Wille Guillaume

Lahondere Didier

Ute Schmidt

Duron Jéromine

Bourrat Xavier

Coupling SEM-EDS and confocal Raman-in-SEM imaging: a new method for identification and 3D morphology of asbestos-like fibers in a mineral matrix

Keywords: situ asbestos diagnosis, Raman-in-SEM imaging, naturally occurring asbestos, 3D analysis, fibrous amphiboles

Asbestos consists in natural minerals crystallized in a specific habit and possessing in particular properties. In the case of Naturally Occurring Asbestos, usual methods applied to the identification of mineral fibers and the determination of their possible asbestiform nature seems not efficient, especially in the case of mineral fibers included in mineral matrix.

We present a new in-situ method based on the use of confocal Raman-in-SEM imaging implemented in a Scanning Electron Microscope as an efficient method for in-situ mineralogy. The limitation of conventional methods is discussed. We applied 2D-Raman imaging to the identification of sub-micrometric fibers included in different mineral matrix. We were able to identify actinolite fibers down to 400 nm in diameter, included in feldspar, quartz and/or calcite matrix.

Moreover, Confocal Raman allows the collection of 3D data that would provide access to critical information on the morphology of the amphibole fibers in the volume, such as aspect ratio, fibers distribution and amphibole volume fraction.

We performed this method on various examples of rocks containing actinolite fibers of mean structural formula is: Na0,04-0,12Mg2,79-3,73Al0,29-0,58K0,01Ca1,79-1,98Mn0,01-0,09Fe 2+ 0,99- 1,91Fe 3+ 0,12-0,25Si7,64-7,73O22(OH)2. We demonstrated that coupling confocal Raman imaging and SEM is a new and efficient in-situ method for identification and morphological characterization of amphibole fibers.

Highlights

New methods are requested for characterizing asbestos fibers in a mineral matrix SEM-Raman imaging is efficient for characterizing mineral fibers in-situ Confocal Raman imaging makes 3D analysis possible 3D analysis provides information on the aspect ratio and volume fraction of asbestos Fibers thinner than 400nm can be identified by confocal Raman in SEM ( = 532 nm)

Introduction

The asbestos group consists of several natural minerals that crystallize according to a specific habit and possess particular properties of flexibility, high tensile strength, and resistance to heat and chemical degradation. Asbestos fibers usually cluster in bundles with separable fibers and/or as long fibers made of small fibrils [1]. The term asbestos refers to a fibrous serpentine called chrysotile and five amphiboles known as the asbestiform varieties of riebeckite (crocidolite), anthophyllite (anthophyllite-asbestos), grunerite (amosite), actinolite (actinolite-asbestos), and tremolite (tremolite-asbestos) [START_REF]Asbestos and other natural mineral fibres, Environmental Health criteria[END_REF][START_REF] Niosh | Asbestos Fibers and Other Elongate Mineral Particles : State of the Science and Roadmap for Revised Edition[END_REF]. In the natural environment, a large set of metamorphic rocks may contain asbestos fibers, such as meta-ophiolitic rocks of basic or ultrabasic compositions (meta-gabbros, meta-basalts, serpentinites) (e.g. [START_REF] Vignaroli | Asbestos fibre identification vs. evaluation of asbestos hazard in ophiolitic rock mélanges, a case study from the Ligurian Alps (Italy)[END_REF][START_REF] Campopiano | Asbestiform minerals in ophiolitic rocks of Calabria (southern Italy)[END_REF][START_REF] Lahondere | Naturally Occurring Asbestos in an Alpine Ophiolitic Complex[END_REF]) and hydrothermally altered dolerites ( [START_REF] Lahondère | TEM and FESEM characterization of asbestiform and non-asbestiform actinolite fibers in hydrothermally altered dolerites (France)[END_REF]), more rarely in talc deposits ( [START_REF] Van Gosen | Using the geologic setting of talc deposits as an indicator of amphibole asbestos content[END_REF]), metamorphosed dolostones and iron formations ( [START_REF] Van Gosen | The geology of asbestos in the United States and its practical applications[END_REF]) and subalkaline plutonic rocks ( [START_REF] Metcalf | Genesis and health risk implications of an unusual occurrence of fibrous NaFe3+-amphibole[END_REF]). Asbestos is listed as a Category 1 human carcinogen, because inhalation of asbestos fibers causes respiratory diseases, in particular asbestosis, lung cancers, and malignant mesothelioma (e.g. [START_REF]Asbestos and other natural mineral fibres, Environmental Health criteria[END_REF][START_REF]IARC monographs on the evaluation of carcinogenic risks to humans: arsenic, metals, fibres and dusts[END_REF][START_REF] Hse | Asbestos: The analysts' guide for sampling, analysis and clearance procedures[END_REF][START_REF] Kazan-Allen | Asbestos and mesothelioma: Worldwide trends[END_REF]). Due to the pathogenicity of asbestos fibers, their use has been banned in most countries around the world. In France, the total ban on asbestos is associated with the decree n° 96-1133 (1997), which notably prohibits the manufacture, processing, sale and import of asbestos fibers. At European level, the ban on asbestos has been the subject of the directive n° 99/77/CE which should be implemented by January, 1 st , 2005 at the latest. The ban applies to natural materials that are quarried and used for public works projects. Compliance with existing rules requires accurate diagnosis of the asbestos or non-asbestos nature of materials likely to be exploited in a quarry or involved in the operation of a construction site.

The use of electron dispersive X-ray spectrometry (EDS) or wavelength dispersive Xray spectrometry (WDS) within a scanning electron microprobe (SEM) or electron probe microanalyzer (EPMA) to analyze asbestiform fibers of serpentines or amphiboles present in a mineral matrix of a mineral mixture is challenging. The main reason is the micrometric to submicrometric diameter of these fibers. In fact, under the usual beam conditions (high voltage = 10 to 20 kV, beam current 5 to 20 nA) used in SEM-EDS or EPMA analysis of silicate minerals, the asbestiform fiber diameter is clearly smaller than the X-ray emission volume (approximately 1.5 -3 µm). As a result, a portion of the X-ray emission detected is generated by the neighboring mineral grains along with the fiber being analyzed (mixed analyses).

The Transmission Electron Microscopy (TEM) is well suited for the determination of morphological and dimensional characteristics, and in conjunction with EDS analysis, for chemical characteristics; and, along with electron diffraction, for crystallographic characteristics of mineral fibers. TEM is considered as the reference method for determining the asbestiform character of fibers in air [START_REF] Steen | Determination of asbestos fibres in air transmission electron microscopy as a reference method[END_REF][START_REF] Harris | Discovering the true morphology of amphibole minerals: complementary TEM and FESEM characterization of particles in mixed mineral dust[END_REF]. However, in the case of mineral fibers contained in a mineral matrix, the use of the TEM requires a complex preparation including the reduction by fragmentation, sampling and milling of a subsample (1 to 2 g), the calcination of the rock powder to remove organic matter and then an acid attack to solubilize the constituents of the matrix such as calcite or gypsum. The test sample (~ 20 mg) should then be mixed with water, sonicated, filtered through a pre-metallized polycarbonate carbon membrane and covered with a second layer of carbon before being examined. The preliminary crushing step causes many difficulties because it can lead to excessive fragmentation of the non-asbestiform three-dimensional amphibole crystals, and then the creation of somewhat elongated fibriform fragments (cleavage fragments) which can be mistakenly identified as asbestos [START_REF] Kauffer | EFFECT OF PREPARATION METHODS ON THE ASSESSMENT OF AIRBORNE CONCENTRATIONS OF[END_REF]. An alternative protocol is based on thinning down a piece of the sample by ionic beam or using a focused ion beam (FIB) in an SEM (SEM-FIB) to sample a thin slice [START_REF] Wirth | Focused Ion Beam (FIB) combined with SEM (FIB/SEM) and TEM: Advanced tools for nano-analysis in Geosciences[END_REF][START_REF] Giffin | Application of BIB-SEM technology to characterize macropore morphology in coal[END_REF][START_REF] Klaver | Application of BIB -SEM technology to characterize microstructure and pores in mudstone at a range of scales[END_REF][START_REF] Seydoux-Guillaume | Nanoscale resetting of the Th/Pb system in an isotopically-closed monazite grain: A combined atom probe and transmission electron microscopy study[END_REF][START_REF] Seydoux-Guillaume | Radiation damage in diopside and calcite crystals from uranothorianite inclusions[END_REF]. These two techniques are efficient but also difficult to achieve and thus costly and not applicable to routine analysis. In addition, only fibers that are elongated parallel to the surface can be investigated. EDS analysis on transmission electron microscope can be used to analyze submicrometric fibers, but quantitative analysis is not as precise as EPMA analysis. Specific methods were developed for analysis of submicrometric objects included in a matrix, such as coupling EDS/WDS measurements with Monte-Carlo simulation [START_REF] Acosta | Monte Carlo simulation of x-ray emission by kilovolt electron bombardment[END_REF] or development of special calculations [START_REF] Armstrong | Quantitative chemical analysis of individual microparticles using the electron microprobe[END_REF], but these methods face a major challenge, which is the knowledge of the shape of the analyzed particle (generally unknown in the case of natural mineral particles/ fibers). Special methods based on the mix of bulk analysis and stratified materials make it possible to produce satisfactory results in specific cases such as grain boundary segregated phases in nickel alloys [START_REF] Tedenac | Identification by Electron Probe Microbeam Analysis of Submicron Borides in Joints of Nickel Base Superalloys[END_REF]. In addition, these methods are not commercially available for routine analysis.

Lowering the high voltage to HV ≤ 5 kV is an analytical method that seems suited to this problem. The use of low voltage is known to notably reduce the analyzed volume down to a submicrometric diameter [START_REF] Small | The analysis of particles at low accelerating voltages (<= 10 kV) with energy dispersive x-ray spectroscopy (EDS)[END_REF][START_REF] Barkshire | High-Spatial-Resolution Low-Energy Electron Beam X-Ray Microanalysis[END_REF]. However, this method encounters/faces major challenges in both EDS and WDS, including the use of L (20<Z<50) and M (Z>50) lines, changes in the peak shape and position in WDS, heightened sensitivity to surface carbon contamination, and uncertainty regarding the Mass Absorption Coefficient (MAC) at low peak energy [START_REF] Merlet | Uncertainty and capability of quantitative EPMA at low voltage -A review[END_REF][START_REF] Gopon | Low-Voltage electron-probe microanalysis of Fe-Si compounds using soft X-rays[END_REF][START_REF] Jonnard | Inter-laboratory comparison of a WDS-EDS quantitative X-ray microanalysis of a metallic glass[END_REF].

Other methods can be applied to the identification of amphiboles, such as X-ray diffraction (XRD), Fourier-transform infrared spectroscopy (FTIR) or thermal analysis [START_REF] Kusiorowski | Thermal decomposition of different types of asbestos[END_REF]. But, these methods do not provide a morphological information needed for the diagnosis, and as mentioned by these authors, since the amosite, crocidolite, actinolite and tremolite are crystallographically very close together, the XRD patterns of the amphibole minerals are similar and not really usable to differentiate amphibole members. For thermal analysis, it can be possible to distinguish the different amphiboles. But, in the case of our samples the objective is to work in-situ on natural rocks (i.e. amphiboles mixed with numerous other minerals), several thermal phenomena could occur on the same thermal range and this can significantly complicate the analysis of the DTA (then it makes DTA not really applicable to « routine » asbestos diagnosis).

We present a new and original in-situ approach of the asbestos diagnostic in natural samples, based on the combined use of conventional petrographic / mineralogical approach and the use of a fully integrated Raman-in-SEM system to perform analyses on the same areas / amphibole fibers. This paper discusses the efficiency of this in-situ analytical method. Conventional petrographic and mineralogical approach use polarized light microscopy (PLM), scanning electron microscopy coupled with SEM-EDS, and electron probe microanalysis (EPMA) for the location, morphological description, and chemical analysis of fibrous or particulate amphiboles. The limits of such conventional methods are discussed. The contribution and advantages of the combination of Confocal Raman integrated into a SEM-EDS are presented as a new in-situ method for the chemical, morphological, and structural identification and description of extremely fine mineral fibers (diameter < 0.5 µm) contained in a natural mineral matrix. We discuss the performance of high-resolution low vacuum Field Emission SEM (LV-FE-SEM) in terms of imaging resolution (down to 1 nm) coupled with EDS qualitative analysis (micrometric resolution), and high-resolution confocal Raman Imaging (resolution close to the diffraction limit, down to 360 nm for a laser wavelength λ = 532 nm [START_REF] Jiruše | Integrating focused ion beam-scanning electron microscope with confocal Raman microscope into a single instrument[END_REF]). We present the data correlation that can be obtained from the results from these techniques in the context of identification, description, and counting of mineral fibers isolated in a natural mineral matrix. We apply this method to multiple samples from various geological environments. We present and discuss 2D (surface) and 3D (volume) analytical results.

Sampling, petrographic and mineralogical study

Sampling

Three samples were collected in France, two from the southern part of the Armorican massif and one from the southwestern part of the Massif Central. The location of the samples is shown on simplified geological map of France (figure 1 after [START_REF] Denner | Les roches dominantes de la France -carte lithologique simplifiée (the dominant rocks of France -simplified lithological map)[END_REF]).

Sample 1 was collected in the southern part of the Armorican massif from a geological formation composed mainly of amphibolites (Pont-Léger and Boufféré Formation [START_REF] Godard G | Explanatory Note, French geological map. France[END_REF]).

These amphibolites are foliated rocks, weakly affected by greenschist recrystallization.

They are locally intersected by feldspathic veins that contain macroscopic crystals of greenish, prismatic amphiboles. Sample 2 comes from the southern part of the Armorican massif from a basic volcanic complex that is more than 1000 m thick and contains underwater flows, breccias, hyaloclastites, tuffs, cinerites, and sills [START_REF] Wyns | Explanatory note and geological map (1/50 000) of Chantonnay (563)[END_REF]. In basalts, hydrothermal overprinting led to crystallization of actinolite at the expense of magmatic pyroxenes [START_REF] Thieblemont | Découverte d'une association de volcanites d'arc et de basaltes de type "MORB" dans la formation paléovolcanique silurienne de La Meilleraie, Vendée[END_REF][START_REF] Thiéblemont | Le magmatisme paleozoique en vendee : apport de la geochimie des elements traces et de la petrologie du metamorphisme a la comprehension du developpement orogenique varisque[END_REF]. Sample 3 was collected in a ten meter thick doleritic sill emplaced within Cambrian schists in the southwestern part of the Massif Central (Monts de Lacaune [START_REF] Demange | Explanatory note and geological map (1/50 000) of Lacaune (987)[END_REF]).

Samples were individually stored in a double bagging to avoid cross-contamination and the spread of fibers in case of the presence of asbestos fibers. Since we have used in-situ methods, the samples were prepared according to the usual methods in petrography, namely the preparation of polished geological thin section. In particular, no grinding was applied and no fiber sampling was performed to avoid any contamination or cleavage fragment creation (that could result in an incorrect diagnosis). Indeed, the grinding of an amphibole sample is likely to form fine and elongated particles, of nanometric diameter, which, with the transmission electron microscope, can easily be confused with true asbestos fibers and for which differentiation criteria are not simple [START_REF] Van Orden | Differentiating Amphibole Asbestos from Non-Asbestos in a Complex Mineral Environment[END_REF].

Petrographic study

Amphibole chemistry (EPMA)

The general chemical formula of the minerals of the amphibole supergroup can be written as A0-1B2C5T8O22W2, where A = Na, K, Ca, Pb, Li; B = Na, Ca, Mn 2+ , Fe 2+ , Mg, Li; C = Mg, Fe 2+ , Mn 2+ , Al, Fe 3+ , Mn 3+ , Cr 3+ , Ti 4+ , Li; T= Si, Al, Ti 4+ , Be and W = (OH), F, Cl, O 2-. Minor elements such as Zn, Ni2+, Co2+, V3+, Sc and Zr may also be observed as C-site cations [START_REF] Hawthorne | Nomenclature of the amphibole supergroup[END_REF]. The amphiboles were analyzed with an electron probe microanalyser (EPMA) CAMECA SXFive (CAMECA, Gennevilliers, France) equipped with five wavelength dispersive X-ray spectrometers (WDS) at BRGM (Orleans, France). Ten chemical elements were analyzed (Si, Ti, Al, Cr, Mn, Fe, Mg, Ca, Na, K).

The operating conditions were accelerating voltage HV=15 kV and beam current BC=12 nA. Samples were coated with carbon under secondary vacuum (thickness = 20 nm) using a CRESSINGTON 308Carbon (Cressington, Watford, UK). K Kα, Ti Kα, Ca Kα and Cr Kα were measured on PET; Fe Kα and Mn Kα, on LiF; and Si Kα, Al Kα, Na Kα and Mg Kα, on TAP. The phi-rho-Z method X-Phi [START_REF] Merlet | An accurate computer correction program for quantitative electron probe microanalysis[END_REF] was used for quantitative calculation. Classification of amphibole chemical analyses was done following the 2012 nomenclature recommended by the International Mineralogical Association [START_REF] Hawthorne | Nomenclature of the amphibole supergroup[END_REF].

Fe3+/ΣFe, Mn3+/ΣMn, OH content, and cation site assignments were determined using the Excel spreadsheet of Locock [START_REF] Locock | An Excel spreadsheet to classify chemical analyses of amphiboles following the IMA 2012 recommendations[END_REF]. Monte-Carlo simulation has shown (CASINO 2.48 software [START_REF] Drouin | CASINO V2.42-A Fast and Easy-to-use Modeling Tool for Scanning Electron Microscopy and Microanalysis Users[END_REF]) that EPMA analyses are possible on fibers whose size exceeds 2 microns in depth / width, i.e. more than 3-4 µm in diameter to account for the polishing effect on fiber dimensions. According to the nomenclature of Hawthorne et al. [START_REF] Hawthorne | Nomenclature of the amphibole supergroup[END_REF], the analyzed amphibole fibers (larger than 3 µm) are classified as calcic amphiboles. Detailed analyses of analyzed amphiboles are given in table 1. Magnesiohornblende and magnesio-ferri-hornblende were mainly identified in sample 1, while actinolite is present in the three samples. Depending of the sample, the actinolite fibers are more or less iron-rich and magnesium-poor. The iron-rich and magnesium-poor actinolite fibers were analyzed in the first sample (15.9-17.8 wt % and 12.5-13.1 wt % respectively). In the others samples, the actinolite fibers have significantly lower iron content (9.8-12.6 wt %) and higher magnesium content (15.9-18.3 wt %). SiO2, Al2O3 and CaO contents do not vary significantly in the three samples. TiO2, MnO, Cr2O3 and Na2O contents are always less than 1 %. Sample 1 contains micron-sized fibers from which acceptable EPMA analyses can be obtained. Conversely, in samples 2 and 3, fibers are mostly extremely thin, with a diameter of less than 1 or 2 μm. For these extremely thin and dispersed fibrils, EPMA analyzes are often unsatisfactory from a stoichiometric point of view, because they most likely incorporate a mixture of the analyzed fiber and matrix minerals. Thus, the EPMA method is quite difficult to use to characterize the chemical composition of asbestiform fibers as defined by EPA [START_REF] Perkins | Method for the Determination of Asbestos in Bulk Building Materials[END_REF]. Figure 3 illustrates these difficulties. In this figure, two fibers from sample 1, one thick (Xa) and the other quite thin (Xf), are contained within an albite matrix. The EPMA analyzes of these two fibers are shown in Table 2. The results obtained from the thick fiber (Xa) correspond to actinolite analyzes. The results obtained from the fine fiber (Xf), on the other hand, show abnormally high contents of SiO2, Al2O3, and Na2O and low contents of MgO, CaO, and FeO. Thus, actinolite can be identified in Xa but not (or only with great difficulty) in Xf.

SEM-EDS and Contribution of Confocal Raman imaging integrated within SEM-EDS (RISE)

Methods

The three samples were observed under SEM using the backscattered detector imaging (BSE) mode. SEM-EDS produces a general description of the sample (matrix, fibers, mineralogical view of the rock), identifies the mineral matrix and areas of interest (areas representative of the sample) for coupled SEM-Raman analysis. SEM-EDS analysis was performed on a LV-FE-SEM TESCAN Mira 3 XMU (TESCAN, Brno, Czech Republic) coupled to an EDAX Team (EDAX, Mahwah, USA) EDS system (EDAX Apollo XPP Silicon Drift Detector).

Raman spectroscopy uses the principle of inelastic scattering of light (monochromatic laser), so-called "Raman effect". Raman is a non-destructive, preparation-free method that can be used to study crystal chemistry and structure. It can be easily applied to thin sections of rock. Because Raman spectra cannot be collected from carbon coated samples, a Raman system integrated in a SEM can be implemented only in a low-vacuum or an environmental SEM [START_REF] Wille | Raman-in-SEM, a multimodal and multiscale analytical tool: Performance for materials and expertise[END_REF]. Information on local chemistry can be obtained at a submicrometric scale using Confocal Raman spectroscopy and Imaging [START_REF] Wille | RISE: Correlative confocal raman and scanning electron microscopy[END_REF][START_REF] Fries | [END_REF]. Amphibole fibers that have the general formula A0-1B2C5T8O22W2 can be identified by confocal Raman spectroscopy using a 532 nm laser wavelength. Several differences are observed on the Raman spectra, depending on the nature of the various cations filling the different sites (A, B, C, T) in the crystal structure [START_REF] Leissner | Exploring the potential of Raman spectroscopy for crystallochemical analyses of complex hydrous silicates: I. Amphiboles[END_REF].

We conducted Raman imaging on three samples in the SEM chamber. The area of interest was located using SEM imaging and EDS microanalysis. Then the SEM stage was used to transfer the sample to the Raman imaging position (automated movement). The advantage of this coupled system is the ability to conduct the different analyses at exactly the same place and on the same fibers (assuming the fact that nanometric fibers cannot been visualized under the optical microscope of a conventional micro-Raman spectroscope). In this way, Raman spectral maps (2 dimensions X,Y (2D)) can be collected and analyzed to identify the mineralogical nature of the fibers. In confocal Raman imaging, Raman spectra are collected point by point in a 2D or 3D array consisting of ten thousands of complete Raman spectra.

Multivariate analysis tools are used to extract information from the acquired spectral arrays [START_REF] Dieing | Software Requirements and Data Analysis in Confocal Raman Microscopy[END_REF]. Raman maps were collected on the three samples in areas previously defined by SEM and optical microscopy. The array size, number of acquired spectra, integration time per spectrum are listed in the corresponding figure captions.

Results

SEM-EDS:

Images of the three samples are presented in figure 4. The images were coupled with qualitative EDS analyses. In fact, as pointed out previously for EPMA analytical resolution, EDS resolution is the same, and the precision of EDS quantitative analysis is much less than WDS (on EPMA). As a result, quantitative EDS analysis is not effective for determining the amphibole nature of a thin fiber. In sample 2 (figure 4 c,d), numerous elongate asbestiform fibers can be observed.

Some fibers occur as bundles that measure several tens of microns wide, and some are isolated in a K-feldspar matrix. Calcite is observed in places.

In sample 3 (Figure 4 e,f), amphibole fibers are observed within different matrix, composed of quartz, epidote, and calcite.

2D confocal Raman imaging:

Figure 5 shows an area of sample 1 analyzed by confocal Raman imaging in SEM (RISE). In this area, actinolite fibers are present within a matrix of albite. The width of the actinolite fibers visible in this area ranges from large fibers of several µm in diameter to thin fibers of about 400 nm in diameter.

An area of sample 2 analyzed by Raman imaging is presented in figure 6. Sample 2 has fine fibers included in a mineral matrix identified as orthoclase by a combination of qualitative EDS and Raman spectroscopy. Calcite is also present in this area.

The area analyzed in sample 3 (figure 7) looks like a loosely organized bundle. Thin fibers are observed in a quartz matrix in combination with calcite particles. The aspect ratio of these fibers is important, then the thinnest fibers can be considered as asbestos fibers.

Contribution of 3D Confocal Raman imaging integrated into a SEM-EDS (RISE)

Introduction and method

Confocal Raman imaging offers the ability to visualize three-dimensional volumes (3D) of the different phases present in a sample. Due to the confocal setup, the focal plane from which Raman spectra are collected can be varied, leading to an optical nondestructive sectioning of the sample. 2D Raman images are collected from the same sample area successively, by changing the focal plane in z-direction. The primary limitation however is the transparency of the matrix to the laser beam.

The 3D volume analysis has several advantages. The first is the ability to describe the particles in the volume, i.e. the true aspect ratio of the particles detected. In fact, The PLO makes it possible to visualize fibers in a transparent volume at micrometric resolution. In contrast, information obtained from the SEM is nanometric, and is only two-dimensional (surface of the sample). As a result, it is not always easy to discriminate between an elongated particle and a true asbestos fiber. 3D mapping produces a precise description of the shape and aspect ratio of the amphibole, and it also offers of accurately identifies the asbestos-containing character of the material. In addition, calculations conducted on the 3D volume can be used to determine the volume fraction of each phase. The results obtained on the three samples are presented below.

The 3D view is obtained by collecting 2D maps equally spaced in Z. The volume is reconstructed using ImageJ software [START_REF] Schindelin | The ImageJ ecosystem: An open platform for biomedical image analysis[END_REF].

Results

Figure 8 presents a 3D view of the area analyzed by 2D RISE (figure 5) in sample 1.

Fibers are visible but the most of them look wide. This can be explained by the presence of a "compact" bundle of fine actinolite fibers or, more likely, the presence of micrometric elongated crystals. In the latter case, a different technique is required to make a determination, for example, either SEM-FIB or TEM observation on an SEM-FIB extracted thin slice. In this case, the 3D view is a useful tool for locating the area of TEM sample extraction.

The Raman volume analysis of sample 2 is presented in figure 9.

The actinolite fibers observed previously in the 2D volume in sample 2 appear to be elongate with a high aspect ratio in the 3D view. This confirms that these are fiber bundles, and the thinnest ones appear to be asbestiform.

The 3D view of the area presented in 2D on figure 10 (sample 3) shows numerous thin fibers with no preferred orientation. The matrix is calcite with quartz. The 3D view shows long fibers and probable fiber bundles, some of which are partially visible at the surface, but others that are not visible. In this case, 3D is useful for revealing the presence of fibers / fiber bundles below the surface of the sample.

Discussion

The identification and morphological characterization of fibrous amphiboles in a mineral matrix is of great interest and of major importance for the extraction and commercial exploitation of stones concerned, for prevention of the naturally occurring asbestos (NOA) hazard, and also for safe extraction and manufacturing of building stones and aggregates at the quarry. The chemical and morphological characterization of very/extremely thin isolated fibers in a mineral matrix requires the use of several microscopic and microanalysis techniques.

To address these considerations, EPMA is an efficient tool that can be used to determine the chemistry of amphibole fibers / particles (diameter > 3 µm).

Unfortunately, the maximum diameter of asbestos fibers of amphiboles is Raman analysis (up to 360 nm @ =532nm [START_REF] Jiruše | Integrating focused ion beam-scanning electron microscope with confocal Raman microscope into a single instrument[END_REF]) let the analyst confirm the mineralogical nature of the fibers [START_REF] Leissner | Exploring the potential of Raman spectroscopy for crystallochemical analyses of complex hydrous silicates: I. Amphiboles[END_REF][START_REF] Petry | Asbestos Mineral Analysis by UV Raman and Energy-Dispersive X-ray Spectroscopy[END_REF] and determine whether this mineral is on the list of regulated asbestos mineral fibers.

RISE microscopy also makes it possible to obtain a 3D view of the distribution of each mineral that is present in a volume of several thousand cubic micrometers. This 3D view can be used to determine the elongation/diameter ratio, a decisive parameter for the "asbestiform" nature of an amphibole particle. This is in fact one of the three diagnostic criteria for asbestos (along with chemistry and crystallography, provided by SEM-EDS and Raman spectroscopy, respectively). Moreover, the 3D view obtained by RISE can reveal the presence of subsurface fibers through a non-destructive analytical determination.

This approach can also be used to meet various needs of the extractive industry. The nature and shape of fibers is a valuable indicator of the fiber emission risk during any step of the aggregate production process (as an example, amphibolite rocks are widely used for roadway construction or as ballast in railroad construction). The 3D view is also the initial step in the determination of the volume content of asbestos fibers in a rock (metal ore, for example).

Conclusion

This paper presents a new in-situ analytical method for the identification of asbestos fibers in their natural state in rocks, based on a combination -within the same analytical system -of SEM-EDS and confocal Raman imaging in the SEM (RISE). The integration of a confocal micro-Raman system in a SEM makes it possible to determine a precise location and to analyze the exact same area by both SEM-EDS a Raman. The spatial resolution of a confocal Raman microscope is much higher than that of EDS and makes it possible to identify mineral fibers thinner than 400 nm in diameter, which is sufficiently accurate for amphibole fibers identification.

The combination of SEM-EDS and confocal Raman imaging leads to a precise in-situ diagnosis of the nature and morphology of amphiboles fibers contained in a mineral matrix, by determination of their mineralogical nature and their asbestiform (or nonasbestiform) morphology, i.e. the determination of the three criteria required to conclude to the presence of asbestos. This diagnosis is performed on polished sections or geological thin sections and any fibers sampling or rock grinding is avoided to eliminate the risk of any contamination and creation of cleavage fragments that could result in an incorrect diagnosis. The use of this combined system makes it possible to perform an accurate in-situ diagnosis on the health risk of rocks that contain amphiboles. This method makes it possible to carry out a diagnosis prior to any operation likely to suspend the asbestos fibers in the atmosphere.

In addition, the 3D volume analysis by Raman can be done in the SEM, due to the confocality of the system. The 3D analysis makes it possible to visualize amphibole fibers below the surface.

  The three sample were observed by polarized light microscopy (PLM) to describe the mineralogy of the samples, and to identify areas of interest containing fibrous / fibrouslike amphiboles. Examples of areas of interest are presented in figure 2. Sample 1 is a dark, banded amphibolite composed of green amphibole (magnesiohornblende), plagioclase (andesine-oligoclase), epidote (clinozoisite), and ilmenite, intersected by a plagioclase-amphibole vein. Matrix amphiboles form large, tridimensional crystals slightly aligned and arranged flat in the banding. Morphological characteristics of fibrous amphiboles associated with plagioclase-bearing veins are shown on figure 2 (S1-A and S1-B). These amphiboles from the vein have fibrous morphologies and form both moderately dissociated elongate particles and extremely thin and sometimes curved fibrils. Sample 2 is a hydrothermally altered basalt intersected by an amphibole-chloritecalcite-K-feldspar vein. Outside the vein, the sample has a pseudomorphic texture characterized by the presence of colorless phenocrysts of clinopyroxene (diopside) partially destabilized in amphibole. Inside the vein, the amphiboles are dispersed and appear completely embedded in calcite and K-feldspar crystals. A view of the contact between the hydrothermally altered basalt and the quartz-calcite-amphibole vein is shown on figure 2 (S2-A). Clinopyroxene crystals are present in the basalt. The amphiboles from the veins form randomly curved bundles with split ends and fibrils, both extremely long and remarkably thin (Figure 2: S2-B). Sample 3 is a dolerite intersected by an epidote-quartz-amphibole vein. Outside the vein, the rock retains a recognizable doleritic texture despite a pervasive greenschist overprint. The matrix consists mainly of crystals of altered plagioclase embedded in an essentially chloritic background. Highly altered and partially chloritized pyroxene crystals are preserved. The polymineralic vein appears zoned, with a quartz-epidote central zone, that grades into an intermediate zone constituted by epidote and amphibole (± chlorite) and then a quartz zone (± amphibole, chlorite) towards the borders. Amphiboles from the vein form brownish, polyfilamentous bundles within the intermediate zone and more or less elongated and thinnest bundles within the quartz crystals of the external zones (Figure 2: S3-A). Fiber bundles regularly exhibit splitted ends and frayed borders (Figure 2: S3-B).

Coupled

  SEM observations / Raman imaging in SEM (RISE) were performed on a Tescan-Witec RISE microscope combining a TESCAN Mira 3 GMU LV-FE-SEM coupled to a WITec Confocal Raman Imaging System (WITec, Ulm, Germany). The RISE confocal Raman imaging system is equipped with a UHTS300 spectrometer, a Zeiss 100x vacuum objective (Numerical Aperture 0.75) mounted inside the SEM chamber, using a 532 nm laser radiation wavelength. Coupled SEM-EDS and Raman imaging were performed on non-coated samples under low vacuum at HV = 15kV under Nitrogen pressure P = 20 Pa.

Figure 4 shows

 4 Figure 4 shows SEM images of different areas of the samples 1, 2, and 3, where amphiboles are present.

Acknowledgements

The authors gratefully acknowledge the BRGM research direction for its help and financial support. This study was supported by the French ministry of ecological and solidarity transition. The RISE acquisitions were performed at the application laboratory of the company WITec in Ulm, with the help and support of Dr. Philippe Ayasse. The authors thank M. Alleki for preparation of geological thin sections at the BRGM thin section lab. The authors also gratefully acknowledge F. Chene and J-P Quinquis for the maps and notices.


Kévin Billon

Morvan Ouisse
email: morvan.ouisse@femto-st.fr

Emeline Sadoulet

Manuel Collet

Pauline Butaud

Gael Chevallier

Abdelkrim Khelif

E Sadoulet-Reboul

Design and experimental validation of a temperature-driven adaptive phononic crystal slab

Keywords: Periodic structures, Metamaterial, Dispersion, Dissipation, Vibroacoutics

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.

INTRODUCTION

A metamaterial is an artificial multiscale architectured material designed to control and manipulate waves in gases, liquids or solids. This paper focuses on elastic wave porpagations. In the case of elastic waves, metamaterials can exhibit particular elastodynamic properties that are not found in a natural material. The term appeared in 1999. However, physics governing their behavior was developed in the 1960s by physicist Viktor Veselago. [START_REF] Veselago | The electrodynamics of substances with simultaneously negative values of epsilon and mu[END_REF] It was not until the 2000s that the first experimental realization 2, 3 was achieved. Mechanical metamaterials have been recently hailed as a new class of structural concept able to bring novel multifunctionalities [START_REF] Kadic | Metamaterials beyond electromagnetism[END_REF] by changes of compliance, shape, or by embedding oscillators or smart material inserts. Some examples (to name a few) are multiscale architecturally structured topologies, [START_REF] Lee | Micro-/nanostructured mechanical metamaterials[END_REF] zig-zag folded sheets, [START_REF] Eidini | Unraveling metamaterial properties in zigzag-base folded sheets[END_REF] pentamodal lattices, [START_REF] Kadic | On anisotropic versions of three-dimensional pentamode metamaterials[END_REF] systems with distributed resonators, [START_REF] Huang | Wave attenuation mechanism in an acoustic metamaterial with negative effective mass density[END_REF][START_REF] Nouh | Wave propagation in metamaterial plates with periodic local resonances[END_REF] smart/magnetic materials, [START_REF] Grima | Smart metamaterials with tunable auxetic and other properties[END_REF] tunable connectivity, 11 phononic stubbed plates, [START_REF] Oudich | Experimental evidence of locally resonant sonic band gap in two-dimensional phononic stubbed plates[END_REF] flat lenses (super lenses), [START_REF] Al-Lethawe | All-angle negative refraction for surface acoustic waves in pillar-based two-dimensional phononic structures[END_REF] 3D tunable phononic crystals, [START_REF] Wang | Mechanically tunable phononic band gaps in three-dimensional periodic elastomeric structures[END_REF] tunable metamaterial beam with shape memory alloy resonators, [START_REF] Candido De Sousa | Adaptive locally resonant metamaterials leveraging shape memory alloys[END_REF][START_REF] De Sousa | Tunable metamaterial beam with shape memory alloy resonators: Theory and experiment[END_REF] auxetic periodic structure (with negative Poisson ratio) [START_REF] Billon | Mechanics and band gaps in hierarchical auxetic rectangular perforated composite metamaterials[END_REF][START_REF] Dalessandro | 3D auxetic single material periodic structure with ultra-wide tunable bandgap[END_REF] and nonlinear auxetic dampers. [START_REF] Ma | A nonlinear auxetic structural vibration damper with metal rubber particles[END_REF] A phononic crystal is a metamaterial with a periodic structure, that exhibits spatial periodicity. Historically, they have been introduced in order to extend to the domain of elastic waves phenomena highlighted during the propagation of electromagnetic waves in photonic crystals. When photonics and phononics effects cross and there is a change in the propagation of elastic and optical waves, the term phoXonic crystal is used. [START_REF] Laude | Simultaneous guidance of slow photons and slow acoustic phonons in silicon phoxonic crystal slabs[END_REF] The control of the elastic waves can be performed by combining Bragg's bandgap [START_REF] Bragg | Bakerian lecture: X-rays and crystal structure[END_REF] (wave interferences), resonant's bandgap (resonance of a component embedded in the unit cell), damping and/or active control. The first occurs when the wavelength is near with the characteristic length of the periodic network. The local resonance phenomenon occurs when the frequency of the wave corresponds to the resonance frequency of the resonator. The periodic structures can stop the wave propagation, but can also lead to other properties such as confining waves or guiding waves in a particular direction or guiding along a chosen path of propagation. These phenomena occur at a longer wavelength than the crystal period. The energy can then be reflected, transmitted, damped, focused or confined in a specific zone of the structure. However, the practical realization of real-life 2D or 3D finite systems may lead to some situations where energy transfers are not in accordance with those predicted by the infinite system considered in the design, because of reflections on the boundaries conditions of the finite structure.

On the other hand, the study of periodic structures has a long history in the field of vibrations and acoustics. [START_REF] Hussein | Dynamics of phononic materials and structures: Historical origins, recent progress, and future outlook[END_REF] This topic has interested researchers over the years, and a growing activity on this field is observed on the last decade, with the objective of designing structures exhibiting properties that conventional ones cannot possess. [START_REF] Billon | Mechanics and band gaps in hierarchical auxetic rectangular perforated composite metamaterials[END_REF][START_REF] Shan | Design of planar isotropic negative Poissons ratio structures[END_REF][START_REF] Slann | Cellular plates with auxetic rectangular perforations[END_REF] The methods currently used are most of the time based on those derived from wave propagation in crystals, [START_REF] Joannopoulos | Photonic crystals: molding the flow of light[END_REF] where almost no dissipation occurs. Reaching the upper scale for structural dynamics implies that damping effects have to be included in the analysis.

In this paper, first, some numerical tools for dispersion analysis of periodic structures are presented. The classical Floquet-Bloch approach is presented, as a reference. This technique uses proper boundary conditions on the unit cell, but dealing with damping is not easy for 2D or 3D cases. Secondly, the shifted-cell operator technique is described. It consists of a reformulation of the partial differential equation (PDE) problem by shifting -in terms of wavenumber-the space derivatives appearing in the mechanical behavior operator inside the cell, while imposing continuity boundary conditions on the borders of the domain. Damping effects can be introduced in the system. This strategy makes it possible to solve the problem with an arbitrary frequency dependency of the physical properties of the cell. A focus is proposed on tools for the post-processing of dispersion diagrams in damped configurations, e.g., group velocity. Third, an adaptive metamaterial based on the combination of metallic parts with a highly dissipative polymeric interface is designed. In order to validate the design and the adaptive character of the metamaterial, results issued from a full 3D model of a finite structure embedding an interface made of a distributed set of the unit cells are presented. After this step, a comparison of the results obtained using the tunable structure simulation and the experimental results is presented. Finally, a supercellbased approach is proposed to handle finite system boundary conditions in order to be able to identify situations in which energy transfer may arise because of reflections on the border of the elastic domain. Computations are performed on a two-cell with adequate boundary conditions. The methodology is described and validated using the full finite model and experimental tests on a 2D metamaterial structure.

NUMERICAL TOOLS FOR EFFICIENT SIMULATIONS OF WAVE PROPAGATION IN DAMPED PERIODIC STRUCTURES

Reference structure

The reference structure used in this paper is presented in reference. [START_REF] Wu | Waveguiding and frequency selection of Lamb waves in a plate with a periodic stubbed surface[END_REF] The system consists of an infinite 3D periodic bidirectional waveguide shown in figure 1a. It is a 1 mm thick plate with periodic cylindrical pillars made of isotropic aluminium 6063-T83 (ν = 0.33, E = 69e9[P a] and ρ = 2700[kg/m 3 ]). The unit cell and the corresponding first Brillouin zone are shown in figure 1b.

Classical method based on Floquet-Bloch theorem

The Floquet-Bloch approach is a method commonly used for the analysis of periodic structures. The material is supposed to be linear, elastic and isotropic. For a 2D periodic structure, the harmonic homogeneous dynamical equilibrium of the system is characterised by the following partial differential equation

ρ(x)ω 2 u(x) + ∇ [C(x).∇ sym (u(x))] = 0, (1)
where u(x) ∈ R 3 is the displacement vector, C(x) is the elastic tensor, ε(x) = ∇ sym (u(x)) = 1 2 (∇u T (x) + u(x)∇ T ) is the strain tensor, ρ is the material density and ω is the circular frequency.

The periodicity is defined on the borders of the domain using the Floquet-Bloch boundary conditions, that write u R = e -jkxr1 u L and v R = e -jkyr2 v L where u R (resp. v R ) is the displacement on the right border and u L (resp. v L ) is the displacement on the left border in x (resp. y) axis, k x and k y are respectively the wavenumbers in the x and y directions [START_REF] Floquet | Sur les équations différentielles linéaires à coefficients périodiques[END_REF][START_REF] Bloch | Über die quantenmechanik der elektronen in kristallgittern[END_REF] as shown in figure 1b. A parametric eigenvalue analysis is performed using the Pardiso solver, [START_REF] Schenk | Solving unsymmetric sparse systems of linear equations with Pardiso[END_REF] two parameters (wavenumbers) are considered, namely k x = [0 π/r 1 ] and k y = [0 π/r 2 ]. The eigenfrequencies are obtained by solving the problem. The wave's dispersion curves of the undamped system are plotted on the whole first Brillouin zone (figure 2a) and on the contours (figure 2b). The frequencies defining the bandgaps can be found by considering only the contour of the irreducible Brillouin zone for non-singular system. [START_REF] Joannopoulos | Photonic crystals: molding the flow of light[END_REF] It is worth noticing that this property is extensively used in open literature, although no formal proof of its validity is available, and therefore the results obtained need to be checked carefully. [START_REF] Craster | Dangers of using the edges of the Brillouin zone[END_REF] So, figure 2b is sufficient to observe partial bandgaps along x direction and complete bandgap in all the directions. It appears that the structure exhibits partial bandgaps for frequencies around 20 and 100 kHz in some specific directions and a complete bandgap between 114 and 143 kHz. In this formulation, the wavenumbers are parameters of the eigenvalue problem, the solutions being the frequencies. Therefore, handling frequency-dependency of the mechanical properties is not easy, since it requires the resolution of a nonlinear and non polynomial eigenvalue problem, or condensation procedures providing ill-conditioned matrices. [START_REF] Manconi | Estimation of the loss factor of viscoelastic laminated panels from finite element analysis[END_REF][START_REF] Mace | Modelling wave propagation in two-dimensional structures using finite element analysis[END_REF] Alternative procedures can be found in the literature, such as ref., [START_REF] Krushynska | Visco-elastic effects on wave dispersion in three-phase acoustic metamaterials[END_REF] where the damping provided by a generalized Maxwell model is included in the stiffness matrix, and ref., [START_REF] Palermo | Extended bloch mode synthesis: Ultrafast method for the computation of complex band structures in phononic media[END_REF] where an EBSM (Extended Bloch Mode Synthesis) with modal reduction is applied for fast calculation. The next section recalls a suitable method for handling frequency-dependency in dispersion analysis.

Shifted-cell operator method

The shifted-cell operator, alternative formulation proposed by, [START_REF] Bensoussan | Asymptotic analysis for periodic structure[END_REF][START_REF] Wilcox | Theory of Bloch waves[END_REF][START_REF] Collet | Floquet-Bloch decomposition for the computation of dispersion of two-dimensional periodic, damped mechanical systems[END_REF] is recalled in this subsection with details on its numerical implementation and the link between left and right eigenvalue problems.

Numerical implementation

The shifted-cell operator [START_REF] Collet | Floquet-Bloch decomposition for the computation of dispersion of two-dimensional periodic, damped mechanical systems[END_REF] technique consists of a reformulation of the PDE problem by shifting -in terms of wavenumber-the space derivatives appearing in the operators inside the cell, while imposing continuity boundary conditions on the borders of the domain. The formulation leads to the following eigenvalue problem :

[(K -ω 2 M) + λ i (L -L T ) -λ 2 i H]φ r i = 0, (2)
where λ i = jk i is the i -th eigenvalue, φ r i is the right eigenvector associated to λ i , M and K are respectively the standard symmetric definite mass and symmetric semi-definite stiffness matrices, L-L T is a skew-symmetric matrix and H is a symmetric semi-definite positive matrix for details). In this formulation, all matrices can depend on ω. A parametric eigenvalue analysis is then performed where the pulsation ω and the wave propagation angle θ (such that k x = k cos θ and k y = k sin θ) are fixed as real parameter, allowing introduction of damping effects. The wavenumbers λ i = jk i and the associated right eigenvectors φ r i are computed by solving the quadratic eigenvalue problem (equation 2).

For the computation of the group velocity that will be described in the next section, both left and right eigenvectors of the non-symmetric eigenvalue problem are required. We propose to rewrite (equation 2), using the state-space, as

0 I d (K -ω 2 M) L φ r i λ i φ r i = λ i I d 0 L T H φ r i λ i φ r i , (3)
which is denoted

A 1 (ω)ψ r i = λ i A 2 (ω)ψ r i , (4)
where

A 1 (ω) = 0 I d (K -ω 2 M) L , (5)
A 2 (ω) = I d 0 L T H , (6)
and

ψ r i = φ r i λ i φ r i . (7)
Conversely, a left-eigenvector for the same eigenvalue satisfies

ψ l i T A 1 (ω) = λ i ψ l i T A 2 (ω). ( 8
)
Equation 8 can be developed to find the link between ψ r i and ψ l i . After a few steps the following relationship is obtained :

ψ l i = λ i (H T φ l i ) -L T φ l i φ l i = λ i (H T φ r -i ) -L T φ r -i φ r -i . (9)
Hence, by solving the right eigenvalue problem, the i-th mode (i ∈ N * ) is obtained, λ i ∈ C being the eigenvalue associated to the right eigenvector φ r i . For each mode i, a mode -i is associated with λ -i such that λ -i = -λ i and φ r -i such that φ r -i = φ l i . So, by solving the right eigenvalue problem, the adjoint solution is found too.

Group velocity

In this section, the expression of the group velocity is derived. It will be used in the next section, on the one hand, as a sorting criterion for distinguishing a propagating wave from an evanescent wave and, on the other hand, as an indicator for branch tracking.

For frequency-dependent systems, the estimation of the group velocity is not trivial. [START_REF] Moiseyenko | Material loss influence on the complex band structure and group velocity in phononic crystals[END_REF][START_REF] Laude | Effect of loss on the dispersion relation of photonic and phononic crystals[END_REF] Equation 4 is differentiated and multiplied by the left eigenvector ψ l i T following the procedure proposed by [START_REF] Moiseyenko | Material loss influence on the complex band structure and group velocity in phononic crystals[END_REF] such that

ψ l i T ∂A 1 (ω) ∂ω ψ r i + ψ l i T A 1 (ω) ∂ψ r i ∂ω = ψ l i T ∂λ i ∂ω A 2 (ω)ψ r i + ψ l i T λ i ∂A 2 (ω) ∂ω ψ r i + ψ l i T λ i A 2 (ω) ∂ψ r i ∂ω . (10)
According to equation 8,

ψ l i T A 1 (ω) ∂ψ r i ∂ω = ψ l i T λ i A 2 (ω) ∂ψ r i ∂ω , (11)
hence

ψ l i T ∂A 1 (ω) ∂ω ψ r i = ψ l i T ∂λ i ∂ω A 2 (ω)ψ r i + ψ l i T λ i ∂A 2 (ω) ∂ω ψ r i . (12)
After several steps, isolating the ∂λi ∂ω expression , the following formulation is obtained :

∂λ i ∂ω = ψ l i T ∂A1(ω) ∂ω -λ i ∂A2(ω) ∂ω ψ r i ψ l i T A 2 (ω)ψ r i . (13)
In this case, the group slowness (see section 6 for details) is expressed as

S g = ∂k i ∂ω = -j φ l i T [-2ωM + ∂K ∂ω -λ i ∂L T ∂ω + λ i ∂L ∂ω -λ 2 i ∂H ∂ω ]φ r i φ l i T [-L + L T + 2λ i H]φ r i . (14)
The complex group velocity is the inverse of the group slowness

C g = 1 ∂ki ∂ω = 1 -j φ l i T [-2ωM + ∂K ∂ω -λi ∂L T ∂ω +λi ∂L ∂ω -λ 2 i ∂H ∂ω ]φ r i φ l i T [-L+L T +2λiH]φ r i . ( 15
)
This expression is consistent with the estimation of the energy velocity v = I/E 40 where I is the flow of energy and E is the total mechanical energy density.

Sorting criteria for distinguishing a propagating wave from an evanescent wave

The problem being damped, all the wavenumbers are complex, and the distinction between "propagative" and "evanescent" waves becomes difficult. This is why sorting criteria are proposed in the following.

First of all, all the waves are shifted to the first irreducible Brillouin zone. Then, criteria are defined as:

• ratio between real and imaginary part of each wavenumbers

C 1 = real(k)/imag(k); (16)
• ratio between real and imaginary part of the velocity of energy transport 40 v = I/E where I is the flow of energy and E the total mechanical energy density, approximated from the kinetic energy

E = 2E c C 2 = real(v)/imag(v); (17)
• ratio between real and imaginary part of the group velocity 38

C 3 = real(C g )/imag(C g ). ( 18
)
Only the waves corresponding to

C 1 > τ 1 ; C 2 > τ 2 et C 3 > τ 3 are considered propagative.
In practice, the thresholds τ 1 , τ 2 et τ 3 are chosen such as τ 1 = τ 2 = 1; τ 3 = 2. This is an arbitrary choice that provides good results for the cases investigated by the authors. However, for others cases, alternative values of the thresholds may be required.

Shifted-cell operator implementation validation for conservative case

The reference structure (figure 1a) is used. Figure 3 presents dispersion curves along the Γ-X (φ = 0 • ) direction obtained with the shifted-cell operator method for conservative case with raw results in black (unsorted results) and sorted results in blue, the real part of wavenumbers is represented on the line from 0 to 1 and the imaginary part of wavenumbers from 0 -1. Sorted results are obtained after the application of sorting criteria describe above. No damping being used in the simulations, consequently, the propagating branches possess purely real wavenumbers hence the blue points along the ordinate axis (imag(k) = 0).

A comparison between the results obtained using the method based on the Floquet-Bloch theorem and the shifted-cell operator is first performed on a conservative structure to validate the implementation of the shiftedcell operator technique. Figure 4a presents the dispersion curves obtained along the Γ -X direction when φ is equal to zero. Shifted-cell operator and Floquet-Bloch results are respectively in blue dots and dotted lines. Both methods lead to similar results and the shifted-cell operator method is thus verified.

It can be noted that around 70 kHz on the red dotted lines, the branches seem to get closer without crossing, there is indeed a crossing from which a method of branch tracking is developed in the next subsection 2.3.5.

Branch tracking

The dispersion diagrams being obtained point by point, the tracking of the branches is required to identify automatically the curves corresponding to each physical mode. The group velocity constitutes a pertinent indicator for the branch tracking from one computational point to another. In order to do that, an algorithm allows to compare the computed group velocity ∂ω ∂ki computed by equation 34 and the post-processed group velocity ∆ω ∆ki which is evaluated using finite difference.

Solutions for associating points on the basis of the nature of the displacement field to identify the branches are proposed in the literature, MAC criteria or eigenvectors orthogonality. [START_REF] Collet | Floquet-Bloch decomposition for the computation of dispersion of two-dimensional periodic, damped mechanical systems[END_REF] These methods need to store the eigenvectors at each step, which may correspond to a lot of data. An alternative is proposed here on the basis of the group velocity value. This indicator ensures the correct plotting of the dispersion diagram, particularly in the case of crossing branches, veering or bifurcation. [START_REF] Morand | Interactions fluides-structures, Issy-les-Moulineaux[END_REF] The indicator operates from left to right (figure 5) : starting from a (ω, k) point with known group velocity ∂ω ∂ki , the routine compares this velocity with all the post-processed group velocities by finite difference of the right-hand points. A minimization is done to identify the point with the post-processed group velocity closest to ∂ω ∂ki . This point is defined as next point and step by step the branch is identified. The sorting algorithm has been validated with damping. For example, in figure 6, around 70 kHz, there is a cross between the light blue branch (n • 6) and the green branch (n • 5). The branches on this diagram are determined from the figure 7, showing the evolution of the group velocity with a color code corresponding to the dispersion diagram of the figure 6. It can be seen that the group velocity goes to zero near the upper or the lower frequency of the band gaps, meaning that the energy distribution becomes stationary.

DESIGN AND VALIDATION OF A TEMPERATURE-DRIVEN ADAPTIVE PHONONIC CRYSTAL SLAB

Description of the concept

The structure presented in section 2.1 is modified in order to obtain an adaptive structure. As shown in figure 8, the pillars are now made of combination between a highly dissipative polymer tBA/PEGDMA [START_REF] Butaud | Contribution à l'utilisation des polymères à mémoire de forme pour les structures à amortissement contrôlé[END_REF] and aluminum 6063-T83.

As it will be shown in the next section, the polymer thickness has been chosen in order to open a resonant bandgap around 40 kHz, which means below the Bragg bandgap frequency. The polymer has a density ρ poly = 1004[kg/m 3 ] and a Poisson ratio ν poly = 0, 37. The elastic properties are highly dependent on the temperature and the frequency.

In the harmonic regime, the complex modulus of the polymer writes

E * poly (ω, T ) = E + jE = E (1 + jtan(δ)), (19)
where tan(δ) = E /E is the loss factor, E is the storage modulus and E is the loss modulus. In this work, a fractional derivative Zener model is used. The expression of the elastic complex modulus is

E * poly (ω, T ) = E 0poly + E ∞poly (jωτ (T )) α 1 + (jωτ (T )) α , (20)
where E 0poly et E ∞poly are respectively the static elastic modulus and the asymptotic value of the dynamical modulus, τ is the relaxation time and α is the order of fractional derivative. The values of the four parameters E 0poly , E ∞poly , α and τ is obtained by the fitting of experimental measurements [START_REF] Butaud | Experimental investigations on viscoelastic properties of a shape memory polymer[END_REF][START_REF] Butaud | Sandwich structures with tunable damping properties: On the use of shape memory polymer as viscoelastic core[END_REF] (tableau 1).

The translation factor a T comes from the Williams-Landel-Ferry law (WLF), [START_REF] Williams | The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids[END_REF] the temperature dependency of a T is described by

log 10 (a T (T )) = -C 0 1 (T -T 0 ) C 0 2 + (T -T 0 ) , (21)
with C 0 1 et C 0 2 the law constants for reference temperature T 0 = 40 • C (C 0 1 = 10, 87 and C 0 2 = 32, 57 K). The relaxation time τ is linked to the translation factor :

τ (T ) = a T (T )τ 0 . (22)
The storage modulus and the loss factor are plotted in figure 9a and figure 9b for two temperatures of interest, namely 25 • C and 90 • C. At ambient temperature, the polymer is stiff and its loss factor is almost equal to zero. At 90 • C, the loss factor is over 1.5 on the frequency range (0 to 100kHz), with a maximum value of 2.5 and the stiffness is very low.

Dynamical properties 3.2.1 Dispersion analysis

The shifted-cell operator technique is used to obtain the dispersion curves along the Γ -X direction. Damping is included in the analysis using fractional derivative Zener model presented above.

Figures 10a to 10c

show dispersion curves along the Γ -X direction obtained with the shifted-cell operator method. A comparison between the results obtained using the reference structure and the adaptive structure is presented. The two states are obtained by changing the temperature of the polymeric interface: at 25 • C, the resonant bandgap is visible around the selected frequency (40 kHz), as defined during the design of the resonator. Above the glass transition of the polymer, the phononic crystal slab tends to behave as an homogeneous damped plate.

The group velocity associated to dispersion curves along the Γ -X direction at ambient temperature 25 • C and 90 • C respectively, using branch tracking shown in the subsection 2.3, is plotted figure 11a and11b. At this step, we have used dispersion analysis to design the temperature-driven adaptive phononic crystal slab, which is expected to follow the physical behaviour described by the diagrams. The purpose of the next section is to check the validity of the design when the crystal is embedded in a finite structure.

Embedding of the crystal in a finite structure : numerical modelling

The crystal (7 × 7 cells) is used as interface between two plates, as shown in figure 12. Cell size is 1 × 1 cm 2 , the total size of the structure is 21 × 7 cm 2 . finite element model of the structure is build (with quadratic Lagrange elements). The structure is free. A piezoelectric patch (transversal excitation) with harmonic voltage (|U | = 100 V ) is included in the model in order to be close to the experimental set up and covers all the frequency range from 0 to 50 kHz. Figure 13 shows numerical frequency responses obtained with the finite elements model for the two temperatures of interest. Squared velocity amplitudes |V z| 2 are averaged on the domain of the input plate (IN) and the output plate (OUT). Blue and red curves are numerical frequency response at ambient temperature 25 • C for the input plate (IN) and the output plate (OUT), respectively. Yellow and purple curves correspond to the same quantities at 90 • C. The grey shape represents the bandgap predicted by the dispersion analysis at ambient temperature. At ambient temperature 25 • C, an output attenuation around 40 kHz is observed. Its width (about 3.1 kHz) is smaller than the frequency range predicted by the bandgap (∆f = 8.9 kHz). At 90 • C, after the glass transition, the output attenuation is not visible anymore.

In order to understand this unexpected bandgap width reduction, the vibration field is investigated at 4 frequencies of interest, illustrated in figure 14 : a first point below the bandgap (34.4 kHz), two points in the expected bandgap (39.2 kHz, 43.3 kHz) and the last one above the bandgap (at 50.0 kHz). The corresponding operational deflections are shown in figures 15a, 15b, 15c and 15d.

As expected, energy can propagate through the lattice at 34.4 kHz (figure 15a) and 50.0 kHz (figure 15d), these two frequencies being outside of the predicted bandgap. At 39.2 kHz (figure 15b), inside the bandgap, the attenuation is clearly visible, all the energy is confined in the input plate (IN) : the stop band effect is obtained. However, at 43.3 kHz (figure 15c), the lattice is not efficient despite the fact that the frequency of interest is located inside the predicted bandgap. The operational deflection field clearly shows that in the central part of the crystal the vibration level is close to zero while the energy propagates along the edges of the periodic lattice : this phenomenon can not be predicted by the dispersion analysis. This phenomena is experimentally investigated in the next section.

Experimental validation

Figure 16 shows the structure which has been tested. Polymer parts are realised by laser cutting and metallic parts by classical mechanical processing. The metamaterial is assembled in two steps by bonding parts together.

The polymer cylinders are glued on the support aluminum plate. After drying and loading processes, at ambient temperature during 24 hours, aluminum cylinders are glued on the polymer cylinders. A piezoelectric patch (sintered ceramic PI, d31 effect, 8 × 7 × 0.2 mm 3 ) provides the harmonic excitation up to 50 kHz.

Figure 17a shows the experimental facility composed by the metamaterial with its mounting bracket, a thermal chamber, a 3D vibrometer, a piezoelectric amplifier and a thermocouple.

The metamaterial is suspended to reproduce free boundary condition as shown in figure 17b. The mounting bracket is introduced in a thermal chamber which presents a glass wall allowing 3D vibrometer measurements. The white noise generator provides random input voltage (5 V ) between 500Hz and 50kHz, this signal is amplified 20 times by the piezoelectric amplifier. Temperature in the thermal chamber is controlled by a thermocouple. A 3D vibrometer is used to measure the velocity, and frequency responses are calculated with H1 estimator. [START_REF] Box | Time series analysis: forecasting and control[END_REF] Experimental results at 25 • C Figure 18a presents numerical and experimental frequency responses at ambient temperature 25 • C. Curves need to be compared by pairs (yellow-blue and purple-red). Yellow and blue curves correspond, respectively, to experimental and numerical spatially averaged squared velocity/voltage amplitude (after ×20 amplification) |V z| 2 /U 2 for the input plate (IN). Purple and red curves correspond, respectively, to experimental and numerical spatially averaged squared velocity/voltage amplitude |V z| 2 /U 2 for the output plate (OUT). In the bandgap region, the attenuation predicted by numerical simulation is around 100dB, which corresponds to a ratio between output and input squared velocity amplitude of the order of 10 5 . Numerical amplitudes in this area are considerably lower than experimental amplitudes taking into account the experimental environment limitations. Reaching such a dynamic range in the measurement is not possible with the used experimental setup. However, the noise level estimation is performed and the green curve shows the background noise generated by the surrounding equipments (acquisition chain, room fan, etc.). It is hence impossible to measure vibration levels below this limit. The general trends (yellow-blue and purple-red) are similar except in the areas where the measurement is close to ambient noise.

Additionally to the velocity/voltage transfer functions presented in figure 18a, transmissibility functions IN/OUT are presented in figure 18b. This transfer is defined, for both experimental and numerical data, as follows:

H = 20 × log 10 Σ out (|V z| 2 /U 2 ) Σ in (|V z| 2 /U 2 ) , (23)
where Σ out (|V z| 2 /U 2 ) and Σ in (|V z| 2 /U 2 ) are, respectively, the average squared velocity amplitude |V z| 2 on the domain of the input plate (IN) and the output plate (OUT).

Red areas show when velocity are in the same order of magnitude as noise. These results confirm that the transmissibility is well estimated by the model, however the bandgap is not clearly observed in the experiments.

Next, the experimental analysis is focused on the 4 frequency points pointed out in the numerical part. Figures 19 show experimental deformed shapes at 33.4 kHz, 38.2 kHz, 42.2 kHz and 50.0 kHz corresponding to numerical deformed shapes (Figure 14). The experimental mesh is not as fine as the finite element one, however the patterns can clearly be identified. Shapes are similar between numerical and experimental results. In particular, it can clearly be observed in figure 19c that the energy is transported along the borders at 42.2 kHz. This observation being experimentally validated, the section 4 aims will propose a new methodology to predict such configurations using cell-based computation.

Experimental results at 90 • C

In order to switch to the second state of the crystal, the structure is heated at 90 • C. Figure 20a presents the corresponding numerical and experimental frequency responses. Curves need to be compared by pairs (yellow-blue and purple-red). Yellow and blue curves correspond, respectively, to experimental and numerical spatially averaged squared velocity/voltage amplitude |V z| 2 /U 2 on the domain of the input plate (IN). Purple and red curves correspond, respectively, to experimental and numerical spatially averaged squared velocity/voltage amplitude |V z| 2 /U 2 on the domain of the output plate (OUT). A good coherence is observed between the numerical and experimental results. Additionally to the velocity/voltage transfer functions presented in figure 20a, transmissibility functions IN/OUT are presented in figure 20b. The effect related to the periodicity is no longer observed at this temperature. The aluminum pillars are completely decoupled from the plate and the polymer still plays a damping role. Frequency response functions are smoothed at this temperature.

The prediction of the resonances at low frequencies is excellent. The discrepancies increase with frequency, the numerical model being stiffer than the real structure (no model updating has been performed). Resonance levels are quite well estimated A poor damping model probably should explains the difference. Anyway, the physics is well captured by the model and the structure behaves as expected.

SUPERCELL ANALYSIS TO HANDLE FINITE BOUNDARY CONDITION

In this section, a supercell analysis is performed in order to predict the propagation of waves along the borders of the finite structure, without requiring the modelling of the full finite structure, which could induce high calculation costs. The principle consists in considering a set of seven connected cells in the analysis, as shown in figure 21. Boundary conditions are as follow:

• in the x direction, classical Floquet-Bloch conditions are used, namely u R = e -jkxr u L where u R is the displacement on the right border and u L is the displacement on the left border in y axis,

• in the y direction, the left and right borders use the effective boundary conditions (free end in our case). Figure 22a shows dispersion curves along the Γ -X path of the Brillouin zone obtained with the shifted-cell operator method at ambient temperature (25 • C) for the 1-cell (classical Bloch analysis) and 7-cell systems. The blue curve corresponds to the 1 cell dispersion curve (figure 10b). The dispersion for the 7-cell system is plotted in yellow, two branches appear in the bandgap and the corresponding deformed shapes are visible in the figures 22b and 22c. Energy can propagate in the x direction along the free border. The interference effect leads to a non vibrating bulk and vibrating edges, which propagate vibration energy.

The proposed methodology hence allows identification of propagating waves along the borders inside the bandgap, cancelling the expected stop band effect. This method can highlight that the attenuation frequency range in finite structures is smaller than predicted by the bandgap (figure 13).

CONCLUSION

In this article, numerical tools for the dispersion analysis in periodic damped structures are used, with a focus on the ability of the shifted-cell operator method to deal with the dissipative behaviour of the system. An adaptive metamaterial based on aluminum pillars glued on a dissipative polymer interface is designed. The mechanical properties of the polymer depend on the frequency and the temperature, they change radically at the glass transition. The viscoelastic behavior of tBA/PEGDMA is approximated by a fractional Zener model. Two states are obtained by changing the temperature of the polymer interface to ambient temperature, the bandgap predicted by the dispersion diagram is visible around the selected frequency. The attenuation in the finite structure is not as important as that predicted by the bandgap in the infinite structure. Energy propagates along each edge of the periodic lattice. This point is confirmed through experimental measurements, and the dispersion analysis can not predict this effect. Finally, a supercell-based approach is used to handle finite system boundary conditions in order to identify situations where energy transfer may arise because of reflections at the border of the domain.

ANNEX: GROUP VELOCITY

In this section, the details for the expression of the group velocity is given. Equation 2.3.2 gives the expression of the group slowness using λ i = jk i ∂k i ∂ω = -j

ψ l i T ∂A1(ω) ∂ω -λ i ∂A2(ω) ∂ω ψ r i ψ l i T A 2 (ω)ψ r i . (24)
The matrices A 1 (ω) and A 2 (ω) are used

∂k i ∂ω = -j ψ l i T 0 0 ∂(K-ω 2 M ) ∂ω -λ i ∂L T ∂ω ∂L ∂ω -λ i ∂H ∂ω ψ r i ψ l i T I d 0 L T H ψ r i . (25)
The expression left-eigenvector (ψ l i ) and right-eigenvector (ψ r i ) are replaced in the following expression

∂k i ∂ω = -j λ i (φ l i T H) -φ l i T L φ l i T 0 0 ∂(K-ω 2 M ) ∂ω -λ i ∂L T ∂ω ∂L ∂ω -λ i ∂H ∂ω φ r i λ i φ r i λ i (φ l i T H) -φ l i T L φ l i T I d 0 L T H φ r i λ i φ r i , (26)
after the multiplication of the matrices, the expression of the slowness is found

The group velocity is the inverse of the group slowness

C g = 1 ∂ki ∂ω = 1 -j φ l i T [-2ωM + ∂K ∂ω -λi ∂L T ∂ω +λi ∂L ∂ω -λ 2 i ∂H ∂ω ]φ r i φ l i T [-L+L T +2λiH]φ r i . (28)
For frequency-independent cases, a simplification is possible

C g = 1 -2jωφ l i T Mφ r i [φ l i T (L-L T )φ r i -2λiφ l i T Hφ r i ] . (29)
In the particular case of on homogeneous cases where the frequency dependency is characterized by a Young's modulus such that E = f (ω)E 0 and a constant Poisson's ratio. Hence K = f (ω)K 0 , H = f (ω)H 0 and L = f (ω)L 0 . The equivalent quadratic eigenvalue problem is

[f (ω)(K 0 + λ i (L 0 -L T 0 ) -λ 2 i H 0 ) -ω 2 M ]φ r i = 0 (30)
In this case, the slowness is expressed as

S g = ∂k i ∂ω = -j φ l i T [-2ωM + ∂f ∂ω (K 0 + λ i (L 0 -L T 0 ) -λ 2 i H 0 )]φ r i φ l i T [f (ω)(-L 0 + L T 0 + 2λ i H 0 )]φ r i . (31)
Using the expression of the quadratic eigenvalue problem

∂f ∂ω (K 0 + λ i (L 0 -L T 0 ) -λ 2 i H 0 ) φ r i = ∂f ∂ω ω 2 M f (ω) φ r i , (32)
so, the slowness can be written as

S g = ∂k i ∂ω = jφ l i T [ω 2 ∂f ∂ω f (ω) -2ω]Mφ r i φ l i T [f (ω)(L 0 -L T 0 -2λ i H 0 )]φ r i , (33)
Therefore, the group velocity is expressed as

C g = 1 ∂ki ∂ω = 1 jφ l i T [ω 2 ∂f ∂ω f (ω) -2ω]Mφ r i φ l i T [f (ω)(L0-L T 0 -2λiH0)]φ r i (34)

Figure 1

 1 Figure 1. a) Infinite reference wave guide. b) Real and reciprocal lattices.

Figure 2

 2 Figure 2. a) Dispersion surfaces on the Brillouin zone. b) Dispersion curves on the contour of the Brillouin zone. Partial bandgaps and complete bandgap respectively in blue and green.

Figure 3 .Γ

 3 Figure 3. Dispersion curves along the Γ -X direction obtained with the shifted-cell operator method for conservative case, real part of wavenumbers is represented on the line from 0 to 1 and the imaginary part of wavenumbers from 0 -1, unsorted and sorted results respectively in black and blue. Grey shapes represent bandgaps.

kωΔω 5 /Δω 7 /Δω 8 /Figure 5 .

 5785 Figure 5. Scheme of the branch tracking routine

Figure 6 .Figure 7 .

 67 Figure 6. Dispersion curves along the Γ -X direction obtained with the shifted-cell operator method for conservative case with branch tracking.

Figure 8

 8 Figure 8. a) Reference structure, 10 mm height aluminum cylinder. b) Metamaterial with 3 mm high polymeric interface between base and 7 mm high aluminum cylinder.Table1. tBA/PEGDMA Zener parameters

10

 10

Figure 9 .

 9 Figure 9. Mechanical properties of the tBA/PEGDMA at 25 • C and 90 • C. a) Storage modulus. b) Loss factor.

Figure 10 .Figure 11 .

 1011 Figure 10. Dispersion curves along the Γ -X direction obtained with the shifted-cell operator method. a) Reference structure, all in aluminum. b) Metamaterial at ambient temperature 25 • C. c) Metamaterial at 90 • C and aluminum plate. Grey shapes represent bandgaps.

Figure 12 .

 12 Figure 12. Finite structure with an interface composed by 7 × 7 unit cells distributed. Piezoelectric patch (PZT) for excitation in the bottom left corner.

Figure 13 .

 13 Figure 13. Numerical frequency responses. Average squared velocity amplitude |V z| 2 on the domain of the input plate (IN) and the output plate (OUT) at ambient temperature 25 • C respectively in blue and red. At 90 • C, respectively in yellow and purple. The grey shape represents the bandgap predicted by the dispersion analysis at ambient temperature.

Figure 14 .Figure 15 .

 1415 Figure 14. Zoom on numerical frequency responses. Average squared velocity amplitude |V z| 2 on the domain of the input plate (IN) and the output plate (OUT) at ambient temperature 25 • C respectively in blue and red. The grey shape represents the bandgap predicted by the dispersion analysis at ambient temperature.

Figure 16 .Figure 17

 1617 Figure 16. Metamaterial after several manufacturing steps.

Figure 18

 18 Figure 18. a) Numerical and experimental frequency responses at ambient temperature 25 • C. Experimental spatially averaged squared velocity/voltage amplitude |V z| 2 /U 2 on the domain of the input plate (IN) and the output plate (OUT) respectively in blue and red. Numerical ones respectively in yellow and purple. The green curve shows the noise measurement. The grey shape represents the bandgap predicted by the dispersion analysis at ambient temperature. b) Numerical and experimental transmissibility functions IN/OUT at ambient temperature respectively in blue and red. Red areas show when velocity are in the same order of magnitude as noise.

Figure 19 .

 19 Figure 19. Experimental deformed shapes a) at 33.4 kHz, b) 38.2 kHz, c) 42.2 kHz and d) 50.0 kHz corresponding to numerical deformed shapes (Figures 14).

Figure 20

 20 Figure 20. a) Numerical and experimental frequency responses at ambient temperature 90 • C. Experimental spatially averaged squared velocity/voltage amplitude |V z| 2 /U 2 on the domain of the input plate (IN) and the output plate (OUT) respectively in blue and red. Numerical ones respectively in yellow and purple. b) Numerical and experimental transmissibility functions IN/OUT at 90 • C respectively in blue and red.

Figure 21 .

 21 Figure 21. Boundary conditions of the 7-cell system.

Figure 22

 22 Figure 22. a) Dispersion curves along the Γ -X path of the Brillouin zone obtained with the shifted-cell operator method for the 1 cell and 7-cell system. b) Deformed shape at 40.1 kHz, c) Deformed shape at 40.2 kHz.

  L + L T + 2λ i H]φ r i .

Table 1

 1

		. tBA/PEGDMA Zener parameters
	E 0poly [MPa] E ∞poly [Mpa]	α	τ 0
	0.67	2211	0.78 0.91

ACKNOWLEDGMENTS

This work was financed by The French National Research Agency under grant number ANR-12-JS09-008-COVIA. It has been performed in cooperation with the Labex ACTION program (ANR-11-LABX-0001-01) and the hybrid microfabrication platform MIFHySTO.


Guillaume Wille

Didier Lahondère

Ute Schmidt

Jeromine Duron

Rostislav Vana

Jérémie Silvent

Xavier Bourrat

Didier Lahondere

Jeremie Silvent

Confocal Raman-in-SEM Imaging: a New Method for 3D Morphology of Asbestos like Fibers in a Mineral Matrix

Asbestos consists of natural mineral fibers crystallized in a specific way with specific properties including flexibility, high tensile strength, resistance to heat and chemical degradation. The term asbestos refers to one fibrous serpentine, and five asbestiform varieties of amphiboles [1] [START_REF] Silvent | Asbestos fibers and other elongate mineral particles: State of roadmap for research[END_REF]. Asbestos is considered a Category 1 human carcinogen (e.g. [1]

[3][4][5][6]
), as inhalation of asbestos fibers causes respiratory diseases, in particular asbestosis, lung cancers and malignant mesothelioma. Due to the pathogenicity of asbestos fibers, their use has been banned in most countries around the world. The classification as "asbestos" comprises three main characteristics: sub-micrometric fibrous morphology (so-called asbestiform), chemistry and crystallography (i.e. mineralogy).

The ban applies to natural materials, which are quarried and used for public works projects. Compliance with existing rules requires accurate diagnosis of the asbestos or non-asbestos nature of the materials likely to be exploited in a quarry or involved in the opening of a construction site. Conventional methods for asbestos diagnosis, mainly based on the use of TEM, are hard to apply to massive materials as the sample preparation protocol is complex, it is likely to release artifacts like cleavage fragments, or sample-piece thinning is costly and hard to achieve and thus not applicable in routine analysis.

The use of a LV-FE-SEM-EDS (Low vacuum field emission gun scannning electron microscope equiped with energy dispersive spectroscopy) coupled to confocal Raman-in-SEM imaging (RISE) is a new and efficient method for identifying the mineral nature of sub-micrometer fibers or fibrous bundles. SEM coupled to EDS provides a qualitative or semi-quantitative composition of the fiber. The crystal structure is determined by confocal Raman spectroscopy (with lateral resolution down to 360 nm [7]). Thus the combination of these two kinds of information allows for accurate identification of both the nature and the morphology of the mineral. The combination of SEM, EDS and confocal Raman imaging in a unique analytical system (RISE) allows precise location of the same area / fiber and identification of its mineral nature. In addition, RISE allows the acquisition of 3D data, which is able to provide morphological information of the mineral distribution in the sample volume and leading to the determination of the aspect ratio, a critical parameter for asbestos (i.e. asbestiform or non-asbestiform). Moreover, this combination of techniques is not destructive. The coupling of SEM-EDS with RISE is a powerful analytical system that simplifies and reinforces existing analytical procedures.

For highest resolution studies of fibers in the sample volume (3 D imaging), the RISE can be connected to a FIB-SEM [7]. Although FIB volume reconstruction is a destructive method, the resolution of the voxel can be extended to below 100 nm [8] a resolution sufficient for visualizing even the thinnest fibers. Furthermore, Raman 3D view is an efficient tool for sampling TEM slices using the FIB.

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