G Montes-Hernandez
email: german.montes-hernandez@ujf-grenoble.fr

F Renard

R Chiriac

N Findling

F Toche

Rapid precipitation of magnesite micro-crystals from Mg(OH) 2 -H 2 O-CO 2 slurry enhanced by NaOH and a heat-ageing step (from ∼ ∼ ∼ ∼20 to 90°C)

Abstract

This study proposes a simple and novel synthesis route for rhombohedral single crystals (<2µm) of magnesite. This synthesis can be summarized by two main sequential reactions: (1) aqueous carbonation of synthetic brucite (Mg(OH) 2 ) by injection of CO 2 in a highly alkaline medium (2Molal of NaOH) at ambient temperature (∼20°C), leading to precipitation of platycompacted aggregates of dypingite ( O H OH CO Mg At laboratory scale, magnesite is typically synthesized at high temperature (>90°C) and its synthesis requires several days or weeks depending on experimental conditions. For this reason, industrial-scale magnesite production has been limited. The proposed magnesite synthesis method, requiring only 48h and moderate temperature, could easily be extrapolated on an industrial scale. Moreover, a simple and novel synthesis route for the production of fine platy particles of hydromagnesite is reported, with synthesis requiring only 5h. Based on their chemical compositions and textural properties, there are potential applications for both minerals, for example as a mineral filler and/or as a flame-retardant.

Introduction

The formation and textural properties of natural mineral Mg-carbonates have already been investigated in the past. However, various questions still remain unanswered concerning their formation in natural systems as well as their production at laboratory and industrial scales. Over the past two decades, the precipitation of Mg-carbonates from Mg-rich solutions or Mg-rich suspensions has been actively investigated with a view to the permanent storage of the captured anthropogenic CO 2 via ex-situ mineral carbonation; i.e., CO 2 (gas)-to-carbonate(solid) transformation (e.g., [1][2][3][4]. Such a carbonation reaction could therefore potentially be part of a Carbon Dioxide Capture and Storage (CCS) system, which attempts to capture CO 2 from industrial sources and store it permanently. The most successful implementation of this mineral carbonation process involves the extraction of magnesium from natural olivine or serpentines and/or Mg-rich solid waste in aqueous solutions and subsequent precipitation of Mg-carbonate under a CO 2 atmosphere (5-10). In addition, the reaction path for the MgO-H 2 O-CO 2 system at ambient temperatures and at atmospheric CO 2 partial pressure is of both geological interest and practical significance. For example, if the reaction path in this system is known, this would lead to a better understanding of the low temperature alteration or weathering of mafic and ultramafic rocks. Moreover, exact knowledge of the reaction path in this system is important in order to assess the performance of geological repositories for nuclear waste where mineral periclase (MgO) and brucite (Mg(OH) 2 ) have been proposed as engineered barriers (e.g., 11).

Generally, in the MgO-H 2 O-CO 2 or Mg(OH) 2 -H 2 O-CO 2 slurry systems a significant number of hydrated and basic (or hydroxylated) carbonates (e.g. nesquehontite, lansfordite, artinite, hydromagnesite, dypingite, pokrosvskite, etc.) can be formed at ambient temperature and at moderate CO 2 pressure (<55 bar) (1, 12-20, this study). All of these minerals are considered to be metastable compounds with respect to magnesite (anhydrous form: MgCO 3 ), which is the most stable form of Mg-carbonate. However, the low temperature precipitation of magnesite is kinetically inhibited by the preferential precipitation of some of the hydrated and/or hydroxylated Mg-carbonates mentioned above, possibly due to the high hydration nature of Mg 2+ ions in solution (21). For this reason, the formation of magnesite at ambient temperature is virtually impossible (1). The minimum reported temperature for magnesite production is about 60-100°C, and its formation also requires a high CO 2 pressure (21)(22)(23). In various cases, the magnesite is obtained by transformation of pre-existing hydroxylated Mg-carbonate such as hydromagnesite (18,24). This transition can be very slow; for example, below 150°C and at moderate pressures, its duration is often in the order of days. Higher temperatures, high salinity, high CO 2 pressure, low magnesium concentration (in a closed system) and the use of organic additives (ex. monoethylene glycol) are known to accelerate the hydromagnesite-to-magnesite transformation (18,25). Reported transformation times are between 2h at 200°C in a solution saturated with NaCl, and over 100 days at 110°C at lower salinity in a closed system. Magnesite formation without any apparent hydromagnesite initially or at an intermediate stage has also been reported under high CO 2 pressures (100-150 bar) and at high temperatures (150-180°C), for experiments combining the dissolution of Mg-silicates with carbonate precipitation (23,(26)(27)(28). The requirement for high temperatures and high CO 2 pressures to precipitate the magnesite has limited its production at industrial scale. Moreover, magnesite used as a filler and pigment in paper, paint, rubber and plastics can sometimes be replaced by synthetic calcite that can be produced in a wide panel of lower temperatures and CO 2 pressures (29)(30)(31). Identifying novel and/or innovative synthesis methods for magnesite at low temperature and low CO 2 pressure still remains a major scientific challenge to obtain a better understanding of its formation in natural systems and to facilitate its production on an industrial scale.

In this context, this study proposes a novel and simple synthesis route for the production of rhombohedral single crystals (<2 µm) of magnesite. This synthesis can be summarized by two main sequential reactions: (1) Here, the solid product contained only about 5% of magnesite, and is called hydromagnesitemagnesite-brucite composite hereafter. The specific P CO2 -T conditions required to precipitate pure hydromagnesite as well as hydromagnesite-eitelite and hydromagnesite-magnesite-brucite composites from Mg(OH) 2 -H 2 O-CO 2 slurry in the presence/absence of NaOH are also reported.

Various analytical tools such as X-ray diffraction (XRD), Field Emission Gun Scanning Electron

Microscopy (FESEM), Thermogravimetric analyses (TGA) and N 2 adsorption isotherms were used to characterize the solid products.

Materials and Methods

The main experimental conditions required to synthesize magnesite, hydromagnesite and Mgcarbonate composites are summarized in Table 1 and the experimental procedures are described in detail in the following sections.

Synthesis of magnesite (MgCO 3 )

One litre of high-purity water with electrical resistivity of 18.2 M1•cm, 2 mol of NaOH and 1 mol of synthetic brucite (Mg(OH) 2 , with chemical purity >95%) were placed in a titanium reactor (autoclave with internal volume of two litres). This reactive suspension was immediately stirred during the reaction by means of a constant mechanical stirring system (400 rpm). The temperature of the suspension increased instantaneously to 26°C due to the exothermic dissolution of NaOH in the system. At this reference temperature, CO 2 (22 mol) was immediately injected in the system at a pressure of 50 bar. The carbonation reaction started instantaneously as attested by the continuous consumption of CO 2 (monitored by a pressure drop in the system) and an increase in temperature during the exothermic carbonation reaction (the maximum temperature reached was 238°C after 1h of reaction (3T212°C)). After 24h of carbonation reaction at ambient (or room) temperature (16-20 °C) (including the exothermic period), a heat-ageing step was performed from ambient temperature to 90°C for a further 24h. Based on preliminary experiments, was found that the NaOH played a catalytic role, i.e., it accelerated brucite carbonation by an increase in carbonate ion concentration with time and promoted the formation of magnesite during the heat-ageing step. More details are provided in the Results and Discussion section.

At the end of the experiment, the autoclave was removed from the heating system and immersed in cold water. The residual CO 2 was degassed from the reactor during water-cooling period. After water cooling at 30°C (for about 15 minutes) the autoclave was disassembled, and the solid product was carefully recovered and separated by centrifugation (30 minutes at 12,000 rpm),

decanting the supernatant solutions. The solid product was washed twice by redispersion/centrifugation processes in order to remove the soluble sodium carbonates formed during the synthesis. Finally, the solid product was dried directly in the centrifugation flasks at 80°C for 48 h. The dry solid product was manually recovered and stored in plastic flasks for subsequent characterization (FESEM, XRD, TGA and N 2 sorption isotherms).

Synthesis of hydromagnesite (Mg 5 (CO 3 ) 4 (OH) 2 .4H 2 O)

Two methods are reported in this study.

(i) Precipitation of hydromagnesite from Mg(OH) 2 -H 2 O-CO 2 slurry. One litre of high-purity water with electrical resistivity of 18.2 M1•cm and 1 mol of synthetic brucite (Mg(OH) 2 ) with chemical purity >95%) were placed in a titanium reactor (autoclave with internal volume of two litres). This reactive suspension was immediately stirred at 400 rpm and heated to 90°C. Once the temperature had stabilized, CO 2 was injected at a pressure of 50 bar and the total pressure in the system was immediately increased from 50 to 90 bar by argon injection. At these T and P conditions, the vapor phase primarily consists of a mixture of Ar + CO 2 , with CO 2 in a supercritical state. The reaction time in this solid-liquid-gas thiphasic experiment (Mg(OH) 2 -H 2 O-CO 2 slurry) was typically 12 days in order to obtain high-purity hydromagnesite. The solid A2 product recovery/drying procedures were similar to those described above for the synthesis of magnesite.

(ii) Knowing the catalytic action of NaOH on brucite carbonation, a faster method was implemented to synthesize hydromagnesite. This simple method required only five hours of carbonation reaction. In fact, this method is very similar to the synthesis method for magnesite (see above) except for the reaction time in each carbonation step: only 3h for carbonation at ambient temperature (including the exothermic period) and 2h for the heat-ageing step at 90°C

were required to obtain high-purity hydromagnesite. The solid product recovery/drying procedures were the same as described above for magnesite synthesis.

Synthesis of Mg-carbonate composites

Hydromagnesite-magnesite-brucite composite. This composite material was synthesized by making a simple modification to the synthesis method for magnesite (see above). More specifically, the carbonation reaction of brucite was carried out in the absence of NaOH, leading to the formation of hydromagnesite (dominant phase) and magnesite (minor phase). These Mg carbonates were intimately dispersed with residual (or unreacted) brucite forming a so-called hydromagnesite-magnesite-brucite composite. The solid product recovery/drying procedures were similar to those described above for the synthesis of magnesite, although in this case the solid product washing procedure to remove soluble Na carbonate was not necessary for this synthesis.

Hydromagnesite-eitelite composite. This composite material was synthesized by making simple modifications to the synthesis method for magnesite (see above). In this case, a heat-ageing step 1B2 at 45°C was performed immediately after CO 2 injection into the system. The reaction time in this Mg(OH) 2 -H 2 O-NaOH-CO 2 system was typically 10 days in order to obtain a hydromagnesiteeitelite binary composite. In this particular case, it was found that the sodium initially contained in NaOH reacted significantly to produce the eitelite mineral (Na 2 CO 3 .MgCO 3 ). So in this case, the NaOH had not just a catalytic affect because sodium was also incorporated in the mineral phase. The solid product recovery/drying procedures were the same as described above for magnesite synthesis.

Characterization of solid products

FESEM observations: magnesite, hydromagnesite and various Mg-carbonate composites were dispersed by ultrasonic treatment in absolute ethanol for five to ten minutes. One or two droplets of the suspension were then deposited directly on an aluminum support for SEM observations, and coated with platinum. The morphology of various selected powders was observed using a Zeiss Ultra 55 field emission gun scanning electron microscope (FESEM) with a maximum spatial resolution of approximately 1nm at 15kV.

XRD measurements: X-Ray Powder Diffraction (XRD) analyses were performed using a Siemens D5000 diffractometer in Bragg-Brentano geometry; equipped with a theta-theta goniometer with a rotating sample holder. The XRD patterns were collected using Cu k4 1 (λ k41 =1.54065) and k4 2 (λ k42 =1.5444 5) radiation in the range 2θ = 10 -70° with a step size of 0.04° and a counting time of 6 seconds per step.

Thermogravimetric analyses: TGA for all Mg-carbonate samples were performed with a Mettler Toledo TGA/SDTA 851e instrument under the following conditions: sample mass of about 10 mg, 150 µl alumina crucible with a pinhole, heating rate of 5°C min -1 , and inert N 2 atmosphere of 50 ml min -1 . Sample mass loss and associated thermal effects were obtained by TGA/SDTA. In order to identify the different mass loss steps, the TGA first derivative (rate of mass loss) was used. The TGA apparatus was calibrated in terms of mass and temperature. Calcium oxalate was used for the sample mass calibration. The melting points of three compounds (indium, aluminum and copper) obtained from the DTA signals were used for the sample temperature calibration.

N 2 sorption isotherms: N 2 sorption isotherms for magnesite, hydromagnesite and Mg-carbonate composites were obtained by using a sorptomatic system (Thermo Electron Corporation). The specific surface area of powdered samples was estimated by applying the Brunauer-Emmet-Teller (BET) equation in the 0.05≤P/P 0 ≤0.35 interval of relative pressure and based on a value of 16.2 Å 2 for the cross-sectional area of molecular N 2 . A non-linear regression by the least-squares method was performed to fit the interval data (n ads vs. P/P 0 ) in the experimental isotherms.

Results and Discussion

Magnesite precipitation: Physicochemical steps and reaction mechanism

The precipitation of magnesite at low temperature is kinetically inhibited by the preferential precipitation of hydrated and/or hydroxylated Mg-carbonates (e.g., nesquehontite, lansfordite, artinite, hydromagnesite, dypingite, pokrosvskite, etc.), possibly due to the high hydration nature of Mg 2+ ions in solution (21). Based on this assumption, the formation of magnesite at ambient temperature is virtually impossible (1).

A simple and novel synthesis route for the formation of rhombohedral single crystals (<2µm) of magnesite is described here. The precipitation process requires only 48h and can be summarized by two main sequential steps (see Fig. 1):

(1) Aqueous carbonation of synthetic brucite (Mg(OH) 2 ) by injection of CO 2 in a highly alkaline medium (2Molal of NaOH) at ambient temperature (∼20°C). This physicochemical step led to the precipitation of platy-compacted aggregates of dypingite ( O H OH CO Mg

2 2 4 3 5 5 . ) ( ) ( )
after 24h of fluid-solid interaction as attested by XRD and FESEM observations on the solid (see Fig. 1 and2). Moreover, a moderate specific surface area (9 m 2 /g) was deduced from the N 2 adsorption isotherm (Table 2).

Assuming that CO 2 absorption in the highly alkaline solution of NaOH is faster and greater than in water via the following exothermic reaction:

O H CO Na CO NaOH aq 2 3 2 ) ( 2 2 + → + (1)
the carbonation of brucite leading to the precipitation of dypingite can be written as follows:

NaOH O H OH CO Mg O H CO Na OH Mg 6 . 1 5 . ) ( ) ( 2 . 0 8 . 0 ) ( 2 2 4 3 5 2 3 2 2 + → + + (2)
These two coupled reactions take place at ambient temperature (∼20°C) for the first 24 hours.

However, as observed in the experiments, these reactions are exothermic, with the suspension temperature reaching a maximum value of 38±1°C after about 1h of reaction. The maximum value of temperature remains constant for about 2h and then it decreases slowly to ambient temperature (∼20°C).

(2) A heat-ageing step from 20 to 90°C was performed to obtain complete transformation of dypingite to magnesite after 24 hours. This assumes a solid state transition implying the simultaneous dehydration and carbonation of a brucitic layer of dypingite precursor coupled with instantaneous formation of magnesite crystals. However, the dissolution of the dypingite precursor coupled with magnesite precipitation cannot be excluded. The general reaction for the dypingite-to-magnesite transformation can be written as follows:

O H NaOH MgCO CO Na O H OH CO Mg 2 3 3 2 2 2 4 3 5 4 . 0 2 . 0 5 . ) ( ) ( 2 . 0 + + → + (3)
In this study, NaOH played a catalytic role, i.e., it accelerated brucite carbonation by an increase in carbonate ion concentration with time. The catalytic role can be verified by adding reactions 1 to 3. The presence of NaOH also promoted magnesite formation during the heat-ageing step. In this way, the precipitation of magnesite, enhanced by NaOH and a heat-ageing step, can be illustrated by the following global carbonation reaction:

O H MgCO CO OH Mg s NaOH aq s 2 ) ( 3
2 ) ( 2 ) ( 2 ) ( +   →  + (4)
This reaction takes place exclusively when NaOH is used as a catalytic agent. Conversely, if the same experiment is carried out in the absence of NaOH, incomplete carbonation of brucite is observed and hydromagnesite is the dominant Mg-carbonate after the heat-ageing step. In such case, the hydromagnesite-magnesite-brucite composite obtained contains only about 5% of magnesite (see Fig. 3).

Synthesis of hydromagnesite

Hydromagnesite (Mg 5 (CO 3 ) 4 (OH) 2 .4H 2 O) is a naturally occurring compound found in magnesium-rich minerals such as serpentine and altered magnesium-rich igneous rocks. It is also produced by alteration of brucite in periclase marbles. In view of its potential use in industrial applications, several scientific studies, technical reports and patents have been published

concerning its synthesis at laboratory scale (e.g., [START_REF] Pohl | European Patent Application[END_REF].

Two different methods for synthesizing hydromagnesite under laboratory conditions are reported here; both methods can be used to synthesize high-pure hydromagnesite as attested by X-ray diffraction observations on the powder products where the experimental XRD patterns successfully match with the ICDD card # 070-0361 (see Fig. 4). Platy fine particles with moderate specific surface area (>20m 2 /g) were obtained for both methods (see Table 2 and FESEM image for synthesis S3 in Fig. 5). However, the synthesis time is drastically reduced from 12 days to 5 hours when NaOH is used as catalyst and/or additive in the Mg(OH) 2 -H 2 O-CO 2 system. Similarly to the synthesis of magnesite, the NaOH-rich solution accelerates brucite carbonation by an increase in carbonate ion concentration with time. For this case, the general reaction for hydromagnesite precipitation, i.e., an intermediate hydroxylated (or basic) Mgcarbonate with respect to magnesite, can be written as follows:

NaOH O H OH CO Mg O H CO Na OH Mg 6 . 1 4 . ) ( ) ( 2 . 0 8 . 0 8 . 0 ) ( 2 2 4 3 5 2 3 2 2 + → + + (5)
As mentioned above, this carbonation reaction of brucite takes only five hours. Conversely, the hydrothermal carbonation of brucite in the absence of NaOH takes about 12 days to obtain hydromagnesite with similar textural properties. The hydrothermal carbonation of brucite to form hydromagnesite at 90°C, 90 bar, and in the absence of NaOH can be described by the following global reaction.

O H OH CO Mg CO OH Mg aq 2 2 4 3 5 ) ( 2 2 4 . ) ( ) ( 2 . 0 8 . 0 ) ( → + (6)
In conclusion, the use of NaOH as a catalyst is a powerful option to accelerate the production of hydromagnesite (only five hours) with potential industrial applications given its advantageous textural properties and composition, as reported in Table 2.

Synthesis of Mg-carbonate composites

As mentioned in the Introduction, magnesite is considered to be the most stable Mg-carbonate compound. However, its precipitation at low temperature is kinetically inhibited by the preferential precipitation of hydrated and/or hydroxylated Mg-carbonates, possibly due to the high hydration nature of Mg 2+ ions in solution (21). Based on these concepts, the formation of anhydrous Mg-carbonates (e.g., magnesite and dolomite) at ambient temperature is virtually impossible (1). However, anhydrous eitelite (Na 2 CO 3 .MgCO 3 ) can be formed at low-temperature [START_REF] Deelman | Neues Jahrbunch Für Mineralogy[END_REF]. This mineral is not found in large quantities in natural sedimentary environments but its anhydrous form is certainly of interest for understanding how Mg in solution can be dehydrated to form anhydrous Mg carbonates at low temperature.

In the present study, the catalytic role of NaOH has been demonstrated for rapid precipitation of magnesite (reactions 1 to 4) and hydromagnesite (reaction 5) (see also Tables 1 and2). However, under specific conditions (see S4 in Table 1), the sodium dissolved during the carbonation process may react significantly after 10 days to form an atypical hydromagnesite-eitelite composite at low temperature (45°C), both minerals were clearly identified by X-ray diffraction on the solid product and the experimental XRD pattern obtained successfully matched the ICDD cards #070-0361 for hydromagnesite and #024-1227 for eitelite (see S4 in Fig. 6).

FESEM observations have revealed irregular fine aggregates (S4 in Fig. 5) and a moderate specific surface area (11 m 2 /g) was deduced from the N 2 adsorption isotherm (see also Table 2).

It is assumed that hydromagnesite is initially formed in this system, according to reaction (5), followed by the precipitation of eitelite. In such a case, two explanations are possible: (i) dissolution of hydromagnesite (so-called precursor) coupled with eitelite precipitation or (ii) a solid state transition implying the simultaneous dehydration and carbonation of a brucitic layer of hydromagnesite precursor coupled with the progressive formation of eitelite crystals. The general reaction for hydromagnesite-to-eitelite transformation can be written as follows:

O H NaOH MgCO CO Na CO Na O H OH CO Mg 2 3 3 2 3 2 2 2 4 3 5 8 . 0 4 . 0 . 2 . 1 4 . ) ( ) ( 2 . 0 + + → + (7)
This reaction cannot be completed because only 1mol of Na 2 CO 3 can be produced in the system, see reaction (1). For this reason, the solid recovered after 10 days of reaction contains mainly hydromagnesite and eitelite minerals, the so-called hydromagnesite-eitelite composite in the present study.

On the other hand, as mentioned in sub-section 3.1, the carbonation of brucite in the absence of NaOH, leads to incomplete brucite carbonation after a heat-ageing step (S2 synthesis) where the mineral composition of the recovered solid was hydromagnesite, magnesite and residual brucite (see Fig. 3 and6). More specific information on the experimental conditions, composition and textural properties are provided in Tables 1 and2. For this case, the hydromagnesite formation is in agreement with reaction [START_REF] O'connor | Final report aqueous mineral carbonation: mineral availability, pretreatment, reaction parametrics, and process studies[END_REF] where a small proportion of hydromagnesite was transformed into magnesite by the following classical reaction:

O H MgCO CO O H OH CO Mg aq 2 3 ) ( 2 2 2 4 3 5 2 . 0 4 . ) ( ) ( 2 . 0 + → + (8)
In general, higher temperatures (>150°C), high salinity, high CO 2 pressure, low magnesium concentration (in a closed system) and the use of organic additives (e.g., monoethylene glycol) are known to accelerate the hydromagnesite-to-magnesite transformation (18,25).

Potential industrial applications

In general, anhydrous Mg-carbonates (e.g., magnesite, eitelite, dolomite, etc.) and intermediate basic magnesium carbonates (e.g., dypingite, hydromagnesite, artinite, pokrosvskite, etc.) with controlled morphology and particle size and moderate-to-high specific surface area, have great potential for use in applications such as mineral filler and pigment in paper, paint, rubber and plastics, and could also be used as flame-retardants. Unfortunately, the production at industrial scale has been limited because high pressures, high temperatures and long production times are frequently required. The present study shows how high-purity magnesite can be produced after 48h of reaction and how high-purity hydromagnesite can be produced after 5h of reaction. Based on its composition and textural properties, the magnesite could be used as a mineral filler in paper and pigments. Moreover, additional applications such as a flame-retardant in electrical and electronic parts, construction materials, etc., can be envisaged for hydromagnesite because their dehydration-dehydroxylation-decarbonation processes consume significant amount of energy in a broad interval of temperature (see Fig. 7). Moreover, the platy fine particles and moderate specific surface area (28 m 2 /g) of hydromagnesite can facilitate its dispersion/distribution when used as a mineral filler or flame-retardant.

Conclusion

This study proposes simple and novel synthesis routes for the production of rhombohedral single crystals (<2µm) of magnesite and platy fine particles of hydromagnesite. It also demonstrates the catalytic action of NaOH during the carbonation process of brucite mineral, leading to fast precipitation of high-purity magnesite after 48h of reaction and fast precipitation of high-purity hydromagnesite after only 5h of reaction. For each case, a global reaction mechanism has been proposed. Finally, it is worth noting that both minerals could have good potential for use as mineral filler and/or as flame-retardant.

  ;(2) complete dypingiteto-magnesite transformation after 24h by a simple heat-ageing step from 20 to 90°C. The dypingite-to-magnesite transformation implies the simultaneous dehydration and carbonation of a brucitic layer of dypingite coupled with instantaneous formation of magnesite crystals. In this study, the NaOH played a catalytic role, i.e., it accelerated brucite carbonation by an increase in carbonate ion concentration with time and it promoted the formation of magnesite during the heat-ageing step as illustrated in the following global reaction:

Figure 1 .

 1 Figure1. Fast precipitation of magnesite from Mg(OH) 2 -H 2 O-CO 2 slurry, enhanced by NaOH and a heat-ageing step from ambient temperature to 90°C. Only 48h were required to produced high-purity magnesite as attested from the XRD pattern (ICDD # 086-2346). Precursor or basic Mg-carbonate was identified as dypingite from the XRD pattern (ICDD # 023-1218). High-purity synthetic brucite was used as solid starting reactant as attested from the XRD pattern (ICDD # 083-0114).

Figure 2 .

 2 Figure 2. FESEM images showing the morphology of the dypingite precursor and magnesite crystals precipitated from the Mg(OH) 2 -H 2 O-NaOH-CO 2 system at ambient temperature and after a heat-ageing step at 90°C, respectively.

Figure 3 .

 3 Figure 3. XRD patterns showing the aqueous carbonation of brucite Mg(OH) 2 in the presence and

Figure 7 .

 7 Figure 7. Thermogravimetric analyses (TGA on top graph and differential TGA on bottom graph)

Table 1 .

 1 Summary of experimental conditions for the synthesis of anhydrous and hydroxylated Mg-carbonates. : Temperature at which the CO 2 gas was injected; P i : Initial CO 2 pressure in the system; t r : total reaction time (including heatageing step); *: Temperature of suspension (including exothermic dissolution of NaOH in water); **: synthesis at constant gas (CO 2 +Ar) pressure (90 bar), the consumption of CO 2 was regulated by automatic argon injection; H: Hydromagnesite; M: Magnesite; B: Brucite; E: Eitelite.

	Exp.	System	Use of	T i (°C)	P i (bar)	Heat-	t r	Isobaric/	Mineral
	Label		NaOH			ageing step		Anisobaric
	S0	Mg(OH)2-H2O-CO2	Yes	Ambient	50	not	24 hours	Anisobaric	Dypingite
				(~26*)
	S1	Mg(OH)2-H2O-CO2	Yes	Ambient	50	From 20 to	48 hours	Anisobaric	Magnesite
				(~26*)		90 °C
	S2	Mg(OH)2-H2O-CO2	Not	Ambient	50	From 20 to	48 hours	Anisobaric	H-M-B composite
				(~20)		90 °C
	S3	Mg(OH)2-H2O-CO2	Not	90	50	not	12 days	Isobaric**	Hydromagnesite
	S4	Mg(OH)2-H2O-CO2	Yes	Ambient	50	From 26 to	10 days	Anisobaric	H-E composite
				(~26* )		45 °C
	S5	Mg(OH)2-H2O-CO2	Yes	Ambient	50	From 38 to	5 hours	Anisobaric	Hydromagnesite
				(~26*)		90 °C
	T

i

Acknowledgements

The authors are grateful to the French National Center for Scientific Research (CNRS) and the University Joseph Fourier in Grenoble for providing financial support.

We proposes simple and novel synthesis routes for the production of rhombohedral single crystals (<2µm) of magnesite and platy fine particles of hydromagnesite. It also demonstrates the catalytic action of NaOH during the carbonation process of brucite mineral, leading to fast precipitation of high-purity magnesite after 48h of reaction and fast precipitation of high-purity hydromagnesite after only 5h of reaction. For each case, a global reaction mechanism has been proposed. 2


D González

A Ammar

E Cueto
email: ecueto@unizar.es

F Chinesta

Towards a high-resolution numerical strategy based on separated representations

Keywords: Curse of dimensionality, separation of variables, high-dimensional problems, kinetic theory

. We focus on an efficient imposition of essential non-homogeneous boundary conditions and the treatment of problems with a very high number of degrees of freedom.

INTRODUCTION

In problems that involve high dimensional spaces, standard discretization techniques, like the finite element method, fail due to an excessive computation time required to perform accurate numerical simulations.

The main aim of this paper is to present a new numerical strategy able to circumvent some of the difficulties due to the multidimensional character of kinetic theory models, among other fields of science. The results will be presented to illustrate its capability for treating highly dimensional spaces by considering different multidimensional Poisson's problems, and an efficient imposition of essential non-homogeneous boundary conditions.

THE METHOD

For simplicity, we start by considering the Poisson problem defined in a space of dimension N with homogeneous boundary conditions,

-∆T = f (x 1 , x 2 , . . . , x N ) in Ω = (-L, L) N . (1)
The problem solution can be written in the form

T (x 1 , x 2 , . . . , x N ) ≈ Q j=1 α j N k=1 F kj (x k ), (2)
where F kj is the jth basis function, with unit norm, and only depends on the kth coordinate. Note that the solution of numerous problems can be accurately approximated using a finite number Q of approximation functions.

The numerical scheme proposed consist of an iteration procedure that solves at each iteration n the following three steps.

Step 1 A projection step of the solution in a discrete basis, where the coefficients α j are computed.

Step 2 Convergence checking.

Step 3 Enrichment of the approximation basis step.

From the coefficients α j just computed the approximation basis can be enriched by adding the new function N k=1 F k(n+1) (x k ) For details of this scheme see [START_REF] Ammar | A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids[END_REF].

IMPOSITION OF NON-HOMOGENEOUS ES-SENTIAL BOUNDARY CONDITIONS

Consider, for simplicity and without loss of generality, the Poisson problem described before, see Eq. ( 1) subjected to the boundary conditions:

T = T D = 0 on Γ = ∂Ω. (3)
Let us assume that we are able to find a function ψ, continuous in Ω, such that -∆ψ ∈ L 2 (Ω) verifying Eq. (3). Then, the solution of the problem given by Eqs. ( 1)-( 3) can be obtained straightforwardly by

T = ψ + z, (4)
where we thus face a problem in the z variable

-∆z = f + ∆ψ in Ω (5) z = 0 on Γ = ∂Ω, (6)
solvable by the method of separation of variables before presented. Following this procedure, the weak form of the problem will be:

Find z ∈ H 1 (Ω) such that for every z * ∈ H 1 0 (Ω) Ω ∇z∇z * dΩ = Ω z * f dΩ - Ω ∇ψ∇z * dΩ (7)
holds.

Several choices for the function ψ exist. In this work we employ R-functions [START_REF] Rvachev | R-functions in boundary value problems in mechanics[END_REF], which constitute an efficient means of constructing ψ functions.

The obtained function ψ, which is generally not product or sum separable, is separated approximately by employing the Singular Value Decomposition (SVD) technique. To this end, we employ singular values up to a given precision, which is normally set to 10 -5 .

NUMERICAL EXAMPLES

In this section we describe some simple numerical experiments that show the potential of the method.

Example: 2D problem

Let us consider now the problem

∆T = 0 in [-1, 1] 2 , (8)
subjected to the boundary conditions

T = x 2 + y 2 on ∂Ω,
that in this case do not verify the partial differential equation ( 8).

In this case, using the before mentioned theory of R-functions, we obtain the following function verifying essential boundary conditions:

ψ = -2 + x 4 + y 4 -2 + x 2 + y 2 (9)
which is represented in Fig. 1. This function is again not sum or product separable, according to [START_REF] Messiter | Product and sum separable functions[END_REF]. Applying SVD, we arrive to the one-dimensional functions depicted in Fig. 2, that approximate the function ψ to the chosen level of precision, 10 -5 in this case. The method before described was applied in meshes composed by 15, 20, 30, 50, 80 and 130 nodes along each spatial direction. The solution was compared to one approximate solution obtained by standard Finite Elements in a mesh composed by 40000 nodes, thus taken as exact. The order of convergence is remarkably high, and has been found to be of the order of 4 (i.e., superconvergent) for all the cases tested.

Extension of the proposed technique to higherdimensional problems

The problem with the straightforward extension of the proposed technique to higher dimensional problems lies in obtaining the appropriate counterpart of the SVD decomposition in high dimensions.

To this end, we have employed the PARAFAC (Parallel Factors) decomposition, also known as CANDECOMP or Canonical Decomposition, provided by the MATLAB tensor toolbox [START_REF] Bader | Efficient MATLAB computations with sparse and factored tensors[END_REF]. This technique provides a diagonal tensor Σ, suitable for the purpose of this work. The r most relevant entries of this tensor are employed in the decomposition, in order to accomplish with a user-predefined tolerance in the representation of the function ψ. In general, we seek for decompositions of our function ψ, whose nodal values are stored in an order n (n = 3 in the example here considered, for simplicity of graphical representation) tensor A, in the form:

A = Σ × 1 U × 2 V × 3 W × 4 . . . ( 10
)
where Σ is a diagonal tensor and U , V , W , ... are the components or factors of the decomposition.

Columns of the factors are orthonormal in standard, two-dimensional SVD, but this is not possible, in general, in tensors of order three or higher. The products involved in the previous decomposition are defined as follows:

B = C × 1 U (11)
is such that

(B) ljk = m i=1 C ijk U li (12)
To analyze the behavior of the technique, we have extended the previous problem to three-dimensional settings:

∆T = 0 in [-1, 1] 3 , (13)
subjected to the boundary conditions

T = x 2 + y 2 + z 2 on ∂Ω.
The higher-order SVD decomposition of the function ψ up to the tolerance 10 -6 provides the functions depicted in Fig. 4. Only four functions along each spatial direction were necessary. With these functions thus calculated, we let the method run to obtain the solution to the problem as depicted along each direction in Fig. 5. The global, three-dimensional, solution to the problem is depicted in Fig. 6. In this case, 18 functions were necessary along each spatial direction to obtain the before mentioned accuracy. The figure depicts the intersection of the solution with the plane x = 0.

A problem with 10 9 degrees of freedom

In order to show the behavior of the proposed method we have analyzed the problem

∆T = 2(1 -y 4 )(1 -z 6 ) + 12(1 -x 2 )y 2 (1 -z 6 )+ + 30(1 -x 2 )(1 -y 4 )z 4 in [-1, 1] 3 , (14)
(the same analyzed in [START_REF] Ammar | A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids[END_REF]), subjected to the boundary conditions T = 0 on ∂Ω, that has been discretized in a regular lattice of 1000 3 elements, thus giving 1001 3 degrees of freedom. This value is on the limit of current capabilities of nowadays computers, and is analyzed here only for the purpose of showing the performance of the method.

The analytical solution to this problem is given by the function T = (1x 2 )(1y 4 )(1z 6 ).

The functions obtained by the method in each spatial direction are depicted in Fig. 7. As can be noticed, the method obtains a very accurate solution in only one iteration, as was the case in [START_REF] Ammar | A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids[END_REF], employing fewer degrees of freedom. The solution obtained is represented in Fig. 8(top), where a downsampling has been applied to the results by employing a coarser mesh. For obvious reasons, it is difficult to depict the results on a mesh of 10 9 nodes. The error obtained is depicted in In this example we do not pursue accuracy, but testing the possibility of employing a huge number of degrees of freedom. What is noticeable about this solution, easily reachable with traditional Finite Elements, is that, despite the extremely high number of degrees of freedom involved, the Matlab code ran in around 1 hour on a PC with four processors (only one was employed, no parallel calculation was allowed) with 1.96 Gb of RAM memory each, running under Linux. This time comprised data file writing, generation of figures, etc.

Figure 1 :FyFigure 2 :

 12 Figure 1: Function ψ for the second problem, Eq. (8).

Figure 3 :

 3 Figure 3: Solution convergence obtained by the proposed technique.

Figure 4 :

 4 Figure 4: Functions obtained along x (a), y (b) and z (c) spatial directions for the boundary conditions of the problem stated in Eq. (13).

Figure 5 :

 5 Figure5: Functions obtained along x, y and z spatial directions for the problem stated in Eq. (13).

Figure 6 :

 6 Figure 6: Solution obtained by the proposed technique for the problem with non-homogeneous boundary conditions in three dimensions. Mesh of 40 elements along each direction.

Figure 7 :

 7 Figure 7: Functions obtained along x, y and z spatial directions for the problem stated in Eq. (14).

Fig. 8 (

 8 Fig. 8(bottom), where a slice on an arbitrary section of the domain has been performed for clarity.

Figure 8 :

 8 Figure 8: Solution of the problem stated in Eq. (7) and a slice on the error map.


I Heulot

K Desnos

I.-F Nezan

M Pelcat

M Raulet

H Yviquel

P.-L Lagalaye

I.-C Le Lann

AN EXPERIMENTAL TOOLCHAIN BASED ON HIGH-LEVEL DATAFLOW MODELS OF COMPUTATION FOR HETEROGENEOUS MPSOC

Keywords: Embedded software, Multicore processing, Data flow computing, Signal processing 1. EXTENDED

A chain of three state-of-the-art tools is demonstrated to generate efficient code for Multi-Processors System-on-Chips (MPSoCs) from a high-level dataflow language. The experimental platform is based on a 5-core Texas Instruments OMAP4 heterogeneous MPSoC running an image processing application.

High-level languages respecting dynamic dataflow Models of Computation (MoCs) are convenient to specify an algorithm in a userfriendly fashion. However, static dataflow MoCs provide more compile-time knowledge of an application parallelism. This knowledge is necessary to produce efficient code for a MPSoC. In this paper, we demonstrate a transformation flow that eases the design of retargetable applications for heterogeneous MPSoC by transforming a high-level specification based on a dynamic dataflow MoC into an MPSoC-optimized application based on a static dataflow MoC. This flow consists of three dataflow-based tools: Modae Studio, Open RVC-CAL Compiler (Orcc), and Preesm. The demonstrated hardware is based on a 5-core Texas Instruments OMAP4 heterogeneous MPSoC.

INTRODUCTION

Modern handheld embedded systems offer an increasing number of functionalities and processing capabilities while respecting a fixed power budget of a few Watts. Recent improvements in embedded systems are due to MPSoCs that combine general purpose cores, dedicated cores, and hardware accelerators within a single chip. Generating efficient code for heterogeneous MPSoCs remains a complex and error-prone task. This demonstration illustrates how dataflow MoCs, having precise semantics, favor interoperability between tools and can be used to program MPSoCs. Three dataflow-based tools (Section 3): Modae Studio, Orcc, and Preesm, are combined to program a heterogeneous MPSoC from a highlevel retargetable code with Kahn Process Network (KPN) MoC semantics [START_REF] Lee | Dataflow process networks[END_REF]. The demonstrator is then presented in Section 4.

MODEL TRANSFORMATION FLOW

The transformation flow presented in Figure 1 is performed by a tool chain composed of: Modae Studio, Orcc, and Preesm. Each of these tools is successively used on a high-level description of the application in order to introduce more behavioral knowledge into its MoC. The transformation flow both offers a high-level front-end to the application designer, and generates an efficient mapping of the application on the targeted architecture.

Modae Studio! is the first tool of this chain and is used as a schematic entry tool: a system is described as a set of interacting processes, communicating via channels, usually depicted as boxes and arrows. Channels link ports point-to-point. Different MoCs can be tagged on channels. Processes are organized as KPNs: processes communicate via infinite FIFOs, with blocking read semantics. The main novelty of Modae approach relies on the resort to Ruby and Python interpreted languages to describe the algorithmic content of the processes. These languages are dynamically typed and userfriendly. Processes are described using an internal Domain Specific Language (DSL), a technique that popularized Ruby-on-rails web 1http://www.modae-tech.com framework. Such an internal DSL facilitates a seamless integration of domain specific concepts into the host language. The objectoriented nature of Ruby and Python also simplifies the design of high-level libraries. Modae Studio provides libraries for image and signal processing, but also for simple probe displays (xy plots, pixel grids, ...). Once this closed system is captured, Modae compiler and simulators retrieve the bit-accurate type of each variable, hence filling the gap between user representations and machine-oriented representations. In the experimental tool chain described here, the transformation engine generates a Dataflow Process Network (DPN) [START_REF] Lee | Dataflow process networks[END_REF] in RYC-CAL dataflow language syntax [START_REF] Wipliez | Compilation infrastructure for dataflow programs[END_REF].

Orcc", the second tool of the transformation flow, is an open source compiler for applications modeled with RYC-CAL and a DPN. When converted into DPNs, processes from KPN are replaced with actors that still communicate via infinite FIFOs. Contrary to KPN processes, actors have a set of firing rules that dictate when an actor is fired and how many data tokens are exchanged [START_REF] Lee | Dataflow process networks[END_REF]. DPNs actors also have the ability to peek in FIFOs, i.e. they can read data-tokens in input FIFOs without consuming them.

The main purpose of Orcc in the flow is to analyze the behavior of the actors generated by Modae Studio and classify their behavior as Synchronous Dataflow (SDF), Cyclo-Static Dataflow (CSDF) [START_REF] Bilsen | Cycle-static dataflow[END_REF], or DPN. CSDF is a restriction of DPN where actor firings follow a cyclic fashion and SDF is a restriction of CSDF where the exchanged tokens are constant over firings. The advantage of CSDF and SDF is that they both have a strong predictability, thus opening the way to efficient compile-time optimizations. If the DPN is classified as CSDF, Orcc is asked to convert the graph into a SDF graph before generating the tool's outputs. Applications classified as DPN are currently not supported by this tool chain. Orcc outputs C files, each implementing the internal behavior of an actor, and a SDF graph interconnecting the actors.

Preesrrr', the last element of the tool chain, is an open source rapid prototyping tool that automatically maps and schedules hierarchical SDF graphs on heterogeneous MPSoCs. Beside the SDF graph provided by Orcc, another input of Preesm is a graphically edited System-Level Architecture Model (S-LAM) of the targeted architecture. Using what is called a scenario, the user can also specify a set of parameters and constraints for the mapping and scheduling tasks, restricting for instance the mapping of an actor on a subset of cores of the architecture.

Before mapping the actors on the heterogeneous MPSoC, Preesm performs a set of conversions on the application model so as to reveal the parallelism embedded in the MoC. Once a static mapping and schedule are obtained, Preesm generates a specific C file for each core of the architecture, handling inter-core communication and synchronization, and containing ordered calls to the C functions of the actors generated by Orcc.

DEMONSTRATOR

Developed by Texas Instruments, Open Multimedia Applications Platform (OMAP) is a family of MPSoC designed for embedded handheld multimedia applications. OMAP devices include a general-purpose ARM processor core with one or more specialized co-processors. The 4th generation OMAP processors are based on a dual-core ARM Cortex-A9 as main processing unit. The OMAP4460 additionally contains two Cortex -M3 microcontrollers 2http://orcc.sourceforge.net 3http://preesm.sourceforge.net Fig. 2. Demonstrator material including a Pandaboard, a computer, a camera, and a projector increasing power efficiency as well as a c64xT Digital Signal Processor (DSP) that can be used to speed-up some DSP-friendly algorithms. Hardware coprocessors are not covered by this demonstration. In this processor, 5 different cores can thus run simultaneously, each running a different program. Inter-core communications are possible through a shared Level-3 memory.

The demonstrator (Figure 2) consists of:

• a camera that captures a video stream, • a Pandaboard ES, including an OMAP4460 MPSoC, that receives the video stream and computes an image processing algorithm distributed on up to 5 cores,

• a handheld projector that displays the resulting video, and

• a PC that generates and reloads code on each of the 5 cores, connected to the Pandaboard ES via Ethernet.

During this demonstration, high-level programming of an OMAP4 MPSoC is shown. A Sobel filter application is described, simulated and exported as RYC-CAL code using the Modae Studio tool. Then, the application MoC is converted into the SDF MoC by the Orcc compiler. The algorithm is automatically mapped and scheduled by the Preesm tool, based on knowledge of the application SDF MoC and of the architecture model. Finally, an executable is built for each MPSoC core, then loaded and run on the OMAP4 processor. The dual-core ARM Cortex -A9 main processing unit of the OMAP4 processor runs a Linux operating system and drives the 3 other cores. The obtained process is highly customizable and, yet, most of the steps are fully automated and performed within seconds. The mapping constraints of actors, as well as the actors themselves, are modified during demonstration, resulting in a different code, mapped and scheduled efficiently on all cores of the OMAP4 processor. Intermediate generated codes and architecture model are demonstrated.

  This work was supported by the ANR COMPA project * I.-C. Le Lann is now with Labsticc-ENSTA Bretagne

Fig. 1 .

 1 Fig. 1. Transformation Flow


Júlio Araújo

Pargo

Nathann Cohen

Frédéric Giroire

Frédéric Havet

Good edge-labelling of graphs

Keywords: edge-labelling, planar graphs, matching-cut, channel assignment

A good edge-labelling of a graph G is a labelling of its edges such that for any two distinct vertices u, v, there is at most one (u, v)-path with non-decreasing labels. This notion was introduced in [3] to solve wavelength assignment problems for specific categories of graphs. In this paper, we aim at characterizing the class of graphs that admit a good edge-labelling. First, we exhibit infinite families of graphs for which no such edge-labelling can be found. We then show that deciding if a graph admits a good edge-labelling is NP-complete. Finally, we give large classes of graphs admitting a good edge-labelling: C 3 -free outerplanar graphs, planar graphs of girth at least 6, subcubic {C 3 , K 2,3 }-free graphs.

Motivation

A classical and widely studied problem in WDM (Wavelength Division Multiplexing) networks is the Routing and Wavelength Assignment (RWA) problem [START_REF] Mukherjee | Optical communication networks[END_REF][START_REF] Ramaswami | Routing and wavelength assignment in alloptical networks[END_REF][START_REF] Bala | Algorithms for routing in a linear lightwave network[END_REF]. It consists in finding routes, and their associated wavelength as well, to satisfy a set of traffic requests while minimising the number of used wavelengths. This is a difficult problem which is, in general, NP-hard. Thus, it is often split into two distinct problems: First, routes are found for the requests. Then, in a second step, these routes are taken as an input. Wavelengths must be associated to them in such a way that two routes using the same fibre do not have the same wavelength. The last problem can be reformulated as follows: Given a digraph and a set of dipaths, corresponding to the routes for the requests, find the minimum number of wavelengths w needed to assign different wavelengths to dipaths sharing an edge. This problem may be seen as a colouring problem of the conflict graph which is defined as follows: It has one vertex per dipath and two vertices are linked by an edge if their corresponding dipaths share an edge. In [START_REF] Bermond | Directed acyclic graphs with unique path property[END_REF], Bermond et al. studied the RWA problem for UPP-DAG which are acyclic digraphs (or DAG) in which there is at most one dipath from one vertex to another. In such digraph the routing is forced and thus the unique problem is the wavelength assignment one.

In their paper, they introduce the notion of good edge-labelling. An edgelabelling of a graph G is a function φ : E(G) → R. A path is increasing if the sequence of its edges labels is non-decreasing. An edge-labelling of G is good if for any two distinct vertices u, v, there is at most one increasing (u, v)path. Bermond et al. [START_REF] Bermond | Directed acyclic graphs with unique path property[END_REF] showed that the conflict graph of a set of dipaths in a UPP-DAG has a good edge-labelling. Conversely, for any graph admitting a good edge-labelling one can exhibit a family of dipaths on a UPP-DAG whose conflict graph is precisely this graph. Bermond et al. [START_REF] Bermond | Directed acyclic graphs with unique path property[END_REF] then use the existence of graphs with a good edge-labelling and large chromatic number to prove that there exist sets of requests on UPP-DAGs with load 2 (an edge is shared by at most two paths) requiring an arbitrarily large number of wavelengths. To obtain other results on this problem, it may be useful to identify the good graphs, which admit a good edge-labelling, and the bad ones, which do not.

Our results

In the following, proofs are omitted. The interested reader is referred to [START_REF] Araújo | Good edge-labelling of graphs[END_REF].

Bad graphs

Bermond et al. [START_REF] Bermond | Directed acyclic graphs with unique path property[END_REF] noticed that C 3 and K 2,3 are bad. J.-S. Sereni [START_REF] Sereni | [END_REF] asked whether every {C 3 , K 2,3 }-free graph (i.e. with no C 3 nor K 2,3 as a subgraph) is good. We answer this question in the negative. We give an infinite family of bad graphs, none of which is the subgraph of another. The construction of this family is based on the bipartite graphs B k , k ≥ 3, also known as 'biwheels', of order 2k + 2, depicted below. and defined as follows :

V (B k ) = {u, v} ∪ {u i | 1 ≤ i ≤ k} ∪ {v i | 1 ≤ i ≤ k}, E(B k ) = {uu i | 1 ≤ i ≤ k} ∪ {u i v i | 1 ≤ i ≤ k} ∪ {v i v | 1 ≤ i ≤ k}, ∪{v i u i+1 | 1 ≤ i ≤ k} with u k+1 = u 1 .
For convenience, we denote by B 2 a path of length 2 with endvertices u and v. All the H k share the following property: for every good edge-labelling, H k has either an increasing (u, v)-path or an increasing (v, u)-path. Hence the graph J i,j,k , i, j, k ≥ 2, obtained from disjoint copies of B i , B j and B k by identifying the vertices u of these three copies and the vertices v of these three copies, is bad, even though {C 3 , K 2,3 }-free. Indeed, any good edge-labelling would have either two increasing (u, v)-paths or two increasing (v, u)-paths, which is a contradiction.

NP -completeness

We now prove that deciding if a graph has a good edge-labelling is NPcomplete. We give a reduction from the NOT-ALL-EQUAL (NAE) 3-SAT Problem [START_REF] Thomas | The complexity of satisfiability problems[END_REF] which is defined as follows:

Instance: A set V of variables and a collection C of clauses over V such that each clause has exactly 3 literals.

Question: Is there a truth assignment such that each clause has at least one true and at least one false literal?

The reduction works as follows. Let V = {x 1 , . . . , x n } and C = {C 1 , . . . , C m } be an instance I of the NAE 3-SAT Problem. We shall construct a graph G I such a way that I has an answer yes for the NAE 3-SAT Problem if and only if G has a good edge-labelling. For each variable x i , 1 ≤ i ≤ n, we create a variable graph V G i defined as follows (See Figure 1.):

V (V G i ) = {v i,j k | 1 ≤ j ≤ m, 1 ≤ k ≤ 4} ∪ {r i,j k | 1 ≤ j ≤ m, 1 ≤ k ≤ 4} ∪{s i,j k | 1 ≤ j ≤ m, 1 ≤ k ≤ 4}. E(V G i ) = {v i,j k v i,j k+1 | 1 ≤ j ≤ m, 1 ≤ k ≤ 3} ∪ {v i,j 4 v i,j+1 1 | 1 ≤ j ≤ m -1} ∪{v i,j k r i,j k | 1 ≤ j ≤ m, 1 ≤ k ≤ 4} ∪ {v i,j k s i,j k | 1 ≤ j ≤ m, 1 ≤ k ≤ 4} ∪{v i,j 4 r i,j 1 | 1 ≤ j ≤ m} ∪ {v i,j+1 k r i,j k+1 | 1 ≤ j ≤ m -1, 1 ≤ k ≤ 3} ∪{v i,j 4 s i,j 1 | 1 ≤ j ≤ m} ∪ {v i,j+1 k s i,j k+1 | 1 ≤ j ≤ m -1, 1 ≤ k ≤ 3}. ... ... v i,j+1 1 v i,j+1 3 v i,j+1 2 v i,j+1 4 r i,j+1 1 r i,j 4 s i,j 4 s i,j+1 1 r i,j 3 s i,j 3 s i,j 2 r i,j 2 v i,j 4 v i,j 3 v i,j 2 v i,j 1 r i,j 1 s i,j 1 v i,j-1 3 v i,j-1 4 r i,j-1 3 r i,j-1 4 s i,j-1 3 s i,j-1 4
Fig. 1. The variable graph V G i

For each clause

C j = l 1 ∨ l 2 ∨ l 3 , 1 ≤ j ≤ m, we create a clause graph CG j defined by V (CG j ) = {c j , b j 1 , b j 2 , b j 3 } and E(CG j ) = {c j b j 1 , c j b j 2 , c j b j 3 } Now for each literal l k , 1 ≤ k ≤ 3, if l k is the non-negated variable x i ,
we identify b j k , c j and b j k+1 (index k is taken modulo 3) with v i,j 1 , v i,j 2 and v i,j 3 , respectively. Otherwise, if l k is the negated variable xi , we identify b j k , c j and b j k+1 with v i,j 3 , v i,j 2 and v i,j 1 , respectively.

Good graphs

We also show large classes of good graphs. To do so, we use the notion of critical graph which is a bad graph, every proper subgraph of which is good.

Clearly, a good edge-labelling of a graph induces a good edge-labelling of all its subgraphs. So every bad graph must contain a critical subgraph. We establish several properties of critical graphs. Firstly, as the union of good edge-labellings of the blocks of a graph is a good edge-labelling of this graph, critical graphs are 2-connected. In particular, every forest is good.

Secondly, a critical graph has no matching-cut, that is an edge-cut which is a matching. Indeed, let M be such an edge cut then if G \ M admits a good edge-labelling then assigning to all the edges of M a label greater than all the labels of G \ M we obtain a good edge-labelling of G. As a consequence, Theorem 2.1 every C 3 -free outerplanar graph is good. Indeed Eaton and Hull [START_REF] Eaton | Defective list colorings of planar graphs[END_REF] showed that every such graph has either a 1vertex or two adjacent 2-vertices and so has a matching-cut.

Secondly, every subcubic {C 3 , K 2,3 }-free graph has a matching cut. Indeed consider a smallest cycle C. Then the edge-cut between C and its complement is a matching. Hence, Theorem 2.2 every subcubic {C 3 , K 2,3 }-free graph is good.

Farley and Proskurowski [START_REF] Farley | Extremal graphs with no disconnecting matching[END_REF] showed that a graph with less than 3 2 |V (G)|-

Conclusions and further research

The average degree of a graph G is Ad

(G) = v∈V (G) d(v) |V (G)| = 2|E(G)| |V (G)| .
The above lower bound on the number of edges of a critical graphs implies that, for any c < 3/2, the number of critical graphs with average degree at most c is finite. Actually, we conjecture that the only ones are C 3 and K 2,3 .

Conjecture 3.1 Let G be a critical graph. Then Ad(G) ≥ 3 unless G ∈ {C 3 , K 2,3 }.
More generally for any c < 4, we conjecture the following. Conjecture 3.2 For any c < 4, there exists a finite list of graphs L such that if G is a critical graph with Ad(G) ≤ c then G ∈ L.

The constant 4 in the above conjecture would be tight because the graph J 2,2,k is critical and Ad(J 2,2,k ) = 8k+8 2k+4 = 4 -4 k+2 . We have shown that a graph with no dense subgraphs is good. On the opposite direction one may wonder what is the minimum density ensuring a graph to be bad. Or equivalently, Problem 3.3 what is the maximum number g(n) of edges of a good graph on n vertices? Clearly, we have g(n) = ex(n, C) where C is the set of critical graphs. As K 2,3 is critical then g(n) ≤ ex(n, K 2,3 ) = 1 √ 2 n 3/2 + O(n 4/3 ) by a result of Füredi [START_REF] Füredi | New asymptotics for bipartite Turán numbers[END_REF]. The hypercubes show that g(n) ≥ 1 2 n log n.

  edges. A well known consequence of Euler's Formula is that every planar graph G of girth at least 6 (the girth is the length of a samllest cycle) has at most3 2 |V (G)| -3 edges. Thus Theorem 2.3 every planar graph of girth at least 6 is good.

	3
	2 2 |V (G)| -3 edges has a matching-cut. Hence a critical graph G has at least 3 2
	Bonsma [4] characterized graphs with no matching-cut and 3 2 |V (G)| -3 2 edges. We use this characterization to show that a critical graph G has at
	least 3 2 |V (G)| -1

2 edges unless G is C 3 or K 2,3 .

Partially supported by the INRIA associated team EWIN between Mascotte and ParGO

Partially supported by the European project FET Aeolus

Email: juliocesar@lia.ufc.br

Email: [Nathann.Cohen,Frederic.Giroire,Frederic.Havet]@sophia.inria.fr


G Montes-Hernandez
email: german.montes-hernandez@ujf-grenoble.fr

F Renard

R Chiriac

N Findling

J Ghanbaja

F Toche

Sequential precipitation of a new goethite-calcite nanocomposite and its possible application in the removal of toxic ions from polluted water

Keywords: Goethite, Calcite, Nanocomposite, Precipitation, Removal, Metalloids, Heavy metals

This study proposes a simple and innovative synthesis route for a goethite-calcite nanocomposite. This synthesis is summarized by three sequential precipitation reactions: (1) precipitation of nanosized acicular goethite (1-FeOOH) using a high OH/Fe molar ratio (=5);

(2) instantaneous precipitation of portlandite (Ca(OH) 2 ) by adding CaCl 2 salt to a goethite alkaline suspension (2NaOH + CaCl 2 1Ca(OH) 2 + 2NaCl) and; (3) sub-micrometric calcite precipitation by injection of CO 2 into a goethite-portlandite alkaline suspension (Ca(OH) 2 + CO 2 1CaCO 3 +H 2 O). The XRD patterns have confirmed the goethite and calcite mineral composition in the composite precipitated at 30 and 70°C. FESEM and TEM observations have revealed the formation of nanosized goethite particles well dispersed with submicrometric calcite particles, leading to an orange-brown colour nanocomposite with high specific surface area of around 92 m 2 /g for a composite synthesized at 30°C and 45 m 2 /g for a composite synthesized at 70°C. Both values were determined using the conventional BET method on N 2 sorption isotherms. Finally, a goethite/calcite weight ratio equal to 0.8 in the composite was determined by thermogravimetric analysis (TGA). Additionally, some adsorption experiments carried out at two different pH values revealed that the goethitecalcite composite has a good sequestration capacity for Cu>Cd>As(III)>Se(IV)>As(V).

Conversely, the Se(VI) did not show any chemical affinity with the goethite-calcite composite under the physico-chemical conditions studied. In practice, the goethite-calcite composite can neutralise acidic wastewater by slight calcite dissolution, enhancing the removal of heavy metals (e.g. Cu and Cd) at the calcite-solution interfaces.

Introduction

Goethite and calcite are two inorganic compounds that are widely studied due to their abundance in nature as minerals (abiotic origin) and biominerals (biotic origin). Both minerals can co-exist in several terrestrial environments such as deep geological formations, water aquifers, soils and aerosols, and play a major role in the fate and transport of several metalloids and heavy metal trace elements and organic molecules at the mineral-fluid interfaces [START_REF] Wolthers | The surface chemistry of divalent metal carbonate minerals[END_REF][START_REF] Paquette | Relationship between surface structure, growth mechanism, and trace element incorporation in calcite[END_REF][START_REF] Aquino | Acid-base properties of a goethite surface model. A Theoretical view[END_REF][START_REF] Cornell | The Iron Oxides[END_REF][START_REF] Ozdemir | Intermediate formation of magnetite during dehydration of goethite[END_REF]. Goethite and calcite minerals have also been identified in several extraterrestrial environments. For example, calcite has recently been discovered in Martian soils by the Phoenix Mars exploration mission [START_REF] Boynton | Evidences for calcium carbonate at the Mars Phoenix landing site[END_REF] and goethite has been suspected as a constituent of Martian dust and dark asteroids [START_REF] Morris | Mösbauer mineralogy of rock, soil and dust at Gusev Crater, Mars[END_REF][START_REF] Beck | Goethite as an alternative origin for the 3.1 micro-m band on dark asteroids[END_REF]. Moreover, goethite and calcite minerals are technologically important materials and widely applied as components in various industrial products, e.g., pigments in the building industry, inorganic dyes, pigments and adsorbents in the paper industry, lacquers or plastics, sorbents in the removal of toxic ions and molecules from polluted water, active ingredient in antacid tablets [START_REF] Núñez | Preparation of high acicular and uniform goethite particles by modified-carbonate route[END_REF][START_REF] Schwertmann | Iron oxides in the laboratory: Preparation and Characterization[END_REF][START_REF] Kosmulski | Morphology of synthetic goethite particles[END_REF][START_REF] Thies-Weesie | Synthesis of goethite as a model colloid for mineral liquid crystals[END_REF][START_REF] Kosmulski | Synthesis and characterization of goethite and goethite-hematite composite[END_REF][START_REF] Garcia-Carmona | Morphological characteristics and aggregation of calcite crystals obtained by bubbling CO 2 through a Ca(OH) 2 suspension in the presence of additives[END_REF][START_REF] Domingo | Calcite precipitation by a high-pressure CO 2 carbonation route[END_REF][START_REF] Domingo | Control of calcium carbonate morphology by precipitation in compressed and supercritical carbon dioxide media[END_REF][START_REF] Montes-Hernandez | Textural properties of synthetic calcite produced by hydrothermal carbonation of calcium hydroxide[END_REF][START_REF] Lee | Fabrication of unusually stable amorphous calcium carbonate in an ethanol medium[END_REF]. In the last few decades, several synthesis routes have been proposed to produce goethite or calcite independently at laboratory or industrial scale, focusing on specific or multiple applications or simply to carry out basic research on the crystal growth processes [START_REF] Domingo | Calcite precipitation by a high-pressure CO 2 carbonation route[END_REF][START_REF] Lee | Fabrication of unusually stable amorphous calcium carbonate in an ethanol medium[END_REF][START_REF] Abou-Hassan | Synthesis of goethite by separation of the nucleation and growth process of ferrihydrite nanoparticles using microfluidics[END_REF][START_REF] Yu | Synthesis of uniform goethite nanotubes with parallelogram cross section[END_REF][START_REF] Schwertmann | Properties of goethite of varying crystallinity[END_REF][START_REF] Mohapatra | pH and temperature dependent facile precipitation of nano-goethite particles in Fe(NO 3 ) 3 -NaOH-NH 3 NH 2 HSO 4 -H 2 O medium[END_REF][START_REF] Hayes | In situ x-ray absorption study of surface complexes: Selenium oxyanions on 1-FeOOH[END_REF][START_REF] Montes-Hernandez | Calcite precipitation from CO 2 -H 2 O-Ca(OH) 2 slurry under high pressure of CO 2[END_REF][START_REF] Montes-Hernandez | Growth of nanosized calcite through gas-solid carbonation of nanosized portlandite particles under anisobaric conditions[END_REF][START_REF] Gower | Calcium carbonate films and helices growth in solutions of poly(aspartate)[END_REF][START_REF] Pastero | The competition between {1014 } cleavage and {0112} steep rhombohedra in gel growth calcite crystals[END_REF][START_REF] Tang | Coprecipitation of chromate with calcite: Batch experiments and x-ray absorption spectroscopy[END_REF]. Studies are continuing with a view to improving existing methods and/or developing innovative routes to obtain well-controlled shapes and sizes of nanometer-to-submicrometer goethite or calcite particles. Moreover, the nucleation and growth processes of goethite and calcite are still open subjects [START_REF] Montes-Hernandez | Calcite precipitation from CO 2 -H 2 O-Ca(OH) 2 slurry under high pressure of CO 2[END_REF][START_REF] Montes-Hernandez | Growth of nanosized calcite through gas-solid carbonation of nanosized portlandite particles under anisobaric conditions[END_REF][START_REF] Montes-Hernandez | Fast precipitation of acicular goethite from ferric hydroxide gel under moderate temperature (30 and 70 C degrees)[END_REF].

Several toxic ions, and particularly so-called oxyanions, are highly mobile in soils and groundwater since most natural minerals have net negative surface charges. Several inorganic oxyanions such as nitrates, chromates, arsenates, molybdates, selenites and selenates, can be toxic to humans or wildlife at µg/L to mg/L concentrations. Other oxyanions such as phosphates or nitrates can disturb the natural environment by amplifying eutrophication processes. Toxic ions can be removed from polluted water by coprecipitation [START_REF] Roundhill | Methods and technique for the selective extraction and recovery of oxoanions[END_REF][START_REF] Harper | Removal of arsenic from wastewater using chemical precipitation methods[END_REF], reverse osmosis [START_REF] Fox | Controlling arsenic, fluoride and uranium by point-of-use treatment[END_REF], chemical reduction to less soluble species [START_REF] Eary | Chromate removal from aqueous waste by reduction with ferrous iron[END_REF], and adsorbing colloid flotation (ACF) methods [START_REF] De Carlo | Recovery of arsenic from spent geothermal brine by flotation with colloidal ferric hydroxide and long chain alkyl surfactans[END_REF][START_REF] Peng | Removal of arsenic from aqueous solution by adsorbing colloid flotation[END_REF]. These methods generally need large facilities due to the various equipment and reagents used in the series of treatments involved. In addition, it is much more difficult to remove toxic oxyanions than cations because anions with a similar structure, such as nitrates, sulphates and phosphates often coexist at high concentrations in nature. Although processes of adsorptive removal of toxic oxyanions from solutions using conventional adsorbents, such as activated carbon [START_REF] Gupta | Arsenic removal by adsorption[END_REF][START_REF] Huang | Treatment of arsenic (V) containing water by the activated carbon process[END_REF], tanning gel [START_REF] Nakano | Adsorption mechanism of hexavalent chromium by redox within condensed-tanning gel[END_REF], zeolites [START_REF] Haggerty | Sorption of chromate and other inorganic anions by organo zeolite[END_REF], iron oxides [START_REF] Zachara | Chromate adsorption on amorphous iron oxyhydroxide in presence of major groundwater ions[END_REF][START_REF] Benjamin | Adsorption and surface precipitation of metals on amorphous iron oxyhydroxide[END_REF][START_REF] Peak | Mechanisms of selenate adsorption on iron oxides and hydroxides[END_REF][START_REF] Gupta | Adsorption of As(III) from aqueous solutions by iron oxide-coated sand[END_REF], and aluminium oxide [START_REF] Peak | Adsorption mechanism of selenium oxyanions at the aluminium oxide/water interface[END_REF], have been reported, they are not selective and sometimes have low effectiveness. Searching effective adsorbents, reagents and/or novel methods for the removal of the toxic ions still represents a major environmental remediation issue.

The present study proposes a simple and innovative synthesis route for a goethitecalcite nanocomposite. To the best of our knowledge, the synthesis of this composite has not been reported in the literature. This new nanocomposite could efficiently remove several metalloids and heavy metals (e.g. Cu(II), Cd(II), As(III), Se(IV), As(V)...) from synthetic polluted water and it offers a number of advantages as an adsorbent compared to pure goethite. For example, nanosized goethite particles generally form colloidal suspensions in mixed reactors and it becomes extremely difficult to separate the solid from the solution after water treatment by sorption. Conversely, in the goethite-calcite nanocomposite, the smaller goethite particles remain adhered to the calcite surfaces, thus providing a simpler and cheaper solid-solution separation process (e.g. by sedimentation and/or filtration). Another practical advantage is that the goethite-calcite composite can neutralise acidic wastewater by slight calcite dissolution during the sorption process, accentuating the removal of heavy metals (e.g.

Cu and Cd) at the calcite-solution interfaces (this study).

This study has a two-fold objective: firstly, propose a simple and innovative synthesis route for a goethite-calcite nanocomposite using a sequential reaction procedure, and secondly, provide some arbitrary adsorption experiments to evaluate its possible application in the removal of toxic ions from polluted water. The goethite-calcite composite was characterised by X-ray powder Diffraction (XRD), Field Emission Gun Scanning Electron Microscopy (FESEM), Transmission Electron Microscopy (TEM), N 2 adsorption isotherms and Thermogravimetric Analysis (TGA/SDTA). The concentrations the elements ([Ca], [Fe],

[Na], [As], [Se], [Cd] and [Cu]) in filtered and acidified solutions from adsorption experiments were determined by inductively coupled plasma optical emission spectrometry (ICP-OES).

Materials and methods

Synthesis of goethite-calcite (stirred reactor)

One litre of high-purity water with electrical resistivity of 18.2 M2•cm, 1 mol of NaOH and 0.2 mol of FeCl 3 .6H 2 O were placed in a titanium reactor (autoclave with internal volume of two litres). This aqueous reactive system was immediately stirred using constant mechanical agitation (400 rpm) during the reaction. The aqueous system was then heated at 30°C or 70°C for 24h using a heating jacket adapted to the reactor. The autoclave was disassembled after 24h and 30g of CaCl 2 salt was added to the goethite alkaline suspension.

The autoclave was rapidly assembled and the heating and agitation systems were re-started.

For this case, the calcium chloride reacts instantaneously with the residual NaOH in the goethite suspension, leading to the formation of portlandite or calcium hydroxide particles (2NaOH + CaCl 2 1Ca(OH) 2 + 2NaCl). When the reaction temperature was stabilised at 30°C or 70°C, CO 2 at 20 bar was injected into the goethite-portlandite suspension. A portlandite carbonation process immediately takes place (Ca(OH) 2 + CO 2 1CaCO 3 +H 2 O). Carbonation equilibrium (monitored by a drop in CO 2 pressure) was reached after about 8h but the reaction was stopped after 24h in both cases (i.e. at 30°C and 70°C).

At the end of the experiment, the autoclave was removed from the heating system and immersed in cold water when the experiment was carried out at 70°C. After water cooling at 30°C (about 10 minutes) the autoclave was disassembled, and the solid product was carefully recovered and separated by centrifugation (30 minutes at 12,000 rpm), decanting the supernatant solutions. The solid product from both syntheses (i.e. at 30°C and 70°C) was washed twice by re-dispersion/centrifugation processes in order to remove the halite (NaCl) co-precipitated during the synthesis. Finally, the solid product was dried directly in the centrifugation flasks at 60°C for 48 h. The dry solid product was manually recovered, weighed and stored in plastic flasks for further characterisation (FESEM, TEM, XRD, TGA and N 2 sorption isotherms).

Characterisation of goethite-calcite composite

FESEM and TEM observations: Pure goethite and goethite-calcite composites were dispersed by ultrasonic treatment in absolute ethanol for five to ten minutes. One or two droplets of the suspension were then deposited directly on an aluminium support for SEM observations, and coated with platinum. Morphological observations of various selected powders were performed using a Zeiss Ultra 55 field emission gun scanning electron microscope (FESEM) that has a maximum spatial resolution of approximately 1nm at 15kV. The composite dispersed by ultrasonic treatment in ethanol was also oriented on carbon Ni grids and then imaged using a JEOL 3010 Transmission Electron Microscope (TEM) equipped with an energy dispersive X-ray analyzer (EDS).

XRD measurements: X-Ray Powder Diffraction (XRD) analyses were performed using a D5000, SIEMENS diffractometer in Bragg-Brentano geometry, equipped with a theta-theta goniometer with a rotating sample holder. The XRD patterns were collected using Cu k1 1 (λ k11 =1.54063) and k1 2 (λ k12 =1.5444 3) radiation in the range 2θ = 10 -70° with a step size of 0.04° and a counting time of 6 seconds per step.

Thermogravimetric analyses: TGA for goethite-calcite composites were performed with a TGA/SDTA 851 e Mettler Toledo instrument under the following conditions: sample mass of about 10 mg, alumina crucible of 150 µl with a pinhole, heating rate of 10°C min -1 , and inert N 2 atmosphere of 50 ml min -1 . Sample mass loss and associated thermal effects were obtained by TGA/SDTA. In order to identify the different mass loss steps, the TGA first derivative (rate of mass loss) was used. TGA apparatus was calibrated in terms of mass and temperature.

Calcium oxalate was used for the sample mass calibration. The melting points of three compounds (indium, aluminium and copper) obtained from the DTA signals were used for sample temperature calibration.

N 2 sorption isotherms: N 2 sorption isotherms for goethite-calcite composites were determined using a sorptomatic system (Thermo Electron Corporation). The specific surface area of powdered samples was estimated by applying the Brunauer-Emmet-Teller (BET) equation in the 0.05≤P/P 0 ≤0.35 interval of relative pressure and using a value of 16.2 Å 2 for the crosssectional area of molecular N 2 . A non-linear regression by the least-squares method was performed to fit the interval data (n ads vs. P/P 0 ) in the experimental isotherms. Additionally, the Barrett-Joyner-Halenda (BJH) method that takes into account the capillary condensation via the Kelvin equation, was used to determine pore size distribution.

Adsorption experiments in a semi-continuous mixed reactor

One litre of solution containing simultaneously two ions (As(III)-Se(IV), As(V)-Se(VI) or Cd(II)-Cu(II): about 50mg/L for each ion) and 3g of goethite-calcite composite was placed in a titanium reactor (autoclave with internal volume of two litres). This suspension was immediately stirred using constant mechanical agitation (400 rpm) during the adsorption process at room temperature (420°C). Subsequently, 15ml of reacting suspension was withdrawn from the reactor as a function of time (0.16, 0.5, 1, 2, 3 and 24h). For this case, the pH was measured in the suspensions at room temperature. The suspensions were then filtered through a 0.2 µm Teflon filter. 10 ml of the resulting filtered solutions was acidified for

Results and discussion

Precipitation reactions

The goethite-calcite nanocomposite synthesis is summarised by three sequential precipitation reactions:

(1) Fast precipitation of nanosized acicular goethite (1-FeOOH) using a high OH/Fe molar ratio (=5);

(2) Instantaneous precipitation of portlandite (Ca(OH) 2 ) by adding CaCl 2 salt to the goethite alkaline suspension, and;

(3) Sub-micrometric calcite precipitation by injection of CO 2 into goethite-portlandite alkaline suspension.

Figure 1 shows a schematic representation of goethite-calcite synthesis, summarising the three sequential precipitation reactions. These three steps are detailed below.

In a previous study [START_REF] Montes-Hernandez | Fast precipitation of acicular goethite from ferric hydroxide gel under moderate temperature (30 and 70 C degrees)[END_REF], it was demonstrated that acicular goethite (1-FeOOH) can be precipitated when a high initial OH/Fe molar ratio (=5) and moderate temperature (30°C and 70°C) are used, without the need to synthesise ferrihydrite and/or amorphous iron hydroxide precursors. The goethite formation described here is characterised by the presence of three successive pH domains, that are specifically associated with (I) the formation of ferric hydroxide gel (FeHgel), leading to acid conditions (pH<2.5); (II) the spontaneous nucleation of goethite from FeHgel, leading to alkaline conditions (pH>11), and (III) the growth of goethite in alkaline conditions (11<pH<13.5). These three steps or pH domains during goethite formation are well correlated with changes in colour [START_REF] Montes-Hernandez | Fast precipitation of acicular goethite from ferric hydroxide gel under moderate temperature (30 and 70 C degrees)[END_REF] and the overall reaction implying the three above steps can be written as:

O H NaCl OH FeO NaOH FeCl 2 3 3 ) ( 3 + + → + (1)
In our system, the goethite grows in alkaline conditions after spontaneous nucleation from ferric hydroxide gel because an excess of sodium hydroxide (NaOH) was initially imposed in the system. The residual NaOH after goethite formation was used to produce portlandite instantaneously by adding the calcium chloride (CaCl 2 ) to the goethite alkaline suspension.

The overall reaction of portlandite precipitation (Ca(OH) 2 ), well known in the laboratory and at industrial scales [START_REF] Domingo | Calcite precipitation by a high-pressure CO 2 carbonation route[END_REF][START_REF] Tomazic | Crystallization of calcium hydroxide from aqueous solution. I. Preliminary study[END_REF], can be expressed as:

NaCl OH Ca CaCl NaOH 2 ) ( 2 2 2 + → + (2)
Calcite precipitation or portlandite carbonation is the final step in the synthesis of goethitecalcite nanocomposites. For this case, CO 2 gas was directly injected into the portlanditegoethite alkaline suspension. CO 2 dissolution in alkaline conditions

( + -+ → + H CO O H CO 2 2 3 2 2
) produces a simultaneous consumption/dissolution of calcium hydroxide (

- + + → OH Ca OH Ca 2 ) ( 2 2
), leading to the precipitation of sub-micrometric calcite until a thermodynamic equilibrium between calcite-solution-CO 2 is obtained [START_REF] Montes-Hernandez | Calcite precipitation from CO 2 -H 2 O-Ca(OH) 2 slurry under high pressure of CO 2[END_REF][START_REF] Montes-Hernandez | Novel Method to estimate the linear growth rate of submicrometric calcite produced in a triphasic gas-liquid-solid system[END_REF].

The overall aqueous carbonation reaction of calcium hydroxide or calcite precipitation can be expressed as:

O H CaCO CO OH Ca 2 3 2 2 ) ( + → + (3)
This neutralisation reaction implies a decrease in pH from 12.2 to 6.5 in the solution which did not promote the dissolution of pre-existing goethite because this mineral is stable in a broad range of pH values (3-13.5) [START_REF] Núñez | Preparation of high acicular and uniform goethite particles by modified-carbonate route[END_REF][START_REF] Schwertmann | Iron oxides in the laboratory: Preparation and Characterization[END_REF][START_REF] Thies-Weesie | Synthesis of goethite as a model colloid for mineral liquid crystals[END_REF].

Characterisation of composite

The XRD measurements confirmed the goethite and calcite mineral composition in the composite precipitated at 30°C and 70°C. For example, the experimental XRD patterns obtained successfully matched with the ICDD cards #005-0586 for calcite and #081-0464 for goethite as shown in Figure 2. These results were also evidenced by TGA. For this case, the 1 st derivative curve, which corresponds to the rate of weight loss, clearly shows two major dehydration episodes, with the desorption of molecular water and the goethite dehydroxylation processes peaking between 55-70°C and between 225-280°C, respectively.

The calcination or de-carbonation of calcite (

2 3 CO CaO CaCO + →
) takes place at higher temperature and two or three decarbonation episodes were identified between 570 and 770°C (Figure 3). Conversely, the pure calcite that was synthesised via reaction (2) and ( 3) was characterised by a single decarbonation episode peaking at a temperature close to 800°C (see Figure 4). This disagreement can be explained by the different size populations of calcite particles in the composite and possibly also by the presence of a small amount of amorphous calcium carbonate with lower thermal stability. Finally, based on TGA, a goethite/calcite weight ratio equal to 0.8 was determined in the composite. Complementary FESEM and TEM observations/measurements revealed nanosized goethite particles well dispersed and/or aggregated with sub-micrometric calcite particles. A small proportion of poorly-crystallized particles (possibly amorphous calcium carbonate) were also observed by TEM. However, no clear evidence of the presence of amorphous calcium carbonate was found. The average particle sizes (length and width for goethite and diagonal dimension for calcite) measured on 50 isolated particles are summarised in Table 1. This mineral mixture forms a homogeneous orange-brown nanocomposite with high specific surface area equal to 92 and 45 m 2 /g for composites synthesized at 30°C and 70°C, respectively (Figure 5). Moreover, both composites synthesised at different temperatures are mesoporous materials as determined from the BJH method. Pore size distribution here was in a short range of pore sizes from 2 to 20 nm (median pore radius = 6nm) for the composite synthesised at 30°C and from 2 to 30nm (median pore radius = 9nm) for the composite synthesised at 70°C.

Toxic ion removal capacity

Additional adsorption experiments in a semi-continuous reactor and ICP-OES measurements revealed that the goethite-calcite composite has a good capacity for removing Cu>Cd>As(III)>Se(IV)>As(V) in the pH range from 6 to 10. Note that only two different pH values were investigated. Conversely, Se(VI) did not present any chemical affinity with the goethite-calcite composite under the physicochemical conditions studied. Pure goethite and calcite were also inefficient adsorbents for removing the Se(VI) from water under the same physico-chemical conditions. Figure 6 summarises the simultaneous removal of three different binary-element systems using the goethite-calcite composite as adsorbent. An adsorption competition and a different chemical affinity for the six toxic ions (selenite, selenate, arsenite, arsenate, cadmium and copper) were clearly observed under the physicochemical conditions investigated. Several studies demonstrated that pH plays a major role in the ion and molecule sorption process because the surface of the adsorbents (e.g. minerals) can be charged positively or negatively as a function of the pH and ionic strength of a given solution [START_REF] Wolthers | The surface chemistry of divalent metal carbonate minerals[END_REF][START_REF] Gaboriaud | Effects of different crystal faces on the surface charge of colloidal goethite (α-FeOOH) particles[END_REF]. In our study, the removal of As(III) was promoted with respect to Se(IV) at alkaline pH (pH=9.4). Conversely, the removal of Se(IV) was promoted with respect to As(III) at slight acidic pH (pH=6) (Figure 7). Finally, it was demonstrated that the ion removal capacity of the goethite-calcite composite was better or equivalent compared to pure goethite (Figure 8), with the main improvement being that the nanocomposite is easier to separate from the fluid because of the aggregation of goethite with calcite particles. This aggregation process is determined by the balance of electrostatic forces, van der Waals forces and steric interactions at the nanoparticle-fluid interfaces. It may be speculated that the aggregation is controlled mainly by electrostatic forces because the surface charges of goethite and calcite change significantly during the three steps of our synthesis route [START_REF] Wolthers | The surface chemistry of divalent metal carbonate minerals[END_REF][START_REF] Gaboriaud | Effects of different crystal faces on the surface charge of colloidal goethite (α-FeOOH) particles[END_REF],

allowing goethite particle aggregation/cohesion on the calcite surfaces (Figure 9). Another major advantage is that the goethite-calcite composite can neutralise acidic wastewater by slight calcite dissolution, thus improving the removal of heavy metals (e.g. Cu and Cd) at the solid-solution interfaces, with a preferential chemical affinity on calcite surfaces, as clearly observed in Fig. 10 for cadmium and copper.

Conclusion

In summary, a simple and novel synthesis route for goethite-calcite nanocomposite is proposed. Remarkably, this nanocomposite offers various advantages as an adsorbent for toxic elements (metalloids and heavy metals) compared to pure goethite. For example, the nanosized goethite particles generally form colloidal suspensions in mixed reactors, and in such case it becomes extremely difficulty to separate the solid from the solution after water treatment by sorption. In our case, the smaller goethite particles adhere to the calcite surfaces, thus allowing a simpler solid-solution separation process (e.g. sedimentation and/or filtration).

Moreover, the goethite-calcite composite can neutralise acidic wastewater by slight calcite dissolution, thereby improving the removal of heavy metals (e.g. Cu and Cd) at the solidsolution interfaces, with a preferential chemical affinity on calcite surfaces. In general, the goethite-calcite nanocomposite was found to have a greater or equivalent capacity for the removal of metalloids and heavy metals compared to the pure goethite.

  measurement of [Ca], [Fe], [Na], [As], [Se], [Cd] and [Cu] by inductively coupled plasma optical emission spectrometry (ICP-OES). In these adsorption experiments only the goethitecalcite composite synthesised at 30°C was used because it has a higher specific surface area (92 m 2 /g).

Figure 3 .

 3 Figure 3. Thermogravimetric analyses for goethite-calcite composites synthesised at 30°C (a)

25

 25

Figure 4 .

 4 Figure 4. First derivative TG curves (DTG) for pure calcite and goethite-calcite composites

Figure 5 .

 5 Figure 5. (a) FESEM image showing nanosized and sub-micrometric particles of goethite and calcite. (b) N 2 experimental sorption isotherms for goethite-calcite composites synthesised at 30°C and 70°C and BET model fitting the experimental data between 0.05 and 0.35 of relative pressure.

Figure 6 .

 6 Figure 6. Simultaneous removal of two toxic ions (i.e. binary-element systems) from

Figure 7 . 29 Figure 8 .

 7298 Figure 7. pH effect on the simultaneous removal of arsenite (As(III) and selenite (Se (IV)

Figure 9 .

 9 Figure 9. FESEM and TEM images showing intimate cohesion/aggregation between goethite

Figure 10 .

 10 Figure 10. Removal of Cd(II) and Cu(II) from solutions by pure goethite and goethite-calcite

Table 1 .

 1 Average size of single goethite and calcite crystals measured on 50 isolated particles from FESEM images and specific surface area of goethite-calcite composites determined from N 2 adsorption isotherms.

			FESEM
	Synthesis	Temperature	Goethite		Calcite	S BET
			Length (nm) Width (nm) Diagonal (nm)
	G-C30C	30°C	240±40	40±8	190±30	92 m 2 /g
	G-C70C	70°C	700±90	75±20	500±40	45 m 2 /g

Acknowledgements

The authors are grateful to the National Research Council (CNRS), France, for providing a financial support for this work.


Khoa Truong
email: truong_khoa.phan@inria.frjoanna.moulierac@inria.fr

Joanna Phan

Moulierac

Ngoc Cuong
email: cuongtran@cse.hcmut.edu.vnnam@cse.hcmut.edu.vn

Nam Tran

Thoai

Xcast6 Treemap Islands -Revisiting Multicast Model

Keywords: C.2.1 [Computer-Communication Networks]: Network Architecture and Design-Network communications Xcast6 Treemap islands, IP Multicast, Overlay Network

Due to the complexity and poor scalability, IP Multicast has not been used on the Internet. Recently, Xcast6 -a complementary protocol of IP Multicast has been proposed. However, the key limitation of Xcast6 is that it only supports small multicast sessions. To overcome this, we propose Xcast6 Treemap islands (X6Ti) -a hybrid model of Overlay Multicast and Xcast6. In summary, X6Ti has many advantages: support large multicast groups, simple and easy to deploy on the Internet, no router configuration, no restriction on the number of groups, no multicast routing protocol and no group management protocol. Based on simulation, we compare X6Ti with IP Multicast and NICE protocols to show the benefits of our new model.

INTRODUCTION

As mentioned in [START_REF] Boivie | Explicit Multicast (Xcast) Concepts and Options[END_REF], IP Multicast encounters a major obstacle in the routing table size in routers when supporting multiple groups at the same time. For this reason, Xcast6 protocol -RFC 5058 [START_REF] Boivie | Explicit Multicast (Xcast) Concepts and Options[END_REF] has been proposed. Because of its simplicity, Xcast6 has been successfully tested in real networks (e.g. Xcast Communication from a Flying Airplane in Japan [7]). Recently, Xcast6 Treemap (X6T) [START_REF] Phan | Treemap -The Fast Routing Convergence Method for Application Layer Multicast[END_REF] has been proposed which works well even without Xcast-aware router on the network. However, there are two disadvantages hindering the deployment of Xcast6 (and X6T): first, the lack of IPv6 world-wide deployment and second, only small multicast sessions can be supported. In this paper, we develop Xcast6 Treemap islands (X6Ti) -a hierarchical architecture Overlay Multicast with the core is X6T. In other words, X6Ti allows interchange (without any configurations) between Overlay Multicast mode (end-hosts duplicate and for-ward data) and X6T mode (Xcast-aware routers duplicate and forward data like IP Multicast). The example in the next section explains how X6Ti works in the two modes.

XCAST6 TREEMAP ISLANDS 2.1 Xcast6 Treemap in an island

We keep the format of the X6Ti packet same as X6T [START_REF] Phan | Treemap -The Fast Routing Convergence Method for Application Layer Multicast[END_REF] but change the forwarding algorithm in X6Ti end-hosts [1]. In summary, each X6Ti packet header includes a list of destination IP addresses, a list of bitmap (bitmap = 1 is to mark the end-hosts which have not received the packet yet) and a treemap (which encodes an overlay tree of end-hosts). It is clear that the packet header contains all necessary information about the multicast group and also the overlay tree, hence the routers do not need to store any additional multicast information. This explains why X6Ti has no restriction on the number of multicast groups. Moreover, following IPv6 standard, X6Ti packets can be handled by both Xcast-aware and normal IPv6 routers on the network. When receiving an X6Ti packet, Xcast router uses its unicast routing table to look up all IP addresses in the list of destinations, then duplicates, changes the bitmap and forwards the packet to appropriate network interfaces (Fig. 1). Therefore, there is no need to develop a new multicast routing protocol. On the other hand, X6Ti packet also set the first unsent IP address in the list of destinations as its destination like a unicast packet. Thus, normal routers can route an X6Ti packet like a normal unicast packet. In addition, when an end-host receives an X6Ti packet, based on the bitmap and the treemap, it knows which end-hosts have not received the packet yet and then forwards the packet as the overlay tree. As shown in Fig. 1, in parts of the network which has Xcast routers (X1 and X4), the overlay tree is not used and packets are duplicated by routers, otherwise, data are multicasted by X6Ti end-hosts based on the treemap.

Xcast6 Treemap islands

Assuming the MTU is 1500 bytes, then the payload length of an X6Ti packet is equal to (1348 -16N ) (N is the number of IP addresses in the packet header) [1]. In the current implementation [1], at most 64 IPv6 addresses can be embedded in each packet header and it is also the maximum number of members in each island (an overlay tree with one branch from the root as shown in Fig. 2). With hierarchical design (2-tiers), many islands are connected using special hosts (X6Ti sub-roots) to form a larger Overlay Network. In fact, this model can be extended to n-tiers in which X6Ti clients can work as X6Ti sub-roots to connect islands. There are several criteria to select an X6Ti sub-root such as outgoing bandwidth, stability of end-host, etc. (more details

SIMULATION AND DISCUSSION

In this section, we compare X6Ti with NICE [START_REF] Banerjee | Scalable Application Layer Multicast[END_REF] and IP Multicast. We use the network topology of France [START_REF] Sndlib | [END_REF] with 25 routers in the backbone. There is a single source and 24 X6Ti sub-roots, each of them connect to a backbone router. Each sub-root manages several tier-2 islands in which X6Ti clients connect to the nearest sub-root. We increase the number of members in each island uniformly so that the total number of hosts in the multicast session is from 200 to 2600. For each test, we select at random 0% (X6Ti-0), 30% (X6Ti-30), 70% (X6Ti-70) and 100% (X6Ti-100) of the routers to be Xcast-aware routers. We collect end-to-end delay (total amount of time a packet is transmitted from the source to a receiver) and link stress (the number of duplicate packets that a physical link has to carry). IP Multicast transmit traffic optimally with low end-toend delay and no traffic redundancy (link stress = 1) (Fig. 3). X6Ti works as pure Overlay Multicast when there is no Xcast-aware router (X6Ti-0 has less link stress than NICE since we choose a good overlay tree as described previously). And obviously, performance of X6Ti is improving when more Xcast-aware routers are deployed on the network. 1 shows some properties in comparision between IP multicast, Overlay Multicast and X6Ti. X6Ti is easy to deploy and no need any router configurations. In addition, with the world-wide IPv6 deployment [4], X6Ti can be deployed at end-hosts first and works like Overlay Multicast. Then, the network operators will deploy Xcast-aware routers when they see it is useful. In fact, X6Ti is even better than Overlay Multicast in term of "Fast route adaptation". When the overlay tree is changed (e.g. hosts join/leave), the X6Ti root or sub-root simply modifies the treemap, there is no need for other hosts to update or store new overlay tree in local like Overlay Multicast. More details and more features on the comparisons can be found in [1].

CONCLUSION

We believe that the X6Ti model overcomes most shortcomings of IP Multicast, Overlay Multicast and Xcast6. In summary, X6Ti solves the problem of difficulty in deployment and state scalability at routers (IP Multicast); multicast group size (Xcast6); traffic redundancy and instability when hosts join/leave (Overlay Multicast). For future work, real applications using X6Ti should be deployed on the Internet to test the feasibility of the protocol.

Figure 1 :Figure 2 : 2 -

 122 Figure 1: Source S sends data to a group of recipients

Figure 3 :

 3 Figure 3: (a) End-to-end delay and (b) Link stress

Table 1 :

 1 A comparison table

		IP multicast	Overlay Multicast	X6Ti
	Efficiency in bw/delay	High	Low-Medium	Medium
	Ease of deployment	Low	High	High
	Fast route adaptation	High	Low	Medium

Table

ACKNOWLEDGMENTS

This work has been partly funded by the ANR JCJC DIMAGREEN.