domain: phys spi

text:
Denis Payan

Guillaume Demol

Thierry Paulmier

Analysis of cathodoluminescence behaviour of fluoropolymers

Keywords: Charging, cathodoluminescence, defects, electron irradiation, material degradation, polymers, spacecraft

This paper presents a study on cathodoluminescence (CL) of different space-used fluoropolymers, such as: Polytetrafluoroethylene (PTFE), fluorinated ethylene propylene (FEP) and Ethylene tetrafluoroethylene (ETFE). This experimental technique allows highlighting radiative energy transitions under electron irradiation and degradation mechanisms under high radiation dose. Samples were irradiated with electron beam (10 keV -1 µA) under vacuum (10 -6 mbar). A parametric study has been performed to analyse especially the effect of temperature and the injected radiation dose on the CL spectra of these different polymers. The CL spectrum for these three materials is composed of three elementary contributions at 2.2 eV, 2.4 eV and 3.75 eV, each of them being associated with specific processes in relation to different chromophores present in the materials or generated by electronic irradiation. A decline on the amplitude of the CL spectra with the increasing temperature or accumulated radiation dose has been observed on these fluoropolymers.

I. INTRODUCTION

Polymer materials are widely used on spacecraft for their thermal stability, high mechanical and electrical properties and relatively low specific weight. They are used on spacecraft as thermal control coatings, adhesives, seals, thermal insulations, electrical insulation or mechanical support. In this study, we will focus the attention on fluoropolymer films, such as FEP (Fluorinated ethylenepropylene), PTFE (polytetrafluoroethylene) and ETFE (ethylene-tetrafluoroethylene) used on spacecraft in OSR (Optical Solar reflector), multilayer insulation MLI, mechanical support or electrical-cable-jacketing.

In space environment, these polymers have to cope with irradiation by energetic particles, especially electrons, that can induce electrical charging and physico-chemical ageing processes. This charging phenomenon can lead to the initiation of electrostatic discharges and electrical arcs on different parts of the spacecraft (for instance solar arrays or electrical cables) that can induce significant damages or electromagnetic disturbances and generate strong risks for the spacecraft mission [START_REF] Catani | Electrostatic behaviour of materials in a charging space environment[END_REF].

It has been shown previously [START_REF] Rose | Recombination Processes in Insulators and Semiconductors[END_REF]- [START_REF] Paulmier | Radiation-Induced Conductivity of Space Used Polymers Under High Energy Electron Irradiation[END_REF], that charging effect is usually smoothed by ionization processes due to high-T. Paulmier and G. Demol are with ONERA/DPHY, Université de Toulouse, The French Aerospace Laboratory, F-31055 Toulouse, -France (corresponding author phone: 33-562.252.947; e-mail: Thierry.Paulmier@ onera.fr) energy electron irradiation that enhances significantly the bulk electric conductivity in these polymers through the creation of electron-holes pairs in the extended states: we speak about Radiation-Induced Conductivity (RIC).

Models based on solid-state physics have been developed at ONERA [START_REF] Pacaud | 1-D Physical Model of Charge Distribution and Transport in Dielectric Materials Under Space Radiations[END_REF], [START_REF] Paulmier | Analysis of Charge Transport and Ionization Effect in Space-Used Polymers Under High-Energy Electron Irradiation[END_REF] and other research groups [START_REF] Tyutnev | Experimental and Theoretical Studies of Radiation-Induced Conductivity in Spacecraft Polymers[END_REF]- [START_REF] Faria | Radiation-induced conductivity of polymers in different gases[END_REF] to describe and predict charge and ionization effects on spaceused polymer materials. These models take into account different physical processes (ionization, trapping, detrapping and recombination) but some physical mechanisms are however not taken into account such as: energy transfer processes and ageing effect (by radical formation and structural modification). Previous results have indeed shown that the high radiation doses received by space used polymers may greatly alter their electrical properties (reduction of RIC and bulk conductivity on Teflon® FEP, PEEK, PEI and silicone with ageing [START_REF] Paulmier | Aging Effect and Induced Electric Phenomena on Dielectric Materials Irradiated With High Energy Electrons[END_REF]- [START_REF] Paulmier | Electrostatic behaviour of space used materials in regard of internal charging met on spacecrafts[END_REF]). This variation has been associated with a dramatic change of the structure of the materials, and band structure as well as formation of new deep traps that lead to an evolution of the traps energetic distribution profile [START_REF] Paulmier | Aging Effect and Induced Electric Phenomena on Dielectric Materials Irradiated With High Energy Electrons[END_REF], [START_REF] Qingquan | Modified isothermal discharge current theory and its application in the determination of trap level distribution in polyimide films[END_REF], [START_REF] Tyutnev | Chemical aspects of the radiation-induced conductivity in polymers[END_REF].

It is therefore necessary to understand the physics steering charge transport, radiation/polymer interactions, and electrical ageing for a realistic predictions of spacecraft charging levels in space environment.

For this purpose, we have developed at ONERA, a cathodoluminescence spectroscopic technique (CL) for polymer analysis. CL corresponds to the emission of photon from materials irradiated, in our study, by energetic electrons (10 keV) that induce energy transfer in the bulk through different physical processes described in the following sections. This technique is relevant for the characterization of the different several physicochemical mechanisms (radical formation, recombination and de-excitation processes in relation with defect and impurities initially present in the material or generated by irradiation) occurring in irradiated materials and therefore to extract parameters related to the physics model used to describe charge transport in space materials (in particular the involved energies). This technique is widely used for the characterisation of inorganic materials [START_REF] Evans | Low-Temperature Cathodoluminescence in Disordered Si02[END_REF], [START_REF] Yacobi | Cathodoluminescence microscopy of inorganic solids[END_REF] but the application of this technique for polymer analysis is quite sparse [START_REF] Qiao | Electroluminescence and cathodoluminescence from polyethylene and polypropylene films: Spectra reconstruction from elementary components and underlying mechanisms[END_REF]. The objective of this study is to use this technique on fluoropolymers to deepen our understanding of ageing, charge transfer and recombination effect in these irradiated materials. The major advantage of this technique is that it provides in-situ and instantaneous analysis of the irradiated polymer with no air exposure that could disturb the physico-chemical state of the materials and screen the ageing processes. This paper presents the experimental characterization of different fluoropolymers by the CL technique. The attention has especially been focused on PTFE, FEP and ETFE polymers because they have a similar chemical structure but quite different charging behaviour under representative space environment [START_REF] Paulmier | Radiation-Induced Conductivity of Space Used Polymers Under High Energy Electron Irradiation[END_REF]. The presence of tri-fluorinated side groups (for FEP) or the presence of ethylene groups within the monomer unit (for ETFE) can lead to very different degradation mechanisms [START_REF] Ebnesajjad | Fluoropolymer applications in the chemical processing industries: the definitive user's guide and handbook[END_REF],which could be characterized by CL measurements. In the first part, we compare the CL spectra of these three fluoropolymers in order to get a better understanding of the physico-chemical mechanisms steering cathodoluminescence in these materials. In the second part, we investigate the effect of material temperature and cumulative radiation dose (ageing) on the cathodoluminescence spectra of these polymer materials. This parametric study allows identifying the nature of the different emitted contributions and to assess the energy involved in the underlying processes.

II. EXPERIMENTAL SETUP

A. Experimental facility

Cathodoluminescence experiments have been performed with the MARCEL facility installed at ONERA (Toulouse, France). It is equipped (Figure 1) with a 10 keV (STAIB) electron gun. The axis of the gun is normal to the sample plane. The distance between electron gun exit and the sample is about 50 cm. It is possible to work in focused or defocused mode for the local or extended irradiation of the sample. A faraday cup installed on the sample holder can measure the electron flux. The temperature of the sample holder (copper) can be controlled in the range 100 -500 K allowing reproducing the temperature variations of materials on flight. A pumping system allows performing experiments at vacuum of 10 -6 mbar.

B. Cathodoluminescence spectroscopy

A CL bench has been developed within the MARCEL facility. The experimental set-up is described in more detail in a previous paper [START_REF] Demol | Cathodoluminescence of aluminum ceramic compounds[END_REF]. A parabolic mirror has been used to enhance light collection. A UV-Vis-NIR spectrometer (StellarNet BLK-C-SR-50 model) was used to analyze the luminescence. The concave holographic grating spectrometer delivers high performance spectral analysis in the UV-Vis-NIR energy range covering 1.20 -6.2 eV (200-1080 nm). The spectral resolution is determined by the grating and the slit width (50 μm) and is typically 0.5 nm. The SpectraWiz software is used for data acquisition. Integrated times of 10 s have been applied to avoid noisy spectra. The resulting spectra have been deconvoluted into Gaussian sub-bands using commercially available software (Origin 9.1, OriginLab Co., Northampton, MA, USA).

C. Sample preparation

In this study, we focused on three polymers belonging to the fluoropolymer family: PTFE, FEP and ETFE. PTFE and ETFE films have been purchased from Goodfellow. FEP films are Teflon® films from Dupont. The main interest is to understand the influence of the chemical structure on the luminescent properties of these polymers and to analyse possible common trends on the CL spectra due to identical chemical groups (chromophores). In addition, the advantage of cathodoluminescence is that it allows highlighting changes in the chemical structure (unsaturation for example) under electronic irradiation. All measurements have been carried out on 100 µm-thick PTFE and FEP samples, and 125 µmthick ETFE films. 15 nm-thick semi-transparent silver electrodes have been deposited on both surfaces of samples for CL measurements. These electrodes have been connected to electrical ground during irradiation in order to get zero potential on the surface and then have a constant incident energy for the electrons along the irradiation. We demonstrated experimentally that the effect of this coating process on light intensity collected on the irradiated samples is very weak (light absorption by this thin metallic coating is very low).

D. Irradiations conditions

Electrons beam (10 keV -1 µA) in focused mode was used to irradiate the samples at room and low temperature (21°C to -100°C). Pt100 thermistors with an accuracy of ± 0.1°C have been used to monitor the temperature of the material surface. The penetration depth of incident electrons (10 keV) has been estimated at ≈ 1.3 µm for these materials using the Casino software [START_REF] Hovington | CASINO: A new monte carlo code in C language for electron beam interaction -part I: Description of the program[END_REF]. Electron beam diameter is about 2.5 mm. The samples were exposed to very high radiation doses (3x10 6 to 7.5 x10 7 Gy). We analysed the influence of radiation doses and temperature (21°C, -50°C and -100°C) on the profile and intensity of the CL spectra emitted by the three fluoropolymers (FEP, PTFE, ETFE).

III. RESULTS

A. CL spectra of PTFE, FEP and ETFE

Figure 3 (a) compares the CL spectra at room temperature (RT) obtained on PTFE, FEP, and ETFE under electronic irradiation (10 keV -1 µA). For sake of clarity, the spectra presented are normalised (Figure 3 (b)) in regard of the maximum intensity for each material. The CL spectra of these three fluoropolymers are very similar suggesting that the origin of cathodoluminescence processes is directly related to the chemical structure of these polymers. Two large emission peaks are present at ∼ 2.2 eV and ∼ 3.7 eV. The main difference between the three polymers is based on the absence of the ∼ 3.7 eV peak contribution for ETFE (Fig 3 .a). We also note that the intense peak at ∼ 2.2 eV is shifted from one polymer to the other. In addition, this peak is asymmetric, which suggests that it contains two contributions. To identify them, each spectrum has been deconvoluted (Fig. 4), the parameters applied for this curve fit being presented in Table 1. The different energies, Full Width at Half Maximum "FWHM", and maximum intensities involved in cathodoluminescence processes could be determined for each material. The spectral similarities (peak positions and FWHM) indicate that the physical processes involved in CL are certainly quite similar between the three materials. We will discuss in more detail the physical origin of these different peaks in section IV following the parametric study as a function of temperature (III.B) and irradiation time (III.C). To summarize, the analysis of the CL curves revealed 3 contributions P1, P2 and P3 present at the following respective energies: ∼ 2.2 eV, ∼ 2.4 eV and ∼ 3.

B. Influence of temperature

In order to further understand the physico-chemical mechanisms steering the luminescence process from these three fluorine polymers, we analysed the temperature dependence on the CL spectra for these polymers. Experiments have been performed at -100°C, -50°C and 21°C. The evolution of the CL spectra as a function of temperature is shown in Fig. 5. For all materials, we can observe mainly a decline of P1 at ∼ 2.2 eV and P2 at ∼ 2.4 eV with the increasing temperature (less pronounced on ETFE). P3 contribution at ∼ 3.7 eV is rather stable with temperature. No significant shift in the peak energy position was observed with temperature variations. To complete these results, we also have examined the temporal/dose evolutions of the CL spectra at these 3 temperatures.

C. Influence of irradiation time / dose

Fig. 6 shows the irradiation time dependence of the CL emission spectra of PTFE and FEP at 21°C. In this section, we will focus only on the CL evolutions of PTFE and FEP because ETFE present very low luminescence efficiency. In general, all CL spectra decrease with irradiation dose. However, the decline in ICL for FEP is steeper than for PTFE. Indeed, after 250s of irradiation, ICL PTFE (∼ 2.2 eV) = 370 cps and ICL FEP (∼ 2.2 eV) = 230 cps which correspond to a decrease of about 53 % and 70 % respectively.

IV. DISCUSSION

The comparison of the CL spectra of the three fluorinated polymers (PTFE, FEP and ETFE) showed that the radiative transitions could be of the same nature. By deconvolution into Gaussian peaks, we suggested in this paper three contributions placed at ∼ 2.2 eV, ∼ 2.4 eV and 3.7 eV on the CL spectra. The slight differences in the position of the peaks from one material to another are probably due to differences in chemical conformation. This energy shift has already been observed in photoluminescence on ETFE and PTFE [START_REF] Peik | Long-lasting photoluminescence in polymers[END_REF]. For a better understanding, the analysis will be detailed peak by peak.

The contribution at ∼ 2.2 eV is the most intense and present in these three materials. This peak cannot be explained by direct transitions σ* → σ (directly related to the C-F or sp 3 type single bonds present initially in the material). These transitions require excitation energy levels in the order of ∼ 5 eV, which is well beyond the incident energies involved in photoluminescence. Indeed, according to studies carried out on photoluminescence by Scanni et al [START_REF] Scanni | Photoluminescence analysis on Teflon bulk and Teflon-like films grown by Ion-beam sputtering[END_REF] and other authors [START_REF] Pugmire | Cross-linking of polytetrafluoroethylene during roomtemperature irradiation[END_REF]- [START_REF] Yu | The nature of color centers in γ-irradiated poly(tetrafluoroethylene)[END_REF], this emission peak could be related to the electronic de-excitation corresponding to π* → π transition (from unsaturated bonds C=C mainly created by irradiation through chain-scission).

Many authors [START_REF] Ebnesajjad | Fluoropolymer applications in the chemical processing industries: the definitive user's guide and handbook[END_REF], [START_REF] Demol | Cathodoluminescence of aluminum ceramic compounds[END_REF], [START_REF] Scanni | Photoluminescence analysis on Teflon bulk and Teflon-like films grown by Ion-beam sputtering[END_REF]- [START_REF] Yu | The nature of color centers in γ-irradiated poly(tetrafluoroethylene)[END_REF] have investigated the changes in the chemical structure of these three fluorinated polymers (PTFE, FEP and ETFE) under irradiation (gamma, ionic or electronic) and have demonstrated the creation of this type of unsaturated group (C=C) under irradiation. To illustrate this point, we present in Fig. 7 potential reaction mechanisms that may lead to the formation of this type of chemical bond. The reaction mechanisms leading to the creation of C=C bonds in the three fluoropolymers are quite similar [START_REF] Forsythe | The radiation chemistry of fluoropolymers[END_REF], [START_REF] Drobny | Ionizing Radiation and Polymers: Principles, Technology, and Applications[END_REF], [START_REF] Nasef | Electron irradiation effects on partially fluorinated polymer films: Structure-property relationships[END_REF] given their close chemical structures. For this reason, we will only detail the degradation mechanisms induced by irradiation in ETFE.

The electronic irradiation of the macromolecular chain of the polymer can lead to two main processes [START_REF] Forsythe | The radiation chemistry of fluoropolymers[END_REF]: 1. the splitting of the C-F and C-H lateral bonds for ETFE: we speak about dehydrofluorination processes. In the case of PTFE, we only observe cleavage of the C-F bond, whereas for the FEP copolymer, C-F or C-CF3 bonds are involved in the cleavage process ; 2. the homolytic cleavage of a C-C bond present on the main chain. These two cases will, in general, lead to different processes. Indeed, as shown in Figure 7, the process (1) corresponding to the phenomena of dehydrofluorination with the loss of fluorine (F) and hydrogen (H) atoms can lead to cross-linking mechanisms by radical recombination. For the second process (2), electrons beam irradiation leads to chain scission (C-C bonds breakage within the main chain), forming radical species (R-CF2-(C • )F2 or R-CH2-(C • )H2. The interaction of these radical species with the main polymer chain will lead to the loss of a hydrogen or fluorine atom and the formation of unsaturated groups like R-CF=CF2 or R-CH=CH2. Thus, the creation of this double bond could be at the origin of the fluorescence processes involved in ∼ 2.2 eV emission peak. Other mechanisms may of course lead to the creation of unsaturation within the macromolecular chain.

Depending on the irradiation conditions and the chemical structure, the polymers will therefore have either: (1) a tendency to cross-linking or (2) a preference for chain scission which can lead to the creation of unsaturation. This has been demonstrated by Rosenberg et al [START_REF] Rosenberg | Low dose γirradiation of some fluoropolymers: Effect of polymer chemical structure[END_REF] who analysed the influence of the H/F atomic ratio on these competing mechanisms in several fluorinated polymers. It was found that the number of crosslinking nodes per dose unit increases with the increase of the H/F ratio: according to the authors, ETFE tends to crosslink more easily than FEP or PTFE. The hypothesis that has been suggested is that the presence of hydrogen atoms facilitates the formation of radicals leading to cross-linking. If it is therefore assumed that crosslinking processes in ETFE predominate in comparison with FEP, this could explain the low luminescence efficiency observed at ~ 2.2 eV for ETFE (because fewer unsaturated groups like C=C would be observed in ETFE).

From different authors [START_REF] Scanni | Photoluminescence analysis on Teflon bulk and Teflon-like films grown by Ion-beam sputtering[END_REF][START_REF] Pugmire | Cross-linking of polytetrafluoroethylene during roomtemperature irradiation[END_REF][START_REF] Khatipov | Color and fluorescence of polytetrafluoroethylene treated by γ-irradiation near the melting point[END_REF][START_REF] Yu | The nature of color centers in γ-irradiated poly(tetrafluoroethylene)[END_REF], this peak is therefore ascribed to fluorescence process induced by the C=C bonds. The evolution of the intensity of this peak with temperature is due to the fact that fluorescence is dependent on temperature variation. Fig. 7: Schematic representation of mechanism of electrons beam irradiation induced reactions in ETFE [START_REF] Park | Effects of Electron Beam Irradiation on Properties of ETFE Insulated Electric Wire[END_REF].

The weak peak at ~ 2.4 eV can be ascribed to two processes. According to Khatipov et al [START_REF] Khatipov | Spectrophotometric and luminescent analysis of polytetrafluoroethylene treated by γ-irradiation near the melting point[END_REF][START_REF] Nurmukhametov | Luminescence and color of radiation-modified polytetrafluoroethylene in blocks[END_REF], the peak located at ~ 2.4 eV could be attributed to the fluorescence of (CF=CF)n groups (with n = 4 -7) produced by chain scission induced by electron irradiation. The evolution of this peak with temperature is again not surprising since fluorescence can be affected by temperature with an increase of the fluorescence intensity when temperature drops down [START_REF] Valeur | Molecular fluorescence: principles and applications[END_REF].

This peak could also be associated with radiative deexcitation mechanisms derived from carbonyl chromophore groups (C=O) [START_REF] Pugmire | Cross-linking of polytetrafluoroethylene during roomtemperature irradiation[END_REF], [START_REF] Charlesby | The Identification of Luminescence Centres in Polyethylene and other Polymers[END_REF]. According to Charlesby et al [START_REF] Charlesby | The Identification of Luminescence Centres in Polyethylene and other Polymers[END_REF], carbonyl group is present in fluoropolymers network in the form of impurities generated during material manufacturing. These groups would probably be located at the end of the macromolecular chain or a side chain. This carbonyl group can induce phosphorescence of the materials which increases with the decline of temperature. A diagram of this radiative de-excitation is illustrated in Fig. 8. This transition is a threestep process: First, electronic irradiation leads to the excitation of a chromophore center by energy absorption: from fundamental state to an excited state (S0 → S1). This step is followed by an energy inter-system crossing from the singlet state S1 to a metastable triplet state T1 in relation to the carbonyl group. Energy relaxation (transition from the triplet state to the fundamental state: T1 → S0) leads to phosphorescence phenomenon and therefore to the emission peak at ~ 2.4 eV. Although this contribution has been attributed in the literature to carbonyl groups present under impurities form, it may well be possible that these species are also created during irradiation by the formation of hydroperoxide groups (R-OOH) [START_REF] Nasef | Electron irradiation effects on partially fluorinated polymer films: Structure-property relationships[END_REF] due to oxygen dissolved in the polymer (in the same way as polyethylene [START_REF] Fotopoulou | Degradation of Various Plastics in the Environment[END_REF]). The origin of the peak at ∼ 3.75 eV remains relatively uncertain, but could be related to other polyene (CF=CF)n chromophores generated by irradiation [START_REF] Yu | The nature of color centers in γ-irradiated poly(tetrafluoroethylene)[END_REF], as suggested by Khatipov et al [START_REF] Khatipov | Color and fluorescence of polytetrafluoroethylene treated by γ-irradiation near the melting point[END_REF]. Indeed, for ethylene-propylene-hexadiene terpolymer, ESR (electron spin resonance) measurements have also shown that the polyenyl radical -(C⋅)H-(CH=CH)n-can be generated and detected for doses higher than 50 kGy [START_REF] Esnouf | Trapped and transient radicals observed in ethylene-propylene-diene terpolymers[END_REF] (which corresponds to the dose levels injected into the materials during the cathodoluminescence tests). This type of polyene group comes mainly from the prior creation of conjugated diene groups [START_REF] Esnouf | Trapped and transient radicals observed in ethylene-propylene-diene terpolymers[END_REF]. During irradiation, this chromophore would be excited from the fundamental state S0 to a singlet excited state S1. The direct de-excitation of S1 → S0 leads to fluorescence phenomena. Different trends observed experimentally allow us to better understand the involved processes. An interesting point concerns the strong differences in intensity observed between ETFE and the other two materials (PTFE and FEP) on this peak at ∼ 3.75 eV. Indeed, as we have seen in Figure 3, this peak does not appear on ETFE unlike PTFE and FEP. This can be explained by the important role, for luminescence, played by polyene structures of the -(CF=CF)n-type (with n between 4 and 7), these structures being generated by irradiation. In the case of PTFE and FEP, given their structure consisting essentially of C-F groups, the concentration of polyenes of this type (-(CF=CF)n-with n between 4 and 7) is high [START_REF] Nurmukhametov | Luminescence and color of radiation-modified polytetrafluoroethylene in blocks[END_REF], which promotes cathodoluminescence processes. In the case of ETFE, on the other hand, the formation of short length polyene -(CF=CF)n-with values of n lower than 3 (limitation due to cross-linking phenomena) is furthered. According to Hudson et al [START_REF] Hudson | Linear Polyene Electronic Structure and Spectroscopy[END_REF], this type of structure has no luminescent properties, which explains why the intensity of the CL peak at ∼ 3.75 eV is very weak or even non-existent for ETFE. The same trend can be observed for the peak at ~ 2.2 and ~ 2.4 eV, likely to be attributed also to the fluorescence of polyene groups (with different n values): the intensity of these peaks is much lower for ETFE which could also be explained by a lower production of fluorescent polyene groups with n>4.

The last point of analysis concerns the changes in the spectrum with the increasing dose: we were able to observe, for the three materials, a gradual decline in the intensity of the CL peaks with the increasing dose. This observation seems to contradict the physical origin proposed for these three peaks, which would be mainly due to the radiation effects and therefore to the dose. However, as mentioned by Khatipov et al [START_REF] Khatipov | Color and fluorescence of polytetrafluoroethylene treated by γ-irradiation near the melting point[END_REF], irradiation induces also the production of optical centers (OC) that can absorb light emitted by the other fluorescent optical centers. The author mentioned, as demonstrated in this current paper, that different types of fluorescent optical centers can be produced by irradiation but also that the concentration of OC of absorption is higher than the concentration of OC producing fluorescence. The density of absorption OC increases with the radiation dose which hinders on the fluorescence process of the irradiated material leading to a gradual decline of the cathodoluminescence peaks. This process is confirmed by the fact that all materials samples presented a change of colour on the irradiated area.

As a summary, these tests show that the luminescence spectra of fluoropolymers are mainly governed by the largestructured polyenes and isolated double bonds generated by irradiation and by possible interaction processes with oxygen dissolved in the material (1. which partly promotes the formation of carbonyl groups, 2. which could interact with triplet states, particularly at room temperature, to inhibit the phosphorescence of the material). In other words, when the sample is irradiated by high-energy electrons, the absorption of energy leads to the excitation of electrons from the fundamental state (S0) to an excited state (S1) or higher excited states. Relaxation processes can take place within these states (rotational and vibrational) leading to the emission of phonons. The return of the electron present in the S1 state to the fundamental state S0 could generate the emission of photons at ~ 2.2, ~ 2.4 and ~ 3.75 eV by fluorescence characteristic of the polyene groups formed during irradiation. However, excited electrons could also relax through the triplet states generated by carbonyl groups (C=O). The relaxation of these triplet states towards the fundamental state generates phosphorescence processes leading to the emission of energy photons at ~ 2.4 eV. Table II summarizes the different luminescent processes involved in fluorinated polymers. that mainly contribute to the fluorescence of these polymers under electron irradiation. Phosphorescence processes (induced by the luminescence of carbonyl groups (C=O)) could also contribute to cathodoluminescence. In addition, we highlight in this paper the different mechanisms that can lead to luminescence inhibition: formation (in the case of ETFE) of short length polyene groups (-(CF=CF)n-type with n < 3) which produces non-radiative transitions and the creation of color centers that can act as optical filters. The analysis of polymers through cathodoluminescence spectroscopy is very important to get a better understanding of the physico-chemical ageing of these materials under irradiation. This CL technique is an interesting tool for in-situ characterization under vacuum and would avoid any air exposure that could induce recovery effects. Finally, as only few studies have been performed on cathodoluminescence of polymers, this paper provides new data and chemical information on the origin of CL spectra of fluorine based polymers.

Fig. 1 :

 1 Fig. 1: Marcel facility

Fig. 3 :

 3 Fig. 3: (a) Comparison of the CL spectra of PTFE, ETFE and FEP at Room temperature. (b) Normalised spectra with respect to the peak at ∼ 2.2 eV.

Fig. 4 :Fig. 5 :

 45 Fig. 4: Emission spectra of the irradiated film samples with the deconvolution process; P1 (dash red) and P2 (dot green curve); The solid blue curve indicates the summation of the components.

Fig. 6 :

 6 Fig. 6: The irradiation time dependence of the CL emission spectra (PTFE and FEP) at room temperature.

Fig. 8 :

 8 Fig. 8: A possible phosphorescence mechanism leading to the emission peak at around 2.4 eV.

  Cathodoluminescence analysis carried out on the various fluorinated polymers have highlighted the predominant role played by chemical processes on the luminescence of these materials. Irradiation leads to the formation of many unsaturated groups, such as the formation of polyene structures of the -(CF=CF)n-type (with n between 4 and 7)

TABLE I :

 I FIT PARAMETERS FOR THE DIFFERENT PEAKS OBSERVED ON THE CL SPECTRA FOR PTFE, FEP AND ETFE.

			Energy Wavelength FWHM	IMax
			(eV)	(nm)	(eV)	(cps)
		P1	2,2	564	0,58	718
	PTFE	P2	2,4	517	0,18	87
		P3	3,75	330	0,88	88
		P1	2,25	551	0,55	739
	FEP	P2	2,45	506	0,15	80
		P3	3,7	335	0,88	80
		P1	2,15	577	0,58	304
	ETFE	P2	2,45	506	0,17	43

TABLE II

 II

	.
	SUMMARY OF THE DIFFERENT POSSIBLE
	SPECTRAL CONTRIBUTIONS OF
	FLUOROPOLYMERS.

ACKNOWLEDGMENT

This work was supported by the Centre National d'Etudes Spatiales (CNES) throughout the R&T programs. The authors thank the development team for CASINO (monte CArlo SImulation of electroN trajectory in sOlids) and for making available their software to the scientific community.


Paul J T Filippi

Pierre-Olivier Mattei

Cédric Maury

A H P Van Der Burgh

C J M De Jong

P.-O Mattei

Sound transmission through a thin baffled plate: validation of a light fluid approximation with numerical and experimental results

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

The response of a thin elastic rectangular ba%ed plate embedded in a gas and excited by an incident acoustic "eld is studied theoretically and experimentally. The model equations are analyzed by a perturbation method based on the expansion of the solution in eigenmodes and a small parameter. The small parameter expresses that the gas has a density which is small with respect to the density of the plate. The approximation obtained, the so-called light -uid approximation, is compared with the numerical solution of the exact boundary integral equations equivalent to the initial system of partial di!erential equations. Finally, the predictions given by the light #uid approximation are compared with experimental results. These two comparisons show the e$ciency and the accuracy of the method.

INTRODUCTION

Problems of sound radiation or transmission by vibrating structures is one of the oldest problems that acousticians have tried to solve. Several very useful books, providing either analytical solutions of canonic problems or developing analytical and numerical methods have been published during the past 30 years. The oldest one to mention is due to Junger and Feit [START_REF] Junger | Sound, Structures, and their Interaction[END_REF], which was published in 1972 by the Massachusetts Institute of Technology. Then a couple of books appeared during the 1980s [START_REF] Fahy | Sound and Structural <ibration[END_REF][START_REF] Lesueur | Rayonnement Acoustique des Structures[END_REF] which proposed a more modern approach of vibro-acoustics: in particular, numerical methods ("nite elements, boundary element methods, the statistical energy analysis method, etc) are widely developed. The most recent textbook has been published at the end of 1998 [START_REF] Ohayon | Structural Acoustics and <ibration[END_REF]: in our opinion, one of its main originalities is the use of functional analysis which is always required for a rigorous mathematical statement of mechanics problems.

A very fundamental review of the various problems raised by the coupling between a vibrating structure and a surrounding #uid is due to Crighton [START_REF] Crighton | [END_REF]. For a thin plate, the author made use of the following two intrinsic units: the inverse of the coincidence frequency (time unit) and the sound wavelength in the #uid at this frequency (length unit). The corresponding equations were called non-dimensional equations: we would prefer reduced equations, instead, because the number of independent variables on which the physical phenomenon depends is reduced to its minimum. The author mentioned that the reduced coupling parameter *the ratio of the #uid density to the plate surface mass, both expressed in this units system*is dimensionless and less than unity for any data of practical interest: he suggested that this could be advantageous for developing numerical approximations. We do not totally agree with this conclusion.

In the present paper, we look for a numerical approximation of the solution as a Taylor-like series in terms of a small parameter truncated at the "rst order term, which is the ratio of the #uid density to the plate surface mass, both expressed in the same coherent system of units. There is a classical controversy on this point: it is often claimed that this parameter has a dimension*the inverse of a length*while is dimensionless. In fact, the parameter that we use is dimensionless too. Indeed, the variables which appear in the mathematical equations which model a physical phenomenon have two di!erent interpretations. The "rst one is that they represent the physical quantities and, thus, they have a dimension. If quantitative results are looked at, a system of units must be adopted: the choice of a coherent system of units*in the present case, a system based on a length unit, a mass unit and a time unit*leads to the same equations for the measures of the physical quantities in that units system. Thus, the corresponding equations are written and solved for dimensionless variables. As a conclusion, we can say that the only di!erence between the reduced equations and the general form used here is the choice of the system of units.

A second remark is that changing the units system has no advantage. If the Taylor-like series in terms of truncated at the "rst order term provides an approximation with a relative error of %, the Taylor-like series in terms of truncated at the "rst order term has exactly the same relative accuracy, whatever the system of units could be: this is due to the fact that the change of units system is a linear operation.

Finally, we will mention a paper by Norris et al. [START_REF] Norris | [END_REF] in which a method much similar to ours is adopted.

In many situations of environmental noise pollution, the physical phenomenon involved is the radiation of an acoustic "eld by a vibrating structure embedded in air: that is, in a #uid with a density small compared to the structure one. This is the case in buildings where noise is transmitted through a solid structure (brick wall, glass, wood doors, etc.) which separates two rooms. Another example is the #ow noise, inside a high-speed vehicle (car, train, plane) generated by the vibrations of its hull which is excited by a #ow-induced external #uctuating wall pressure.

It is well known that the dynamic behaviour of a structure embedded in a gas is very close to the corresponding in vacuo one. Thus, in most situations, the presence of the surrounding gas is a small perturbating factor. The main in#uence of this factor is an increase of the damping of the in vacuo resonance modes of the structure. For the prediction of this vibro-acoustic behaviour, it is possible to take advantage of the existence of a small factor by using a perturbation method.

In a recent paper [7], the general problem of a #uid-loaded plate has been studied in some detail. Mainly three types of representation of solutions were considered: boundary integral expressions, #uid-loaded eigenmode series representation and resonance mode series expansion. For each of the corresponding equations, a perturbation method has been developed. A rigorous presentation of the eigenmode and resonance mode approaches, mathematically justi"ed and with a physical interpretation was developed in reference [8] for the case of a three-dimensional elastic body. It is assumed here that these results remain true for the thin-plate approximation of the elasticity equations.

In the present paper, we consider the eigenmode series representation of the solution. In section 2, the problem is stated, the eigenmode series representation is brie#y recalled and the perturbation method which gives the #uid-loaded eigenmodes and eigenfrequencies is summarized. The section 3 is devoted to the comparison between the light #uid approximation and the numerical solution of the exact boundary integral equations. The section 4 is devoted to the comparison of the perturbation approximation with experimental results: the experiments have been conducted on a stainless-steel plate mounted in a wall which separates two large-size anechomK c rooms.

STATEMENT OF THE PROBLEM

Consider a thin elastic plate, with thickness h, occupying the domain of the plane z"0 in a three-dimensional space. Along its boundary * , an external unit normal vector n can be de"ned almost everywhere. The ba%e, which occupies the complement of M " 6* , is perfectly rigid. The plate thickness is negligible compared to the wavelengths involved. The two half-spaces >(z'0) and \(z(0) contain a perfect gas.

The mechanical characteristics of the plate are: E"Young's modulus, "the Poisson ratio, D"Eh/12(1! )"rigidity, "plate mass per unit area. The plate is assumed to be clamped along * . The damping of the plate material can easily be introduced by an imaginary part in the Young's modulus: this is a correct modelization of elastic waves absorption in many standard materials, in particular in metals for which the relative value of the Young's modulus imaginary part is O(10\). The #uid is characterized by a density and a sound speed c . The system is excited by a harmonic acoustic source, S\(Q)e\ SR, located in \.

GOVERNING EQUATIONS

Let u(M)"u(x, y) be the plate displacement, positive in the z'0 direction; and let p>(Q)"p>(x, y, z) and p\(Q)"p\(x, y, z) denote the acoustic pressure "elds, respectively, in > and \. The pressure step P(Q)"P(x, y) across the plate is de"ned by

P(x, y)"lim m [p>(x, y, n)!p\(x, y, !n), n'0.
The functions u(M), p>(Q) and p\(Q) satisfy the following system of equations:

( #k)p>(Q)"0, Q3 >, ( #k)p\(Q)"S\(Q), Q3 \, (D ! )u(M)#P(M)"0, M3 , * X p>(M)"* X p\(M)" u(M) 0, M3 , M3 , u(M)"* n u"0, M3* (1)
and a Sommerfeld condition on p> and p\.

GREEN'S REPRESENTATION OF THE PRESSURE FIELDS AND THE

INTEGRO-DIFFERENTIAL EQUATION FOR THE PLATE DISPLACEMENT

Let G S (Q, Q) be the Green function of the Helmholtz equation which satis"es a homogeneous Neumann condition on z"0 and the Sommerfeld condition at in"nity; that is

G S (Q, Q)"! e IP //Y 4 r(Q, Q) ! e IP //Y \ 4 r(Q, Q \ ) ,
where the co-ordinates of the points Q and Q \ are, respectively, (x, y, z) and (x, y, !z).

The pressure "elds can be written as

p>(Q)" u(M)G S (Q, M)d (M), Q3 >, p\(Q)"p\ (Q)! u(M)G S (Q, M)d (M), Q3 \, with p\ (Q)" \ G S (Q, Q)S\(Q)dQ. (2)
By introducing equation ( 2) into the third of equations ( 1), one gets the well-known integro-di!erential equation for the plate displacement:

(D ! )u(M)#2 u(M)G S (M, M)d (M)"p\ (Q), M3 (3)
with the boundary conditions of clamping. As done in reference [7], this equation can be replaced by its weak form (energy form), i.e.,

a(u, v)! +(u, v)! S (u, v),"1p\ , v2 (4)
with

"2( / ) 1u, v2" u(M)v*(M)d (M), a(u, v)"D u v*#(1!v) 2 *u *x*y *v* *x*y ! *u *x *v* *y ! *u *y *v* *x d (M) S (u, v)" u(M)G S (M, M)v*(M)d (M)d (M).
Here v(M) is an appropriate test function, and v*(M) is its complex conjugate. One can recall that if v is replaced by u, the "rst integral is proportional to the kinetic energy of the plate, the second one is its potential energy, and the third one is proportional to the radiated energy.

The de"nition of the appropriate test function space is classical (see any elementary textbook on functional analysis or reference [START_REF] Ohayon | Structural Acoustics and <ibration[END_REF]). Simply recall that the functions v must have the same regularity as the plate displacement u and satisfy the same boundary conditions: the most classical set of test functions is generated by the in vacuo resonance modes of the plate.

EIGENMODE SERIES REPRESENTATION OF THE SOLUTION

AND LIGHT FLUID APPROXIMATION

REPRESENTATION OF THE SOLUTION AS A FLUID-LOADED PLATE EIGENMODE SERIES

The eigenmodes ; L and eigenvalues L of the #uid-loaded plate are de"ned by the eigenvalue problem

a(; L , v)" L +1; L , v2! S (; L , v),, (5)
where ; L is sought in a suitable functional space: in particular, every function ; of this space satis"es the boundary conditions of clamping

;(M)"* n ;(M)"0, M3 .
It can be shown that the ; L satisfy the following orthogonality relationships:

1; L , ;* K 2! S (; L , ;* K )"0 for mOn,
or, equivalently a(; L , ;* K )"0 for mOn.

A pressure "eld can be associated with each eigenmode:

p L (Q)" ; L (M)G S (Q, M)d (M), Q3 $ . ( 6
)
For simplicity, it is assumed that only simple eigenvalues are present. The response of the system can, thus, be expanded into the following series of ; L and p L :

u(M)" L L L ! 1p\ , ;* L 2 a(; L , ;* L ) ; L (M), M3 , p>(Q)" L L L ! 1p\ , ;* L 2 a(; L , ;* L ) p L (Q), Q3 >, p\(Q)"p\ (Q)! L L L ! 1p\ , ;* L 2 a(; L , ;* L ) p L (Q), Q3 \, (7)
These series are de"ned for any real angular frequency , because the eigenvalues L have a non-zero imaginary part. When an eigenvalue has a multiplicity order equal to or greater than 2, the modi"cation of the former formulas is performed in the classical way (see, for example, reference [START_REF] Courant | Methods of Mathematical Physics[END_REF]).

LIGHT FLUID APPROXIMATION OF THE EIGENMODE SERIES

Because the #uid is a gas, the ratio "2 / is, in general, a small parameter, and an appropriate perturbation method can be used. The eigenvalues and eigenmodes are expanded with respect to : it is hoped that the sum of the "rst two terms (zero and "rst orders) provide a good approximation. The authors have followed the method developed in reference [START_REF] Nayfeh | Perturbation Methods[END_REF] called the Rayleigh}SchroK dinger method by Nayfeh. An approach to the perturbation methods, much similar to Nayfeh's one, is presented in reference [START_REF] Sanchez-Hubert | <ibration and Coupling of Continuous Systems[END_REF] where the formalism of functional analysis is emphasized.

Let us brie#y recall the basic principle. The eigenvalues L and the eigenmodes ; L are expanded as follows:

; L "; L # ; L # ; L #2 L " L # L # L #2 .
This expansion is introduced into equation ( 5) and the terms with like power of are made equal. This leads to a countable set of equations, the zero and "rst order ones being

a(; L , v)" L 1; L , v2, a(; L , v)" L 1; L , v2! L S (; L , v)# L 1; L , v2. (8)
The solution of the zeroth order equation is the nth in vacuo eigenvalue and eigenmode. By setting v";* L in the second equation ( 8), it easily follows that the "rst order equation leads to

L " L S (; L , ;* L ) 1; L , ;* L 2 " L ; L (M)G S (M, M); L (M)d (M)d (M) ; L (M)d (M) , (9)
;

L (M)"! O O O L L O ! L S (; L , ;* O ) 1; O , ;* O 2 ; O (M) "! O O O L L O ! L ; L (M)G S (M, M); O (M)d (M)d (M) ; O (M)d (M) ; O (M). (9a)
The corresponding light #uid approximation of the system response is given by

u(M)K L L # L L # L ! 1p\ , ;* L # ;* L 2 a(; L # ; L , ;* L # ;* L ) [; L (M)# ; L (M)], ( 10
)
p>(Q)K L L # L L # L ! 1p\ , ;* L # ;* L 2 a(; L # ; L , ;* L # ;* L ) pL L (M), p\(Q)Kp\ (Q)! L L # L L # L ! 1p\ , ;* L # ;* L 2 a(; L # ; L , ;* L # ;* L ) pL L (M), ( 10a
) with pL L (M)" [; L (M)# ; L (M)]G S (M, M)d (M).
This result can be compared with the type of approximation*outer and inner expansions*which is developed in reference [START_REF] Norris | [END_REF]. Away from any in vacuo resonance frequency of the plate, can be made equal to 0 in expressions [START_REF] Nayfeh | Perturbation Methods[END_REF] and (10a), and one obtains the "rst two terms of the outer expansion. If is made equal to 0, these expressions are not de"ned at each in vacuo resonance frequency of the plate, as is the outer expansion of the cited article. In the neighborhood of in vacuo resonance frequency of the plate, these expressions correspond to the inner expansion.

Let us assume now that there exists one in vacuo eigenvalue with multiplicity order 2: for example P " P> . The corresponding eigenmodes ; P and ; P> are independent functions which can be assumed to be orthogonal. Upon using the perturbation expansions of these eigenmodes and two di!erent perturbation expansions of the double eigenvalue, it appears that equations (8) It must be mentioned that this approximation is valid under restrictive conditions: the "rst one is that the parameter is small compared to unity ( (0)4m\ seems quite correct; it is less obvious to give a criterium in reduced units). A second condition of validity is that the driving frequency must be away from the critical frequency (frequency for which the wavenumber in the plate is equal to the wavenumber in the #uid). In reference [7] it was suggested that the validity conditions of the light #uid approximation do not depend on the plate geometry and/or on the boundary conditions, and are the same for a bounded or an unbounded plate. If this assumption is adopted, the validity conditions to account for are those which apply for the simple situation of an in"nite plate.

EFFICIENCY OF THE LIGHT FLUID APPROXIMATION

A computer program has been developed for the case of a rectangular clamped ba%ed plate [START_REF] Jong | [END_REF]. The in vacuo eigenfrequencies and eigenmodes used in the light #uid approximation are calculated by Warburton's method [START_REF] Leissa | <ibration of Plates[END_REF].

The accuracy and e$ciency of the light #uid approximation is validated in two di!erent ways: in both cases, the system is excited by an incident spherical wave. In the "rst case, one compares the approximate sound level attenuation as obtained by the light #uid approximation with the numerical solution of the exact equations. In the second case, the approximate sound level attenuation is compared with experimental results.

COMPARISON BETWEEN THE LIGHT FLUID APPROXIMATION AND THE NUMERICAL SOLUTION OF THE EXACT EQUATIONS

The exact equations are solved by the method proposed in reference [START_REF] Mattei | [END_REF]. The system of partial di!erential equations is transformed into a system of boundary integral equations for the plate displacement u, the pressure step P and two-layer 0)50 m) in the plane z"0; source at x"0, y"! 1 ) 3m, z"! 7 ) 3m; "rst &&microphone'' at x"0, y"0, z"! 0 ) 5 m; second &&microphone'' at x"0, y"0, z"0)25 m. densities along * to account for the boundary conditions. The unknown functions are approximated by truncated series of Techebyche! polynomials. The coe$cients are solutions of a collocation system of linear algebraic equations which is equivalent to a Ritz}Galerkin system.

The plate material has the following characteristics: E"2)1;10 Pa, "0)33, "15)6 kg/m. The #uid physical properties are c "340 m/s, "1)29 kg/m. The wavelength at the coincidence frequency is A "0)0586 m. The plate dimensions are: (!0)77 m(x(0)77 m, !0)55 m(y(0)55 m); in reduced units, one has: ( !13)14 A (xN (13)14 A , !9)386 A (yN (9)386 A ). The coupling parameter is equal to 0)165 m\, or, in reduced units, to 0)0970 \ A . The point source is located at (x"0, y" !1)3m, z" ! 7)3 m). The di!erence between the sound levels at points (x"0m,y"0m,z"0)25 m) and at (x"0m, y"0m, z" ! 0)5 m) is calculated by both programs. The results are presented in Figure 1.

The frequency step is 1 Hz in the regular parts of the curves and 0)1 Hz around the peaks. The agreement between the two curves is excellent: the relative di!erence between resonance frequencies is O(10\) and the level di!erence is less than 1 dB (let us recall that the relative accuracy of Warburton's approximation of the resonance frequencies is O(10\)). Other calculations have been conducted for various examples and show a similar agreement.

COMPARISON BETWEEN THE LIGHT FLUID PREDICTIONS AND EXPERIMENT

The experiment has been conducted in the twin anechoic rooms of the Laboratoire de MeH canique et d'Acoustique. A large anechoic room is connected to a smaller semi-anechoic room by an aperture in which the plate is clamped. The wall between the two rooms is bare on the semi-anechoic side (almost perfectly rigid surface) and covered with glass-wool wedges on the other side (almost perfectly absorbent ba%e). The sound source is located in the semi-anechoic room. Thus, the experimental conditions are somewhat di!erent from the model problem of a perfectly rigid ba%e on both sides. But it can easily be seen analytically that the transmitted sound "eld close to the central part of the plate is not very much in#uenced by the absorbing properties of the ba%e.

Indeed, the transmitted pressure "eld has the following Green's representation,

p>(Q)" u(M)G S (Q, M)d (M)# * XY p(M)G S (Q,M)d (M),
where the function * XY p(M) is determined by a boundary integral equation expressing, for example, that a Robin condition is satis"ed along :

* XY p(M)# ik * XY p(M)G S (Q, M)d (M)" ! ik u(M)G S (Q,M)d (M).
The "rst term in the representation of p>(M) can be considered as an incident "eld, the source being the vibrating plate. The second term is a di!racted "eld, the incident "eld being the plate radiation: it corresponds to a single-layer potential with a source lying on the ba%e. As a consequence, close to the plate (energy source), the "rst term must be predominant, at least because the secondary source corresponding to the second term lies in the exterior of the plate. Furthermore, the former boundary integral equation shows that this term is a secondary radiation: the function * XY p(M) can be calculated by a classical convergent iterative procedure which leads to * XY p(M)"0 as zero order approximation. This conclusion is, of course, independent of the #uid density.

The experimental plate is made of stainless steel with the dimensions used in the former subsection. Its thickness has been measured at more than 25 points and a mean value h"0)0019 m has been deduced (the thickness #uctuations are about $10\ m).

The rigidity has been determined experimentally by the following procedure. The "rst 10 resonance frequencies of a circular plate with free boundary are measured and the rigidity is adjusted to get the best "t with the theoretical ones: this leads to D"122 N;m. Practically, the resonance frequencies of the circular plate are measured in air. Nevertheless, they are considered as equal to the in vacuo ones, because the main in#uence of the air embedding is the damping of the resonances, the shifting of the real part being a few percent (much less than the error due to the lack of the material isotropy or the thickness #uctuations). The Poisson ratio has been taken equal to 0)33.

The sound source is located at (x"0)26 m, y" !0)17 m, z" !3)0 m); a "rst microphone, which is located at (x" !0)26 m, y" !0)17 m, z" !0)25 m), records the sum of the incident and re#ected "elds; a second microphone, which is located at (x" !0)26 m, y" !0)17 m, z"0)25 m), records the transmitted "eld.

The transfer function between the second and the "rst microphones are measured and computed as functions of the frequency. Figure 2 shows the result obtained with a frequency step of 1 Hz and Figure 3 presents a third-octave mean transfer function.

CONCLUSION AND FINAL REMARKS

The comparison between the light #uid approximation and the experimental results shows the e$ciency of this theoretical approach. The "rst two sources of error in the numerical model are the non-isotropy and the thickness variations of the plate. A third source is that the plate is not an absolutely plane surface: it is, in fact, a very shallow shell (the central de#ection is about 1 cm), with boundaries slightly constrained. These discrepancies between the theoretical model and the experimental device can explain the shifts between the resonance frequencies. The resonance peaks of the computed curve are, in general, sharper and higher than the experimental ones: this is due to the fact that the stainless steel has been considered as perfectly elastic. If a small damping is introduced (for example by using a complex Young's modulus E(1#i ) with 10\( (10\) the peaks levels are much reduced. But, for this particular experiment, it does not seem very useful to work out a much more accurate theoretical modelling; indeed, from a practical point of view, the most signi"cant acoustical properties of the plate are described by its third-octave mean response which is not very much a!ected by all these modelling errors.

In the present work, the light #uid approximation has been developed for the representation of the solution as a series of the #uid-loaded plate eigenmodes. As shown in reference [7], a light #uid approximation can be established for any representation of the solution of the vibro-acoustics problem. In practice, such an approximation is based on the analytical or numerical solutions of two disjoint problems: on the one hand, the response of the in vacuo structure; on the other hand, the non-homogeneous Neumann problem for the Helmholtz equation. At each step of the perturbation method, these two problems are solved alternatively.

In our opinion, the perturbation method is e$cient if the "rst correcting order term is su$cient to provide the accuracy required for the real-life situation under consideration. Indeed, terms of higher orders imply to compute integrals over the structure surface which involve the iterated Green's kernel. Thus, to our knowledge, it is faster to solve the exact equations than to compute the approximate solution up to order 2. Nevertheless, at least for some simple geometries, it is, perhaps, possible to develop an integration algorithm which, being well adapted to the functions to integrate, will be particularly e$cient. But this is a matter of numerical techniques that the authors have not entered in, up to now.

Figure 1 .

 1 Figure 1. Comparison between the light #uid approximate transfer function (} ) } ᭛} ) }) and the &&exact'' one (**): steel plate with thickness 2 mm, occupying "( !0)77 m(x(0)77 m, !0)50 m(y(0)50 m) in the plane z"0; source at x"0, y"! 1 ) 3m, z"! 7 ) 3m; "rst &&microphone'' at x"0, y"0, z"! 0 ) 5 m; second &&microphone'' at x"0, y"0, z"0)25 m.

Figure 2 .

 2 Figure 2. Comparison between the measured transfer function (**) and its light -uid approximation (} ) } ) }).

Figure 3 .

 3 Figure 3. Comparison between the measured mean (third octaves) transfer function and its light -uid approximation ).