Kateryna Omelchuk

Moncef Stambouli

Alexandre Chagnes

Investigation of aggregation and acid dissociation of new cationic exchangers for liquid-liquid extraction

Keywords: Acidic dissociation constant, cationic exchanger, dimerisation, physicochemistry, trimerisation, solvent extraction

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A C C E P T E D M A N U S C R I P T

2 Solvent extraction is a mature technology that is used in industry for the recovery of metals from leaching solutions. In solvent extraction, the aqueous leaching solution is contacted with an organic phase containing an extractant or a mixture of extractants diluted in an aliphatic or an aromatic diluent. Both immiscible phases are mixed and settled in mixerssettlers. Afterwards, metals in the loaded organic phase are stripped with an appropriate aqueous solution before performing further treatments such as precipitation/crystallisation or electrolysis.

Extractants used in solvent extraction are divided into three families: (i) cationic exchangers such as bis-(2-ethylhexyl)phosphoric acid (D2EHPA), bis-(2,4,4trimethylpentyl)phosphinic acid (Cyanex 272), etc., (ii) anionic exchangers such as tri-noctylamine (Alamine® 336), Trioctylmethylammonium chloride (Aliquat® 336),etc. and (iii) solvating agents such as tri-n-octylphosphine oxide (TOPO), tri-n-butylphosphate (TBP), etc.

[1]. The choice of the solvent depends on the nature of the leaching solution, the nature of the metals to be extracted and the nature of the impurities present in the leaching solution. The search for new extractants is a real challenge in the development of liquid-liquid extraction processes since it is mandatory to treat more and more complex ores or secondary resources in an efficient and economical way. Cationic exchangers are used in many applications such as the recovery of cobalt from acidic sulfate or chloride media [2][3][4], uranium from concentrated phosphoric acid [5,6], etc.

Although solvent extraction seems easy to implement, its physicochemistry is complex because the system contains two phases in which many equilibria occur as well as an interphase. Furthermore, the non-ideal behaviour of the system due to high concentration in aqueous phase also increases the complexity of the physicochemistry of liquid-liquid extraction.

In solvent extraction chemistry, the knowledge of the acidity constants and the interaction constants between the components of an extracting system is essential to clarify processes occurring in solution and to ascertain the stoichiometry of the extracted species [7,8]. In this paper, the physicochemical properties of new synthesized and commercial organophosphorus cationic exchangers have been investigated (Table 1). These extractants could be used for the recovery of cobalt, nickel, and manganese from acidic sulfate or chloride media. They were also studied for the recovery of uranium from concentrated phosphoric acid providing that they are used in mixture of solvating agents such as tri-n-octylphosphine or tri-n-

Material and Methods

Synthesis of organophosphorus acids

Phosphorus trichloride (Fisher Scientific, purity 99%), pyridine (Fisher Scientific, purity 99%), 1-octanol (Fisher Scientific, purity 99%), 2-methyl-1-propanol (Fisher Scientific, purity 99%), 1-butanol (Fisher Scientific, purity 99,5%), lithium aluminium hydride (Fisher Scientific, purity 95%, powder), iodine (Fisher Scientific, purity 99%), sodium (Sigma Aldrich purity ≥99.8%), ethylene glycol butyl ether (Sigma Aldrich, purity ≥99%), 6undecanone (TCI, purity >98 %), 2-ethyl-1-hexanol (TCI, purity >99,5%) and 1,3-dichloro-2propanol (TCI, purity>98%) were used as delivered.

The synthesis of BiDiBOPP was performed according to the procedure previously published by Beltrami et al. [6] (Scheme 1).

Scheme 1

The 1,3-dibutoxy-2-propanol (2a) was synthesized by the reaction of n-butanol and 1,3dichloro-2-propanol in the presence of sodium metal. The reaction of 2a with phosphorus trichloride (PCl3) at room temperature led to bis(1,3-dibutoxypropan-2-yl) phosphite (3a) with 78% yield. After purification, 3a was oxidised by I2 in pyridine-water mixture with ratio of 9:1 to form bis(1,3-dibutoxypropan-2-yl) phosphoric acid (BiDiBOPP, 4a) (Scheme 1).

A C C E P T E D M A N U S C R I P T

4

The cationic exchangers were synthesized by using the same procedure as for BiDiBOPP except for the synthesis of bis(undecan-6-yl) phosphoric acid (4d) (Table 1). Indeed, the corresponding alcohol 2d used for the synthesis of 4d was synthesized by the reduction of undecan-6-on with lithium aluminium hydride LiAlH4 at 0 °C in anhydrous THF according to the following reaction:

The reaction yields are reported in Table 2.

The purities of the extractants synthesized in this work varied between 90% and 95% (Table 3). They were determined by 1 H, 13 C, 31 P NMR spectroscopy, molecular mass spectroscopy (MS) and electrospray ionisation molecular mass spectroscopy (ESI-MS). For a sake of comparison, the purities were also determined by 31 P NMR. The 31 P NMR spectra of UPA and EHPA showed the presence of an impurity characterized by peaks located at -14 ppm and -13 ppm, respectively, attributed to the compounds reported in Table 4 [9]. The presence of such an impurity in UPA was confirmed by molecular mass-spectrometry.

The 31 P NMR spectra of the synthesized extractants also showed the presence of two peaks located at 0.5 and 20 ppm corresponding to phosphorus impurities whose the total concentration was estimated to be lower than 5%. This range of chemical shifts can be attributed to phosphonates (5-30 ppm), phosphine oxides (20-50 ppm) and phosphoric acid esters (0-15 ppm). Molecular mass-spectrometry analyses confirmed the presence of phosphoric acid and phosphonate esters whereas no trace of phosphine oxide was observed.

Impurities analyses showed that Cyanex® 272 contained 11% phosphine oxide and 4.6%

impurities bearing an acid functional group. Cyanex® 301 contained 10% phosphine oxide whereas D2EHPA and Ionquest® 801 contained only 1% and 3% acids and/or esters as impurities [10].

Given that phosphine oxide is not a Brønsted acid and the concentrations of acid impurities are low, it was considered that the presence of these impurities did not significantly impact the values of the dissociation constants and the aggregation constants of the extractants determined by potentiometric measurements and Vapour Pressure Osmometry (VPO). The values of pKa in water/2-propanol mixtures were determined by potentiometric titrations with a glass electrode (Metrohm® 913) at 25 °C under nitrogen atmosphere (inert conditions).

pKa measurements

A C C E P T E D M A N U S C
It was not possible to use water-ethanol mixtures due to solubility issues of the extracting agents synthesized in the present work. During the potentiometric titrations, the ionic strength of the water/propan-2-ol mixtures was kept constant at 0.1 mol.L -1 by using KCl.

The electrode calibration was performed in buffer solutions containing different concentrations of 2-propanol in water according to the procedure described by Wilson et al. and Mussini et al. [11,12]. As the literature reported only buffer solutions in water/ethanol mixtures, it was assumed that pH values of the buffer solutions in water/2-propanol mixtures were the same as those in water/ethanol solutions reported in Table 5. Such an assumption relies on the fact that the chemicals structures of ethanol and 2-propanol are similar, the same types of interactions occurs in water/ethanol and water/2-propanol mixtures and these solvents exhibit similar values of dielectric constants (20.2 and 25.2 at 20 °C for 2-propanol and ethanol, respectively) and the same dipolar moments (1.69 D). This assumption was checked by determining the pKa of D2EHPA in water-ethanol and water-2-propanol mixtures by potentiometric measurement assuming that water/2-propanol buffer solutions have the same pH as water/ethanol buffer solutions reported in Table 5. Similar values of pKa of D2EHPA in water were deduced from potentiometric measurements in water/ethanol and water/2-propanol media, i.e pKa=2.3 and 2.6, respectively. Although a difference is observed between both pKa values, it has been considered in the present paper that the difference is sufficiently small given the large discrepancies of pKa reported in the literature [See supporting information of ref.

3].

Measurement of phoshorus content in aqueous phases

Solubility of organophosphorus compounds in aqueous phase was determined by the colorimetric method after digestion with concentrated sulphuric acid as reported by Biswas et al. [14]. In this method, orthophosphate was formed, during the digestion of the organophosphorus compounds. Orthophosphate concentration was estimated by the molybdenum-blue method at 830 nm [15].

Vapor Pressure Osmometry measurements

A C C E P T E D M A N U S C R I P T 6
The organic solutions of tri-n-octylamine (TOA, Aldrich, purity=99%) and organophosphorus acids diluted in n-heptane were prepared by weight so that extractant concentrations were comprised between 0.05 and 0.5 mol kg -1 . Vapor Pressure Osmometry (VPO) measurements

were carried out at 25 °C using a Knauer K-7000 Vapour Pressure Osmometer.

Prior to VPO measurements, the thermocouple head was cleaned thoroughly, and the osmometer was calibrated with TOA diluted in n-heptane. Calibration was performed using TOA diluted in n-heptane because TOA was expected to be monomeric in n-heptane in the concentration range of interest.

The difference of resistance (R) was measured as a function of the concentration for both standard and organophosphorus acids. The measured property R may be expressed as a function of the total extractant concentration m HL 0 as:

R = kcal m HL 0 (1)
where kcal is the calibration constant, which can be obtained by measuring R for different TOA solutions since TOA exists only as monomers in n-heptane in the range of investigated concentrations. In order to obtain reliable R values, the drop size was kept as constant as possible and equal in both thermistors, and the measurements were performed by reading R five minutes after placing the drops on the thermistors. Five readings were recorded for each sample and the average value was used in all calculations. In this manner, the precision of R were maintained below 2%. The calibration constant deduced from VPO measurements with TOA in n-heptane at 25 °C was kcal=2015,5 Ω.kg.mol -1 .

Results and Discussions

Acid dissociation constants

The low water solubility of long chain organophosphorus acids prevents the direct determination of pKa in water by potentiometric measurements. Therefore, the acid dissociation constants of the organophosphorus acids were determined by potentiometry in water/propan-2-ol mixtures. The pKa values of the organophosphorus compounds in water/propan-2-ol mixtures were determined by using the Gran equation:

𝑉 𝑏 • 10 -𝑝𝐻 = 𝛾 𝐻𝐿 𝛾 𝐿 - 𝐾 𝑎 (𝑉 𝑒 -𝑉 𝑏 ) (2)
where Vb is the volume of sodium hydroxide used for the titrations, 𝛾 𝐻𝐿 and 𝛾 𝐿 -are the activity coefficients of the protonated form (HL) and the dissociated form (L -) of the Acid dissociation can be described by the following equilibrium [16]:

HL + H 2 O [L ⋯ H ⋯ H 2 O] ≠ L -+ H 3 O + (3)
where

[L ⋯ H ⋯ H 2 O] ≠ is an activated complex.
The solvent effect of ion-ion, ion-dipole and dipole-dipole interactions can be estimated by means of the Kirkwood equation [17,18]:

𝑝𝐾 𝑎 = 𝑝𝐾 𝑎 0 + 1 2,303 [ 𝑒 2 2𝑘𝑇 ( 1 𝑟 𝐿 - + 1 𝑟 𝐻 3 𝑂 + ) + 3 4𝑘𝑇 ( 𝜇 ≠ 2 𝑟 ≠ 3 - 𝜇 𝐻𝐿 2 𝑟 𝐻𝐿 3 - 𝜇 𝐻 2 𝑂 2 𝑟 𝐻 2 𝑂 3 )] 1 𝜀 (4)
where  is the dielectric constant of the solvent, 𝑝𝐾 𝑎 0 is the 𝑝𝐾 𝑎 value at infinite dielectric constant, e is equal to 1.6 10 -19 C, k is the Boltzmann constant (1.38 10 -23 m 2 kg s -2 K -1 ), T is the temperature in Kelvin, r is the radius of charges species (in m), 𝜇 is the dipole moment (in D).

This equation shows a linear relationship between pKa values and 1/. Such a relationship is interesting as it allows the determination by extrapolation of pKa of compounds poorly soluble in water by using solvent mixtures in which the compound is soluble. In the present case, mixtures of water and propan-2-ol was utilized since the synthesized and commercial organophosphorus compounds exhibit a very low solubility in water and high solubility in water/propan-2-ol mixtures.

In order to calculate the pKa values of the organophosphorus compounds in water, their pKa in several mixtures of water and propan-2-ol have been determined by potentiometric measurements and the Gran method [Eq. ( 2)]. Figure 1 demonstrates the validity of Eq. (4).

In Figure 1, the values of the dielectric constants of the water/propan-2-ol mixtures have been calculated by using Eq. ( 5) deduced from the Redlish-Kister equation [19] KCl deduced by extrapolation at r=78.5 (dielectric constant of water at 25 °C) with the linear relationship found in Figure 1.

The pKa values determined in the present work for the commercial extractants are in accordance with those reported in the literature (Table 6). The difference in pKa is likely due to the difference in ionic strength and experimental methods used to determine the pKa (examination of pKa data from the literature shows discrepancies depending on the methods used for determining pKa of hydrophobic compounds [21]).

It can be observed in Table 6 that an increase of the number of oxygen atoms in the hydrophobic chains and an increase of the number of P-O bonds in the structure of the organophosphorus acids decrease pKa values likely due to attractive inductive effects, which are responsible for an increase of O-H bond polarization. Such an observation was also reported by Fu et al. and Binghua et al. [22,23]. Furthermore, there is no influence of the nature of the hydrophobic chains on the pKa values, excepting for BiDiBOPP and IPA. In the latter cases, branching of the hydrophobic chains is responsible for a slight increase of pKa as it has ever been reported by Martinez et al. [24]. Furthermore, the acidity increases when an oxygen atom is replaced by a sulphur atom because of an increase of the proton lability in the presence of sulphur atom resulting from the difference in electronegativity [25].

Aggregation constants

Figure 2 shows R deduced from VPO measurements as a function of the total concentration of the organophosphorus acids at 25 °C [Eq. ( 1)]. The average molecular weight in g.mol -1 of the extractants can be deduced by the following equation:

𝑀 ̅ = 𝑘 𝑐𝑎𝑙 𝑘 𝑒𝑥𝑡𝑟𝑎𝑐𝑡𝑎𝑛𝑡 (6)
where kcal is the calibration constant (kcal=2015,5 Ω.kg.mol -1 for TOA diluted in n-heptane)

and kextractant in Ω.kg.g -1 is the slope of the linear relationship R = kextractant m HL 0 plotted in Figure 2 where m HL 0 is the total concentration of organophosphorus acid in n-heptane. The values of kextractant and 𝑀 ̅ are given in Table 7 as well as the average number of aggregates of the organophosphorus acids, which were calculated by using Eq. ( 7).

𝑛 ̅ = 𝑀 M𝑚𝑜𝑛𝑜𝑚𝑒𝑟 (7)
where Mmonomer is the molecular weight of the monomeric forms of organophosphorus acids.

A C C E P T E D M A N U S C R I P T 9

Table 7 shows that the synthesized organophosphorus acids exhibit a weak tendency to form aggregates under the present experimental conditions. However, the organophosphorus acids seem to exist as dimers and they could eventually form trimers.

The aggregation of the organophosphorus acid compounds can be expressed as:

2𝐻𝐿 (𝐻𝐿) 2 (8) 3𝐻𝐿 (𝐻𝐿) 3 (9)
Assuming ideal behavior of organophosphorus acids, the stoichiometric constant K2 and K3

for the dimerization and the trimerisation equilibria, respectively, can be written as:

𝐾 2 = 𝑚 (𝐻𝐿) 2 𝑚 (𝐻𝐿) 2 (10)
and

𝐾 3 = 𝑚 (𝐻𝐿) 3 𝑚 (𝐻𝐿) 3 (11)
The total concentration of the monomeric form of organophosphorus acid (m HL 0 ) can be expressed as:

m HL 0 = m HL + 2K 2 m HL 2 + 3K 3 m HL 3 (12)
The apparent concentration of organophosphorus acid (m HL app ), i.e. the total concentration of monomeric and associated organophosphorus acids, is defined as:

m HL app = m HL + K 2 m HL 2 + K 3 m HL 3 (13)
The following equation is deduced from Eq. ( 13);

dm HL app dm HL = 1 + 2K 2 m HL + 3K 3 m HL 2 (14)
By combining Eqs. ( 12) and ( 14), the following equation can be written:

𝑑𝑚 𝐻𝐿 𝑎𝑝𝑝 𝑑𝑚 𝐻𝐿 = 1 + 2𝐾 2 𝑚 𝐻𝐿 + 3𝐾 3 𝑚 𝐻𝐿 2 = m HL 0 m HL (15)
The average number of aggregates n given in Eq. 8 can be rewritten as:

A C C E P T E D M A N U S C R I P T 10 𝑛 = 𝑚 𝐻𝐿 0 𝑚 𝐻𝐿 𝑎𝑝𝑝 (16)
Therefore, Eq. ( 15) can be rewritten as follows:

dm HL app dm HL = n m HL app m HL (17)
The following expression deduced from Eq. ( 17) can be used to calculate the concentration of the monomeric form of the organophosphorus acid (𝑚 𝐻𝐿 ):

𝑙𝑛(𝑚 𝐻𝐿 ) -𝑙𝑛(m HL 0 ) = ∫ 1 𝑛 𝑑 (𝑙𝑛(𝑚 𝐻𝐿 𝑎𝑝𝑝 )) 𝑚 𝐻𝐿 𝑎𝑝𝑝 m HL 0 ( 18
)
with

m HL 0 = w(extractant) • 1000 w(solvent) • MW(extractant) 𝑚 𝐻𝐿 𝑎𝑝𝑝 = ∆𝑅 k 𝑐𝑎𝑙
where w(extractant) and w(solvent) are the weight of extractant and solvent, respectively, and MW is the molecular weight of the extractant in g.mol -1 . R and kcal denote the difference in resistance and the calibration constant in Ω.kg.mol -1 measured during VPO experiments , respectively.

Finally, the dimerization and trimerisation constants can be calculated by plotting the following equations:

m HL 0 𝑚 𝐻𝐿 -1 𝑚 𝐻𝐿 = 2K 2 + 3K 3 𝑚 𝐻𝐿 (19) 𝑚 𝐻𝐿 𝑎𝑝𝑝 𝑚 𝐻𝐿 -1 𝑚 𝐻𝐿 = K 2 + K 3 𝑚 𝐻𝐿 (20)
The corresponding values of K2 and K3 are gathered in Table 7. These values have been used

to calculate the average number of aggregation (𝑛) as a function of extractant concentration

Partition coefficient

The partition coefficients of the organophosphorus compounds (P) between 1 mol.L -1 HCl and HL diluted in n-dodecane is defined by the following equation:

HL HL P = [HL] [HL] (21)
where [HL] ̅̅̅̅̅̅ and [HL] are the concentrations of the monomeric form of the organophosphorus acid in the organic phase and in the aqueous phase, respectively.

[HL] can be expressed as:

[𝐻𝐿] = 𝑆 𝐻𝐿 (1 + 𝐾 𝑎 [𝐻 + ] ) ( 22
)
where SHL is the solubility of the organophosphorus acid in the aqueous phase determined by colorimetric titration.

[𝐻𝐿 ̅̅̅̅ ] can be calculated by solving the following equation:

[HL] 0 = [HL] + 2 [(HL) 2 ] + 3 [(HL) 3 ] = [HL] + 2K 2 [(HL)] 2 + 3K 3 [(HL)] 3 (23)
where [HL] 0 , K 2 and K 3 are the initial concentration of the organophosphorus acid in the organic phase, the dimerization constant and the trimerization constant, respectively.

Table 8 gathers the partition coefficients of the monomeric form of the organophosphorus compounds calculated by using Eqs. ( 22) and ( 23). The partition coefficients of the monomeric form of the synthesized extractants are between 10 and 100 times greater than

A C C E P T E D M A N U S C R I P T

12 those obtained for commercial extractants. The hydrophobicity of the synthesized extractants is obviously higher since the length of the hydrocarbon chains are longer.

Conclusion

The chemical structure of the organophosphorus extractants influences the acidic properties and the aggregation behavior. The values of pKa are strongly affected by the presence of oxygen atoms in the alkyls chains. It appears that pKa values increase in the following order: phosphoric acid<phosphonic acid<phosphonic acid and the presence of sulphur increases the acidity of the molecules. Conversely, there is no significant influence of the hydrophobilicty of the extractants on their acidity strength. Agregation characterization shows that bis(undécan-6-yl) phosphoric acid (UPA) exhibits a weak association while the two isomers, bis(1,3-dibutoxypropan-2-yl) phosphoric acid (BiDiBOPP) and bis(1,3diisobutoxypropan-2-yl) phosphoric acid (IPA) form more dimers and trimers. All syhnthesized extractants exhibit very low aggregation constants compared to D2EHPA likely because of the high steric hindrance, which may not favor hydrogen bonds and may destabilize the aggregates.

5.

Omelchuk, P. Szczepański, A. Shrotre, M. Haddad, A. Chagnes, Effects of Structural Changes of New Organophosphorus Cationic Exchangers on a Solvent Extraction of Cobalt, Nickel and Manganese from Acidic Chloride Media, RSC Adv. between an organic phase containing 0,1 mol.L -1 HL in n-dodecane and 1 mol.L -1 HCl (temperature = 25 °C). Partition coefficients have been calculated by using Eq. 22 and constants reported in Tables 6 and7. 7.

HL

  use of these extractants instead of more classical ones such as D2EHPA or Cyanex 272 may be beneficial since the addition of oxygen atoms in the hydrophobic chains could advantageously lead to a decrease of the pKa values, and therefore an increase of metal extraction efficiency at low pH, while the increase of the steric hindrance of the molecule may improve the selectivity of the extractants and improve metal separation in solvent extraction processes. In particular, aggregation behaviour and acidic dissociation of the molecules have been determined by means of potentiometric and vapour pressure osmometry (VPO) measurements combined with appropriate models.

  A C C E P T E DM A N U S C R I P T 7 organophosphorus compounds, respectively. 𝐾 𝑎 denotes the acid dissociation constant and 𝑉 𝑒 is the equivalence volume. It is expected that the ratio 𝛾 𝐻𝐴 𝛾 𝐴 -=1 at constant ionic strength.

  w/prop, w and prop are the dielectric constants of the water-ethanol mixture, water (w=78.5) and 2-propanol (w=18.6), respectively, and xprop denotes the mole fraction of 2propanol in the mixture. the pKa values of the organophosphorus acids in aqueous solution of 1 mol.L -1

  . A good agreement has been found between calculated and experimental values of𝑛.Aggregation diagrams of the extractants as a function of the total concentration of extractants diluted in n-heptane have been calculated by using K2 and K3 (Figure4). No trimer was observed in the extraction solvent containing UPA and EHPA while dimers and trimers exist in BiDiBOPP and IPA diluted in n-heptane. The dimerization and trimerization constants are very low compared to those determined by Buch et al. for D2EHPA in n-heptane (𝐾 2 = 115, 𝐾 3 = 3286) likely because of the high steric hindrance of the organophosphorus acids synthesized in the present work, which may not favor hydrogen bonds and may destabilize the aggregates

  , G. Cote, H. Mokhtari, B. Courtaud, B. A. Moyer, A. Chagnes, A review on the recovery of uranium(VI) from wet phosphoric acid by solvent extraction processes, Chemical Reviews 114 (24) (2014) 12002-12023. A C C E P T E D M A N U S C R I P T 13 D. Beltrami, G. Cote, H. Mokhtari, B. Courtaud, A. Chagnes, Modelling of the extraction of uranium (VI) from concentrated phosphoric acid by synergistic mixtures of bis. Tebby, Handbook of Phosphorus-31 Nuclear Magnetic Resonance DataCRC Press, 1 st edition (1991). R.K. Biswas, M.A. Habib, M.G.K. Mondal, Kinetics and Mechanism of Stripping of Mn(II)D2EHPA Complex by Sulphuric Acid Solution, Hydrometallurgy 80 (3) (2005) 186-195. G.S. Wilson, R.P. Buck, S. Rondinini, A.K. Covington, F.G.K. Baucke, C.M. Brett, K.W. Pratt, Measurement of pH. Definition, standards, and procedures, Pure and Appl. Chem. 74 (11) (2002) 2169-2200. 12. . 13. D.D. Perrin, B. Dempsey, Buffers for pH and metal ion control, Springer Editor (

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Variation of pKa as a function of 1/ for (a) Bidibopp, (b) IPA, (c) UPA, (d) TPA, (e) Cyanex 272, (f) D2EHPA in water/2-propanol mixtures (temperature = 25 °C).

  Acidic properties and aggregation behaviour of extractants have been determined by potentiometry and VPO measurements.-Steric hindrance destabilizes the aggregates -The presence of oxygen atoms in the alkyl chains of the extractants is responsible for a decrease of the pKa values.

Table 2 :

 2 Reaction yields for the synthesis of new extractants.

	G.T. Castro, O.S. Giordano, S.E. Blanco, Determination of the pKa of Hydroxy-
	Benzophenones in Ethanol-water Mixtures. Solvent Effects, Journal of Molecular
	Structure: Theochem 626 (1-3) (2003) 167-78.
	17. J.G. Kirkwood, Theory of Solutions of Molecules Containing Widely Separated
	Charges with Special Application to Zwitterions, The Journal of Chemical Physics 2
	(7) (1934) 351-61.
	18. K.J. Laidler, K.J, Chemical Kinetics, 3 rd edition, New York: Harper Collins (1987).
	19.
	
	20. J. B. Hasted, Water: A Comprehensive Treatise 2 (1973) 405.

Table 3 :

 3 Purities (%P) of the extractants synthesized in the present work determined by molecular Mass Spectroscopy (MS), Ionization Electrospray molecular Mass Spectroscopy (ESI-MS), 31 P NMR, 13 C NMR and 1 H NMR spectroscopy.

Table 4 :

 4 Impurities identified in UPA and EHPA.

	Extractants	Chemical shift (ppm)	Impurity structure
	UPA	-14
	EHPA	-13

Table 5 :

 5 Buffer solutions in water/ethanol mixtures for the determination of pKa values by potentiometric titration[12, 13].

	Ethanol, %(w/w)	pH (25 °C)	Buffer solutions Concentration, Compounds mol.kg -1
		4.973	KHPh	0.05
	40.0	7.597	KH2PO4 Na2HPO4	0.025 0.025
		5.395	HOAc NaOAc	0.05 0.05
		2.79	NaCl HCl	0.008840 0.002210
	52.0	5.40	NaCl HOAc	0.00553 0.0221
		6.92	TEA TEA-HCl	0.00221 0.00442
		2.88	NaCl HCl	0.009352 0.002338
			NaCl	0.00585
	73.4	6.06	HOAc	0.0234
			NaOAc	0.0117
		6.75	TEA TEA-HCl	0.00234 0.00468
		3.01	NaCl HCl	0.009688 0.002422
			NaCl	0.00605
	85.4	6.65	HOAc	0.0242
			NaOAc	0.0121
		6.74	TEA TEA-HCl	0.00242 0.00484

Table 6 :

 6 Experimental pKa values of organophosphorus acids in aqueous solution of 1 mol.L - 1 KCl. a : pKa values from[3, 25, 26], b : this work.

	Extractants pKa (lit.) a	pKa (exp.) b
	Cyanex® 301	2,84	2.9
	Cyanex® 272	3,73	4.2
	Ionquest® 801	3,30	3.0
	D2EHPA	2,75	2.6
	Bidibopp	-	2.4
	IPA	-	2.5
	TPA	-	2.3
	UPA	-	3.5

Table 7 :

 7 Values of Kextractant in Ω.kg.g -1 , average molecular weight 𝑀 ̅ in g.mol -1 , average aggregation number 𝑛 ̅ of the extractants, dimerization constant (K2) and trimerization constant (K3) deduced by VPO. Partition coefficients (P) of the monomeric form of organophosphorus acids(HL)

	Extractant	Kextractant, Ω.kg.g -1	𝑴 ̅ , g.mol -1	𝒏 ̅	K2, L.mol -1	K3, L 2 .mol -2
	Bidibopp	2.8147	699	1.50±0.23	3	9
	IPA	2.9646	680	1.50±0.19	6	10
	UPA	4.17	483	1.20±0.14	2	0
	EHPA	2.3785	847	1.20±0.12	4	0


Jean-Florent Raymond
email: jean-florent.raymond@mimuw.edu.pl

Ignasi Sau
email: ignasi.sau@lirmm.fr

Dimitrios M Thilikos

An edge variant of the Erdős-Pósa property *

published or not. The documents may come

Introduction

Typically, an Erdős-Pósa property reveals relations between covering and packing invariants in combinatorial structures. The origin of the study of such properties comes from the Erdős-Pósa Theorem [START_REF] Erdős | On independent circuits contained in a graph[END_REF], stating that there is a function f : N → N such that for every k ∈ N and for every graph G, either G contains k vertex-disjoint cycles, or there is a set X of f (k) vertices in G meeting all cycles of G. In particular, Erdős and Pósa proved this result for

f (k) = O(k • log k).
An interesting line of research aims at extending Erdős-Pósa Theorem for packings and coverings of more general graph structures. In this direction, we say that a graph class G satisfies the Erdős-Pósa property if there exists a function f G : N → N such that, for every graph G and every positive integer k, either G contains k mutually vertex-disjoint subgraphs, each isomorphic to a graph in G, or it contains a set S of f G (k) vertices meeting every subgraph of G that is isomorphic to a graph in G. When this property holds for a class G, we call the function f G the gap of the Erdős-Pósa property for the class G. In this sense, the classic Erdős-Pósa Theorem says that the class containing all cycles satisfies the Erdős-Pósa property with gap O(k • log k).

Given a graph J, we denote by M(J) the set of all graphs containing J as a contraction. Robertson and Seymour proved the following proposition, which in particular can be seen as an extension of the Erdős-Pósa Theorem.

Proposition 1. Let J be a graph. The class M(J) satisfies the Erdős-Pósa property if and only if J is planar.

A proof of Proposition 1 appeared for the first time in [START_REF] Robertson | Graph Minors. V. Excluding a planar graph[END_REF]. Another proof can be found in Diestel's monograph [START_REF] Diestel | Graph Theory[END_REF]Corollary 12.4.10 and Exercise 40 of Chapter 12]. In view of Proposition 1, it is natural to try to derive good estimations of the gap function f M(J) in the case where J is a planar graph. In this direction, the recent breakthrough results of Chekuri and Chuzhoy imply that f M(J) (k) = k • log O (1) k [START_REF] Chekuri | Large-treewidth graph decompositions and applications[END_REF] and, even more, that f M(J) = (k + |V (J)|) O(1) [START_REF] Chekuri | Polynomial bounds for the grid-minor theorem[END_REF]. Before this, the best known estimation of the gap for planar graphs was exponential, namely f M(J) (k) = 2 O(k log k) , and could be deduced from [START_REF] Leaf | Treewidth and planar minors[END_REF] using the proof of [START_REF] Robertson | Graph Minors. V. Excluding a planar graph[END_REF]. Moreover, some improved polynomial gaps have been proven for particular instantiations of the graph J (see [6-9, 15, 16]). Another direction is to add restrictions on the graphs G that we consider, which usually allows to optimize the gap f M(J) . In this direction, it is known that f M(J) = O(k) in the case where graphs are restricted to some non-trivial minor-closed class [START_REF] Fomin | Strengthening Erdős-Pósa property for minorclosed graph classes[END_REF].

We consider the edge counterpart of the Erdős-Pósa property, where packings are edgedisjoint (instead of vertex-disjoint) and coverings contain edges instead of vertices. We say that a graph class G satisfies the edge variant of the Erdős-Pósa property if there exists a function f G such that, for every graph G and every positive integer k, either G contains k mutually edge-disjoint subgraphs, each isomorphic to a graph in G, or it contains a set X of f H (k) edges meeting every subgraph of G that is isomorphic to a graph in G. Recently, the edge variant of the Erdős-Pósa property was proved in [START_REF] Kawarabayashi | Edge-disjoint odd cycles in 4-edge-connected graphs[END_REF] for 4-edge-connected graphs in the case where G contains all odd cycles.

In this paper we concentrate on the case where G = M(J) for some graph J. We find it an interesting question whether an edge-analogue of Proposition 1 exists or not. To our knowledge, the only case for which M(J) satisfies the edge variant of the Erdős-Pósa property is when J = K 3 , i.e. when the class of graphs G contains all cycles. This result is the edge-counterpart of the Erdős-Pósa Theorem and appears as a (hard) exercise in [4, Exercice 23 of Chapter 7]. For every r 2, let θ r be the graph containing two vertices and r multiple edges between them (see Figure 1). The results of this paper can be stated as follows:

Theorem 1. The edge variant of the Erdős-Pósa property holds for M(θ r ) with gap f M(θr) , with

f M(θr) (k) = O(k 2 r 3 polylog kr) and f M(θr) (k) = O(k 4 r 2 polylog kr).
Theorem 1 is the edge-counterpart of the main result of [START_REF] Fomin | Quadratic upper bounds on the Erdős-Pósa property for a generalization of packing and covering cycles[END_REF]. The proof is presented in Section 3 and contains three main ingredients. The first is a reduction of the problem to graphs of bounded degree, presented in Section 3.1. The second is an application of recent results of [START_REF] Chekuri | Large-treewidth graph decompositions and applications[END_REF] to obtain bounds on the treewidth of the graphs we consider (Section 3.2) and the last is a extension of the techniques in [START_REF] Fomin | Strengthening Erdős-Pósa property for minorclosed graph classes[END_REF] fitting our needs, which is presented in Section 3.3. Section 2 contains definitions and preliminary results and Section 4 discusses further research about the problem investigated in this paper.

Definitions and preliminaries

For any graph G, V(G) (respectively E(G)) denotes the set of vertices (respectively edges) of G. Even when dealing with multigraphs (i.e. graphs where more than one edge is allowed between two vertices) we will use the term graph.

A graph G is a subgraph of a graph G if V(G ) ⊆ V(G) and E(G ) ⊆ E(G), and we denote this by G ⊆ G. If X is a subset of V(G) (respectively E(G)), we denote by G[X]
the subgraph of G induced by X, i.e. the graph with vertex set X (respectively ∪ e∈X e) and edge set {{x, y} ∈ E(G), x ∈ X and y ∈ X} (respectively X). If S is a subset of vertices or edges of a graph G, the graph G \ S is the graph obtained from G after the removal of the elements of S. For every vertex v ∈ V(G) the neighborhood of v in G, denoted by N G (v), is the subset of vertices that are adjacent to v, and its size is called the degree of v in G, written deg G (v). The maximum degree ∆(G) of a graph G is the maximum value taken by deg G over V(G). Given a non-negative integer k, a triple (V 1 , S, V 2 ) is called a k-separation triple of a graph G if |S| k and {V 1 , S, V 2 } is a partition of V(G) such that there is no edge between a vertex of V 1 and a vertex of V 2 . Unless otherwise stated, logarithms are binary. For any two integers a, b such that a b, the notation a, b stands for the set of integers {a, a + 1, . . . , b}. In a tree T , rooted at a vertex r ∈ V(T ), a vertex u ∈ V(T ) is said to be a descendant of a vertex v = u if the path in T from r to u contains v. The set of descendants of v is denoted by desc T (v).

Minors and models. In a graph G, a contraction of an edge e = {u, v} ∈ E(G) is the operation that removes e from G and identifies the vertices u and v. In this paper, we keep multiple edges that may appear between two vertices after a contraction (for instance, contracting an edge in a triangle gives a graph with two vertices connected by two edges). For any graph J, let M(J) denote the class of contraction models (models for short) of J, i.e. the class of graphs that can be contracted to J. We say that a graph J is minor of a graph G (denoted by J m G) if a subgraph of G is a model of J (J-model for short), or, equivalently, if J can be obtained from G by a series of vertex deletions, edge deletions, and edge contractions.

Packings and coverings. Let G and J be graphs. We denote by pack v J (G) the maximum number of vertex-disjoint models of J in G and by cover v J (G) the minimum size of a subset S ⊆ V(G) (called J-vertex-hitting set) that meets the vertex sets of all models of J in G. These invariants are widely studied in the context of the classic Erdős-Pósa property.

Similarly, we write pack e J (G) for the maximum number of edge-disjoint models of J in G and cover e J (G) for the minimum size of a subset S ⊆ E(G) (called J-edge-hitting set) that meets the edge sets of all models of J in G. Obviously, for every two graphs G and J, the following inequality holds:

pack e J (G) cover e J (G).
A graph J is said to satisfy the (vertex-)Erdős-Pósa property for minors (vertex-Erdős-Pósa property for short) if there is a function

f J : N → N, called vertex-Erdős-Pósa gap of J, satisfying cover v J (G) f J (pack v J (G)
). The research of this paper is motivated by the course of detecting graphs J for which there is a function h J : N → N satisfying the following inequality for every graph G:

cover e J (G) h J (pack e J (G)). (1)
Such graphs are said to satisfy the edge variant of the Erdős-Pósa property for minors (or, in short, the edge-Erdős-Pósa property) and the function h J is called the gap of the edge-Erdős-Pósa property for J (edge-Erdős-Pósa gap for short). This definition is an edge-counterpart to the existing Erdős-Pósa property and (vertex-)Erdős-Pósa gap.

Treewidth. A tree decomposition of a graph G is a pair (T, V) where T is a tree and V a family (V t ) t∈V(T ) of subsets of V(G) (called bags) indexed by the vertices of T and such that

(i) t∈V(T ) V t = V(G);
(ii) for every edge e of G there is an element of V containing both endpoints of e; and

(iii) for every v ∈ V(G), the subgraph of

T induced by {t ∈ V(T ) | v ∈ V t } is connected.
The width of a tree decomposition T is defined as max t∈V(T ) |V t | -1 (that is, the maximum size of a bag minus one). The treewidth of G, written tw(G), is the minimum width of any of its tree decompositions.

A tree decomposition (T, V) of a graph G is said to be a nice tree decomposition if (i) every vertex of T has degree at most 3;

(ii) T is rooted at one of its vertices r whose bag is empty (V r = ∅); and (iii) every vertex t of T is

• either a base node, i.e. a leaf of T whose bag is empty (V t = ∅) and different from the root;

• or an introduce node, i.e. a vertex with only one child t such that V t = V t ∪ {u} for some u ∈ V(G);

• or a forget node, i.e. a vertex with only one child t such that V t = V t ∪ {u} for some u ∈ V(G);

• or a join node, i.e. a vertex with two child t 1 and t 2 such that

V t = V t 1 = V t 2 .
It is known that every graph has an optimal tree decomposition which is nice [START_REF] Kloks | Computations and Approximations[END_REF].

The graph θ r and the Erdős-Pósa property. The vertex-Erdős-Pósa property of θ r received some attention, in particular in [START_REF] Fiorini | Optimal Erdős-Pósa property for pumpkins[END_REF][START_REF] Fomin | Quadratic upper bounds on the Erdős-Pósa property for a generalization of packing and covering cycles[END_REF][START_REF] Joret | Hitting and harvesting pumpkins[END_REF]. For instance the main result of [START_REF] Fomin | Quadratic upper bounds on the Erdős-Pósa property for a generalization of packing and covering cycles[END_REF] is the following estimation of the vertex-Erdős-Pósa gap for M(θ r ).

Proposition 2 ([9]

). For every r ∈ N * , θ r has the vertex-Erdős-Pósa property with gap O(k 2 ).

By a careful analysis of the size of a M(θ r )-hitting set presented in [START_REF] Fomin | Quadratic upper bounds on the Erdős-Pósa property for a generalization of packing and covering cycles[END_REF] (c.f. Lemma 6), the estimation of the gap of Proposition 2 can be made quadratic in both k and r. From this, we can derive a O(k 3 r 3 ) edge-Erdős-Pósa gap for M(θ r ) (Corollary 2) by using our Lemma 7 that makes possible to translate a M(θ r )-vertex-hitting set into a M(θ r )-edge-hitting set.

However, Theorem 1 gives better estimations of this gap, either in k or in r.

Patterns in graphs of big treewidth. In the following section, we will use several propositions asserting that every graph G of treewidth at least c H contains some fixed graph H as a minor, where the constant c H depends on H. For instance, we will show in Lemma 6 a simple relation between the constant c k•θr and the vertex-Erdős-Pósa gap for θ r . These propositions as stated thereafter.

Proposition 3 ([17, Lemma 3.2]). For every integer r 1 and graph G, if tw(G) 2r -1 then G m θ r .

Proposition 4 ([9, Lemma 1], see also [START_REF] Birmelé | Brambles, prisms and grids[END_REF]). Let k and r be two positive integers. For every graph G, if tw(G) 2k 2 r 2 then G contains at least k vertex-disjoint models of θ r .

Proposition 5 ([2, Theorem 1.1]).

There is a function f Prop5 (t) = O(polylog t) such that, for every graph G and every positive integers h and p, if hp 2 tw(G)

f Prop5 (tw(G)) , there is a partition G 1 , . . . , G h of G into vertex-disjoint subgraphs such that tw(G i ) p for each i ∈ 1, h . Proposition 6 ([2, Theorem 1.2]).
There is a function f Prop6 (t) = O(polylog t) such that, for every graph G and every positive integers h and p, if

h 3 p tw(G) f Prop6 (tw(G)) then there is a partition G 1 , . . . , G h of G into vertex-disjoint subgraphs such that tw(G i ) p for each i ∈ 1, h .
3 The edge-Erdős-Pósa property for graphs θ r

Bounding the degree

In the sequel, we deal with graphs in which some vertices are marked. If G is a graph and m : V(G) → {0, 1} is a function, we say that (G, m) is a graph marked by m. A vertex v of G such that m(v) = 1 is said to be marked. We denote by µ the function that, given a graph, returns its number of marked vertices. We now define an r-good partition. Given a positive integer r, a marked tree (T, m) is said to have an r-good partition of root v if there is a pair ((T 1 , m 1 ), (T 2 , m 2 )) of marked trees such that:

(i) T 1 and T 2 are subtrees of T such that (E(T 1 ), E(T 2 )) is a partition of E(T ); (ii) r µ ((T 1 , m 1 )) 2r; (iii) v ∈ V(T 2 ); and
(iv) every vertex that is marked in (T, m) is either marked in (T 1 , m 1 ) or marked in (T 2 , m 2 ), but not in both. In other words, for every u ∈ V(T ),

• if v ∈ V(T 1 ) ∩ V(T 2 ) then m(v) = 1 ⇔ m 1 (v) = 1 or m 2 (v) = 1 but not both;
• otherwise, let i ∈ {1, 2} be the integer such that v ∈ V(T i ). Then we have

m(v) = m i (v).
We remark that because of (iv), µ(T ) = µ(T 1 ) + µ(T 2 ). If for every v ∈ V(T ), (T, m) has an r-good partition of root v, then T is said to have an r-good partition.

Lemma 1. For every integer r > 0 and every marked tree (T, m), if µ(T ) 2r then (T, m) has an r-good partition.

Proof. Let r > 0 be an integer. We prove this lemma by induction on the size of the tree.

Base case: |V(T )| = 0. Since 2r 2 > |V(T )| , T does not have 2r marked vertices and we are done. Induction step: Assume that for every integer n < n, every marked tree (T , m ) on n vertices and satisfying µ((T , m )) 2r has an r-good partition (induction hypothesis).

Let us prove that every marked tree on n vertices has a r-good partition if it has at least 2r marked vertices. Let (T, m) be a tree on n vertices and let v be a vertex of T. We assume that µ((T, m)) 2r. We distinguish two cases.

• µ((T, m)) = 2r:

Let T 1 = T, let m 1 = m, let T 2 = ({v}, ∅), and let m 2 : V(T 2 ) → {0, 1} be the function equal to 0 on every vertex of T 2 . Remark that (E(T 1 ), E(T 2 )) = (E(T ), ∅) is a partition of E(T ), T 2 contains v, and as (T, m) contains (exactly) 2r marked vertices, so does (T 1 , m). Consequently ((T 1 , m 1 ), (T 2 , m 2 )) is an r-good partition of (T, m).

• µ((T, m)) > 2r:

We distinguish different cases depending on the degree of the root v in T. (T ,m ) has an r-good partition ((T 1 , m 1 ), (T 2 , m 1 )) of root u. We extend it to T by setting T 1 = T 1 and T 2 = (V(T 2 ) ∪ {v}, E(T 2 ) ∪ {v, u}) . Notice that T 2 contains v. As the subtree T 1 contains at least r and at most 2r marked vertices (induction hypothesis), so does T 1 . Also, remark that (E(T 1 ), E(T 2 )) is a partition of E(T ) and that since u ∈ T 2 , the graph T 2 is connected. Therefore the pair (T 1 , T 2 ) is an r-good partition of T.

Case 2: deg(v) = d > 1.
Let u 1 , . . . , u d be the neighbors of v in T and for every i ∈ 1, d , let C i be the connected component of T \ {v} that contains u i . We also define, for every i ∈ 1, d , the restricted marking function

w i = m |V (C i ) .
Subcase (a): there exists i ∈ 1, d such that µ((C i , w i )) > 2r.

Let

T = (V(C i ) ∪ {v}, E(C i ) ∪ {u, v}) and let m = m | V(T ) . Remark that |V(T )| < |V(T )| and µ((T , m )) > 2r.
According to the induction hypothesis, (T , m ) has an r-good partition ((T 1 , m 1 ), (T 2 , m 2 )) of root u i . Similarly as before, we can extend it into an r-good partition ((T 1 , m 1 ), (T 2 , m 2 )) of (T, m). This is done by setting:

T 1 = T 1 , m 1 = m 1 , T 2 = (V(T 2 ) ∪ {v}, E(T 2 ) ∪ {v, u i }), and
m 2 : v → 0 u ∈ V(T 2 ) \ {v} → m 2 (u) . Subcase (b): there exists i ∈ 1, d such that r µ((C i , w i )) 2r. Let T 1 = C i and T 2 = T [E(T ) \ E(T 1 )] . In this case, (E(T 1 ), E(T 2 )
) is a partition of E(T ) and T 2 is connected since it contains v, the vertex which is adjacent to the C j 's. Thus, if we set m 1 = m |V (T 1 ) and m 2 = m |V (T 2 ) , ((T 1 , m 1 ), (T 2 , m 2 )) is an r-good partition of (T, m).

Subcase (c): for all i ∈ 1, d , µ((C i , w i )) < r.

Let j = min j ∈ 2, d , j i=1 µ((C i , w i )) r . We set:

T 1 = (∪ i∈ 1,j V(C i ) ∪ {v}, ∪ i∈ 1,j (E(C i ) ∪ {v, u i })), m 1 : v → 0 u ∈ V T 1 \ {v} → m(u) , T 2 = T [E(T ) \ E(T 1 )] ,
and

m 2 = m |V (T 2 )
.

By definition of j, µ((T 1 , m 1 )) r and as for every i ∈ 1, d , µ((C i , w i )) < r we also have µ((T 1 , m 1 )) < 2r. As before, the pair ((T 1 , m 1 ), (T 2 , m 2 )) is an r-good partition of (T, m).

In conclusion, we proved by induction that for every integer r, every tree having at least 2r marked vertices has an r-good partition.

In the sequel we will deal with packings of the graph θ r , for r > 1. The following remark is important.

Remark 1. If G is not biconnected, the number of edge-disjoint models of θ r in G is equal to the sum of the number of edge-disjoint models of θ r in every biconnected component of G. This enables us to treat biconnected components separately. Lemma 2. Let k > 0, r > 0 be two integers, and let G be a biconnected graph with ∆(G) 2kr. Then G contains k edge-disjoint models of θ r .

Proof. As G is biconnected, the removal of a vertex v of maximum degree gives a connected graph. Let T be a minimal tree of G \ {v} spanning the neighborhood N G (v) of v. We mark the vertices of T that are elements of N G (v): this gives the marking function m for T. Let us prove by induction on k that (T, m) has k edge-disjoint marked subtrees (T 1 , m 1 ), . . . , (T k , m k ), each containing at least r marked vertices. If we do so, then we are done because {{v}, T i } i∈ 1,k is a collection of k edge-disjoint θ r models. In fact, as for every i ∈ 1, k , T i contains r r vertices adjacent to v in G, contracting the edges of T i in G[{v} ∪ V(T i )] gives the graph θ r . Let r > 0 be an integer. Base case k = 1: Clear. Induction step k > 1: Assume that for every k < k, every tree with at least 2k r vertices marked has k edge-disjoint subtrees, each with at least r marked vertices. Let (T, m) be a marked tree such that µ((T, m))

2kr. According to Lemma 1, (T, m) has an r-good partition ((T 1 , m 1 ), (T 1 , m 1 )) such that r µ((T 1 , m 1 )) 2r and µ((T 1 , m 1 )) = µ((T, m))µ((T 1 , m 1 )) 2(k -1)r. By induction hypothesis, (T 1 , m 1 ) has k -1 edge-disjoint marked subtrees (T 2 , m 2 ), . . . , (T k , m k ) each containing at least r marked vertices. Remark that as all these trees are subgraphs of T 1 , which is edge-disjoint from T 1 in T , they are edge-disjoint from T 1 as well. Consequently, (T 1 , m 1 ), (T 2 , m 2 ), . . . , (T k , m k ) is the family of edge-disjoint subtrees we were looking for. Proof. Let G be a graph and k, r be two positive integers. By Proposition 4, if tw(G) 2k 2 r 2 , then G contains k vertex-disjoint models of θ r . Therefore, we only have to consider the case where tw(G) < 2k 2 r 2 .

Bounding the treewidth

We also define A and α to be two positive real numbers such that f Prop5 (x) A log α x holds for large enough x (cf. Proposition 5 for the definition of f Prop5 ). Finally, let h r (k) = 4kr 2 • A log α (2k 2 r 2 ). Remark that h r (k) = O(kr 2 polylog kr). We will show that graphs whose treewidth is at least h r (k) contain k vertex-disjoint models of θ r . Let us assume that tw(G) h r (k). We thus have:

tw(G) 4kr 2 • A log α (2k 2 r 2 ) 4kr 2 • A log α (tw(G)) (as we assume tw(G) < 2k 2 r 2 ) tw(G) A log α (tw(G)) 4kr 2 (because A log α (tw(G)) is positive) tw(G) f Prop5 (tw(G)) k(2r) 2 (for big enough tw(G)).
Notice that k and 2r meet the conditions of Proposition 5. Consequently, there is a partition G 1 , . . . , G k of G into vertex-disjoint subgraphs such that ∀i ∈ 1, k , tw(G i ) 2r. By Proposition 3, each of these subgraphs contains a model of θ r . Consequently, G contains k vertex-disjoint models of θ r , as required.

A very similar proof can be used to show the following lemma, using Proposition 6.

Lemma 4. There is a function h r (k) = O(k 3 r polylog kr) such that for every k and r positive integers and every graph G, if tw(G) h r (k) then G contains at least k vertexdisjoint models of θ r .

From vertices to edges

In this section, we show how an estimation of a vertex-Erdős-Pósa gap can be derived from the bound on the treewidth obtained in Section 3.2. The proof of the two following lemmata are inspired from the proof of [10, Lemma 2].

Lemma 5 (adapted from Lemma 2 of [START_REF] Fomin | Strengthening Erdős-Pósa property for minorclosed graph classes[END_REF]). Let k, r be two positive integers and G a graph such that Let (T,V) be an optimal nice tree decomposition of G. For all t ∈ V(T ), let H t be the subset of V(G) equal to t ∈desc T (t) V t \ V t , that is, informally, the subset of vertices that are in bags below V t but not in V t . We also define the function p : V(T ) → N as:

pack v θr (G) = k. Then G has a (tw(G) + 1)-separation triple (V 1 , S, V 2 ) such that 1 3 k pack v θr (G[V 1 ]) 2 3 k and pack v θr (G[V 1 ]) + pack v θr (G[V 2 ]) k. Proof.
∀t ∈ V(T ), p(t) = pack v θr (G[H t ]
), which counts the number of vertex-disjoint models of θ r in the subgraph of G induced by H t .

Remark 2. The function p is nondecreasing along every path from a vertex of T to the root of T, because if a vertex t ∈ V(T ) is a child of a vertex t ∈ V(T ), then H t ⊆ H t , and thus

pack v H (G[H t ]) pack v H (G[H t ]).
Remark 3. As T is a nice decomposition of G, its vertices can be of four different types.

We make remarks about the value taken by p depending on the type of the vertices: Let t ∈ V(T ) be a node such that p(t) > 2 3 k and such that for every child t of t, p(t ) 2 3 k. Let us make some claims about t.

Base
Claim 1: such a t exists.

Proof of Claim 1. The value of p on the root r of T is k (because G[H r ] = G) and according to the previous remark, the value of p on base nodes is 0. As p is nondecreasing on a path from a base node to the root (see Remark 2), such a vertex t exists. ♦ Claim 2: t is unique.

Proof of Claim 2. To show that t is unique, we assume by contradiction that there is another t ∈ V (T ) with t = t and p(t ) > 2 3 k, and such that for every child t of t , p(t ) 2 3 k. Three cases can occur:

(i) t is a descendant of t. However, p is nondecreasing along any path from a vertex to the root (Remark 2) and p(t ) 2 3 k, whereas the value of p for each child of t is at most 2 3 k: this is a contradiction.

(ii) t is a descendant of t . The same argument applies (symmetric situation).

(iii) t and t are not in the above situations. Let v be the least common ancestor of t and t . As p is nondecreasing along any path from a vertex to the root, the child v t (respectively v t ) of v whose t (respectively t ) is descendant of should be such that

p(v t ) > 2 3 k (respectively p(v t ) > 2 3 k). By definition of v, we have v t = v t . As v is a join node, p(v) = p(v t ) + p(v t ) > 4
3 k, which is impossible.

♦

Claim 3: t is either a forget node or a join node.

Proof of Claim 3. By definition of t, the value p(t) is different from the value(s) taken by p over the child(ren) of t. This can only occur in the cases of a join node or a forget node. ♦

We now present a (tw(G) + 1)-separation triple (V 1 , S, V 2 ) of G with the required properties. Case 1: t is a forget node with t as child.

Let

S = V t , V 1 = H t , and E 2 = V(G) \ (V 1 ∪ S).
Case 2: t is a join node with t 1 , t 2 as children.

As

2 3 k < p(t) = p(t 1 ) + p(t 2 ) (Remark 3), there is i ∈ {1, 2} such that p(t i ) k 3 . Let S = V i , E 1 = H t i , and E 2 = V(G) \ (V 1 ∪ S).
In both cases, we have:

(i) 1 3 k pack v θr (G[V 1 ]) 2 3 k by definition of V 1 and t; (ii) (V 1 , S, V 2 ) is a partition of V(G); (iii) there is no edge between a vertex in V 1 and a vertex of V 2 (intuitively, S separates V 1 and V 2 ); (iv) k = pack v θr (G) pack v θr (G[V 1 ]) + pack v θr (G[V 2 ]
), because V 1 and V 2 are disjoint subsets of V(G); and (v) |S| tw(G) + 1, because S is a bag of an optimal tree decomposition of G.

To see why (iii) is true, assume by contradiction that there are two vertices u ∈ V 1 and v ∈ V 2 such that {u, v} ∈ E(G). Let s 0 ∈ V(T ) be the child of t such that S = V s 0 (cf. the two different cases above). By definition of V 1 there is a vertex s 1 ∈ V(T ) of T in desc T (s 0 ) whose bag V s 1 contains u. By definition of V 2 , the vertex v does not belong to the bag V s 0 nor to a bag of a descendant of s 0 . Let s 2 be a vertex of T containing u and which is, according to the previous remark, not the bag of a descendant of s 0 nor s 0 .

As (T, V) is a tree decomposition of G and {u, v} ∈ E(G), we have the following:

• there is a vertex s ∈ V(T ) whose bag contains both u and v;

• the subgraph of T induced by vertices whose bags contain u (respectively v) is connected.

Consequently there is a path in T from s 1 to s (respectively from s 2 to s) whose each bag contains u (respectively v). As s is on the (only) path of T linking s 1 to s 2 , one of u, v belongs to the bag V s . But this contradicts the fact that (V 1 , S, V 2 ) is a partition of V(G).

We conclude that (V 1 , S, V 2 ) is a (tw(G) + 1)-separation triple of G with the required properties.

Lemma 6 (adapted from Lemma 2 of [START_REF] Fomin | Strengthening Erdős-Pósa property for minorclosed graph classes[END_REF]). Let r ∈ N * and let h r : N → N be a function such that:

• ∃k 0 ∈ N, ∃c ∈ ]0, 1[, ∀r ∈ N * , ∀k > k 0 , 2h r 2 3 k c • h r (k); and • for every graph G, if tw(G) h r (k) then G m (k + 1) • θ r .
Then there is a function f r (k) = O(h r (k)) such that for every graph we have:

cover v θr (G) f r (pack v θr (G))
Proof. Let r 2 be an integer. We show by induction on k the following statement:

if pack v θr (G) = k then cover v θr (G) h r (k) • log 3/2 k i=0 c i . (2)
Base case k = 0: Clear. Induction step k > 0: We assume that (2) holds for every positive integer k < k. Let G be a graph such that pack v θr (H) = k. First, remark that tw(G) < h r (k), otherwise by definition of h r we would have pack v θr (G) > k. Thus, by Lemma 5 G contains a h r (k)-separation triple

(V 1 , S, V 2 ) such that 1 3 k pack v θr (G[V 1 ]) 2 3 k and pack v θr (G[V 1 ]) + pack v θr (G[V 2 ]) k. Remark 4. Since pack v θr (G[V 1 ]) + pack v θr (G[V 2 ]) pack v θr (G), there is a real number α ∈ 1 3 , 2 3
, for which we have:

pack v θr (G[V 1 ]) = α • pack v θr (G) pack v θr (G[V 2 ]) (1 -α) • pack v θr (G).
Notice that max{α, 1 -α} 2 3 .

The triple (V 1 , S, V 2 ) is a partition of V(G), so the following holds:

cover v θr (G) cover v θr (G[V 1 ]) + cover v θr (G[V 2 ]) + |S| cover v θr (G[V 1 ]) + cover v θr (G[V 2 ]) + h r (k) h r (αk) • log 3/2 αk i=0 c i + h r ((1 -α)k) • log 3/2 (1-α)k i=0 c i + h r (k) (induction hypothesis) (h r (αk) + h r ((1 -α)k)) • log 3/2 2 3 k i=0 c i + h r (k) (Remark 4) 2 • h r 2k 3 • log 3/2 2 3 k i=0 c i + h r (k) (Remark 4) c • h r (k) • log 3/2 2 3 k i=0 c i + h r (k) (definition of h r ) h r (k) • log 3/2 k i=1 c i + h r (k) h r (k) • log 3/2 k i=0 c i .
This concludes the induction step. Now, remark that since c ∈ ]0, 1[, the series ∞ i=0 c i converges to 1 1-c . Hence, for every r ∈ N * the following inequality holds for every graph G:

cover v θr (G) 1 1 -c • h r (pack v θr (G)),
as required.

Corollary 1. Let f r be the vertex-Erdős-Pósa gap of M(θ r ). Then we have

• f r (k) = O(k 2 r 2 polylog kr); • f r (k) = O(k 4 r polylog kr).
These estimations follow from Lemma 6 and Propositions 5 and 6.

The following lemma shows how to translate a vertex-Erdős-Pósa gap into an edge-Erdős-Pósa gap in the case of θ r . The main idea of the proof is that if the considered graph has small maximum degree, a small edge-hitting set can be constructed from a small vertex-hitting set. On the other hand, a big maximum degree forces a large packing of θ r -models. Lemma 7. If f r is the vertex-Erdős-Pósa gap of M(θ r ), then the edge-Erdős-Pósa gap of M(θ r ) is less than 2kr • f r (k).

Proof. Let G be a graph, let r 2 be an integer and let f r is the vertex-Erdős-Pósa gap of M(θ r ). We want to prove that if G contains less than k edge-disjoint models of θ r , then it has a θ r -edge-hitting set of size less than 2kr • f r (k).

According to Remark 1, we can assume that G is biconnected. If it is not the case, we consider its biconnected components separately (if it has no biconnected component then the lemma is trivial).

First, remark ∆(G) < 2kr, otherwise by Lemma 2 G would contain at least k edgedisjoint θ r -models.

Notice that if G does not contain k edge-disjoint θ r -models, it does not contain k vertexdisjoint θ r -models either. Consequently, there is a set X ⊆ V(G) meeting every θ r model of G and such that |X| f r (k). Let us consider the set Y ⊆ E(G) of edges incident to vertices of X, i.e. Y = {{u, v} ∈ E(G), u ∈ X}. Remark that as ∆(G) < 2kr, we have |Y | 2kr • f r (k). Now, assume that there is a θ r -model in G not having edges in Y. None of its vertices is in X, which is contradictory. So Y is a θ r -edge hitting set of the required size. This concludes the proof.

Further research

The main question, initiated in this paper, is whether for every planar graph J, the class M(J) satisfies this edge variant of the Erdős-Pósa property. As for the vertex version, it is easy to see that the planarity of J is necessary. For instance, if J = K 5 , consider as graph G an n-vertex toroidal wall, which is a 3-regular graph embeddable in the torus that contains K 5 as a minor. One can check that G does not contain two edge-disjoint models of K 5 , but Ω( √ n) edges of G are needed in order to hit all its K 5 -models. Moreover, a second question is: when this property holds, does it hold with a polynomial gap for all graphs? Also, finding lower bounds on this gap for specific graphs is another interesting and complementary question. Let us mention that, as it is the case for the vertex version (see [START_REF] Erdős | On independent circuits contained in a graph[END_REF][START_REF] Fiorini | Excluded forest minors and the Erdős-Pósa property[END_REF]), for any non-acyclic planar graph J for which the edge variant of the Erdős-Pósa property holds for M(J), we have that f M(J) (k) = Ω(k log k). Indeed, let G be an n-vertex cubic graph with treewidth Ω(n) and girth Ω(log n) (such graphs are well-known to exist). Since J is planar, the treewidth of any graph excluding J as a minor is bounded by a constant [START_REF] Robertson | Graph Minors. V. Excluding a planar graph[END_REF], hence any set of edges of G meeting all models of J has size Ω(n) (as the removal of an edge may decrease the treewidth by at most two). On the other hand, since J contains a cycle and the girth of G is Ω(log n), any model of J in G contains Ω(log n) edges (assuming that J does not have isolated vertices), and therefore G contains O(n/ log n) edge-disjoint models of J (here we have used that the degree of G is bounded), easily implying that f M(J) (k) = Ω(k log k). In particular, it holds that f M(θr) (k) = Ω(k log k) for any r 2, so a first avenue for further work in this direction is to optimize the gap function f M(θr) (k) given in Theorem 1.

Finally, when the graphs G (in which the packings or coverings are taken) are restricted to classes of bounded degree, the proof of Lemma 7 can easily be adapted to prove that the bound of the vertex version also holds for the edge version.

Figure 1 :

 1 Figure 1: The graph θ 5 .

Case 1 :

 1 deg(v) = 1. Let u be the neighbor of v in T, let T = T \ {v}, and m = m | V(T ) . Remark that µ((T , m )) 2r and |V(T )| = |V(T )| -1. By induction hypothesis,

Lemma 3 .

 3 There is a function h r (k) = O(kr 2 polylog kr) such that for every k and r positive integers and every graph G, if tw(G) h r (k) then G contains at least k vertexdisjoint models of θ r .

Corollary 2 .

 2 An edge-gap of O(k 3 r 3 ) for M(θ r ) can be derived from Proposition 2.Proof of Theorem 1. It follows from the application of Lemma 7 to the estimations of the vertex-Erdős-Pósa gap of θ r given in Corollary 1.

  node t: p(t) = 0, because since t has no descendant, H t = ∅; Introduce node t with child t : as the unique element of V t \ V t cannot appear in the bags of desc T (t ) (by definition of a tree decomposition), H t = H t and then p(t) = p(t ); Join node t with children t 1 and t 2 : H t = H t 1 ∪ H t 2 , but H t 1 and H t 2 are disjoint and there is no edge between the vertices of H t 1 and of H t 2 in G[H t ] (otherwise the set V t 1 = V t 2 would contain an endpoint of this edge, which also belongs to H t 1 or H t 2 , and this is contradictory). Thus there is no θ r -model in G[H t ] that uses (simultaneously) vertices of H t 1 and of H t 2 , and therefore p(t) = p(t 1 ) + p(t 2 ).

* This work was supported by the ANR project AGAPE (ANR-09-BLAN-0159) and the Languedoc-Roussillon Project "Chercheur d'avenir" KERNEL. The third author was co-financed by the European Union (European Social Fund ESF) and Greek national funds through the Operational Program "Education and Lifelong Learning" of the National Strategic Reference Framework (NSRF) -Research Funding Program: ARISTEIA II.