Benyoub Nassima

Benhamou Abdallah

Debab Abdelkader

PEOPLE'S DEMOCRATIC REPUBLIC OF ALGERIA MINISTRY OF HIGHER EDUCATION AND SCIENTIFIC RESEARCH UNIVERSITY OF SCIENCE AND TECHNOLOGY MOHAMED BOUDIAF -ORAN-Synthesis and modification of a mesoporous material type MCM-41 by an amine for the adsorption of organic pollutants: anionic and cationic dyes

The main objective of this work was to synthesize the MCM-41 material from optimized protocols. In the second step, the pore size and the specific surface area of the parent material was increased by the incorporation of swelling agents (long chain carbon amines N-N, Dimethyl-dodecylamine « DMDDA ») in post-synthesis. Then the selective extraction of the amine and the calcination allowed us to obtain materials with pore sizes and even larger surfaces than those of the starting materials (parent materials). The adsorbents, identified as MCM-41 (P), DMDDA-41 (A), DMDDA-41 (B), DMDDA-41 (C) and MCM-41 (P/C), were characterized regarding their texture, mesoscopic ordering and chemical surface and finally their adsorption capacity was evaluated using two diffrent dyes (anionic: Orange II sodium and cationic: Janus Green B). The results obtained during the adsorption study show the efficiency of these materials, in particular the "amine and calcined" materials for decolorizing aqueous media contaminated with organic dyes. The kinetic studies and the adsorption isotherms were carried out to clarify the method of fixing each of the two dyes on the two materials tested. The experiments showed that the amine material had a maximum capacity for fixing Orange II (221.06 mg / g) (anionic dye), whereas for a maximum adsorption capacity (455.23 mg / g ) Of the Janus GB (cationic dye), the calcined material is more efficient.

1-Introduction :

Since the 1990s, the chemistry of inorganic nanostructured materials has developed considerably with the introduction of soft sol-gel chemistry. Thus, the scientists of the Mobil company proposed the first syntheses of mesostructured silicates, ie materials with an organized porous system made up of mesopores [START_REF] Mccusker | Micoporous and Mesoporous Materials[END_REF][START_REF] Lebiau | Adsorption, Surface Area and Porosity[END_REF]. Since then, many research groups have patented new families of materials with different pore structures, pore sizes and synthesis methods: several methods are needed in the manufacturing process of new porous materials organized "MPO". At present, a new family of ordered mesoporous solids is very widely studied by many researchers from different horizons for various applications including adsorption [3,[START_REF] Benhamou | Aminofunctionalized MCM-41 and MCM-48 for the removal of chromate and arsenate[END_REF] and catalysis. In the field of adsorption, mesoporous materials such as MCM-41, HMS, SBA-15, SBA-1 have been functionalized by various groups for the adsorption of metal ions and various organic pollutants (synthetic dyes) [START_REF] Barclay | Waste minimization guide for the textile industry, a step towards cleaner production[END_REF]. The main objective of this work is to synthesize a mesoporous material of type MCM-41 (hexagonal), to modify it by: addition of amine, selective extraction with ethanol and calcination and then d To evaluate the adsorption capacity of these mesoporous materials with respect to two dyes, anionic sodium orange II and Janus Green B with a cationic character. A detailed study is presented defining the different methods of characterization, as well as a kinetic study and that of adsorption isotherms will also be discussed.L'objectif principal de ce

2-Synthesis and Characterization

2.1.

Materials : The principal reagents used are summarized in a table which follows, where their origin, formula and possibly their purity are specified (Table 1). In all cases, the solvent employed is distilled water and, the syntheses are carried out at ambient temperature, in an open container. All are made with stirring.

2.2.S ynthesis:

In a beaker, the water / TMAOH mixture is stirred vigorously for 5 minutes, then the CTAB is added in small quantities without stopping the stirring which will continue for 30 minutes, and finally the SiO.sub.2 is added in order to have A gel after about 2 hours (the speed of agitation is less strong). Si-MCM-41, called material P, is added to an emulsion consisting of water and N-N-Dimethyl-dodecylamine (DMDDA) previously stirred for 5 minutes. This mixture is left under stirring for 30 minutes, then transferred to a Teflon reactor, which is placed in an oven set at 120°C. for 3 days. The material obtained is filtered and washed several times with distilled water and then dried at ambient temperature: it is the « Si-MCM-41/A » amine material. The selective extraction of the amine is carried out using a specific solvent which is the ethanol, using a soxhlet. The material obtained is the « Si-MCM-41/B » deaminated material. The material Si-MCM-41 / P is calcined in 550°C during 7 hours giving rise to a material called « Si-MCM-41-P / C » parent / calcined material. This step is carried out at 550°C. for 7 hours with a heating rate of 1°C./min, giving rise to a material called « Si-MCM-41/C » calcined material.

2.3.Characterization techniques:

This characterization aims in a very thorough way, the study and the identification of the textural properties, structural and superficial, as well as the electrochemical properties of the surface of the obtained materials.

A combination of physicochemical (X-ray diffraction, BET, ATG, ATD and Zetrametry) and spectroscopic (FTIR) techniques is realized and their understandings could allow a good exploitation of these materials for specific applications.

3-Results & discussion :

The X-ray diffraction at low angles is used to demonstrate the arrangement of the channels created by the micelles of surfactants. Figure 1 shows the diffractograms measured between 0.5 and 6 degrees (2θ) of the DMDDA-41A, DMDDA-41B and DMDDA-41C materials, respectively the DMDDA amine material, the previous material after deamination, and Calcined material. These four materials are compared to the MCM-41/P which is the parent material. The main peak is observed at 2θ = 2.28° followed by three less intense peaks between 3.2 <2θ <5.4°. This figure can be indexed in a crystallographic system and it is possible to attribute the peaks to diffractions on reticular planes indexed by Miller indices "hkl": the planes (100), (110), (200) and 210) as shown in the figure .  The diffractograms of the materials (amine, deaminated and calcined) have the same peaks as those observed on the purely silicic material MCM-41/P, which shows that the structure of the material is preserved after modifications made more than they all present an arrangement 2D hexagonal (p6mm) of the pores (Figure 2). A peak slippage (MCM-41P and DMDDA-41B) is observed at higher diffraction angles [START_REF] Showkat | Analysis of Heavy Metal Toxic Ions by Adsorption onto Amino-functionalized ordered Mesporous Silica[END_REF][START_REF] Abry | Surface molecular engineering applied to the design of heterogeneous bio-inspired catalysts[END_REF][START_REF] Lecren | Organizations of molecular magnets in inorganic networks and coordination: Synthesis, structure and properties[END_REF], which results in smaller inter-reticular distances as shown in the table.

Table 2 summarizes the structural data derived from the diffractograms for all materials prepared from MCM-41. For the MCM-41P, an increasing slope is observed for low relative pressures 0 <(P / P0) <0.3 corresponding to a monolayer filling of the surface [START_REF] Abry | Surface molecular engineering applied to the design of heterogeneous bio-inspired catalysts[END_REF], followed by a steep slope of the curve for the Relative pressures between 0.3 <(P / P0) <0.8; Due to the capillary condensation of the nitrogen inside the mesopores. And towards (P / P0)> 0.8 there is a multilayer adsorption at the surface of the parent material.

For the DMDDA-41A material, at relative pressures P / P0 <0.4, a monolayer filling of the surface is obtained and for P / P0> 0.5 there appears a plateau of a multilayer adsorption. The same observations noted for the parent material are noted for DMDDA-41B and DMDDA-41C. For both materials, the relative low pressure plateau is almost the same (P / P0 <0.5), but the distinction is in the slopes of the following plateau: steep slope to the calcined material (up to 1 in P / P0) in contrast to the deaminated material (0.5 <P / P0 <0.8).

In the case of these two materials, the desorption does not follow the adsorption creating an H1 type hysteresis which abruptly closes at P / P0 = 0.5 for the material (DMDDA-41C) and at P / P0 = 0.4 For the DMDDA-41B material, this suggests a phenomenon of capillary condensation in the mesopores. The width of the hysteresis increases (DMDDA-41C) indicating the pore size distribution is much wider in the calcined material (Figure 4) [START_REF] Lecren | Organizations of molecular magnets in inorganic networks and coordination: Synthesis, structure and properties[END_REF][START_REF] Badsi | Synthesis and modification of a mesoporous material type MCM-41 by a series of amines for the adsorption of anionic dyes[END_REF][START_REF] Bahloul | Synthesis and modification of a mesoporous material type MCM-41 by N-N dimethyl-n-octylamine: Application to the adsorption of 2,4,6 trichlorophenol[END_REF]. The pore diameter distributions shown in figure 4 show that the materials prepared have different pore diameters. The traces are all centered indicating a better uniformity of the pore diameter. As the pore diameter distribution widens, this indicates the formation of a less homogeneous network of pores.

The analysis of these data makes it possible to observe great differences in the parameters of the four materials. The results of surface area, pore diameter and pore volume decrease remarkably for the amine material (the surface and part of the pore volume are occupied by the amines), in contrast to MCM-41P.

Afterwards, deamination increases these parameters again, while calcination leads to higher values. The IRTF spectra of the MCM-41 materials prepared are shown in figure 5 where it is found that some vibration bands are present in the materials: they are similar to that of amorphous silica. In addition to these bands, the amine material (DMDDA-41A) records two new bands characteristic of the amine groups, the first located at 2900 cm-1 characterizes the vibrations (CN), while the second one is observed at 1480 cm -1 Resulting from the deformation vibration of the bonds (NH). [START_REF] Jacoby | Lipase Mucor miehei immobilized in silicated porous materials: bioreactors for the synthesis of methyl esters from vegetable oils[END_REF][START_REF] Calmettes | Design of a catalyst by grafting a complex of ruthenium in a confined environment; Application to selective oxidation[END_REF][START_REF] Charmot | New strategies for the insertion of iron oxide particles in organized mesostructures, characterization and analytical activity[END_REF]] And for the calcined material, it has positive potentials between 2 <pH <3.6 and negative potentials from pH = 3.6 [START_REF] Sing | IUPAC reporting physisorption data for GAS /SOLID systems; with Special Reference to the Determination of Surface Area and Porosity[END_REF][START_REF] Calmettes | Design of a catalyst by grafting a complex of ruthenium in a confined environment; Application to selective oxidation[END_REF][START_REF] Badsi | Synthesis and modification of a mesoporous material type MCM-41 by a series of amines for the adsorption of anionic dyes[END_REF][START_REF] Bahloul | Synthesis and modification of a mesoporous material type MCM-41 by N-N dimethyl-n-octylamine: Application to the adsorption of 2,4,6 trichlorophenol[END_REF].

4-Application to adsorption

Our study consists in developing techniques for low -cost water discolouration while recovering industrial waste: Discoloration of synthetic solutions: Orange 2 sodium with anionic character [START_REF] Onelia | Adsorption of Synthetic Orange Dye Wastewater in Organoclay[END_REF] and Janus Green B with a cationic character. Their characteristic data are summarized in the following table. First, a study of their UV-visible spectra at wavelengths between 200 and 800 nm was carried out to determine the wavelengths which correspond to the maximum adsorption and to establish the calibration curves. Thus the 483 nm, 618 nm waves were respectively for the Orange II Sodium Salt and the Janus Green B. We then studied the influence of different physicochemical parameters on the coloring interactions -synthesized materials (agitation time, Initial dye concentration, adsorption capacity test).

It is noted that in each experiment, the tests were carried out in flasks covered with aluminum foil, in order to avoid degradation of the pollutant in the presence of light.

The study of the adsorption of the two dyes on the different materials is represented in the histograms of the quantities adsorbed by the various materials synthesized (figure 8)

We note that the two materials "parent and amine" possess the greatest adsorption capacity of the anionic dye "Orange II". On the other hand, the two materials, namely the calcined parent and the calcined modifier, possess the highest adsorption rates of the cationic dye "Janus GB". The DMDDA-41 B exhibits in both dyes an average adsorption This difference in the adsorption rates of the different materials is due to two reasons:

1. Differences in the physico-chemical properties of materials (porosity, S BET , ...) 2. Differences in surface loads of different materials. The parent material has only the silanol (Si-OH) groups and quaternary ammonium groups of the surfactant, which explains its ability to fix the anionic dye more than the cationic. Moreover, the amine material has, in addition to the preceding groups [START_REF] Bouras | Adsorption properties of organophilic bridged clays: Synthesis and characterization[END_REF], amine groups (NH.sub.2): the charge of their surface is more positive at the maximum capacity for fixing the Orange II.

For calcined materials (DMDDA-41 P / C and DMDDA-41 C), they have only the silanol groups (Si-OH) which make the charge of their surface more negative, which makes their adsorption capacity of the Janus GB maximum. The adsorption kinetics tests are carried out on the amine and calcined matrix, since they possess, respectively, the greatest adsorption capacity of the anionic and cationic micropollutant. The results are presented by the evolution of the adsorbed quantities of the two dyes by the two materials (Qe) as a function of the contact time (t) in FIG. 9 (A andB).

In the two figures (A and B), two distinct phases of adsorption are observed: the first between 0 and 40 minutes which is rapid and takes place at the sites accessible on the surface, followed by a slow evolution of the adsorbed amount before to reach the second phase, which is in the form of an equilibrium plateau, due to the diffusion inside the pores of the material. The balance is reached after 60 minutes.

A Fiigures 10 and 11 represent the linearization of the two kinetic models for the two organic pollutants and Table 8 groups together the values of the parameters of the two kinetic models applied to their elimination. According to the values of the kinetic constants, the kinetics of the dyes follows the pseudo second order model (R 2 = 0.999), rather than that of the pseudo-first.

B Table 8: Parameters of the two kinetic models applied to the removal of the two organic pollutants "Janus Green B & Orange II Sodium"

The plots of the curves: Qt = f (t1 / 2), of the intraparticle diffusion (FIG. 12), are not linear and we notice three distinct sections associated with three successive steps:

• A first linear portion of the first step corresponds to a rapid diffusion in the boundary layer of the molecules of the solute, the adsorbate migrating from the solution to the external surface of the adsorbent.

• The second section, the curvature, is attributed to the intraparticle diffusion, which determines the rate of control of the adsorption mechanism.

• The plateau corresponds to a state of equilibrium: the intra-particle diffusion slows down, leading to a maximum of adsorption and a very low concentration of adsorbate in the solute [START_REF] Hickey | Pharmaco Complexity[END_REF] Using the classification of adsorption isotherms, those obtained are of type L "Langmuir". This type is characterized by a decreasing slope as the equilibrium concentration increases, due to the decrease in the number of adsorption sites, following the gradual covering of the surface of the various materials. In this type of adsorption, there is no interaction between the adsorbed molecules: this is a physisorption [START_REF] Hickey | Pharmaco-Complexity[END_REF]. The representation of the adsorption isotherms is markedly better by the Sips model in the two dyes (R 2 > 0.994) and especially by the Langmuir model in the fixation of the cationic pollutant Janus Green B (R 2 > 0.995).

The study of the linearizations of the Langmuir, Freundlich and Sips models (Langmuir-Freundlich), defined above, gives the various parameters of the three models that we have gathered in Table 10. The essential characteristic of the Langmuir isotherm can be expressed as a function of a dimensionless constant separation factor which is given by the following expression:

R L = [1 / (1+K L * C 0 )]
The evolution of the separation factors as a function of the initial concentration at 25 ° C is shown in figure 14. For both materials and the two dyes, the values of this factor are between 0 <RL <1 : This shows that the adsorption of the two dyes by the different materials is a favorable process, and the 1 / n values indicate a favorable adsorption especially for the low concentrations [START_REF] Achour | Possibility of retention of humic substances by adsorption on bentonite[END_REF][START_REF] Bouras | Adsorption properties of organophilic bridged clays: Synthesis and characterization[END_REF].

Conclusion :

The main objective of this work was to synthesize the MCM-41 material from optimized protocols. It was then possible to increase the size of the pores and the specific surface area of the parent material by the post-synthesis incorporation of blowing agents (long chain carbon amines DMDDA).

The results of X-ray diffraction characterization of the different matrices show that for the five types of morphology the diffractogram obtained has three indexable peaks in (100), ( 110) and (200) in hexagonal symmetry (space group ) . For amine and calcined materials, the presence of indexable peaks in hexagonal network always shows that the structure is always preserved after the postsynthesis addition of the amine emulsion or by a high treatment temperature.

The BET analysis of the materials shows a remarkable decrease in pore diameter and the specific surface area for each of the amines and deaminates, unlike the MCM-41C which has the highest surface area and pore diameter.

The thermogravimetric analysis shows a thermal stability of the materials with a greater loss of mass for the amine material, which again confirms the incorporation of the amines inside the pores. The results of the infrared spectra show the incorporation of the NH 2 groups and for those of the zeta-metry show that the surface charge differs from one material to another, which suggests their applications in the adsorption domain of the various micropollutants .

The results obtained during the adsorption study show the effectiveness of these materials, in particular "the amine material" and "the calcined material" for decolorizing aqueous media contaminated with organic dyes. The kinetic studies and the adsorption isotherms were carried out to clarify the method of fixing each of the two dyes on the two materials tested. The experiments showed that the amine material had a maximum capacity for fixing Orange II (221.06 mg / g) (anionic dye), whereas for a maximum adsorption capacity (455.23 mg / g ) of Janus GB (cationic dye), the calcined material is more efficient.

The second order kinetic model applies well in the case of the adsorbent / adsorbate systems studied (R 2 = 0.999). The adsorption isotherms of the two dyes by the two adsorbents are satisfactorily described by the Langmuir model in the case of fixation of the cationic pollutant (R 2 = 0.995). However, for the adsorption of the anionic pollutant, the Sips model is more favorable (R 2 = 0.994).

The RL separation factor values for the Langmuir model of the two dyes is between (0 <RL <1): this shows that the adsorption in this case is a favorable process.

The adsorption of the two micropollutants (Janus Green B and Orange II Sodium) is favored at low concentrations (1 / n <1).

Figure 1 :Figure 2 :

 12 Figure 1: Si-MCM-41 diffraction pattern (parent, amine, deaminated, calcined)

Figure 3 :

 3 Figure 3: Isotherm of adsorption / desorption of Si-MCM-41

Figure 4 :

 4 Figure 4 : Distribution of pore diameters of materials MCM-41

  The broad band between 3450 cm -1 , characteristic of elongation of the bond (O-H), water and silanol groups of the surface. Another vibrating band (O-H) noted around 1650 cm -1 testifying to the water present in the materials. At 1100 cm -1 , an asymmetric (O-Si-O) elongation junction of the SiO 4 tetrahedral entities was observed, whereas about 950 and 750 cm -1 , two bends characteristic of the asymmetric elongation of the (Si-O) bond of these same entities present. We note also the presence of a band characterizing the deformation of the angle (O-Si-O) of the tetrahedral entities SiO 4 , this towards 450cm -1 .[START_REF] Jacoby | Lipase Mucor miehei immobilized in silicated porous materials: bioreactors for the synthesis of methyl esters from vegetable oils[END_REF][START_REF] Sing | IUPAC reporting physisorption data for GAS /SOLID systems; with Special Reference to the Determination of Surface Area and Porosity[END_REF][START_REF] Toumi | Synthesis of Micros-Mesoporous Composite Materials from ZSM-5 Zeolite[END_REF]

Figure 5 :

 5 Figure 5: IRTF Spectrum of Si-MCM-41

Figure 6 :

 6 Figure 6 : Thermogravimetric analysis A-(TGA) & Differential thermal analysis B-(DTA) of the different materials

Figure 7

 7 Figure 7 shows the curves ζ = f (pH) of the various MCM-41 materials in the pH range from 2 to 11. Each point ζ = f (pH) is the mean of the results obtained on the measurements of the electrophoretic mobility of 100 to 500 p at a zeta potential of between 5 and 8 mV. The pH for which the zeta potential is zero (no displacement of the particles under the effect of the electric field) is called the isoelectric point (PIE). [19, 20] Parent material MCM-41 has positive potentials between 2 <pH <5 and negative potentials from pH = 5.5. For the DMDDA-41 A, it has positive potentials between 2 <pH <8.5 and negative potentials from pH = 8.5. Then, DMDDA-41 B has positive potentials between 2 <pH <6 and negative potentials from pH = 6.1.

Figure 8 :

 8 Figure 8 : The retention quantities of the two dyes by the various materials MCM-41

Figure 9 :

 9 Figure 9 : Kinetics of elimination of Janus Green B by DMDDA-41C (A) and Orange II Sodium by DMDDA-41A (B)

Figure 10 :Figure 11 :

 1011 Figure 10 : Application of the pseudo-first order (A) and pseudo-second order (B) model to the elimination of Orange II Sodium by DMDDA-41A

Figure 12 :

 12 Figure 12: Intraparticular diffusion of Orange II Sodium (A) and Janus Green B (B) Table 9: Kinetic parameters of the intraparticular diffusion of the two dyes "JanusGreen B & Orange II Sodium"

Figure 13 :

 13 Figure 13: Adsorption isotherms of Orange II Sodium by DMDDA-41A (A) and Janus Green B by DMDDA-41C (B)

Table 1 : Main reagents used

 1

	Reagents	Formula	Origin
	Cetyltrimethylammonium bromide 99% (CTAB) C 19 H 42 Br	Biochem
	Smoked silica 98 %	SiO 2	Alfa Aesar
	Tetramethylammonium hydroxide (TMAOH)	(CH 3 ) 4 N-OH	Biochem
	N-N, Dimethyl-dodecylamine (DMDDA)	CH 3 (CH 2 ) 11 N(CH 3 ) 2 Alfa Aesar
	Ethanol 96%	C₂H₅OH	Biochem
	Janus Green B (Phenazinium)	C 30 H 31 Cl 3 N 6	Alfa Aesar
	Orange II Sodium Salt	C 16 H 12 N 2 O 4 S-Na	Sigma-Aldrich

Table 2 : X-ray diffraction data of MCM-41P and modified materials.

 2

	Material	2	d 100 (nm)	a 0 (nm)
	MCM-41 P	2,28	3,87	4,47
	DMDDA-41A	1,97	4,39	5,07
	DMDDA-41B	2,10	4,21	4,86
	DMDDA-41C	2,02	4,48	5,17

The adsorption-desorption isotherms of the parent MCM-41 and the prepared materials are shown in figure

3

. These are type IV according to the IUPAC classification and Hysteresis H1 which is representative of structured mesopores.

[START_REF] Kresge | Ordered mesoporous molecular sieves synthesized by a liquid-crystal template mechanism[END_REF][START_REF] Dada | Langmuir, Freundlich, Temkin and Dubinin-Radushkevich Isotherms Studies of Equilibrium Sorption of Zn2+ Unto Phosphoric Acid Modified Rice Husk[END_REF][START_REF] Lecren | Organizations of molecular magnets in inorganic networks and coordination: Synthesis, structure and properties[END_REF][START_REF] Brunauer | Adsorption of Gases in Multimolecular Layers[END_REF]

Table 3 : Comparison of nitrogen adsorption- desorption data at 77 ° K of MCM-41P and modified materials.

 3

	Material	S BET (m 2 /g)	d P (nm)	V p (cm 3 /g)
	MCM-41 P	1147	3,22	0,85
	DMDDA-41A	74	-	0,28
	DMDDA-41B	387	5,84	1,07
	DMDDA-41C	1246	10,6	1,86

Table 4 : Thermogravimetric (TGA) and Thermal Differential (ATD) data for the different materials Sample Mass loss (%) 1 st Zone 2 nd Zone 3 rd Zone

 4

	MCM-41 P	8,23	48,41	1,43
	DMDDA -41 A	3,51	70,67	3,59
	DMDDA -41 B 15,67	45,97	1,63
	DMDDA -41 C	7,02	2,76	0,87

Table 5 : Values of the isoelectric points of the different materials.

 5

	Material	PIE (isoelectric point)
	MCM-41 P	5,6
	DMDDA -41 A	8,7
	DMDDA -41 B	5,8
	DMDDA -41 C	3,6

Table 6 : Physico-chimical characteristics of the two dyes

 6

		Janus Green B	Orange II Sodium Salt
	IUPAC	3-
	name

Diethylamino-7-(4- dimethylaminophenylazo)-5- phenylphenazinium chloride 4-(2-Hydroxy-1- naphthylazo)benzenesulfonic acid sodium salt Formula C 30 H 31 CLN 6 C 16 H 12 N 2 O 4 S-Na Developed formula Solubility Soluble Soluble Stability Stable in NTPs Stable in NTPs Appearance Blue-green powder Orange powder Molecular weight 511.06 g/mol 350.32 g/mol Absorbance ʎ max 1 = 395 nm, ʎ max 2 = 660 nm ʎ max = 483 nm

Table 7 : Retention quantities of Orange II Sodium and Janus Green B by the different materials

 7

	Material
	MCM-41	P	P/C	A	B	C
	Q max (mg/g)
	« Orange II	44.34 4.96 44.80 33.98 4.28
	Sodium »
	Q max (mg/g) « Janus GB »	7.

05 49.80 19.13 35.70 49.90

Table 9 : Kinetic parameters of the intraparticular diffusion of the two dyes "Janus Green B & Orange II Sodium"

 9

			Intraparticular diffusion
	Material	Q e (exp) (mg/g)	(mg/g.min 1/2 ) K id	(mg/g) I	R 2
	DMDDA-	44.24	2.009	22.044 0.740
	41A
	DMDDA-	182.64	6.748	108.741 0.605
	41C
	The adsorption isotherms of the anionic
	and cationic dye at 25 ° C. by the materials
	DMDDA-41A and DMDDA-41C are
	shown in figure 13.

Table 10 : Parameters of the adsorption isotherms applied to the removal of the two organic pollutants "Janus Green B & Orange II Sodium"

 10


Clémentine Courtès

Convergence for PDEs with an arbitrary odd order spatial derivative term

We compute the rate of convergence of forward, backward and central finite difference θ-schemes for linear PDEs with an arbitrary odd order spatial derivative term. We prove convergence of the first or second order for smooth and less smooth initial data.

Introduction

We study in this paper linear partial differential equations with an arbitrary odd order spatial derivative term, which read

∂tu + ∂ 2p+1 x u = 0, (1)
with p ∈ N. The particular case p = 0 corresponds to the advection equation with a unit constant speed ∂tu + ∂xu = 0 and describes the passive advection of scalar field carried at constant speed. The case p = 1 leads to Airy equation ∂tu + ∂ 3 x u = 0 that models the propagation of long waves in shallow water [START_REF] Witham | Linear and Nonlinear Waves[END_REF] and derives from a linearization of the Korteweg-de Vries equation [START_REF] Craig | Linear dispersive equations of Airy type[END_REF]. We especially focus on the initial value problem where (1) is considered with the initial condition u | t=0 = u0. We deal with the numerical approach of this Cauchy problem and study the convergence of several finite difference schemes. Our concern here is to find a rate of convergence without assuming the smoothness of the initial data. For this purpose, we use the finite difference method to discretize [START_REF] Courtès | Étude de l'équation d'Airy sous contrainte[END_REF] in R × [0, T ]. We choose to deal with a uniform time and space discretization. Let ∆t > 0 and ∆x > 0 be the time and space steps, we note t n = n∆t for all n ∈ {0, ..., N } where N = T ∆t and xj = j∆x for all j ∈ Z. We denote by v n j (j,n) the discrete unknowns defined by

     v n+1 j + θ∆t D 2p+1 • v n+1 j = v n j -(1 -θ)∆t D 2p+1 • v n j , ∀(j, n) ∈ Z × {0, .., N }, v 0 j = 1 ∆x
x j+1

x j u0(y)dy, ∀j ∈ Z,

with

D 2p+1 • v n j = D 2p+1 + v n j = 2p+1 k=0 2p+1 k (-1) k ∆x 2p+1 v n p-k+j+1 (forward scheme), ( 3a
)
or D 2p+1 • v n j = D 2p+1 - v n j = 2p+1 k=0 2p+1 k (-1) k ∆x 2p+1 v n p-k+j (backward scheme), ( 3b
)
or D 2p+1 • v n j = D 2p+1 c v n j = 1 2 D 2p+1 + v + D 2p+1 - v n j (central scheme). ( 3c
)
The parameter θ belongs to [0, 1] and we recover the explicit scheme for θ = 0 and the implicit scheme for θ = 1. Notations 1. We denote by H s (R) (with s > 0) the Sobolev space defined with the norm ||u||

H s (R) = R 1 + |ξ| 2 s | u (ξ)| 2 dξ 1 2
, where u is the Fourier transform of u. Moreover, we use the standard ∞ 0, N ;

2 ∆ (Z) space whose norm is ||v|| ∞ (0,N; 2 ∆ (Z)) = sup n∈{0,..,N } j∈Z ∆x|v n j | 2 . Lastly, we note A B when A ≤ CB
where C is a constant independent of ∆x and ∆t.

1 Laboratoire de Mathématiques d'Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France.

2 Order of accuracy for an initial datum in H 4p+2 (R)

We hereafter find some condition on θ, ∆t and ∆x for the schemes to be consistent and stable, to conclude the convergence study according to the Lax-Richtmyer theorem [START_REF] Lax | Survey of the stability of linear difference equations[END_REF].

Consistency estimate

In Sect. 2, we suppose the initial datum regular enough to compute all the needed derivatives and the Taylor expansions up to the desired order. Indeed, supposing u0 regular is sufficient to ensure the same regularity for u(t, .) for all t ∈ [0, T ] because of the following result.

Remark 1. Let u be a solution of (1), then by linearity of the equation all the derivatives of u verify (1) too and by Fourier transform, the L 2 -norm of all its derivatives are conserved :

||∂ k x u(t, .)|| L 2 (R) = ||∂ k x u0|| L 2 (R) , for all k ∈ N. Thus, u0 ∈ H 4p+2 (R) implies u(t, .) ∈ H 4p+2 (R), ∀t ∈ [0, T ]. Definition 1. For all (j, n) ∈ Z × {0, .., N }, we note (u∆) n j = 1 ∆x x j+1
x j u(t n , y)dy with u the exact solution of the Cauchy-problem (1) from u0. For all (j, n) ∈ Z × {0, .., N }, the consistency error is defined as

n j = (u∆) n+1 j -(u∆) n j ∆t + θ D 2p+1 • u∆ n+1 j + (1 -θ) D 2p+1 • u∆ n j , with D 2p+1 • defined by (3a)-(3c). Proposition 1. Assume u0 ∈ H 4p+2 (R) (and u0 ∈ H 6p+3 (R) if θ = 1
2 ) then, for the forward or backward finite difference schemes (3a) and (3b), the following consistency inequality holds

|| || ∞ (0,N; 2 ∆ (Z)) ∆t 1 2 -θ ||∂ 4p+2 x u0|| L 2 (R) + ∆x||∂ 2p+2 x u0|| L 2 (R) + ∆t 2 ∂ 6p+3 x u0 L 2 (R) .
For the central finite difference scheme (3c), the consistency inequality is as follows

|| || ∞ (0,N; 2 ∆ (Z)) ∆t 1 2 -θ ||∂ 4p+2 x u0|| L 2 (R) + ∆x 2 ||∂ 2p+3 x u0|| L 2 (R) + ∆t 2 ∂ 6p+3 x u0 L 2 (R) .
Before proving the previous result, we state a useful lemma.

Lemma 1. For all and p in N, there exists ξ ∈] -p, p + 1[ such that

2p+1 k=0 2p + 1 k (-1) k (p -k + 1) =          0 if < 2p + 1, ! if = 2p + 1, ! ( -2p)! ξ -2p-1 if > 2p + 1.
Proof of Lemma 1. Let (xj-p, ..., xj+p+1) be 2p + 2 points regularly spaced of h, we recall the divided difference of order 2p + 2 of a smooth function f :

(2p + 1)!f [xj-p, ..., xj+p+1] = 2p+1 k=0 2p+1 k (-1) k f (x p-k+j+1 ) h 2p+1 . (4)
Moreover, we recall the existence of ξ ∈]min(xj-p, ..., xj+p+1), max(xj-p, ..., xj+p+1)[ such as

(2p + 1)!f [xj-p, ..., xj+p+1] = f (2p+1) (ξ).
For more details, please refer to [START_REF] Demailly | Analyse numérique et équations différentielles[END_REF]. Lemma 1 is a consequence of the two previous equations with f : y → y , h = 1, j = 0 and xi = i for i ∈ Z.

Proof of Proposition 1. For u0 ∈ H 4p+2 (R) and for the forward finite difference scheme, one has

D 2p+1 • u∆ n j = D 2p+1 + u∆ n j = 2p+1 k=0 2p+1 k (-1) k ∆x 2p+1 1 ∆x x j+1 x j u(t n , y + (p -k + 1)∆x)dy .
Using a Taylor expansion (in space) up to order 2p + 2 and exchanging the two sums inside leads to

D 2p+1 + u∆ n j = 1 ∆x x j+1 x j 2p+1 =0 ∂ x u(t n , y) ! 2p+1 k=0 2p+1 k (-1) k (p -k + 1) ∆x ∆x 2p+1 dy + (R+) n j , (5)
where

(R+) n j = 1 ∆x x j+1 x j y+(p-k+1)∆x y ∂ 2p+2 x u(t n , z) (2p + 1)! 2p+1 k=0 2p+1 k (-1) k (y + (p -k + 1)∆x -z) 2p+1 ∆x 2p+1 dzdy.
For simplicity, we will only use

||R n + || 2 ∆ ∆x||∂ 2p+2
x u(t n , .)|| L 2 (R) . Equation ( 5) is simplified thanks to Lemma 1. Eventually, we obtain

D 2p+1 + u∆ n j = 1 ∆x x j+1 x j ∂ 2p+1
x u(t n , y)dy + (R+) n j . Similarly, by adapting the previous computation, one has

D 2p+1 + u∆ n+1 j = 1 ∆x x j+1 x j ∂ 2p+1
x u(t n , y)dy +

x j+1

x j ∆t∂t∂ 2p+1

x u(t n , y)dy

+ x j+1 x j t n+1 t n ∂ 2 t ∂ 2p+1 x u(s, y)(t n+1 -s)dsdy + (R+) n+1 j .
In order to compute the difference

(u ∆ ) n+1 j -(u ∆ ) n j ∆t
that appears in Definition 1, we perform a Taylor expansion (in time) up to order 3. Gathering all those results together yields

n j = ∆t 2∆x x j+1 x j ∂ 2 t u(t n , x)dx + 1 ∆t∆x x j+1 x j t n+1 t n ∂ 3 t u(s, y) (t n+1 -s) 2 2 dsdy + θ∆t ∆x x j+1 x j ∂t∂ 2p+1 x u(t n , y)dy + θ ∆x x j+1 x j t n+1 t n ∂ 2 t ∂ 2p+1 x u(s, y)(t n+1 -s)dsdy +θ (R+) n+1 j + (1 -θ) (R+) n j .
The conclusion comes from the relation ∂tu(t, x) = -∂ 2p+1

x u(t, x), the Cauchy-Schwarz inequality and the conservation of the L 2 -norm (cf. Remark 1).

Remark 2. The regularity

H 4p+2 (R) (or H 6p+3 (R) if θ = 1
2 ) comes from the Taylor expansion in time and is essential in this proof. Remark 3. We follow exactly the same guidelines for the backward finite difference scheme. For the central finite difference scheme, we need to perform a Taylor expansion in space up to order 2p + 3 to obtain

D 2p+1 c u∆ n j = 1 ∆x x j+1 x j ∂ 2p+1 x u(t n , y)dy + (Rc) n j , with ||R n c || 2 ∆ ∆x 2 ||∂ 2p+3 x u0|| L 2 (R) .

Stability

We note, for all v n j j∈Z and ξ

∈ [0, 1], V n (ξ) = k∈Z v n k e 2iπkξ in L 2 ([0, 1]
) with the equivalence of the norms

: j∈Z ∆x|v n j | 2 = ∆x 1 0 V n (ξ) 2 dξ. Eventually, we define the shift operator S by S v n = (v n j+ ) j∈Z thus, S V n = e -2iπ ξ V n . Definition 2. A scheme is said to be stable in 2 ∆ (Z), if there exists a constant C independent of ∆t and ∆x such that, for v n j (j,n) verifying Equation (2), v n+1 2 ∆ (Z) ≤ (1 + C∆t) ||v n || 2 ∆ (Z) , ∀n ∈ {0, ..., N }.
Proposition 2. For small ∆t and ∆x, the stability under the Courant-Friedrich-Lewy condition (in short CFL cond.) is explained in Table 1.

The following computation will simplify the proof of Proposition 2.

Lemma 2. One has, for all ξ ∈ [0, 1],

2p+1 k=0 2p+1 k (-1) k e -2iπ(p-k+1)ξ = e -iπξ (-2isin(πξ)) 2p+1 .
p even p odd p = 0 p = 0 (Advection) Forward stable under the CFL cond.

unconditionally unstable stable under the CFL cond. scheme

∆t(1 -2θ) ≤ -∆x ∆t(1 -2θ) ≤ ∆x 2p+1 2 2p
Backward stable under the CFL cond.

stable under the CFL cond.

unconditionally unstable scheme

∆t(1 -2θ) ≤ ∆x ∆t(1 -2θ) ≤ ∆x 2p+1 2 2p
Central stable under the CFL cond.

stable under the CFL cond.

stable under the CFL cond. scheme

∆t(1 -2θ) ≤ 2C∆x 2 ∆t(1-2θ) ≤ 2C ∆x 4p+2 2 4p ∆t(1-2θ) ≤ 2C ∆x 4p+2 2 4p
Table 1: Stability results for finite difference θ-schemes

Proof of Lemma 2. A proof may be found in Lemma 1.1 of [START_REF] Courtès | Étude de l'équation d'Airy sous contrainte[END_REF].

Proof of Proposition 2. The forward finite difference scheme (3a) leads to

U n+1 (ξ) 1 + θ∆t (∆x) 2p+1 2p+1 k=0 2p + 1 k (-1) k e -2iπ(p-k+1)ξ = U n (ξ) 1 - (1 -θ)∆t (∆x) 2p+1 2p+1 k=0 2p + 1 k (-1) k e -2iπ(p-k+1)ξ , for any ξ in [0, 1].
The two sums are simplified thanks to Lemma 2. We finally obtain U n+1 (ξ) = A+(ξ) U n (ξ), with A+ the amplification coefficient defined by, ∀ξ ∈ [0, 1]

A+(ξ) = 1 -(1-θ)∆t (∆x) 2p+1 e -iπξ (-i) 2p+1 (2sin(πξ)) 2p+1 1 + θ∆t (∆x) 2p+1 e -iπξ (-i) 2p+1 (2sin(πξ)) 2p+1 . (6)
We are looking for a condition ensuring

|A+(ξ)| 2 < (1 + C∆t) 2 for any ξ in [0, 1].
Case 1 : Assume that the parameter p of the spatial derivative is even. If p = 0, the stability condition leads to

2 2p ∆t (∆x) 2p+1 (sin(πξ)) 2p (1 -2θ) ≤ -1, (cf. [1]
) which is impossible for all ξ ∈ [0, 1] : thus the forward finite difference scheme is unconditionally unstable for p even and non zero. On the contrary, assuming p = 0 means that the forward finite difference scheme is stable under CFL condition : ∆t(1 -2θ) ≤ -∆x (which implies θ > 1 2 ). Case 2 : In this case, the parameter p of the spatial derivative is odd, then the sufficient condition becomes ∆t (∆x) 2p+1 (2sin(πξ)) 2p (1 -2θ) ≤ 1 (cf. [START_REF] Courtès | Étude de l'équation d'Airy sous contrainte[END_REF]). Then the forward finite difference scheme is stable under the CFL condition ∆t(1

-2θ) ≤ ∆x 2p+1 2 2p
. Table 1 is a straightforward consequence.

Remark 4. For the backward finite difference scheme, the only difference in the amplification coefficient is e iπξ instead of e -iπξ (in both the numerator and denominator). The parity needed for the stability changes because of that difference. For the central finite difference scheme, e -iπξ is replaced with cos(πξ) in the numerator and the denominator of the amplifiaction coefficient.

Error estimates

We define the convergence error as follows.

Definition 3. For all j ∈ Z and n ∈ {0, ..., N }, for u the analytical solution of (1) from u0 and (v n j ) (j,n) the numerical solution of (2), the convergence error is denoted by e n j and defined by e n j = 1 ∆x x j+1

x j u(t n , y)dy-v n j . We are now able to state the main result of this section.

Theorem 1. For an initial datum u0 ∈ H 4p+2 (R) (and u0 ∈ H 6p+3 (R) if θ = 1
2 ), the error estimate of the forward finite difference scheme (3a) (if p is odd) or of the backward finite difference scheme (3b) (if p is even) satisfies

||e|| ∞ (0,N; 2 ∆ (Z)) ∆t 1 2 -θ ∂ 4p+2 x u0 L 2 (R) + ∆x ∂ 2p+2 x u0 L 2 (R) + ∆t 2 ∂ 6p+3 x u0 L 2 (R) .
For the central finite difference scheme (3c), the convergence rate becomes

||e|| ∞ (0,N; 2 ∆ (Z)) ∆t 1 2 -θ ∂ 4p+2 x u0 L 2 (R) + ∆x 2 ∂ 2p+3 x u0 L 2 (R) + ∆t 2 ∂ 6p+3 x u0 L 2 (R) .
All those results are gathered in Table 4.

Proof. We suppose p odd, so we work with the forward finite difference scheme. The case p even, with the backward scheme is similar. The definition of the convergence error implies

e n+1 (ξ) = A+(ξ) e n (ξ) + ∆t 1 + θ∆t ∆x 2p+1 e -iπξ (-i) 2p+1 (2sin(πξ)) 2p+1 n (ξ).
One has

1 1+ θ∆t ∆x 2p+1 e -iπξ (-i) 2p+1 (2sin(πξ)) 2p+1 L ∞ ([0,1])
≤ 1 and the stability condition gives ||A+|| L ∞ ([0,1]) ≤ 1 + C∆t. Thus, we obtain the following estimate, by discrete Grönwall lemma

||e n+1 || 2 ∆ ≤ (1 + C∆t)||e n || 2 ∆ + ∆t|| n || 2 ∆ ≤ ... ≤ e CT ||e 0 || 2 ∆ + ∆te CT n k=0 || k || 2 ∆ .
The initial condition v 0 j , Eq. ( 2), together with the consistency error conclude the proof.

Remark 5. As expected, for the particular case θ = 1 2 (the so-called Crank-Nicolson case), the rate of convergence in time is better as illustrated in Table 4, provided u0 ∈ H 6p+3 (R) (and not only in H 4p+2 (R)).

Less smooth initial data

The previous order of accuracy is obtained for initial data u0 at least in

H 4p+2 (R) (or H 6p+3 (R) if θ = 1
2 ). In this section, our aim is to relax this hypothesis to obtain rates of convergence for non-smooth initial data, for example, u0 ∈ H m (R) with m > 0. We detail only the case θ = 1 2 but state the Crank-Nicolson results in Table 4.

Initial datum in H

m (R) with m ≥ 2p + 2
As explained previously (Remark 2), the regularity of u0 is determined by the Taylor expansion in time. A first step is then to deal with the time term in error estimates. The following proposition provides that the time error prevails until u0 ∈ H 2p+2 (R), for which the spatial error becomes predominant. Proposition 3. Assume u0 ∈ H m (R) with m ≥ 2p + 2, and let us fix M = min(m, 4p + 2), then the error estimate for the forward (respectively backward) finite difference scheme, if p is odd (respectively even), yields

||e|| ∞ (0,N; 2 ∆ (Z)) ∆t M 4p+2 ||∂ M x u0|| L 2 (R) + ∆x||∂ 2p+2 x u0|| L 2 (R) .
For the central difference scheme, we suppose m ≥ 2p + 3, and one has (for the same M )

||e|| ∞ (0,N; 2 ∆ (Z)) ∆t M 4p+2 ||∂ M x u0|| L 2 (R) + ∆x 2 ||∂ 2p+3 x u0|| L 2 (R) .
Before proving this result, we introduce a regularization of u0 thanks to mollifiers ϕ δ δ>0 . Let χ be a C ∞ -function such that

• 0 ≤ χ ≤ 1, • χ ≡ 1 in [-1 2 , 1 2 ] and supp(χ) ⊂ [-1, 1] (where supp is its support), • χ(-ξ) = χ(ξ), ∀ξ ∈ [-1, 1].
Let ϕ be such as ϕ (ξ) = χ (ξ) and for all δ > 0, we define ϕ δ such that ϕ δ (ξ) = χ (δξ), which implies ϕ δ = 1 δ ϕ . δ . Eventually, • let u δ be the solution of (1) with u δ 0 = u0 ϕ δ as initial data, where stands for the convolution product.

• We denote then ((v δ ) n j ) (n,j)∈{0,...,N }×Z the numerical solution obtained by applying the numerical scheme (2) from u δ 0 . • The unknowns u and (v n j ) (n,j)∈{0,...,N }×Z are always the exact and numerical solutions starting from the initial data u0. Lemma 3. Assume u0 ∈ H r (R) with r > 0 then the following upper bound holds, for 0

≤ ≤ r ≤ s, u0 -u δ 0 H (R) δ r-||u0|| H r (R) and u δ 0 H s (R) 1 δ s-r ||u0|| H r (R) .
Proof. Lemma 3 is proved in [START_REF] Courtès | Rate of convergence for finite difference scheme on the Korteweg-de Vries equation[END_REF] and follows from very classical arguments (see also [START_REF] Després | Finite volume transport schemes[END_REF]).

Proof of Proposition 3. We are now able to prove Proposition 3. The triangular inequality applied to the convergence error gives

||e n || 2 ∆ E1 + E2 + E3 with E1 =   j∈Z ∆x 1 ∆x x j+1 x j u(t n , x) -u δ (t n , x)dx 2   1 2 , (7)
E2 =   j∈Z ∆x 1 ∆x x j+1 x j u δ (t n , x)dx -(v δ ) n j 2   1 2 , ( 8
) E3 =   j∈Z ∆x (v δ ) n j -v n j 2   1 2 . ( 9
)
Cauchy-Schwarz inequality together with the conservation of the L 2 -norm (Remark 1) yield E1 ≤ ||u0 -

u δ 0 || L 2 (R) δ M ||∂ M x u0|| L 2 (R) .
The latest inequality comes from Lemma 3 with = 0 and r = M . For the E2-term, we use the previous section (Sect. 2). Indeed, E2 corresponds to the convergence error for a smooth initial data u δ 0 . Hence, one has

E2 ∆t||∂ 4p+2 x u δ 0 || L 2 (R) + ∆x||∂ 2p+2 x u δ 0 || L 2 (R) ∆t δ 4p+2-M ||∂ M x u0|| L 2 (R) + ∆x||∂ 2p+2 x u0|| L 2 (R) ,
where the latest inequality comes from Lemma 3 with (s, r) = (4p + 2, M ) and (s, r) = (2p + 2, 2p + 2).

Finally, the stability of the scheme gives the following estimate for E3 :

E3 = ||(v δ ) n -v n || 2 ∆ ≤ ||u δ 0 -u0|| L 2 (R) .
Thus, the convergence error is upper bounded by

||e n || 2 ∆ δ M ||∂ M x u0|| L 2 (R) + ∆t δ 4p+2-M ||∂ M x u0|| L 2 (R) + ∆x||∂ 2p+2 x u0|| L 2 (R) .
Proposition 3 comes from the optimal choice for δ : δ = ∆t

1 4p+2 .
Remark 6. The result for the central finite difference scheme is proved exactly in the same way, with the same s, r, and δ.

Initial datum in

H m (R) with m ≥ 0
The main result of this paper is summarized in the following theorem where the initial data u0 has any Sobolev regularity.

Theorem 2. Assume that u0 ∈ H m (R) with 0 ≤ m, then, the forward (respectively backward) finite difference scheme if p is odd (respectively even) has the following error-estimate

||e|| ∞ (0,N; 2 ∆ (Z)) ∆t min(m,4p+2) 4p+2 ||∂ min(4p+2,m) x u0|| L 2 (R) + ∆x min(m,2p+2) 2p+2 ||∂ min(2p+2,m) x u0|| L 2 (R) .
The previous inequality becomes for the central finite difference scheme

||e|| ∞ (0,N; 2 ∆ (Z)) ∆t min(m,4p+2) 4p+2 ||∂ min(4p+2,m) x u0|| L 2 (R) + ∆x 2 min(m,2p+3) 2p+3 ||∂ min(2p+3,m) x u0|| L 2 (R) .
The previous results are summarized in Table 4.

Proof. Here again, we suppose that p is odd, we thus detail the proof for the forward finite difference scheme.

We have already proved the case m ≥ 4p + 2 in Sect. 2 and the case 2p + 2 ≤ m ≤ 4p + 2 in Subsect. 3.1.

Let us now focus on the case 0 ≤ m ≤ 2p + 2.

The proof of Theorem 2 follows the same guidelines as the proof of Proposition 3. Let u0 ∈ H m (R), we regularize this initial data thanks to mollifiers ϕ δ δ>0 whose properties are listed in Subsect. 3.1. This involves introducing the same new unknowns u δ , u δ 0 and ((v δ ) n j ) (j,n) . The convergence error e n j (j,n) is upper bounded by the same E1, E2 and E3, defined in ( 7)-( 9). Lemma 3 with = 0 and r = m leads to

E1 + E3 δ m ||∂ m x u0|| L 2 (R)
.By definition u δ 0 ∈ H k (R), ∀k > 0, therefore Proposition 3 applies with k = 2p + 2 for example and M = min(k, 2p + 2) = 2p + 2. It gives the following estimate for E2 : E2 ∆t

p+1 2p+1 ||∂ 2p+2 x u δ 0 || L 2 (R) + ∆x||∂ 2p+2 x u δ 0 || L 2 (R) ∆t p+1 2p+1 + ∆x ||∂ 2p+2 x u δ 0 || L 2 (R)
. We then apply Lemma 3 with s = 2p + 2 and r = m. Finally, it yields

||e n || 2 ∆ δ m ||∂ m x u0|| L 2 (R) + ∆t p+1 2p+1 + ∆x δ 2p+2-m ||∂ m x u|| L 2 (R) .
The conclusion comes from the optimal choice δ = ∆t

p+1 2p+1 + ∆x 1 2p+2 .
Remark 7. The backward scheme, with p even, is very similar. The central finite difference scheme is proved with the same method except for the variable k, which is taken k = 2p + 3 for that scheme.

Numerical results

In order to illustrate numerically the previous results, we perform two sets of examples : on the one hand, we compute the numerical rate of convergence of various equations for a fix initial data and one the other hand, the equation is fixed and we test different initial data.

In all examples, the computational domain is set to [0, 50] subdivided into J cells with J ∈ {800, 1600, 3200, 6400, 12800, 25600, 51200, 102400}

and the numerical simulation is performed up to time T = 0.1. Not to have a too restricted Courant-Friedrich-Lewy condition, we implement the implicit scheme (θ = 1) and impose ∆t = ∆x. The convergence error is computed between the solution with J cells and a 'reference' solution with 2J cells in space.

Since the indicator function belongs to H s (R) for all s < 1 2 , we build test functions in H s (R) with s < 1 2 + k by integrating k-times the indicator function. Such functions will be denoted in H

1 2 3 2 -or H 5 2-

 135 +k-(R).The first test consists of fixing u0 in H and compute the convergence rate for p = 0 (advection equation), p = 1 (Airy equation) and p = 2. The numerical results are gathered in Tables2 and 3and correctly match with the expected theoretical rates. For the second sample of examples, the equation is fixed (p = 0 for Fig.1-left and p = 1 for Fig.1-right) whereas the Sobolev regularity of the initial data is fluctuating. As shown in Fig.1, the theoretical rates are represented by the line and the numerical rates correspond to the dot. The exponent of the Sobolev regularity of u0 is shown in the x-axis. Again, the different rates match very well, which tends to indicate that the convergence orders we have proven are optimal.

Figure 1 :For θ = 1 2 For θ = 1 2 (

 1122 Figure 1: Numerical versus theoretical orders-left : Advection equation (p = 0), right : Airy equation (p = 1)

Table 2 :

 2 For a Sobolev regularity H

		p = 0		p = 1		p = 2
	∆x	(Advection)	(Airy)	(Fifth-order derivative)
		L 2 -error	order	L 2 -error	order	L 2 -error	order
	6.250.10 -2					2.985.10 -3
	3.125.10 -2					2.757.10 -3 0.115
	1.563.10 -2					2.441.10 -3 0.175
	7.813.10 -3 1.194.10 -4		1.348.10 -3		2.176.10 -3 0.166
	3.906.10 -3 7.381.10 -5 0.694	1.125.10 -3 0.261	1.961.10 -3 0.149
	1.953.10 -3 4.471.10 -5 0.723	9.670.10 -4 0.219
	9.766.10 -4 2.664.10 -5 0.747	7.968.10 -4 0.279
	4.883.10 -4 1.585.10 -5 0.749	6.572.10 -4 0.278
	theoretical		0.750		0.250		0.150
					3 2 -
		p = 0		p = 1		p = 2
	∆x	(Advection)	(Airy)	(Fifth-order derivative)
		L 2 -error	order	L 2 -error	order	L 2 -error	order
	6.250.10 -2					3.388.10 -2
	3.125.10 -2					3.639.10 -2 0.093
	1.563.10 -2					3.032.10 -2 0.263
	7.813.10 -3 2.586.10 -3		1.011.10 -2		2.528.10 -2 0.262
	3.906.10 -3 1.347.10 -3 0.940	7.507.10 -3 0.430	2.138.10 -2 0.242
	1.953.10 -3 6.873.10 -4 0.971	6.267.10 -3 0.261
	9.766.10 -4 3.437.10 -4 0.999	4.401.10 -3 0.510
	4.883.10 -4 1.719.10 -4 1.000	3.074.10 -3 0.520
	theoretical		1.000		0.417		0.250

Table 3 :

 3 For a Sobolev regularity H

	5 2 -

Table 4 :

 4 Error estimates for u

	(R)
	H m
	0 ∈


Pascal Casari

Laurent Gornet

Characterization of residual stresses in a composite curved sandwich beam

Keywords: Sandwich structure, B. Residual stress, C. Finite element analysis, D. Mechanical testing

Introduction

The aim of this study is to investigate the thermomechanical response of a sandwich composite shell, which exhibits a double curvature. In this kind of heterogeneous structure, the skins and the core play very different roles. The skins are stiff and their coefficient of thermal expansion (noted CTE) is low. The foam core has stiffness 1000 times lower than the stiffness of the layers in the direction of the fibers and expands 10 times more. Residual stress control is required to ensure the geometrical stability of the structure, and for this reason it is important to measure the order of magnitude of these stresses. For single laminates, it has been shown in [START_REF] Bogetti | Process-induced stress and deformation in thick-section thermoset composite laminates[END_REF][START_REF] White | Process modeling of composite materials: residual stress development during cure. Part II. Experimental validation[END_REF][START_REF] Ridgard | Accuracy and distortion of composite parts and tools: causes and solutions[END_REF][START_REF] Radford | Measurement of manufacturing distortion in flat composite laminates[END_REF][START_REF] Johnston | A plane strain model for process induced deformation of laminated composite structures[END_REF][START_REF] Fernlund | Experimental and numerical study of the effect of cure cycle, tool surface, aspect ratio, and the lay-up on the dimensional stability of autoclave-processed composite parts[END_REF][START_REF] Gigliotti | Development of curvature during the cure of AS4/8552 [0/90] unsymmetric composite plates[END_REF] that the significant phenomena responsible for residual strain are chemical and thermal shrinkage, and frictional effects in combination with moulds whose constitutive materials have different coefficients of thermal expansion to the manufactured composites. In those published studies, modelling of residual stresses has progressed, but they concern mostly laminates. Only a few studies have been performed on sandwich structures, and essentially on flat panels [START_REF] Matsunaga | Assessment of a global higher-order deformation theory for laminated composite and sandwich plates[END_REF][START_REF] Matsunaga | Interlaminar stress analysis of laminated composite and sandwich circular arches subjected to thermal/mechanical loading[END_REF][START_REF] Matsunaga | A comparison between 2-D single-layer and 3-D layerwise theories for computing interlaminar stresses of laminated composite and sandwich plates subjected to thermal loadings[END_REF]. In the present work, a set of experiments has been performed going from the scale of the structure down to the scale of its constitutive parts. First, tests carried out on the sandwich structure provide an order of magnitude of residual stresses. Then curved beams, easier to test, are cut from the sandwich shell, and are subjected to the removal of edges and outer skin. Thus, materials or structures can be characterized in the range of thermal and mechanical loadings encountered in the structure's life. Once material properties have been identified (or collected from datasheets), a thermo-mechanical finite element model (noted FEMo) has been built and is used to simulate the heating of a beam simply supported at the ends.

Measurement of residual stresses and materials properties

Sandwich panel

The sandwich shell under consideration is part of a radar antenna (see Fig. 1), It has been designed mostly with process constraints, and no calculations were run before manufacture. It has a double curvature and is approximately 2 m wide, 0.9 m high and 50 mm thick. After manufacturing, this structure shows a slight twist, and a residual deflection, which tends to hollow out the panel. This effect can be attributed to the thermal shrinkage of both skins and core during polymerization and to the difference of coefficients of thermal expansion of skins and core when cooling down from 80 8C to room temperature. The cure cycle of the structure is 8 h at 80 8C. The initial state of the structure can be considered as the non-stressed state at curing temperature. From then on, residual stresses remain in the constituents. The main idea proposed in this study is to give an order of magnitude of residual stresses by cutting a piece from the sandwich panel (Fig. 2). Strains released from the cutting are measured by means of triaxial strain gages bonded on both faces. Under elastic hypotheses, the corresponding stresses are given in Fig. 3 (Table 1 for the description of the skin stacking sequence). The first remark is that strains remain low but significant. On both sides, the plies are predominantly in tension in the width direction of the panel. An opposite global shear deformation appears between the two skins, which leads to the confirmation of a global twist which has been observed on the shell. Regarding stresses in each ply, they are low and match the global equilibrium, so that the core is consequently in compression in the width direction. Shear in the plies is much lower than tensions or compressions in the fiber directions. This means that the stacking sequence seems convenient for storing energy in the principal directions of the plies and not in shear. In fact, the symmetry of the structure associated with the viscoelasticity of the matrix tends to reduce the residual stresses. In addition to this first step of characterization, a finite element model has been constructed with NISAe. It consists in modelling the panel with linear rectangular shell elements for the skins and three-dimensional elements for the core as detailed in Fig. 4. The inputs are both mechanical and thermal properties, measured on coupons and presented later. The predicted maximum deflection of the panel is 2.6 mm but the geometrical report on the shape of a manufactured panel gave a value of 0.6 mm. This measurement of deflection had been completed several days after curing. Numerical and experimental results were not comparable: a significant mismatch was appearing in the strains, stresses and deflection results. This led to the need of accurate modelling and characterization of each component of the sandwich material in order to explain this gap.

Sandwich beam

A second set of experiments has been conducted on a sandwich beam extracted from the structure. The aim was to find out which components of the sandwich are responsible for the residual stresses. The removal operations are illustrated in Fig. 5. The beam is 50 mm wide. The displacement field is then measured by means of a testing device, which simply supports the beam (Fig. 6) on both ends. The first step consists of cutting the beam from the panel. The residual stress field then becomes unidirectional. Through the second step, the effect of the edges on the shape of the beam is analyzed. This stems from a study conducted in [START_REF] Cheng | Reduction of the concentrated residual stress on the edges of a sandwich plate[END_REF] in which some sharp corners in the sandwich have been found to cause very high stress concentrations at the beam's tips. Here, the removal of the edges has no significant effect in terms of deflection of the beam. The last step is the removal of the outer skin in order to release residual stresses included in this component. Deflection variations (Fig. 7) are taken in a regular 300 mm-long part of the beam (Fig. 6). The graph in Fig. 7 shows that the most important effect is due to the skin removal: the beam tends to flatten out. This leads to the conclusion that internal stresses probably remain in tension and in compression in the skins and in shear in the foam core. A possible interpretation of the curvature is to calculate the corresponding bending moment that must be applied in order to return to the initial shape (before cutting). The result is presented in Table 2. In addition, it may be noted that the sandwich structure is not symmetrical through the thickness. The beam deflection due to heating will be compared to the Fig. 5. Removal operations on a sandwich beam in order to release residual stresses.

response of FEMo in the 'finite element modelling' section below.

Material properties

Thermal expansion and mechanical tests have been conducted on the constitutive parts of the sandwich in order to quantify the properties required for the FEMo analysis. The identification has been made on the two different plies used in the lamination of the sandwich shell. These are a twill woven carbon fabric and a stitched [G45] made of two unidirectional plies. The properties of the core have been taken from the manufacturer's datasheet. The estimated material properties required for modelling are listed in Table 3. The 08 direction of the panel and of the laminates is the x axis plotted in Fig. 5.

Skins

In order to characterize the constitutive plies of the skins, two specific plates (called twill and stitched) have been manufactured with the same process as that used for the sandwich shell. The twill plate is made of a twill balanced fabric and has a [0,G30] S stacking sequence. This ensures a mirror symmetry and produces a quasi-isotropic composite laminate. In the stitched plate, six layers have been stacked in the 08 direction. A mirror symmetry cannot be exactly used here, and the direction of maximum stiffness is 458. From each plate, seven coupons have been cut every 158 from 0 to 908 (Fig. 8).

From the stress-strain curves obtained, the elastic modulus, the Poisson's ratios and the coefficients of thermal expansion could be measured. Results confirm that the twill laminate can be considered as a quasiisotropic material. For the stitched laminate, these properties are plotted in Fig. 9 and show that the behaviour cannot be considered as quasi-isotropic. Moreover, classical laminate theory cannot match completely these results because, due to its constitution, the laminate should have the same properties in the directions (q), (Kq), and (p/2Kq). A compromise has been found in the choice of the properties of this layer: the evolution of elastic and thermal parameters versus angle is plotted in addition to experimental results. The discrepancy is probably due to movement of the stitched layers during lamination: dry layers are very deformable and the specific angle of 458 can vary rapidly by 5-108 under a stretching action. Nevertheless, elastic properties are well simulated. Only the evolution of the coefficient of thermal expansion is not satisfactory because the variations around the average vary from negative to positive values.

Core

The foam core is a very soft material compared to the skins. Its properties are recalled in Table 3 from the manufacturer's datasheet. A special feature of the core properties is the difference in stiffness from tension to compression. This particularity could not be taken into account in modelling, but it may have a significant effect on the stress and strain fields in the sandwich shell and beam. The core is considered here as a linear elastic and isotropic material, and its behaviour has to be included as three-dimensional in modelling with tensile Young's modulus. Indeed, because of the sandwich curvature, the thermal expansion of the core through the thickness leads to a variation of this curvature.

Modelling of the sandwich beam

Modelling is now focused on the behaviour of the beam extracted from the shell in order to simulate a heating experiment applied on the beam. In this test, the beam is simply supported at the ends and heated with a temperature-controlled airflow. The upper and lower displacements in the mid-section are recorded. Deflection and thickness variation of the beam can be deduced after an increase of 60 8C, representing the heating of the structure from room to curing temperature.

Thermo-mechanical analysis

After heating, the core has forced the curved composite shell to expand through the thickness and the skins have also expanded; this leads to a new internal equilibrium and the existence of internal stresses. This effect can be taken into account in the finite element calculations if the core is modelled with three-dimensional elements. The skins are considered in a plane stress state with the same stacking sequence as for the panel (Table 1). The finite element code Cast3M capability for coupled thermo-mechanical heat transfer analysis is used in order to model heat conduction in the multi-layer composite sandwich structure and to determine the elastic strain and stress induced by temperature variations. Both steady-state and transient heat transfer finite element capabilities have been used for computations, but here only the first one is taken into consideration. Predictions are made with both composite shell and second order tetrahedral and full Lagrange heat conduction brick elements (Fig. 10). Classical thermal boundary conditions such as the surface heat flux or the properties of the surface convection must be specified. In order to model the experimental facilities, only surface convection boundary conditions are used.

Results

At first, deflection results have been calculated versus the CTE value of the stitched layers, varying from K1!10 K6 to 7!10 K6 8C K1 as mentioned in Fig. 9. Resulting deflection and thickness variation of the beam are plotted in Fig. 11 and compared to experimental values. The calculated deflection is sensitive to this parameter, and the value of CTE for the stitched layers which fits the experiment is close to 4.7!10 K6 8C K1 . This choice of a constant CTE must be done because its evolution shown in Fig. 9 cannot be reproduced in numerical simulations. Thickness variation from the test and the simulation are very close, which indicates that the CTE of the core is satisfactory. Results in terms of stresses in the skins are calculated from the model with the above-mentioned CTE value and given in Table 4. A significant stress level (maximum of 28.1 MPa in a twill layer of the outer skin) is created by the internal equilibrium and shows that the highest stresses are located in the outer skin and due to the difference of CTE between the four constitutive layers.

Compared to the stress level measured on the panel from strain analysis (maximum of 7 MPa in one of the stitched plies), it seems that this increase of 60 8C generates more stresses than the ones remaining in the structure after manufacturing. These stress levels are not completely comparable, but this order of magnitude is a typical one that may create creep deformations and affect the geometrical stability of the structure.

Conclusion

The approach presented in this article is based on a procedure of characterization of residual stresses that can be applied to any sandwich structure for which destructive tests are possible. Data collected among displacements, curvatures or strains have been selected and consistently compared with calculations. These specific results have given some orders of magnitude of residual stresses, but have also underlined the need for accurate mechanical and thermal material properties. Indeed, some parameters like the evolution of the CTE of the stitched layers are not explained by classical laminate theory, and require more accurate investigations. This approach is not comprehensive yet, because only a few data and measurements are available for the core. Moreover, there is no up-to-date measurement technique for stress analysis and thermal characterization and only the thickness variation of the beam with temperature could be validated. Work on the thermomechanical and viscous properties of the core are probably the next aspects to study.

Fig. 1 .

 1 Fig. 1. Sandwich panel and mismatch between theoretical and manufactured shapes.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Piece of sandwich cut out from the structure, and direction of the strain gages.

Fig. 4 .

 4 Fig. 4. FEMo of the panel and transverse displacement field.

Fig. 6 .

 6 Fig. 6. Test device for curvature evolution on sandwich beams.

Fig. 7 .

 7 Fig. 7. Deflection of the beam after part removal.

Fig. 8 .

 8 Fig. 8. Geometry and direction of the tensile coupons.

Fig. 10 .

 10 Fig. 10. FEMo of the sandwich beam.

Fig. 11 .

 11 Fig. 11. Deflection of the beam and thickness variation vs CTE of the stitched layer.

Table 1

 1 Properties of the sandwich panel

	Component	Thickness (mm)	Stacking sequence
	Outer skin	1.6	1 Twill fabric K308
			2 Stitched fabrics 08
			1 Twill fabric C308
	Inner skin	0.7	1 Twill fabric K308
			1 Twill fabric 08
			1 Twill fabric C308
	Core	50

Table 3

 3 Thickness 0.23 mm E x ZE y Z40 GPa, n xy Z0.1, G xy Z3 GPa 3 Stitched ply Carbon T300, 2 layers at [G45] 440 g/m 2 , Thickness 0.45 mm E x ZE y Z56 GPa, n xy Z0.04, G xy Z3.3 GPa

	Material properties
	Constitutive materials properties of the sandwich	Geometric	Mechanical	CTE!10 6 m/m
	Twill fabric	Carbon T300, [0;90]	220 g/m 2 , 3
	Resin	Aralditee LY5052-HY5052	Fiber volume fraction:
			40!V f !46
	Core	Klegecelle R55	55 kg/m 3	E

compression Z77 MPa, E tension Z51 MPa, GZ27 MPa, nZ0.32 35 Fig. 9. Mechanical and thermal behaviour of the stitched layers.

Table 4

 4 FEMo results of stress-state in the plies of the beam after 60 8C heating

		Thickness (mm)	Stiffness matrix		Stresses (MPa) (in the coordinate system of each ply)
						Ply	s xx	s yy	s xy
	Inner skin	0.7	0	3:28 1:16 0	1	Twill [K30]	7.3	4.4	0.1
	Outer skin	1.6	B @ 0	1:16 3:28 0 0 0 1:06 3:06 2:16 0	C A !10 4 1	Twill [0] Twill [30] Twill [K30]	6.1 4.6 26.2	0.7 K2.9 28.1	K0.1 0.1 K1.2
			B @	2:16 3:06 0 0 0 2:20	C A !10 4	Stitched [45] Stitched [45] Twill [K30]	K5.5 K6.4 23.1	K6.5 K8.6 19.6	0.1 0.1 1.0