Rahma Abid

Gérard Delahay

Hassib Tounsi
email: hassibtounsi@yahoo.fr

Preparation of LTA, HS and FAU/EMT intergrowth zeolites from aluminum scraps and industrial metasilicate

Keywords: Aluminum scraps, Sodium metasilicate, Zeolite LTA, Intergrowth FAU/EMT, Hydrothermal synthesis

Na-LTA zeolite crystals were synthesized via hydrothermal method using aluminum scraps and industrial sodium metasilicate. The obtained hydrogel with molar composition: 3,741 Na 2 O: Al 2 O 3 :1.450 SiO 2 :175.520 H 2 O was aged at room temperature for 14 hours and heated at 80°C for 6 hours. The use of aluminum scraps provides well crystallized cubic Na-LTA zeolites with truncated edges and apexes. When H 2 O/Al 2 O 3 molar ratio was decreased: 3.741 Na 2 O: Al 2 O 3 :1.450 SiO 2 :92.50 H 2 O there is formation of an intergrowth of FAU/EMT phases contaminated with LTA zeolite after 6h of crystallization. The prolongation of the crystallization time to 48 hours leads the formation of hydoxysodalite zeolite. On the other hand, the increase of aging time to 96 hours, the crystal sizes of zeolite decrease.

Introduction

The use of aluminum has a good interest in the industrial sector, but it produces a large amount of waste which is harmful to the environment and human health. This waste inhibits root growth, which in turn affects the ability of the plant to adsorb water and nutrients [START_REF] Panda | Aluminum Stress Adaptation in Plants[END_REF]. Also exposure to aluminum can affect health, including nervous system, Alzheimer disease and other neurodegenerative disorders [START_REF] Niu | Neurotoxicity of Aluminum[END_REF]. For that, many researchers have been focused in the recycling aluminum waste to product materials. Abdelkader et al [START_REF] Abdelkader | Preparation and characterization of mesoporous γ-Al 2 O 3 recovered from aluminum cans waste and its use in the dehydration of methanol to dimethyl ether[END_REF] Used waste aluminum beverage cans for the preparation of mesoporous gamma alumina by a simple precipitation method using two different precipitating agents (NaOH or NH4OH).Jraba et al [START_REF] Jraba | Valorization of Aluminum Chips into γ-Al2O3and η-Al2O3with High Surface Areas via the Precipitation Route[END_REF] valorize aluminum chips into γ-Al 2 O 3 and η-Al 2 O 3 by acidic precipitation of sodium aluminate obtained via an alkaline attack of the scraps. M. Selim et al. [START_REF] Selim | Preparation and characterization of Na-A zeolite from aluminum scrub and commercial sodium silicate for the removal of Cd2+from water[END_REF] reported the preparation of LTA zeolite from aluminum scrubs and commercial sodium silicate solution at 110°C for 8h. Tounsi et al [START_REF] Tounsi | Preparation and characterization of Na-LTA zeolite from Tunisian sand and aluminum scrap[END_REF] prepared zeolite Na-LTA, NaX and HS from Tunisian sand and aluminum scraps. Furthermore, Abdelrahman [START_REF] Abdelrahman | Synthesis of zeolite nanostructures from waste aluminum cans for efficient removal of malachite green dye from aqueous media[END_REF] used waste aluminum cans to prepare different zeolites (LTA, ANA, SOD, and NaP1) via a low-cost hydrothermal method. R. Sánchez-Hernández et al. [START_REF] Sánchez-Hernández | A. Ecofriendly bench-scale zeolitization of an Al-containing waste into gismondine-type zeolite under effluent recycling[END_REF] prepared NaP1 zeolite from Al-containing waste from metallurgical industry by recycling the effluents with zero liquid discharge. A. Meshram et al [START_REF] Meshram | Recovery of valuable products from hazardous aluminum dross: A review[END_REF] used aluminum dross to synthesis zeolite FAU type X.

Zeolites are crystallized inorganic solids characterized by a 3-D framework constituted via the arrangement of tetrahedral units [SiO 4 ] 4-and [AlO 4 ] 5-linked by their oxygen atoms [START_REF] Breck | Ion exchange reactions in zeolites[END_REF][START_REF] Ali | Controllable synthesis of NaP zeolite and its application in calcium adsorption[END_REF]. The assembly of these tetrahedral units leads to the formation of large microporous networks consisting of channels and cages distributed periodically in space. Negative charges carried by [AlO 4 ] -tetrahedra are compensated generally by alkaline (Na + , K + …) and alkaline earth metal cations (Ca 2+ , Mg 2+ …). These cations are surrounded with water molecules and are located in the cavities and channels of the zeolite framework [START_REF] Ali | Controllable synthesis of NaP zeolite and its application in calcium adsorption[END_REF][START_REF] Xu | Chemistry of zeolites and related porous materials: synthesis and structure[END_REF]. Due to their fascinating properties i.e. physical and hydrothermal stabilities, high surface areas [START_REF] Wang | Zeolite-zeolite composite fabricated by polycrystalline y zeolite crystals parasitizing ZSM-5 zeolite[END_REF], non-toxicity, framework and compositional flexibilities, zeolites found applications in many fields. Thus, zeolites found applications in the fields of ion exchange, adsorption, catalysis, separation and molecular sieves [START_REF] Koohsaryan | Nanosized and hierarchical zeolites: A short review[END_REF]. On the other hand, nanosized zeolites were used in innovative applications like holographic sensors, medicine and photovoltaic [START_REF] Zaarour | Progress in zeolite synthesis promotes advanced applications[END_REF][START_REF] Rabo | Early discoveries in zeolite chemistry and catalysis at Union Carbide, and follow-up in industrial catalysis[END_REF]. Among all synthetic zeolites, the Na-LTA is the most produced and used in the industry. It is largely used for the formulation of phosphorus free detergents as water softener and for alkane/alkene separation [START_REF] Granato | Molecular simulation of propane-propylene binary adsorption equilibrium in zeolite 4A[END_REF]. Na-LTA has a pore diameter of 4 Å, which can be modified either to 5 Å or 3 Å by ion exchange with calcium or potassium, respectively.

Zeolites are generally synthesized under hydrothermal conditions using an aluminosilicate gel in the presence of a mineralizing agent (OH -, F -) in an aqueous solution.

The aluminosilicate gel is heated in an autoclave from a few hours to several days at temperature between ambient and 200°C under autogenous pressure [START_REF] Breck | Ion exchange reactions in zeolites[END_REF]18]. The hydrothermal synthesis of zeolites usually used expensive materials such as silica, alumina chemical sources and structure directing agents [START_REF] Cundy | The hydrothermal synthesis of zeolites: Precursors, intermediates and reaction mechanism[END_REF]. So when the chemical composition or sources of reagent were modified they cause the formation of a different kind of zeolite [START_REF] Oonkhanond | The effect of composition on the growth and morphology of zeolite A in solution[END_REF].

Therefore, inexpensive preparations have been reported in literature using industrial wastes.

Chen et al [START_REF] Chen | Synthesis and characterization of zeolites NaA and NaX from coal gangue[END_REF] prepared zeolite NaX and NaA from coal mining solid waste, coal gangue, by alkali fusion followed by hydrothermal treatment. Zhang et al [START_REF] Zhang | Utilization of iron ore tailing for the synthesis of zeolite A by hydrothermal method[END_REF] utilized iron ore tailing for the synthesis of zeolite A by hydrothermal method. Khoshbin and Karimzadeh [START_REF] Khoshbin | Synthesis of mesoporous ZSM-5 from rice husk ash with ultrasound assisted alkali-treatment method used in catalytic cracking of light naphtha[END_REF] succeeded to prepare free template nanostructured ZSM-5 zeolite from rice husk ash using ultrasound radiation. Maatoug et al. [START_REF] Maatoug | Valorization of vitreous China waste to EMT/FAU, FAU and Na-P zeotype materials[END_REF] prepared Na-X, Na-P, SOD and EMT/FAU zeotype materials using vitreous China waste by conventional hydrothermal route and alkaline fusion prior to hydrothermal synthesis. Also G. Garcia et al. study the effect of NaCl on the selective synthesis of FAU-zeolites from diatomite [START_REF] Garcia | Selective synthesis of FAUtype zeolites[END_REF]. On the other hand, inexpensive preparations of zeolites derived from low-cost geological materials have been reported in the literature [START_REF] Abdullahi | A review on sustainable synthesis of zeolite from kaolinite resources via hydrothermal process[END_REF][START_REF] Bortolatto | Synthesis and characterization of Y zeolites from alternative silicon and aluminium sources[END_REF][START_REF] Belviso | State-of-the-art applications of fly ash from coal and biomass: A focus on zeolite synthesis processes and issues[END_REF][START_REF] Kunecki | Synthesis of faujasite (FAU) and tschernichite (LTA) type zeolites as a potential direction of the development of lime Class C fly ash[END_REF]. In this study we investigated the valorization of aluminum scraps into LTA, HS and FAU/EMT intergrowth zeolites using industrial sodium metasilicate as silicon source. The starting hydrogel with the following molar composition "3.741 Na 2 O: Al 2 O 3 :1.450 SiO 2 :175.520 H 2 O" was prepared. Others sources of aluminum were also tested for the sake of comparison. Furthermore, the influence of aging time and H 2 O/Al 2 O molar ratios were also investigated.

Materials and Methods

Materials

Sodium metasilicate (Na 2 SiO 3 .5H 2 O, PQ CORPORATION) was used as silicon source in the preparation of LTA zeolites. The conventional aluminum sources were aluminum nitrate (Al(NO 3 ).9H 2 O, MERCK 95%), aluminum chloride (AlCl 3 .6H 2 O, FLUKA 99 %) aluminum powder, synthesis grade (SCHARLAU 99%), aluminum sulfate LR (Al 2 (SO 4 ) 3 .16H 2 O, SUVCHEM 98 %). The aluminum scraps was collected from metal manufacturing industry in the region of Sfax; South-eastern of Tunisia which has the following composition (Al (wt. %) = 99.77±0.02; Fe (wt. %) = 0.207±0.006 and Cu wt. %) = 0.008±0.001). Sodium hydroxide pellets (NaOH, SIGMA ALDRICH 99%) were used as mineralizing agent.

Synthesis of zeolite Na-LTA

Firstly, we dissolved the NaOH pellets in the required quantity of distilled water. Then we divided the solution to equal volumes in polypropylene bottles. The desired quantity of aluminum scraps (aluminum powder) was dissolved in one half of hydroxide solution volume and then filtered to remove the impurities. Sodium metasilicate was added to the NaOH solution and mixed in capped bottle until clear. Aluminum nitrate, aluminum chloride, and aluminum sulfate sources were dissolved in distilled water. In order to obtain the aluminosilicate hydrogel, the silicon solution was poured to the aluminum solution quickly.

The obtained thickly gel was stirred during a period of 10 min and then was aged for 14 h at room temperature (RT). Finally, the aged gel was heated at 80°C for 6h. After the crystallization was achieved, the obtained solids were filtered, washed with distilled water until pH around 9, and then dried at 80°C for 24h.

It should be noted that the final molar composition of the gel used for the preparation In order to determine the specific area of the prepared Na-LTA, Na + cations were exchanged with Ca 2+ and NH 4 + . The exchange was carried out at RT by stirring 6g of the parent Na-LTA zeolite in 250 mL of 0.5M CaCl 2 .2H 2 O or 0.5M NH 4 Cl solutions for 24h. The suspensions were filtered and rinsed with distilled water to remove excess of adsorbed species and then dried at 100°C for 24h.

Characterizations and analyses

The samples obtained were characterized by a variety of conventional techniques. Powder Xray diffraction (XRD) patterns were obtained using D8 ADVANCE BRUKER 40 Kv with Cu Kα radiation. The diffraction pattern was collected in the 2θ range of 5-50° at a scan speed of 0.02. The crystallinity C (%) of the prepared samples was estimated by XRD analysis using commercial zeolite Na-LTA (VALFOR ® 100) as standard (Eq. 1). The method consists of comparing the total of six current selected peaks in the angular range 15-30 in 2θ as imposed by the standard test method for the determination of relative crystallinity of zeolite Na-LTA (ASTM D5357-03).

Eq. 1

IR spectra were recorded by a Perkin-Elmer FTIR spectrometer with a resolution of 4 cm -1 using KBr method. The scanning electron microscopy (SEM) images were taken on a Hitachi S 4800 operated at 5 kV. Some samples were analyzed using Energy-dispersive X-ray spectroscopy on AZTEC-EDS apparatus. Nitrogen sorption experiments were carried out at -196°C on a micrometrics ASAP 2020 after degassing the sample at 250°C. The micropore volume (V micro ) and external area (S ext ) were determined by the plot method micrometrics Tristar Surface Area and Porosity analyzer. The BET surface areas (S BET ) and pore size distributions were determined by nitrogen adsorption-desorption on an ASAP2020 instrument (Micromeritics Instrument Corporation, GA), the pore size distributions were calculated by the advanced Barrett-Joyner-Halenda (BJH) method using the adsorption-desorption branches of the isotherms. Prior to these measurements, the samples were degassed at 250 °C in vacuum.

Results and discussions

Effect of aluminum sources

To [START_REF] Bosnar | Influence of anions on the kinetics of zeolite A crystallization: A population balance analysis[END_REF] showed that the presence of inorganic anions (NO 3-, SO 4 2-, Cl -) slows down the germination and crystallization processes by increasing the steric and electrostatic interactions between the inorganic anions and the aluminosilicate anions.

It should be mentioned that the zeolites prepared from aluminum powder and aluminum scraps have slightly enhanced relative crystallinities than the commercial one.

On the other hand, the chemical analysis performed by EDS technique on different spots of the surface of Z NaA.Sc 6h-80 , Z CaA.Sc 6h-80 and Z NH4A.Sc6h-80 samples are reported in Table 1. Figure 2 showed the XRD patterns of the Z NaA.Sc 6h-80 exchanged with calcium (Z Ca- A.Sc 6h-80 ) and ammonium (Z NH4-A.Sc 6h-80 ). The structure of the parent zeolite Na-LTA does not change after the introduction of Ca 2+ and NH 4 + in exchange sites. Indeed, the XRD profiles of Z NH4-A.Sc 6h-80 and Z Ca-A.Sc 6h-80 samples showed fine lines characteristic of LTA zeolites.

Nevertheless, the relative crystallinities decrease from 110% for Z NaA.Sc 6h-80 to 88% and 69%

for Z NH4A.Sc6h-80 and Z Ca-A.Sc 6h-80 , respectively. Price and al. [START_REF] Price | Local and Average Structural Changes in Zeolite A upon Ion Exchange[END_REF] showed that the exchange with NH 4 + ions compromises some of the long-range order of the zeolite LTA crystals due to the loss of some framework aluminum. Whereas, no alteration to the long-range crystal order is observed when divalent Ca 2+ ions was exchanged. In our case, we think that the good crystallinity of Z NH4A.Sc6h-80 compared to Z Ca-A.Sc 6h-80 , is due to the NH 4 + partial exchange in sample Z NH4A.Sc6h-80 . Indeed, the remaining sodium cations in Z NH4A.Sc6h-80 sample are about 7.16 wt. % whereas they are about 1.68 wt. % for Z Ca-A.Sc 6h-80 .

The SEM micrographs of Z NaA.Sc 6h-80 , Z Ca-A.Sc 6h-80 and Z NH4A.Sc6h-80 were displayed in figure 3. The presence of cubic crystals with truncated edges and apexes with an average diameter less than 1µm attested the formation of LTA zeolite type. Some crystals presented twinned morphology which is rather common in natural and synthetic zeolites. It is the result of mirror plane that occurs inside the structure. The twinning phenomenon in LTA zeolite is the result of a rotation along threefold axis [111] at an angle of 180° [START_REF] Porcher | Epitaxial growth of zeolite X on zeolite A and twinning in zeolite A: structural and topological analysis[END_REF]. It was shown that crystal twinning could stabilize a crystal polymorph that is otherwise not the most stable form [START_REF] Davey | Stabilization of a metastable crystalline phase by twinning[END_REF].

According to Kosanovic et al [START_REF] Kosanović | Chemically controlled particulate properties of zeolites: Towards the face-less particles of zeolite A. Part 1. Influence of the batch molar ratio [SiO2/Al2O3]b on the size and shape of zeolite A crystals[END_REF] the reaction mixture contributes to a high supersaturation which leads to a high surface nucleation and the formation of cubic crystal with rounded edges and apexes [START_REF] Kosanović | Chemically controlled particulate properties of zeolites: Towards the face-less particles of zeolite A. Part 1. Influence of the batch molar ratio [SiO2/Al2O3]b on the size and shape of zeolite A crystals[END_REF].

On the other hand, Kosanovic et al. [START_REF] Kosanović | Chemically controlled particulate properties of zeolites: Towards the face-less particles of zeolite A. Part 1. Influence of the batch molar ratio [SiO2/Al2O3]b on the size and shape of zeolite A crystals[END_REF] suggested that the formation of cubic crystals with truncated edges (formation of {011} crystal faces) was a result of the decrease of crystal growth in this direction caused by distortion of terminal SiO 4 tetrahedrons, positioned at crystal edges and apexes, in the presence of excess of aluminate ions.

The introduction of Ca 2+ and NH 4 + into the Na-LTA zeolite is accompanied by noticeable changes in the surface morphology, as shown in figure 3. The cubic automorphism of the original zeolite-LTA was not destroyed, but its surface morphology was completely changed from the original smooth surface. there is a steep in the nitrogen uptake attributed to the filling of micropores. The presence of hysteresis loop from 0.45 to 0.95 arises from the packing of zeolite nanocrystals [START_REF] Bendenia | Adsorptive properties of X zeolites modified by transition metal scation exchange[END_REF]. The low S BET = 10.8 m 2 /g value of zeolite Na-LTA is due to the limited zeolite channel size (Ф = 4.2 Å) preventing access to nitrogen molecules (Ф = 4.54 Å). So, the measured S BET corresponds to the external surface of the sample. According to Breck et al [START_REF] Breck | Ion exchange reactions in zeolites[END_REF] the presence of Na + cations near octagonal openings prevents the diffusion of nitrogen molecules. The exchange of Na-LTA with Ca 2+ cations modified the pore diameter of zeolite from 4 Å to 3 Å which increased the S BET from 10.8 m 2 /g to = 550.0 m 2 /g.

Crystallization kinetic of zeolite LTA at 80°C

Figure 5 shows the XRD patterns of the zeolites obtained at different crystallization times from 1 to 6 hours at 80°C using aluminum scraps and sodium metasilicate. All samples were aged for 14h at RT before crystallization. After 1 hour of crystallization, the sample (Z NaA.Sc1h- 80 ) is still amorphous. After 2 hours, fine peaks characteristic of the LTA zeolite appear and increase in intensity up to 6 hours. From the results in figure 5, the optimal crystallization time of the zeolite LTA is 6 hours where the crystallinity is about 113%.

Effect of aging time

The effect of aging time on the obtained products was also investigated. All samples were hydrothermally crystallized for 6h at 80°C after an aging time at RT for 0h (Z-0h), 24 h (Z-24h); 48 h (Z-48h); 72 h (Z-72h) and 96 h (Z-96h). The XRD patterns in figure 6 reveal that all products leads to zeolite Na-LTA, with relative crystallinities equals to 92.6 %, 96.3%, 80.2%, 101.2% and 85.2% for aging times of 0h, 24h, 48h, 72h and 96h, respectively.

The SEM micrographs of the obtained samples aged at different times are displayed in figure 7. The SEM results confirm the synthesis of zeolite LTA with the presence of truncated cubic crystals. For the no aged sample Z-0h, the crystallization is not completed and one can see amorphous gel between the cubic crystals of LTA zeolite. We note that when the aging time increases the size of the crystals decreases. This can be explained by the fact that at longer aging time, there is formation of more pre-nuclei [START_REF] Nibou | Adsorption of Zn2+ ions onto NaA and NaX zeolites: Kinetic, equilibrium and thermodynamic studies[END_REF]. In general, aging at low temperature promotes nucleation process and increases the number of nuclei in the synthesis system because it offers the time required to achieve the formation of nuclei, which leads to the formation of smaller crystals [START_REF] Huang | Synthesis of hierarchical porous zeolite NaY particles with controllable particle sizes[END_REF][START_REF] Valtchev | Transmission Electron Microscopy Study of the Formation of FAU-Type Zeolite at Room Temperature[END_REF].

Effect of H 2 O/Al 2 O 3 molar ratio

To investigate the influence of H 2 O/Al 2 O 3 molar ratio on the nature of obtained products, four hydrogel compositions have been prepared: As can be seen in figure 8, the increase of water quantity does not favors the crystallization of the hydrogel and the obtained solids are amorphous (Z A.Sc 6-h80-H2Ox2 , Z A.Sc 6h- 80-H2Ox3 ). It is established that with the increase of water quantity, there is a decrease of the alkalinity of the system. The dilution diminishes the Si and Al primary units' concentrations which slows the nucleation process. On the other hand, the decrease of the H [START_REF]Synthesis of Phase-Pure Zeolite Sodalite from Clear Solution Extracted from Coal Fly Ash[END_REF]. On the other hand, it is well known that zeolite formation is a kinetically controlled process and the reaction is generally stopped when the desired zeolite has formed.

Ø Z ASc
Extended reaction time at high temperatures and/or high pressures usually results in dense phases.

Concluding remarks

In this study, zeolites type LTA, EMT/FAU intergrowth and HS have been prepared using cheap starting materials; aluminum scraps and industrial sodium metasilicate.

  of LTA zeolite is (3.741 Na 2 O: Al 2 O 3 :1.450 SiO 2 :175.520 H 2 O [30]) which is the sequential optimization by the simplex method of the standard composition of Thomson and Franklin (3.165 Na 2 O :Al 2 O 3 :1.926 SiO 2 :128 H 2 O [31]) to obtain 95.4% of relative crystallinity.

Figure 4

 4 Figure 4 presents the nitrogen adsorption-desorption isotherms of the samples Z NaA.Sc

Figure 2 :Figure 3 :Figure 4 :Figure 5 :

 2345 Figure 2: XRD of samples Z NaA.Sc 6h-80 , Z Ca-A.Sc 6h-80 and Z NH4A.Sc6h-80

Figure 6 :

 6 Figure 6: XRD patterns of LTA zeolites obtained by hydrothermal method at different aging time.

Figure 7 :

 7 Figure 7: SEM micropgraphs of samples aged at different times Z-0h (A), Z-48h (B) and Z-96h (C).

Figure 82 :Figure 9 :

 829 Figure 82 : XRD patterns of variation molar ratio H 2 O/Al 2 O 3 ; Z NaA.Sc 6h-80 H2O/2 , Z NaA.Sc 6h-80 -H2Ox2 ; and Z NaA.Sc 6h-80-H2Ox3 for 6h at 80°C

Figure 5 Erreur!

 5 Figure 1

Figure 6 Figure 7 2 Intensity

 672 Figure 6

2 O: Al 2 O 3 :1.450 SiO 2 :175.520 H 2

  Despite, the prolongation of the crystallization time to 12 h of the aluminosilicate gels prepared from nitrate , chloride, and sulfate, only amorphous materials were obtained. It seems that the increase of the reaction time does not initiate the germination stage to achieve the growth and crystallization of LTA zeolite. Bosnar et al.

investigate the influence of aluminum sources, several preparations have been studied by varying the aluminum precursors; aluminum scraps (Z NaA.Sc 6h-80 ), aluminum powder (Z NaA.P 6h-80 ), aluminum nitrate (Z NaA.N 6h-80 ), aluminum chloride (Z NaA.C 6h-80 ) and aluminum sulfate (Z NaA.S 6h-80 ). The obtained hydrogels with final molar compositions of "3.741 Na O" were aged 14 h at RT and then were crystallized at 80°C for 6 h. From figure 1, one can see that aluminum scraps (Z NaA.Sc6h-80 ) and aluminum powder (Z NaA.P6h-80 ) provide well crystallized Na-LTA zeolite (|Na 94.75 (H 2 O) 39.17 | [Si 96 Al 96 O 384 ] (ICDD No.01-089-8015)

[START_REF] Treacy | Collection of simulated XRD powder patterns for zeolites fifth (5th)[END_REF]

. On the other hand, the use of nitrate (Z NaA.N 6h-80 ), chloride (Z NaA.C 6h-80 ) and sulfate (Z NaA.S 6h-80 ) aluminum precursors leads to amorphous solids.

  , zeolite LTA crystals have four particular morphologies i) regular cubic crystal with sharp edges and apexes when 2 ≤SiO 2 /Al 2 O 3 ≤ 2.2 and Na 2 O/H 2 O ≤ 0.0136, ii) cubic crystals with truncated edges when SiO 2 /Al 2 O 3 ≤ 2.2 and 0.0136 ≤Na 2 O/H 2 O ≤0.0162, iii) cubic crystals with rounded edges and apexes when 1.3 ≤SiO 2 /Al 2 O 3 ≤ 2 and 0.016 ≤ Na 2 O/H 2 O ≤0.024 and iv) pseudo-spherical obtained when SiO 2 /Al 2 O 3 ≤ 1.3 and Na 2 O/H 2 O ≥ 0.037. In our case, we obtained cubic crystal with truncated edges (molar ratio SiO 2 /Al 2 O 3 = 1.45 and Na 2 O/H 2 O= 0.021). A high alkalinity of

.741 Na 2 O: Al 2 O 3 : 1.450 SiO 2 : 92.50 H 2 O

  2 O/Al 2 O 3 molar ratio (Z A.Sc 6h-80-H2O/2 ) leads to the formation of an intergrowth of FAU/EMT contaminated with LTA zeolite. The XRD pattern shows broad peaks which confirmed the ultra-small sized zeolite particles. It is established that FAU ([Na 2 Al 2 Si 4 8H 2 O] (ICDD No.00-039-1380) and EMT ([Na 19.36 (Al 22.27 Si 73.73 O 192 ) (ICDD No. 00-046-0566)) zeolites exhibit a strong tendency to form coherent intergrowth since they are constructed from the same building units. EMT is the structural twin of FAU, i.e. both are composed of D6R connected sodalite cages, but the location of connections is different, resulting in cubic FAU and hexagonal EMT structure, respectively [42-44].On the other hand, the kinetic study of the crystallization of the gel composition: 3(Z A.Sc6h80 ; H2O/2 ) was realized at 80 °C from 6 to 48 h (figure9). As shown before, the XRD profile of the hydrogel after 6 h of crystallization showed FAU/EMT intergrowth contaminated with LTA zeolite. With the increase of crystallization time to 12h, LTA zeolite became the abundant phase. The further prolongation

	of crystallization time to 24 h (Z A.Sc24h80 H2O/2 ), there is appearance of small quantity of
	hydroxysodalite (HS) zeolite (|Na 8 (H 2 O) 8 |[Si 6 Al 6 O 24 ] ICDD No.11-0401) besides LTA and
	FAU/EM. After 48h of crystallization (Z A.Sc 48h-80 H2O/2 ), the product contains mainly HS
	zeolite. From these results, one can conclude that the transformation of the metastable phase
	(zeolite Na-LTA) to more thermodynamically stable phase zeolite HS at longer reaction time,
	illustrated Oswald's rule of successive transformations. The nucleation and crystallization of
	the new phase occur in the supersaturated solution throughout the dissolution of the former
	phase

The optimized hydrogel molar composition 3,741 Na 2 O: Al 2 O 3 :1,450 SiO 2 :175,520 H 2 O which

	leads to
	LTA zeolite has been used. It was shown that aluminum scraps and aluminum powder provide
	well crystallized Na-LTA zeolites with truncated edges and apexes after hydrothermal
	treatment at 80°C for 6h. On the other hand, nitrate, chloride and sulfate aluminum precursors
	lead to amorphous solids. When the water quantity was decreased (3.741 Na 2 O: Al 2 O 3 :1.450
	SiO 2 :92.50 H 2 O) there is formation of an intergrowth of FAU/EMT phases contaminated

with LTA zeolite after 6h of crystallization. The prolongation of the crystallization time to 48 hours leads to the formation of the hydoxysodalite zeolite.

Table captions Table 1 :

 captions1 Chemical analysis, BET surface area (S BET ) and microporous volume (µV ) of Z NaA.Sc 6h-80, Z CaA.Sc 6h-80 and Z NH4A.Sc6h-80 samples.

Table 1 :

 1 Chemical analysis, BET surface area (S BET ) and microporous volume (µV ) of Z NaA.Sc 6h-80, Z CaA.Sc 6h-80 and Z NH4A.Sc6h-80 samples.

	Samples	Si (wt. %)	Al (wt. %)	Na (at. %)	Ca (wt. %)	Si/Al	S BET	µV
						(mol/mol)	(m 2 /g)	(cm 3 /g)
	Z NaA.Sc6h-80	45.74 ± 0.90	37.30 ± 0.98	16.96 ± 1.75	-	1.187 ± 0.010	10.8	0.0026
	Z CaA.Sc6h-80	39.05 ± 1.21	32.67 ± 0.53	1.68 ± 0.16 26.60 ± 1.70	1.148 ± 0.020	550.0	0.2337
	Z NH4A.Sc6h-80	52.24 ± 0.56	40.60 ± 0.15	7.16 ± 0.63		1.236 ± 0.010	85.9	0.0275


Gianluca Rizzi

Manuel Collet

Félix Demore

Bernhard Eidel

Patrizio Neff

Angela Madeo

Exploring metamaterials' structures through the relaxed micromorphic model: switching an acoustic screen into an acoustic absorber

Keywords: mechanical metamaterials, wave-propagation, metamaterials, meta-structure, relaxed micromorphic model

While the design of always new metamaterials with exotic static and dynamic properties is attracting deep attention in the last decades, little effort is made to explore their interactions with other materials. This prevents the conception of (meta-)structures that can enhance metamaterials' unusual behaviours and that can be employed in real engineering applications. In this paper, we give a first answer to this challenging problem by showing that the relaxed micromorphic model with zero static characteristic length can be usefully applied to describe the refractive properties of simple meta-structures for extended frequency ranges and for any direction of propagation of the incident wave. Thanks to the simplified model's structure, we are able to efficiently explore different configurations and to show that a given meta-structure can drastically change its overall refractive behaviour when varying the elastic properties of specific meta-structural elements. In some cases, changing the stiffness of a homogeneous material which is in contact with a metamaterial's slab, reverses the structure's refractive behaviour by switching it from an acoustic screen (total reflection) into an acoustic absorber (total transmission). The present paper clearly indicates that, while the study and enhancement of the intrinsic metamaterials' properties is certainly of great importance, it is even more challenging to enable the conception of meta-structures that can eventually boost the use of metamaterials in real-case applications.

Introduction

The last decade has seen the birth of a true research outburst on so-called mechanical metamaterials which are able to show exotic mechanical properties both in the static and dynamic regime. Theoretical, experimental and numerical studies have flourished all around the world providing new insights in the domain of material properties manipulation which, only few years ago, was thought far from being prone to possible ground-breaking evolutions. We are today assisting to the conception and subsequent realization of new materials which, simply thanks to their internal architecture, go beyond the materials' mechanical properties that we are used to know and which, for this reason, are called mechanical metamaterials. Already in the late 1980s, it was proven that some foams with special internal architecture can give rise to 'negative Poisson' effects, i.e., they fatten when stretched, contrarily to what happens to the great majority of known materials which experience a reduction in the cross-section when submitted to tensile loads [START_REF] Lakes | Foam structures with a negative Poisson's ratio[END_REF]. More recently, the frontiers of metamaterials' conception are rapidly moving forward, giving rise to the manufacturing of always new metamaterials with more and more impressive properties. It is thus possible today to see 3D-printed pyramids connected by hinges giving rise to a block that is hard like a brick on one side but soft like a sponge on the other [START_REF] Bilal | Intrinsically polar elastic metamaterials[END_REF], "unfeelability" cloaks hiding to the touch objects put below them [START_REF] Kadic | metamaterials with independently tailored bulk modulus and mass density[END_REF][START_REF] Milton | Which elasticity tensors are realizable[END_REF], plastic cubes made out of smaller plastic cubes giving rise to bizarre deformations when squeezed [START_REF] Coulais | Combinatorial design of textured mechanical metamaterials[END_REF], or even metamaterials exploiting microstructural instabilities to change their mechanical response depending on the level of externally applied load [START_REF] Kochmann | Exploiting microstructural instabilities in solids and structures: from metamaterials to structural transitions[END_REF]. When considering the dynamical behaviour of mechanical metamaterials, things become, if possible, even more impressive, given the unusual responses that such metamaterials can provide when coming in contact with elastic waves [START_REF] Barchiesi | Mechanical metamaterials: a state of the art[END_REF][START_REF] Deymier | Acoustic metamaterials and phononic crystals[END_REF][START_REF] Hussein | Dynamics of phononic materials and structures: Historical origins, recent progress, and future outlook[END_REF]. It is today possible to find researchers working on metamaterials exhibiting band-gaps [START_REF] Bilal | Architected lattices for simultaneous broadband attenuation of airborne sound and mechanical vibrations in all directions[END_REF][START_REF] Celli | Bandgap widening by disorder in rainbow metamaterials[END_REF][START_REF] El Sherbiny | Discrete and continuous aspects of some metamaterial elastic structures with band gaps[END_REF][START_REF] Goh | Inverse metamaterial design for controlling band gaps in scalar wave problems[END_REF][START_REF] Koutsianitis | Conventional and star-shaped auxetic materials for the creation of band gaps[END_REF][START_REF] Liu | Locally resonant sonic materials[END_REF][START_REF] Wang | Harnessing buckling to design tunable locally resonant acoustic metamaterials[END_REF][START_REF] Zhu | Study of anomalous wave propagation and reflection in semi-infinite elastic metamaterials[END_REF], cloaking [START_REF] Bückmann | Mechanical cloak design by direct lattice transformation[END_REF][START_REF] Misseroni | Cymatics for the cloaking of flexural vibrations in a structured plate[END_REF][START_REF] Misseroni | Omnidirectional flexural invisibility of multiple interacting voids in vibrating elastic plates[END_REF][START_REF] Norris | Active elastodynamic cloaking[END_REF], focusing [START_REF] Cummer | Controlling sound with acoustic metamaterials[END_REF][START_REF] Guenneau | Acoustic metamaterials for sound focusing and confinement[END_REF], channelling [START_REF] Bordiga | Prestress tuning of negative refraction and wave channeling from flexural sources[END_REF][START_REF] Kaina | Slow waves in locally resonant metamaterials line defect waveguides[END_REF][START_REF] Miniaci | Valley-based splitting of topologically protected helical waves in elastic plates[END_REF][START_REF] Tallarico | Edge waves and localization in lattices containing tilted resonators[END_REF][START_REF] Wang | Channeled spectrum in the transmission of phononic crystal waveguides[END_REF], negative refraction [START_REF] Bordiga | Prestress tuning of negative refraction and wave channeling from flexural sources[END_REF][START_REF] Lustig | Anomalous energy transport in laminates with exceptional points[END_REF][START_REF] Morini | Negative refraction in quasicrystalline multilayered metamaterials[END_REF][START_REF] Srivastava | Metamaterial properties of periodic laminates[END_REF][START_REF] Willis | Negative refraction in a laminate[END_REF][START_REF] Zhu | Study of anomalous wave propagation and reflection in semi-infinite elastic metamaterials[END_REF], etc., as soon as they interact with mechanical waves.

Notwithstanding the massive research efforts deployed to unveil new metamaterials' performances, researchers have just begun to understand the underlying mechanisms, so that "many designs so far have relied on luck and intuition" [START_REF] Bertoldi | Flexible mechanical metamaterials[END_REF].

In order to provide deeper theoretical insight into the mechanisms which allow to tailor metamaterials' mechanical properties, so-called homogenization techniques have been developed which provide rigorous predictions of the macroscopic metamaterials' mechanical behaviour, when knowing the properties of the base materials and their spatial distribution. Homogenization techniques have proven their effectiveness for the description of metamaterials' bulk behaviour in the static and quasi-static regime [START_REF] Allaire | Homogenization and two-scale convergence[END_REF][START_REF] Bensoussan | Asymptotic analysis for periodic structures[END_REF][START_REF] Bouchitté | Homogenization of a soft elastic material reinforced by fibers[END_REF][START_REF] Camar-Eddine | Determination of the closure of the set of elasticity functionals[END_REF][START_REF] Geers | Multi-scale computational homogenization: Trends and challenges[END_REF][START_REF] Hashin | A variational approach to the theory of the elastic behaviour of multiphase materials[END_REF][START_REF] Hill | Elastic properties of reinforced solids: some theoretical principles[END_REF][START_REF] Miehe | Computational homogenization analysis in finite plasticity simulation of texture development in polycrystalline materials[END_REF][START_REF] Milton | The Theory of Composites[END_REF][START_REF] Pideri | A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium[END_REF][START_REF] Sánchez-Palencia | Non-homogeneous media and vibration theory[END_REF][START_REF] Suquet | Elements of Homogenization for Inelastic Solid Mechanics, Homogenization Techniques for Composite Media[END_REF][START_REF] Willis | Bounds and self-consistent estimates for the overall properties of anisotropic composites[END_REF] as well as, more recently, in the dynamic regime [START_REF] Andrianov | Higher order asymptotic homogenization and wave propagation in periodic composite materials[END_REF][START_REF] Bacigalupo | Second-gradient homogenized model for wave propagation in heterogeneous periodic media[END_REF][START_REF] Boutin | Large scale modulation of high frequency waves in periodic elastic composites[END_REF][START_REF] Chen | A dispersive model for wave propagation in periodic heterogeneous media based on homogenization with multiple spatial and temporal scales[END_REF][START_REF] Craster | High-frequency homogenization for periodic media[END_REF][START_REF] Hu | Nonlocal homogenization model for wave dispersion and attenuation in elastic and viscoelastic periodic layered media[END_REF][START_REF] Sridhar | A general multiscale framework for the emergent effective elastodynamics of metamaterials[END_REF][START_REF] Srivastava | On the limit and applicability of dynamic homogenization[END_REF][START_REF] Srivastava | Evanescent wave boundary layers in metamaterials and sidestepping them through a variational approach[END_REF][START_REF] Willis | Exact effective relations for dynamics of a laminated body[END_REF][START_REF] Willis | Effective constitutive relations for waves in composites and metamaterials[END_REF][START_REF] Willis | The construction of effective relations for waves in a composite[END_REF].

While homogenization methods are effective to describe bulk metamaterials' behaviours, they are intrinsically unsuitable to deal with metamaterials of finite size, because the 'average operations', on which they are built, make strong use of projection functions (e.g., Bloch-Floquet ones) that are defined in unbounded function spaces [START_REF] Sridhar | Homogenization of locally resonant acoustic metamaterials towards an emergent enriched continuum[END_REF][START_REF] Srivastava | Evanescent wave boundary layers in metamaterials and sidestepping them through a variational approach[END_REF].

As a result of this gap, the response of finite-size metamaterials' structures is today mostly explored via direct Finite Element (FEM) simulations that implement all the details of the involved microstructures (e.g., [START_REF] Krushynska | Coupling local resonance with Bragg band gaps in single-phase mechanical metamaterials[END_REF]). Despite the precise propagation patterns that these direct numerical simulations can provide, they suffer from unsustainable computational costs. Therefore, it is impossible today to explore large-scale meta-structures combining metamaterials and classical-materials bricks of different type, size and shape.

A first answer to this problem has been given by the introduction of the so-called relaxed micromorphic model without curvature contribution that has recently proven its effectiveness for the description of the mechanical behaviour of a specific finite-size band-gap metamaterial with tetragonal symmetry [START_REF] Agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF].

When compared to other techniques, this micromorphic approach shows its advantages at the considered macroscopic scale. In particular, with respect to the direct finite element implementation, it allows a rapid calculation (few hours vs. some weeks for the FEM simulation) of the metamaterials slab's refractive properties for all angles of incidence and considered frequency ranges. With respect to dynamical and high-frequency homogenization, the presented approach allows to effectively deal with the finite-size of the metamaterial's slab thanks to the introduction of well posed interface conditions guaranteeing the uniqueness of the searched solution.

Finally, other enriched continuum models such as couple stress or strain gradient models cannot describe band-gap metamaterials because they feature the same kinematics of a classical Cauchy continuum (only the displacement field). This implies that they can at best describe some dispersion in the acoustic curves, but cannot reproduce higher frequency optic modes.

In the present paper, we want to move beyond these first encouraging results and show how the relaxed micromorphic model without curvature contribution can be used to characterize many other tetragonal band-gap metamaterials that can be used for acoustic applications. To this aim, we will apply the inverse fitting procedure presented in [START_REF] Agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF][START_REF] Neff | Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization[END_REF] to different metamaterials thus providing their mechanical description via the relaxed micromorphic model without curvature contribution. 1 We will then explore how the behaviour of these metamaterials can be profoundly changed by simply coupling them to classical homogeneous materials, realizing what we will call meta-structures, which can have very different mechanical properties when compared to those of the original metamaterial. The exploration of these new meta-structures is made possible thanks to the simplified structure of the relaxed micromorphic model without curvature contribution that allows quick computations of different structures obtained by embedding the selected metamaterial in different homogeneous materials. We show how the simple fact of changing the elastic properties of the external homogeneous material allows us to switch the structure's behaviour from total reflection to total transmission and vice-versa. It is clear that this new possibility of exploring different combinations of finite-size metamaterials and classicalmaterials opens concrete perspectives for the true employment of metamaterials in engineering design. Indeed, metamaterials' reflection and transmission properties have been analysed so far, always referring to the particular arrangement of their internal architecture so as to modify the metamaterial's reflection/transmission behavior when the metamaterial is embedded, e.g., in air. Although such approaches can lead to the design of tunable interfaces allowing total transmission or total reflection depending on the topological microstructure's properties [START_REF] Park | Zero-reflection acoustic metamaterial with a negative refractive index[END_REF][START_REF] Zhao | Tunable transmission and deterministic interface states in double-zero-index acoustic metamaterials[END_REF][START_REF] Zhou | Complete transmission through a periodically perforated rigid slab[END_REF], they are not suitable to explore the behaviour of the same metamaterials when they are used as building blocks of more complex structures that also contain other metamaterials and/or classical-materials elements, especially when these structures are widely extended in space.

As a matter of fact, these metamaterials that are conceived to act as perfect screens or perfect absorbers, could drastically change their response when embedded in homogeneous media or combined to other metamaterials. Little attention is given today to the need of exploring the response of "combined" classical-materials/metamaterials structures and this is preventing us from designing realistic metastructures that control elastic waves and recover energy. Many authors have recognized the interest of studying the refractive behaviour of metamaterials when combined to homogeneous materials to enhance unorthodox responses such as wave filtering [START_REF] Bigoni | Statics and dynamics of structural interfaces in elasticity[END_REF] or negative refraction [START_REF] Brun | Dynamics of structural interfaces: filtering and focussing effects for elastic waves[END_REF][START_REF] Willis | Negative refraction in a laminate[END_REF]. In a similar spirit, we present in this paper a simple meta-structure composed of a tetragonal metamaterial embedded in a homogeneous medium and we explore its wave-filtering response. The generality of our approach would allow us to study the response of the other structures cited so far [START_REF] Bigoni | Statics and dynamics of structural interfaces in elasticity[END_REF][START_REF] Brun | Dynamics of structural interfaces: filtering and focussing effects for elastic waves[END_REF][START_REF] Misseroni | Omnidirectional flexural invisibility of multiple interacting voids in vibrating elastic plates[END_REF][START_REF] Willis | Bounds and self-consistent estimates for the overall properties of anisotropic composites[END_REF][START_REF] Willis | Exact effective relations for dynamics of a laminated body[END_REF][START_REF] Willis | Negative refraction in a laminate[END_REF] and to highlight how the structure's response could be modified when modifying simple parameters like the outer medium elasticity or the metamaterial's characteristic size.

In this paper we show how the simple fact of assembling a finite-size metamaterial together with blocks of classical-materials can dramatically change the response of the metamaterial itself in such a way that the same structure may act as a complete absorber (total transmission) or a complete screen (total reflection). This switch of the structure's dynamical properties is obtained by keeping the same geometry and the same metamaterial, while changing the homogeneous material in which the metamaterial itself is embedded. These results have been made possible thanks to the use of the relaxed micromorphic model without curvature contribution that, drastically reducing the computational time of the associated numerical simulations, could open the effective exploration of these new meta-structures. We clearly show that, while the interest of studying the intrinsic metamaterial's properties by engineering its microstructure is certainly of great importance, it is even more important to unveil the effects of its interactions with other finite-size metamaterials and classical-materials bricks. It is indeed based on these interactions that it is possible to unfold new meta-structures which can further enhance the properties of the base metamaterials thus opening the way to realistic applications.

Notation

We recall here the notation that we will use throughout the paper. Let R 3×3 be the set of all real 3 × 3 second order tensors which we denote by capital letters. A simple and a double contraction between tensors of any suitable order is denoted by • and : respectively, while the scalar product of tensors of suitable order is denoted by •, • . 2 The Einstein sum convention is implied throughout this text unless otherwise specified. The standard Euclidean scalar product on R 3×3 is given by X, Y = tr(X • Y T ) and consequently the Frobenius tensor norm is X 2 = X, X . The identity tensor on R 3×3 will be denoted by 1; then, tr(X) = X, 1 . We denote by B L a bounded domain in R 3 , by δB L its regular boundary and by Σ any material surface embedded in B L . The outward unit normal to δB L will be denoted by ν as will the outward unit normal to a surface Σ embedded in B L . Given a field a defined on the surface Σ, we define the jump of a through the surface Σ as:

a = a + -a -, with a -:= lim x∈B - L \Σ x→Σ a,
and a + := lim

x∈B + L \Σ x→Σ a, (1)
where B - L , B + L are the two subdomains which result from splitting B L by the surface Σ.

Classical gradient ∇ and divergence Div operators are used throughout the paper. 3 The subscript , j indicates derivation with respect to the j-th component of the space variable, while the subscript , t denotes derivation with respect to time. 4Given a time interval [0, t 0 ], the classical macroscopic displacement field is denoted by u(x, t) ∈ R 3 , with x ∈ B L , t ∈ [0, t 0 ]. In the framework of enriched continuum models of the micromorphic type, extra degrees of freedom are added through the introduction of the micro-distortion tensor P denoted by P (x, t) ∈ R 3×3 , with x ∈ B L , t ∈ [0, t 0 ]. This tensor can be related to micro-deformation mechanisms that take place at the scale of the unit cell and can account for dilation/compression-, shear-and rotationlike vibration modes, when different frequency levels are considered.

2 Equilibrium equations, constitutive relations, and energy flux

Isotropic Cauchy continuum

The equilibrium equations for the Cauchy continuum are

ρ u ,tt = Div [σ] , (2)
where σ is the Cauchy stress tensor. In the isotropic case, it takes the constitutive form σ = 2µ sym∇u + λ tr (sym∇u) 1, where λ and µ are the Lamé parameters and sym∇u is the strain tensor.

When dissipative phenomena can be neglected, the following flux equation must hold:

E ,t + DivH = 0 , ( 3
)
where E is the total energy of the system and H is the energy flux vector, whose explicit expression is given by (see e.g. [START_REF] Aivaliotis | Low-and high-frequency stoneley waves, reflection and transmission at a cauchy/relaxed micromorphic interface[END_REF] for a detailed derivation)

H = -σ • u ,t . (4)

Relaxed micromorphic continuum with zero static characteristic length

The equilibrium equations are obtained by looking for stationary points of the following action functional:

A = t0 0 B L (J -W ) dXdt ( 5
)
where J is the kinetic energy density and W is the strain energy density of the considered micromorphic continuum.

In particular, the expression of the kinetic energy density takes the form [START_REF] Agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF][START_REF] Madeo | Relaxed micromorphic modeling of the interface between a homogeneous solid and a band-gap metamaterial: New perspectives towards metastructural design[END_REF][START_REF] Madeo | Modeling phononic crystals via the weighted relaxed micromorphic model with free and gradient micro-inertia[END_REF][START_REF] Romano | Micromorphic continua: non-redundant formulations[END_REF]]:

5 J (u ,t , ∇u ,t , P ,t ) = 1 2 ρ u ,t , u ,t + 1 2 J micro sym P ,t , sym P ,t + 1 2 J c skew P ,t , skew P ,t + 1 2 T e sym∇u ,t , sym∇u ,t + 1 2 T c skew∇u ,t , skew∇u ,t , (6)
where u is the macroscopic displacement field, P ∈ R 3×3 is the non-symmetric micro-distortion tensor, ρ is the macroscopic apparent density, and J micro , J c , T e , T c are 4th order micro-inertia tensors whose form will be specified in the following subsection.

The relaxed micromorphic continuum contains curvature terms connected to CurlP . Here, we use the relaxed micromorphic continuum without curvature contribution. The expression of the strain energy density without curvature contribution ( [START_REF] Agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF][START_REF] Madeo | Relaxed micromorphic modeling of the interface between a homogeneous solid and a band-gap metamaterial: New perspectives towards metastructural design[END_REF][START_REF] Madeo | Modeling phononic crystals via the weighted relaxed micromorphic model with free and gradient micro-inertia[END_REF][START_REF] Romano | Micromorphic continua: non-redundant formulations[END_REF]:

µ L 2 c 2 CurlP 2 = 0, L c = 0) is
W (∇u, P ) = 1 2 C e sym (∇u -P ) , sym (∇u -P ) + 1 2 C micro sym P, sym P + 1 2 C c skew (∇u -P ) , skew (∇u -P ) , (7)
where C e , C micro , and C c are 4th order tensors whose characteristic will be given in Sec. 2.3. The minimization of the Action functional, eq. ( 5), while using eq. ( 6)-( 7) provides the following equilibrium equations ρ u ,tt -Div ( σ ,tt ) = Div ( σ) ,

(J micro + J c ) P ,tt = σ -s, (8)
The flux equation for the relaxed micromorphic continuum is formally the same as eq. ( 3), but H has now the following expression (see [START_REF] Aivaliotis | Relaxed micromorphic broadband scattering for finite-size meta-structures-a detailed development[END_REF] for more details):

H = -( σ + σ) T • u ,t . (10)

Particularization of the relaxed micromorphic model to plane strain and tetragonal symmetry

We now focus on finding solutions in a plane strain framework. This means that we constrain the displacement field u and the micro-distortion P to depend only on the first two components x 1 and x 2 of the space variable x:

u(x 1 , x 2 ) =   u 1 (x 1 , x 2 ) u 2 (x 1 , x 2 ) 0   , P (x 1 , x 2 ) =   P 11 (x 1 , x 2 ) P 12 (x 1 , x 2 ) 0 P 21 (x 1 , x 2 ) P 22 (x 1 , x 2 ) 0 0 0 0   . ( 11
)
Given the metamaterials targeted in this paper (see Section 4), we particularise the equilibrium equations to the tetragonal case. This means that the elastic and micro inertia tensors appearing in eq. ( 6)-( 7) can be represented in the Voigt form as:

C e =      λ e +
• • µ * e      , C c =    • • . . . . . . • . . . 4µ c    , C micro =      λ micro + 2µ micro λ micro . . . • λ micro λ micro + 2µ micro . . . • . . . . . . . . . • • µ * micro      , (12)
J micro = ρ      L 2 3 + 2L 2 1 L 2 3 . . . • L 2 3 L 2 3 + 2L 2 1 . . . • . . . . . . . . . • • • • L * 2 1      , J c = ρ    • • . . . . . . • . . . 4L 2 2    , T e = ρ       L 2 3 + 2L 2 1 L 2 3 . . . • L 2 3 L 2 3 + 2L 2 1 . . . • . . . . . . . . . • • L * 2 1       , T c = ρ    • • . . . . . . • . . . 4L 2 2    , (13)
where only the coefficients involved in a plane strain problem are reported (the dots represent components acting on out-of plane variables and are not specified here). 6In the definition (13) of the micro-inertia tensors appearing in the kinetic energy [START_REF] Andrianov | Higher order asymptotic homogenization and wave propagation in periodic composite materials[END_REF], it is underlined the fact that they introduce dynamic internal lengths that can be directly related to the dispersion behaviour of the metamaterial at very small (in the limit vanishing) wavenumbers (J micro ,J c ), as well as at very large (in the limit infinite) wavenumbers (T e , T c ).

3 Boundary conditions for a finite-size relaxed micromorphic slab embedded between two Cauchy half-spaces

Two half-spaces made up of a homogeneous Cauchy material are separated by a micromorphic slab of finite width h. The three materials are in perfect contact with each other: the material on the top of the first interface is a classical linear elastic isotropic Cauchy medium, the material in the middle is an anisotropic relaxed micromorphic medium, while the material on the bottom of the second interface is again a classical isotropic Cauchy medium (see Fig. 1). As it can be seen in [START_REF] Aivaliotis | Low-and high-frequency stoneley waves, reflection and transmission at a cauchy/relaxed micromorphic interface[END_REF][START_REF] Aivaliotis | Microstructure-related stoneley waves and their effect on the scattering properties of a 2d cauchy/relaxed-micromorphic interface[END_REF][START_REF] Madeo | Reflection and transmission of elastic waves in non-local band-gap metamaterials: a comprehensive study via the relaxed micromorphic model[END_REF] there are two boundary conditions which can be imposed at a Cauchy/relaxed-micromorphic interface if the static characteristic length (L c ) is zero (our case here): the continuity of displacement and continuity of generalized traction.

q x 2 x 1 +¥ -¥ +¥ -¥ k h/2 h/2
In the considered 2D case, there are then eight sets of scalar conditions, four on each interface. The finite slab has width h and we assume that the two interfaces are positioned at x 1 = -h/2 and x 1 = h/2, respectively (see Fig. 1). The continuity of displacement conditions to be satisfied at the two interfaces of the slab are:

u - c = u s , on x 1 = - h 2 , u s = u + c , on x 1 = h 2 , ( 14
)
where u - c and u + c are the displacement of the "minus" (x 1 < 0) and "plus"(x 1 > 0) Cauchy half-space, respectively. As for the continuity of generalized traction, we have:

t - c = t s , on x 1 = - h 2 , t s = t + c , on x 1 = h 2 , ( 15
)
where t ± c = σ ± • ν ± are classical Cauchy tractions, t s = ( σ + σ) • ν is the generalized traction in the relaxed micromorphic medium, with ν being the outward unit normal to the surface considered (see [START_REF] Aivaliotis | Frequency-and angle-dependent scattering of a finite-sized meta-structure via the relaxed micromorphic model[END_REF][START_REF] Agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF] and eq. ( 9) for details about the definitions of generalized tractions).

2D tetragonal microstructures for acoustic control

In view of the conception of meta-structures for applications in acoustic control, we consider here three tetragonal unit cells which give rise to three different metamaterials at the macroscopic scale. These metamaterials will then be characterised through the relaxed micromorphic model, thus widening the set of tetragonal microstructures that have been characterised so far with this new model (see [START_REF] Aivaliotis | Frequency-and angle-dependent scattering of a finite-sized meta-structure via the relaxed micromorphic model[END_REF][START_REF] Agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF] for the characterization of an ultrasound microstructure). The selected microstrutures are shown in Fig. 2. They all show band-gaps for relatively low frequencies, especially the one presented in Fig. 2(a) which completely falls in the acoustic frequencies range (see dispersion curves Fig. 34). Given the particular distribution of voids in the unit cells, these microstructures result to be by far stiffer in compression than in shear. This can be observed in Fig. 34, by remarking that for a wave propagating along the horizontal direction (θ = 0), the acoustic "pressure" wave is by far steeper than the "shear" wave (see Fig. 3(a)). This difference is lost for other directions of propagation (see, e.g., Fig. 3

(b)-3(c)).
All the simulations that are reported in this section and in the next one have been carried on with the software Mathematica for what concerns the relaxed micromorphic semi-analytical solutions, and with the software Comsol for the detailed discrete numerical solutions(see [START_REF] Aivaliotis | Frequency-and angle-dependent scattering of a finite-sized meta-structure via the relaxed micromorphic model[END_REF] for more details).

Dispersion curves and calibration of the relaxed micromorphic elastic parameters

In this section we provide the relaxed-micromorphic characterization of the three metamaterials previously introduced (Fig. 2) by means of the fitting procedure developed in [START_REF] Agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF][START_REF] Neff | Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization[END_REF]. This fitting procedure is based on two different steps aimed at separately characterizing the metamaterial on the static and dynamic regime [START_REF] Agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF][START_REF] Neff | Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization[END_REF]. In particular, the static parameters are identified by remarking that the relaxed micromorphic model tends to a macroscopic equivalent Cauchy medium of stiffness λ macro , µ macro , and µ * macro , when considering the long-wave limit (small frequencies and small wave numbers). These macroscopic parameters can be identified by classical numerical homogenization. They are obtained by imposing periodic boundary conditions that mimic an infinitely extended structure [START_REF] Agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF][START_REF] Neff | Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization[END_REF]. On the other hand, the micro-parameters λ micro , µ micro , and µ * micro are identified by imposing kinematic uniform boundary conditions on different but equivalent and symmetry-preserving unit cells [START_REF] Neff | Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization[END_REF]. It is important to underline that the micro-parameters so identified estimate is just a lower bound. The parameters λ e , µ e , µ * e are uniquely identified as a combination of the macro-and micro-elastic parameters thanks to the homogenization formulas derived in [START_REF] Agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF][START_REF] Neff | Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization[END_REF]:

µ e = µ macro µ micro µ micro -µ macro , µ * e = µ * macro µ * micro µ * micro -µ * macro , κ e = κ macro κ micro κ micro -κ macro , with    κ i = 2µ i + 3λ i 3 i = {e, micro, macro} . (16)
As for the dynamic parameters, four of them can be computed by considering the limiting case k → 0 (vanishing wavenumbers). In particular, the parameters L 1 , L 2 , L 3 , and L * 1 can be determined by imposing the cut-off of the relaxed micromorphic model to be equal to the corresponding numerical values obtained, e.g., by Bloch-Floquet analysis. Indeed, the cut-offs frequencies of the relaxed micromorphic model are found to be given by [START_REF] Agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF]:

ω r = µ c ρ L 2 2 , ω s = µ e + µ micro ρ L 2 1 , ω * s = µ * e + µ * micro ρ L * 2 1 , ω p = µ e + λ e + µ micro + λ micro ρ (L 2 1 + L 2 3 )
.

The remaining dynamic parameters L 1 , L 2 ,L 3 , and L * 1 are found to have a strong effect on the dispersion curves when k → ∞ and they are determined by inverse approach to reach the best possible fitting of the Bloch-Floquet dispersion curves [START_REF] Aivaliotis | Frequency-and angle-dependent scattering of a finite-sized meta-structure via the relaxed micromorphic model[END_REF][START_REF] Agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF].

Relaxed micromorphic characterization of the metamaterial MM1

As a result of the fitting procedure briefly summarized before, the metamaterial MM1 (see Fig. 2(a)) results to be characterized via the relaxed micromorphic parameters give in Table 1 (a). Table 1 (b) provides the corresponding values of the Cauchy medium obtained as the long wave limit of the relaxed micromorphic medium of Table 1 Table 1: Panel (a) shows the values of the relaxed micromorphic static and dynamic parameters for the metamaterial MM1 determined via the fitting procedure given in [START_REF] Aivaliotis | Frequency-and angle-dependent scattering of a finite-sized meta-structure via the relaxed micromorphic model[END_REF][START_REF] Agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF]. The apparent density ρ is computed based on the titanium microstructure of Fig. 2(a). Panel (b) shows the values of the equivalent Cauchy continuum elastic coefficients corresponding to the long-wave limit of MM1 computed with the procedure explained in [START_REF] Neff | Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization[END_REF]. Figure 3 shows the comparison of the dispersion curves obtained via the relaxed micromorphic model with those issued via Bloch-Floquet analysis, for three different directions of propagation.

L 1 [m] L 2 [m] L 3 [m] L * 1 [m] 0.100 1.24908 × 10 -3 2.02572 × 10 -2 2.44985 × 10 -2 L 1 [m] L 2 [m] L 3 [m] L * 1 [m] 4.5639 × 10 -

Relaxed micromorphic characterization of the metamaterials MM2 and MM3

Following the aforementioned procedure, the metamaterials MM2 and MM3 are characterized through the relaxed micromorphic parameters given in Table 2 and Table 3, respectively. The resulting fitting of the dispersion curves is shown in Fig. 4. 5.9 × 10 9 5.984 × 10 9

L 1 [m] L 2 [m] L 3 [m] L * 1 [m] 0.0272027 0.000209403 0.021232 0.0258829 L 1 [m] L 2 [m] L 3 [m] L * 1 [m] 0.0000368211 0 0.0000368211 0.0000260365 ρ [kg/m 3 ] µ c [Pa]
3840.77 100000 Table 2: Panel (a) shows the values of the relaxed micromorphic static and dynamic parameters for the metamaterial MM2 determined via the fitting procedures given in [START_REF] Aivaliotis | Frequency-and angle-dependent scattering of a finite-sized meta-structure via the relaxed micromorphic model[END_REF][START_REF] Agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF]. The apparent density ρ is computed based on the titanium microstructure of Fig. 2(c). Panel (b) shows the values of the equivalent Cauchy continuum elastic coefficients corresponding to the long-wave limit of MM2 computed with the procedure explained in [START_REF] Neff | Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization[END_REF].

L 1 [m] L 2 [m] L 3 [m] L * 1 [m] 0.0353782
3.62851 × 10 -7 0.0129766 0.028261 Table 3: Panel (a) shows the values of the relaxed micromorphic static and dynamic parameters for the metamaterial MM3 determined via the fitting procedures given in [START_REF] Aivaliotis | Frequency-and angle-dependent scattering of a finite-sized meta-structure via the relaxed micromorphic model[END_REF][START_REF] Agostino | Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model[END_REF]. The apparent density ρ is computed based on the steel microstructure of Fig. 2(e). Panel (b) shows the values of the equivalent Cauchy continuum elastic coefficients corresponding to the long-wave limit of MM3 computed with the procedure explained in [START_REF] Neff | Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization[END_REF].

L 1 [m] L 2 [m] L 3 [m] L * 1 [m] 9.30984 × 10 -

Metastructure's refractive behaviour

In this section we will show how the relaxed micromorphic model can be suitably used to describe the reflective properties of a metamaterial's slab embedded in a homogeneous material (see Fig. 5). In what follows, we will restrict ourselves to the microstructure defined in Fig. 2(a), since the results for the other microstructures are analogous. We will start by considering the simpler case in which the external homogeneous material is the same as the one used for the metamaterial MM1 (see Fig. 2(a) for its elastic characteristics) and we will then explore how the reflective metamaterial's behaviour changes when changing the outer Cauchy material properties. As we will see, this will allow us to explore the effect of the wavelength of the incident wave on the performances of the relaxed micromorphic model. We will then study how the meta-structure's behaviour as well as the performances of the relaxed micromorphic model vary when increasing the number of unit cells in the metamaterial's slab.

q x 2 x 1 +¥ -¥ +¥ -¥ k h/2 h/2
Figure 5: Schematic representation of a wave with wavenumber k hitting at angle θ a microstructured material slab of thickness h embedded between two isotropic Cauchy media.

Dependence of the metastructure's reflective behaviour on the elastic properties of the outer Cauchy materials

We will show in this subsection that the refractive behaviour of the used metamaterial can be drastically modified by simply changing the elastic properties of the outer Cauchy materials. More precisely, the simple fact of changing the properties of the external material can actually "reverse" the metamaterial slab's refractive behaviour (from total reflection to total transmission and vice-versa). This drastic change can be engineered for an extended range of frequencies and angles of incidence. In order to drive the exploration of the more performing structures, we took advantage of the computational performance of the relaxed micromorphic model that allowed us to test different structures in an otherwise unreachable limited time. We show in Fig. 6 the acoustic dispersion curves of the internal metamaterial, as compared to those of three different "outer" Cauchy materials. We can see from Fig. 6 that, in all considered cases, the external Cauchy material is relatively stiffer than the internal metamaterial in the long-wave limit, especially with the reference to the shear modulus. This can be inferred by recalling that the slope of the acoustic curves at the origin represents the speed of propagation (c p and c s ) of the corresponding waves (see Table 4) and that these speeds are directly related to the metamaterial's elastic properties in the long wave limit (see Table 1).

Macro wave speed of MM1 [m/s]

c 0 p = λ macro + 2µ macro ρ macro c 0 s = µ * macro ρ macro c 45 p = λ macro + µ macro + µ * macro ρ macro c 45 s = µ macro ρ macro 927.28 [m/s] 18.04 [m/s] 662.36 [m/s] 649.20 [m/s]
Table 4: Wave speed expressions for the Cauchy material which is the long-wave limit of MM1. The expressions for the pressure and shear waves are explicitly given for the two directions of propagation θ = 0 and θ = π/4. Such speeds are the slopes of the tangents at the origin of the acoustic dispersion curves in Fig. 3(a) and Fig. 3(b) and are computed based on the macro parameters given in the table in Fig. 2 The three outer Cauchy materials have been chosen starting from the same material as the one constituting the metamaterial MM1 (titanium) and then lowering the propagation speeds c p and c s so as to widen the range of frequencies for which the relaxed micromorphic model gives quantitatively good results. 7 Indeed, with reference to Figs. 7-8-9, we can remark that the frequency interval for which the relaxed micromorphic model is predictive of the microstructure's reflective behaviour is larger for the "softer" outer Cauchy material CM3 (Fig. 9). Indeed, the fact of considering a "softer" outer material is equivalent to say that, at any given frequency, the corresponding wavelength of the travelling incident wave is smaller than the one of the wave propagating in the "stiffer" material. It is known that, as far as a homogenized model is concerned, its accuracy for the study of a problem of the type presented in this paper, may depend on three different characteristic lengths, namely:

• the wavelength of the travelling incident wave;

• the thickness of the metamaterials's slab;

• the characteristic size of the unit cell.

Having fixed the unit cell's dimensions for applications in acoustic control, we do not discuss here the influence of the third characteristic length. As for the influence of the thickness of the metamaterial's slab, we refer to Section 5.2.

Here, we limit ourselves to remark that, given the intrinsic simplifications associated to a continuum model, a threshold value t for the wavelength of the incident wave exists, below which the model starts loosing its predictive capabilities. For the meta-structures of Fig. 7-8 this threshold value is reached already for frequencies slightly higher than the metamaterial's band-gap. When considering the "softer" outer Cauchy material CM3, the wavelength of the incident wave remains lower than the threshold t for a larger frequency range that exceeds the band-gap (see Fig. 9). Being aware of the existence of such threshold t is essential for a correct use of enriched continuum models over the appropriate frequency ranges.

Dependence of the metamaterial's reflective behaviour on the thickness of the slab

In this subsection, we show to which extent the reflective behaviour of the metamaterial's slab is influenced by the thickness of the slab itself. At the same time, we are able to show that the performances of the relaxed micromorphic model increases when increasing the thickness of the slab. Indeed, by comparison of Fig. 10 with Fig. [START_REF] Bacigalupo | Second-gradient homogenized model for wave propagation in heterogeneous periodic media[END_REF], it is possible to infer that the fact of increasing the slab's thickness mainly acts on the number and amplitude of oscillations that occur in the reflection coefficient for frequencies lower and higher than the band-gap. Moreover, it can be seen from these figures that the performances of the relaxed micromorphic model is improved when increasing the number of unit cells constituting the metamaterial slab embedded in the Cauchy material CM1. Nevertheless, some mismatch can still be observed for frequencies higher than the band-gap, also for relatively high number of unit cells. This high-frequency mismatch is related to the fact that the wavelength of the incident wave exceeds the threshold value t as discussed before in Subsection 5.1.

To improve the higher-frequency micromorphic description of the structure in this case, a substantial constitutive extension of the relaxed micromorphic model is needed so as to account for higher frequency modes that presumably play an important role in this frequency range. Similar arguments are valid for the slab embedded in the Cauchy material CM2, as shown by Fig. 11 and Fig. 8. As for the slab embedded in the Cauchy material CM3, we already observed in Fig. 9 that its reflective behaviour is better caught by the relaxed micromorphic model than in the previous case, even at higher frequency.

This means that, for this structure, the relaxed micromorphic model presented is sufficient for its correct description in the considered frequency range and that its generalization can be avoided in this case, also for relatively high frequencies. Indeed, in this last case, the fact of increasing the number of unit cells does not significantly improve the description of the slab's refractive behaviour (see Fig. 12). The slight differences between the reflection patterns obtained via the relaxed micromorphic model and those obtained via the full simulations (see Fig. 12) can hence be attributed uniquely to a constitutive enhancement of the relaxed micromorphic model to include extra degrees of freedom and higher modes.

Reflection coefficient of the tetragonal metamaterial slab as a function of the angle of incidence θ and of the wave-frequency

Since the predictive capability of the relaxed micromorphic model here proposed has been assessed in the previous sections, in this subsection, we show the behaviour of the reflection coefficient as a function of the frequency of the incident wave and the angle of incidence for the structure of Fig. 2(a) with outer Cauchy material CM1.

We start noticing that for the case of an incident "pressure" wave the structure's refractive behaviour is relatively unaffected by the value of the angle of incidence (see Fig. 13(a) and (c)).

In particular, when considering fewer unit cells in the metamaterial slab (Fig. 13(a)), we can observe almost total reflection occurring in a wide frequency range (extending outside the band-gap) and for all angles of incidence. Few frequencies can be identified around which total transmission occurs. When increasing the number of cells in the metamaterial slab, the frequencies around which total transmission occurs increase in number. We can thus remark that the simple fact of considering a finite-size metamaterial with a different number of unit cells modifies the structure's behaviour in a significant way. Things are even more interesting when considering "shear" incident waves, since the structure's behaviour starts being significantly affected by the angle of incidence of the travelling wave.

In particular, a critical angle exists (see Fig. 13(b) and (d)) such that all waves hitting the interface with an angle included between normal incidence and this critical value are almost completely reflected for any frequency (even outside the band-gap). For angles beyond the critical value the structure's behaviour becomes similar to that observed for incident "pressure" waves.

We analyse the same meta-structure of Fig. 2(a) by now considering the material CM2 as "outer" Cauchy material. By direct comparison of Figs. 14 with Figs. 13, it can be easily inferred that the meta-structure's behaviour is somehow reversed with respect to the previous structure.

First of all, we can identify a "critical angle region" for incident "pressure" waves instead than for "shear" ones. Moreover, we can remark that an almost total transmission occurs for angles smaller than this critical value instead than a total reflection in the previous case.

For "shear" incident waves the behaviour is still different because different zones are identified depending on the value of the incident angle. In particular, two critical angles exist in this case such that total transmission occurs between these two critical values, while total reflection occurs otherwise.

In summary, we can see how the fact of simply changing the properties of the "outer" Cauchy material reverses the meta-structure's behaviour in specific frequency and angle-of-incidence ranges, both for "pressure" and "shear" incident waves.

Conclusions

In this paper we use the relaxed micromorphic model to characterize three different 2D tetragonal metamaterials that can be used for applications in acoustic control. The reduced structure of the relaxed micromorphic model allows us to efficiently explore different meta-structural configurations in which a metamaterial's slab is embedded in a homogeneous Cauchy material. As a result, we are able to show that the metamaterial's refractive behaviour can be drastically changed by simply acting on the stiffness of the homogeneous material. In this way, the same structure can be adapted so as to act as a total screen or a total absorber in specific frequency and angle-of-incidence ranges. The results presented so far, clearly show that the study of the mechanical behaviour of metamaterials cannot be disjoined by the study of their interactions with other materials, if one wants to enable the realistic conception of new engineering meta-structures. By presenting our results, we also outline that, as any model, also enriched models have limitations that have to be identified "a priori" to avoid their inappropriate use.

In particular, we underline that the model's performances may depend on three characteristic lengths that are related to i) the unit cell's size, ii) the metamaterial's slab thickness and iii) the wavelength of the travelling wave. Depending on the relative proportions of these three characteristic lengths the relaxed micromorphic model will be more or less efficient in the description of the meta-structure's behaviour over an extended frequency range. Indeed, while the relaxed micromorphic model will always be predictive of this behaviour in the long wave limit, more or less marked differences may emerge for higher frequencies and lower wavelengths. This calls for the formulation of a new enriched model, including extra degrees of freedom and suitable dynamic internal lengths, so as to extend its efficiency to higher-frequency/smaller-wavelength regions for a wide panel of external excitations.

  where σ := T e sym∇u + T c skew∇u, s := C micro sym P, σ := C e sym (∇u -P ) + C c skew (∇u -P ) .

Figure 1 :

 1 Figure 1: Schematic representation of a wave with wavenumber k hitting at angle θ a relaxed micromorphic slab of thickness h embedded between two isotropic Cauchy media.

Figure 2 :

 2 Figure 2: (a) unit cell giving rise to the metamaterial 1, or in short MM1. (b) Table geometry and material properties of the unit cell: ρ Ti , E Ti , and ν Ti stand for the density, Young modulus and Poisson's ratio of titanium, respectively. (c) unit cell giving rise to the metamaterial 2, or in short MM2. (d) Table geometry and material properties of the unit cell: ρ Ti , E Ti , and ν Ti stand for the density, Young modulus and Poisson's ratio of titanium, respectively. (e) unit cell giving rise to the metamaterial 3, or in short MM3. (f) Table geometry and material properties of the unit cell: ρ St , E St , and ν St stand for the density, Young modulus and Poisson's ratio of steel, respectively.

  4 2.28195 × 10 -3 1.44323 × 10 -3 4.84074× 10 -3 ρ [kg/m 3 ] µ c

Figure 3 :

 3 Figure 3: Dispersion curves for the metamaterial MM1 corresponding to three different direction of propagation (a) θ = 0 • , (b) θ = 45 • , and (c) θ = 60 • .

  λ e [Pa] µ e [Pa] µ * e [Pa] 6.35565 × 10 7 4.96092 × 10 9 1.13113 × 10 7 λ micro [Pa] µ micro [Pa] µ * micro [Pa] 6.553 × 10 9

× 10 7 λ

 7 micro [Pa] µ micro [Pa] µ * micro [Pa] 2.5 × 10 9 17 × 10 9 1.075 × 10 1 0

  8 4.16349 × 10 -8 9.30984 × 10 -8 0.0000294403ρ [kg/m 3 ] µ c [Pa]

Figure 4 :

 4 Figure 4: Dispersion curves for the metamaterial MM2 (a)-(b) and MM3 (c)-(d) corresponding to two direction of propagation (a)-(c) θ = 0 • and (b)-(d) θ = 45 • .

Figure 6 :

 6 Figure 6: Acoustic curves of the metamaterial MM1 (continuous lines) as compared to the dispersion curves of three different Cauchy materials (dotted lines). It is highlighted that two dispersion curves coincide since c p2 = c p3 , while a third one is almost superimposed since its value c s2 is close to c p2 and c p3 (see Tables 5-7-6 for numerical values of these speeds).

Figure 7 :

 7 Figure 7: Comparison of the microstructure's (black line) and micromorphic (red line) reflection coefficient as a function of frequency for a 20 unit cells slab of MM1 and embedded in CM1. (a) "pressure" normal incident wave with respect to the slab's interface. (b) "pressure" 45 • incident wave with respect to the slab's interface. (c) "shear" incident wave normal to the slab's interface. (d) "shear" 45 • incident wave with respect to the slab's interface.

Figure 8 :

 8 Figure 8: Comparison of the microstructure's (black line) and micromorphic (red line) reflection coefficient as a function of frequency for a 20 unit cells slab of MM1 and embedded in CM2. (a) "pressure" normal incident wave with respect to the slab's interface. (b) "pressure" 45 • incident wave with respect to the slab's interface. (c) "shear" incident wave normal to the slab's interface. (d) "shear" 45 • incident wave with respect to the slab's interface.

Figure 9 :

 9 Figure 9: Comparison of the microstructure's (black line) and micromorphic (red line) reflection coefficient as a function of frequency for a 20 unit cells slab of MM1 and embedded in CM3. (a) "pressure" normal incident wave with respect to the slab's interface. (b) "pressure" 45 • incident wave with respect to the slab's interface. (c) "shear" incident wave normal to the slab's interface. (d) "shear" 45 • incident wave with respect to the slab's interface.

Figure 10 :

 10 Figure 10: Comparison of the microstructure's (dashed black line) and micromorphic (red line) reflection coefficient as a function of frequency for a 30 (a)-(b)-(c)-(d) and a 60 (e)-(f)-(g)-(h) unit cells slab of MM1 and embedded in CM1. (a) and (e) "pressure" normal incident wave with respect to the slab's interface for a 25 and 30 cells respectively. (b) and (f) "pressure" 45 • incident wave with respect to the slab's interface for a 25 and 30 cells respectively. (c) and (g) "shear" incident wave normal to the slab's interface for a 25 and 30 cells respectively. (d) and (h) "shear" 45 • incident wave with respect to the slab's interface for a 25 and 30 cells respectively.

Figure 11 :

 11 Figure 11: Comparison of the microstructure's (black line) and micromorphic (red line) reflection coefficient as a function of frequency for a 30 unit cells slab of MM1 and embedded in CM2. (a) "pressure" normal incident wave with respect to the slab's interface. (b) "pressure" 45 • incident wave with respect to the slab's interface. (c) "shear" incident wave normal to the slab's interface. (d) "shear" 45 • incident wave with respect to the slab's interface.

Figure 12 :

 12 Figure 12: Comparison of the microstructure's (black line) and micromorphic (red line) reflection coefficient as a function of frequency for a 30 unit cells slab of MM1 and embedded in CM3. (a) and (e) "pressure" normal incident wave with respect to the slab's interface for a 25 and 30 cells respectively. (b) and (f) "pressure" 45 • incident wave with respect to the slab's interface for a 25 and 30 cells respectively. (c) and (g) "shear" incident wave normal to the slab's interface for a 25 and 30 cells respectively. (d) and (h) "shear" 45 • incident wave with respect to the slab's interface for a 25 and 30 cells respectively.

Figure 13 :

 13 Figure 13: Simulation of the reflection coefficient with the obtained relaxed micromorphic model for a 20 (a)-(b) and a 100 (c)-(d) unit cells thick slab made up of MM1 material and embedded in CM1 Cauchy as function of the angle of incidence and of the wave-frequency -(a) and (c): incident pressure wave; (b) and (d) incident shear wave.

Figure 14 :

 14 Figure 14: Simulation of the reflection coefficient with the obtained relaxed micromorphic model for a 20 (a)-(b) and a 100 (c)-(d) unit cells thick slab made up of MM1 material and embedded in CM2 Cauchy as function of the angle of incidence and of the wave-frequency -(a) and (c): incident pressure wave; (b) and (d) incident shear wave.

  (a).

	λ e [Pa]	µ e [Pa]	µ * e [Pa]
	1.00884 × 10 8	2.52771 × 10 9	1.25592 × 10 6
	λ micro [Pa]	µ micro [Pa]	µ * micro [Pa]
	1.832 × 10 8	4.50125 × 10 9	2.698 × 10 8

Table 5 :

 5 Wave speed expressions for the Cauchy material CM1.

	(a).
	13

Table 6 :

 6 Wave speed expressions for the Cauchy material CM2.

	Wave speed of CM3 [m/s]
	c p3 =	λ Ti + 2µ Ti 13ρ Ti	c s3 =	µ Ti 13ρ Ti
		1735.98	854.75

Table 7 :

 7 Wave speed expressions for the Cauchy material CM3.

The relaxed micromorphic model can be used to qualitatively describe the mechanical response of

2D and 3D metamaterials whose class of symmetry can be represented via fourth order elastic tensors. The only class of symmetry that is currently excluded is the hexagonal one, for which the introduction of higher order elastic tensors will be needed.

For example, (A • v) i = A ij v j , (A • B) ik = A ij B jk , , (C • B) ijk = C ijp B pk , (C : B) i = C ijp B pj , v, w = v • w = v i w i , A, B = A ij B ij , etc.

The operators ∇, curl and Div are the classical gradient, curl and divergence operators. In symbols, for a field u of any order, (∇u) i = u ,i , for a vector field v, (curlv) i = ijk v k,j and for a field w of order k > 1, (Div w) i 1 i 2 ...i k-1 = w i 1 i 2 ...i k ,i k .

Being reserved to the time variable, the index t is treated separately and does not comply with Einstein notation.

The presence of curvature terms is essential to catch size-effects in the static regime that are not the target of the present paper.

When performing the calibration of the inertia parameters, we could establish that the characteristic lengths L 1 , L 2 , L 3 , and L * 1 are related to the high-frequency/large-wavelength metamaterial's response, in particular they directly intervene in the expressions of the cut-off frequencies. On the other hand, the characteristic lengths L 1 , L 2 , L 3 , and L * 1 have a significant effect on the high-frequency/short-wavelength metamaterial's response, since they visibly increase dispersive behaviours associated to the asymptotic part of the dispersion curves.

We lowered the propagation speeds cp and cs by directly changing the values of the the stiffness of the outer metamaterial or equivalently by increasing the density as shown in Tables 5-6-7 so as to meet these propagation speeds. This means that different materials CM1, CM2, and CM3 can be found that have the same speeds as in Tables 5-6-7. All these materials would give rise to the meta-structure's behaviours presented in this section.