V Orozco-Montes

J Bigarre

T Lecas

P Brault

A Caillard

PdPt nanocatalyst synthesis via liquid medium sputtering

Keywords: Magnetron Sputtering, liquid substrate, Nanoparticles, Catalysts

PdPt nano-catalysts were synthesized via magnetron sputtering method in vacuum on vegetal glycerin as liquid substrate. The nanoparticles formation and their dispersion in the liquid phase have been studied by varying deposition parameters as the argon pressure, the magnetron power and glycerin temperature. Pt-based nano-catalysts are incorporated into electrochemical cells and their catalytic activities towards oxygen reduction reaction (ORR) are evaluated.

Introduction

Pt-based nanoparticles (NPs) are widely used as catalysts for oxygen reduction reactions (ORR) in electrochemical systems such as proton exchange membrane fuel cell (PEMFC). Further enhancements of the Pt catalytic activity while reducing its amount inside the fuel cell is one of the main issues for practical application of these systems due to the limited abundance of Pt on Earth. Since ORR is a surface phenomenon, a high fraction of surface atoms is desirable to achieve efficient Pt utilization and so small NPs with size ranging from 2 to 4 nm must be preferred [1].

Depending on the size range and the future applications, several techniques exists to prepare metallic NPs. These are mainly categorized into "bottom up" and "top down" methods, which utilize solution-phase colloidal chemistry and conventional physical methods on solid substrates [2]. The chemical methods are very versatile in terms of controlling NPs shape and size by varying the reaction conditions. However, NPs chemical synthesis require additives, which generate by-products difficult to remove and NPs with limited purity. On the other hand, physical methods on solid substrates avoid to use additives allowing to produce pure metallic NPs with the same starting material composition [3].

Among physical techniques to prepare Pt-based nanocatalyst, magnetron sputtering has produced active Pt based nano-catalysts on carbon porous electrode [4], but the control of the NPs properties and the production of large quantity of materials is difficult using the sputtering onto such substrate. A solution consists to synthesize Pt nano-catalysts directly in liquid phase using vacuum magnetron sputtering technique [3,5]. Such substrate is known to form colloidal solution made of NPs dispersed in the liquid substrate but thin film can also appears on the surface of this liquid. This innovative method takes advantages of the extremely low vapor pressure of liquid such as ionic liquid, vegetable oil and silicon oils compatible to vacuum techniques [6].

In this context, the present study aims to synthesize PdPt based NPs in vegetal glycerin using a plasma sputtering process, to obtain stable colloidal NPs solutions compatibles with the preparation of catalytic ink. The NPs properties (dispersion and morphology) as function of the deposition parameters (liquid temperature, magnetron power and gas pressure) were investigated. Finally, these NPs transferred into catalytic inks were tested in a electrochemical cell in order to qualify their activity towards to the ORR.

Experimental details

The experimental setup displayed in Fig. 1 consists of a HV cross-shaped sputter chamber equipped with a DC 2 inches in diameter DC magnetron and a Pd0.95Pt0.05 target. The vacuum system consists of a primary and turbomolecular pump allowing to achieve a vacuum of 10 -4 Pa in the deposition chamber without liquid in the chamber deposition. 4 ml of vegetal glycerin contained in a glass vial is placed on a sample holder. The glycerin with a vapor pressure inferior to 1.33 Pa (20 °C) is sufficiently low to a base pressure of 10 -3 Pa in the deposition chamber without any visible transformation of the liquid. However, the pumping speed was adjusted to obtain a pressure of 10 -2 Pa by a gate valve placed between the deposition chamber and the turbomolecular pump to limit the glycerin evaporation. The distance between the glycerin surface and the target surface (PdPt or Pt) was fixed at 12 cm. Prior to sputtering experiments, the target surface was sputter cleaned in argon plasma during 5 min with the presence of a shutter. The sputtering of the metallic target was performed in pure Ar (40 sccm) and three series of PdPt NPs depositions were performed. For the first series, magnetron power and Ar pressure was fixed at 100 W and 1 Pa respectively, while the surface temperature of vegetal glycerin was modulated (from 27°C to -6°C) using a circulation chiller. The temperature at the glycerin surface was measured by using a thermocouple immersed in the liquid. In the second series, a constant 100W DC power was delivered to the cathode and the Ar pressure was adjusted to 1, 4 and 9 Pa. At last, for the third series, the Ar pressure was fixed at 1 Pa and the magnetron power was adjusted to 13, 50 and 100 W. In all cases, the sputtering deposition was carried out for 20 min.

During sputtering, atoms that are ejected from the target material condense on the glycerin surface, form NPs (which could be assembled in a network) and then diffuse within the glycerin and form a colloidal solution. This mechanism is still under investigations and discussions [3].

The NPs were then transferred to carbon NPs to form NPssupported carbon. For that, Vulcan XC-72 previously treated under N2 at 400°C for 4 hours, was directly added to the colloidal solution in order to obtain a mass metal/carbon ratio of 0.1. After, this mixture was successively filtered and washed several times with ultrapure water and acetone. Finally, a process dry was carried out at 70°C in an oven. The obtained PdPt/C catalytic powders were used for the catalytic ink preparation.

The catalytic powder was investigated by a Transmission Electron Microscope (TEM -Philips CM20) operating at 200 kV and equipped with a Lab6 filament. The powder was previously dissolved in ethanol and a droplet was deposited onto a TEM grid (Cu with a holey carbon foil).

The electrochemical measurements have been performed in a conventional three-electrode electrochemical cell at room temperature. The working electrode was prepared by deposition of the catalytic ink onto a glassy carbon disc (0.71 cm -2 geometric surface area). Cyclic voltammograms of PdPt/C were recorded in a N2-saturated 0.1 M NaOH and 0.5M H2SO4.

Results

As shown on pictures of Fig. 2, a homogeneous black solution is obtained at 27°C (a) which seems to indicate that the metal is dispersed in the liquid phase while thin films on the liquid are observed at -6°C (d). This film is not continuous and has been broken in at least 5 parts (during the deposition or during the air venting). For intermediate temperature, 19°C and 5°C, the presence of a film on the surface and materials in the liquid are observed. In most cases, the film on the surface tends to disappear/diffuse on the liquid phase after few hours. Such observation could be related to the glycerin viscosity which tends to increase with the temperature decrease. More investigation are required to clearly explain this phenomenon, but in any case, it seems that the glycerin surface temperature drive the NPs formation and one parameter which can control the NPs growth. the NPs are well-dispersed but for 9 Pa the NPs seems to form a ramified network. Due to the low resolution of this TEM, the precise measurement (regarding the structure/morphology/size) is rather difficult. However, the mean NPs size could be estimated to about 5 nm for 4 Pa and 10 nm for 9 Pa. Such increased of the NPs size is typical for magnetron sputtering when a solid substrate is used [7]. Consequently, the argon pressure is another parameter (after the liquid temperature) affecting the NPs properties and this parameter could be one of the parameters used to control the NPs size. The influence of the sputtering pressure on the NPs formation could be related to the energy of the sputtered atoms when arriving on the liquid surface. A low pressure, the atoms have a long mean free path in the gas phase and consequently high kinetic energy at the liquid surface which lead to the formation of dense agglomerate. In contrast, for high pressure, the atoms have a low kinetic energy which lead to porous NPs based thin film which can be easily dispersed in liquid. This could explain the formation of agglomerates at low pressure and dispersed NPs at high pressure.

In addition, there are several sputtering parameters (discharge voltage, sputtering time, inert gas pressure, target temperature and liquid target-distance) and liquid environment properties (surface or bulk) that can control the grow mechanisms of sputtered NPs [4]. To date the NPs grow kinetic in a liquid environment is not defined and it is still a topic of discussion as already mentioned.

Conclusions

PdPt based NPs were synthesized by a plasma sputtering process on vegetal glycerin. We have shown that the glycerin surface temperature and the argon pressure strongly affect the NPs formation. At high temperature, a colloidal solution is obtained whereas a thin films on liquid are obtained at low temperature. A argon pressure of 4 Pa induces the formation of well-dispersed NPs on carbon whereas at lower and higher pressure, large agglomerates and ramified networks of NPs are observed.

We have shown that the PdPt catalysts is active for the ORR, but a full investigation of all catalysts is necessary and will be presented in the ISPC24 conference. We planned to measure the efficiency of the catalyst in PEMFC

Fig. 1 .

 1 Fig. 1. Schematic of the experimental set-up allowing the growth of NPs inside vegetal glycerine by plasma magnetron sputtering.

Fig. 2 .

 2 Fig. 2. Sample pictures immediately after the PdPt sputter deposition onto vegetal glycerine at a) 27°C, b) RT: 19°C, c) 5°C and d)-6°C

Fig. 3 .

 3 Fig. 3. TEM images corresponding to PdPt NPs supported on carbon NPs synthesized at 100 W and different Ar pressure: a) 1Pa, b) 4Pa and c) 9 Pa. On TEM images of Fig 3a corresponding to 1 Pa, the NPs seems to be assembly in agglomerates with a size of few 10's of nanometres, whereas on TEM images of Fig. 3b (4 Pa) and 3c (9 Pa), PdPt NPs are clearly visible. For 4 Pa, the NPs are well-dispersed but for 9 Pa the NPs seems to form a ramified network. Due to the low resolution of thisTEM, the precise measurement (regarding the structure/morphology/size) is rather difficult. However, the mean NPs size could be estimated to about 5 nm for 4 Pa and 10 nm for 9 Pa. Such increased of the NPs size is typical for magnetron sputtering when a solid substrate is used[7]. Consequently, the argon pressure is another parameter (after the liquid temperature) affecting the NPs properties and this parameter could be one of the parameters used to control the NPs size. The influence of the sputtering pressure on the NPs formation could be related to the energy of the sputtered atoms when arriving on the liquid surface. A low pressure, the atoms have a long mean free path in the gas phase and consequently high kinetic energy at the liquid surface which lead to the formation of dense agglomerate. In contrast, for high pressure, the atoms have a low kinetic energy which lead to porous NPs based thin film which can be easily

Fig. 4

 4 Fig.4displays the cyclic voltammogram of PdPt/C synthesized at 50 W and 1 Pa in 0.1 M NaOH (red) and 0.5M H2SO4 (black). A single reduction peak corresponding to the Pt-oxide reduction peak at potentials ranging from 0.6 to 0.8V (vs.RHE) is observed in both medium. The presence of this peak clearly shows that the PdPt catalyt is active for the ORR. The comparison of all catalysts is under investigation and the calculation of the electrochemical active surface area (EASA) will be obtained by the integration of the reduction peak.

Fig. 4 .

 4 Fig. 4. Cyclic voltamograms recorded on a PdPt/C catalyst (scan rate: 10mV s -1 , N2 saturated 0.5M H2SO4 and 0.1M NaOH)

conditions, after the incorporation of the catalysts at the PEM/electrode interface.

Acknowledgements

This work has been supported by The Region Centre Valde-Loire through the Lavoisier INPACT program. The authors thank A. Sauldubois for TEM analysis (Centre de Microscopie Electronique, Université d'Orléans) and S. Baranton for the useful discussions about catalytic inks preparation and the electrochemical measurements.


D Magliacano
email: dario.magliacano@univ-fcomte.fr

M Ouisse

A Khelif

S De Rosa

F Franco

N Atalla

VALIDATION OF SHIFT CELL APPROACH FOR THE MODELLING OF ACOUSTIC PROPERTIES OF POROUS MATERIALS EMBEDDING PERIODIC INCLUSIONS

The design based on periodic elements can be a powerful strategy for the achievement of lightweight sound packages and can represent a useful solution for manufacturing aspects. The inclusion of vibroacoustic design rules, at early stage of the product development, is a feasible and interesting research target; in fact, the use of porous media with periodic inclusions can improve the dynamic filtering effects thus offering several possibilities of applications in the engineering fields where both weight and space, as well as vibroacoustic integrity and comfort, still represent critical issues. Indeed, although porous materials are commonly used for vibroacoustic applications, they suffer from a lack of absorption at low frequencies. This difficulty is usually overcome by a multi-layering approach. However, while reducing the impedance mismatch at the air-material interface, the efficiency of such devices relies on the allowable thickness. A more efficient way to enhance the low frequency performances of sound packages consists in embedding periodic inclusions in a porous layer. If the radius of these periodic inclusions is comparable with the acoustic wavelength, then an increase of the acoustical performances can be observed since the excitation of additional acoustic modes helps in dissipating more acoustic energy. Therefore, even if this procedure still relies on the dimension of the inclusions, it can provide an acoustical performance improvement, which is way more efficient than just increasing the thickness. This paper starts with the description of the shift cell technique, which allows the description of the propagation of all existing waves

from the description of the unit cell. The shift cell technique provides a reformulation of Floquet-Bloch periodic conditions and its major advantage is that it allows the implementation of any frequency dependence and damping in the problem: this is a new advantage when computing the dispersion curves of a porous material modelled as an equivalent fluid. The second part of this work shows the improvements of acoustic properties (in terms of transmission loss

) that are obtained due to the periodic nature of the system and some qualitative considerations. The test case herein presented involves a square unit cell of porous material with a circular inclusion, for which the results are quite encouraging and promising.

LIST OF SYMBOLS

• 𝜔 = angular frequency; • 𝑥, 𝑦, 𝑧 = space variables;

• 𝑗 = imaginary unit;

• 𝑝 0 = amplitude of the excitation mode (incident pressure);

•

INTRODUCTION

The design based on the inclusion of vibroacoustic design rules at early stage of products development, through the use of porous media with periodic inclusions which exhibit proper dynamic filtering effects, is a powerful strategy for the achievement of lightweight sound packages and represents a convenient solution for manufacturing aspects. The main advantage of designing sound packages with periodic arrangements is that they can provide a combination of absorption effects, resonance effects and wave interferences effects. This offers different applications in transportation (aeronautics, space, automotive, railway), energy and civil engineering sectors, where both weight and space, as well as vibroacoustic integrity and comfort, still remain as critical issues. Indeed, although porous materials are commonly used for vibroacoustic applications, they suffer from a lack of absorption at low frequencies compared to their efficiency at higher ones. This difficulty is usually overcome by multi-layering. However, while reducing the impedance mismatch at the air-material interface, the efficiency of such devices relies on the allowable thickness. Instead, a more efficient way to enhance the low frequency performances of sound packages consists in embedding periodic inclusions in a porous layer. If the radius of these periodic inclusions is comparable with the acoustic wavelength, then an increase of the acoustical performances can be observed. In order to develop efficient numerical techniques in order to handle the problem, the shift cell operator technique here is presented, providing details on its implementation [START_REF] Magliacano | The shifted cell operator technique applied to equivalent fluids models for the computation of dispersion diagrams of periodic porous materials[END_REF]. Essentially, the shift cell technique consists of a reformulation of the Floquet-Bloch partial differential problem, in which the phase shift of the boundary conditions related to wave propagation is integrated into the derivation operator. Consequently, the periodicity is included in the overall behaviour of the structure while the continuity conditions are imposed at the edges of the unit cell. Its major advantage is that it allows to implement any frequency dependence and damping in the problem; this is essential, if one needs to compute the dispersion curves of a porous material modelled as an equivalent fluid. In detail, it allows the description of the propagation of all existing waves from the description of the unit cell through the resolution of a quadratic eigenvalue problem. This is done through the 𝑘(𝜔) method, that allows to compute dispersion curves for frequency-dependent problems, instead of using the 𝜔(𝑘) one that leads to non-linear eigenvalue problems. Until now, this type of model has never been implemented to handle porous material models with periodic inclusions [START_REF] Collet | Floquet-Bloch decomposition for the computation of dispersion of two-dimensional periodic, damped mechanical systems[END_REF], [START_REF] Billon | Composites periodiques fonctionnels pour l'absorption vibroacoustique large bande[END_REF]. This will render possible to overcome the limits of existing approaches, by a more specific design of the system through a process of optimization and testing of different inclusions, in order to obtain a device whose frequency efficiency outperforms existing designs. Dispersion curves (calculated with the shift cell technique) and acoustical characteristics (computed using Floquet-Bloch periodic conditions) for different numerical test cases are shown. In particular, they are obtained for a 3D melamine unit cell, with and without inclusion. The behaviour of this porous material is described by JCA model in the following pages, but one can identically use any other equivalent fluid model.

FLOQUET-BLOCH THEORY

Floquet-Bloch (hereafter F-B) theory provides a strategy to analyse the behaviour of systems with a periodic structure. Floquet's seminal paper deals with the solution of 1D partial differential equations with periodic coefficients; in solid state physics, Bloch generalizes Floquet's results to 3D systems and obtains the description of the wave function associated with an electron traveling across a periodic crystal lattice. This wave function is a solution of the Schrödinger equation with a periodic potential and Bloch showed that it is the product of a simple plane wave multiplied by a periodic function with the same periodicity of the lattice. In the literature dealing with wave propagation problems in mechanical systems the theory is referred to as Floquet-Bloch theory or, simply, Floquet theory [START_REF] Floquet | Sur les équations différentielles linéaires à coefficients périodiques[END_REF]. In layered systems, due to the heterogeneity of the relevant elastic properties, to particular geometric features, or to both, only certain wave modes can physically propagate inside the structure. Each of these modes can be identified by a determined (generally nonlinear) function relating the time frequency and the spatial frequency (or wave number). These relationships are called dispersion curves and they summarize all the oscillatory behaviour of the system. Dispersion curves provide estimations of the elastic and geometrical parameters from experiments or numerical models and offer a better perspective to explain the wave field behaviour inside bodies.

SHIFT CELL OPERATOR TECHNIQUE

Considering a porous layer as an equivalent fluid [START_REF] Doutres | Evaluation of the acoustic and non-acoustic properties of sound absorbing materials using a three-microphone impedance tube[END_REF], the starting equation for developing the associated shift cell formulation is

(∇ + 𝑗𝒌) 𝑇 ( 1 𝜌 (∇ + 𝑗𝒌)𝒑) = - 𝜔 2 𝐾 𝒑, (1)
with 𝜌 = 𝜌(𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙, 𝜔) and 𝐾 = 𝐾(𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙, 𝜔).

By further developing the latter equation and considering 𝒑 = 𝑝𝑒 𝑗𝒌𝒙 where 𝒌, for a 3D application (Figure 1), is (3)

𝒌 = 𝑘𝝓, 𝝓 = ( 𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜙 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜙 𝑠𝑖𝑛𝜃 ), (2)

Weak formulation

The weak formulation is calculated from the following equation, where 𝑝 ̃ is a weighting test function:

∫ 𝑝 ̃𝑒-𝑗𝒌 𝑇 𝒙 (∇ 𝑇 ∇(𝑝𝑒 𝑗𝒌𝒙 ) 𝜌 + 𝑗𝒌 𝑇 (2 ∇(𝑝𝑒 𝑗𝒌𝒙 ) 𝜌 + 𝑝𝑒 𝑗𝒌𝒙 ∇ 1 𝜌 ) -𝒌 𝑇 𝒌 𝑝𝑒 𝑗𝒌𝒙 𝜌 + 𝜔 2 𝐾 𝑝𝑒 𝑗𝒌𝒙 ) 𝛺 𝑑𝛺 = 0(4) ∫ ( 1 𝜌 ∇ 𝑇 𝑝 ̃∇𝑝 -𝑗𝒌 𝑇 2 𝜌 (𝑝 ̃∇𝑝 -∇𝑝 ̃𝑝) + 𝒌 𝑇 𝒌 4 𝜌 𝑝 ̃𝑝 -𝜔 2 1 𝐾 𝑝 ̃𝑝) 𝑑𝛺 𝛺 = 0 (5)
Considering that 𝝋 is the eigenvector, the equation can be written in its matrix form (𝑲 + 𝑗𝑘𝑳 + 𝑘 2 𝑯 -𝜔 2 𝑴)𝝋 = 0 (6) with the following matrices:

• 𝑲 → ∫ 1 𝜌 ∇𝑝 ̃∇𝑝 𝛺 𝑑𝛺; • 𝑳 → ∫ 2 𝜌 (∇𝑝 ̃𝑝 -𝑝 ̃∇𝑝) 𝛺 𝑑𝛺; • 𝑯 → ∫ 4 𝜌 𝑝 ̃𝑝 𝛺 𝑑𝛺; • 𝑴 → ∫ 1 𝐾 𝑝 ̃𝑝 𝛺 𝑑𝛺.

Right and left eigenvalue problems

The latter formulation leads to the following right eigenvalue problem:

[(𝑲 -𝜔 2 𝑴) + 𝜆 𝑖 𝑳 -𝜆 𝑖 2 𝑯]𝝋 𝑖 𝑟 = 0 (7) where 𝜆 𝑖 = 𝑗𝑘 𝑖 is the i-th eigenvalue, 𝜑 𝑖 𝑟 denotes the right eigenvector associated to 𝜆 𝑖 , 𝑴 and 𝑲 are respectively the standard symmetric definite mass and symmetric semi-definite stiffness matrices, 𝑳 is a skew-symmetric matrix and 𝑯 is a symmetric semi-definite positive matrix. In this formulation, all matrices are frequency dependent. For frequency-dependent systems, the estimation of the group velocity is not trivial. The equation [START_REF] Allard | Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials[END_REF]

Classifying criteria to distinguish propagative and evanescent waves

Applying the shift cell operator technique to a sample modelled by an equivalent fluid, all the wave numbers are complex; consequently, there are not purely propagative solutions anymore. All waves are evanescent, with an evanescence rate that may be used to classify the branches in two categories: those that will be rapidly damped and those that will be slowly damped in space, with the latter that could be classified as propagative ones.

The distinction between the two is difficult and thus some classifying criteria are required:

• the ratio between the real and the imaginary parts of every wavenumber → 𝐶 1 = 𝑟𝑒𝑎𝑙(𝑘)/𝑖𝑚𝑎𝑔(𝑘); • the ratio between the real and the imaginary parts of the energy transport speed 𝑣 = 𝐼/𝐸, where 𝐸 is approximated by 𝐸 = 2𝐸 𝑘 → 𝐶 2 = 𝑟𝑒𝑎𝑙(𝑣)/𝑖𝑚𝑎𝑔(𝑣); • the ratio between the real part and the imaginary parts of the group velocity → 𝐶 3 = 𝑟𝑒𝑎𝑙(𝐶 𝑔 )/𝑖𝑚𝑎𝑔(𝐶 𝑔 ).

Note that

𝐸 𝑘 = 1 2 𝑚𝐶 𝑝 2 .
Only the waves corresponding to 𝐶 1 > 𝜏 1 , 𝐶 2 > 𝜏 2 and 𝐶 3 > 𝜏 3 are considered propagative. In practice, for the purpose of the following analysis, the thresholds 𝜏 are chosen such as 𝜏 1 = 𝜏 2 = 𝜏 3 = 1. This is an arbitrary choice and these values are not meant to be considered as universal: for each different case, one may need to tune them [START_REF] Billon | Composites periodiques fonctionnels pour l'absorption vibroacoustique large bande[END_REF].

Validation of the method

The validation analysis is carried out in the frequency range 0 -17000 [Hz]; this range of frequencies assures that the wavelength is much larger than the pore size, which is a necessary condition in order to use equivalent fluid models. The 2D and 3D unitary cells are respectively constituted by a 2 [cm] square with a 0.5 [cm] radius circular hole and by a 2 [cm] cube with a 0.5 [cm] radius cylindrical hole. For similar cases, some results are available in literature in terms of absorption coefficient [START_REF] Groby | Absorption of a rigid frame porous layer with periodic circular inclusions backed by a periodic grating[END_REF].

In order to validate the implementation, a first calculation is made to compare shift cell results with those obtained using Floquet-Bloch periodic conditions, using (non-dissipative) air as material. Both methods are used to obtain the dispersion diagram along the direction that corresponds to 𝜙 = 0° and 𝜃 = 0° (for the 3D case) in the first Brillouin zone. Stepped (red) lines in Figure 2 correspond to the results obtained with the Floquet-Bloch method, while (blue) points correspond to the results obtained with the shift cell operator method. The comparison shows a perfect agreement between the results of the two methods.

Meaning and behaviour of band gaps for inclusions of increasing size

The imaginary part of dispersion curves is linked to dissipation, while the real part represents the propagative behaviour.

Figure 3 shows what happens to the band gap size, in the previously defined 2D unit cell, when the size of the inclusion (that has perfectly rigid walls) changes. The radius of the inclusion for the three analysed cases are respectively equal to , where 𝑠 is the side length. After this, instead of using the adiabatic value (142 [kPa]) for the bulk modulus of air, one can artificially add a frequency-constant imaginary part to it (for example: 142+𝑗12 [kPa]); doing so, one can simulate a band gap behaviour similar to that obtained when using equivalent fluids, as shown later. Indeed, a complex bulk modulus prevents the presence of proper band gaps in dispersion curves; one can clearly see that the gap is opening but, because of the damping, 𝑘 𝑥 is no longer purely imaginary. So, the real part of 𝑘 𝑥 does not disappear anymore, but remains low (compared to imaginary one), which means that the wave will be strongly spatially attenuated, and this is exactly the expected behaviour for band gaps in dissipative media. Figure 3: Dispersion curves for an air 2D cell, with increasing inclusion size (from left to right) and with normal (second raw) and modified (third raw) bulk modulus.

DISPERSION CURVES

For each dispersion curve plot, three eigenvectors are reported in terms of acoustic pressure field (Figures 4 and5 One may notice that there are some discontinuous curves: this is probably due to the fact that, as said, all waves are evanescent with different rates: a non-perfect tuning of the sorting criteria could lead to lines that disappears and reappears on the plots.

TRANSMISSION LOSS

While dispersion curves are computed for an infinite repetition of unit cells, transmission loss is calculated for a finite repetition of 5 unit cells, using the same domain and boundary conditions of the infinite periodic system. This, in a first approximation, allows to compare the dispersion relations and the acoustical characteristics of the equivalent finite medium. Indeed, a further increasing in the number of repeated cells would lead to a change in the mean value of absorption coefficient and transmission loss respectively below 2% and 20% respect to the usage of a repetition of 5 unit cells. The transmission loss is numerically computed as 𝑇𝐿 = 10 log 10

𝛱 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝛱 𝑡𝑟𝑎𝑛𝑠𝑚𝑖𝑡𝑡𝑒𝑑 (17)
where 𝛱 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 and 𝛱 𝑡𝑟𝑎𝑛𝑠𝑚𝑖𝑡𝑡𝑒𝑑 represent the incident and transmitted power, respectively. For our plane wave configuration, the latter is compared, for homogeneous flat configurations, with the Transfer matrix method [START_REF] Allard | Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials[END_REF]: 𝑇𝐿 = 10 log (

For the inhomogeneous configuration, the validation is obtained using an implementation of the plane wave forced response of the periodic cell accounting for fluid loading [START_REF] Atalla | NOVAFEM User's guide[END_REF]. Concerning the case with the inclusion, one can notice that an improvement of transmission loss properties, respect to the homogeneous case, is shown at all frequencies, in particular in correspondence of a peak at a frequency around 7 [kHz], in which it is equal to about 15 [dB], and at high frequencies. Note that, for the sake of comparison with the related dispersion curves, only their 1 st branch is meaningful due to the fact that the correspondent mode is the only one that is actually excited during these transmission loss simulations. Indeed, the improvement peak exactly corresponds to the frequency range of the 1 st branch of dispersion curves in which the wave is strongly spatially attenuated. This is definitely encouraging, for the purpose of deriving the equivalent acoustic properties of the unit cell from its dispersion characteristics.

CONCLUSIONS

The shift cell technique has been presented, providing details on its numerical formulation. It has been necessary to introduce some classifying criteria and, consequently, the derivation of the group velocity expression. A validation, made by a comparison with Floquet-Bloch periodic conditions, has been shown before explaining the meaning and behaviour of band gaps in dispersion curves. Dispersion curves and transmission loss plots have then been computed for a JCA-modelled melamine unit cell. Further developments of the work will include the estimation of the computational efficiency between the shift cell and the Floquet-Bloch approaches, the study of a case for actual low-frequency performance improvement and the implementation of the shift cell technique using Biot model, for both 2D and 3D geometries.

  𝜌 0 = density of the interstitial fluid (air); • 𝑐 0 = sound speed in the interstitial fluid (air); • 𝑍 0 = characteristic impedance of the interstitial fluid (air); • 𝑘 0 = wave number in the interstitial fluid (air); • 𝜌 = density of the material; • 𝑍 𝑐 = characteristic impedance of the material; • 𝑘 = wave number in the material; • 𝑝 = pressure; • 𝐾 = bulk modulus; • 𝜃, 𝜙 = angles of incidence; • 𝑝 * = 𝑐𝑜𝑛𝑗(𝑝); • 𝛺 = poro-elastic volume; • 𝛤 = domain boundary; • 𝐼 = flow of energy; • 𝐸 = total energy; • 𝐸 𝑘 = kinetic energy; • 𝑚 = mass; • 𝐶 𝑝 = particle velocity; • 𝐶 𝑔 = group velocity; • 𝑠 = side length; • 𝛱 𝑑𝑖𝑠𝑠 = total dissipated power; • 𝛱 𝑡ℎ𝑒𝑟𝑚 = thermal dissipated power; • 𝛱 𝑣𝑖𝑠𝑐 = viscous dissipated power; • 𝛱 𝑖𝑛𝑐 = incident power; • 𝛱 𝑖𝑛𝑝𝑢𝑡 = input power; • 𝛱 𝑡𝑟𝑎𝑛𝑠 = transmitted power; • 𝑆 = surface interested by incident pressure; • 𝑑 = thickness; • 𝜏 ∞ = transmission coefficient; • 𝑇𝐿 = transmission loss.

Figure 1 : 2 ∇

 12 Figure 1: Reciprocal lattice vector in a 3D unitary cell [3].

Figure 2 :

 2 Figure 2: Comparison between dispersion curves obtained with Floquet-Bloch and shift cell techniques on a 2D (on the left) and a 3D (on the right) air unit cell.

  ). Only the real parts are shown, the imaginary parts being null. They are all extracted at the frequency of 8500 [Hz] (half of the range) and along the direction that corresponds to 𝜃 = 𝜙 = 0° in the first Brillouin zone. Their branches are ordered as: at increasing frequencies, 1st is represented by the first real part that reaches the unitary value, 2nd is the second and so on. The fundamental acoustic parameters of the tested porous material have been experimentally determined in GAUS laboratory at University of Sherbrooke (Canada) and are: porosity = 0.99, tortuosity = 1.02, resistivity = 8430 [ 𝑃𝑎 * 𝑠 𝑚 2 ], viscous characteristic length = 0.138 [𝑚𝑚], thermal characteristic length = 0.154 [𝑚𝑚], density = 5.73 [ 𝑘𝑔 𝑚 3 ].

Figure 4 :

 4 Figure 4: Dispersion curves for a homogeneous melamine unit cell and, from top left to bottom, real parts of the 1 st , 2 nd and 3 rd branch eigenvectors.

Figure 5 :

 5 Figure 5: Dispersion curves for a melamine unit cell with inclusion and, from top left to bottom, real parts of the 1 st , 2 nd and 3 rd branch eigenvectors.

Figure 6 :

 6 Figure 6: Transmission loss computed for a 3D repetition of 5 melamine unit cells.

  In the resolution of the right eigenvalue problem, the 𝑖-th mode (𝑖 𝜖 𝑁 + ) is defined by its 𝜆 𝑖 ≥ 0 and its eigenvector 𝝋 𝑖 𝑟 . For each mode 𝑖, a mode -𝑖 is associated with 𝜆 -𝑖 ≤ 0 such that 𝜆 -𝑖 = -𝜆 𝑖 and 𝝋 -𝑖 𝑟 = 𝝋 𝑖 𝑙 . by solving the right eigenvalue problem, the left solution is found too. The starting equation is now differentiated and multiplied by the left eigenvector such that

	can be rewritten as
			𝑨 1 (𝜔)𝝍 𝑖 𝑟 = 𝜆 𝑖 𝑨 2 (𝜔)𝝍 𝑖 𝑟	(8)
	with
	• 𝑨 1 (𝜔) = ( 𝑲 -𝜔 2 𝑴 𝑳 0 𝑰 𝒅 • 𝑨 2 (𝜔) = ( 𝑰 𝒅 0 ); 0 𝑯 • 𝝍 𝑖 𝑟 = ( 𝑟 𝝋 𝑖 𝜆 𝑖 𝝋 𝑖 𝑟 ).	);
	where 𝑰 𝒅 is the identity matrix.
	Conversely, a left-eigenvector for the same eigenvalue satisfies
		𝝍 𝑖 𝑙 𝑇 𝑨 1 (𝜔) = 𝜆 𝑖 𝝍 𝑖 𝑙 𝑇 𝑨 2 (𝜔), with 𝝍 𝑖 𝑙 = ( 𝑩 𝑨 ). { 𝑩 = 𝝋 -𝑖 𝑙 𝑟 = 𝝋 𝑖 𝑨 𝑇 = 𝜆 𝑖 𝝋 -𝑖 𝑟 𝑇 𝑯 -𝝋 -𝑖 𝑟 𝑇 𝑳 = 𝜆 𝑖 𝝋 𝑖 𝑙 𝑇 𝑯 -𝝋 𝑖 𝑙 𝑇 𝑳	(9) (10)
	𝜕𝑨 1 (𝜔) 𝜕𝜔	𝝍 𝑖 𝑟 + 𝑨 1 (𝜔) 𝝍 𝑖 𝜕𝝍 𝑖 𝑟 𝜕𝜔 𝑙 𝑇 𝜕𝑨 1 (𝜔) = 𝜕𝜆 𝑖 𝜕𝜔 𝜕𝜔 𝝍 𝑖 𝑨 2 (𝜔)𝝍 𝑖 𝑟 + 𝜆 𝑖 𝑟 + 𝝍 𝑖 𝑙 𝑇 𝑨 1 (𝜔) 𝜕𝑨 2 (𝜔) 𝜕𝜔 𝜕𝝍 𝑖 𝑟 = 𝝍 𝑖 𝑟 + 𝜆 𝑖 𝑨 2 (𝜔) 𝜕𝜔 = 𝝍 𝑖 𝑙 𝑇 ( 𝜕𝜆 𝑖 𝜕𝜔 𝑨 2 (𝜔) + 𝜆 𝑖 𝜕𝑨 2 (𝜔) 𝜕𝜔 ) 𝝍 𝑖 𝑟 + 𝝍 𝑖 𝑙 𝑇 𝜆 𝑖 𝑨 2 (𝜔) 𝑟 𝜕𝝍 𝑖 𝜕𝜔	𝑟 𝜕𝝍 𝑖 𝜕𝜔	(11) (12)
	Considering that
			𝝍 𝑖 𝑙 𝑇 𝑨 1 (𝜔) = 𝜆 𝑖 𝝍 𝑖 𝑙 𝑇 𝑨 2 (𝜔)	(13)
	one obtains
			𝜕𝜆 𝑖 𝜕𝜔	=	𝝍 𝑖 𝑙 𝑇	[	𝜕𝑨 1 (𝜔) 𝜕𝜔 𝝍 𝑖 𝑙 𝑇 𝑨 2 (𝜔)𝝍 𝑖 -𝜆 𝑖 𝜕𝑨 2 (𝜔) 𝜕𝜔 𝑟	𝑟 ]𝝍 𝑖	(14)
	which gives the expression of the group slowness using 𝜆 𝑖 = 𝑗𝑘 𝑖 : 𝜕𝑘 𝑖 𝜕𝜔 = -𝑗 𝝋 𝑖 𝑙 𝑇 [-2𝜔𝑴+ 𝜕𝑲 𝜕𝜔 +𝜆 𝑖 𝜕𝑳 𝜕𝜔 -𝜆 𝑖 2 𝜕𝑯 𝜕𝜔 𝑟 ]𝝋 𝑖 𝝋 𝑖 𝑙 𝑇 [-𝑳+2𝜆 𝑖 𝑯]𝝋 𝑖 𝑟	(15)
	The group velocity is the inverse of the group slowness:
		𝐶 𝑔 =	𝜕𝜔 𝜕𝑘 𝑖	=	𝑗𝝋 𝑖 𝑙 𝑇 [-𝑳+2𝜆 𝑖 𝑯]𝝋 𝑖 𝑟 𝑙 𝑇 [-2𝜔𝑴+ 𝝋 𝑖 𝜕𝑲 𝜕𝜔 +𝜆 𝑖 𝜕𝑳 𝜕𝜔 -𝜆 𝑖 2 𝜕𝑯 𝜕𝜔	]𝝋 𝑖 𝑟	(16)

ACKNOLEDGEMENTS

This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 675441.