Guy Didier Fanou

Mamadou Traore
email: traore@lspm.cnrs.fr

Benjamin Kouassi Yao

Andrei Kanaev

Khay Chhor

Khay

Photocatalytic activity of TiO2-P25@n-TiO2@HAP composite films for air depollution

Keywords: TiO 2, hydroxyapatite, composite nanoparticles, photocatalysis. 1.Introduction

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

The industrial, domestic and military activities are increasingly rejecting toxic materials, especially in recent decades [START_REF] Jacobson | Atmospheric pollution: history, science, and regulation[END_REF][START_REF] Atkinson | Atmospheric chemistry of VOCs and NO x[END_REF][START_REF] Jia | Elaboration des matériaux composites nanostructurés Ag, Au/TiO 2 pour la dépollution des effluents gazeux avec une activation par plasma[END_REF]Bryne et al. 1998;[START_REF] Guo | Identification of a plant nitric oxide synthase gene involved in hormonal signaling[END_REF]. The water and air quality is decreasing which represents a serious threat for all living species. Among the possible techniques for the destruction of pollution traces are the Advanced Oxidation Technologies (TOA), the most important of which are ozonation, the use of ultraviolet (UV), plasma degradation and photocatalysis. From 2000 to 2018, the number of publications on TOA increases by more than fourfold and reaches 7000 publications [START_REF] Giwa | Recent advances in advanced oxidation processes for removal of contaminants from water: A comprehensive review[END_REF]. [START_REF] Fujishima | TiO 2 photoelectrochemistry and photocatalysis[END_REF] suggested the use of TiO 2 excited by ultraviolet radiation to decompose organic pollutants.. The principle of photocatalysis involves photogeneration of electron-hole pairs. Under light excitation, the electrons (e -) of the valence band (BV) of the TiO 2 pass to the conduction band (BC), leaving a positive charge or hole (noted h + ) in the BV. These electron-hole pairs can recombine by releasing heat, or react separately with pollutant molecules of the environment. The recombination of the photoinduced charges served to be the main shortcoming of the photocatalytic process, which can be reduced by doping of the active component and/or its integration in a composite medium. In this connection, the combination of nano-TiO 2 and HAP may be an interesting solution.

Because of a large band gap energy ~7.7 eV (λ = 160 nm), measured e.g. by [START_REF] Feldbach | Luminescence Spectroscopy of Caapatites under VUV Excitation[END_REF] and calculated by [START_REF] Tsukada | Band gap and photocatalytic properties of Ti-substituted hydroxyapatite: comparison with anatase-TiO 2[END_REF], appreciable photocatalytic activity of this material cannot be expected. However, HAP was capable forming oxygen defects under UV illumination serving as active centers [START_REF] Nishikawa | Photocatalytic activity of hydroxyapatite for methyl mercaptane[END_REF] and some photocatalytic activity of HAP has been observed on the reduction of methyl mercaptan and dimethyl sulfide [START_REF] Tanaka | FTIR studies of adsorption and photocatalytic decomposition under UV irradiation of dimethyl sulfide on calcium hydroxyapatite[END_REF]. The use of hydroxyapatite (HAP) as an adsorbent material is of increasing interest because of its remarkable property of ion exchange, bonding with organic molecules, low solubility in water [START_REF] Iliev | Influence of the size of gold nanoparticles deposited on TiO 2 upon the photocatalytic destruction of oxalic acid[END_REF][START_REF] Krestou | Mechanism of aqueous uranium (VI) uptake by hydroxyapatite[END_REF][START_REF] Ma | Effects of aqueous Al, Cd, Cu, Fe (II), Ni, and Zn on Pb immobilization by hydroxyapatite[END_REF][START_REF] Qian | Removal of Fe3+ from Aqueous Solution by Natural Apatite[END_REF][START_REF] Bahdod | Adsorption of phenol from an aqueous solution by selected apatite adsorbents: Kinetic process and impact of the surface properties[END_REF]. Hydroxyapatite also has catalytic properties that have been studied in several fields [START_REF] Monma | Catalytic behavior of calcium phosphates for decompositions of 2propanol and ethanol[END_REF]Matsumura et al. 1984;[START_REF] Joris | [END_REF][START_REF] Kibby | Tracer studies of acid-catalyzed reactions. XI. Stereoselectivity in alcohol dehydration over hydroxyapatite and alumina catalysts[END_REF][START_REF] Bett | The microcatalytic technique applied to a zero order reaction: The dehydration of 2-butanol over hydroxyapatite catalysts[END_REF][START_REF] Matsumura | Partial Oxidation of Methane to Carbon-Monoxide and Hydrogen with Molecular-Oxygen and Nitrous-Oxide over Hydroxyapatite Catalysts[END_REF][START_REF] Matsumura | Catalytic oxidation of carbon monoxide over stoichiometric and non-stoichiometric hydroxyapatites[END_REF]Chlala 2016): dehydrogenation, dehydration, combustion, oxidation, reforming, etc. Several studies have focused on the use of the adsorption-photocatalytic coupling of the TiO 2 / HAP composite for the removal of gaseous pollutants (Azouani et al. 2010a;Azouani et al. 2010b;[START_REF] Azouani | Stability and growth of titanium-oxo-alkoxy Ti x O y (OiPr) z clusters[END_REF][START_REF] Benmami | High photocatalytic activity of monolayer nanocoatings prepared from non-crystalline titanium oxide sol nanoparticles[END_REF]). Thus, NO adsorption was observed by [START_REF] Nonami | Apatite-Coated Titanium Dioxide Photocatalyst for Air Purification[END_REF], [START_REF] Mitsionis | Hydroxyapatite/titanium dioxide nanocomposites for controlled photocatalytic NO oxidation[END_REF] as well as [START_REF] Nasr-Esfahani | Alumina/TiO2/hydroxyapatite interface nanostructure composite filters as efficient photocatalysts for the purification of air[END_REF] in photocatalytic studies.

However, the improvement of the gas phase photocatalytic activity due to the presence of hydroxyapatite on the surface of TiO 2 is controversial; a negative or no effect can be observed with regard to the process procedure and the UV wavelengths used.

The coupling between TiO 2 and HAP and the overall process efficiency are expected to depend on the process procedure. In this work, we applied a new approach to the photocatalyst synthesis by using titanium oxo-alkoxy (TOA) nanoparticles as a link between TiO 2 P25 and HAP components. The photocatalytic activity of the prepared TiO 2 -P 25 @ n-TiO 2 @HAP materials has been evaluated.

Material and Methods

Elaboration of TiO 2 -P 25 @ n-TiO 2

The general approach to synthesize the TiO 2 -P 25 @n-TiO 2 has been proposed by [START_REF] Fanou | Elaboration of novel nanoparticulate TiO 2 -P25@nTiO 2 composite for Catalysis[END_REF]. The sol nanoparticles are prepared in the sol-gel reactor using titanium tetraisopropoxide (TTIP) as a precursor. Two stock solutions: 50 cm 3 of a TTIP/propan-2-ol solution and 50 cm 3 of a H 2 O/propan-2-ol solution, was synchronously injected under nitrogen into the turbulent mixing zone (Re = 4500) of sol-gel reactor (Azouani et al. 2010b) where the oxo-TiO 2 nanoparticles were generated from hydrolysis condensation reactions.

The particle size can be tuned by adjusting the hydrolysis ratio H = C H2O /C TTIP , where C H2O and C TTIP are respectively water and titanium precursor concentrations. We have prepared colloids using hydrolysis ratio H = 2, which corresponds to one of stability domains of the sol-gel process and enables particles size of 5.2 nm [START_REF] Azouani | Stability and growth of titanium-oxo-alkoxy Ti x O y (OiPr) z clusters[END_REF]). The suspended oxoparticles solution was subsequently transferred to a glove box LABstar MBraun where they were combined slowly with TiO 2 -P25 powder under stirring. The ratio 66 % of TiO 2 -P25/ n-TiO 2 was the same for all prepared samples. A stable nanocoating, with a specific area of 72 m 2 /g, was prepared by immobilization of nanoparticles on glass bed by dip-coating.

Elaboration of TiO 2 -P 25 @ n-TiO 2 @HAP

Two types of composites were prepared by i) crystallization on the surface of the TiO 2 -P25 @ n-TiO 2 of hydroxyapatite in a physiological solution SBF (CPS / HAP-SBF) ii) incorporation of hydroxyapatite in the matrix TiO 2 -P25 @ n-TiO 2 (CPS / HAP).

Co-precipitation method

The chemicals (Sigma Aldrich) involved in this experiment are calcium nitrate Ca(NO 3 ) 2 ,4H 2 O (purity 99%), ammonium dihydrogen phosphate NH 4 H 2 PO 4 (purity ≥ 98%) and Ammonia (purity 99,5%). 125 ml of calcium nitrate (0.5 mol.L -1 ) and 75 ml of ammonium phosphate (0.5 mol.L -1 ) are mixed with vigorous stirring. Then, concentrated ammonia is added dropwise to pH≈10-10.5 to give a white precursor precipitate of HAP.

After stirring for 1 hour, the suspension is filtered and stored in a hermetically sealed beaker in order to avoid carbonation due to atmospheric carbon dioxide. HAP is then mixed with the colloid suspension TiO 2 -P 25 @n-TiO 2 so as to obtain different mass percentage. The mixture of the hydroxyapatite and composite particles are immobilized by dip-coating on the 1 mm diameter glass beads for photocatalytic tests. Prepared materials are noted as CPS / HAP.

Simulated Body Fluid (SBF) method

This preparation method is advocated by [START_REF] Li | Induction and morphology of hydroxyapatite, precipitated from metastable simulated body fluids on sol-gel prepared silica[END_REF]. The chemicals were purchased from Sigma Aldrich in quantities indicated in Table 1.

In general, 250 ml of distilled water were poured into a beaker and placed in a water bath at 36.5 °C with magnetic stirring. Each reagent was then subsequently added to the reaction medium during the synthesis. The pH was adjusted to 7.25 using 1N HCl. At the end, the volume of the solution was completed to 500 mL with distilled water and stored in a refrigerator at 5-10 °C. The crystallization of hydroxyapatite on TiO 2 -P25 @ n-TiO 2 composite was carried out by placing the composite samples and the SBF solution in contact under slight agitation. The crystallization time varies from 30 min to 5h. Longer calcination times were also used in the preparation and specified if applicable. Prepared materials are noted as CPS / HAP-SBF.

Photocatalyst Characterization

The prepared samples were structurally characterised by X-ray diffraction XRD (INEL XRG 3000) using CuKα radiation with Nickel filter. The surface morphology of photocatalyst was examined by SEM JEOL JSM 64400F with acceleration voltage of 10 kV. Raman spectra were measured at 516 nm using micro-Raman high-resolution HR800 installation (HORIBA JobinYvon) with the spectral and spatial resolution respectively 0.25 cm -1 and 5 μm. The scattered light is collected by Peltier cooled CCD camera in a backscattering configuration.

The photocatalytic activity of the prepared samples was tested in a continuous-flow fixed-bed reactor on ethylene decomposition. A gas flow of pollutant (120 ppm) mixture with dry air and flow rate of 7.5 mL/min passes through a reactor tube of 6-mm diameter and 35 cm length made of glass transparent in the UV-A spectral range [START_REF] Benmami | Supported nanometric titanium oxide sols as a new efficient photocatalyst[END_REF][START_REF] Tieng | New homogeneously doped Fe(III)-TiO 2 photocatalyst for gaseous pollutant degradation[END_REF]. The glass beads coated photocatalyst samples were filled the reactor tube of 15 cm length around the middle. Inside the tube, just after the glass bead, a chromel / alumel thermocouple monitors the temperature. The tube is surrounded at a radial distance of 3 cm by six 8-W lamps of 29 cm length emitting at 362 nm (Δλ hwfm = 22 nm). During the experiments, the reactor is cooled by compressed air stream. Ethylene concentrations before (C in ) and after (C out ) the photocatalytic reactor were monitored by online gas chromatography (Varian 191 CP 3800) equipped with a capillary column (HP-PLOT/Q) and a flame ionization detector FID. Before turn on the UV lamps, one hour period is applied for the stabilization of Ethylene concentrations and the gas inlet stream temperature at 25 °C. Two injection loops of 250 μL and heated at 80 °C allow measurements of pollutant concentration in continuous mode. The reactor yield (or ethylene conversion) is calculated according to the formula: η(%) = (C in -C out )/C in ×100. The column temperature and flow rate of the carrier gas (N 2 ) were respectively 50 °C and 5 mL/min.

Characterization

3-1. X-ray diffraction

The X-ray plots of pure hydroxyapatite (figure 1) show well-defined peaks indicating that the solids are well crystallized.

In the range of 2θ between 25 and 35 °, they reveal clear diffractions of the (002) planes at 25.7 ° and those of the ( 211), (300), and (202) planes at 31.6, 32, 7 ° and 33.9 ° respectively.

All these planes can be indexed in the hexagonal space group system P63 / m [START_REF] Nathanael | Mechanical and photocatalytic properties of hydroxyapatite/titania nanocomposites prepared by combined high gravity and hydrothermal process[END_REF]. The average size of the crystallites was estimated using the Sherrer formula [START_REF] Smith | Modern Powder Diffraction[END_REF]) and the Bragg reflection of the (002) planes. These parallel planes with the c⃗ axis represent the preferential direction of growth of the hydroxyapatite.

= (2 ) cos

d is the diameter of the crystallites (Å), k is a constant (0.9), λ is the incident wavelength (Å), θ is the Bragg angle (degree) and b is the half width. recorded peak height expressed in radians.

The calculated value of d is close to 21 nm for the sample treated at 100 ° C. and 22.5 nm for that calcined at 450 ° C.

The X-ray diffraction patterns of the samples of the TiO 2 -P25 @ n-TiO 2 composite containing a dispersion of HAPs are shown in Figure 2.

The spectra show a presence of crystallized HAP in the material. The intensity of the most intense HAP peaks around 31.5 ° increases with the increase in its mass content in the material. They appear clearly in the sample diagram with 30% PAH.

3-2. Raman spectroscopy

The vibrational study by Raman scattering was recorded on the powders of the following samples: i) Pure hydroxyapatite treated at 450 ° C.

ii) TiO 2 -P25 @ n-TiO 2 / PAH noted CPS / HA with a mass content equal to 3 and 5% of incorporated PAH.

The spectrum of hydroxyapatite is shown in Figure 3.

The frequency range between 400 and 1200 cm -1 characterizes the diffusion bands of the (PO 4 ) 3-group. The peaks observed around 430 and 449 cm -1 can be attributed to the deformation vibration υ2. The Raman bands at 580, 592 and 610 cm -1 correspond to the triple degenerate vibrational mode υ4. The symmetrical elongation vibration mode υ 1 of the (PO 4 ) 3- group appears as a very intense peak located at 962 cm -1 . This diffusion band is characteristic of the structure of hydroxyapatite [START_REF] Smeulders | Insoluble organic compounds in the Bayer process[END_REF]. The antisymmetric vibration υ 3 can be associated with the diffusion bands observed at 1032, 1048 and 1079 cm -1 . Figure 4 shows the Raman spectra of the TiO2-P25 @ n-TiO2 / HAP (CPS / HAP) composite with a mass content of 3 and 5% PAH.

The spectra obtained clearly show three Raman bands at 399, 516 and 642 cm -1 of the anatase of the composite. As for the hydroxyapatite, because of its low concentration in the matrix (3 and 5%) the intensity of the diffusion bands is very weak making the observation difficult.

The only detectable peak is 962 cm -1 attributed to the vibrational mode υ 1 of the (PO 4 ) 3- group.

Photocatalytic activity

4-1. Co-precipitation method

The photocatalytic tests were measured on the samples of 0.5% and 1% by weight of hydroxyapatite with the references CPS / HAP-0.5 and CPS / HAP-1 respectively. Figure 5 shows the evolution of photocatalytic activity of ethylene degradation.

The conversion of ethylene to CPS / HAP-0.5, CPS / HAP-1 shows a rapid increase for 20 min under UV radiation. As Tieng et al have already observed (2011), the warm up time of UV lamps was too short (Figure 6a) to account for this behavior. The temperature increases from 25 °C to 46 ± 2 °C in about 3 min. Moreover no photolysis of ethylene is observed (Figure 6b). The delay in the achievement of the asymptotic ethylene conversion has been previously explained by the formation of non-volatile intermediate reaction products, assigned to the oxidized polymeric hydrocarbons: furan and furfural oligomers Ol N with N=6÷8 [START_REF] Tieng | Major non-volatile intermediate products of photo-catalytic decomposition of ethylene[END_REF].

After this step, the photocatalytic yield reaches a stable steady state corresponding to a conversion value η greater than that of the reference composite: 12% for CPS / HAP-1 and 6% for CPS / HAP-0.5. The first order rates of the pollutant decomposition in Figure 7, which reflects the intrinsic photocatalyst activity [START_REF] Fanou | Elaboration of novel nanoparticulate TiO 2 -P25@nTiO 2 composite for Catalysis[END_REF], showed a linear proportionality to the HAP content. It begins however from a non-zero value, characteristic of the pure titania activity, increasing by ~50% at 1wt% of HAP. This enhancement might be associated with: i) strong adsorption capacity of hydroxyapatite with respect to the pollutant and the secondary reaction products likely to form during the photocatalytic reactions [START_REF] Hauchecorne | Photocatalytic degradation of ethylene: An FTIR in situ study under atmospheric conditions[END_REF], ii) better availability of active sites because of elimination of poisons by adsorbance on the composite surface , and iii) charge separation efficiency enhancement in the composite. [START_REF] Mitsionis | Hydroxyapatite/titanium dioxide nanocomposites for controlled photocatalytic NO oxidation[END_REF] have reported that the formation of OH o radicals is greater onto HAP / TiO 2 composite than TiO 2 alone. In the same time, the band gap energy of HAP/TiO 2 composite E g =3.08 eV [START_REF] Kamat | Photophysical, photochemical and photocatalytic aspects of metal nanoparticles[END_REF] has been reported to be insignificantly lower compared with pure titania (3.12 eV), whereas that of HAP reported by [START_REF] Feldbach | Luminescence Spectroscopy of Caapatites under VUV Excitation[END_REF] and [START_REF] Lopez | Effect of substrate on surface morphology and photocatalysis of large-scale TiO 2 films[END_REF] was significantly larger: 7.7 eV and >6 eV respectively. Accordingly, synergistic effect between adsorbing and active sites could be responsible for the enhanced activity of the composite. The following scenario highlighting the synergistic action of these two components, composite and adsorbent, can be described as follows: the molecules of the pollutant are additionally adsorbed by HAP and then migrate along the composite surface to the active nanoparticles where they react. The secondary products, if formed, can be advantageously captured by HAP discharging active sites to be decomposed on a later stage.

4-2. SBF method

Figure 8 shows the results of the photocatalytic activity of the films prepared with a crystallization time of 30 min.

The obtained results indicate that the ethylene conversion η increases rapidly for about 20 minutes of irradiation to reach a maximum of 3% to 5% higher than that of the reference (TiO 2 -P25 @ n-TiO 2 ). We also explained this behavior by the thermal activation of the material following the rise of temperature. It then shows a slow decrease to a lower value than that of the untreated sample after about 40 minutes. This behavior is fairly reproducible for the three samples. Different reasons can explained this behavior : (i) scattering of the incident light by the surface particles (ii) e -/h + charge transfer blocking at the compositehydroxyapatite interface of the composite to the adsorbed ethylene [START_REF] Feldbach | Luminescence Spectroscopy of Caapatites under VUV Excitation[END_REF][START_REF] Tsukada | Band gap and photocatalytic properties of Ti-substituted hydroxyapatite: comparison with anatase-TiO 2[END_REF], iii) poisoning of sites material assets by the presence of products from the elaboration stage.

In order to confirm these hypotheses, we studied the influence of the crystallization time on the photocatalytic activity shown in Figure 9. The results show the similar behavior observed on Figure 8 and the maximum ethylene conversion is also obtained at the same time (30-40 min).

We observe that the prolonged crystallization time decreases the photocatalytic activity of the composite material. This reduction in photocatalytic performance is important and varies from 5% (30 min) to 75% for a crystallization time of 5 h. These results support the explanations we have advanced above: hydroxyapatite on the surface, increase the scattering of the incident light and reduce the charges transfer at the interface and the crystallization products can poison the photocatalyst.

The different performances of the two prepared TiO 2 -P 25 @ n-TiO 2 @HAP materials (Figures 5 and8) can be explained by their probable structure. Hydroxyapatite crystals deposited on the surface of TiO 2 -P 25 @ n-TiO 2 is obtained by the SBF method. In this case, an increase of crystallization time causes a decrease of the photocatalytic activity. Whereas, when hydroxyapatite crystals are incorporated in the TiO 2 -P 25 @ n-TiO 2 matrix, it is dispersed in the composite and can create a synergistic effect of hydroxyapatite on the photooxidation properties of the photocatalyst.

Conclusion

Two families of new composite photocatalysts were prepared by crystallization on the surface of the TiO 2 -P25 @ n-TiO 2 of the hydroxyapatite in a physiological solution SBF and by incorporation of hydroxyapatite in TiO 2 -P25 @ n-TiO 2 matrix. Both HAP and TiO 2 conserve own crystallinity in the material. The photocatalytic yield (gas phase conversion of ethylene) of the material obtained by co-precipitation was increased with an increase of the HAP content, indicating the synergy between two components. On the other hand, the prolonged crystallization time of the composite in SBF has a negative effect on the photocatalytic yield.

Fig. 1 :

 1 FIGURES

Fig

  Fig. 2: X-ray diffractograms of samples of TiO2-P25 @ n-TiO2 / HAP composite treated with different mass compositions of hydroxyapatite (a) 0%, (b) 2% (c) 15% and (d) 30%.

Fig. 3 :Fig. 4 :

 34 Fig. 3: Raman spectrum of hydroxyapatite treated at 450 ° C recorded with excitatory

Fig. 5 :

 5 Fig. 5: Evolution of the photodegradation of ethylene by CPS / HAP-0.5 and CPS / HAP-1 as a function of the irradiation time. The conversion rate of ethylene by the untreated composite is given for comparison. ( λ = 360 nm, ethylene 150 ppm, flow rate 10 mL.min -1 ).

Fig. 6 :

 6 Fig. 6: Temperature evolution during photocatalytic tests (a) and Ethylene concentration evolution under UV irradiation without photocatalysts (b).

Fig. 7 :Fig. 8 :

 78 Fig. 7: Photocatalytic activity as a function of HAP doping.

Fig. 9 :

 9 Fig. 9: Photodegradation of ethylene by CPS / HAP-1SBF as a function of the crystallization

Acknowledgements :

Not applicable

Funding :

Not applicable

Availability of data and materials :

The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.

Declarations


Marco Runkel
email: marco.runkel@tu-berlin.de

Mireille Chiroleu-Assouline
email: mireille.chiroleu-assouline@univ-paris1.fr

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction to IIPF 2020 special issue in ITAX: reflections on the interactions between environmental economics and public finance

Introduction to IIPF 2020 special issue in ITAX -Reflections on the interactions between environmental economics and public finance

This special issue of International Tax and Public Finance contains selected papers presented at the 76th Annual Congress of the International Institute of Public Finance (IIPF), which was hosted by the University of Iceland on August 19-21, 2020. Due to the Covid-19 pandemic, for the first time this congress was held in an online format. The number of presentations at the congress was slightly lower than at past physical IIPF congresses despite an initially higher number of submissions (588). But with 290 papers presented in contributed sessions covering all topics within public economics, it is still remarkable high, given the circumstances during the pandemic and the fact that it was the first online congress of the IIPF.

The theme of the conference was "Public Finance, Natural Resources and Climate Change". It had been chosen in 2019, long before the Covid-19 crisis erupted, and reflected the need to move forward on policies to achieve the long-term climate goals outlined in the 2016 Paris Agreement. The health crisis, however, did not undermine the need to address the climate crisis. On the contrary, the Covid-19 crisis -similar to global warming -shows the dramatic vulnerability of mankind to changes in its natural environment. Climate and health crises, together with their possible interactions, therefore represent a 'twin crisis' that belongs to the largest challenges of the 21 st century. For this reason, the scientific committee of the congress decided to adhere to the overall conference theme, where possible and suitable complemented by some aspects of the Covid-19 pandemic.

The climate issue has traditionally been the domain of environmental and natural resource economics, but the treatment of environmental quality or the habitability of the planet as global public goods, as well as that of pollution or GHG emissions as externalities, and -last but not least -the potential correction by taxes, subsidies, emission trading systems or other kinds of regulation makes public economics the parent discipline to refer to. In previous decades, both disciplines -environmental and resource economics, on the one hand, and public economics, on the other hand -have taken their own development with their own scientific associations, conferences and field journals. What does a congress of the IIPF with the climate issue as the main conference theme tell us about the way the relationship between environmental economics and public economics is evolving? Before introducing the papers published in this special issue, we will address this question by briefly discussing the keynote lectures as well as some (very) selective papers presented in the contributed sessions of the congress and not published in the special issue.

Keynotes and Selective Presentations

Four inspiring keynote speakers have been invited to talk on the central conference theme Ottmar Edenhofer went back to the origins by painting a comprehensive picture of the obstacles to the implementation of Pigouvian carbon taxation, but also of its successes, and his talk provided a number of stimulating avenues of research. He based his arguments on both a sound scientific basis and excellent knowledge of the scientific literature as well as comprehensive practical experiences in policy consulting, a combination that turned out to be highly fruitful. Among the implementation difficulties he mentioned was the problem of assessing the social cost of carbon, and this was precisely the issue addressed by Rick van der Ploeg. He particularly insisted on the presence of multiple sources of uncertainties and on the importance of the choice of the discount rate, a subject that gains a lot of attention in environmental and resources economics not only since the release of the Stern-Review on the economics of climate change2 and, at the same time, is familiar to researchers in public finance when the policy envisaged has long-term objectives. But perhaps one of the most original points for the usual IIPF conference audience was his proposed incorporation of a model of temperature variation to account for the reciprocal interactions between climate and economic activity, in line with the work of William Nordhaus, who was awarded the Bank of Sweden prize in 2018.

The other two keynote lectures dealt with the power generation and passenger car sectors, which were among the first to be regulated in the EU to reduce their GHG emissions, but for which much remains to be done and effective regulation requires a thorough understanding of the underlying mechanisms. Starting from the recognition that long-term climate goals can only be met if major efforts are made to decarbonize electricity generation and electrify other energy uses, Karen Palmer focused on demonstrating the usefulness of market-specific instruments: since the integration of renewables into the energy mix poses supply-demand matching challenges, automation and innovative tariff design can help facilitate efficient demand shifting, with direct and indirect climate benefits. Finally, Anna Alberini presented a study, using German data, of the effects of an annual registration tax that increases with a vehicle's CO 2 emission rate.3 She showed that this type of tax is effective in shifting new cars to lower-emission models, but that these effects seem to disappear over time. Finally, drivers seem to integrate the information they receive on the future evolution of the cost of using their vehicle in different ways, depending on the type of tax and the way it is calculated. Both the talk of Karen Palmer as well as the talk of Anna Alberini show the importance of carefully analyzing the usefulness of basic public finance instruments in the design of a suitable climate policy, that is central in environmental and resource economics.

Beyond these four plenary lectures, the IIPF 76 th Annual congress was the occasion for a notable opening to works in environmental economics. Among all accepted papers, there were 42 papers classified with the topic 'Environment', divided into subgroups 'Climate change policy', 'Natural resources', 'Environmental tax reforms', 'Mobility and congestion', 'Public investment and public procurement', 'Energy taxation' and 'Green bonds'. From these papers, we formed three sessions with the general title 'Environmental Economics', three sessions named 'Environmental Regulation' and one specific session on 'Distributional Effects of Environmental Policies'. In addition, there were single papers on environmental issues included in broader sessions on, for example, 'Local Public Finance' and 'Growth and Fiscal Policies'. Among the many exciting papers presented at this conference, we would like to mention a few that may be representative of new research avenues for the future, some of which have since been published in other journals.

The first strand of papers which we would like to highlight discusses environmental policy in a framework where not all market participants are solely self-interested, but also have some kind of social or green preferences. These papers within the field of environmental and resource economics therefore emphasize the importance of behavioral aspects in the design of environmental and climate policy, an aspect that have gained general interest in the field of public economics over the last decades ('behavioral public economics').

The paper "Can Subsidies Rather Than Pollution Taxes Break The Trade-Off Between Economic Output And Environmental Protection?" by Thomas Renström, Luca Spataro and Laura Marsiliani, 4 revisits the debate on the choice between taxing polluting emissions and subsidizing the abatement activities of perfectly competitive firms in which investors can be socially responsible. This social responsibility objective of investors is represented by a warm-glow mechanism a la Andreoni, so that they feel partly responsible for the pollution content of their portfolio and, at equilibrium, demand a "pollution premium" in order to hold "dirty assets". Firms are incentivized to engage in cleanup activities by investors' portfolio choices, as higher pollution also implies a higher cost of capital in the capital markets. Using a continuous time model, the authors compare the effectiveness of the two types of fiscal policy in improving environmental quality and the conditions under which they can also improve the economic performance. They show that while both can be successful in reducing pollution, their effects on the performance of the economy can be quite different: while the pollution tax always reduces per capita consumption and the capital installed in the economy, the subsidy can increase both. Indeed, in this model the pollution tax corresponds to a tax on profits, which reduces the marginal productivity of capital and induces firms to reduce capital investment and thus reduce pollution flows and the pollution premium. In contrast, the pollution abatement subsidy reduces firms' production costs, so that it induces an increase in firm size, and thus in installed capital, output, and consumption per capita, but also in the resources that firms devote to pollution abatement, thereby reducing pollution as well as the pollution premium.

The second paper with behavioral market participants was presented by Laura Birg and Jan S. Voßwinkel entitled "Green Consumers, Emission Taxes, and Firm Relocation". 5 This paper lies at the intersection of two strands of literature, introducing the risk of firms relocating abroad into the analysis of the effect of environmental policy in the presence of green consumers in a vertical duopoly with two types of firms, a 'green' firm with high environmental quality and a 'brown' firm with low environmental quality. The authors shed new light on the pollution haven hypothesis by emphasizing the impact of relocation on firms' environmental quality choices. They show that the combination of the environmental tax with the private incentives provided by consumers' willingness to pay for green products produces complex effects, with offshoring decisions resulting from the emissions tax affecting the quality levels of both firms. If the green firm does not relocate, both the green firm and the brown firm provide better environmental quality products. At the same time, if the green firm does not outsource but the brown firm does, the difference in quality is the highest, implying that competition between the firms is the lowest. The welfare-maximizing environmental policy then changes with firm offshoring.

The second strand of papers is on the trade-off between environmental policy, on the one hand, and economic growth and employment, on the other. We would like to emphasize these environmental economics papers, because they are related to the question of equity and inequality, which has become one of the major topics in public economics over the last decades (with a large number of separate sessions at the IIPF 2020 congress on topics like, e.g, 'Distributional Effects of Taxation', 'Inequality' and 'Optimal Tax-Transfer Policy').

One relevant paper in this strand is "Environmental Policies in a Stagnant Economy" presented by Masako Ikefuji and Yoshiyasu Ono Ikefuji. They consider a stagnant economy that faces involuntary unemployment due to a shortage of aggregate demand. 6 Using a theoretical model, they examine the effects of environmental policies, such as an emissions tax or a public pollution control policy, on consumption, pollutant emissions, and welfare. They find that, in such an economy, environmental regulation reduces pollution but more importantly creates new jobs, reduces inflation and stimulates consumption. A double dividend can thus be obtained thanks to the unused labor force until full employment is achieved.

In a similar spirit, Marc Hafstead and Roberton Williams presented a paper entitled "Distributional Effects of Environmental Policy across Workers: A General-Equilibrium Analysis" in which they use a computable general equilibrium model to study the sectoral decomposition and dynamics of the effects of environmental policy on workers' employment, wages and unemployment duration. They find that the significant negative impacts mainly affect workers initially employed in a few carbon-intensive industries. Finally, they show that pre-announcement and phase-in of policies can significantly reduce shortterm negative employment impacts, both overall and for the most affected groups of workers.

As a third and final strand of papers, we would like to mention papers concerned with the political economy of environmental policy. As for all kinds of policies analyzed in public economics, political economy issues are key to the question whether suitable climate policies can be implemented in practice. Many of the argument's in Ottmar Edenhofer's keynote lecture circled around this question and also the paper by Pauli Lappi, published in this special issue and refereed to below, is on the political economy of environmental policy.

In addition, we would like to showcase the paper "Climate Policy and Inequality in Two-Dimensional Political Competition", presented by Waldemar Marz. Interestingly, this paper is relevant for both the second and third strands of papers that we have highlighted here, since it refers to the inequality issue as well as to political competition in the presence of climate policy. More specifically, the author proposes an innovative theoretical framework to analyze the reasons for the rejection of environmental policies in economies marked by income inequality. He models a two-dimensional political competition on an environmental policy with distributional implications, the carbon tax, and the redistributive policy through the proportional income tax at the same time. He shows that the greater the pre-tax income inequality or the more importance voters place on the issue of redistribution, the greater the polarization of carbon tax proposals across parties -even if the income tax is used to counteract rising income inequality. Remarkably, this result does not depend on the progressivity of the carbon tax revenue recycling mechanism.

Overall, this brief review of selected papers presented at the congress reveals some important and fruitful common points of interest between environmental economics and public economics. Nevertheless, some blind spots remain. For instance, in spite of the title of the conference, the subfield of natural resource economics was not represented by a substantial number of papers. The same is true for the various subfields dealing explicitly with the interactions between nature and economics (biodiversity conservation, sustainability issues, sustainability theory). This is both understandable and regrettable. It is understandable because the methods of analysis used in these fields are very specific, often based on formalization and a high degree of technicality, in which only researchers in the field invest, under the strong pressure of the increasing specialization of research in economics. But this is unfortunate because the investigation of these complicated relationships between the economy, natural resources, whether exhaustible or renewable, and nature may benefit public economics in a similar way as public economics has influenced -and in the future for sure continues to influence -the analysis of topics in environmental and resource economics. Despite the lack of corresponding papers presented at the IIPF 2020 congress, we hope that including topics from environmental and resource economics into the main theme of a public finance congress will nevertheless stimulate further research on the intersection of these two important research fields.

Content of this Special Issue

Two of the keynote speakers decided to write a paper based on their plenary talk and submit it to the special issue. These two contributions, which offer complementary and overarching views on carbon pricing, will undoubtedly become major references in the literature and we thank their authors again for submitting them to this special issue.

In his paper "Pigou in the 21 st . A tribute on the occasion of the 100th anniversary of the publication of The Economics of Welfare", jointly written with Max Franks and Matthias Kalkuhl, Ottmar Edenhofer critically reviews the widespread view that Pigouvian pricing is well-founded from a theoretical point of view, but connected with a lot of implementation problems and therefore infeasible in practice. The authors identify a number of theoretical and political challenges, but also point to recent success stories in implementing the Pigouvian principle in practice. Finally, the authors briefly identify some possible ways to go for future research on the Pigouvian approach. Overall, the paper gives a very informative and differentiated view on Pigouvian pricing which is stimulating for both further research and political discussions on this central topic in climate policy.

The application of the Pigouvian principle relies in fact on the evaluation of the social cost of carbon, defined as the present value of the damage caused to the society by the emission of a ton of carbon today. Based on his keynote lecture, Rick van der Ploeg discusses in his paper "Carbon pricing under uncertainty" the sensitivity of this social cost of carbon to different types of uncertainty -climatic and economic -and to the risk of economic disasters and of climatic tipping points. He argues that the effect of these uncertainties, combined with the choices to be made about the discount rate for future damages, is to significantly widen the confidence interval of the social cost of carbon, which justifies the more pragmatic approach adopted by policy makers at the Paris agreement in 2016. The choice of a temperature cap implies a cap on cumulative global emissions, which the paper shows is sensitive to the uncertainty of temperature sensitivity and risk tolerance, requiring the price of carbon to grow over time at a rate different from the rate of interest.

Next to these two papers based on the IIPF 2020 keynote lectures, the special issue contains five papers that were presented in contributed sessions of the congress and that deal with a variety of topics from public economics.

Closely related to the main theme of the congress is the paper "Lobbying for size and slice of the quota" by Pauli Lappi. The author presents a two-stage common agency framework in which several principals (countries or firms) give contribution payments to a single agent (regulator) that sets the overall size and distribution of an emissions quota. The basic insight is that lobbying leads to an inefficiently high number of permits. However, if the environmental damage taken into account by the agent just reflects the sum of damages of the principals, the emission quota is efficient, even though each principal takes into account only his own damage. The reason is that the principals' contributions sum up and thereby give the agent the incentive to choose the quota as if it maximizes social welfare. Overall, the important contribution of the paper is to show that the efficiency implications of lobbying depend on the aggregated damages taken into account by principals and agents.

The following two papers deal with the consequences of certain international tax rules on profit shifting incentives, thus contributing to a topic that was very prominent at the conference.

Entitled "Designing international tax reform: lessons from TCJA", the paper by Mindy Herzfeld analyzes the motivations for and effects of the tax law changes enacted in the United States by the Tax Cuts & Jobs Act (TCJA) in terms of their impact on cross-border trade and investment by comparing them to the goals of international tax reform that were widely supported prior to the law's enactment. She finds that in fact the changes in the law seem to have been far from achieving the desired results. In particular, the new system has neither eliminated incentives for profit shifting nor leveled the playing field between US and foreign multinationals. This failure appears to be due in large part to an inadequate appreciation of the consequences of the interactions between the complex new rules and existing law.

The paper "Tariff Elimination versus Tax Avoidance: Free Trade Agreements and Transfer Pricing" by Hiroshi Mukunoki and Hirofumi Okoshi uses a monopoly model of a multinational enterprise in order to identify the so-called Rules of Origin (ROO) within a Free Trade Agreement (FTA) as a means to reduce profit shifting. Under the ROO, exporters have to identify the origin of exports to be eligible for a preferential tariff rate in the FTA. Under the value-added version of this rule, the difference between the exported goods' output and input prices, relatively to the output price, has to be higher than a certain threshold value. If the multinational manufactures the input good in a tax haven outside the FTA, the ROO gives the multinationals an incentive to reduce the transfer prices of the input good and, thus, profit shifting to the tax haven. If the incentive is strong enough, it is even possible that the direction of profit shifting is reversed. Because of higher tax revenues inside FTA countries, the ROO may render the FTA welfare-increasing.

The next two articles in this Special Issue are devoted to the impacts of the value added tax design on growth. This is explicitly the case for the paper entitled "The Value Added Tax and Growth: Design Matters" by Santiago Acosta-Ormaechea and Atsuyoshi Morozumi. It examines, for OECD countries over the 1970-2018 period, the effects on long-term growth of a value-added tax (VAT) increase that does not affect total tax revenues because it is offset by a decrease in income tax, depending on how the VAT is increased. In most of these countries, there is a system of exemptions and reduced rates on various grounds that lowers the amount of VAT actually levied. The results of this paper suggest that a broadening of the VAT base through fewer exemptions and a more uniform rate structure with fewer reduced rates is more growth-enhancing than simply raising the standard rate.

In their paper "VAT threshold and small business behavior: evidence from Thai tax returns", Athiphat Muthitacharoen, Wonma Wanichthaworn and Trongwut Burong show that the growth of small firms in a country like Thailand, characterized by high VAT informality (defined as the presence of non-VAT registered firms), is constrained by the existence of a size-dependent VAT registration threshold. In particular, their results clearly reveal a bunching response of firms to the threshold and that non-VAT-registered firms have a significantly lower growth rate than a propensity score-matched group of firms that voluntarily registered. These effects are stronger for firms near the threshold, which is likely due to misreporting.

In particular, the papers on profit shifting and the VAT published in this special issue show that the range of topics discussed at the IIPF congress 2020 was much broader and not only confined to public finance issues of environmental and resource policies. Similar to previous IIPF conferences, while the 2020 congress addressed the main conference theme in keynote lectures and a substantial number of contributed sessions, it was also open to papers on all topics discussed in public economics. In order to reflect the resulting variety of topics presented in contributed sessions, the editors of International Tax and Public Finance agreed on contributing to the special issue. Their paper offers a view on some general developments in public economics revealed by the congress and closes this special issue.

Prizes and Awards

The 2020 congress also featured a number of prizes and awards.

Conclusion and Acknowledgments

Despite the unfavorable conditions due to the pandemic, the 2020 IIPF Congress was a great success and we are grateful to all the authors submitting their papers to the congress and being prepared to present the papers in the online format. The same thanks go to the keynote speakers and all the discussants of papers in the contributed sessions. We also thank the many authors that decided to submit their papers to the IIPF special issue. The congress itself and, therefore, also the publication of articles in this special issue would not have been possible without the help of the members of the Scientific Committee. We are grateful for their great effort and advice in reviewing hundreds of papers submitted to the congress. We are also thankful for the assistance we received by the editors and staff of International Tax and Public Finance during the publication process of the special issue. Last but not least, we would like to express our deep gratitude to Barbara Hebele of the IIPF and to the local organizers in Reykjavik chaired by Þórólfur Geir Matthíasson (University of Iceland, Iceland). Working together with Barbara as well as with Þórólfur and his friendly team was always fun and we think that together we managed to make this first online IIPF congress an important experience and a memorable event!

  :Ottmar Edenhofer (Potsdam Institute for Climate Impact Research and Technische Universität Berlin), on "Pigou in the Post-Corona Era", Karen Palmer (Resources for the Future) on "Flexible Electricity Demand and Decarbonization: Policy, Prices and Automation", Rick van der Ploeg (University of Oxford) on "Climate Policy, Natural Resources and Public Finance Post Covid-19" and Anna Alberini (University of Maryland) on "All Car Taxes are Not Created Equal: Evidence from Germany". The keynote lectures allowed both to deepen the relationship between the fields of public finance and climate policy and to provide conference participants with specific insights into aspects that are often overlooked by non-specialists. 1

  The 2020 Peggy and Richard Musgrave Prize for the best paper contributed by authors under the age of 40 was awarded to to Jósef Sigurðsson (Bocconi University, Italy; now Norwegian School of Economics in Bergen, Norway) for his paper "Labor Supply Responses and Adjustment Frictions: A Tax-Free Year in Iceland". In addition, the IIPF Young Economists Awards were given to the following researchers: Ashley Cooper Craig (University of Michigan, USA) for his paper "Optimal Income Taxation with Spillovers from Employer Learning", Andreas Gerster (University of Mannheim, Germany) and Michael Kramm (Technical University of Dortmund, Germany) for their paper "Correct Me if You Can -Optimal Non-Linear Taxation of Internalities" and Arezou Zaresani (University of Melbourne, Australia) for her paper "Adjustment Cost and Incentives to Work: Evidence from a Disability Insurance Program". The 2020 ITAX PhD Student Award was presented to Thiago Scot (UC Berkeley, USA) for the paper "Corporate Taxation and Evasion Responses: Evidence from a Minimum Tax in Honduras" which was co-authored with Felipe Lobel (UC Berkeley, USA) and Pedro Zúniga (Servicio de Administración de Rentas, Honduras). More than 80 papers applied for the IIPF awards, and more than 60 for the ITAX award.

All keynote lectures are available in video on the permanent conference website: https://www.iipf.org/cng20.htm.

Stern, N. (2006), Stern Review on the Economics of Climate Change. UK Government Economic Service, London. www.sternreview.org.uk.

Published as Alberini, A., & Horvath, M. (2021). All car taxes are not created equal: Evidence from Germany. Energy Economics, 105329.

Published as Renström, T. I.,Spataro, L., & Marsiliani, L. (2021). Can subsidies rather than pollution taxes break the trade-off between economic output and environmental protection? Energy Economics, 95, 105084.

Published as Birg, L., & Voßwinkel, J. S. (2021). Emission taxes, firm relocation, and product differentiation. European Journal ofLaw and Economics, 51(2),

Published as Ikefuji, M., & Ono, Y. (2021). Environmental policies in a stagnant economy. Economic Modelling, 105574.


Yifan Zhu
email: yifan.zhu@univ-lorraine.fr

Aurélien Merkel

Krupali Donda

Shiwang Fan

Liyun Cao

Badreddine Assouar
email: badreddine.assouar@univ-lorraine.fr

Nonlocal acoustic metasurface for ultra-broadband sound absorption

Keywords: Nonlocal acoustic metasurfaces, Sound absorption, Acoustic metamaterials

Classical meta-absorber designs usually have a trade-off between bandwidth, efficiency and thickness. Here, we introduce the concept of nonlocal acoustic metasurface absorber by using a bridge structure connecting resonating unit cells to improve the performances of the meta-absorber.

By utilizing the coupling effect between the adjacent unit cells, ultra-broadband sound absorption is achieved with deep-wavelength thickness. The physical mechanism of the nonlocal acoustic metasurface absorber is investigated and uncovered by developing analytical models. We theoretically and numerically study the nonlocal metasurface with connecting bridge and the traditional metasurface without bridge. The nonlocality can introduce three specific effects; the optimization of effective acoustic impedances, the shift of Fabry-Perot resonant frequencies, and the enhancement of the coupling effects between adjacent unit cells. These effects contribute to improve the bandwidth and the efficiency of the acoustic meta-absorber. We numerically and experimentally achieve an average absorption coefficient larger than 0.9 within the ultrabroadband bandwidth extending from about 600 Hz to 2600 Hz, with a metasurface of 6.8 cm, viz., λ/9 for the lowest frequency. Our finding demonstrates the advantage of non-local acoustic metasurface to conceive subwavelength sound meta-absorber.

I. Introduction

Conventional acoustic metasurfaces have the ability to manipulate acoustic wavefront based on generalized Snell's law and the phase discontinuities between adjacent unit cells. The phase shifts at the interface of metasurface can lead to various wavefront manipulations, such as extraordinary refraction/reflection, acoustic focusing, non-diffraction beams, sound absorption, and so on [1]. In the past several years, acoustic metasurfaces have been employed to design sound absorbers [2][3][4][5][6][7][8][9][10][11][12] with optimized performances, which is difficult to be realized by traditional sound absorbing materials. For example, the thickness of acoustic metasurface absorber can be largely decreased to deep subwavelength dimension with highly resonant unit cells [2][3][4][5][6], especially for low-frequency regime. Furthermore, the bandwidth of acoustic absorber can be improved via combination of multiple unit cells with different resonant frequencies, to achieve broadband sound meta-absorber [7][8][9][10][11][12]. The design of absorbing metasurface is actually dictated by the trade-off between bandwidth, efficiency and thickness.

On the other hand, the concept of nonlocality has been widely studied in both optics [START_REF] Eringen | Nonlocal Continuum Field Theories[END_REF] and optical metasurface [START_REF] Kwon | [END_REF][15]. Nonlocality can be seen as a more exact theoretical description of the effective medium, that considers nonlocal responses from transverse coupling. Drawing inspiration from their optical counterparts, recently, the mechanism of nonlocal metasurface has been extended to acoustics [16][17][18][19], which take advantages of the nonlocal response due to the transverse coupling between adjacent acoustic metastructure unit cells. This physical mechanism is completely different from the one on which are based conventional acoustic metasurfaces. For example, the latter can hardly realize perfect extraordinary refraction/reflection when the incident angle is large, due to the fact that the high-order diffractions could influence the acoustic field distributions near the surface of the metasurfaces. The concept of nonlocal acoustic metasurface [16] can overcome this problem. For instance, optimized meta-grating structures based on nonlocality have been designed to realize high-efficient extraordinary reflection [16] refraction [17] and beam splitting [18]. The nonlocal metasurface is further designed to realize hyperbolic sound propagation [19], via connection between the adjacent unit cells of the metasurface, showing the unique advantages of the nonlocal acoustic metasurfaces.

In this research, we propose the concept of nonlocal metasurfaces to conceive a sound absorber, showing the interest of nonlocal couplings which interestingly increase their efficiency.

We will investigate and analyze our concept based on theoretical, numerical and experimental approaches, and discuss in details the involved physical mechanism.

II. Model of nonlocal metasurface and theory

We first compare two conventional designs of acoustic meta-absorbers with multiple unit cells based on Fabry-Perot (FP) resonances as schematically shown in Fig. 1(a) and 1(b), whose mechanism are similar to previous two designs in Refs. 7 and 11, respectively. The supercell of the first one consists of 22 straight channels with high filling ratio at the surface [7], and large height under the surface. Then, in order to gain space, the supercell of the second one consists of 22 folded channels with low filling ratio [11] at the surface, and smaller height under it. The comparison between the two designs indicates that there is always a trade-off between the absorption efficiency and the structure thickness, since the second structure is less efficient than the first one in term of absorption. When the surface filling ratio change from 0.7 to 0.2, the structure thickness is decreased from 14.6 cm to 6.6 cm, but the averaged absorption efficiency is decreased from 0.92 to 0.56. Based on the previous theory [7], the absorption efficiency is directly determined by the filling ratio. Here comes the question: Can we get a high absorption with a low surface filling ratio in the meta-absorber? Indeed, we will show that if we introduce the nonlocality effect in the second design, we can improve the absorption efficiency and bandwidth, even with a very low surface filling ratio. In order to introduce the nonlocal effect, a bridge structure is deliberately designed to connect the unit cells, as a key element to make the supercell totally continuous. Therefore, the efficiency and bandwidth can be improved. We compare three particular cases, that is, the original folded metasurface without bridge, the metasurface with cavities attached on the channel, and the metasurface with optimized bridge attached on the folded channels to connect all the unit cells, respectively. The acoustic metasurface with optimized bridge is defined as a nonlocal metasurface.

By comparing nonlocal acoustic metasurface with the local one, the mechanism of nonlocal acoustic metasurface absorber (NAMA) is investigated and revealed via developing and analyzing effective circuit models. The coupled resonances between the unit cells lead to high efficiency and continuous frequency band. We explore the advantage of nonlocal acoustic metasurface to conceive subwavelength sound absorber, which could have practical application for noise control, environmental acoustics, and so on.

Two classical acoustic absorbing metasurfaces based on multiple FP resonant units are shown in Figs. 1(a) and 1(b). The supercell is marked by red dashed rectangle with a surface size of 4.5 cm×1.5 cm (3w×w). Both structures shown in Figs. 1(a) and 1(b) contain three supercells. The first classical design shown in Fig. 1(a) contains 22 straight channels with channel lengths L(n) ranging from 4.7 cm to 14.6 cm as the black curve shown in Fig. 1(c). The width of the channels is w0=1.4 cm. The thickness of the channels is t1=0.15 cm, about 20 to 30 times of the thickness of thermalviscous boundary layer [11]. The second one shown in Fig. 1(b) contains 22 folded channels with same effective channel lengths. The folding feature makes the all the folded channels have limited heights (H(n)<6.6cm). The width of the fold channels is w1=0.4 cm. Since the channels with different effective lengths correspond to different resonant absorption frequencies, this type of meta-absorber can realize broadband sound absorption. If we considering a Lorentzian form of resonance and all order resonance modes, the effective acoustic impedance of the meta-absorber can be expressed as [2,7,20]: (1) where Z0=ρ0c0 is the air impedance, ρ0=1.21 kg/m 3 is air density and c0=343 m/s is the sound speed in the air, ω=2πf is the angular frequency, Ωmn=2πf(m,n) denotes the m-order resonant angular frequency of nth straight/folded channel, β is the dissipation coefficients of the acoustic metastructure, and Z(n) is the acoustic impedance of nth channel. The filling ratio γ is defined as the ratio of the surface area occupied by channels' cross sections to the effective surface area exposed to sounds. The total acoustic impedance of the supercell is defined as Z≡p/v, p being the sound pressure modulation and v the displacement velocity in response to the pressure at the surface of the supercell.

The absorption coefficient is calculated by:

(2) For the ideal case, when the channels number is large enough and their eigenfrequencies are gradient-varying, the impedance and the absorption can be approximated as [7]:

(3) (4)

1 1 22 22 0 2 2 1 0 1 1 , ( ) () 4 mn n m n mn i Z Z i Zn p W w W w bw g - - ¥ = = = ae ö ae ö = º ç ÷ ç ÷ -- è ø è ø åå å 2 0 0 1 Z Z A Z Z - = - + 1 0 1 1 [ 2 tanh )] Z Z i p g p W w - = - ( / 2 1 11 1 11 (1 ) 2 tanh / ) 1 (1+ ) 2 tanh ) i A i g p g W w g p g W w - - - + ( = - - ( /
Here, Ω11=2πf(1,1) is the cutoff (lowest) resonant angular frequency of straight/folded channel.

The absorption calculated by Eq. ( 4) for the ideal case is plotted by dashed lines in Fig. 1(d). The averaged absorption within 600 Hz to 2600 Hz for the two structures with γ0=0.7 and γ1=0.2 are Aideal0=0.97, and Aideal1=0.56, respectively, as shown in Eq. 4. The absorption depends directly on the surface filling ratio, the low γ value will induce impedance mismatch at the interface. The approximated relation can be given as:

(

The simulated absorptions for the two structures are plotted by solid lines in Fig. 2(d). Throughout the paper, the numerical simulations are accomplished by COMSOL Multiphysics 5.5a. The averaged simulated absorptions for the two structures within 600 Hz to 2600 Hz are A0=0.92, and A1=0.56, respectively, which agree well with the analytical approach. A perfectly flat absorption requires an infinite number of resonance modes, which is not possible in a real system. Thus, the absorption spectrum of our systems with a finite number of resonance modes are not perfectly flat.

In the following, we will introduce the nonlocal effect in the second design with folded channel to achieve a NAMA. The design goal is to have a high efficient absorption even with a low filling ratio γ1=0.2, as well as a continuous bandwidth with an optimized profile. for the first and second meta-absorber, respectively.

In order to reduce the impedance mismatch of the folded meta-absorber shown Fig. 2(a), we design an array of cavities to be attached on the folded channels, as shown in Fig. 2(b). The height of cavities is h1=1.2 cm, and width is w2=1 cm. The two absorbers shown in Figs. 2(a-b) are both traditional local acoustic metasurface, with absence of connections between adjacent unit cells.

A nonlocal metasurface requires obvious coupling between adjacent unit cells [19]. In order to introduce the nonlocal effect in the metasurface absorber, a coupling bridge structure is designed to be attached on the channel array, which connects all the channels as shown in Fig. 2(c). The height of the optimized bridge is gradient varying from h2=1.2 cm to h3=2.4 cm (h(x)=0.012+0.267x). The fact that the different channels are connected behind the surface of the metasurface results in a nonlocal surface impedance. The metasurface with internal bridge shown in Fig. 3(c) is defined as a nonlocal acoustic metasurface.

In order to analytically analyze the above different metasurfaces, effective circuit models are built for the three models: without bridge (O), with cavity (C), and with bridge (B) as illustrated in Figs. 2(d-f), respectively. The acoustic impedance of n th branch consists of two parts, that is, the acoustic impedances ZO(n), ZC(n), ZB(n), of n th unit cell, and the acoustic impedances ZOM(n), ZCM(n), ZBM(n), induced by the transverse coupling of the neighboring unit cells.

For the O/C/B cases, the acoustic impedance of the nth unit cell shown in Fig. 2(d-f) can be written with the same form of Eq. ( 1) with ZO/C/B(n)=Z(n), expanded as:

(6)
Then, the total impedance of an effective circuit can be expressed as: (7) In the following, we will show the different mechanisms for the physical quantities of γO/C/B, fO/C/B(m, n), and ZOM/CM/BMM(n) for O/C/B cases, respectively.

The first effect of the inserted cavity or bridge is the improvement of the acoustic impedance matching, which can be also regarded as the increasing of the effective filling ratio, although the

1 0 O / C / B O/C/B 2 2 0 O/C/B O/C/B 2 ( , ) ( ) [2 ( , )] ) 4 m i Z f m n n f m n i Z p p w p w b w g - ¥ = ae ö = ç ÷ -- è ø å 1 22 O/C/B 1 O/C/B OM/CM/BM 1 = ( )+ ( ) n n Z n Z Z - = ae ö ç ÷ è ø å
original filling ratio for O/C/B cases are all 0.2. For the O case, the unit cell is a folded channel without any attachment. The filling ratio is original γO=S1/S=(t1×w1)/(t×w)=0.2. In this case, the predicted ideal absorption by Eq. ( 5) is only AidealO=0.56. For the C case, the inserted cavities can improve the acoustic impedance matching between the unit cells and the free space. We characterize it as a change of effective filling ratio, which is related to the volume filling ratio of the cavity, expressed as (8) where H=2.8 cm is the maximal height of the cavity or bridge, as marked in Fig. 2(g). We get the value of γC=0.41. The ideal absorption is AidealC=0.82 deduced by Eq. ( 5). For the B case, the effective filling ratio is also related to the volume filling ratio, calculated as:

Since the bridge is designed with a shape of h(x)=0.012+0.267x, we get the value of γB=0.61. Thus, the predicted ideal absorption is increased to AidealB=0.93.

The second effect of the inserted cavity or bridge is the shifts of the FP resonant frequency.

For an original channel array for the O case, the m th order resonant frequency for n th unit cell is

[21] ( 10
)
where L(n) comprises the length of the channel [Fig. 1(c)] and an additional length induced by the radiation at the opening [START_REF] Kinsler | Fundamentals of Acoustics[END_REF]. However, for the C case, the insert cavity influences the original resonant frequency, the m th order resonant frequency for n th unit cell becomes: (11) in which the height of the cavities is considered to decrease the effective length of the channel. For the B case, the m th order resonant frequency for n th unit cell becomes: (12) in which the local height of the bridge is considered to apparently decrease the effective length of the channel.

1 1 2 C 1 t w t w H twH h g + » 1 , 3 1 1 0 1 2 ( ) , w B t w h t w x w x H t H d g + » ò 0 O (2 -1) ( , )= , 4 ( ) m c n L n f m
The third effect is that the inserted bridge can strengthen the transverse coupling between neighboring channels. For the O/C/B cases, the acoustic impedance induced by the coupling effects at the opening of the channel is expressed as [START_REF] Johansson | [END_REF][24]: [START_REF] Eringen | Nonlocal Continuum Field Theories[END_REF] where ZM(n-1)(n) is the mutual impedance between (n-1) th and n th unit cells. The mutual impedance between the unit cells is defined as the average sound pressure on one of the channel openings, generated by a unit volume velocity on the other [START_REF] Johansson | [END_REF]. The schematic diagram for calculation of the mutual impedance is shown in Fig. 2(g). The mutual impedance ZM(n-1)(n) is calculated as: [START_REF] Kwon | [END_REF] where lac≈0.425t1 is the effective distance from the opening of the channel to the assumed monopole sound source [START_REF] Johansson | [END_REF]. t1=0.15 cm is size of the channel in x-direction, t=0.2 cm is the distance between two neighboring channels.

As shown in Fig. 2(g), the coupling effects for the O and C cases only exist at the openning of the channel, while for the B case, the coupling effects exist at not only the opening of the channel,

0 C 1 (2 -1) ( , )= , 4 ( )- m c m n L n h f 0 B (2 -1) ( , )= , 4 ( )-4 ( ) m c L n h f m n n 2 2 M( -1)( ) M( )( 1) OM/CM/BM0 O/C/B O/C/B ( ) , ( -1)
( 1)

n n n n+ Z Z Z n = Z n Z n+ + 2 2 1 1 ( ) /2 0 0 2 2 /2 1 1 ( 1)( ) 1 1 , 4 2

III. Numerical simulation and analysis

We, afterwards, have numerically investigated the performance of the O/C/B structures. In COMSOL simulations, the multiphysics module of "Acoustic-Thermovisous Acoustic Interaction, Frequency Domain" is used. The regions of the folded channels are built in "Thermovisous Acoustics, Frequency Domain" module. The regions of cavities, bridges and the outside air are built in "Pressure Acoustics, Frequency Domain" module. The boundaries of the folded channel are set as sound hard boundaries. Boundary layers meshes are built in the channel for the acoustic absorption simulation. cm, the total thickness of the NAMA is 6.8 cm, that is λ/9 of the lowest working frequency.

We have then analyzed different modes within the curves in Figs. Observing the acoustic field distributions of O1, C1, B1 and B2, the strong resonance of n th channel can also induce weak resonance in (n-1) th and (n+1) th channel. This is owing to the Lorentzian form of the resonance of FP channel that has a small bandwidth. Thus, for the O2 and C2 cases, the incident frequency can induce the resonances in 22 nd , 1 st and 2 nd channels, while there is no transverse coupling between 22 nd and 1 st channels since they are geometrically separated. Therefore, the inherent decoupling between 22 nd and 1 st channels, and the small 3(e). The acoustic pressure amplitude of O3, C3, B3 are 14.8, 19.6, and 22.9, respectively. The corresponding sound absorptions showed in Fig. 3(b) are 0.46, 0.79 and 0.95, respectively. These results suggest that the 2-order mode for B case has the highest sound intensity inside the FP channel, similar to 1-order mode, which lead to the highest sound absorption.

The modes of O4(2880 Hz), C4(2880 Hz), and B4(2880 Hz) are located at an interesting frequency at high frequency region. The acoustic field distributions are shown in Fig. 3(f). O4 and C4 modes can induce 2-order and 3-order modes in the supercells, while B4 mode can only induce 2-order mode, which agree with the mode distributions in Fig. 3(a). An abnormal phenomenon [marked by Abnormal 2 in Fig. 3(a)] is observed indicating that the analytical solution for the C case for 3-order mode does not agree with the simulated one. This is due to the fact that standing wave is excited between adjacent unit cells [25] for C4 case as shown in Fig. 3(f). Another abnormal phenomenon [marked by Abnormal 3 in Fig. 3(b)] indicates that at 2880Hz, the absorptions for O4, C4, and B4 cases are 0.55, 0.99, 0.36, respectively. The C case has the highest absorption, much larger than the O and B ones. This phenomenon is different from the range of [600 Hz, 2600 Hz], which is due to the fact that the standing wave in the resonators can significantly increase the sound absorption [4,26], and the mode density at C4 (consists of both 2-order and 3-order modes) is higher than B4. The absorption coefficient, sound pressure amplitude in the resonant channels, and sound pressure amplitude in their adjacent channels for all interested modes are shown in Fig. 3(g). The results suggest that the sound absorption usually increases with the sound pressure amplitude in the resonant channels and their adjacent ones. Based on the above theoretical and numerical results and analysis, we have obtained a summary for original straight channel and the O/C/B cases as shown in Table 1. We list the expressions or calculated values of real filling ratio, effective filling ratio, ideal absorption, simulated averaged absorption, and FP resonant frequency for different cases.

In the following, we discuss the loss effect in the proposed metastructure. Ignoring the mutual impedances in Eq. ( 13) and observing Eq. ( 1), the loss in a single channel is decided by the dissipation coefficients β and the effective filling ratio γ. The parameter of β is related to the channel thickness (t1=0.15cm) and the thermal viscous boundary layer thickness [24], expressed as:

Based on Eq. ( 15), dvisc values for the frequency range of [600 Hz, 2600 Hz] are [0.09 mm, 0.043 mm], which is about 1/17 to 1/35 of the channel thickness. The dissipation coefficient β is related to this size comparison, and can be usually extracted from the experimental results [2,20]. Also, the sound absorption at FP resonance [21] is much larger than the non-resonant case, which means FP resonance has a sufficiently ideal β for high absorption. Therefore, the absorption coefficient for a single channel is mainly decided by the effective filling rate γ, viz., the acoustic impedance matching ability between incident wave and the inner channels. In the Then, we have also investigated the sound absorption for oblique incidence case, to study the effect of the incident angle on the absorption.

IV. Experimental evidence

Finally, we show here the experimental demonstration of the proposed models of the O/B/C cases, respectively. Three samples have been fabricated by 3D printing as shown in Fig. 5(a

V. Conclusion

We have conceived and experimentally demonstrated a nonlocal acoustic metasurface absorber showing high broadband absorption coefficient with a very low surface filling ratio. This breaks the limitation of conventional quarter-wavelength resonators based meta-absorber.

Theoretical models have been developed to theoretically analyze the proposed meta-structures. By comparing the proposed nonlocal metasurface (B case) with two traditional metasurfaces (O and C cases), we show that the nonlocality can introduce three specific effects, that is, the optimization of the effective acoustic impedances (characterized as effective filling ratio), the shift of the Fabry-Perot resonant frequencies, and the enhancement of the coupling effects between adjacent unit cells. This can improve the efficiency and the bandwidth of sound absorption. We have evidenced an ultra-broadband absorption of NAMA over 2 octaves with A>0.8. The average absorption is about 0.9 within [600 Hz, 2600 Hz] frequency range. The nonlocal acoustic metasurface is subwavelength with a thickness of λ/9. The proposed strategy can facilitate the design of acoustic absorbers based on multiple resonant units with improved bandwidth and flexible design, which could have practical applications in noise control and acoustic sink.

FIG. 1 .

 1 FIG. 1. The comparison of two classical meta-absorbers. (a) The structure of the first meta-

  FIG. 2. The structures of the supercell (a) without bridge (O, local metasurface), (b) with cavity

Figure 3 (

 3 Figure 3(a) shows all the existing resonant modes within the range of [500 Hz, 3000 Hz] for

Figure 3 (

 3 Figure 3(b) exhibits the numerical sound absorption coefficients within 500 Hz to 3000 Hz

  3(a) and 3(b). The modes of O1 (1000 Hz), C1 (1060 Hz), and B1 (1280 Hz) are corresponding to 1-order resonances at 8 th unit cells. The acoustic pressure distributions for O1, C1, B1 are shown in Fig. 3(c). The acoustic pressure amplitude for O1, C1, B1 are 28.3, 32.2, and 35.3, respectively. The corresponding sound absorptions in Fig. 3(b) for O1, C1, B1 are 0.72, 0.80 and 0.89, respectively. These results suggest that the B case has the highest sound intensity inside the FP channel, which lead to the highest sound absorption. The modes of O2 (1580 Hz), C2 (1580 Hz), and B2 (1580 Hz) are located at an interesting frequency close to the mode gaps for O [1590 Hz, 1640 Hz] and C [1650 Hz, 1670 Hz] case.Figure3(b) shows that two absorption valleys are observed at 1580 Hz for the O and C cases with low absorption of 0.54 and 0.68 [marked by Abnormal 1 in Fig. 3(b)], while no valley is observed for the B case at 1580 Hz with the absorption of 0.90. The acoustic field distributions in the supercells are shown in Fig. 3(d).

  frequency gaps of [1590 Hz, 1640 Hz] and [1650 Hz, 1670 Hz] lead to the absorption valley in the O and C cases. While for the B curve, there is no apparent absorption valley, because the small frequency overlapping region [1960 Hz, 2140 Hz] compensates the decoupling effect between 22 nd and 1 st channels. The modes of O3 (1980 Hz), C3 (2005 Hz), and B3 (2395 Hz) correspond to 2-order resonances at 17 th unit cells. The acoustic pressure distributions for O2, C2, B2 are shown in Fig.

FIG. 3 .

 3 FIG. 3. (a) 1-order, 2-order and 3-order modes within the range of 500 Hz to 3000 Hz for O/B/C

Figure 4 (FIG. 4 .

 44 FIG. 4. (a) The simulated absorption for the O/C/B cases with 0, 30, 45, and 60 degree incidences

  ) with the same size of 4.8 cm×4.8 cm×6.8 cm (considering the wall thickness). They are made of polylactic acid (PLA) material with Young's modulus of EPLA=3.2×10 9 Pa, mass density of ρPLA=1250 kg/m 3 , and Poisson's ratio of νPLA=0.35. The experimental setup is shown in Fig. 5(b), illustrating a lab-made acoustic impedance tube with a size of 5 cm×5 cm×22 cm to measure the absorption spectrum. The simulated and experimental results are shown in Fig. 5(c). The simulated average absorptions within the frequency of [600 Hz, 2600 Hz] for the O/C/B cases are 0.56, 0.78, and 0.90, respectively. The corresponding measured average absorptions are 0.65, 0.81 and 0.94, respectively. The measured results are slightly higher than the simulated ones due to the imperfect sound insulation of the sample at the back wall. The metastructure for the B case is the desired NAMA design, whose simulated absorption is larger than 0.8 within [640 Hz, 2554 Hz] frequency range, and the measured absorption is larger than 0.8 within [640 Hz, 2828 Hz]. The numerical and experimental bandwidths (A>0.8) are both larger than 2 octaves. These results suggest that the bandwidth of the NAMA is wider than the previous designs [11-12], and the absorption curve is plain enough, showing the advantage of the nonlocal effect for low-frequency acoustic absorption.

FIG. 5 .

 5 FIG. 5. (a) The photograph of the 3D printed samples for O/C/B cases with a same size of 4.8

  Table, the effective filling ratios for C and B cases are larger than their real filling ratios. The ideal average absorption is calculated by Eq. (5) with A=4γ/(1+γ)2 , whose results agree well the simulated average absorption in the Table, which validate above analysis for a single channel. The final absorption curves are also related to FP resonant frequencies distributions and the coupling effects from mutual impedances.

	d	visc	=	0.22mm 100 / n f ´

TABLE .

 . The results for cases of Straight Channel (S), Without Bridge (O), With Cavity (O),

	and With Bridge (B)
		S	O	C	B
	Real filling ratio	γ0=0.7	γ1=0.2	γ1=0.2	γ1=0.2
	Effective filling ratio	γ0=0.7	γO =0.2	γC=0.41	γB=0.61
	Ideal Absorption	Aideal0=0.97	AidealO=0.56	AidealC=0.82	AidealD=0.94
	Simulated Absorption	A0=0.92	AO=0.56	AC=0.79	AB=0.90
	FP Resonant Freq.

Acknowledgments

This work is supported by the Air Force Office of Scientific Research under award number FA9550-18-1-7021, and by la Région Grand Est and Institut Carnot ICEEL.

Reference