtained by mixing an organic halide salts with metal salts or a hydrogen bond donor (HBD). These components are capable of self-association, often through hydrogen bond interactions, to form eutectic mixture with a melting point lower than that of each individual component. Most of DESs are liquid between room temperature and 70°C [START_REF] Zhang | Deep eutectic solvents: syntheses, properties and applications[END_REF]. These solvents exhibit similar physico-chemical properties to the traditionally used ILs, while being much cheaper and environmentally friendlier. In fact, they are easily produced at low cost and in high purity. Furthermore, they are non-toxic and most importantly, being made from biodegradable components [START_REF] Abbott | Deep Eutectic Solvents Formed between Choline Chloride and Carboxylic Acids: Versatile Alternatives to Ionic Liquids[END_REF][START_REF] Leron | Solubility of carbon dioxide in a choline chloride-ethylene glycol based deep eutectic solvent[END_REF].

For these reasons, DESs are now of growing interest in many fields of research such as biocatalysis [START_REF] Gorke | Hydrolase-catalyzed biotransformations in deep eutectic solvents[END_REF][START_REF] Lindberg | Deep eutectic solvents (DESs) are viable cosolvents for enzyme-catalyzed epoxide hydrolysis[END_REF], electrochemistry [START_REF] Cojocaru | Using deep eutectic solvents to electrodeposit CoSm films and nanowires[END_REF],pharmaceuticals [START_REF] Morrison | Characterization of thermal behavior of deep eutectic solvents and their potential as drug solubilization vehicles[END_REF], liquid-liquid extraction [START_REF] Li | Extraction desulfurization process of fuels with ammonium-based deep eutectic solvents[END_REF],

gas (NH3, NO, SO2 and CO2) absorption [START_REF] Zhong | Solubilities of ammonia in choline chloride plus urea at (298.2-353.2) K and (0-300) kPa[END_REF][START_REF] Zhong | Phenol-Based Ternary Deep Eutectic Solvents for Highly Efficient and Reversible Absorption of NH3[END_REF][START_REF] Duan | Solubility and thermodynamic properties of NH3 in choline chloride-based deep eutectic solvents[END_REF][START_REF] Dou | Mechanistic Study of Selective Absorption of NO in Flue Gas Using EG-TBAB Deep Eutectic Solvents[END_REF][START_REF] Chen | Highly Efficient and Reversible Capture of Low Partial Pressure SO2 by functional Deep Eutectic Solvents[END_REF][START_REF] Yang | Efficient Absorption of SO2 by EmimCl-EG Deep Eutectic Solvents[END_REF][START_REF] Sun | Efficient SO2 absorptions by four kinds of deep eutectic solvents based on choline chloride[END_REF][START_REF] Deng | Investigation of solubilities of carbon dioxide in five levulinic acid-based deep eutectic solvents and their thermodynamic properties[END_REF][START_REF] Trivedi | Deep eutectic solvents as attractive media for CO2 capture[END_REF][START_REF] Xie | Solubilities of CO2 , CH4 , H2 , CO and N2 in choline chloride/urea[END_REF][START_REF] Dietz | PC-SAFT modeling of CO2solubilities in hydrophobic deep eutectic solvents[END_REF][START_REF] Ghaedi | Investigation of various process parameters on the solubility of carbon dioxide in phosphoniumbased deep eutectic solvents and their aqueous mixtures: Experimental and modeling[END_REF][START_REF] Liu | Solubilities and Thermodynamic Properties of Carbon Dioxide in Guaiacol-Based Deep Eutectic Solvents[END_REF][START_REF] Ghaedi | CO2 capture with the help of Phosphonium-based deep eutectic solvents[END_REF][START_REF] Ji | Phase equilibria of high pressure CO2 and deep eutectic solvents formed by quaternary ammonium salts and phenol[END_REF][START_REF] Dietz | PC-SAFT modeling of CO2solubilities in hydrophobic deep eutectic solvents[END_REF][START_REF] Tatar | Accurate prediction of CO2solubility in eutectic mixture of levulinic acid (or furfuryl alcohol) and choline chloride[END_REF] and other chemical and industrial processes.

To carry out the solubility of CO2 in the solvent, a static apparatus with on-line analysis of the vapor phase was designed. Once the apparatus was validated, the solubility of CO2 in monoethanolamine, MEA, diethanolamine, DEA, or N-methyldiethanolamine, MDEA, at 30 mass % in DES medium (choline chloride and ethylene glycol at 1:2 mole ratio) was determined at three different temperatures 298.1, 313.1 and 333.1 K and pressures from 2 Pa up to 800 kPa. CO2 absorption isotherms were correlated using the model of Gabrielsen et al.

[44] which in turn was based on the model developed by Posey et al. [START_REF] Posey | A simple model for prediction of acid gas solubilities in alkanolamines[END_REF]. The model was previously established for aqueous solutions. Its transposition to the DES media was successfully carried out in the case of primary and secondary amine. The model proposal for the tertiary amine correlates satisfactorily the experimental measurements up to 35 kPa.

Experimental section

Materials

The chemicals, monoethanolamine (MEA), Diethanolamine (DEA), N-methyldiethanolamine (MDEA), choline chloride (ChCl), ethylene glycol (EG) and hydrochloric acid (HCl, 5N), were purchased from Sigma-Aldrich and were used without further purification. Their purity and source are given in Table 1. High purity deionized water (conductivity = 18 MΩ.cm using a "Purelab" Classic water purification module) was used. The CO2 was supplied by Air Liquide with mole fraction purity greater than 0.999.A digital balance (Mettler-Toledo AG204) having an accuracy of 0.0001 g was used.

Apparatus

The realized apparatus is composed of a glass equilibrium cell of a volume of 396 cm 3 with a double envelope for thermoregulated water circulation which ensures a constant temperature of the mixture, Fig. 1. The autoclave withstands pressure up to 10 bars, and in order to avoid any overpressure, the lid is equipped with a safety capsule which serves as a valve. In addition, a pumping system allows a vacuum in the entire apparatus when necessary. A calibrated copper-constantan thermocouple, introduced in a glove finger, monitors the equilibrium temperature (absolute uncertainty of ± 0.1K). Total pressure of the solution is delivered by a Keller type pressure sensor (range: 0 to 10 bars; relative uncertainty of 0.5%).

To cushion the effect of CO2 pressure into the solution, a stainless-steel reserve is placed at the outlet of the gas cylinder. The pressure of CO2 in the reserve is determined by a supplementary pressure sensor (measurement range: 0 to 10 bar).

The solution is homogenized with a self-aspirating hollow stirrer whereas the gaseous phase is submitted to a continuous loop circulation thanks to a peristaltic pump. At the exit of the cell (autoclave), the gaseous phase passes through a sampling loop of 2 mL connected to a gas chromatograph, GC for on-line analysis. As for the liquid phase, an offline volumetric analysis (described below) is carried out. To avoid any vapor condensation upstream GC analysis, the sampling line and the injection valve are maintained at 120°C.

Isotherms of absorption are determined by introducing successive additions of CO2.

Preparation of the absorbent

The amine and the DES, (ChCl and EG, 1:2 mole ratio respectively) are prepared separately in a balloon and degassed by heating and submitting each liquid to vacuum. Due to its high hygroscopicity, the DES is prepared inside a glove box. Some argon is finally added in the DES balloon to protect it from ambient air, the time to transport the solution from the glove box to the cell. An exact amount (determined by weighing) of the DES and the amine is introduced into the glass cell. The stirred, homogenous and transparent liquid of about 250 mL is finally subjected to moderate heating and purges in order to evacuate the dissolved argon.

Experimental protocol

Before any new measurement, a vacuum is created in the whole apparatus to ensure that there is no air or CO2 in the device and to check off an eventual leakage of the system. The equilibrium cell is maintained at constant temperature. The previously prepared solution is then introduced by aspiration into the autoclave via the tube provided for this purpose and stirred (2000 rpm), Fig. 1. Once the temperature becomes constant, some CO2 is added from the one liter thank (containing CO2 at 3 bars and at room temperature) by opening V2 and V3

valves. Despite a vigorous mixing of the liquid, the phase equilibrium is reached within 12 hours due to the slow absorption kinetic and the viscosity of the solutions of DES/amine. It is monitored by the pressure sensor equipped with online data acquisition system.

Prior to the gas phase on-line analysis, in order to determine the partial pressure of CO2, the GC was calibrated using different compositions of CO2 and N2 mixtures. In addition, a sample of the liquid phase is withdrawn and the CO2 loading, α, is determined by volumetric titration with hydrogen chloride solution, Eq. ( 1):

= ℎ ℎ (1)
When the equilibrium pressure of CO2 in the autoclave is under the atmospheric pressure, nitrogen gas (slightly above 1 bar) is introduced into the cell to facilitate the collect of the liquid sample and to transport the vapor phase towards the gas chromatograph via the sampling loop. The next increment of CO2 is carried out after purging the vapor phase of the cell.

Results and discussion

Validation of the experimental method

To validate the experimental protocol previously described, solubility of CO2 in aqueous MEA at 30% (by weight) was determined at three temperatures (298.1, 313.1 and 333.1 K) and at large partial pressures of CO2 ranging from 1 Pa to 800 kPa, Table 2. The obtained results were compared with literature data [START_REF] Jou | The solubility of CO2in a 30 mass percent monoethanolamine solution[END_REF][START_REF] Shen | Solubility of carbon dioxide in aqueous mixtures of Monoethanolamine with Methyldiethanolamine[END_REF][START_REF] Aronu | Solubility of CO2 in 15, 30, 45 and 60 mass% MEA from 40 to 120°C and model representation using the extended UNIQUAC framework[END_REF], Figs. 234. A very good agreement is obtained with Jou et al. [START_REF] Jou | The solubility of CO2in a 30 mass percent monoethanolamine solution[END_REF] and Aronu et al. [START_REF] Aronu | Solubility of CO2 in 15, 30, 45 and 60 mass% MEA from 40 to 120°C and model representation using the extended UNIQUAC framework[END_REF] data whereas a slight deviation is observed with Shen and Li points [START_REF] Shen | Solubility of carbon dioxide in aqueous mixtures of Monoethanolamine with Methyldiethanolamine[END_REF].

Solubility of CO2 in the solvent MEA/DES, DEA/DES and MDEA/DES

A protocol identical to that of aqueous MEA was used to determine CO2 in the vapor phase (by GC) and CO2 in the liquid phase (by volumetry).

The solubility of CO2 in MEA or DEA or MDEA (30 wt%) dissolved in the DES (ChCl : EG

(1:2)) are reported in Tables 345. The experimental values expressed as the partial pressure of CO2 function of the loading, α, are represented in Figs. 567.

As expected, for a given partial pressure of CO2, the solubility of the gas in term of α decreases with increasing temperature. At the physical absorption range, α > 0.5, the influence of temperature on the solubility is negligible for the three amines, the curves overlap. In the same way, for a given temperature and partial pressure of CO2 (example PCO2 = 1 kPa, T= 313.1K), the solubility of CO2 in term of molality, decreases from MEA (primary amine) to MDEA (tertiary amine). The solubility of CO2 in the DES (without amine) studied by Leron and Li [START_REF] Leron | Solubility of carbon dioxide in a choline chloride-ethylene glycol based deep eutectic solvent[END_REF], remains much lower than that of MDEA+DES, Fig. 8, due to the absence of chemical absorption when the amine is missing.

On the other hand, the solubilities of CO2 in the DES/ MEA (15% wt), reported in Table 6, are almost similar with those obtained in the DES/ MEA (30wt%) media, Fig. disagreement with the present study where α is above 0.6 for both amine at the quoted partial pressure of CO2.

Correlation model for CO2 absorption in nonaqueous media-Absorption enthalpy of CO2

Correlation

The studied solvent consists of two entities: choline chloride totally ionized and stablishing strong coulombic electrostatic interactions with charged solutes; monoethyleneglycol, like all alcohols, or water ..., is an ampholyte molecular solvent (BH) whose autoprotolysis equilibrium is as follows:

2 BH ⇆ H + (2)
The equilibrium constant is:

= H ! ! (3)
It is known that ampholytic solvents could dissolve salts in two solvated forms: free ions (anions A #$% and cations C #$% ) and ion pairs (A , C ) #$% .

The molar proportions of these species depend on several parameters (dielectric constant of the solvent, radius and charge of the ions, temperature, solute-solvent interactions ...). As a first approximation when the solvent has a dielectric constant, * + , higher than 40, the free ions predominate and vice versa. In our case the solvent is a mixture of MEG * + = 37.7 at 298 K)

and Choline Chloride. The dielectric constant of the latter is unknown, but probably very weak like most ionic species (example * + (KCl) = 4.86 at room temperature). The dielectric constant that results from mixing the two constituents of the solvent should lead to the coexistence of free species , -./ and -./ , and ion pairs (A , C ) -./ . However, it is reasonable to assume that the presence of the choline chloride, establishing strong coulombic interactions with charged solutes, favors the predominance of free ions , -./ and -./ . This is what we admit in the present study.

Kortunov et al. [START_REF] Kortunov | In Situ Nuclear Magnetic Resonance Mechanistic Studies of Carbon Dioxide Reactions with Liquid Amines in Non-aqueous Systems: Evidence for the Formation of Carbamic Acids and Zwitterionic Species[END_REF] studied the mechanism of the reaction between primary and secondary amine with CO2 in nonaqueous media using 1 H and 13 C NMR. They confirmed that the same mechanism occurs in aqueous and nonaqueous media with however a difference in the stability of the carbamic acid formed intermediately. In nonaqueous medium, carbamic acid and carbamate, coexist at the chemical equilibrium but their relative proportion depends on many parameters (the concentration, pressure, molecular structure of the amine, nature of the solvent, temperature, partial pressure of CO2 ...). The great similarity of the experimental absorption isotherms of MEA and DEA in aqueous and non-aqueous medium, Figs. 10 and 12, leads us to assume that at the absorption equilibrium, carbamate species predominant:

2 R 1 R NH + ⇆ R 1 R NH + R 1 R NCO (4)
Assuming that the activity coefficient is equal to the unity, the equilibrium constant is as follows:

′ 56 7 = 88′9: ! 88′9 ! ! 88′9:! (5)
At equilibrium and by neglecting ions from the autoprotolysis of the solvent:

88′9: ! = 88′9 ! = ; (6) 88′9:! = (1 -2 ) ; (7)
where:

α: is the CO2 loading rate a0: the initial concentration of amine

The solvated CO2, [CO2]sol, follows Henry law, where KH is the constant:

> 56 7 = ? ! -./ (8)
By neglecting the physical absorption of CO2 in the solvent, Eqs. ( 5)-( 8) leads to the following expression:

> 56 7 = 56 7 @ 7 (1 @) 7

CO2 absorption by MDEA in non-aqueous media (ethanol) has been studied by Kierzkowska-Pawlak and Zarzycki [START_REF] Kierzkowska-Pawlak | Solubility of carbon dioxide and nitrous oxide in water + methyldiethanolamine and ethanol + methyldiethanolamine solutions[END_REF]. In the absence of water, the authors expected a limited absorption of CO2 according Henry's law (physisorption). Actually, the authors observed a solubility of CO2 much higher than that of NO2 under the same experimental conditions. NO2 absorption in water is known to be only physical (do not present chemical reaction). The authors attribute this important solubility of CO2 to "additional" interactions due to hydrogen bonds with the solvent and also evoke a possible interaction of the nitrogen atom of the amine with CO2.

Ethylene glycol being an amphiprotic solvent, HB (like water, alcohol…), we propose the chemical reaction in nonaqueous media,(Eq. ( 11)). In the same way as for aqueous media, the chemical absorption range is comprised between α = 0 and α = 1 (different from the case of MEA and DEA where the absorption range is below α K 0.5).

DEF, + + : ⇄ DEF, : + (11) DEF, : and are respectively the lyonium and lyate ions. As mentioned previously, we admit that the solvated free ions are predominant compared to the ion pairs.

The equilibrium constant of the reaction ( 10) is:

′ 56 7 = DEF,: ! ( ) ! DEF,! ! (12)
According to the same hypothesis as for the primary and secondary amine:

P BC 7 = K BC 7 α a ; (1 -α) (13)
In the same way as for primary and secondary amine, K BC 7 include K′ BC 7 and Henry constant KH.

The model used is an extension towards the nonaqueous media of the one proposed by Posey et al. [START_REF] Posey | A simple model for prediction of acid gas solubilities in alkanolamines[END_REF] and later used by Gabrielsen et al. [START_REF] Gabrielsen | A Model for Estimating CO2 Solubility in Aqueous Alkanolamines[END_REF] for the absorption of CO2 by the three classes of amine in aqueous phase. The influence of the temperature on the constant of equilibrium is given by the following relation, Eq. ( 14): For primary and secondary amines the expression is truncated to the first two adjustable parameters. Therefore the partial pressure, PCO2, is independent of the initial concentration of the amine, as for the aqueous phase.

ln K BC 7 = A + B T + C * a ; * α + DUa ; * α (
The minimized objective function is as follows (Eq. ( 15)):

V = W X Y 56 7 Z[/, 56 7 \]^, _` ab c1 (15)
N is the number of experimental point.

The parameters of Eq. ( 14) are reported in Table 7. As shown in Figs. [START_REF] Rochelle | Amine scrubbing for CO2 capture[END_REF] and 14, the model satisfactorily reproduces the experimental points obtained with MEA and DEA. In the case of MDEA, the model fits well the experimental data for α < 0.5 (Fig. 15); beyond this value the model deviates from the experimental points. This phenomenon could be explained by the non-negligible quantity of ion pairs formed during the CO2/MDEA reaction.

Heat of absorption of CO2

From Eq. ( 14), enthalpy of CO2 absorption is deduced, using Gibbs-Helmholtz relation:

d e ef g ∆i f jk b = - ∆: f (16)
Where ΔG, Gibbs free energy of the reaction, is related to the equilibrium constant KCO2 by: ∆i = -8f ln 56 7

Assuming that the effect of pressure on KCO2 is negligible, from Eq. ( 16) and ( 17) we deduce KCO2 as a function of the temperature:

(ln 56 7 ) l 1 m n = - ∆: 8 (18)
The resolution of the derivative function of temperature of equation of Eq. ( 14) into Eq. ( 18), gives (-∆: o p q p r ):

∆: [s-.+^t .u [v\+[w\ = -8 (19)
Where R is the universal gas constant.

The calculated ∆H xy#$z{|}$~ were then compared with those obtained in the aqueous solutions of amines reported by different bibliographic sources, Tables 8.

The enthalpy of CO2 absorption by MEA (30wt%) in (1 ChCl : 2 EG) solution is significantly lower than that of MEA in aqueous medium. The use of MEA in (1 ChCl : 2 EG) would allow an energy saving in the regenerator of the order of 20%. With regard to DEA the nature of the solvent does not have a significant effect on the CO2 absorption enthalpy. In the case of MDEA, the type of solvent plays an important role: in (1 ChCl : 2 EG) medium the value is clearly lower than in aqueous medium, resulting in a possible saving of energy in the regenerator of almost 40%.

Conclusion

The three different classes of amine (MEA, DEA and MDEA) were chosen in order to compare their behavior in the DES medium with that in water. With this aim a static apparatus with on-line analysis of the vapor phase was developed. Three temperatures (298.1; 313. Relative standard deviation: ur(PCO2)= 5% for PCO2< 1 kPa and ur(PCO2)= 1% for PCO2> 1 kPa. Relative standard deviation: ur(PCO2)= 5% for PCO2< 1 kPa and ur(PCO2)= 1% for PCO2> 1 kPa. Relative standard deviation: ur(PCO2)= 5% for PCO2< 1 kPa and ur(PCO2)= 1% for PCO2> 1 kPa. Relative standard deviation: ur(PCO2)= 5% for PCO2< 1 kPa and ur(PCO2)= 1% for PCO2> 1 kPa.

9 .

 9 In the same way, the values of α observed in the aqueous solution of MEA or DEA overlap those obtained in the DES/ amine (30wt%) media, Fig.10and Fig.12. This phenomenon is not observed for MDEA. For example, the solubility of CO2 at 313.1 K in the DES/MDEA, α=0.4, is obtained at a higher pressure (PCO2 = 100 kPa) whereas in the aqueous solution for the same α=0.4, PCO2 < 10 kPa, Fig.11.Uma Maheswari and Palanivelu[START_REF] Uma Maheswari | Carbon Dioxide Capture and Utilization by Alkanolamines in Deep Eutectic Solvent Medium[END_REF] carried out the determination of α in the amine (MEA, DEA)/DES (ChCl:EG, molar ratio of 1:2) at 298.1 K and PCO2 = 200 kPa. The obtained results, α = 0.492 in the case of MEA and α = 0.301 in the case of DEA, are in total

( 9 )

 9 K BC 7 includes K′ BC 7 and Henry constant KH It is commonly admitted that the absorption of CO2 by a tertiary amine, in this case MDEA, in an aqueous medium follows the reaction below where the solvent participates in the reaction: DEF, + + : ⇄ DEF,: + : H

  14) α: is the loading rate (mol of CO2/mol of amine) a0: initial concentration of amine in the DES KCO2: combined Henry's law and chemical equilibrium constant for CO2 partial pressure A, B, C and D: adjustable parameters The model is valid only in the field of chemical absorption: α <0.5 for primary and secondary amines (MEA and DEA) and α<1 for tertiary amines (MDEA). The first two parameters A and B take into account the influence of the temperature on the equilibrium constant. The last parameters C and D, allows a global correction of non-ideality.

1 and 333. 1 K

 1 ) with various loading of CO2 in each solution were explored. The range of the partial pressure of CO2 investigated is particularly large, from 2 Pa to approximately 800kPa.The study shows that the substitution of water by DES solvents leads to almost the same capacity of CO2 absorption by the amines except for MDEA where a lower solubility of CO2 is observed comparing to the aqueous medium.The model for primary and secondary amine fits quite well the experimental values obtained with the MEA and DEA. In the case of MDEA, the model deviates from the experimental points when the loading rate is above α > 0.5. In the hypothesis of a use of the DES+amine solvent for CO2 capture in post-combustion, a decrease of the vapor pressure of the solvent (comparing to that of water+amine) has an advantage from the point of view losses by vaporization in the absorber. The second advantage is most likely a lower effect of equipment corrosion, the third positive point is a lower enthalpy of absorption of MEA and MDEA in (1ChCl : 2 EG) comparing to aqueous medium, resulting in a possible saving of energy in the regenerator of almost 40%. However these positives are counter balanced by the increase of the viscosity and by a slower kinetic of the absorption reaction.

Fig. 1 .

 1 Fig.1. Apparatus for CO2 absorption.

Fig. 2 .

 2 Fig. 2. Solubility of CO2 in aqueous solution of MEA (30 wt %) at 298.1 K.

Fig. 3 .Fig. 4 .Fig. 5 .

 345 Fig. 3. Solubility of CO2 in aqueous solution of MEA (30 wt %) at 313.1 K.

Fig. 6 .

 6 Fig. 6. CO2 solubility in DEA 30 wt% + DES (1 ChCl: 2 EG) at three different temperatures.

Fig. 7 .

 7 Fig. 7. CO2 solubility in MDEA 30 wt% + DES (1ChCl:2 EG) at three different temperatures.

Fig. 8 .

 8 Fig. 8. Comparison of CO2 solubility at 313.1 K in several solvents: (1 ChCl: 2 EG) + MEA ( ) or + DEA ( ) or + MDEA ( ); and in (1 ChCl: 2 EG): ( ), [20]. Percentage of amine in DES = 30 wt%.

Fig. 9 .Fig. 10 .

 910 Fig. 9. CO2 solubility in MEA (15 or 30%) + DES (1 ChCl : 2 EG) at 313.1 K.

Fig. 11 .

 11 Fig. 11. Comparison of CO2 solubility at 313.1 K in aqueous and nonaqueous MDEA.

Fig. 12 .

 12 Fig. 12. Comparison of CO2 solubility at 298.1 K in aqueous and nonaqueous DEA.

Fig. 13 .

 13 Fig. 13. Comparison between experimental and calculated values (solid lines) of CO2 solubility in 30 wt% MEA + DES (1 ChCl : 2 EG).

Fig. 14 .

 14 Fig. 14. Comparison between experimental and calculated values (solid lines) of CO2 solubility in 30 wt% DEA+ DES (1 ChCl : 2 EG).

Fig. 15 .

 15 Fig. 15. Comparison between experimental and calculated values (solid lines) of CO2 solubility in 30 wt% MDEA+ DES (1 ChCl : 2 EG).

Table 1

 1 Chemicals used in this work.

	Chemical	Acronym	CASNumber	Purity (mass fraction)	Purity GC	Source
	Carbon dioxide	CO2	124-38-9	0.999	0.998	Air Liquide
	Ethanolamine	MEA	141-43-5	≥0. 990	0.990	Sigma Aldrich
	Diethanolamine	DEA	111-42-2	≥ 0.980	0.981	Sigma Aldrich
	N-Methyldiethanolamine	MDEA	105-59-9	≥ 0.990	0.992	Sigma Aldrich
	Choline Chloride	ChCl	67-48-1	0.990	-	Sigma Aldrich
	Ethylene glycol	EG	107-21-1	≥ 0.995	0.990	Sigma Aldrich

Table 2

 2 Solubility of CO2, α (mol of CO2/ mol of amine), in aqueous 30% (w/w) monoethanolamine solution.

			P CO2/ kPa
	α	T = 298.1 K	T = 313.1 K	T = 333.1 K
	0.118	0.0019	0.0028	0.0195
	0.233	0.0032	0.0145	0.0816
	0.361	0.0158	0.0710	0.580
	0.460	0.146	0.558	2.41
	0.514	1.66	3.01	13.48
	0.633	44.75	80.33	174.8
	0.707	132.6	243.2	420.1
	0.802	297.2	566.7	795.9

Table 3

 3

	Solubility of CO2, α (mol of CO2/ mol of amine), in DES (1 ChCl : 2 EG) + MEA 30%(w/w) media.
			P CO2/ kPa
	α	T = 298.1 K	T = 313.1 K	T = 333.1 K
	0.145	0.0066	0.0122	-
	0.172	0.0071	0.0146	0.0421
	0.204	0.0079	0.0202	-
	0.279	0.0141	0.0547	0.34
	0.378	0.0677	0.35	2.66
	0.437	0.35	1.75	10.13
	0.466	0.62	2.91	16.69
	0.532	3.43	11.25	42.81
	0.591	27.60	55.10	122.9
	0.619	102.6	132.8	175.9
	0.645	274.4	315.8	369.6
	0.667	472.9	521.3	583.7
	0.676	616.9	669.7	731.0
	Relative standard deviation: ur(PCO2)= 5% for PCO2< 1 kPa and ur(PCO2)= 1% for PCO2> 1 kPa.

Table 4

 4 Solubility of CO2, α (mol of CO2/ mol of amine), in DES (1 ChCl : 2 EG) + DEA 30% (w/w).

			P CO2/ kPa
	α	T = 298.1 K	T = 313.1 K	T = 333.1 K
	0.159	0.0393	0.16	0.83
	0.238	0.18	0.79	4.34
	0.376	1.20	5.16	22.20
	0.497	10.75	33.96	105.2
	0.605	33.01	78.38	141.2
	0.710	153.0	184.0	236.0
	0.782	337.3	375.8	431.0
	0.834	536.4	575.4	629.05
	0.874	736.5	777.5	822.9

Table 5

 5 Solubility of CO2, α (mol of CO2/ mol of amine), in DES (1 ChCl : 2 EG) + MDEA 30% (w/w).

			P CO2/ kPa
	α	T = 298.1 K	T = 313.1 K	T = 333.1 K
	0.092	0.98	2.85	9.06
	0.205	2.75	7.68	24.03
	0.210	4.19	11.67	35.26
	0.269	7.48	20.72	61.48
	0.344	28.77	77.90	143.3
	0.472	69.09	151.6	186.0
	0.660	230.8	327.4	410.0
	0.664	230.0	308.4	385.7
	0.851	535.2	638.2	725.3
	0.920	643.1	740.2	824.7

Table 6

 6 Solubility of CO2, α (mol of CO2/ mol of amine), in DES (1 ChCl : 2 EG) + MEA 15% (w/w).

	α (313.1K)	PCO2 /kPa
	0.145	0.0112
	0.303	0.107
	0.304	0.105
	0.436	1.56
	0.486	6.84
	0.606	58.29
	0.635	156.5
	0.674	359.2
	0.699	450.7
	0.716	689.1

Table 7

 7

	Regressed Parameters of KCO2*
		A		B	C	D
	MEA-CO2	25.45	-8355	-1.79	0.00
	DEA-CO2	29.18	-8655	-1.95	0.00
	MDEA-CO2	20.04	-4576	0.02	3.50
	* : ln	= + +	+

Table 8

 8 Enthalpy of CO2 absorption in aqueous and nonaqueous (1 ChCl : 2 EG)amine 30wt% -Comparison with literature data.

	ΔH Abs (kJ mol -1 )

a : Calorimetry; b : Calculated from solubility data, c predicted value


Clment Dombry
email: clement.dombry@math.univ-poitiers.fr

Christian Mazza
email: christian.mazza@unifr.ch

Some remarks on Betti numbers of random polygon spaces

Keywords: Configuration space, Betti number, Poincaré polynomial, random polygonal linkage, random manifold AMS Subject classification. Primary: 60B05, 55R80

Polygon spaces like

) or they three dimensional analogues N ℓ play an important rle in geometry and topology, and are also of interest in robotics where the l i model the lengths of robot arms. When n is large, one can assume that each l i is a positive real valued random variable, leading to a random manifold. The complexity of such manifolds can be approached by computing Betti numbers, the Euler characteristics, or the related Poincaré polynomial. We study the average values of Betti numbers of dimension p n when p n → ∞ as n → ∞. We also focus on the limiting mean Poincaré polynomial, in two and three dimensions. We show that in two dimensions, the mean total Betti number behaves as the total Betti number associated with the equilateral manifold where l i ≡ l. In three dimensions, these two quantities are not any more asymptotically equivalent. We also provide asymptotics for the Poincaré polynomials

Introduction 1.Background

In this note, we consider a question raised by M.Farber in [START_REF] Farber | Topology of random linkages[END_REF]. We study closed planar n-gons whose sides have fixed lengths l 1 , • • • , l n where l i > 0 for 1 ≤ i ≤ n. The set of polygonal linkage in R 2 [START_REF] Farber | Topology of random linkages[END_REF] parametrizes the variety of all possible shapes of such planar n-gons with sides given by ℓ = (l 1 , • • • , l n ). The unit vector u i ∈ C indicates the direction of the i-th side of the polygon. The condition l i u i = 0 expresses the property of the polygon being closed. The rotation group SO(2) acts on the set of side directions (u 1 , • • • , u n ) diagonnaly.

M ℓ = {(u 1 , • • • , u n ) ∈ S 1 × • • • S 1 ; n i=1 l i u i = 0}/SO
Polygon spaces play a fundamental rle in topology and geometry, as illustrated for example by Kempe Theorem which states that "Toute courbe algébrique peut être tracée à l'aide d'un système articulé", see e.g. [START_REF] Lebesgue | Leons sur les constructions gomtriques, professes au Collège de France en 1940-1941[END_REF]. [START_REF] Farber | Topolgy of robot motion planning[END_REF] provides other examples of such universality results in topology. Polygon spaces generated an active research area in geometry (see e.g. [START_REF] Farber | Homology of planar polygon spaces[END_REF], [START_REF] Hausmann | Sur la topologie des bras articulés In Algebraic Topolgy Poznan[END_REF], or [START_REF] Thurston | The mathematics of the three-dimensional manifolds[END_REF]), but are also of strong interest in applications like robotics where each l i models the length or a robot arm (see e.g. [START_REF] Farber | Topology of random linkages[END_REF], [START_REF] Farber | Betti numbers of random manifolds Homology[END_REF], [START_REF] Farber | Homology of planar polygon spaces[END_REF] and [START_REF] Farber | Topolgy of robot motion planning[END_REF]).

We can also point out potential applications in polymer science where such polygons model proteins. In systems composed of a large number n > > 1 of components, the l i are usually only partially known, so that we can assume that each l i ∈ R + is random. We denote by µ n the distribution of ℓ. We will obtain our results under the following assumption:

(H) µ n is a product measure µ n = µ ⊗n with µ a diffuse measure on (0, ∞) such that e ηx µ(dx) < ∞ for some η > 0.

Notice that M tℓ and M ℓ are equal when t > 0, so that the measure µ n might be seen as a probability measure on the unit simplex △ n-1 .

To get some idea on the nature of the random manifold M ℓ , one can study the stochastic behavior of invariants like Betti numbers, the Euler characteristics or the total Betti number (see below). Here, we focus on the Betti numbers b p (M ℓ ), for dimensions p = p n growing with n. We recall the result of [START_REF] Farber | Homology of planar polygon spaces[END_REF] describing Betti numbers of planar polygon spaces as functions of the length vector ℓ. In what follows,

[n] denotes the set {1, • • • , n}. A subset J ⊂ [n] is called short if j∈J l j < j / ∈J l j . It is called median if j∈J l j = j / ∈J l j . Let 1 ≤ i 0 ≤ n be such that l i 0 is maximal among l 1 , • • • , l n .
p (M ℓ ) = a p (ℓ) + ãp (ℓ) + a n-3-p (ℓ), (1)
so that the Poincaré polynomial of the random manifold is given by

p M ℓ (t) = n-3 p=0 b p (M ℓ )t p = q(t) + t n-3 q( 1 t ) + r(t), (2)
where

q(t) = n-3 k=0 a k t k and r(t) = n-3 k=0 ãk t k ,
see [START_REF] Farber | Topology of random linkages[END_REF]. The total Betti number B(M ℓ ) defined by

B(M ℓ ) = n-3 p=0 b p (M ℓ ) = p M ℓ (1),
provides ideas on the size or on the complexity of the manifold M ℓ . We will study the asymptotic behavior of B(M ℓ ) when n is large and ℓ is random. We first give some examples following [START_REF] Farber | Homology of planar polygon spaces[END_REF].

In the equilateral case where each l i is equal to some l > 0, it turns out that one can give exact formulas for the various Betti numbers, and therefore for B(Ml):

assume n = 2r + 1 odd. Then b k (Ml) = n-1 k when k < r -1, b k (Ml) = 2 n-1 r-1 when k = r -1 and b k (Ml) = n-1
k+2 when k > r -1. The related total Betti number is then given by p Ml (1) = 2 n-1 -n-1 r . For arbitrary large n, one has (see [START_REF] Farber | Homology of planar polygon spaces[END_REF])

p Ml (1) = B n = 2 n-1 (1 - 2 πn + o(n -1/2 )) , n → ∞. (3)
For pentagons, that is when n = 5, the moduli space M ℓ is a compact orientable surface of genus not exceeding 4.

The vector length ℓ is said to be generic when n i=1 l i ε i = 0, for any ε = (ε i ) 1≤i≤n , where , ∀i ε i ∈ {-1, +1}. When n is even, equilateral weights with l i ≡ l are not generic. [START_REF] Farber | Homology of planar polygon spaces[END_REF] proved that for generic ℓ, the total Betti number B(M ℓ ) is bounded by 2B n-1 , so that the explicit formulas obtained for equilateral n-gons provide bounds for the maximum over ℓ of B(M ℓ ).

Results

[3] considered the special case where µ is the uniform probability measure on the unit interval with µ n = µ ⊗n , and the case where µ n is the uniform measure on the simplex ∆ n-1 . It was proven that for fixed p ≥ 0, the average p-dimensional betti number

µ n [b p (M ℓ )] = b p (M ℓ )µ n (dℓ)
is asymptotically equivalent to n-1 p , the difference going to zero at an exponential speed. The techniques use exact formulas for the volume of the intersection of a half space with a simplex. We will avoid such formulas to treat general diffuse probability measures using probabilistic techniques, since in fact such volume formulas do not exist for arbitrary measures. Next, [START_REF] Farber | Topology of random linkages[END_REF] consider both planar and spatial polygon spaces, and proved, under an admissibility condition on µ n similar results for mean Betti numbers and also for higher moments, again for fixed dimensions p. As an open question, the author raises the issue of computing the average total Betti number

µ n [B(M ℓ )] = B(M ℓ )µ n (dℓ).
We will consider more generally the mean Poincaré polynomial

pM ℓ (t) = µ n [p M ℓ (t)] = p M ℓ (t)µ n (dℓ), with pM ℓ (1) = µ n [B(M ℓ )].
As the author notices, the knowledge of the individual average Betti numbers µ n [b p (M ℓ )] for large n and fixed p can't help since the terms cannot simply be added up. We will therefore consider the asymptotic behavior of high dimensional Betti numbers µ n [b pn (M ℓ )], where p n goes to infinity when n → ∞ (see Proposition 3.1).

We will obtain our results for product measure µ n satisfying assumption (H) and assume throughout the paper that this hypothesis is satisfied. We prove in Proposition 4.1 that the mean total Betti number is such that

pM ℓ (1) = µ n [B(M ℓ )] ∼ 2 n-1 , (4)
This shows that equilateral polygons (see [START_REF] Farber | Betti numbers of random manifolds Homology[END_REF]) are representative of the emerging average manifold as n >> 1, as suggested in [START_REF] Farber | Topology of random linkages[END_REF]. We will also consider the mean Poincaré polynomial as n is large, and show that pM ℓ (t) ∼ (1 + t) n-1 when 0 < t < 1, and that pM ℓ (t) ∼ (1 + t) n-1 t -2 when t > 1.

Further moments are also considered and their asymptotic is given in Proposition 4.3. We next consider spatial polygon spaces

N ℓ = {(u 1 , • • • , u n ) ∈ S 2 × • • • S 2 ; n i=1 l i u i = 0}/SO(3).
In this case, for generic length vector ℓ, [START_REF] Hausmann | Cohomology rings of polygon spaces[END_REF] proved that the even Betti numbers are given by

b 2p (N ℓ ) = p j=0 (â j (ℓ) -ân-j-2 (ℓ)), (5)
where âj (ℓ) denotes the number of short subsets J ⊂ [n] of cardinality |J| = j + 1 containing n. The Betti number of odd dimensions vanish. Furthermore, [START_REF] Hausmann | Cohomology rings of polygon spaces[END_REF] proved that the related Poincaré polynomial is given by

p N ℓ (t) = 1 1 -t 2 J∈Sn t 2(|J|-1) -t 2(n-|J|-1) ,
where J ∈ S n if and only if {n} ⊂ J ⊂ [n] and J is median or short. If ℓ is generic, there is no median set and this is equivalent to

p N ℓ (t) = 1 1 -t 2 n-1 j=0 âj (t 2j -t 2(n-j-2) ) = 1 1 -t 2 q(t 2 ) -t 2(n-2) q(t -2 ) , (6)
where

q(t) = n-1 j=0 âj t j .
In the equilateral case where l i ≡ l, [START_REF] Klyachko | Spatial polygons and configurations of points in the projective line Algebraic geometry and its applications[END_REF] proved that the 2p-dimensional Betti number b 2p (Nl) is given by

b 2p (Nl) = p i=0 n -1 i ,
when n = 2k + 1 ≥ 3, so that the Euler characteristics or total Betti number is explicitely given as

p Nl (1) = k-1 i=0 2k i (k -i), with p Nl (1) ∼ n 2π 2 n-2 .
We will study the asymptotic behavior of the mean Poincaré polynomial

pN ℓ (t) = µ n [p N ℓ (t)] = p N ℓ (t)µ n (dℓ),
in the large n limit by providing large deviations estimates. We will see that

pN ℓ (1) = µ n [B(N ℓ )] ∼ n2 n-2 > > p Nl (1).
Furthermore, we will see in Proposition 4.2 that the mean Poincaré polynomial exhibits asymptotically a singular behavior in the neighborhood of t = 1, that is

pN ℓ (t) ∼ (1 + t 2 ) n-1 (1 -t 2 ) when 0 < t < 1,
and

pN ℓ (t) ∼ (1 + t 2 ) n-1 (t 2 -1)t 2 when t > 1.
This shows that equilateral configuration spaces are not representative of the random manifold in dimension 3 when n is large.

Preliminaries

We introduce here the main technical tool used in our analysis of the Betti numbers of random polygon spaces: a probabilistic interpretation of formulas ( 1) and ( 5) in terms of random permutations and stopping times. We first introduce some notations.

For any length vector ℓ ∈ (0, ∞) n , we define l obtained from ℓ by the following permutation of the coordinates : let i 0 be the minimal index such that l i 0 is maximal among the l i , 1 ≤ i ≤ n, and define l = ( l1 , • • • , l1 ) by ln = l i 0 , li 0 = l n , and li = l i if i / ∈ {i 0 , n}.

We denote by σ a random permutation of Σ n-1 with uniform distribution

U Σ n-1 .
The stopping time τ σ (ℓ) is defined by

τ σ (ℓ) = min 0 ≤ t ≤ n -1 ; t i=1 l σ(i) + l n - n-1 i=t+1 l σ(i) ≥ 0 .
We use also the notation τ (ℓ) = τ Id (ℓ) and τ (ℓ) = τ Id ( l). Please note that these stopping times are well-defined and that τ ≤ n -1 and τ ≤ n -2.

We denote by k a random variable with binomial distribution B n-1,q with parameters n -1 and q ∈ [0, 1].

First consider the planar case.

Lemma 2.1 The number a p (ℓ) of short sets is given by

a p (ℓ) = n -1 p U Σ n-1 [τ σ ( l) > p]
The number of median sets ãp (ℓ) vanishes µ n -almost surely. Hence, the planar Betti numbers are given µ n -almost surely by

b p (M ℓ ) = U Σ n-1 n -1 p 1 {τσ( l)>p} + n -1 p + 2 1 {τσ ( l)>n-p-3}
and have expected value

µ n [b p (M ℓ )] = µ n n -1 p 1 {τ (ℓ)>p} + n -1 p + 2 1 {τ (ℓ)>n-p-3}
In the spatial case, the following representation holds.

Lemma 2.2

The coefficients âp are given by

âp (ℓ) = n -1 p U Σ n-1 [τ σ (ℓ) > p]
Hence, the even spatial Betti numbers are given µ n -almost surely by

b 2p (N ℓ ) = 2 n-1 (U Σ n-1 ⊗ B n-1,1/2 ) 1 {τσ (ℓ)>k;0≤k≤p} -1 {τσ (ℓ)>k;n-p-2≤k≤n-2}
and have expected value

µ n [b 2p (N ℓ )] = 2 n-1 (µ n ⊗ B n-1,1/2 ) 1 {τ (ℓ)>k;0≤k≤p} -1 {τ (ℓ)>k;n-p-2≤k≤n-2}
Proof of Lemmas 2.1 and 2.2

We consider the planar case and prove the first lemma. The second lemma corresponding to the spatial case is proved with a very similar analysis.

From the definition of the coefficient a p (ℓ), we have

a p (ℓ) = J⊂[n-1] ; |J|=p 1 A J ( l)
where

A J = {ℓ ; j∈J l j + l n -j / ∈J l j < 0}. From the definition of τ σ , it is easily seen that ℓ ∈ A {σ(1),••• ,σ(p)} if and only if τ σ (ℓ) > p. Furthermore, for each subset J ⊂ [n -1] such that |J| = p, there are p!(n -1 -p)! permutations σ ∈ Σ n-1 such that J = {σ(1), • • • , σ(p)}. As a consequence, the coefficient a p (ℓ) rewrites a p (ℓ) = 1 p!(n -1 -p)! σ∈Σ n-1 1 {τσ ( l)>p} = n -1 p U Σ n-1 [τ σ ( l) > p].
From the definition of the coefficient ãp (ℓ), we have

ãp (ℓ) = J⊂[n-1] ; |J|=p 1 B J ( l)
where B J = {ℓ ; j∈J l j + l nj / ∈J l j = 0}. But it is easily seen that since µ is diffuse, the sum j∈J l j + l nj / ∈J l j is also diffuse and µ n (B J ) = 0 for any J ⊂ [n -1]. Hence ãp (ℓ) is almost surely equal to zero.

The formula for the Betti number b p (ℓ) is then a reformulation of equation [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF]. Thanks to the invariance of µ n under the action of the permutation group, the distribution of τ σ ( l) under µ n does not depend on σ ∈ Σ n-1 and hence is equal to the distribution of τ . The result for the expected value µ n [b p (M ℓ )] follows.

As will be clear in the sequel, the asymptotic behavior of the Betti numbers is strongly linked with the asymptotic behavior of the random variables τ (ℓ) and τ (ℓ). This is the point of the following lemma.

Lemma 2.3

The following weak convergences hold under µ n as n → ∞:

1. weak law of large numbers:

n -1 τ ⇒ δ 1/2 ,

central limit theorem:

n -1/2 τ - n 2 ⇒ N (0, σ 2 τ ),
where σ τ = σ 2m , m = IE(l) and σ 2 = Var(l).

large deviations: for any

ε > 0, lim sup n -1 log µ n (|n -1 τ -1/2| ≥ ε) < 0
The same results also hold for τ instead of τ with the same variance σ τ = σ τ .

The proof is postponed to the appendix.

3 High dimensional Betti numbers

Planar polygons

The following proposition gives the asymptotic of average high dimensional Betti numbers.

Proposition 3.1 Let (p n ) n≥1 be a sequence of integers.

1. If lim sup n -1 p n < 1/2, then µ n [b pn (M ℓ )] ∼ n-1 pn as n → ∞. 2. If lim inf n -1 p n > 1/2, then µ n [b pn (M ℓ )] ∼ n-1 pn+2 as n → ∞. 3. If lim n -1/2 (p n -n/2) = α, then µ n [b pn (M ℓ )] ∼ 2 πn e -2α 2 2 n-1 as n → ∞.
Applying Proposition 3.1 with a specific choice of the sequence p n , we deduce the following corollary. The asymptotic of the binomial coefficient is obtained with Stirling's formula.

µ n [b pn (M ℓ )] = n -1 p n µ n (τ > p n ) + n -1 p n + 2 µ n (τ > n -3 -p n ).
When lim sup n -1 p n < 1/2, the weak law of large numbers provided in Lemma 2.3 implies that

µ n (τ > p n ) → 1 and µ n (τ > n -3 -p n ) → 0
as n → ∞, and from large deviations estimates, the convergence speed to zero is exponential. The first point in Proposition 3.1 follows since

b pn (n, µ n ) = n -1 p n µ n (τ > p n ) + (n -p n -1)(n -p n -2) (p n + 1)(p n + 2) µ n (τ > n -3 -p n ) ∼ n -1 p n .
Similarly, when lim inf n -1 p n > 1/2,

µ n (τ > p n ) → 0 and µ n (τ > n -3 -p n ) → 1
as n → ∞ where the convergence to zero is exponentially fast. The second point in Proposition 3.1 follows.

Finally, in the case lim n 1/2 (p nn/2) = α, the central limit Theorem from Lemma 2.3 yields that as n → ∞

µ n (τ > p n ) → 1 -F N (α/σ τ ) and µ n (τ > n -3 -p n ) → 1 -F N (-α/σ τ ),
where F N is the repartition function of the standard normal distribution. Furthermore, from the local limit theorem for the binomial distribution,

n -1 p n ∼ n -1 n -3 -p n ∼ 2 πn e -2α 2 2 n-1 ,
as n → ∞. These estimates yield the last point in Proposition 3.1 since

1 -F N (α/σ τ ) + 1 -F N (-α/σ τ ) = 1.

Spatial polygons

We perform a similar study in the spatial case. The asymptotic behavior of average high dimensional Betti numbers is given by the following Proposition.

Proposition 3.2 Let (p n ) n≥1 be a sequence of integers.

1. If lim sup n -1 p n < 1/2, then µ n [b 2pn (N ℓ )] ∼ pn k=0 n-1 k
as n → ∞.

If lim inf n

-1 p n > 1/2, then µ n [b 2pn (N ℓ )] ∼ n-pn-3 k=0 n-1 k as n → ∞. 3. If lim n -1/2 (p n -n/2) = α, then µ n [b 2pn (N ℓ )] ∼ C(α)2 n-1 as n → ∞, with C(α) = ∞ 2|α| e -u 2 2 √ 2π P(|Z| < um σ )du,
where m = µ(l), σ 2 = Var(l), and Z is standard normal.

Proof of Proposition 3.2:

Recall from Lemma 2.2 that the expected Betti number µ n [b 2p (N ℓ )] is given by

µ n [b 2p (N ℓ )] = 2 n-1 (µ n ⊗ B n-1,1/2 ) 1 {τ >k;0≤k≤pn} -1 {τ >n-1-k;1≤k≤pn+1}
(we use here the fact that k and n -1k have the same distribution under B n-1,1/2 ). Consider first the case lim sup n -1 p n < 1/2 and write

µ n (τ > p n )B n-1,1/2 (0 ≤ k ≤ p n ) ≤ (µ n ⊗B n-1,1/2 ) 1 {τ >k;0≤k≤pn} ≤ B n-1,1/2 (0 ≤ k ≤ p n ).
Using the weak law of large numbers n -1 τ → 1/2 under µ n and the asymptotic for p n , wee see that µ n (τ > p n ) → 1 as n → ∞. Hence the equivalent

(µ n ⊗ B n-1,1/2 ) 1 {τ >k;0≤k≤pn} ∼ B n-1,1/2 (0 ≤ k ≤ p n ).
In the same way,

0 ≤ (µ n ⊗B n-1,1/2 ) 1 {τ >n-1-k;1≤k≤pn+1} ≤ µ n (τ > n-1-p n )B n-1,1/2 (1 ≤ k ≤ p n +1))
and a large deviations argument shows that µ n (τ > np n ) converges exponentially fast to zero, so that this last term is of smaller order than

B n-1,1/2 (0 ≤ k ≤ p n ).
This proves the first point.

Consider now the case lim inf n -1 p n > 1/2. It appears that many terms cancel out and we have for large n

µ n [b 2pn (N ℓ )] = 2 n-1 (µ n ⊗ B n-1,1/2 ) 1 {τ >k;0≤k≤pn} -1 {τ >k;n-pn-2≤k≤n-2} = 2 n-1 (µ n ⊗ B n-1,1/2 ) 1 {τ >k;0≤k≤n-3-pn} -1 {τ >k;pn+1≤k≤n-2} ∼ B n-1,1/2 (0 ≤ k ≤ n -3 -p n ),
where the equivalent is proved just as above.

Finally, consider the case p n = n/2 + α n √ n with α n → α. We use the central limit Theorem and write

µ n [b 2pn (N ℓ )] = 2 n-1 µ n ⊗ B n-1,1/2 ) 1 {τ >k;k≤pn} -1 {τ >k;n-pn-2≤k≤n-2} = 2 n-1 µ n ⊗ B n-1,1/2 ) 1 {n -1/2 (τ -n/2)>n -1/2 (k-n/2);n -1/2 (k-n/2)≤αn} -1 {n -1/2 (τ -n/2)>n -1/2 (k-n/2);-αn-2n -1/2 ≤n -1/2 (k-n/2)≤n -1/2 (n/2-2)} ∼ 2 n-1 E[1 {στ G 1 >G 2 /2;G 2 /2<α} -1 {στ G 1 >G 2 /2;G 2 /2>-α} ]
with G 1 and G 2 independent standard Gaussian random variables. The constant C(α) corresponds to the expectation in the last line. Using symetry properties for the distribution of (G 1 , G 2 ), we easily verify the announced formula for C(α). This ends the proof of Proposition 3.2

Asymptotic behavior of the Poincaré polynomial 4.1 Planar polygons

We will here consider the random Poincaré polynomial p M ℓ (t) as given in (2) in the large n limit. We first give a representation of this invariant in terms of random permutations and stopping times.

p M ℓ (t) = (1 + t) n-1 (U Σ n-1 ⊗ B n-1, t 1+t ) 1 {τσ( l)>k} + t -2 1 {τσ( l)>n-1-k} .
As a consequence,

pM ℓ (t) = (1 + t) n-1 (µ n ⊗ B n-1, t 1+t ) 1 {τ >k} + t -2 1 {τ >n-1-k} .
Thanks to this lemma, we prove the following Proposition giving the asymptotic of the average Poincaré polynomial. Proposition 4.1 Let pM ℓ (t) be the mean Poincaré polynomial. When t > 0,

1. If 0 < t < 1, then pM ℓ (t) ∼ (1 + t) n-1 . 2. If t > 1, then pM ℓ (t) ∼ (1 + t) n-1 t -2 .
3. If t = 1, then the mean total Betti number satisfies pM ℓ (1) ∼ 2 n-1 .

Proof of Lemma 4.1 Equation (2) together with Lemma 2.1 yield

q(t) = n-1 k=0 n -1 k t k U Σ n-1 (τ σ ( l) > k) = (1 + t) n-1 n-1 k=0 n -1 k t 1 + t k 1 1 + t n-1-k U Σ n-1 (τ σ ( l) > k) = (1 + t) n-1 (U Σ n-1 ⊗ B n-1, t 1+t ) τ σ ( l) > k .
Please note that in the sum the terms corresponding to k = n -2 and k = n -1 vanish. Finally, Lemma 4.1 follows from the relation p M ℓ (t) = q(t) + t n-3 q(t -1 ) + r(t), with r(t) µ n -almost surely vanishing and from the fact that the distribution of k under

B n-1, 1 1+t is equal to the distribution of n -1 -k under B n-1, t 1+t .
We use once again the invariance property of µ n under the action of the symetric group to simplify the expression of the average Poincaré polynomial µ n [p M ℓ (t)].

Proof of Proposition 4.1

We use the representation of the average Poincar polynomial given in Lemma 4.1 together with weak convergence for (τ , k) under µ n ⊗ B n-1, t 1+t to study the asymptotic behavior .

The weak law of large number for τ (see Lemma 2.3) and a standard weak law of large numbers for binomial distribution imply that (n -1 τ , n -1 k) converges weakly under µ n ⊗ B n-1, t 1+t to (0, t 1+t ). The continuous mapping theorem implies that for 0 < t < 1 or t > 1, the following weak convergence holds under µ n ⊗ B n-1, t 1+t :

1 {τ >k} + t -2 1 {τ >n-1-k} ⇒ 1 { 1 2 > t 1+t } + t -2 1 { 1 2 >1-t 1+t } = min(1, t -2 ).
Integrating this (bounded) convergence yield the result for t = 1.

For t = 1, the continuous mapping theorem does not hold no longer since the map (τ , k) → 1 {τ >k} is not continuous at point (1/2, 1/2). We need here the central limit Theorem. From Lemma 2.3 and standard results for binomial distribution, (n -1/2 (τn/2), n -1/2 (kn/2)) converges weakly under µ n ⊗ B n-1,1/2 to N (0, σ 2 τ ) ⊗ N (0, 1/4). The continuous mapping Theorem yields

1 {τ >k} + 1 {τ >n-1-k} = 1 {n -1/2 (τ -n/2)>n -1/2 (k-n/2)} + 1 {n -1/2 (τ -n/2),n -1/2 (n/2-1-k)} ⇒ 1 {σ τ G 1 >G 2 /2} + 1 {σ τ G 1 >-G 2 /2}
with G 1 and G 2 independent standard Gaussian random variables. We integrate this (bounded) convergence and remark that E(

1 {σ τ G 1 >G 2 /2} ) = E(1 {σ τ G 1 >-G 2 /2} ) = 1/2.
Remark: we can use large deviations results to estimate the speed of convergence in Proposition 4.1 when t = 1. For example for 0 < t < 1, write

µ n (1 + t) -(n-1) p M ℓ (t) -1 = (µ n ⊗ B n-1, t 1+t ) 1 {τ >k} -1 + t -2 1 {τ >n-1-k} = (µ n ⊗ B n-1, t 1+t ) 1 {n -1 (τ -k)≤0} + t -2 1 {n -1 (τ +k)≥1} . Now large deviations for n -1 (τ , k) under (µ n ⊗ B n-1, t 1+t
) will give the speed of convergence to 0 in a logarithmic scale. For t > 1, we have

µ n (1 + t) -(n-1) (p M ℓ (t)) -t -2 = (µ n ⊗ B n-1, t 1+t ) 1 {n -1 (τ -k)>0} + t -2 1 {n -1 (τ +k)<1} ,
and we can use the same method.

Spatial polygons

We use the same strategy in the spatial case and use formula [START_REF] Hausmann | Sur la topologie des bras articulés In Algebraic Topolgy Poznan[END_REF] giving the Poincar polynomial for generic vector length. Since µ is diffuse, µ n -almost every vector length is generic and equation ( 6) holds. The related total Betti number is obtained by taking the t → 1 limit in ( 6)

p N ℓ (1) = lim t→1 1 1 -t 2 q(t 2 ) -t 2(n-2) q(t -2 ) = (n -2)q(1) -2q ′ (1) = (n -2) n-1 j=0 âj -2 n-1 j=0 jâ j (7)
We use the following representations for the Poincar polynomial:

Lemma 4.2
The Poincar polynomial is given µ n -almost surely by

p N ℓ (t) = (1 + t 2 ) n-1 1 -t 2 (U Σ n-1 ⊗ B n-1, t 2 1+t 2
) 1 {τσ(ℓ)>k}t -2 1 {τσ (ℓ)>n-k} , for 0 < t < 1 or t > 1, and by

p N ℓ (1) = n2 n-1 (U Σ n-1 ⊗ B n-1,1/2 ) ( n -2 n - 2k n )1 {τσ (ℓ)>k} for t = 1.
Proposition 4.2 Let pN ℓ be the mean Poincaré polynomial associated with random spatial polygons. When t > 0,

1. If 0 < t < 1, then pN ℓ (t) ∼ (1+t 2 ) n-1 1-t 2 .
2. If t > 1, then pN ℓ (t) ∼ (1+t 2 ) n-1 t 2 (t 2 -1) . 3. If t = 1, then the total Betti number satisfies pN ℓ (1) ∼ n2 n-2 .

Remark: In the case of spatial polygons, the Poincar polynomial is an even function. Hence its asymptotic mean behavior for t < 0 follows directly from Proposition 4.2.

Proof of Lemma 4.2

The proof is very similar to the proof of Lemma 4.1. Equation ( 6) together with Lemma 2.2 yield

q(t) = (1 + t) n-1 (U Σ n-1 ⊗ B n-1, t 1+t ) [τ σ (ℓ) > k] .
The case t = 1 follows from the relation

p N ℓ (t) = 1 1 -t 2 q(t 2 ) -t 2(n-2) q(t -2 )
and from the fact that the distribution of k under

B n-1, 1 1+t 2 is equal to the distribu- tion of n -K under B n-1, t 2 1+t 2 . Hence µ n (n -1/2 ln > ε) = (1 -F µ (εn 1/2 )) n ,
and the exponential Markov inequality implies 1 -F µ (εn 1/2 ) ≤ exp(-ηεn 1/2 ) e ηx µ(dx), η > 0.

This implies the weak convergence n -1/2 ln to zero. Conditionnaly to ln = u, the other components (l i ) 1≤i≤n-1 are i.i.d. with conditional distribution given by

µ n ( li ≤ x | ln = u) = F µ (x ∧ u) F µ (u) .
Denote by m(u) and σ 2 (u) the related conditionnal expectation and variance. From the central limit theorem for independent variables, conditionnaly to ln = u, the quantity n -1/2 ( pn i=1 lin-1 i=pn+1 li ) converges weakly to a gaussian distribution of mean 2αm(u) and variance σ 2 (u). Hence the conditionnal probability

µ n   n -1/2 ( pn i=1 li - n-1 i=pn+1 li ) ≥ 0| ln = u   converges to F N (2αm(u)/σ(u))
as n → ∞. We now have to integrate this with respect to ln . Taking into account that ln converges weakly to l max = inf{x ∈ R; F µ (x) = 1} ∈ (0, +∞] as n → ∞ and that (m(u), σ(u)) → (m, σ) as u → l max , we see that

µ n   n -1/2 ( pn i=1 li - n-1 i=pn+1 li ) ≥ 0   → F N (2αm/σ).
This proves the central limit theorem for τ .

We now prove the large deviation estimate. Since

µ n (τ ≤ (1/2 -ε)n)) = µ n  l n + [(1/2-ε)n] i=1 li - n-1 i=[(1/2-ε)n] li ≥ 0   ,
we will provide large deviations estimates for the random sum

S n = ln + [(1/2-ε)n] i=1 li - n-1 i=[(1/2+ε)n]
li .

For t ∈ R, the logarithmic moment generating function is defined by

Λ n (t) = log(µ n (exp(tS n ))).
Using Laplace method, we see that as n → ∞, n -1 Λ n (t) converges to Λ(t) = (1/2ε) e ty µ(dy) + (1/2 + ε) e ty µ(dy).

Using Grtner-Ellis theorem, see e.g. [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF], we deduce a large deviations principle for the sum n -1 S n of speed n and of good rate function I being the Fenchel-Legendre transform of Λ. The exact form of I is irrelevant here but it is important to see that I is strictly positive on [0, ∞). Standard arguments from large deviations theory (see [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF]) give that I vanishes only at (1/2ε)m -(1/2 + ε)m = -2εm < 0, and hence the action I is negative on [0, ∞). As a consequence, the large deviations principle states that lim sup n -1 log µ n (τ

≤ (1/2 -ε)n) ≤ -inf [0,∞) I < 0.
The same technique is used to deal with µ n (τ ≥ (1/2 + ε)n) and this proves the Lemma.

Lemma 4. [START_REF] Farber | Betti numbers of random manifolds Homology[END_REF] The following bivariate Central Limit Theorem holds under µ n ⊗U ⊗2 Σ n-1 :

n -1/2 (τ σ 1 (ℓ)n/2, τ σ 2 (ℓ)n/2) ⇒ N (0, σ 2 τ ) ⊗2 .

It also holds for τ

Proof of Lemma need a bivariate central limit Theorem for (τ σ 1 ( l), τ σ 2 ( l)) under Let p n,i = n 2 + α n,i √ n with α n,i → α as n → ∞ for i = 1, 2. By the definition of τσ i , We know from the proof of Lemma 2.3 that n -1/2 ln converges weakly to zero. It remains to check that n -1/2 ( p n,i j=1 lσ i (j) -n-1 j=p n,i +1 lσ i (j) ) i=1,2 satisfies a bivariate central limit theorem. Let θ i , i = 1, 2 be real numbers, and consider the linear combination (θ 1 ε n,1 (j) + θ 1 ε n,2 (j)) lj , where we set ε n,i (j) = 21 {σ i (j)≤p n,i } -1. Conditionnaly to ln = u, the components lj are i.i.d. with mean m(u) and variance σ(u), and hence the above sum is a linear triangular array of independent variables with random coefficients (θ 1 ε n,1 (j) + θ 1 ε n,2 (j)) 1≤j≤n-1 . The coefficients are almost surely bounded and satisfy a weak law of large numbers under U ⊗2

(µ n ⊗ U ⊗2 Σ n-1 ) (τ σ i ≤ p n,i ; i = 1..2) = (µ n ⊗ U ⊗2 Σ n-1 )   n -1/2 ln + n -1/2 (
Σ n-1 n -1 n-1 j=1 (θ 1 ε n,1 (j) + θ 1 ε n,2 (j)) 2 → θ 2 1 + θ 2 2 .
(note that the empirical distribution 1 n-1 n-1 j=1 δ (ε n,1 (j),ε n,1 (j)ε n,2 (j)) converges weakly to the uniform distribution on {(±1, ±1)}). As a consequence, conditionaly to ln = u, the above sum converges to a gaussian random variables of mean 2(α 1 θ 1 + α 2 θ 2 )m(u) and variance (θ 2 1 + θ 2 2 )σ 2 (u). Integrating with respect to ln we obtain that the sum converges weakly to a gaussian random variables with mean 2(α 1 θ 1 + α 2 θ 2 )m and variance (θ 2 1 + θ 2 2 )σ 2 . This proves the bivariate central limit theorem with asymptotic independent components.

  Denote by a p (ℓ) the number of short subsets J ⊂ [n] of cardinality |J| = p + 1 and containing i 0 . Denote by ãp (ℓ) the number of median subsets J ⊂ [n] of cardinality |J| = p + 1 and containing i 0 . Then one has for p = 0, 1, . . . , n -3 b

Corollary 3 . 1

 31 Let p ∈ (0, 1) and p n = [np]. Then, lim n→∞ n -1 log µ n [b pn (M ℓ )] = -p log p -(1p) log(1p) Proof of Proposition 3.1: From Lemma 2.1, the average Bett numbers is given by

Lemma 4 . 1

 41 For any t > 0, the Poincar polynomial is given µ n -almost surely by

2 i=1θ

 2 i n -1/2 ( p n,i j=1 lσ i (j) -n-1 j=p n,i +1 lσ i (j) = n -1/2 n-1 j=1

In the case t = 1, equation [START_REF] Hausmann | Cohomology rings of polygon spaces[END_REF] and Lemma 2.2 imply

Proof of Proposition 4.2 The case 0 < t < 1 and t > 1 are easily deduced from Lemma 4.2 using the following law of large numbers: under

n -1 (k, τ ) converges weakly to (1/2, t 2 1+t 2 ) as n → ∞. Details are omitted since they are as in the proof of Proposition 4.1.

In the case t = 1, the central limit Theorem from Lemma 2.3 states that (n

whith G 1 and G 2 independent standard Gaussian random variables.

Remark: In order to estimate the speed of convergence in Proposition 4.2 when t = 1, we can use for 0 < t < 1 the expression

)

and for t > 1

)

Large deviations results for n

) would give the speed of convergence in a logarithmic scale.

Higher moments

We consider here the higher moments of the Poincar polynomial and prove that their asymptotic behavior is given by the first moment. To this aim, we prove a weak law of large numbers for the renormalized Poincar polynomial.

We begin with the case of planar polygon.

Proposition 4.3 For any t > 0, the following weak convergence holds under

As a consequence, for any t > 0 and ν ∈ N,

Proof of Proposition 4.3 Proposition 4.1 states that the expectation under µ n of (1 + t) -(n-1) p M ℓ (t) converges to min(1, t -2 ) as n → ∞. Hence, weak convergence will be proved as soon as we show that the variance under µ n of (1 + t) -(n-1) p M ℓ (t) goes to zero. We use the representation of the Poincar polynomial from Lemma 4.1 and the replica trick to compute the second moment

We need to show that the two factors of Prod are asymptotically independent in the limit n → ∞. This would yield

and hence the variance of p M ℓ (t) (1+t) n-1 would converge to zero as n → ∞. We now prove asymptotic independance of the two factors. When 0 < t < 1 or t > 1 the aymptotic independence follows from the weak law of large numbers obtained in Lemma 2.3, both factors converging weakly to min(1, t -2 ) (note that the distribution of τ σ ( l) under µ n ⊗ U Σ n-1 is equal to the distribution of τ (ℓ) under µ n ). When t = 1, we use the bivariate central limit Theorem stated in Lemma 4.3 in the Appendix. Weak convergence is proved.

The convergence of the moments is a direct consequence of the weak convergence once we remark that the renormalized Poincar polynomial (1 + t) -(n-1) p M ℓ (t) is µ n almost surely bounded by 1 + t -2 (this is clear from the representation given in Lemma 4.1).

We consider now the higher moments of the Poincar polynomial for spatial polygons spaces. The results and methods are very similar to one of the planar case and are based on Lemma 4.2. Hence we give only the main lines of the proof. Proposition 4. [START_REF] Farber | Homology of planar polygon spaces[END_REF] The following weak convergence holds under µ n as n → ∞,

As a consequence, for any t > 0 and ν ∈ N,

Proof of Proposition 4.4

The proof is similar to the proof of Proposition 4.3 with the following expression of the renormalized Poincar polynomial deduced from Lemma 2.2: for 0 < t < 1 or t > 1

) 1 {τσ (ℓ)>k}t -2 1 {τσ(ℓ)>n-k} , and for t = 1

Convergence of the expectation was proved in Proposition 4.2. The variance is computed using thanks to the replica trick and is shown to converge to zero because of the asymptotic independence of

Appendix

Proof of Lemma 2.3

The weak law of large number is a consequence of the central limit theorem that we prove now. Let We now prove that n -1/2 ln converges weakly to zero and that n -1/2 ( pn i=1 lin-1 i=pn+1 li ) satisfies a central limit theorem. To see this, we denote by F µ the repartition function of µ, and remark that the distribution of ln is given by µ n ( ln ≤ x) = F µ (x) n .