Olivier Dalstein

Dr Davide R Ceratti

Cédric Boissière

David Grosso

Andrea Cattoni
email: andrea.cattoni@lpn.cnrs.fr

Marco Faustini
email: marco.faustini@upmc.fr

Pr C D Boissiere

Dr M Grosso

Faustini

Dr A Cattoni

Pr D Grosso

M2NP Pr D Grosso

Nano-imprinted, Sub-micrometric, MOF-based 2D Photonic Structures: Toward Easy Selective Vapors Sensing by a Smart-Phone Camera

In this work we describe a soft-lithographic approach to fabricate sub-micrometer MOFbased 2D photonic structures. Nanometric Zeolitic Imidazole Framework material ZIF-8 (zinc) was chosen as the sensible MOF material because of its chemical stability and its vapor selective adsorption properties. Two different systems were fabricated: nanopatterned colloidal ZIF-8 homo-and ZIF-8/TiO 2 hetero-structures. Several features (stripes, squares, etc.) with dimension as small as 200 nm, were replicated on different substrates such as silicon, flexible plastics and even on aluminum cans, over relatively large surfaces (up to 1 cm 2 ). In addition, we demonstrate here that these photonic MOF-heterostructures can be used as very low-cost sensing platforms compatible with smart-phone technology. This method relies on the evaluation of the change in diffraction efficiency of the photonic MOF-patterns, induced by the MOF refractive index variation, which is simply detected by a CCD camera, as those integrated in simple smart-phones, without need of complex optical instrumentations for transduction data processing. Performances of the sensors were first evaluated using isopropanol adsorption/desorption cycling as a model case. In addition, a "real" environmental issue was tackled. Selective detection of styrene in presence of interfering water was demonstrated at concentration below the human permissible exposure limit. In-situ ellispometric analyses were also carried out in order to confirm the sensor performances and to propose a mechanism for styrene uptake into the nanoMOFs.

1.

Introduction

Metal Organic Frameworks (MOFs) have attracted immense attention because of their large, tunable porosity and versatile chemistry [1] that makes them appealing for many applications, including storage, separation, [2] sensing [3] and catalysis. [4] However these materials are generally synthetized under powder form that is not always suitable for direct application. In order to fulfill their promises, MOFs need to be processed, shaped and integrated in functional devices. [5] This is especially the case for the utilization of MOFs in the domain of photonics. Some recent works demonstrated the integration of MOFs into photonic components such as thin films, [6] Fabry-Pérot, [7] 1D Bragg stacks, [8] or 3D colloidal crystals. [9] In all these examples the materials were exploited as chemical sensors, [10] for which the signal transduction relies on the change of refractive index in response to the analyte adsorption into the porosity. [6] On the other hand, the integration of MOFs into 2D photonic components haven't been explored yet. Two dimensional periodical systems are common architectures of many, sophisticated nanophotonic devices such as 2D photonic crystals, gratings, waveguides, plasmonic antenna [11] or even metamerials. [12] A simple method for the integration of MOFs into 2D photonic components would pave the way for advanced optical devices in which the original properties of MOFs (high porosity, sorption selectivity..) and of the periodic structures (light confinement, diffraction, guiding..) are rationally combined. The main technological challenge to further develop such devices is that they require fabrication of nonscattering, patterned films at the sub-micrometer scale. With such a perspective, the seemingly simple step aiming at patterning MOFs materials by lithographic approaches has become a very active research domain in the last years and several strategies have been proposed. [13] MOF crystals have been 2D spatially localized by soft-lithographic approaches, [14] digital microfluidics, [15] localized electrochemical, [16] growth, [17] seedmediated [18] deposition, Dip-Pen Lithography, [19] Deep X-Ray Lithography, [20] UV Lithography [21] combined with hot embossing. [22] However, none of these approaches was suitable for fabricating MOFs-based 2D photonic devices since crystals size and/or pattern periodicity were in the micrometer range. In addition, most of the reported patterning methods were not adapted to large-scale fabrication on different substrates, a critical aspect to enable their utilization as sensing platform for instance.

Here we propose a simple and versatile approach to fabricate sub-micrometer MOF-based 2D photonic structures, and we demonstrate their utilization as very low-cost sensing platforms compatible with smart-phone technology. Soft Nano-Imprinting Lithography was exploited as patterning technology because of its easiness, versatility and because it allows fabrication at throughputs necessary for many realistic applications. Zeolitic Imidazole Framework material ZIF-8 (zinc) was chosen as the sensible MOF material. [23] In general, ZIFs are chemically and thermally stable materials, a fundamental requirement for any practical device. Study of the gas adsorption and thermal and chemical stability of ZIF-8 (Zn) demonstrated its permanent, high thermal stability (up to 550°C), and remarkable chemical resistance to boiling alkaline water and organic solvents. [24] In addition, due to its great surface area and "large porosity", ZIF-8 (Zn) is the most used in chemical sensing because of the excellent selective adsorption properties. [23] In order to obtain optical quality patterns and overcome light-scattering, the material was processed as colloidal solution in which the crystal sizes were inferior to 50 nm. Two different systems were fabricated: nanopatterned colloidal ZIF-8 homo-and ZIF-8/TiO 2 hetero-structures. Both patterned systems were obtained by either nanoimprinting directly a sessile drop of nano-MOF suspension, or by depositing this suspension onto a nanoimprinted sol-gel based titania films. Several features (stripes, squares, etc.) with dimension as small as 200 nm, were replicated on different substrates such as silicon, flexible plastics and even on aluminum cans, over relatively large surfaces (up to 1 cm 2 ). The resulting systems exhibited angle-dependent structural colors characteristic of 2D diffraction gratings. A new sensing approach based on the variation of optical properties of MOF patterns, induced by the selective adsorption of organic solvent vapors, was investigated. This method relies on the evaluation of the adsorption-induced-color-luminance change, induced by the MOF refractive index variation, which is simply detected by a CCD camera, as those integrated in simple smart-phones, without need of complex optical instrumentations for transduction data processing. Proof of concept and performances were first demonstrated using isopropanol adsorption/desorption cycling as model system. It was then applied to the real industrial problem of the production of polystyrene latex prepared by oil in water emulsions of styrene, for which the detection of reactor leaks and quantitative analysis of a highly toxic styrene in presence of high concentration of interfering water is a "real" safety issue. We demonstrated that styrene could be selectively detected and quantified by evaluating its adsorption kinetic by the patterned MOF layer. Concentrations as small as 57 ppm, which is below the human permissible exposure limit, could be detected. In all cases, the styrene adsorption was evaluated by optical transduction using the associated luminance variation of the gratings probed by a simple CCD camera, and double checked by direct analysis of the optical properties of a non-patterned film through conventional spectroscopic ellipsometry.

2.

Results and discussion

Fabrication of the patterns

The nanopatterning strategy of porous ZIF-8 materials is based on the utilization of preformed ZIF-8 nanocrystals as illustrated in Figure 1. ZIF-8 nanoparticles suspension was prepared by following the procedure previously reported by Cravillon et al. [25] The reaction leads to a stable and homogeneous colloidal solution of microporous ZIF-8 nanocrystals in ethanol. The crystal average size was measured equal to 41± 8 nm by scanning electron microscopy (Fig. S1). Thin films exhibiting high optical quality could be obtained by dip coating or spin coating the colloidal solution on silicon substrates. The crystal structure of the particles films was accessed by Grazing-Incident Wide-Angle X-ray Scattering (GI-WAXS) that shows the characteristic pattern of ZIF-8 materials (Fig. S2). The refractive index of the colloidal films was measured to be equal to 1.2 ± 0.05 depending on particle's packing, in agreement with previous findings. [6a] In order to characterize the porosity and the vapor sorption selectivity of the colloidal ZIF-8 films, water and isopropanol adsorption experiments have been carried out at room temperature (Fig. S3). The adsorption experiments were performed by environmental ellipsometric porosimetry (EEP) on a non-patterned film having a thickness of 155 nm obtained by dip-coating the ZIF colloidal solution at 5 mm.s -1 onto a silicon substrate. The hydrophobicity of the ZIF-8 microporosity was confirmed since no significant variation in refractive index is observed except for very high vapor pressure (P/P 0 >0.9), for which water condensation occurs in the inter-particle voids. On the other hand, isopropanol (i-PrOH) adsorption curve shows a double steps increase of refractive index at low and high relative pressures, due to i-PrOH adsorption into the micropores of the ZIF-8 framework (17% of the total film volume) and the inter-particular mesopores (39% of the total film volume) as observed in a previous study. [6a] The quantitative analysis of the adsorbed volume fraction was obtained through the BEMA (Bruggeman Effective Medium Approximation) model assuming the refractive index of the dense ZIF-8 material being equal to 1.54. [6a] In addition, the ellipsometric investigation indicated that capillary condensation into the ZIF-8 micropores is associated with a weak contraction of the film (around 2%, data not shown).

Concerning nanostructuration of the ZIF-8 nano crystals films, two routes were selected. In the first one, soft nanoimprint lithography was applied directly on a drop-casted ZIF-8 colloidal suspension (see Figure 1a). In the second one, ZIF-8 colloids are deposited onto a pre-patterned TiO 2 layer (see Figure 1b). For the first route, a 10 µL-droplet of colloidal solution was first applied on the substrate. The solution droplet was immediately covered by the patterned PDMS stamp without applying any external pressure. The liquid layer was homogeneously spread beneath the PDMS stamp while the stamp's nanocavities were naturally filled by capillarity. In this configuration, the evaporation of the solvent takes place through the permeable PDMS stamp. Replication of the features and homogeneous packing of the colloidal network were obtained after complete evaporation and de-molding. Finally, the system was consolidated at 130°C for 10 minutes.

An example of replication is shown on the optical photograph in Figure 2 applied to the direct embossing of sol-gels derived films. [26] The system was stabilized at 110°C for 5 minutes before demolding, and before being heat-treated at 450°C for 5 minutes to ensure complete densification and crystallization of TiO 2 network. The nanopatterned TiO 2 film is thus composed by a dense network of sub-20 nm anatase particles as characterized by SEM and GI-WAXS and shown in Fig. S4 The cross-section view of a typical nanoimprinted TiO 2 film with periodicity of 400 nm is shown in Fig. 3 In order to better visualize the whole system a band of the ZIF-8 layer was wiped away from the surface (see Figure 3 uncovered TiO 2 nanogrooves and (c) higher magnification of the deposited zone.

MOF-based diffraction grating sensors

The MOF patterned surfaces have been used as reflective diffraction gratings enabling easy sensing by a simple smart-phone's camera. The principle is illustrated in Figure 4 heterostructure, three decreases in luminance are observed: a first one at low P/P 0 that is attributed to the vapor uptake into the ZIF-8 microporosity (transition I!II); an additional decrease below P/P 0 = 0.7 (transition II!III) probably due to the i-PrOH condensation at the ZIF-8/TiO 2 interface; and a significant decrease in luminance at P/P 0 > 0.9 (transition III!IV) due to isopropanol capillary condensation within the ZIF-8 interparticles voids. The desorption branch follows the adsorption one except at low pressures where a hysteresis is depicted. This can be attributed to a slower desorption due to the stabilization of the adsorbate within the MOF microporosity. The great difference in luminance between low and high P/P 0 exposure is highlighted in photographs taken by smart phone in Figure 5 (c) for the ZIF-8/TiO 2 sample. One must underline here the better stability of the ZIF-8/TiO 2 heterostructure resulting from the encapsulation of the MOF particles into the hard TiO 2 grooves. Differently from the nanopatterned ZIF-8 system, here the patterned material is TiO 2 that is insensitive to the i-PrOH exposure. The interparticle condensation occurs in the top ZIF-8 layer but after drying the system recovers its initial morphology (Fig. S9). For this reason, only the results of the ZIF-8/TiO 2 gratings will be presented from this point hereafter. In order to probe the reversibility of the i-PrOH adsorption/desorption in the ZIF-8/TiO 2 system, the variation of luminance was investigated during cycling in i-PrOH enriched atmosphere between P/P 0 = 0 and P/P 0 = 0.5 in air at 25°C as shown and detailed in Fig. S5.

The variation of luminance upon the adsorption/desorption cycles is reproducible, indicating that the material is able to withstand cyclic exposure to the vapor stream. The same results

were obtained by cycling i-PrOH on nanopatterned ZIF-8 gratings (at P/P 0 between 0 and 0.5, irreversible collapsing does not take place) as shown in Fig. S6.

Selective Styrene Detection

The effectiveness of the detection mode described above for isopropanol vapors is now transposed to a more industrially relevant application such as the selective detection of toxic and volatile styrene monomer in presence of water. Styrene is a building block for synthesis of rubbers, plastics, resins, and polyesters in synthetic industry. In particular, the styrene/water biphasic mixture is the main component of the widely used synthesis of polystyrene beads by oil in water emulsion polymerization. [27] a potential human carcinogen, styrene vapors may lead to irritation of the skin, eyes and the respiratory system.

Similar issues have been discussed in the literature: luminescent Eu-MOFs were used to selectively detect explosive 2,4,6-trinitrophenol (TNP) in water or in vapor phase. [3a, 28] Several materials have been also exploited to detect styrene (and propanol) by adsorption including conductive polymer composites, [29] hydrogen-bond acidic polymers, [30] organoplatinum coatings, [31] monolayer-protected gold nanoparticles, [32] ZnO nanowires [START_REF] Li | Micro-Nano Technology Xiv[END_REF] or Pd-gate metal-oxide-semiconductor. [START_REF] Pandey | [END_REF] However in all these cases the sensitive materials were applied on different detection platform such as surface acoustic wave (SAW) or resistive sensors. In addition they were not very porous and/or not compatible with a simple optical detection. Due to its high porosity and selectivity, a ZIF-8 material emerges as a strong candidate for ambient optical detection of styrene in presence of water.

The adsorption/desorption behavior of nanoZIF-8 exposed to styrene/water vapors was first investigated on planar films by in-situ ellipsometry as illustrated and detailed in Figure S7.

The generation of vapors was obtained by mixing a controlled proportion of dry air, and air enriched with styrene and water vapors in order to control the composition of the atmosphere, indicated in term of relative vapor pressures (1, 2, 5, 10 %) of styrene and water.

The evolution of the refractive index of a ZIF-8 film exposed to styrene/water vapors generated from liquid mixtures composed of 1, 2, 5, and 10% styrene is shown in Figure 6(a).

As expected, the refractive index increases due to the selective styrene uptake into the ZIF-8 particles. As a blank, the same experiment was performed only with water vapors and no variation of the refractive index was observed. Heating the coating 5 min at 130°C in dry air allows to fully desorb the molecules and recover the initial state. The increase in refractive index is relatively slow and the system doesn't reach equilibrium even after 35 minutes of experiment. On the other hand, the styrene uptake rate varies with the vapor pressure, being faster at 10% and slower at 1%. The styrene uptake variations (expressed as normalized refractive index) were then plotted with respect to the square root of time in Figure 6(b). Each uptake is characterized by two successive linear curves, suggesting that the uptake can be described with two successive phenomena that are both limited by diffusion. An adsorption and diffusion mechanism is proposed and illustrated in Fig. S8. The first faster uptake (regime i) could be attributed to the styrene adsorption on the surface of ZIF-8 nanoparticles. This regime would be limited by the diffusion of styrene from the atmosphere to the surface of the material and it ends when equilibrium is reached. Being in equilibrium, the concentration of styrene on the surface is directly related to the styrene relative vapor pressure. The second uptake behavior (regime ii) is very probably due to the progressive styrene penetration into the microporosity towards the core of the ZIF-8 particles. Regime ii would be limited by the diffusion of styrene from the surface into the microporosity. 1.17

i ii ellipsometry and luminance measurements, the concentration-dependent uptake rate can be determined from the slopes of the straight tendencies of regime ii. In order to verify if this approach could be generalized to other organic molecules, the experiment shown in Figure 6 (b) was performed with other organic solvents (Fig. S10). In particular, a ZIF-8 film was exposed to styrene, toluene, cyclohexane and xylene vapors at the same concentration of 5% P/P 0 . As shown in Figure S10 (left), for all the systems, the two regimes could be identified. The amplitude of the refractive index and the variation of the uptake rates determined by fitting the slope of regimes (i) and (ii) for each solvent are reported in the graph in Fig. S10 (right). A significant variation in uptake rates in both regimes is observed for the four solvents. The adsorption rate of the four volatile hydrocarbons are found in the following order for regime ii: cyclohexane > xylene > styrene > toluene. This difference can be attributed to a certain number of factors including a molecular sieving effect and the different chemical affinity between the adsorbed solvent and the ZIF-8 particle surface (regime i) and the inner microporosity (regime ii). This solvent-dependent behavior suggests that a strategy based on the determination of the adsorption rates can be envisioned to evaluate the contribution of each component in the case of mixed solvents. This could be achieved by approaches of combinatorial sensing by the integration of different sensing materials in a single platform as reported in the literature. [8c] 3

Conclusions

In this work we demonstrated an versatile approach to fabricate MOF-based 2D photonic structures by soft NanoImprinting Lithography. ZIF-8 homo and ZIF-8/TiO 2 heterostructures, exhibiting several features with dimensions as small as 200 nm, were replicated on different substrates over relatively large surfaces (up to 1 cm 2 ). The resulting systems exhibited iridescence characteristic of 2D diffraction gratings. A new sensing approach based on the variation of the optical properties of MOF patterns, induced by the selective adsorption of organic solvent vapors, was studied. This method relies on the evaluation of the adsorptioninduced-color-luminance change, induced by the MOF refractive index variation, which is simply detected by a CCD camera, as those integrated in simple smart-phones. Detection performances of the sensing platforms were demonstrated using isopropanol as a model adsorbate and a biphasic styrene/water mixture as a relevant environmental contaminant system. Vapors sensing was successfully demonstrated by using ZIF-8 as sensible matrix integrated into 2D diffraction gratings. Interestingly detection of styrene can be based on the dynamic variation of optical properties of the films and not on their absolute values. In this simple proof-of-concept, we didn't test vapors concentrations below 1 % of styrene; in addition some improvements can be envisioned to improve the transduction method. Even in the case of the less concentrated vapors mixture (1%), the content of styrene in the atmosphere was calculated being equal to 57 ppm, a value that is below the Permissible Exposure Limit in the US for employees working in chemical industry; [35] other styrene detection protocols mainly based on gas chromatography [35] ensures better sensitivity but requires the utilization of bulky, expensive instrumentations. Importantly, this approach based of MOF diffraction gratings allows an easy optical detection by using a simple and accessible tool such as the camera of a smart phone that can be potentially interfaced with image analysis applications for real-time air-quality monitoring.

Master and PDMS-based stamp fabrication

The silicon master mold were fabricated by electron beam lithography at 100 KeV (Vistec 5000+) and reactive ion etching. An A7 PMMA positive-tone resist (495PMMA A Resists, solids: 7% in Anisole) was spin-coated on a 2-inch Si(100) wafer and baked for 45 min at 160 °C. The sample was exposed at 10 nA with a dose of 1300 µC/cm2, developed for 45 sec in a solution of Methyl-isobutyl-ketone (MIBK) and Isopropanol (mixture 1:3), rinsed in Isopropanol and gently dried with pure nitrogen gas. The pattern was transferred by reactive ion etching using a SF6/CHF3 gas mixture.

The anti-sticking treatment was performed using 1H,1H,2H,2H-Perfluorooctyltriethoxysilane (POTS) by chemical vapor deposition method. A sealed glass Petri dish containing the Silicon master and several drops of POTS was heated in an oven at about 120 °C for 1 h to enable the reaction between the OH groups on the Si substrate surfaces and the POTS and then maintained at about 150 °C for 2 h to remove the unreacted POTS molecules. [36] A bi-layer hard-PDMS/PDMS stamp was then replicated first by spin-coating a thin hard-PDMS layer on the silicon master and secondly by casting a mixture of commercial available (RTV615 from GE) two components solution (1:10) on top. The bi-layer hard-PDMS/PDMS stamp was degassed and soft-baked at 60 °C for 48 h. Finally, after being peeled off from the silicon master, the bi-layer stamp was treated with trichloromethylsilane TMCS. [37] ZIF-8 homo-and ZIF-8/TiO 2 hetero structures fabrication

The synthesis of the ZIF-8 nanoparticles followed the conditions reported by Cravillon et al. [25] A solution of Zn(NO 3 ) 2 .6H 2 O (2.933g, 9.87 mmol ; 98% Sigma-Aldrich) in 200 mL of methanol (100% VWR Chemicals) is poured in a solution of 2-methylimidazole (6.489g, 79.04 mmol ; 99% Sigma-Aldrich) in 200 mL of methanol under stirring at room temperature.

After 1h, the nanocrystals are separated from the milky solution by centrifugation at 20000 rpm for 15 min. Nanoparticles were redispersed in absolute ethanol and centrifuged 3 times to remove the unreacted species.

For the patterning of ZIF-8 nanostructures, a 10 µL-droplet of colloidal solution was first applied on the substrate. The solution droplet was immediately covered by the patterned PDMS stamp without applying any external pressure. The liquid layer was homogeneously spread beneath the PDMS stamp while the stamp's nanocavities were naturally filled by capillarity. In this configuration, the evaporation of the solvent takes place through the permeable PDMS stamp. Replication of the features and homogeneous packing of the colloidal network were obtained after complete evaporation and de-molding. Finally, the system was consolidated at 130°C for 10 minutes.

For the fabrication of ZIF-8/TiO 2 heterostructures, titania thin films were first prepared from a solution composed of TiCl 4 /H 2 O/Ethanol (TiCl 4 99% Sigma-Aldrich) with a respective molar ratio of 1/6/50. After complete dissolution and homogenization, a layer of solution was deposited on silicon wafers by spin-coating at 2500 rpm for 10s under a relative humidity of 40%. In the same time, the PDMS stamp was pumped 10 min under vacuum and then readily applied on the as-deposited titania thin film. After 1 min at room temperature the thin film surrounded by the PDMS stamp was heated at 100°C for 5 min on a hot plate. The PDMS stamp was then removed and the titania thin film was finally calcinated at 450°C for 5 min on a hot plate. ZIF-8 coating was deposited on nanoimprinted TiO 2 gratings by dip-coating with a withdrawal speed of 8 mm/s.

Characterization

Ellipsometry measurements were performed on a UV-visible (from 240 to 1000 nm) variable angle spectroscopic ellipsometer (VASE -2000U Woollam), and the data analyses were performed with the Wvase32 software using Cauchy models. Environmental ellipsometry porosimetry involves a dynamic in situ monitoring of the refractive index and the thickness of the film upon variation of the relative vapor pressures between 0% and 100% inside the analysis chamber. Microscope images showing the film structure were collected using a field-emission gun scanning electron microscopy (FEG-SEM -Hitachi SU7000 instrument).

Grazing incidence wide-angle X-ray scattering (GI-WAXS) was recorded using a Rigaku S-MAX 3000 with simultaneous small angle X-ray scattering (SAXS) capabilities, on an image plate.

Detection experiments

The photocamera-assisted detection experiment is schematically illustrated in Figure 4 and it was performed in a closed black box in order to avoid interferences with ambient light.

Specific windows and connections were built for irradiation and detection and to allow vapors injection into the closed box. The diffracting grating samples were irradiated by a white LED (ZEISS CL 6000 LED) perpendicularly to the substrate. For each experiment, the intensity of the incident light was tuned between 0,1 and 3 W/cm 2 in order to achieve reasonable diffracted light avoiding saturation in the diffracted images. A commercially available smartphone I-Phone 3GS (photo camera resolution 3 MegaPixel) was used for the i-PrOH detection while a steady photo-camera Manta G125CIRC (resolution 1.3 MegaPixels) was used for the longer styrene adsorption detection. The photo camera was place at an angle of 28° as respect of the incident light and at distance of 30 cm from the sample.

Figure 1

 1 Figure 1 Soft-lithographic approaches for the fabrication of (a) nanopatterned ZIF-8 and (8) ZIF-8/TiO 2 heterostructures.

  photos in Figure2(f) and Figure2(g) (and schematically represented in Figure1 (a)). The periodicity of the grooves array was equal to 780 nm while both width and height of the lines ranged between 200 and 240 nm (corresponding to 5-6 colloids packed together). A 200nmthick residual layer composed by a continuous film of ZIF-8 colloids remained at the bottom.The thickness of the residual layer could be partially tuned by modifying the concentration of the initial ZIF-8 colloidal solution. Figure2(h) shows the cross-section SEM micrograph of a sample obtained from the same replication process but by diluting twice the initial colloidal solution. In this configuration, grooves array that are free of residual layer are obtained. Other pattern morphologies such as square arrays, as displayed in SEM micrograph in Figure2(c), could be obtained using appropriated PDMS stamps. Being based on the replication of flexible stamp, the method could be extended on a wide range of substrates such as a curved aluminum can (Fig.2b) or a transparent, flexible polypropylene film (Fig2c). In these cases the morphology and the quality was similar to those observed on silicon in Figure2(d).

Figure 2

 2 Figure 2 Photograph of the diffracting ZIF-8 nanopatterned surfaces applied on (a) a silicon wafer, (b) an aluminum can and (c) a flexible polypropylene film. FEG-SEM micrographs of

Figure 1 (

 1 Figure 1(b) displays the second fabrication route for the preparation of nanostructured TiO 2 /ZIF-8 multilayer coatings. The fist step consisted in the deposition of sol-gel based titania layer by spin-coating on silicon. Immediately after evaporation of the volatile solvent, the uncondensed titania precursors layer was embossed by a flexible hard-PDMS/PDMS stamp without applying any pressure using a technique called Degassing Assisted Patterning

  (a). The nanopatterned TiO 2 films were then homogenously covered by ZIF-8 colloids by dip-coating. The morphology of the final nanostructured TiO 2 /ZIF-8 system is shown on the SEM photos of Figure 3(b) and (c).

Figure 3

 3 Figure 3 FEG-SEM micrographs of (a) typical patterned TiO 2 film (cross-section view) and ZIF-8 /TiO 2 hetero-structures: (b) scratched zone exhibiting both ZIF-8 covered and

  (a). When the periodical structure is irradiated by a white light, a part of the reflected light is split by diffraction into several beams travelling in different directions. The directions of the beams depend on the spacing of the grating and the wavelength. When light is normally incident on the gratings, the relationship between the grating spacing d and the angle of the diffracted beams of light 𝜃 ! is given by the following Bragg's equation(1):𝑑 sin 𝜃 ! = 𝑚𝜆(1)where m is the diffraction order and 𝜆 the wavelength of each component of the light. The top image in Figure 4(b) shows the photograph of the ZIF-8 gratings having d-spacing of 1000 nm taken by the photo-camera of commercially available smart-phone. The image was taken at angle 𝜃 ! equal to 28° as respect to incident beam. In this configuration, the color of the pattern looks blue since the 1 st order diffracted light has a wavelength equal to 469 ± 5 nm according to equation 1. The photo taken by the camera is then converted into gray-scale format. The process consists in the conversion of the RGB values to Luminance Intensity for each pixel. The image analysis of a small Zone Of Interest (ZOI) gives an average value of Luminance Intensity that can be directly attributed to the efficiency of the diffracting platform. If one assumes that the height of the nanostructures does not vary, the intensity of the diffracted light depends on the refractive index contrast between the patterned material and the surrounding medium. Opposite tendencies are thus expected for the two ZIF-8 based systems illustrated in Figure 4 (c) while exposed to organic vapors. In the case of directly patterned ZIF-8 colloids, the MOF layer plays the role of "active" diffraction grating. One expects an increase in diffraction luminance created by the increase of the optical index contrast between the medium (n air =1 : constant) and the grating material (n MOF = 1.2 :increasing with the organic vapor adsorption). In this case, the modification of the structure height is negligible, since we detected a contraction of only 2% in volume of the MOF layer by ellipsometry at high vapor pressure. In the case of ZIF-8/TiO 2 heterostructure, the dense TiO 2 bottom layer plays the role of "passive" diffraction grating. The uptake of organic vapors by the ZIF-8 porous crystals provokes thus a decrease in optical index contrast between the two components and a decrease in luminance intensity.

Figure 4

 4 Figure 4 (a) Illustration of the experimental set-up for the detection by a simple camera; (b) photographs of the diffracting grating in colors and gray-scale taken a smart phone photocamera; illustration of the patterned ZIF-8 and ZIF-8/TiO 2 heterostructure with opposite refractive index evolution as function of the vapor adsorption.

Figure 5

 5 Figure 5 (a) Isopropanol uptake measured by luminance difference; (b) illustration of the phenomena taking place during adsorption ; (c) photographs of the ZIF-8/TiO 2 heterostructure at 0 and 100% P/P 0 of isopropanol, taken by a smart phone at an angle of 28°.

Figure 6

 6 Figure 6 (a) Evolution of the refractive index of ZIF-8 colloidal film with respect to time for styrene/water vapors adsorption/desorption at different concentration. (b) Plot of the normalized refractive index as a function of the square root of the time for styrene/water adsorptions at different vapors concentrations.

Figure 7 (

 7 b) shows the plot of uptake rates as a function of the vapors composition obtained from ellipsometry and diffraction grating (DG) experiments. A good agreement in term of tendency is found indicating that sensing of styrene can be performed by using an extremely sensitive technique like ellipsometry as well as a simple photo-camera.

Figure 7

 7 Figure 7 (a) Plot of the luminance of the ZIF-8/TiO 2 diffraction grating with respect to the square root of time for styrene/water adsorptions at different vapors concentration. (b) Plot of the uptake rates versus vapor concentration obtained by diffraction grating (DG) and ellipsometry.

Acknowledgements

We thank David Montero for his support for electronic microscopy (Institut des Matériaux de Paris Centre FR2482). We acknowledge the financial support from CNano Region Ile-de-France and the Labex Matisse.

Supporting Information

Supporting Information is available from the Wiley Online Library or from the author.


Marc Rebillat
email: marc.rebillat@ensam.eu

Kerem Ege
email: kerem.ege@insa-lyon.fr

Maxime Gallo

Jérôme Antoni

M Rébillat

Repeated exponential sine sweeps for the autonomous estimation of nonlinearities and bootstrap assessment of uncertainties

Keywords: Nonlinear vibrating structure, nonlinear system identication, parallel Hammerstein models, bootstrap procedure, uncertainty quantication

come

Repeated exponential sine sweeps for the autonomous estimation of nonlinearities and bootstrap assessment of uncertainties

Introduction

Vibrating structures are generally assumed to behave linearly and in a noisefree environment. This is in practice not perfectly the case. First, nonlinear phenomena such as jump phenomenon, hysteresis or internal resonance appear when the transverse vibration of a bi-dimensional structure exceeds amplitudes in the order of magnitude of its thickness [START_REF] Touzé | Asymmetric non-linear forced vibrations of free-edge circular plates. part 1: Theory[END_REF]. Second, the presence of plant noise is a natural phenomenon that is unavoidable for all experimental measurements.

In order to perform reliable measurements of vibrating mechanical structures one should thus keep in mind these two issues and care about them. The rst one, respectively nonlinearities , can be considered as a deterministic process in the sense that in the absence of noise the output signal depends only on the input signal. The second one, respectively noise , is purely stochastic: in the absence of an input signal, the output signal is not null and cannot be predicted at any arbitrary instant. At rst, these two issues can be thought of as independent and solved by means of two distinct procedures. However, it turns out that they are actually coupled. Indeed, all the noise that is not correctly removed from the measurements could be misinterpreted as nonlinearities, thus polluting measurements. And if nonlinearities are not accurately estimated, they will end up within the noise signal and information about the structure under study will be lost. In this paper, we thus try to estimate nonlinearities while quantifying the uncertainties on these estimations that result from noise.

The underlying idea consists in extracting the maximum of available linear and nonlinear deterministic information from measurements without misinterpreting noise.

The rst problem addressed here is related to the estimation of nonlinear models of vibrating structures [START_REF] Pearson | Discrete-Time Dynamic Models[END_REF][START_REF] Kerschen | Past, present and future of nonlinear system identication in structural dynamics[END_REF]. Some approaches are based on a physical modeling of the structure whereas some perform without any physical assumption (black-box models). As nonlinear mechanisms in structures are complex and various and as we do not intend to build a model for each case, we choose to rely on black-box models. Among these black-box approaches, some assume a given form for the selected model (block-oriented models [START_REF] Chen | Modeling and identication of parallel nonlinear systems: Structural classication and parameter estimation methods[END_REF][START_REF] Janczak | Identication of Nonlinear Systems using Neural Networks and Polynomial Models: A Block-Oriented Approach[END_REF][START_REF] Giri | Block-Oriented Nonlinear System Identication[END_REF][START_REF] Schoukens | Structure discrimination in block-oriented models using linear approximations: A theoretic framework[END_REF]) whereas some do not put constraints on the model organization. Because block-oriented models can be interpreted easily, this class of models has been retained. A class of block-oriented models that is particularly interesting is the class of parallel Hammerstein models (see Fig. 1). It belongs to the class of Sandwich models [START_REF] Chen | Modeling and identication of parallel nonlinear systems: Structural classication and parameter estimation methods[END_REF] and is shown to possess a good degree of generality [START_REF] Pearson | Discrete-Time Dynamic Models[END_REF]. Moreover, thanks to exponential sine sweeps [START_REF] Muller | Transfer-function measurement with sweeps[END_REF][START_REF] Stan | Comparison of dierent impulse response measurement techniques[END_REF][START_REF] Majdak | Multiple exponential sweep method for fast measurement of head-related transfer functions[END_REF], nonparametric versions of such models can be very easily and rapidly estimated [START_REF] Rébillat | Identication of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF][START_REF] Novak | Nonlinear system identication using exponential swept-sine signal[END_REF]. The procedure developed recently for the nonparametric estimation of parallel Hammerstein models [START_REF] Rébillat | Identication of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF][START_REF] Novak | Nonlinear system identication using exponential swept-sine signal[END_REF] will thus be extended here in order to be able to take into account stochastic plant noise. This procedure has already proven to be very useful to study vibrating structures in various contexts [START_REF] Ege | Vibroacoustics of the piano soundboard: (non)linearity and modal properties in the low-and mid-frequency ranges[END_REF][START_REF] Novak | Nonlinear acoustic characterization of micro-damaged materials through higher harmonic resonance analysis[END_REF][START_REF] Rébillat | Nonlinear structural damage detection based on cascade of hammerstein models[END_REF]. The second problem addressed here is related to the estimation of uncertainties caused by the presence of noise in the context of nonlinear system estimation. Of prime importance for this issue is the design of the excitation signals.

The use of special types of periodic excitations has been proved particularly advantageous in this respect [START_REF] Pintelon | System Identication: A Frequency Domain Approach[END_REF]. A typical example is provided by multisines (i.e. a Fourier series with random phases) where some harmonics are voluntary removed. The detection of energy at the non-excited frequencies in the system response then provides a clear evidence of the presence of nonlinearites and also enables their quantication and, to some extent, their qualication (e.g. whether they correspond to odd or even orders). Another property of multisines is to allow the separation of nonlinearities from measurement noise. Several strategies towards this aim are described in reference [START_REF] Pintelon | System Identication: A Frequency Domain Approach[END_REF]. They usually require the use of several multisine excitations with varying gains and/or randomly diering phases. In a recent work, a fast method was proposed to identify nonlinearities based on the use of a nonstationary excitation with an underlying periodic structure [START_REF] Zhang | Fast detection of system nonlinearity using nonstationary signals[END_REF]. The idea was to progressively stear the system outside its linear range and at the same time to benet from the properties of a multisine excitation to quantify the emergence of nonlinearities and separate out plant noise. The aim is here to demonstrate that the use of such a procedure based on multiple exponential sine sweeps will allow for a more robust and ecient estimation of nonlinear models of vibrating structures.

The aim of this paper is thus to provide a methodology that allows for the autonomous estimation of nonlinearities and uncertainties by bootstrap on a given vibrating structure. Nonlinearities are estimated by means of a blockoriented nonlinear model approach based on parallel Hammerstein models and on exponential sine sweeps [START_REF] Rébillat | Identication of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF][START_REF] Novak | Nonlinear system identication using exponential swept-sine signal[END_REF]. Estimation uncertainties are simultaneously assessed using repetitions of the input signal (multi exponential sine sweeps) as the input of a bootstrap procedure. Mathematical foundations and practical implementation of the method are discussed on an experimental example. The experiment chosen here consists in exciting a steel plate under various boundary conditions with exponential sine sweeps and at dierent levels in order to assess the evolutions of nonlinearities and to estimate uncertainties over a wide range of frequencies and input amplitudes. The paper is organized as follows. The original method developed for the nonparametric estimation of parallel Hammerstein models [START_REF] Rébillat | Identication of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF][START_REF] Novak | Nonlinear system identication using exponential swept-sine signal[END_REF] is rst rapidly described in Sec. 2. Then, the benets of the use of multiple exponential sine sweeps for the estimation of nonlinear models of vibrating structures is discussed in Sec. 3. Performances of the whole method are then illustrated experimentally on a vibrating plate in Sec. 4.

Identication of parallel Hammerstein models

This section aims at resuming the identication of parallel Hammerstein models by means of exponential sine sweeps, as initially proposed in Refs. [START_REF] Rébillat | Identication of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF][START_REF] Novak | Nonlinear system identication using exponential swept-sine signal[END_REF]. It introduces the quantities and notations that will be used later in the paper.

Parallel Hammerstein models

Volterra series are a convenient and general tool that provides an analytical expression of the relationship between the input e(t) and the output s(t) of a Hammerstein models based on exponential sine sweeps models [START_REF] Rébillat | Identication of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF][START_REF] Novak | Nonlinear system identication using exponential swept-sine signal[END_REF]. A: Generation of an exponential sine sweep e(t) from parameters (f 1 , f 2 , Tm). B: Excitation of the nonlinear system and recording of the output signal s(t). C: Generation of an inverse lter y(t) from parameters (f 1 , f 2 , Tm). D: Convolution of the inverse lter y(t) with the output signal s(t). E: Extraction of intermediate values {gn(t)} by means of temporal windowing. Computation of {hn(t)} from {gn(t)} using the matrix C.

weakly non-linear system [START_REF] Boyd | Fading memory and the problem of approximating nonlinear operators with volterra series[END_REF]. A system is considered here as weakly nonlinear if it does not exhibit any discontinuous nonlinearity (such as a hard saturation for example). The class of weakly nonlinear systems is thus rather general and encompasses many real-life applications. Parallel Hammerstein models constitute an interesting subclass of Volterra systems [START_REF] Gallman | Iterative method for identication of nonlinear-systems using a Uryson model[END_REF][START_REF] Pearson | Discrete-Time Dynamic Models[END_REF]. This class of model is very appealing as it is at the same time easy to estimate and to interpret as well as being mathematically still rather general.

In a parallel Hammerstein models [START_REF] Gallman | Iterative method for identication of nonlinear-systems using a Uryson model[END_REF][START_REF] Pearson | Discrete-Time Dynamic Models[END_REF], each branch is composed of one nonlinear static polynomial element followed by a linear one, h n (t), as shown in Fig. 1. Mathematically, the relation between the input e(t) and the output s(t) of such a system is given by Eq. ( 1), where * denotes the convolution product:

s(t) = N n=1 (h n * e) n (t). (1)
In this model, each impulse response h n (t) is convolved with the input signal raised to its n th power and the output s(t) is the sum of these convolutions. The rst impulse response h 1 (t) represents the linear response of the system. The other impulse responses {h n (t)} n∈{2•••N } model the nonlinearities. The family {h n (t)} n∈{1•••N } will be referred to as the kernels of the model. Any parallel Hammerstein model is fully represented by its kernels.

Exponential sine sweeps

Estimating each kernel h n (t) of a parallel Hammerstein models is not a straightforward task. A simple estimation method that has been proposed previously [START_REF] Rébillat | Identication of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF][START_REF] Novak | Nonlinear system identication using exponential swept-sine signal[END_REF] for this purpose is briey recalled here (see Fig. 2 for an overview of the method). To experimentally cover the frequency range over which the system under study has to be identied, cosines with time-varying frequencies are commonly used. When the instantaneous frequency of e(t) = cos[φ(t)] is increasing exponentially from f 1 to f 2 in a time interval T , this signal is called an Exponential Sine Sweep (see Fig. 2.A). It can be shown [START_REF] Rébillat | Identication of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF][START_REF] Novak | Nonlinear system identication using exponential swept-sine signal[END_REF] that by choosing

T m = (2mπ -π/2)ln( f2 f1 )/2πf
∀n ∈ N * , cos [nφ(t)] = cos [φ(t + ∆t n )] with ∆t n = T m ln(n) ln( f2 f1 )
.

(

) 2
Equation ( 2) states that for any exponential sine sweep of duration T m , multiplying the phase by a factor n yields the same signal, advanced in time by ∆t n .

Kernel recovery in the time domain

If an exponential sine sweep is presented at the input of parallel Hammerstein models, by combining Eq. ( 1) and Eq. ( 2) and by using properties of Chebyshev polynomials (see Appendix), one obtains the following relation (see Fig. 2.B):

s(t) = N n=1 (g n * e) n (t + ∆t n ) with g n (t) = N n=1 ck,n h k (t) (3)
where g n (t) can be interpreted as the contribution of the dierent kernels to the n th harmonic and ck,n is the coecient (n, k) of the matrix C. Details of the computation of the matrix C are provided in the Appendix.

In order to identify each kernel h n (t) separately, a signal y(t) operating as the inverse of the input signal e(t) in the convolution sense can be built as shown in [START_REF] Rébillat | Identication of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF][START_REF] Novak | Nonlinear system identication using exponential swept-sine signal[END_REF] (see Fig.

(y * s)(t) = N n=1 g n (t + ∆t n ). ( 4
)
Because ∆t n ∝ ln(n) and f 2 > f 1 , the higher the order of nonlinearity n, the more advanced is the corresponding g n (t). Thus, if T m is chosen long enough, the dierent g n (t) do not overlap in time and can be separated by simply windowing them in the time domain (see Fig. 2.E). Using Eq. ( 3), the family {h n (t)} n∈{1•••N } of the kernels of the parallel Hammerstein models under study can then be fully extracted as

[h 1 (t) • • • h N (t)] T = C[g 1 (t) • • • g N (t)] T . (5)

Advantages and limitations of the method

The method described here [START_REF] Rébillat | Identication of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF][START_REF] Novak | Nonlinear system identication using exponential swept-sine signal[END_REF] thus easily provides a direct mathematical access to all the kernels of a parallel Hammerstein model. The main advantage of this exponential sine sweep based method is to be fast and simple: using only one exponential sine sweep one can have direct access to the kernels of an arbitrary vibrating device without the need of any complicated signal processing procedure (see Fig. 2).

The method presented here is not exactly the same as the standard sine sweep method [START_REF] Muller | Transfer-function measurement with sweeps[END_REF][START_REF] Stan | Comparison of dierent impulse response measurement techniques[END_REF]. It can be thought as similar as the input signal being used is still a sine sweep. However, the sine sweep used here has to satisfy some specic additional requirements. First of all, the input sine sweep must have its frequency varying exponentially with time. Secondly, the length of the exponential sine sweep cannot be set arbitrarily and must be chosen among a given set of admissible values. Finally, the two methods do not have the same purpose. With the standard sine sweep method, the aim is to estimate the linear impulse response of a given system while being robust to nonlinearities.

With the proposed method, the aim is to estimate a whole nonlinear model of the system under study. The method presented here can thus be thought as an extension of the standard sine sweep method.

However, this method still needs to be improved to some extent. The main limitation lies in its unability to distinguish between nonlinearities and experimental noise. This results in an overestimation of the nonlinear behavior as estimated kernels also include some plant noise. Another limitation of this method is related to the parameter N that denotes the number of kernels (or equivalently the number of branches) to be estimated in the parallel Hammerstein models. For the moment this parameter is chosen arbitrarily or by using some empirical rules and eorts have to be done to estimate it automatically. These questions constitute the main topics of this paper and will be addressed in the following section.

Finally, recent studies [START_REF] Torras-Rosell | A new interpretation of distortion artifacts in sweep measurements[END_REF][START_REF] Ciric | On the eects of nonlinearities in room impulse response measurements with exponential sweeps[END_REF] have shown that distortion artifacts can appear in the causal part of the impulse responses estimated by means of exponential sine sweep when the system under study is not perfectly modeled by parallel Hammerstein blocks. Indeed, in practical applications one will never face a system that is exactly a parallel Hammerstein model as this kind of system corresponds to a mathematical idealization. In this situation the method proposed here will thus unavoidably return some sort of artefact present in the causal part of the estimated impulse responses. However, as mentioned by references [START_REF] Torras-Rosell | A new interpretation of distortion artifacts in sweep measurements[END_REF][START_REF] Ciric | On the eects of nonlinearities in room impulse response measurements with exponential sweeps[END_REF] those artifacts remain of very small amplitude.

Estimation uncertainties

The identication method described in the previous section originally used a single sweep for the excitation signal, as described by Eq. ( 2). Although this is enough to decompose the system output into nonlinear contributions of dierent orders, it does not allow for the estimation of uncertainties. This may be troublesome when weak nonlinearities typically related to higher-order kernels are highly contaminated by plant noise. Inspired from Refs. [START_REF] Pintelon | System Identication: A Frequency Domain Approach[END_REF][START_REF] Zhang | Fast detection of system nonlinearity using nonstationary signals[END_REF], a simple solution to alleviate this situation is to repeat the same exponential sine sweep several times so as to excite the system with a periodic signal. By taking advantage of repeated experiments, it is then possible to estimate the contribution of plant noise by synchronous averaging. The use of the synchronous average on repeated experiments is not a novel contribution as such; yet, when combined in a certain manner with the exponential sine sweep method, it provides rather unique and simple ways 1) to automatically test for the determination of the eective number of kernels to estimate and 2) to assess estimation uncertainties on the nonlinear kernels by use of the bootstrap. These issues are discussed in the following subsections.

Noise estimation by synchronous averaging

The principle is to repeatedly excite the system with the same sine sweep e 0 (t). The input signal then reads e(t) = K-1 k=0 e 0 (t -kT m ), with K periods of duration T m indexed by k (see Sec. 2.2 for the denition of T m ). A property of non-linear systems described by parallel Hammerstein branches is to respond with a periodic output sharing the same period T as the input (PISPO system: Period In Same Period Out). The same property obviously holds for the deconvolved signal. Specically, let

x(t) = g(t) + n(t) (6)
denote the noisy counterpart of Eq. ( 4) where g(t) = (y * s)(t) = N n=1 g n (t + ∆t n ) has now been periodized such that g(t) = g(t -T m ) and n(t) stands for plant noise.

The aim is now to separate g(t) from n(t). Since the system output is theoretically periodic in the absence of noise, a natural estimate of it is provided by the synchronous average

ĝ(t) = 1 K K-1 k=0 x(t -kT m ) (7)
and, by subtraction, an estimate for plant noise is

n(t) = x(t) -ĝ(t). (8)
Note that these estimates hold whatever the probability and power spectral distributions of the noise provided that it is uncorrelated from one period to another. In addition, it is readily checked that they are unbiased. Given a number K of periods of the same sweep, the variance of ĝ(t) can be shown to be equal to σ 2 n /K with σ 2 n the variance of plant noise of which an estimate at time t is returned by

σ2 n (t) = 1 K K-1 k=0 n(t -kT m ) 2 . (9)
Therefore, the noise-to-signal ratio (NSR) of the synchronous average decreases proportionally with K. Specically, by dening

NSR(K ) = 20 log 10 RMS(ĝ -g) RMS(ĝ) , (10)
the NSR based on K averages (wherein RMS(z) stands for the Root-Mean-Square value of signal z(t)), one has the simple relationship

NSR(K ) = NSR(1) -3 • log 2 (K) (11)
where NSR(1) = RMS(n)/RMS(x) is the initial NSR. Therefore, the NSR decreases by 3 dB when doubling the number of averages. However, even if this formula may suggest that noise disappears when increasing the number of averages, one should keep in mind that noise will never be completely eliminated from nite-length measurements.

Using the synchronous average ĝ(t) in place of g(t) in the identication method of section 2 will obviously decrease the uncertainties on the estimation of the non-linear kernels h n (t). However, despite the NSR being decreased, the complete removal of noise is not possible (as long as K < ∞) and it still remains to assess the limit where weak nonlinearities can be statistically distinguished from noise and the corresponding uncertainties estimated. These two issues are addressed in the following two subsections.

Autonomous determination of the number of kernels

The principle of the exponential sine sweep based method is rooted on the detection of the n th order harmonic contributions g n (t), n = 1, ..., N in the timedomain. As illustrated in Fig. 2.E, it might not be obvious at rst to determine how many g n (t)'s have been excited when higher-order ones are likely to be masked by noise.

However, the determination of the correct number N of harmonic contributions is crucial to return unbiased estimates [START_REF] Rébillat | Identication of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF]. This is due to the fact that the kernels h n (t) to be estimated are depending on all identied harmonic contributions g k (t), k = 1, ..., N , as seen by inverting the second formula in Eq. ( 3). Too small a number would result in underestimating the nonlinearities in the system. On the contrary, too large a number would considerably increase estimation noise.

An autonomous procedure is proposed that determines the eective number of kernels which can be distinguished from plant noise. It is based on the following statistical F-test. Let's consider the synchronous average ĝ(t) in a short time interval I n centered around the expected occurrence of the n-th kernel, i.e. t ∈ I n = [t n,1 , ..., t n,2 ] with ∆t n ≤ t n,1 and t n,2 < ∆t n-1 . The question is whether ĝ(t) signicantly protrudes from background noise in that interval and therefore originates from a n-th nonlinearity or whether it is mainly noise (see Fig. 6). This corresponds to two alternative hypotheses: H0: ĝ(t), t ∈ I n is plant noise only H1: ĝ(t), t ∈ I n contains an n th order harmonic contribution g n (t).

Following the analysis of section 3.1 and making use of Eq. ( 9), an estimate of the variance in interval I n under H0 is returned by

σ2 In|H0 = 1 t n,2 -t n,1 + 1 tn,2 t=tn,1 1 K σ2 n (t) (12)
(note the division by K since the level of noise on the synchronous average is K times as small as on the initial signal according to formula [START_REF] Rébillat | Identication of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF]). Therefore, assumption H0 is to be rejected if the current variance

σ2 In = 1 t n,2 -t n,1 + 1 tn,2 t=tn,1 ĝ(t) 2 (13)
is found statistically greater than σ2

In|H0 . This can be easily formalized with a F-test for the equality of two variances: H0 is rejected if the ratio σ2

In /σ 2 In|H0 is greater than F (α) K-1,K-1 , the critical value of the F distribution with K -1 and K -1 degrees of freedom and a signicance level of α.

The highest-order harmonic contribution g N (t) that is found signicant through this procedure then determines the model order N .

Bootstrap assessment of uncertainties

Eventually, the use of repeated exponential sine sweeps also gives access to the uncertainties on the non-linear kernels h n (t). Even if a full analytic treatment of the question is possible in the case of Gaussian noise, a resampling strategy based on the bootstrap is preferred due to its simplicity and versatility with respect to unknown noise properties [START_REF] Zoubir | Bootstrap Techniques for Signal Processing[END_REF].

The bootstrap is a technique to articially produce random repetitions of the experiment from the available data. The aim is to obtain an histogram for the estimated kernels h n (t) from which condence intervals (or any other measure of statistical dispersion related to uncertainties) can easily be constructed. This is achieved by producing a series of virtual repetitions of the same experiment.

Let x k (t) = x(t -kT m ), 0 ≤ t < T m , denote the k-th cycle of signal x(t) (as dened in Eq. ( 6)) and π a random draw of the integers 0, ..., K -1 (i.e. each element π(i), i = 0, ..., K -1, is randomly assigned a value of set 0, ..., K -1 with uniform probability 1/K). A new virtual measurement x (b) (t) (where (b) denotes the bootstrap index and is not to be confused with an exponentiation) is then produced by concatenating K random draws with replacement,

x π(0) (t); ...; x π(K-1) (t) . [START_REF] Novak | Nonlinear acoustic characterization of micro-damaged materials through higher harmonic resonance analysis[END_REF] from which the synchronous average

ĝ(b) (t) = 1 K K-1 k=0 x π(k) (t) (15)
is computed. By means of an example with K = 3, one possible bootstrap draw, say b = 1, would return {π(0); π(1); π(2)} = {2, 1, 1} and therefore ĝ(1) (t) =

U (b) n (f ) = H (b) n (f ) - 1 B B-1 b=0 H (b) n (f ). ( 16
)
The mean square uncertainty on the n-th kernel based on K averages may then be dened as

MSU n (f, K) = 1 B B-1 b=0 |U (b) n (f )| 2 . ( 17
)
Finally, the uncertainty-to-signal ratio (USR) can be dened as:

USR n (K) = MSU n (f, K)df 1 B B-1 b=0 |H (b) n (f )| 2 df ( 18
)
where the integrals are taken over the full frequency band of interest.

This index can be interpreted as a compact way to assess the quality of the estimated kernels and can be compared to NSR(K ) as dened in Eq. [START_REF] Rébillat | Identication of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF]. NSR(K ) stands for the ratio of the energy of the noise versus the energy of the estimated signal for K repetitions of the exponential sine sweep. USR n (K) denotes the ratio of the energy of the uncertainty on the n th kernel versus the energy of the estimated n th kernel for K repetitions of the exponential sine sweep. As the uncertainty on the n th kernel may be linked with the variance of noise, a correlation between both indexes is to be expected.

Note that the bootstrap is a computer intensive method. Yet it was found reasonably fast to run for values of B up to the order of a few hundreds.

4 Application to a vibrating plate

Experimental setup

In order to illustrate the performance of the proposed extended exponential sine sweep method, vibratory measurements have been done on a simple mechanical system. The chosen structure is a thin rectangular steel plate of 540 × 640 × 1 mm 3 dimensions. An electrodynamic mini-shaker (B&K Type 4810) is used to excite the plate at a point chosen arbitrarily (see Fig. 3). Acceleration of the plate is measured at two positions: at the driving point through an impedance head (PCB Type 288D01) and at several centimeters from it using a second accelerometer (B&K Type 4508). Two congurations have been tested: in the rst one (see Fig. 3(a)) the plate is clamped at edges with four metal bars (of 2 cm width) screwed to the top ledges of a cavity of 500 × 600 × 700 mm 3 dimensions.

in the second conguration (see Fig. The extended exponential sine sweep method presented here has been applied to these two dierent congurations. The aim is not to compare the amount of nonlinearities for dierent test cases or to discuss the relative results of dierent experimental congurations, but simply to show how the methodology can be easily applied to examples of common engineering problems which may arise in the vibroacoustic community: a typical lightly damped clamped structure and a highly damped free-free one. These two systems with dierent boundary conditions and dierent amounts of damping exhibit dierent levels of nonlinearities from dierent origins (geometrical nonlinearities, contacts, . . . ).

For each conguration the plate is excited with 20 exponential sine sweeps of 30 sec each, repeated periodically. The instantaneous frequency of the sweep increases from 20 Hz to 1 kHz. The sampling frequency is xed at 25.6 kHz in order to allow for the estimation of high order kernels. Several measurements at dierent gains have been performed in order to assess the evolutions of nonlinearities and of the signal to noise ratio over a wide range of frequencies and input amplitudes.

Assessment of estimation uncertainties

This subsection illustrates the assessment of estimation uncertainties by means of the bootstrap technique described in Sec. 3.3. The signal under study corresponds to the second accelerometer (glued on the plate several centimeters away from the impedance head) for the rst conguration where the plate is clamped (see Fig. 3(a)). It contains a maximum of K max = 20 repetitions of the response of the system to exponential sine sweeps. During the rst repetition the system is considered to exhibit a transient response while it reaches its stationary regime when the second repetition of the sweep is starting up. For that reason, the rst repetition has been removed from the analysis. The objective is here to demonstrate the USR (see Eq. ( 18)) improvement caused by the synchronous averaging process. for the estimation of non-linear kernels up to order N = 6. The estimated kernels are displayed in each gure together with the corresponding mean square uncertainty MSU(f, K) of Eq. ( 17) as obtained from bootstrap. One can clearly see the benet of reducing the noise by synchronous averaging, especially for the high-orders kernels for which the initial USR without averaging is very low (-11 dB for the 6 th order kernel for example) and becomes acceptable after averaging (-23 dB for the 6 th order kernel). Moreover, comparison of Figs. 4(a) and 4(b) displays a reduction of 13 dB of the estimation uncertainties for each kernel order, which is very close to the value predicted by Eq. [START_REF] Rébillat | Identication of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF]. Figure 5 then illustrates the improvement with respect to K of both the NSR as given by Eq. ( 11) and the USR as given by Eq. ( 18). As expected, the curves evidence an excellent correlation between both indexes.

Figure 5: Improvement with respect to K of both the NSR as given by Eq. [START_REF] Rébillat | Identication of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF] and the USR as given by Eq. ( 18).

Determination of the eective number of kernels

This section illustrates the automatic determination of the eective number of kernels N opt as explained in Sec. 3.2. Measurements are now performed on the second conguration (suspended damped plate, see Fig. 3(b)). The signal studied here corresponds to the second accelerometer, after synchronous averaging on all periods except the rst one (transient regime). Four gains for the excitation signal are successively tested G = 0.1 -1 -5 and 10 in order to study the robustness of the determination method to predict the proper number of kernels to estimate. For the fourth gain (G = 10), the excitation force delivered from the shaker is one hundred times higher than for the rst gain (G = 0.1). For example, in terms of displacement, at a resonance frequency around 130 Hz and at the point of measurement of the second accelerometer (several centimeters away from the shaker), the gain G = 0.1 creates a displacement of about 5 µm whereas the gain G = 10 generates a displacement of about 0.5 mm, that is half the plate thickness (1 mm). It is obvious that with such large-amplitude vibration motions, geometrical nonlinearities appear strongly with high order kernels being excited. In Fig. 6, the determination of the eective number of kernels (using the methodology presented in Sec. 3.2) is illustrated in the time-domain on the signal g(t) = (y * s)(t) (output signal after deconvolution, see Eq. ( 4)). The thresholds dened in Sec. 3.2 (through F-test at 0.995 and with a security parameter chosen equal to 6) is displayed in red lines in Fig. 6 together with the intervals I n which are materialized by the vertical shaded areas. These thresholds allow estimation of the eective number of kernels to retain for each excitation level. Green windows are displayed for retained kernels and red windows for rejected ones.

As expected, the optimal model order N opt of retained kernel increases with the level of excitation: N opt = 1-3-8 and 12 respectively. These estimated kernels are presented in Fig. 7 for each gain. One can note that the level of the kernels increases with the excitation gain. Moreover, the frequency content of the rst kernel (corresponding to the linear part) is very similar for each measurement, thus assessing the reproducibility and precision of the measurements. To sum it up, the methodology presented here is a useful tool to select the proper model order N opt and a major improvement of such nonlinearity estimation methodologies.

Conclusion

The aim of this paper is to provide a methodology that allows for the autonomous estimation of nonlinearities and of uncertainties by bootstrap on a given vibrating structure. Nonlinearities are estimated by means of a blockoriented nonlinear model approach based on parallel Hammerstein models and on exponential sine sweeps. Estimation uncertainties are simultaneously assessed using repetitions of the input signal (multi-sine sweeps) as the input of a bootstrap procedure. Mathematical foundations and practical implementation of the method are discussed on an experimental example. The experiment chosen here consists in exciting a steel plate under various boundary conditions with exponential sine sweeps and at dierent levels, in order to assess the evolutions of nonlinearities and of the signal-to-noise ratios over a wide range of frequencies and input amplitudes.

The major improvements provided in this article are: the fact that the synchronous average allows the design of a simple statistical test to automatically determine the eective number of non-linear kernels in the exponential sine sweep method (in a manner that is actually quite unique to this method), the fact that the synchronous average is combined with the bootstrap to propose a versatile technique for assessing the uncertainties on any quantity of interest by taking advantage of repeated measurements. Subsequently, it can easily be shown that they satisfy the recurrence relation given in Eq. [START_REF] Torras-Rosell | A new interpretation of distortion artifacts in sweep measurements[END_REF].

k = 0 T 0 (x) = 1 (20a) k = 1 T 1 (x) = x (20b) k > 1 T k+1 (x) = 2xT k (x) -T k-1 (x) (20c)
Then, by writing the polynomials as in Eq. ( 21) , one can obtain Eq. ( 22), using Eq. ( 20), and nd the coecients of the matrix A.

T k (x) = k i=0 A(i, k)x i (21)
i = 0 A(0, k + 1) = -A(0, k -1)

(22a) 0 < i < k A(i, k + 1) = 2A(i -1, k) -A(i, k -1) (22b) i k A(i, k + 1) = 2A(i -1, k) (22c)
The linearisation of the polynomials can now be rewritten in a matrix form, as in Eq. ( 23). The matrix C, necessary to access to {h n (t)} n∈[1,N ] , is the matrix A without the rst column and the rst row, as seen in Eq. ( 5). To avoid the implementation of the recurrence, the C matrix of order 8, which is sucient for practical use, is given in Eq. ( 25).

Figure 1 :

 1 Figure 1: Representation of parallel Hammerstein models

Figure 2 :

 2 Figure 2: Schematic representation of the estimation procedure for parallel

  2.C). After convolving the output of Parallel Hammerstein models s(t) given in Eq. (3) with y(t), one obtains Eq. (4) (see Fig.2.D) :

n

  (t), b = 1, ..., B, from which a histogram can nally be calculated. Similarly, histograms are available on any transform of the estimated kernels, in particular on their Fourier transforms as displayed in Sec. 4. Specically, by denoting H (b) n (f ) the Fourier transform of the bootstrap estimate of the n-th kernel h (b) n (t) from a random draw b, the uncertainties in the frequency domain are evaluated by the dierence

Figure 3 :

 3 Figure 3: Overview of the experimental setup. (a) First conguration: steel plate coupled to a cavity (clamped boundary conditions). (b) Second conguration: suspended damped plate (free boundary conditions).

Figures 4 (

 4 Figures 4(a) and 4(b) display the eect of the synchronous average over K = 19 repetitions of the exponential sine sweep as compared to K = 1

Figure 4 :

 4 Figure 4: Illustration of the USR improvement implied by synchronous averaging. a) Frequency gains of kernels estimated on one period of the excitation signal together with the mean square uncertainty obtained classically. b) Frequency gains of kernels estimated by synchronously averaging K = 19 periods of the excitation signal together with the mean square uncertainty MSU n (f, 19) obtained by bootstrap (B = 150 random draws with replacement).

Figure 6 :

 6 Figure 6: Illustration of the determination of the eective number of kernels for the second conguration (suspended damped plate, see Fig.3(b)) for four dierent excitation gains (G = 0.1 -1 -5 -10). The signal |g(t)| = |(y * s)(t)| (output signal after deconvolution, see Eq. (4)) in the time domain is displayed in dB in blue and the selection thresholds in red. Retained kernels are displayed with green windows, and rejected ones with red windows.

Figure 7 :

 7 Figure 7: Estimated kernels for the second conguration (suspended damped plate, see Fig.3(b)) after determination of the eective number of kernels for each of the four excitation levels.

6

 6 Acknowledgments Since 2011, the Laboratoire Vibrations Acoustique is part of the LabEx CeLyA of Université de Lyon, operated by the French National Research Agency (ANR-10-LABX-0060/ ANR-11-IDEX-0007). Part of this work was presented at the NOVEM 2015 conference (NOise and Vibration: EMerging technologies) [23]. Appendix: Computation of the matrix C Chebyshev polynomials {T k [cos(φ)]} k∈N are dened by Eq. (19). ∀k ∈ N, cos(kφ) = T k [cos(φ)]

  [START_REF] Gallo | A multi-sine sweep method for the characterization of weak non-linearities; plant noise and variability estimation[END_REF] results in Eq. (24) which gives explicitly the C matrix.C = A -1 (24)

  1 with m ∈ N * one obtains the following property:

  (t) is obtained by following the procedure of Sec. 2. This is repeated B times such as to collect B estimates h

	1 3 (x 2 (t)+x 1 (t)+x 1 (t)); another draw would return {π(0); π(1); π(2)} = {0, 2, 0} and therefore ĝ(2) (t) = 1 3 (x 0 (t) + x 2 (t) + x 0 (t)), etc . . .
	Next, for each bootstrapped synchronous average ĝ(b) (t) a new estimate
	h (b)

n