Minh Duy Le

Mickael Matrat

Arij Ben

Amara Ifp

Energies Nouvelles

Fabrice Foucher

Bruno Moreau

Yi Yu Prisme

Exploring and Modeling the Chemical Effect of a Cetane Booster Additive in a Low-Octane Gasoline fuel

Recent internal combustion (IC) engine developments focus on gasoline fuel. This requires a better understanding of fuel reactivity at different thermodynamic conditions. Gasoline fuel reactivity control by additives is an efficient method to get better IC engine performances. 2-Ethylhexyl nitrate (EHN) promoting effect (0.1 -1% mol.) on combustion has been investigated experimentally and numerically. Rapid compression machine (RCM) experiments were carried out at equivalence ratio 0.5 at 10 bar, from 675 to 995 K. The targeted surrogate fuel is a mixture of toluene and n-heptane in order to capture the additive effect on both cool flame and main ignition. A kinetic model was developed from literature data assembly and validated upon a large set of variations including species profiles and ignition delays of pure compounds as well as mixtures. At the experimental conditions, it was found that the EHN reduces the ignition delay time (IDT) of the surrogate fuel in the whole temperature range. EHN effectiveness tends to be minimum around 705 K and increases with temperature. The results also indicate that EHN effect increases nonlinearly with EHN doping levels. Numerical analyses revealed that the EHN effect is linked to NO2-NO loops, which enhances fuel reactivity. The methodology proposed here enable to simulate the EHN effect with simple compounds rather than the full EHN chemistry set. This strategy could simplify the consideration of additive effect when computational fluid dynamics (CFD) simulations are performed on engine. Finally, the study also highlights the EHN effectiveness on several thermodynamic conditions as well as equivalence ratios. The objective is to assess its performance upon large operating conditions which appears to be of interest with novel combustion systems targeting low temperature as well as lean combustion.

Introduction

Worldwide transport is actually powered nearly entirely by IC engines. Their contribution in this sector is expected to be still very high in the next few decades. Novel combustion systems, such as spark assisted compression ignition (SACI) [START_REF] Manofsky | Bridging the Gap between HCCI and SI: Spark-Assisted Compression Ignition[END_REF] or gasoline compression ignition (GCI) [START_REF] Kalghatgi | Gasoline compression ignition approach to efficient, clean and affordable future engines[END_REF] are being developed for better performances. These combustion systems aim at increasing the engine efficiency. Their operation in lean combustion is a lever to reach this objective. This combustion mode operates in low temperature, which can result in low nitrogen oxides (NOx) and soot emissions. The main disadvantages of lean combustion are over-frequent misfire and engine instability [START_REF] Wang | Experimental study on factors affecting lean combustion limit of S.I engine fueled with compressed natural gas and hydrogen blends[END_REF]. In addition, the novel combustion systems currently investigated tend to explore new thermodynamic conditions for the combustion of gasoline fuels. Thus, reactivity control is one of the most difficult challenges regarding these engine developments. This task requires both experimental and simulations results. CFD simulations are a reliable method of engine simulation [START_REF] Puduppakkam | Predictive Combustion and Emissions Simulations for a High Performance Diesel Engine Using a Detailed Fuel Combustion Model[END_REF]. This kind of simulations require a simple but robust kinetic model to reduce computational cost.

To replace fossil fuels, renewable energy sources such as biofuel or natural gas have been studied and used in different kind of engines [START_REF] Mccormick | Properties of Oxygenates Found in Upgraded Biomass Pyrolysis Oil as Components of Spark and Compression Ignition Engine Fuels[END_REF][START_REF] Ansari | Experimental investigation into effects of high reactive fuel on combustion and emission characteristics of the Diesel -Natural gas Reactivity Controlled Compression Ignition engine[END_REF]. In the near future, taking advantage of all petroleum fractions is one of the most efficient way in using petroleum fuels. In this scope, low octane gasoline fuel is of interest. The benefits of this kind of fuel link to its low cost thanks to simple treatment in petroleum refineries. This fuel can be used in many moderns IC engines presented previously such as SACI or GCI.

Fuel additives have been used to control the fuel reactivity in many types of engines [START_REF] Seyferth | The Rise and Fall of Tetraethyllead. 2[END_REF][START_REF] Hashimoto | Fuel Ignition Quality Control by Addition and Catalytic Decomposition of Cetane Improver[END_REF]. EHN is a conventional additive used to enhance the reactivity of diesel fuels. The EHN efficiency has been investigated in many types of engines operating at low temperature [START_REF] Kaddatz | Light-Duty Reactivity Controlled Compression Ignition Combustion Using a Cetane Improver[END_REF][START_REF] Dempsey | Effect of Cetane Improvers on Gasoline, Ethanol, and Methanol Reactivity and the Implications for RCCI Combustion[END_REF]. Ghosh et al. [START_REF] Ghosh | Predicting the Effect of Cetane Improvers on Diesel Fuels[END_REF], by carrying out cetane number (CN) measurements in ignition quality tester (IQT), observed that EHN effectiveness in CN boosting increases with the base fuel reactivity. In order to fundamentally understand EHN effect, previous academic studies investigated the EHN effect on the autoignition of gasoline fuels having a different reactivity [START_REF] Hartmann | Experiments and modeling of ignition delay times, flame structure and intermediate species of EHN-doped stoichiometric n-heptane/air combustion[END_REF][START_REF] Goldsborough | Experimental and modeling study of fuel interactions with an alkyl nitrate cetane enhancer, 2-ethyl-hexyl nitrate[END_REF]. These studies underline a complex promoting effect of EHN that depends on the thermodynamic conditions, the doping level and the fuel chemical composition. Since EHN decomposes to an alkoxy radical and NO2 at relatively low temperature [START_REF]Thermal decomposition of isooctyl nitrate[END_REF], the chemical effect of EHN depends on both the derived radicals and the nitrogen chemistry. Hartmann et al. [START_REF] Hartmann | Experiments and modeling of ignition delay times, flame structure and intermediate species of EHN-doped stoichiometric n-heptane/air combustion[END_REF] and Goldsborough et al. [START_REF] Goldsborough | Experimental and modeling study of fuel interactions with an alkyl nitrate cetane enhancer, 2-ethyl-hexyl nitrate[END_REF] developed kinetic models to predict EHN behavior in mixture with n-heptane and two reference fuels (n-heptane/ iso-octane/ toluene mixtures) respectively. These modeling works reveal that two mains items need to be improved: (1) the prediction of the base fuel reactivity and (2) the coupling of nitrogen chemistry with hydrocarbons. Indeed, a confident prediction of base fuel reactivity is an important step to ensure a better investigation of EHN effect. Moreover, detailed mechanisms regarding nitrogen chemistry interactions with large hydrocarbons still need to be explored, especially for lean mixtures.

In order to assess the effect of a nitrogen containing additive under different conditions, this study aims at: (1) investigating experimentally and numerically the EHN effect at typical dopant amounts (0.1 -1% mol.) on the autoignition behavior of a low-octane gasoline surrogate fuel in lean combustion; (2) understanding the chemical effect of EHN on the fuel reactivity at different thermodynamic conditions and (3) investigating numerically EHN effect in various fuels, whose reactivity are different depending on the operating conditions. The employed fuel surrogate in experiments is a mixture of n-heptane and toluene with a research octane number (RON) of 84 which falls into a low octane gasoline range. The mixture also offers the possibility to evaluate both the additive effect on high temperature and low temperature with two stage ignitions due to a nonnegligible amount of alkane. In this study, IDT were measured in a RCM and a detailed kinetic mechanism containing nitrogen chemistry for fuel/EHN mixture was developed thanks to the assembly of literature data.

Experiments

The single piston RCM of PRISME laboratory was used in this study. The details of this RCM can be found elsewhere [START_REF] Pochet | Experimental and numerical study, under LTC conditions, of ammonia ignition delay with and without hydrogen addition[END_REF]. A creviced piston was employed to limit the formation of a roll-up vortex during the compression and maintain the post-compression charge homogeneity. The main characteristics of the RCM of PRISME laboratory are summarized in the table 1. The intake pressure was measured by a Keller PAA-33X/80794 piezoresistive transducer. The in-cylinder pressure history was registered thanks to a piezoresistive AVL QH32C transducer. The temperatures including intake temperature and piston initial temperature were controlled by K thermocouples. The Brooks Cori-Flows M13V101 were used to regulate the mass flow rates. The experimental uncertainties of intake pressure, in-cylinder pressure, intake temperatures and mass flowrate are ±1 mbar, ±1%, ±2 K, ±1%, respectively.

The high purity liquid fuels used in this study are from Sigma-Aldrich: toluene (99.8%), n-heptane (99%), EHN (97%). To fulfill the desired gas mixture, the synthetic air (21% O2, 79% N2) and high purity nitrogen (> 99.999%) from Air Liquide are used. The base fuel contains toluene (65% vol.) and n-heptane (35% vol.). These components present different reactivity in the low temperature range (T < 1000 K) and were also used as fuel surrogate in literature due to the adequate comprehension of their chemistry. The doping level of EHN in the base fuel varies from 0.1 (mixture 2) to 1% mol. (mixture 3). The composition of gas mixtures used in this study is shown in table 2. The pressure at top dead center (Pc = 10 bar) and temperatures at top dead center (Tc) are achieved by adjusting the initial temperature (T0), from 55 to 120°C, and the intake pressure (P0), from 227 to 600 mbar. Tc are calculated by using the adiabatic core model. The gas mixtures were prepared in a reservoir on the day of manipulation. The reservoir was heated to 80°C to ensure the total vaporization of the fuel. The homogeneity of each gas mixture was ensured by a mechanic agitation during 30 minutes.

As presented in figure 1, the main IDT is defined as the time between the end of the compression and the combustion Maximum Pressure Rise Rate (MPRR), which is the maximum value of dP/dt calculations.

In the case of two-stage ignition, the 1 st -stage IDT is defined as the time between the end of compression and the first distinguishable peak of dP/dt calculations. The measurable IDT by this RCM ranges from 1 to 200 ms.

Kinetic Modeling

A kinetic model was developed to simulate the EHN effect on the fuel surrogate. This model contains three sub-mechanisms describing chemistry of the base fuel, of EHN and of nitrogenous molecules. The sub-mechanism of the base fuel was adopted from the model of Lawrence Livermore National Laboratory (LLNL) [START_REF] Mehl | Kinetic modeling of gasoline surrogate components and mixtures under engine conditions[END_REF], which contains detailed chemistry of toluene and n-heptane. The toluene submechanism was then updated following mostly the recent study of aromatic molecules by Yuan et al. [START_REF] Yuan | Investigation on the pyrolysis and oxidation of toluene over a wide range conditions. II. A comprehensive kinetic modeling study[END_REF]. The updated reactions include the elementary reactions of toluene and its derived radicals and molecules, e.g., phenyl, benzyl, methylphenyl, cresoxy radicals, phenol, cresol and small unsaturated compounds C2-C4. The importance of 1,3-cyclopentadiene (C5H6) and cyclopentadienyl radical (C5H5) chemistry toward the toluene reactivity is highlighted by Yuan et al. [START_REF] Yuan | Investigation on the pyrolysis and oxidation of toluene over a wide range conditions. II. A comprehensive kinetic modeling study[END_REF]. Therefore, C5H6 sub-mechanism was updated according to recent data [START_REF] Oleinikov | Oxidation of cyclopentadienyl radical with molecular oxygen: A theoretical study[END_REF][START_REF] Ghildina | Reaction mechanism, rate constants, and product yields for unimolecular and H-assisted decomposition of 2,4cyclopentadienone and oxidation of cyclopentadienyl with atomic oxygen[END_REF][START_REF] Galimova | Reaction mechanism, rate constants, and product yields for the oxidation of Cyclopentadienyl and embedded five-member ring radicals with hydroxyl[END_REF].

Concerning the reaction pathway of EHN, previous studies agree on the main decomposition steps [START_REF] Hartmann | Experiments and modeling of ignition delay times, flame structure and intermediate species of EHN-doped stoichiometric n-heptane/air combustion[END_REF][START_REF] Goldsborough | Experimental and modeling study of fuel interactions with an alkyl nitrate cetane enhancer, 2-ethyl-hexyl nitrate[END_REF][START_REF] Bornemann | Thermal decomposition of 2-ethylhexyl nitrate (2-EHN)[END_REF]. First, the weakest bond (O-N bond) of EHN molecule is broken to release NO2. The 2ethylhexyloxy (EHO) radical can then undergo a β-scission to form formaldehyde and 3-heptyl radical. This reaction pathway of EHN is adopted in this study. As EHN releases NO2 during its decomposition, nitrogen chemistry and its interactions with hydrocarbons are included in this kinetic model. NOx sub-mechanism containing small molecules (C1-C2) was adopted mainly from the recent review of Glarborg et al. [START_REF] Glarborg | Modeling nitrogen chemistry in combustion[END_REF]. The reactions of nitroethane proposed by Zhang et al. [START_REF] Zhang | Experimental and Kinetic Modeling Study of Nitroethane Pyrolysis at a Low Pressure: Competition Reactions in the Primary Decomposition[END_REF] were included in the model. The same reaction scheme was adapted to describe the reactivity of the two nitropropane isomers (1-nitropropane and 2-nitropropane). The NOx interactions with heavier hydrocarbons (C4-C7) were based on the work of Anderlorh et al. [START_REF] Anderlohr | Modeling of autoignition and NO sensitization for the oxidation of IC engine surrogate fuels[END_REF] and Chai et al. [START_REF] Chai | Rate coefficients for fuel + NO 2: Predictive kinetics for HONO and HNO 2 formation[END_REF].

The validation of the kinetic model was carried out using both literature [START_REF] Glänzer | Thermische Zerfallsreaktionen von Nitroverbindungen in Stosswellen. III: Dissoziation von 1-Nitropropan und 2-Nitropropan[END_REF][START_REF] Rasmussen | Sensitizing effects of NOx on CH4 oxidation at high pressure[END_REF][START_REF] Dagaut | Mutual Sensitization of the Oxidation of Nitric Oxide and Simple Fuels Over an Extended Temperature Range: Experimental and Detailed Kinetic Modeling[END_REF][START_REF] Nelson | Hydrocarbon-NOx interactions at low temperatures-1.Conversion of NO to NO2 promoted by propane and the formation of HNCO[END_REF][START_REF] Dubreuil | HCCI combustion: Effect of NO in EGR[END_REF][START_REF] Moréac | Nitric oxide interactions with hydrocarbon oxidation in a jetstirred reactor at 10 atm[END_REF] and measurements performed in this work. The target data include IDT and species profiles measurements. Experimental conditions cover a wide range of temperature and pressure (500 -1200 K, 1 -50 bar) as well as reactor types (shock tube, plug flow reactor and perfectly stirred reactor). This step validates the mechanism for the combustion of mixtures of toluene and n-heptane as well as for interactions between NOx and hydrocarbons.

In this study, the RCM results are simulated by using the 0-D closed homogenous reactor model in CHEMKIN-PRO [START_REF]Reaction Design[END_REF]. During compression process , a frozen chemistry assumption is employed.

End of compression

This assumption could be justified by the low reactivity of the surrogate fuel in the investigated experimental conditions. The heat loss during the combustion is taken into account by using an "effective volume", which is calculated by adopting the adiabatic core hypothesis. To obtain the volume profiles, non-reactive experiments with N2 were first performed at the same initial pressure and temperature as for each of the targeted experiment.

Results and Discussions

The experimental and modeling results are presented as follows. First, the measured and simulated IDT of the base fuel with/without EHN in RCM experiments are introduced. Then the EHN effect on the reactivity of the base fuel is chemically discussed thanks to the kinetic model. Finally, the impact of fuel reactivity, pressure and equivalence ratio on EHN efficiency are numerically investigated.

RCM Experiments Results and Simulations

Figure 2 presents the experimental results obtained in this study and the simulations of the 1 st -stage and main IDT of the surrogate fuel with and without addition of EHN. The measurements were conducted at 10 bar from 675 to 1000 K for an equivalence ratio (Φ) of 0.5. A twostage ignition is observed at the lowest temperatures (T < 750 K). This feature is reasonably captured by the model. Globally, the kinetic model predicts correctly the surrogate fuel reactivity. The most remarkable differences between experimental and modeling results are observed at low temperatures (T < 700 K), for which the model underestimates the IDT of the fuel. The considered 0-D simulation method in this study could be used to model reliably the RCM results for single-stage ignition fuel [START_REF] Goldsborough | Experimental and modeling study of fuel interactions with an alkyl nitrate cetane enhancer, 2-ethyl-hexyl nitrate[END_REF]. However, due to the complex multi-dimensional effects inside RCM initiated by boundary layer, roll-up vortex, non-uniform heat release and piston crevice, the multi-stage ignition processes are highly perturbed. The measured IDT can be increased up to 25% [START_REF] Mittal | CFD modeling of twostage ignition in a rapid compression machine: Assessment of zero-dimensional approach[END_REF]. The application of 0-D simulation method is consequently under debate for multi-stage ignition fuels [START_REF] Mittal | CFD modeling of twostage ignition in a rapid compression machine: Assessment of zero-dimensional approach[END_REF]. A computational fluid dynamics (CFD) simulation as presented by Mittal et al. [START_REF] Mittal | CFD modeling of twostage ignition in a rapid compression machine: Assessment of zero-dimensional approach[END_REF] could be used to get a better agreement between experiments and simulations when a two-stage ignition is observed however this is beyond the scope of the study. In addition, most of toluene reactions in this model are adopted from the model by Yuan et al. [START_REF] Yuan | Investigation on the pyrolysis and oxidation of toluene over a wide range conditions. II. A comprehensive kinetic modeling study[END_REF]. However, there is a large spread in predictions of IDT of toluene at low temperature (T < 873 K) by recent models in the literature, including Yuan et al.'s model, as it has been observed by Zhang et al. [START_REF] Zhang | Probing the antagonistic effect of toluene as a component in surrogate fuel models at low temperatures and high pressures. A case study of toluene/dimethyl ether mixtures[END_REF]. In fact, due to the low reactivity of toluene and to experimental device limitations, the available data for validation at low temperature are limited. Further studies would be required in order to get more reliable simulation results.

As presented in figure 2, at 10 bar and Φ of 0.5, EHN shows a promoting effect at all investigated temperatures for both 1 st -stage and main ignition of the surrogate fuel. At doping level of 1% mol., EHN suppresses the cool flame of the surrogate fuel ignition at 750 K. This feature is captured by the model. When two-stage ignitions are observed, the time between 1 st -stage ignition and the main ignition is not affected by EHN. It can be concluded that EHN shows its effect only on the 1 st -stage IDT of the two-stage ignition. The EHN promoting effect increases nonlinearly with EHN doping levels. For a given doping level, EHN shows the smallest effect around 705 K. EHN reduces more significantly IDTs of the surrogate fuel as temperature increases. The model underestimates slightly EHN effect at the upper range of temperature (T > 850 K). Overall, the model successfully predicts the EHN effect on the IDT of the surrogate fuel over the whole range of examined temperatures and doping levels.

In addition to IDT, the model simulates reasonably well the pressure profiles of RCM experiments. Figure 3 introduces the measured and simulated pressure profiles of measurements with and without EHN 0.1% at 705 K, 10 bar and Φ of 0.5. The heat release governs directly the pressure profile. Experimental results indicate that EHN does not influence the heat release magnitude of the 1 st -stage ignition. This feature is well predicted by the kinetic model developed in this study. In all conditions, the model predicts reliably the peak pressure of the 1 st -stage ignition. Additionally, the simulated peak pressures of main ignition are about 4 bar in average higher than the measured ones. This fact could be associated to the heat loss of combustion to the piston, to the mass loss of the fuel during the experiment preparation, which is estimated about 3% of initial fuel mass and to the uncertainty of pressure transducer (±1%).

EHN Effect Evaluation via Kinetic Modelling

The simulation results indicate that EHN decomposes totally within less than 1 ms for initial concentrations at all considered temperatures (675 -995 K) in RCM experiments. EHN influences the reactivity at a very early time. The products of EHN decomposition are EHO and NO2. An analysis by species substitution is carried out to investigate the importance of each product. In this analysis, a reactive compound is replaced by a non-reactive one called "fictive compound" having the same thermodynamic properties. For example, to determine NO2 effect formed by the reaction EHN = EHO + NO2, NO2 in this reaction is replaced by a non-reactive molecule "NO2-fictive". The simulated IDT in presence of NO2-fictive is IDTNO2. A new parameter named Impact Ratio (IRN), with N being the compound replaced by the corresponding fictive compound, is defined as:

𝐼𝑅 𝑁 = 𝐼𝐷𝑇 𝑁 -𝐼𝐷𝑇 𝑛𝑜𝑟𝑚𝑎𝑙 𝐼𝐷𝑇 𝑛𝑜𝑟𝑚𝑎𝑙 × 100% (2)
IDTnormal is the simulated IDT without any fictive compound. IRN can take negative or positive value. If IRN is positive, the compound N has a promoting effect for the fuel reactivity and vice versa, the compound N has an inhibitor effect if IRN is negative. Bigger the absolute value of IRN is, the corresponding compound N has higher impact on the reactivity of the fuel.

Figure 4 presents the impact ratios of EHO and NO2 for EHN effect considering the RCM experimental conditions investigated in this study. While IRNO2 varies with temperature, IREHO is nearly constant. The absolute value of IRNO2 is generally larger than the one of IREHO.

It can be concluded that NO2 conducts globally the EHN effect in the considered experimental conditions especially at doping level of 1% in the upper range of temperature (T > 850 K). Additionally, NO2 has a very small inhibiting effect on the surrogate fuel reactivity at very low temperature (T < 700 K). One application of kinetic model is to be used in CFD simulation to get reliable engine simulation results. In order to reduce the computational cost, the kinetic model is required to be as simple as possible. Therefore, it is important to investigate whether EHN could be replaced by a simpler molecule. The results presented figure 4 suggest that only NO2 could be used to replace EHN in simulation especially at temperature above 850 K. This feature is presented in figure 5. NO2 promoting effect is generally similar to EHN promoting effect. The simulated IDT in the presence of NO2 is about 30% higher than the one in the presence of EHN at the same doping level (1% mol.). In RCM experiments of this study, it is found that EHN promoting effect reaches a minimum around 705 K and increases with temperature. To understand this trend, rate of production (ROP) analyses at 1 ms are carried out for different EHN doping levels (0, 0.1 and 1% mol.). The simulation conditions are: 10 bar, Φ = 0.5 and two temperatures: 705 K and 995 K.

At 705 K, EHN effect depends more on EHO, which decomposes to form an isomer of n-heptyl radical (C7H15-3). This n-heptyl radical contributes to the n-heptane oxidation as presented in figure 6. This feature enhances the reactivity of the surrogate fuel. In addition to EHO, NO2 can contribute to promote the surrogate fuel reactivity by enhancing toluene oxidation. Figure 7 represents the schematic oxidation of toluene at 705 K: The new reaction (R2) leads to the formation of C6H5CH2O radical, which is more reactive than C6H5CH2OO. C6H5CH2O radical can easily break into C6H5 and CO by many H atom release steps. Additionally, reaction R3 forms OH radicals. These features enhance the consumption of toluene within the fuel.

At 995 K, EHN effect depends importantly on NO2 which has a high impact on toluene oxidation. The schematic oxidation of toluene is presented in figure 8. Consequently, EHN is more efficient at high temperature (995 K) than at low temperature (775 K). This trend is well observed in RCM experiments. These ROP analysis results reveal the importance of nitrogen chemistry in EHN promoting effect. It is also observed that EHN effect takes part in the initiation and propagation steps of combustion.

Evaluation of EHN Effect for Several Operating Conditions

The validated kinetic model is used to investigate numerically EHN effect in various conditions. As presented in figure 2, the best agreement between experiments and simulations is obtained in the range of temperature from 700 to 850 K. The EHN effect prediction focuses in this range of temperature. The 0-D simulations in this part are carried out by using the closed homogenous reactor having a constant volume in CHEMKIN-PRO [START_REF]Reaction Design[END_REF]. The EHN effectiveness (Reff) is defined as shown in the equation (1). Higher Reff is, more effective is EHN in reducing IDT.

𝑅 𝑒𝑓𝑓 = 𝐼𝐷𝑇 𝑢𝑛𝑑𝑜𝑝𝑒𝑑 -𝐼𝐷𝑇 𝑑𝑜𝑝𝑒𝑑 𝐼𝐷𝑇 𝑢𝑛𝑑𝑜𝑝𝑒𝑑 × 100% (1)

Variation of RON

Figure 9 represents EHN effect in various fuels, which are mixtures of toluene and n-heptane. The details of these fuels are presented in table 3. RON value of these fuels were measured by Badra et al. [START_REF] Badra | A methodology to relate octane numbers of binary and ternary n-heptane, iso-octane and toluene mixtures with simulated ignition delay times[END_REF] in a single-cylinder Cooperative Fuels Research (CFR) engine. The high RON value corresponds to the high reactive fuel and vice versa. The simulation conditions are: 713 K, Φ = 0.7, 21.1 atm, which are close to the conditions of IQT tests carried out by Bogin et al. [START_REF] Bogin | Ignition Quality Tester (IQT) Investigation of the Negative Temperature Coefficient Region of Alkane Autoignition[END_REF].

The simulation results presented in figure 9 indicate that EHN is more effective with more reactive fuel. EHN effectiveness tends to a limit when the doping level increases. These features are coherent with experimental results by IQT test of Ghosh et al. [START_REF] Ghosh | Predicting the Effect of Cetane Improvers on Diesel Fuels[END_REF]. In fact, the fuel combustion is a series of free radicals reactions consisting of four steps: initiation, propagation, branching and termination [START_REF] Ghosh | Predicting the Effect of Cetane Improvers on Diesel Fuels[END_REF]. The branching step is the most important step of combustion, which dominates the fuel reactivity. EHN influences the initiation and propagation step of combustion. In the case of low reactive fuel, the branching step is limited and not affected by the presence of EHN. Conversely, the highly reactive fuel takes more advantage of EHN effect: more reactive the fuel is (low RON), more remarkable is the EHN promoting effect.

Conclusions

In order to evaluate the EHN efficiency on gasoline fuel reactivity control in different conditions, the EHN promoting effect (0.1-1% mol.) is investigated experimentally and numerically. RCM experiments were carried out at an equivalence ratio Φ of 0.

Figure 1 .

 1 Figure 1. Typical pressure (blue line) and dP/dt (red line) profiles in RCM experiment. Experimental condition: surrogate fuel, Φ = 0.5, 705 K, 10 bar.

Figure 2 .

 2 Figure 2. Effect of EHN for two dopant concentrations: 0.1% (green) and 1% (red), on the reactivity of the surrogate fuel (blue) at 10 bar and Φ = 0.5. Experiments: symbol. Simulation: line. Main IDT: solid lines and filled symbols. 1 st -stage IDT: dashed lines and unfilled symbols.

Figure 3 .

 3 Figure 3. Measured (solid lines) and simulated (dashed lines) pressures profiles in the RCM at 10 bar, 705 K and Φ = 0.5. Undoped surrogate fuel: blue, surrogate fuel with 0.1% EHN: green.RCM data validation step confirms the good performance of the new kinetic model, which predicts reliably the EHN effect on the reactivity of the surrogate fuel in lean combustion.

Figure 4 .

 4 Figure 4. Impact ratios of EHO (black) and NO2 (red) in EHN effect at two doping levels: 0.1% mol. (dashed lines) and 1% mol. (solid lines) for the surrogate fuel reactivity. Experimental conditions: Φ = 0.5, 10 bar.

Figure 5 .

 5 Figure 5. Simulated IDT of the surrogate fuel in this study in presence of additives. Conditions simulations: Φ = 0.5, 10 bar. Neat surrogate: blue solid line. Surrogate with EHN (1% mol.): red solid line. Surrogate with NO2 (1% mol.): red dashed line.

Figure 6 .

 6 Figure 6. Main oxidation pathway of n-heptane at 705 K.

Figure 7 .

 7 Figure 7. Main oxidation pathway of toluene at 705 K. At 705 K, toluene forms principally benzyl radical by H-abstraction reaction. This radical then reacts with O2 to form benzyl peroxy radical (C6H5CH2OO), which is has a low reactivity due to the limitation of intramolecular isomerization. EHN enhances the reactivity by a loop NO2-NO-(1) as: EHN = EHO + NO2 (R1) C6H5CH2 + NO2 = C6H5CH2O + NO (R2) NO + HO2 = NO2 + OH (R3)

Figure 8 .

 8 Figure 8. Main oxidation pathway of toluene at 995 K. At high temperature, toluene reacts with OH to form both benzyl radical and methylphenyl radical. Additionally, toluene can form benzene and methyl radical by reacting with H radical. Methyl radical can react with NO2 by a similar way as benzyl radical. As temperature increases, the reaction between HO2 and NOx is favored. The loop NO2-NO-(2) representing EHN promoting effect becomes: EHN = EHO + NO2 (R1) C6H5CH2 + NO2 = C6H5CH2O + NO (R2) CH3 + NO2 = CH3O + NO (R*) HO2 + NO2 = HONO + O2 (R**) HONO = NO + OH (R***) NO + HO2 = NO2 + OH (R3) Compared to the loop NO2-NO-(1), the loop NO2-NO-(2) generates more OH radical thanks to the reactions (R*), (R**) and (R***).Consequently, EHN is more efficient at high temperature (995 K) than at low temperature (775 K). This trend is well observed in RCM experiments. These ROP analysis results reveal the importance of nitrogen chemistry in EHN promoting effect. It is also observed that EHN effect takes part in the initiation and propagation steps of combustion.

Table 1 .

 1 RCM main characteristics.

	Compression time	35 ms
	Time Pmax/2 to Pmax (t50)	4 ms
	Piston acceleration/deceleration time 15/5 ms
	Crevice volume	9.94 cm 3
	Compression ratio	8.9; 12.9; 20.7
	Stroke, bore	300 mm, 50 mm

Table 2 .

 2 Mole fraction of the different gas mixtures in RCM experiments.

	Mix. Toluene (%) n-Heptane (%) EHN (ppm) O2 (%) N2 (%)
	1	0.78	0.31	0	20.77	78.14
	2	0.78	0.31	10.86	20.77	78.14
	3	0.77	0.3	108.61	20.77	78.14

Table 3 .

 3 Characteristic of fuels used in simulations.

	Fuel	Composition (% mol.) Toluene n-Heptane	RON
	1	0.65	0.35	84
	2	0.50	0.50	66
	3	0.20	0.80	28

  Figure 11 introduces the influence of EHN in fuel reactivity by varying equivalence ratio. Two fuels having RON values of 28 and 84 were examined. Similarily to the variation of pressure, EHN effectiveness is predicted to increase with equivalence ratio. EHN is always more effective in the more reactive fuel at all equivalence ratios. The EHN influence is more sensitive to equivalence ratio in the case of low reactivity fuel than in the case of high reactivity fuel. For low reactivity fuel (RON = 84), as equivalence ratio increases from 0.7 to 1.3, EHN effectiveness increases from 43 to 68%. For high reactivity fuel (RON = 28), EHN efficiency changes slightly, from 68 to 78%.

	Figure 10. Simulated EHN effectiveness in two fuels having RON of 28 (blue)
	and RON of 84 (red) with various pressures. Simulations conditions: 713K, Φ
	= 0.7, 1% mol. EHN.
	Variation of Equivalence Ratio
		50
		(%)
		30 R eff
		10
		20	40	60	80
			RON value of neat fuel
		Figure 9. Simulated EHN effectiveness in different fuel with various doping
	Figure 11. Simulated EHN effectiveness in two fuels having RON of 28 (blue)	levels: 0.1% mol. (green), 0.5% mol. (blue) and 1% mol. (red). Simulations
	and RON of 84 (red) with various equivalence ratios. Simulations conditions:	conditions: 713K, Φ = 0.7, 21.1 atm.
	713K, 21.1 atm, 1% mol. EHN.
		Variation of Pressure
		Figure 10 introduces EHN effect (1% mol.) at different pressures while
		temperature and mixture composition remain unchanged. Two fuels
		(RON values of 28 and 84) which present very different reactivity
		were examined. EHN is more effective in the more reactive fuel (RON
		= 28) at all considered pressures. In addition, EHN effectiveness
		increases with pressure. While pressure increases, EHN impact is more
		sensitive to the low reactivity fuel. In the case of low reactivity fuel
		(RON = 84), as pressure increases from 20 to 30 atm, EHN
		effectiveness raises from 59 to 68%. Meanwhile, for high reactivity
		fuel (RON = 28), EHN effectiveness remains nearly constant, from 68
		to 71%.

  5, a pressure of 10 bar and a temperature range from 675 to 995 K. The targeted fuel is a low-octane gasoline surrogate fuel (RON = 84) comprising both toluene and n-heptane. A kinetic model is developed from literature data assembly. Three sub-mechanisms describing chemistry of the base fuel, of EHN and of nitrogenous molecules are included in the model. The validated kinetic model is used to predict EHN effect in various conditions.Experiments results indicate that EHN reduces the IDT of the surrogate fuel at all examined temperatures and doping levels. At 750 K, EHN suppresses the cool flame ignition of the surrogate fuel. The EHN effectiveness tends to a minimum around K and increases with temperature. The results also indicate that EHN effect increases nonlinearly with EHN doping levels.The model reasonably predicts the EHN effect over the whole range of examined temperatures and doping levels. Numerical analyses indicate that EHN is effective in the very early stage of combustion thanks to a rapid decomposition. EHN effect depends majorly on NO2 generated by its decomposition, i.e., on nitrogen chemistry. EHN promoting effect is linked to NO2-NO loops, which are temperature dependent. These loops enhance fuel reactivity by forming new pathways of fuel consumption and by giving a supplementary source of OH radicals, thereby, enhancing the combustion. In order to capture the EHN effect on other kinetic mechanism, this study also demonstrates NO2 addition could be used to simulate most of the doping effect for simulations over a temperature range from 850 to 1000 K.Simulations by the validated kinetic model in this study indicate that EHN efficiency increases with fuel reactivity, pressure and equivalence ratio.This study provides insight into the effect of a cetane booster additive (EHN) on the reactivity of fuels. It emphasizes the underlying mechanisms responsible of the reactivity variation at different thermodynamic conditions representative of engine conditions. These results open up the use of kinetic modelling to understand additive kinetics and improve fuel / engine adequacy.

		70
	(%)
	R eff	50
		10		20 Pressure (atm)		30
		70
	(%)
	R eff	50
		30
		0,6	0,8	1 Equivalence ratio	1,2	1,4

Contact Information

mickael.matrat@ifpen.fr

Definitions/Abbreviations


M F Pucci

Benoît Duchemin

M Gomina

J Breard

J Breard Identification

J Bréard

IDENTIFICATION OF KEY PARAMETERS IN DYNAMIC WETTING AND DEWETTING PHENOMENA WITH MOLTEN POLYMERS FOR COMPOSITE PROCESSING

Keywords: dynamic wetting, capillary effects, molten polymers

whether they are published or not. The documents may come

Introduction

The main current research interest in the field of composite materials consists in minimizing the environmental impact of both materials and manufacturing. Having a good control of (bio)composite manufacturing implies a good knowledge of wetting phenomena occurring during impregnation of (natural) fibres by thermoset or thermoplastic polymers. It is known that capillary effects, and particularly capillary parameters like molten polymer surface tension and fibre surface energy, play an important role on impregnation. These effects coupled with the viscous ones are dominant during composite manufacturing [START_REF] Bréard | Analysis of dynamic flows through porous media. Part I: Comparison between saturated and unsaturated flows in fibrous reinforcements[END_REF][START_REF] Pucci | Capillary effects on flax fibers-modification and characterization of the wetting dynamics[END_REF]. The visco-capillary balance in dynamic wetting can be described by a hydrodynamic approach. In the literature, the de Gennes and Cox-Voinov theories [START_REF] Snoeijer | Moving Contact Lines: Scales, Regimes, and Dynamical Transitions[END_REF] were used to fit experimental data (obtained mainly by optical methods) of liquid drops spreading on model substrates at standard conditions (room temperature). This procedure is not suitable in the case of fibres wetted by molten polymers for composite applications. Moreover, the influence of temperature and polymer molecular weight still remain unknown. The aim of this study was to set an experimental procedure that can be applied to fibres and polymers to investigate wetting dynamic and identify physical parameters such as a slip length, that can be inserted into numerical models to predict flow process [START_REF] Yeager | Numerical model of fiber wetting with finite resin volume[END_REF].

The experimental procedure was validated on cellulosic films with different liquids. Paraffin oils are totally wetting liquids and they were used in order to evaluate the influence of temperature on wetting dynamics. Polyethylene glycols (PEGs) are partially wetting liquids; they were used to evaluate the effect of molecular weights (Mn) on wetting dynamics of molten polymers. An independent characterisation of the solid and of the liquids was carried out as well as solid/liquid dynamic wetting tests. Hydrodynamic models were used to fit the dynamic contact angle data. The results are meaningful for a better understanding of the influence of temperature, viscosity and liquid molecular weight on the dynamic contact angles and on capillary phenomena existing during the impregnation of fibrous reinforcements in composite processing.

Materials and experimental methods

Cellulosic films, referenced as NatureFlex TM 23 NP and gracefully provided by Innovia Films, were used in this study. Films have a thickness of 23 µm and are manufactured from biodegradable cellulose, which is derived from wood pulp. In order to consider cellulosic films as model substrates, such as ideal smooth and chemically homogeneous surfaces, morphology and surface chemistry of films were analysed by atomic force microscopy (AFM), Fourier-transform infrared spectroscopy (FTIR) and Wide-angle X-ray diffraction (WAXD).

Liquids used in this study were two paraffin oils, named (1) and ( 2) and three PEGs with Mn = 1450, 3350 and 10000 g/mol. Rheological tests were carried out with a Physica MCR 501 rheometer from Anton Paar to measure viscosity as a function of temperature. A Krüss DSA100 tensiometer equipped with a thermalized syringe and chamber was used to measure surface tension as a function of temperature via the pendant drop method (Figure 1-left) and density via the sessile drop method.

A Krüss K100SF tensiometer with a temperature controller was used to perform dynamic wetting tests between the cellulosic films and all the liquids (Figure 1-right). Regarding paraffin oils, three temperatures were considered for dynamic tests: 25, 50 and 75°C. Dynamic wetting tests of PEGs were carried out at T=75°C. Different velocities were imposed on the films for liquid immersion (wetting) and withdrawal (dewetting). This protocol produced dynamic contact angles for a large range of capillary numbers Ca.

Results and discussion

It was found that hydrodynamic models fit quite well experimental results of advancing contact angles (θ a ) for paraffin oils at different temperatures as a function of capillary number above Ca ≈ 10 -3 (Figure 2). Varying the temperature modifies both surface tension and viscosity, changing the capillary number regime. This temperature variation does not remarkably affect the slip length, which is therefore a characteristic of the liquid in this hydrodynamic approach. Furthermore, the experimental data follow a same trend at various temperatures. For molten PEG at T=75°C, different trends were obtained for different Mn. This result suggests that the slip length depends on the chain length, which in turn affects the dynamic wetting. These points will be detailed in the communication.

Figure 1 :

 1 Figure 1: Left: Pendant drop capture for surface tension measurement. Right: Immersion of a cellulosic film in a molten polymer at the K100SF tensiometer.

Figure 2 :

 2 Figure 2: Advancing contact angles as a function of capillary number and fits according to the Cox-Voinov law.


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

M Ouisse

A Khelif

S De Rosa

F Franco

A LITERATURE REVIEW FOR THE ANALYSIS OF VIBORACOUSTIC PROPERTIES OF PERIODIC INCLUSIONS IN POROUS MATERIALS

The design based on periodic elements is a powerful strategy for the achievement of lightweight sound packages and represents a convenient solution for manufacturing aspects. An interesting research target is 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; 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. This work is a literature review which mainly focuses on three aspects concerning periodic inclusions in porous materials: the concept of substitution of a fluid layer for a porous layer, the Transfer Matrix Method and the descriptions of two alternatives to multi-layering.

SUBSTITUTION OF A FLUID LAYER FOR A POROUS LAYER 1.Description of sound propagation in porous media

On a microscopic scale, sound propagation in porous materials is generally difficult to study because of the complicated geometries of the frames. Only the mean values of the quantities involved are of practical interest. The averages must be performed on a macroscopic scale, on a homogenization volume with sizes large enough to become meaningful. At the same time, these sizes must be much smaller than the acoustic wavelength. The description of sound propagation in porous material can be complicated by the fact that sound also excites and moves the frame of the material. If the frame is motionless, in a first step, the air inside the porous medium can be replaced at the macroscopic scale by an equivalent free fluid. This equivalent fluid has a complex effective density ρ and a complex bulk modulus K. The wave number k and the characteristic impedance Zc of the equivalent fluid are also complex [START_REF] Allard | Propagation of Sound in Porous Media[END_REF].

An empirical model provided by Delany and Bazley

The k and Zc have been measured by Delany and Bazley (1970) for a large range of frequencies in many fibrous materials with almost unit porosity. According to these measurements, the quantities k and Zc depend mainly on the angular frequency ω and on the flow resistivity σ of the material. Good fits for k and Zc have been obtained:

𝑍 𝑐 = 𝜌 0 𝑐 0 [1 + 0.057𝑋 -0.754 -𝑗0.087𝑋 -0.732 ]. (1)
𝑘 = 𝜔 𝑐 0 [1 + 0.0978𝑋 -0.700 -𝑗0.189𝑋 -0.595 ]. (2)
where ρ0 and c0 are the density of air and the speed of sound in air; X is a dimensionless parameter which is suggested to be valid within the range 0.01 < 𝑋 < 1.00. These relations will not provide perfect predictions of acoustic behaviour of all the porous materials in every frequency ranges. Nevertheless, the laws of Delany and Bazley are widely used and can provide reasonable orders of magnitude for Zc and k. With fibrous materials, which are anisotropic, the flow resistivity must be measured in the direction of propagation for waves travelling in either the normal or the planar direction. The case of oblique incidence is more complicated. It should be pointed out that after the work by Delany and Bazley, several authors suggested slightly different empirical expressions of k and Zc for specific frequency ranges and for different materials [START_REF] Allard | Propagation of Sound in Porous Media[END_REF].

Mesostructure based models

In the case of common porous materials, an analytical description of sound propagation that takes into account the complete geometry of the microstructure is not possible. This explains why the models of sound propagation in these materials are mostly phenomenological and provide a description only on a large scale [START_REF] Allard | Propagation of Sound in Porous Media[END_REF]. In 1987, Johnson Koplik and Dashen proposed a semiphenomenological model to describe the complex density of an acoustical porous material with a motionless skeleton having arbitrary pore shapes; 4 parameters are involved in the calculation of this dynamic density: the open porosity ϕ, the static air flow resistivity σ, the high frequency limit of the tortuosity α∞ and the viscous characteristic length Λ. In 1991, Champoux and Allard introduced an expression for the dynamic bulk modulus for the same kind of porous material based on the previous work by Johnson et al.; 2 parameters are involved in the calculation of this dynamic bulk modulus: the open porosity ϕ and the thermal characteristic length Λ'. In Biot-Allard model, which includes the description of the skeleton movement, a material is characterized by a number of conventional mechanical parameters (density of the skeleton, Young's modulus of the skeleton in vacuum, Poisson's coefficient of the skeleton in vacuum, structural damping), as well as by specific parameters called Biot parameters (porosity, resistivity, tortuosity, viscous characteristic length, thermal characteristic length).

MULTILAYERED SYSTEMS WITH POROUS MATERIALS MODELED USING THE TRANSFER MATRIX METHOD

The description of the acoustic field in a porous layer is complicated by the presence of the shear wave and the two longitudinal waves. In a layered medium with porous layers, elastic solid layers and fluid layers, a complete description can become very difficult. A matrix representation of sound propagation well consolidated in literature is described here. The stratified media are assumed laterally infinite. They can be of different nature: elastic solid, thin plate, fluid, rigid porous, limp porous and poroelastic. However, the different media are assumed to be homogeneous and isotropic. Figure 1 illustrates a plane acoustic wave impinging upon a material of thickness h, at an incidence angle θ. Various types of waves can propagate in the material, according to their nature. The x1 component of the wave number for each wave, propagating in the finite medium, is equal to the x1 component kt of the incident wave in the free air:

𝑘 𝑡 = 𝑘 𝑠𝑖𝑛 𝜃.
(

) 3
being k the wave number in free air. Sound propagation in the layer is represented by a transfer matrix [T] such that

𝑽(𝑀) = [𝑇]𝑽(𝑀 ′ ). (4)
Where M and M' are set close to the forward and the backward face of the layer, respectively, and where the components of the vector V(M) are the variables which describe the acoustic field at a point M of the medium. The matrix [T] depends on the thickness h and the physical properties of each medium [START_REF] Allard | Propagation of Sound in Porous Media[END_REF].

Figure 1. Plane wave impinging on a domain of thickness h.

ALTERNATIVES TO MULTI-LAYERING

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 airmaterial interface, the efficiency of such devices relies on the allowable thickness [START_REF] Weisser | Acoustic behavior of a rigidly backed poroelastic layer with periodic resonant inclusions by a multiple scattering approach[END_REF].

Embedding rigid inclusions

One way allowing to enhance the low frequency efficiency of sound packages consists in embedding periodic rigid inclusions in a porous layer [START_REF] Groby | Acoustic response of a periodic distribution of macroscopic inclusions within a rigid frame porous plate[END_REF]. If the radius of these periodic inclusions is comparable with the acoustic wavelength, then an increase of the absorption coefficient can be observed. In [START_REF] Groby | Acoustic response of a periodic distribution of macroscopic inclusions within a rigid frame porous plate[END_REF], the influence of the periodic inclusions on the absorption coefficient was explained by excitation of additional acoustic modes which dissipate acoustic energy. When the porous layer is backed by a flat rigid surface and when only one inclusion per unit cell grating is embedded, an additional trapped mode can be excited. This results in a quasi-total absorption peak at a frequency below that of the usual quarter-wavelength resonance in the homogeneous layer case. Other interesting studies related to volume heterogeneities are in [START_REF] Olny | Acoustic wave propagation in double porosity media[END_REF][START_REF] Groby | Absorption of a rigid frame porous layer with periodic circular inclusions backed by a periodic grating[END_REF][START_REF] Borchers | Acoustic protection on payload fairings of expendable launch vehicles[END_REF][START_REF] Sugie | Effect of inserting a Helmholtz resonator on sound insulation in a double-leaf partition cavity[END_REF][START_REF] Fang | Ultrasonic metamaterial with negative modulus[END_REF].

Embedding Helmholtz resonators

The homogenization theory cannot be applied to cellular material made of large periodic unit-cell.

In this case, a low frequency solution to improve the acoustic efficiency of passive open-cell porous materials is to embed Helmholtz resonators (HR) in the porous matrix. Doing so, at the Helmholtz resonance frequency, the transmission loss is greatly improved and the sound absorption of the host material is decreased if it is made of a highly sound absorbing material. One of the first works describing such structure is a patent filed in by Borchers et al. [START_REF] Borchers | Acoustic protection on payload fairings of expendable launch vehicles[END_REF]. Much later, Sugie et al. [START_REF] Sugie | Effect of inserting a Helmholtz resonator on sound insulation in a double-leaf partition cavity[END_REF] proposed a similar heterogeneous material made of a fibrous sound absorbers with resonant inclusions. More recently, the acoustic community had shown a keen interest in the equivalent material (also called effective material or metamaterial) presenting a negative bulk modulus at the HR resonance frequency [START_REF] Fang | Ultrasonic metamaterial with negative modulus[END_REF]. In [START_REF] Lagarrigue | Absorption of sound by porous layers with embedded periodic arrays of resonant inclusions[END_REF] the sound absorption efficiency is investigated in case of rigid backed acoustic foams with resonant split hollow cylinder inclusions. Under the assumption that the HR periodicity is much smaller than the acoustic wavelength, the resonant materials are usually modelled as homogenized equivalent material with modified bulk modulus to account for the presence of the resonant inclusion [START_REF] Fang | Ultrasonic metamaterial with negative modulus[END_REF]. However, as already stated, cellular resonant material with large periodicity obviously prevents the use of the homogenization method. In this case, the orientation of the HR neck may have a strong influence on the sound absorption behaviour of the resonant material [START_REF] Lagarrigue | Absorption of sound by porous layers with embedded periodic arrays of resonant inclusions[END_REF][START_REF] Doutres | Modeling and experimental validation of cellular porous material with large resonant inclusions[END_REF].


Louvel Sylvain
email: louvel@esitc-paris.fr

Audo Mariane

Sebaibi Nassim

Boutouil Mohamed

Gomina Moussa

Alkali activated mortars formulation optimized by an experimental design

Keywords: Alkali activated concrete, untreated soils, ground granulated blast furnace slag (GGBFS), experimental design

The worksites of "Grand-Paris" will generate high amounts of excavated lands, without few recovery routes. One of them could be to use those excavated lands into alkali activated materials, based on ground granulated blast furnace slag. The study herein presented concerns the feasibility of using various untreated soils originating from Normandy (France) as the granular skeleton in alkaline activated granulated blast furnace slag-based mortars. The use of experimental designs allowed to optimize the binding phase, by varying the composition of the activation solution, for sand and slag-based mortars. The impact of hardened properties of materials was thus studied.

1| INTRODUCTION

Alkali activated concretes based on ground granulated blast furnace slag are today one of the most preferred alternatives to Portland cement-based concrete. Indeed, these new materials have two main advantages, allowing to reduce their carbon impact: the use of industrial by-products and the reduction of the use of Portland cement, whose production is highly CO2 emitting [START_REF] Duxson | Geopolymer technology: the current state of the art[END_REF], [START_REF] Provis | Alkali Activated Materials: State-of-the-Art Report[END_REF]. Moreover, the use of local resources for material formulation would further reduce these environmental impacts.

The study herein presented concerns the feasibility of using various untreated soils originating from Normandy (France) as granular skeleton in alkaline activated blast furnace slagbased mortars. The use of experimental designs allowed to optimize the binding phase by varying the composition of the activation solution (concentration, molar ratios, etc.) [START_REF] Box | Multi-factor experimental designs for exploring response surfaces[END_REF], [START_REF] Fernández-Jiménez | Alkali-activated slag mortars Mechanical strength behaviour[END_REF]. After optimizing the formulation for sand and slag-based mortars, the impact of hardened properties of materials was thus studied.

2| MATERIALS

Ground blast furnace slag (GBFS) were provided by ECOCEM. GBFS composition is given in table 1.

Table 1 | GBFS composition

SiO2

Al2O3 CaO MgO wt% 37 11 43 9 0-2 mm siliceous sand provided by SACAB was used in the studied mortars. Silica (>99% purity), supplied by NORD-HUMIDITE, and sodium hydroxide (>98% purity), supplied by LABBOX, were used as chemicals for the activation solution production. X-ray diffraction of silica is presented on figure 1 and bulging around 20° 2θ confirm its amorphous state. Osmoseis-purified water was produced by an ELGA PURELAB apparatus.

3| METHODS

Preparation of the sodium silicate solution

Activation solutions were prepared by adding sodium hydroxide to osmosis-purified water. After partial cooling, ground silica was added. The mixture was kept under continuous stirring for at least 24 hours until perfect dissolution of the silica and equilibrium state between species was obtained [START_REF] Favier | Mécanisme de prise et rhéologie de liants géopolymères modèles[END_REF].

Realization and conservation of specimens

900 g of GGBFS and the activation solution (quantity defined by the experimental design, see paragraph 3.3) were mixed for 90 s at small speed. 2700 g of sand was then added and mixed for 90 s at small speed, followed by 90 s at high speed. After 24 hours, half of the specimen series continued to be stored at 20 °C and 95% of relative humidity, while the other half was stored underwater at 45 °C.

For each series, compressive strengths were measured at 24 hours, 3 days and 7 days after the specimens being cast.

Description of the experimental design

The experimental design is a centered composite factorial experimental design. It was selected to study the impact of first-and second-order parameters and their interactions [START_REF] Box | Multi-factor experimental designs for exploring response surfaces[END_REF]. It was chosen to study three parameters with three levels for each parameter, directly related to the activation solution:

-The mass percentage of solid in the solution (wt%); -The molar ratio Na2O/SiO2 in the solution; -The mass ratio H2O/GBFS.

Table 2 presents the equivalence between the normalized values and the effective values. The detailed design of experiments is presented in table 3. The α-value was set to 1.682 and the center point was repeated six times in order to evaluate the repeatability of the experiment [START_REF] Box | Multi-factor experimental designs for exploring response surfaces[END_REF].

Table 2 shows the results of the mechanical compression tests at 3 days. Each value represents the average compressive strength of three pieces.

The statistical analysis of the results was carried out using Xlstat and Matlab softwares.

4| RESULTS AND DISC USSION

3-days compressive strength results showed an efficient activation of the GGBFS, as strengths as high as 77 MPa could be obtained, with a 45 °C curing. This will allow a large range of applications for the geopolymer binder. As many studies have already shown, the results presented in table 3 show that thermal curing resulted in an increase of mechanical strength for most formulations. The analysis of the compressive strength results allowed the determination of a model equation at 3 days for both storage conditions (Eq. 1 and Eq. 2). Figure 3 and figure 4 present the distribution of experimental values compared to the predicted values at 3 days and for both curing conditions.

σ45°Ccure=59.76+4.22x+0.59y+11.67z-5.99x²-3.00y²-13.87z²-5.78xy+12.60xz-10.02yz (Eq. 1)

σ20°Ccure= 46.93+3.81x+8.25y+4.12z-4.94x²+4.19y²+10.66z²+7.60xy+0.37xz-4.46yz (Eq. 2)

A quite good fitting between experimental and calculated data is observed. Indeed, R² are 0.83 for storage at room temperature and 0.90 for storage at 45°C respectively, which highlights the relevance of the studied parameters. At a level of confidence of 95%, the statistically influent parameters (p<0.05) are, for both equation z, x², z², and yz.Those results are not surprising, because one of Fernãndez-Jiménez conclusion is that the concentration of the solution is a significant parameter on compressive strength [START_REF] Fernandez-Jimenez | Durability of alkali-activated fly ash cementitious materials[END_REF]. Moreover, concerning the z parameter, Kamalloo et al. mentioned that high water quantities in metakaolin-based geopolymers conduct to bad mechanical strenghts [START_REF] Kamalloo | Modeling of compressive strenght of metakaolin based geopolymer by the use of neuronal network[END_REF]. This phenomenon is also well-known for Portland cement concretes. To validate the model, three randoms formulations was tested and there formulations are presenting in table 1. Table 4 presents predicted value and mechanical results for formulations XXI, XXII and XXIII. The results demonstrate that further away the formulation is from the central point (x=y=z=0), the more false the predicted values are, with a maximum of relative error for for the formulation XXIII of -187%. However this is not surprising, considering the outlier values presented before.

In addition, other parameters than compressive strengths must be studied in order to get the optimal GGBFS-based geopolymer: fresh state properties and environmental impacts. Indeed, the activation solutions can present high viscosity depending on concentration and SiO2/Na2O molar ratio [START_REF] Yang | The viscosity properties of sodium silicate solutions[END_REF], which can be prejudicial for fresh-state geopolymers properties. Also, even if the GBFS present a low environmental impact (as they are made of industrial by-products), the activation solution is based on chemicals with a higher environmental impact. Nevertheless, it must be kept in mind that activation solution environmental impact can be easily reduced by using by-products such as waste glass or rice hull [START_REF] Vafaei | High strength geopolymer binder based on waste-glass powder[END_REF], [START_REF] Mellado | Carbon footprint of geopolymeric mortar: study of the contribution of the alkaline activating solution and assessment of an alternative rtoute[END_REF]. Thus, a balance between sufficient workability, high-enough compressive strength and low environmental-impact should be obtained to get the optimal material.

5| CONCLUSIONS

The method of optimizing a formulation of an alkaliactivated material by using an experimental design limited the number of casts to only 20, when a complete design would have required 125 (5 3 ). This process was used to obtain quickly and easily, equations for a predictive model. However, the model equations remain difficult to apply outside the values of the experimental design, with outlier values at the limits of the models. This methodology was firstly applied to 3-days compressive strength. Later, it will also be applied to other parameters, such as fresh-state fluidity, 5-days and 7-days compressive strength, and CO2-eq emissions. This method made it possible to determine the optimal formulation of an activation solution, allowing a binding phase with good mechanical properties. Subsequently, a new granular skeleton based on untreated soil will be used to observe the differences between this material and the optimized material. The use of that type of resource could then be applied on sites such as those in "Grand Paris", which are likely to generate large quantities of excavated land, currently without any recovery routes.

Fig. 1 |

 1 Fig. 1 | X-ray diffraction pattern of used silica obtained on Xpert Cu-Kα, presented as a funtion of 2θ.

Fig. 2 |

 2 Fig. 2 | Latin square model for specimens sawing

Fig. 3 |

 3 Fig. 3 | Experimental compressive stregnth as a function of model prediction after curing at 20°C for 3-day. The grey color curves delimits 95% confidence domain.

Fig. 4 |

 4 Fig. 4 | Experimental compressive stregnth as a function of model prediction after curing at 45°C for 3-day. The grey color curves delimits 95% confidence domain.

Figure 5

 5 Figure 5 and figure 6 show visualizations of the mechanical results at 3 days, according to the 3 parameters of the experimental design. Although this modeling allows a better visualization of the interactions between the 3 parameters, outlier values can be observed at the edges of the model, for example in figure 3, -61.6 MPa for a point with x=1.65, y=-1.65 and z=1.65. Nevertheless, this model allows to predict the experimental conditions for maximizing the compressive strength. If we reduce our confidence to the bounds -1, 1 for each parameters, this maxima should be obtained for x=1, y=1 and z=-0.6 for 20 °C cure and 45 °C cure, for theoretical compressive strengths of 61.68 MPa and 57.43 MPa respectively.

Fig. 5 |

 5 Fig. 5 | Predictive model after 3 day at 20°C.

Fig. 6 |

 6 Fig. 6 | Predictive model after 3-day at 45°C.

Table 2 |

 2 Equivalence between normalized notation and chemical solution parameters

		-α	-1	0	+1	+α
	wt%	16.59	20.00 25.00	30.00	33.41
	nSiO2/nNa2O	0.16	0.50	1.00	1.50	1.84
	mH2O/mBGFS	0.33	0.40	0.50	0.60	0.67

Table 3 |

 3 Chemical solution properties and mechanical results for all formulations at 3 days

	Order of fabrication	Formulation	Chemical solution properties wt% nSiO2/ nNa2O mH2O/ mBGFS	Mechanical result at 3 days σ45°C σ20°C
			x	y	z	MPa	MPa
	20	I	-1	-1	-1	15.55 12.88
	5	II	+1	-1	-1	4.03	7.09
	6	III	-1	+1	-1	52.04 30.20
	17	IV	+1	+1	-1	24.93 53.76
	7	V	-1	-1	+1	30.45 22.32
	1	VI	+1	-1	+1	76.89 16.94
	16	VII	-1	+1	+1	16.18 11.00
	10	VIII	+1	+1	+1	46.30 32.99
	2	IX	0	0	0	60.66 47.72
	15	X	0	0	0	65.56 46.93
	19	XI	0	0	0	60.32 42.56
	3	XII	0	0	0	62.36 44.68
	12	XIII	0	0	0	52.18 47.94
	4	XIV	0	0	0	57.08 51.36
	13	XV	-α	0	0	33.89 30.09
	9	XVI	+α	0	0	38.15 54.14
	18	XVII	0	-α	0	30.24 60.37
	11	XVIII	0	+α	0	42.27 44.53
	8	XIX	0	0	-α	2.11	2.76
	14	XX	0	0	+α	33.85 40.79
	21 22 23	XXI XXII XXIII	1.56 0.02 0.29	-0.23 0.42 1.61	-0.55 0.06 1.45	Presented in table 4

Table 4 |

 4 Predicted value and mechanical result for tested formulations at 3 days


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

M Ouisse

A Khelif

S De Rosa

F Franco

N Atalla

Computation of wave dispersion characteristics in periodic porous materials modeled as equivalent fluids

This paper starts with the presentation of the shift cell technique, which allows the description of the propagation of all existing waves starting from the unit cell through a quadratic eigenvalue problem. Its major advantage is that it allows the implementation of any frequency dependence and damping in the problem: this is a fundamental advantage when computing the dispersion curves of a porous material modeled as an equivalent fluid. The second part of this work concerns the investigation of the link between the dispersion curves and the acoustic properties of the material. Deriving the equivalent acoustic properties of the unit cell from its dispersion characteristics, indeed, could be a very efficient approach for designing the sound packages with a simple a preliminary eigenvalue analysis.

List of symbols

 𝜔 = angular frequency;  𝑥, 𝑦, 𝑧 = space variables;  𝑗 = imaginary unit;  𝑝 0 = amplitude of the excitation mode (incident pressure);  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;  𝐸 𝑝 = potential energy;  𝑣 = istantaneous local velocity;  𝑣 𝐸 = energy transport speed;  𝐶 𝑔 = group velocity;  𝑠 = side length;  𝛱 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 = incident power;  𝛱 𝑡𝑟𝑎𝑛𝑠𝑚𝑖𝑡𝑡𝑒𝑑 = transmitted power;  𝑆 = surface interested by incident pressure;  𝑑 = thickness;  𝜏 ∞ = transmission coefficient;  𝑇𝐿 = transmission loss.

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 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 behavior 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.

A validation of the implementation of this model in order to handle porous materials with periodic inclusions, together with a detailed explication of the meaning and the behavior of band gaps, can be found in literature [START_REF] Magliacano | Validation of shift cell approach for the modelling of acoustic properties of porous materials embedding periodic in NOVEM[END_REF].

Dispersion curves and acoustical characteristics for different numerical test cases are shown. In particular, they are obtained for a 3D melamine unit cell, with and without inclusion. The behavior of this porous material is described by JCA model in the following pages, but one can identically use any other equivalent fluid model.

The second part of this work concerns the investigation of the link between the dispersion curves and the acoustic properties of the material. In particular it's demonstrated that, starting from the wave numbers obtained as an output from the quadratic eigenvalue problem, it's possible to compute an equivalent transmission loss curve that, if compared to those obtained using classical methods, shows a very good agreement. Deriving the equivalent acoustic properties of the unit cell from its dispersion characteristics, indeed, could be a very efficient approach for designing sound packages with a simple a preliminary eigenvalue analysis.

All the results shown are related to a 3D melamine unit cell constituted by a cube with side equal to 2 [cm] (homogeneous case) and with a 0.5 [cm] radius cylindrical rigid inclusion (case with inclusion). The analyses are carried out in the frequency range 0 -17000 [Hz]; this range of frequencies is interesting for acoustic applications and assures that the wavelength is much larger than the pore size, which is a necessary condition in order to use equivalent fluid models. Dispersion diagrams are computed along the direction that corresponds to 𝜙 = 0° and 𝜃 = 0° in the first Brillouin zone.

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).

Figure 1: Reciprocal lattice vector in a 3D unitary cell [START_REF] Billon | Composites periodiques fonctionnels pour l'absorption vibroacoustique large bande[END_REF].

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

𝒌 = 𝑘𝝓, 𝝓 = ( 𝑐𝑜𝑠𝜃𝑐𝑜𝑠𝜙 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜙 𝑠𝑖𝑛𝜃 ), (2)
one can obtain

∇ 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 = 0.

(3)

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 ( 7) can be rewritten as

𝑨 1 (𝜔)𝝍 𝑖 𝑟 = 𝜆 𝑖 𝑨 2 (𝜔)𝝍 𝑖 𝑟 ( 8
)
with

 𝑨 1 (𝜔) = ( 0 𝑰 𝒅 𝑲 -𝜔 2 𝑴 𝑳 );  𝑨 2 (𝜔) = ( 𝑰 𝒅 0 0 𝑯 );  𝝍 𝑖 𝑟 = ( 𝝋 𝑖 𝑟 𝜆 𝑖 𝝋 𝑖 𝑟 ).
where 𝑰 𝒅 is the identity matrix.

Conversely, a left-eigenvector for the same eigenvalue satisfies 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 [START_REF] Billon | Composites periodiques fonctionnels pour l'absorption vibroacoustique large bande[END_REF].

After some steps, one obtains 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 ]𝝋 𝑖 𝑟 𝝋 𝑖 𝑙 𝑇 [-𝑳+2𝜆 𝑖 𝑯]𝝋 𝑖 𝑟 (11)
The group velocity is the inverse of the group slowness [START_REF] Moiseyenko | Material loss influence on the complex band structure and group velocity in phononic crystals[END_REF]:

𝐶 𝑔 = 𝜕𝜔 𝜕𝑘 𝑖 = 𝑗𝝋 𝑖 𝑙 𝑇 [-

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:

1. the ratio between the real and the imaginary parts of every wavenumber → 𝐶 1 = 𝑟𝑒𝑎𝑙(𝑘)/𝑖𝑚𝑎𝑔(𝑘); 2. the ratio between the real parts of the energy transport speed 𝑣 𝐸 = 𝐼/𝐸 and the goup velocity 𝐶 𝑔 , where 𝐼 is the flow of energy and 𝐸 = 𝐸 𝑘 + 𝐸 𝑝 = ∫ Only the waves corresponding to 𝐶 1 > 𝜏 1 and 𝐶 2 > 𝜏 2 are considered as propagative ones. In practice, for the purpose of the following analysis, the thresholds 𝜏 are chosen such as 𝜏 1 = 1 and 𝜏 2 = 0.7. 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]. In order to better appreciate the behavior of each branch in the frequency range of study in the following plots, dispersion and 𝐶 𝑔 curves are also colorized with a scale of colors that indicates the level of "propagativeness": the value 0 means that the wave at that specific frequency is totally spatially attenuated, while the value 1 represents a properly propagative behavior.

Calculation of group velocity and branch-tracking algorithm

In a dispersion diagram there is a set of points, forming branches, that one may wish to connect and follow according to the nature of each branch. Some solutions are proposed in literature, such as a MAC sorting criterion [START_REF] Collet | Floquet-Bloch decomposition for the computation of dispersion of two-dimensional periodic, damped mechanical systems[END_REF], but these methods require to store many data at every iteration. Instead, the group velocity constitutes a relevant indicator in order to follow the branches from a point of calculation to the next one [START_REF] Billon | Composites periodiques fonctionnels pour l'absorption vibroacoustique large bande[END_REF]. The proposed technique consists in comparing 𝐶 𝑔 𝑖 (𝑓) and 𝑪 𝒈 (𝑓 + ∆𝑓): from the group velocity associated to a starting point, the routine compares the initial 𝐶 𝑔 𝑖 with all the group velocities at the next frequency step and a minimization is made in order to identify the point at 𝑓 + ∆𝑓 to which is associated the closest value of 𝐶 𝑔 . This point is defined as the new starting one and so on, step by step, the branch is identified.

Figure 2: Evanescent and propagative (on the top) and branch-tracked (on the bottom) group velocity diagrams, for the case of a homogeneous unit cell.

Figure 3: Evanescent and propagative (on the top) and branch-tracked (on the bottom) group velocity diagrams, for the case of an unit cell with an inclusion.

Dispersion curves

For each dispersion curve plot, three eigenvectors are reported in terms of acoustic pressure field (Figures 5 and7). 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. 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.

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 (13) 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 1st 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 1st branch of dispersion curves in which the wave is strongly spatially attenuated. This is definitely encouraging, for the purpose of deriving more equivalent acoustic properties of the unit cell from its dispersion characteristics.

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 branch-tracking criterion has been described, in order to clearly identify each branch in the dispersion diagram. Dispersion curves and transmission loss plots have then been computed for a JCAmodelled melamine unit cell. An equivalent transmission loss curve, obtained starting from the eigenvalues, has been compared to those obtained with classical methods, showing a very good agreement. Further developments of the work will include an estimation of the computational efficiency between the shift cell and the classical Floquet-Bloch approaches, as well as the implementation of the shift cell technique using Biot model.

  energy → 𝐶 2 = 𝑟𝑒𝑎𝑙(𝑣 𝐸 )/𝑟𝑒𝑎𝑙(𝐶 𝑔 ).

  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: Evanescent and propagative (on the left) and branch-tracked (on the right) dispersion curves for a homogeneous melamine unit cell.

Figure 5 :

 5 Figure 5: From left to right, real parts of the 1st, 2nd and 3rd branch eigenvectors for the homogeneous case.

Figure 6 :

 6 Figure 6: Evanescent and propagative (on the left) and branch-tracked (on the right) dispersion curves for a melamine unit cell with an inclusion.

Figure 7 :

 7 Figure 7: From left to right, real parts of the 1st, 2nd and 3rd branch eigenvectors for the case with an inclusion.

Figure 8 :

 8 Figure 8: Transmission loss computed for a 3D repetition of 5 melamine unit cells.

Acknowledgements

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.