Dr Farah Kawtharani

Dr Svetlana Mintova

Dr Richard Retoux

Mehran Mostafavi

Dr Guy Buntinx

Dr Vincent De Waele
email: vincent.de-waele@univ-lille.fr

Dr Olivier Poizat

Pr M Mostafavi

Hot-electron photodynamics in silver-containing BEA-type nanozeolite studied by femtosecond transient absorption spectroscopy

Keywords: pump-probe spectroscopy, plasmonic, zeolite, silver nanoparticle, hot electron

Supporting information for this article is given via a link at the end of the document: BET, XRD, HRTEM and additional UV-vis characterizations, Additional sets of transient data and fits of the pump-probe decays.

Introduction

Zeolites are microporous crystalline solids constituted by a regular 3-dimensional arrangement of TO4 groups (T = Si, Al) that form cages and channels with a size lower than 2 nm. The zeolite materials offer unique possibility to tailor the properties of photoactive guest chemical species [1][2][3][4] , organo-metallic complexes [5,6] , metal clusters and nanoparticles [7,8] by confining them inside the pores and aligning along their 3D architecture [9,10] . Metal-containing zeolites are notably investigated for photocatalysis [11][12][13] and one attractive route of development consists coupling of zeolites with plasmonic nanostructure to enhance their photocatalytic activity [14][15][16] . Plasmonic chemistry [17][18][19][20] is an emerging area of the photocatalysis that aims to exploit the photoexcitation of the electrons of metal nanoparticle to trigger chemical reactions at the surface of the metal nanoparticles. These processes occur in femtosecond-picosecond time-scale and they have been investigated by transient absorption spectroscopy [21][22][23][24][25] . The dynamics of the hot-electrons generated upon the absorption of photons has been characterized for metal nanoparticle of different sizes [21,24,26,27] and shapes, in weak and high perturbation regime, and different surrounding and surface interactions [28][29][START_REF] Bauer | EQEC '05. European Quantum Electronics Conference[END_REF] .

In view of these emerging plasmonic applications, we are interested in nanosized zeolites [START_REF] Awala | [END_REF]32] with a size of 5-100 nm, that can be stabilized in transparent colloidal suspensions or assembled in thin films for light-driven applications. Nanozeolites were successfully functionalized by the incorporation of metallic [33,34] and semiconductor [35] nanoparticles (CdS, ZnS, CdSe) with different shapes and size using radiolysis [33,36,37] , chemical [34,38] and photochemical [37,39] approaches. For most of the systems investigated, the synthesis conditions can be optimized to favor the stabilization of relatively small (1-3 nm) metal and semiconductor nanoparticles. Such very small supported nanoparticles are interesting for plasmonic applications because, in principle, the smaller the particles, the higher the density of energy acquired by the absorption of light. We verified this assumption in our previous study [40] , where the hot electron dynamics of 1 nm-sized silver nanoparticles (Ag-NPs) stabilized in LTL type nanozeolite assembled in transparent thin films was studied. This study was performed on template-free LTL nanozeolite dehydrated under high vacuum. We demonstrated that highly excited electrons can be photogenerated in the zeolite film under low light illumination. In the case of small metallic nanoparticles supported in zeolites, the interactions at the surface of the metals with the zeolite or the molecules confined inside their porous volume, can impact the hot electron relaxations.

In the present work, the synthesis and the characterization of silver nanoparticles (Ag NPs) in two zeolite samples with a Si:Al ratio of 13 and 7.5 stabilized in aqueous colloidal suspensions is presented. The ultrafast optical response of these Ag-BEA samples to characterize the impact of the zeolite on the hot electron lifetime was elucidated. The crystallinity of zeolite samples BEA-1 and BEA-2 synthesized with two different Si/Al ratios was studied by X-ray diffraction (XRD). The XRD patterns of the samples are depicted in Figure 1. Both samples exhibit the Bragg reflections corresponding to the BEA-type framework structure, consisting of a broad peak composed of two peaks at 7.25°and 7.7° 2θ with Miller indices (hkl) of ( 100) and ( 101), respectively and a peak at 22.5° with hkl (302). [START_REF] Treacy | Collection of Simulated XRD Powder Patterns for Zeolites Fifth (5th) Revised Edition[END_REF] The broad peak at 7.7° 2θ indicates the presence of polymorphs A and B with an estimated ratio of 45:55. [START_REF] Kadgaonkar | [END_REF] No signatures of framework destruction or amorphization are detected after ion-exchange of the samples with silver (Figure 1). The SEM pictures show that the nanocrystals with a size of 10 nm are aggregated resulting in particles with an average size of 60-120 nm. The DLS curves reveal a monomodal particle size distribution with a mean hydrodynamic diameter of the aggregates of 114 nm (Ag + -BEA-1) and 90 nm (Ag + -BEA-2). The zeta potential values (ς) of -50mV and -45 mV for samples Ag + -BEA-1 and Ag + -BEA-2, respectively confirmed the high stability of negatively surface charged nanoparticles. The chemical composition of samples Ag + -BEA-1 and Ag + -BEA-2 determined by ICP is summarized in Table 1, while the amount of water and organic templates as determined by TG are presented in Table 1 and Figure 2. The weight losses of the samples are described by considering 4 temperature ranges similarly presented for pure BEA [43][44][45] . For Ag + -BEA-1, in region 1 (25-150 °C) a peak assigned to desorption of zeolitic water is present. The decomposition of the organic template (TEA + ) from zeolite Beta is represented by the peak in the temperature interval (150-350 °C), Region 2. In region 3 (350-500 °C), a peak situated at 353°C and a less intense peak at 430°C, are present. These peaks are assigned to the degradation of the template located in the channels compensating the negative charges of the framework The peak observed in region 4 (500-750 °C) located at 550 °C is assigned to the burning of the residues formed in region 3 and tcomplete removal of the organics. For sample Ag + -BEA-2, similar results were obtained, however the small variation of the TG profile is an indication for the different amount of TEA + species, and consequently of the distribution and the chemical environment of Ag + cations inside the zeolite crystals. Samples Ag + -BEA-1 and Ag + -BEA-2 are different in the Ag + content due to the different Si/Al ratios of the BEA zeolites and they contain different amount of template and water (Figure 2). The molar concentration of silver in the suspensions Ag 0 -BEA-1 and Ag 0 -BEA-2 are 7.1×10 -5 mol.l -1 and 2.25×10 -5 mol.l -1 , respectively; inset pictures (g) and (j) correspondingly). Inset pictures (e) and (f) show samples Ag + -BEA-1 (0.8wt%) and Ag 0 -BEA-1 (0.8wt%), correspondingly. Insets (h) and (i) show pictures of samples Ag + -BEA-2 (0.8wt%) and Ag 0 -BEA-2 (0.8wt%), correspondingly.

Results

Characterization of the non-reduced Ag-containing zeolite suspensions

The silver cations (Ag + ) were reduced to metal nanoparticles (Ag 0 ) in suspensions Ag + -BEA-1 and Ag + -BEA-2 subjected to reduction procedures A and B (see experimental section). The reduction process in the presence of triethylamine (Et3N) has been directly followed by UV-vis absorption owing to the transparency of the colloidal zeolite suspensions and to the strong plasmon absorption band of silver nanoparticles in the visible spectral range. For both samples, an excess amount of Et3N (procedure A) was used for both Ag + -BEA-1 and Ag + -BEA-2 samples; well-defined absorption spectra containing a strong band at 410 nm and a second low intensity absorption band bellow 350 nm are presented in Figure 3. As a visual indication, the color of the suspensions turns from white-pale grey to yellow. In both samples, the initial pale-grey color is associated with a broad and weak absorption/diffusion spectrum corresponding to the ionexchanged samples which is more intense for Ag + -BEA-2.

The reduction kinetics for samples Ag + -BEA-1 and Ag + -BEA-2 differs noticeably (Figure 3). In sample Ag 0 -BEA-1, the plasmon band at 410 nm and the width of the band is slightly diminishing during the formation of the Ag 0 nanoparticle, as it is usually reported for nmsized spherical Ag-NPs [46] . A similar spectral evolution for Ag 0 BEA-2 is observed but with an asymmetry of the plasmon band featuring more intense tail in the red-part of the spectra. That second contribution cannot be assigned directly. This feature is expected for large (>20 nm) or non-spherical nanoparticles, or in case of inter-particles interactions resulting from the confinement of Ag NPs in the Beta zeolite matrix. Alternatively, this asymmetry might reflect specific electronic interactions between the metal nanoparticles and their micro-environment.

The stability of the zeolite framework was further confirmed by recording the N2 sorption isotherms, XRD patterns and TEM images.

Nitrogen adsorption-desorption measurements confirmed that the microporosity of the BEA zeolite is preserved after the chemical reduction (see SI, Figure 1). The XRD patterns show that the crystallinity of the Ag 0 -BEA-1 and Ag 0 BEA-2 samples is preserved (see SI-Figure 2). In addition, the TEM images of the reduced samples reveal the crystalline fringes of BEA zeolite (SI-Figure 3). In summary, the BEA zeolite samples under the mild reduction conditions stayed intact.

To confirm the presence of Ag NPs in the BEA zeolite matrix, STEM and HAADF images were recorded (Ag 0 BEA-2). The contrast of the HAADF image is determined by the atomic number (Z-contrast) of the elements in the sample. The Ag NPs are evidenced as bright dots (due to their high Z) randomly distributed in the zeolite sample (low Z-contrast, due to lighter elements) that appears as more or less grey/background. The silver nanoparticles embedded within the zeolite matrix are highly dispersed (Figure 4). The chemical composition of two selected areas of sample Ag 0 -BEA-2 are presented in Figure 4. Two measurements are performed; on bright zone of the sample labelled as zone A1, and the other darker zone labelled as zone A2 (Figure 4). The EDS spectrum of the bright zone (zone A1) confirms the Ag nanoparticles; the Si peak which is the main constituent of the zeolite structure (except C and O), and another peak of Ag are present. On the contrary, the grey/dark zone pointed by zone A2 contains only a peak of Si, and no Ag peak was measured revealing the pure zeolite. Additional HRTEM images of Ag 0 -BEA-1 and Ag 0 -BEA-2 (see SI-Figure 3) confirmed the formation of nm-sized Ag NPs (1-2 nm) in the BEA zeolite matrix. The as-prepared suspensions are transparent over the visible spectral range and stable for months (SI-Figure 5). Alternatively, the procedure B was applied for preparation of Ag NPs. Et3N was added drop-by-drop in both suspensions; the UV-vis spectra recorded from the samples after each addition of Et3N are shown in SI-Figure 4. During the drop-by-drop reduction process, the plasmon band for sample Ag-BEA-2 broaden significantly. The maximum of the plasmon band is shifting to the red in the course of the reduction while its red-tail shows a continuous shift upwards. This spectral shape and evolution of the plasmon band indicate the formation of large Ag NPs (> 20 nm); the red-component is a contribution of the higher multipoles in the Mie scattering formulation. The same spectral evolution for sample Ag + -BEA-1 was obtained applying the procedure B. The formation of silver nanoparticles with a diameter of 30 nm aggregated at the surface of zeolite crystals is observed (SI-Figure 4). This last sample was further investigated by pump-probe spectroscopy and was named Ag 0 -BEA-1b. The zeolite suspensions reduced by method B are stable for few days only.

Ultrafast electrons-phonons photodynamics in sample Ag 0 -BEA-1 ; weaker absorption contributions at 500 nm and 320 nm are observed. This transient signal is decaying within 5 ps without any significant evolution of its spectral profile. In the red-part of the plasmon band a rapid shift and narrowing of the absorption band is observed. While in the blue part below 400 nm, the absorption and the bleach contributions decay concomitantly with an isobestic point located at nm. At longer pump-probe delays (Figure 5, t = 50 ps), the initial signal disappeared and replaced by a weak but clearly distinguishable absorption component bellow 340 nm, a bipolar-like signal with a negative lob at 370 nm and a positive one at 440 nm. This long-term transient absorption signal is slowly decaying within several hundreds of picoseconds (not shown). These two consecutive steps can be assigned as follow: the first short-lived contribution up to 5 ps agrees well with the transient spectra reported for photoexcited Ag NPs (see the review of Hartland [25] and references therein). The spectral components above 350 nm reflect the modification of the plasmon resonance resulting from formation of hot-electrons, while the component observed at higher energy (bellow 350 nm) is due to the perturbation of the inter-band optical transitions. The latest corresponds notably to the novel optical absorption resulting from the depletion of energy level below the Fermi energy (Ef). The lifetime of this signal (less than 5 ps) is characteristic with a decay governed by electron-phonon coupling following a photoexcitation under high-excitation regime [27,47] . The second long-term contribution (t = 50 ps and longer), features the blue shift of the plasmon band and a small depletion of the conduction band bellow Ef. Similar transient spectra have been assigned to the change of the dielectric constant induced by a local increase of the temperature around the nanoparticles [48] . In our experiments, the pump intensity was kept low enough to avoid any permanent change resulting from a strong heating of the Ag NPs or of its surrounding, as that was observed in photothermal or optical limitation [49] experiments.

Based on the transient spectra, we verified that the photoexcitation of Ag 0 -BEA-1 induces the formation of hot electrons that dissipate their energy within a few picoseconds into the metal lattice that induces a limited transient perturbation of the surrounding media. Because this behavior is essentially similar to that observed for bare Ag NPs, we concluded that Ag NPs stabilized in the Ag 0 -BEA-1 sample are exhibiting essentially a metallic character. In order to go beyond the qualitative characterization of the process, we were interested in revealing the influence of the zeolite microenvironment on the electronic properties of the embedded Ag NPs. In particular, one point we wanted to address concerns the possible perturbation of the hot electron decay. Indeed, such a perturbation might be interpreted formally as a signature of the interaction of the hot-electrons with the electronic levels the zeolite framework containing the organic template (TEA + ), Et3N and water molecules. The transient spectra of Ag 0 -BEA-1 were measured following the method already applied for characterization of silver nanoparticle [24,26,47] . The samples were excited at 400 nm with different pump intensity varying from I0 to I0/20, with I0 = 1.2 mJ/cm 2 (the intensity of the incident light on the sample). The electrons-phonons dynamics has been followed at the probe wavelength of 380 nm (3.25 eV). That probe energy lies well below the threshold of the inter-band transition of silver (322 nm, 3.85 eV) and it is in quasi-resonance with the plasmon. At this probe wavelength, for silver, the transmission changes are considered as being roughly proportional to the excess of internal energy of the gas of electrons. [47] The transmission changes recorded upon different laser intensities at probe=380 nm are plotted in Figure 6-a and in SI-Fig- 7. The amplitude of the transient signal and the hot electrons lifetime are decreasing when the intensity of the pump is reduced. This behavior is qualitatively in good agreement with a thermal exchange of energy between the excited hot electrons and the metal lattice. The data were fitted using a sum of exponential functions (See SI 2.2). All the decays are well reproduced by a major contribution (1 ~ 1-3 ps) counting for more than 80 % of the signal and by a second minor component (2 ~ 10 ps). The use of two exponential decays in the fit function (see SI) reflects the dependency of the heat capacity of the electron with the temperature. From the fitted curves, we deduced the hot-electron apparent lifetime as the the time t(1/e) where the intensity of the transient signal has decreased by the factor 1/e. The value t(1/e) allows us to compare the kinetics between the different samples and excitation conditions. At the highest pump intensity, the Ag NPs with a size 1-2 nm can absorb several electron-volt per nanometercube, and consequently the electron temperature, Te, can reach thousands of Kelvin. This corresponds to a hot electron lifetime in the range 2-3 ps. At the lowest laser intensity (I0 = 30 µJ/cm 2 ), Te is only a few hundreds of Kelvin and a decay time t(1/e) = 0.9 ps is measured that is close to the electrons-phonons coupling time-constant limit reported for silver nanoparticles (0.8-1.1 ps) [24,47,48] .

Ultrafast electrons-phonons photodynamics in samples Ag 0 -BEA-2 and Ag 0 -BEA-2b

The same pump-probe measurements have been performed on samples Ag 0 -BEA-2 and Ag 0 -BEA-2b. For sample Ag 0 -BEA-2, the time-resolved spectra (SI, Figure 6) are qualitatively similar to the data reported for Ag 0 -BEA-1 (Figure 5), attesting for the formation and decay of hot-electrons in the metal nanoparticles. In Figure 6-a, the decays were recorded for pump-probe conditions (intensities and overlap) identical to that of sample Ag 0 -BEA-1 (Figure 6-a). At high intensity (I0), we can notice that the decay is longer for the Ag 0 -BEA-2 while at lower intensity, the difference between the two samples is less significative. However, for both samples, we are pointing out the fact that reducing the laser intensity from I0/4 to I0/10 has a relatively weak impact on the decay time. This effect is usually observed in case of coupling between the hot-electrons and its surrounding [29,50] .

The transient spectra of sample Ag 0 -BEA-2b are presented in SI-Figure -6. While a significant difference in the steady state absorption was found (Figure 3 and SI-Figure 4) in comparison with the two other samples, the transient spectra are qualitatively similar to those obtained for Ag 0 -BEA-1 and Ag 0 -BEA-2. We noticed however the absence of a residual contribution at long time-scale, which suggests that less perturbation of the surrounding of the nanoparticles occurs during the cooling process of the metal lattice. Regarding the photodynamic of sample Ag 0 -BEA-2b,(Figure 6-b, SI-Figure 7 and 10), the hot-electrons lifetime measured under high excitation intensity (I0 = 1 mJ/cm 2 ) is significantly longer than for the two other samples. In comparison with Ag 0 -BEA-1 and Ag 0 -BEA-2, we clearly observe the acceleration of the decay while the pump intensity is decreasing as expected for the silver nanoparticles with a size of 10-20 nm

Discussion

The results reported in the previous sections have allowed us to demonstrate: (1) the stabilization of Ag NPs in nanosized BEA zeolite matrix and ( 2) the metallic character of these Ag NPs evidenced by the formation of hot-electrons upon photo-excitation. In this section, we will discuss the effects of the preparation on the ultrafast optical and electronic response of the Ag-BEA-1 and Ag-BEA-2 samples.

As mentioned in the introduction, zeolites are materials of choice for the stabilization of metal clusters and metal nanoparticle due to the spatial constraints of their framework. In addition, the presence of organic templates, TEA + in the channels of BEA zeolite can also influence the growing process of silver nanoparticles in two ways. First, by controlling the initial distribution of Ag + introduced inside the zeolite, and second by hindering the diffusion of oligomers and nascent nanoparticles. The amount of silver cations introduced in both Ag + -BEA-1 and Ag + -BEA-2 samples by ion-exchange was relatively low (Table 1), with an average ratio of 0.15-0.2 Ag + /Al. So, only a small part of the charge compensating TEA + cations initially incorporated in the zeolite have been ion-exchanged with the silver whose final concentration in the zeolite is 1.2 Ag + /u.c. and 0.6 Ag + /u.c for samples Ag + -BEA-1 and Ag + -BEA-2, respectively. After the reduction, the plasmon band at 410 nm for both samples was measured; the samples were stable for months (SI-Figure -4). We estimated the amount of reduced silver based on the optical density at the maximum of the plasmon band using the extinction coefficient of 9200 Lmol -1 cm -1 for silver atoms [46] and the concentration of Ag + in the samples determined by ICP. Within the accuracy of the extinction coefficient, which can vary with the size and shape of Ag NPs, we found that the percentage of reduced silver is in the range 90-95 %. It shows the high efficiency of our reduction procedure, and a relatively high accessibility of the Ag + cations. In our experiments, the formation of silver nanoparticles was taking place on the time scale of hours, however the reduction kinetics for the two samples was different. This can be explained with the different cation distribution, mobility or even chemical interactions for the two samples. The TG/DTG measurements (Figure 2) provide us the first evidence of different micro-environment. The TEA + and water content in samples Ag + -BEA-1 and Ag + -BEA-2 are 14.6 wt% / 1.0 wt% (TEA / water) and 17.3 wt% / 4.7wt%, respectively. Sample Ag + -BEA-1 is more hydrophobic, and contains less organic template and water in comparison to sample Ag + -BEA-2, that can be explained with the different amounts of defects (surface and bulk) (Figure 2). For Ag + -BEA-2, an increase of the total mass lost, mainly associated with a decrease of the water content, and an increase of the amine residues is observed. This corresponds probably to the incorporation of Et3N, replacing partially or totally the water molecules. Interestingly, the presence of two new additional sharp peak at 220 and 226 °C was noticed (Figure 2-d) that we tentatively assigned to the Et3N molecules in the vicinity of the Ag NPs. Another important finding is to report the formation of Ag NPs in zeolite matrix with remarkable distinction in size using two reduction procedures (see procedures A and B). HRTEM images have revealed the formation of small Ag nanoparticles with a size typically 1-2 nm and dispersed homogeneously in the zeolite matrix using the procedure A. While the reduction of Ag + -BEA-1 and Ag + -BEA-2 under drop-wised addition of Et3N (procedure B) favored the formation of unstable larger Ag NPs (20-30 nm). More than one hour was necessary to achieve the complete reduction of cationic Ag + into Ag NPs (Figure 2). By following the procedure A, reduction in excess of Et3N, the concentration of reduced silver atom is kept at its maximum during the process so that the rapid formation of dispersed nucleation centers is favored. In opposite, following procedure B, the kinetics of formation of Ag NPs is comparable to the diffusion time, and the Ag NPs can escape and coalesce as extra-framework nanoparticles. This observation is in a good agreement with the mechanism of growth of colloidal metal nanoparticles for which the faster is the creation of nucleation centers, the smaller are the formed nanoparticles. [36,51,52] The growth of metal nanoparticles in zeolite follows the same principle and the final shape and size of the NPs is the results of the initial stabilization of the small oligomeric clusters [8,34,35] tailored by the channels and porous volume followed by a slow diffusion and structural rearrangement [35] . For Ag-BEA, it appears that the rapid formation of the initial nucleation centers is a determining step to keep the final size of the Ag-NPs in the 1-2 nm range.

In the last part of the discussion we will address the consequences of the different method of preparation employed in this work on the plasmonic response of Ag NPs in BEA zeolite. The steady state and time-dependent optical response of metal nanoparticles are governed by the collective properties of the electrons that depend directly on the density of electronics states (DOS) of the metal and its surrounding. The consequence is that the photodynamic of the electrons can reflect the electronic properties and interaction of the metal nanoparticles embedded in the zeolite. The measured hot electrons decays time t(1/e) versus the pump laser intensity for the three samples are depicted in Figure 7. In metal nanoparticles, the energy absorbed by the electrons is transferred to the metal lattice by electron-phonon coupling. In this process, the hot electron lifetime increases with the amount of absorbed energy due to the temperature dependency of the electronic heat capacity. Our three samples qualitatively follow this trend, but we clearly observe distinct behaviors in both the profile of the curves and in the values of . First, the hot electron lifetimes recorded for the highest pump intensities (I > 250 µJ/cm 2 ) are significantly longer for Ag 0 -BEA-2b (2-4 ps) than for Ag 0 -BEA-1 (1.6-2.85 ps) and Ag 0 -BEA-2 (1.6-2.64 ps). This observation must be associated with the fact that in Ag 0 -BEA-2b the Ag NPs exhibit a larger size (>20 nm) in comparison with the two other samples for which the size is limited in the range 1-2 nm. The variation of the hot electron decay with the size of noble nanoparticles can be attributed to two main processes. The first explanation is based on the hypothesis of a higher thermal conductance, or vibrational coupling, at the interface of the metal which favor the transfer of energy from the metal lattice to its surrounding [24] . In this case, the observed lifetime would represent the time for the thermal transfer of energy from the hot electron to the surrounding of the metal. This hypothesis is supported by the transient spectra recorded after 10 ps for samples Ag 0 -BEA-1 and Ag 0 -BEA-2 shown in Figure 5 and SI Figure 5. A long-living signal typical of the perturbation of the plasmon band induced by a local change of the temperature is expected [48] . The second possible explanation is based on the hypothesis of ultrafast interactions (< 100 fs) between the surface plasmon, by chemical interface damping, or between the highly excited electrons, by charge transfer, with the molecular species at the surface of the Ag NPs. [18,19,[START_REF] Bauer | EQEC '05. European Quantum Electronics Conference[END_REF] Both mechanisms result in the transfer of a part of the energy of the photon and in the subsequent reduction of the energy deposited in the hot electron. Consequently, the decay time is also reduced. Theoretical calculations 59 showed that the close proximity between the energy level of the Ag atoms and of the zeolite matrix support this scenario.

To further validate that the difference in the hot electron lifetime between samples Ag 0 -BEA-1 and Ag 0 -BEA-2, and the sample Ag 0 -BEA-2b is due to a size effect, we also performed a direct comparison with the plasmonic response of bare Ag NP (10 nm) stabilized in PVA/water solution. The hot electron dynamics was measured for this sample (Ag-PVA) and the Ag 0 -BEA-1 under rigorously equivalent laser intensity and pump-probe overlap. The results show clearly the acceleration of the decay in the Ag 0 -BEA-1 sample (Figure 8). This confirms that in Ag 0 -BEA-1 and Ag 0 -BEA-2 the hot electron dynamics is faster than in bulk silver nanoparticle. This suggests that the hot-electrons decays is not proceeding only by electron-phonon coupling but it is governed by the interaction at the interface of the metal. At the lowest excitation, we notice the reverse situation: the hot-electron lifetime is longer for the small Ag NPs (Ag 0 -BEA-1 , Ag 0 -BEA-2) than for the Ag-BEA-2b. Bauer and co-workers [29,[START_REF] Bauer | EQEC '05. European Quantum Electronics Conference[END_REF]50] reported that the gold nanoparticles in strong interactions (chemisorption) with sulfate molecules show hot electron decay time in weak perturbation regime equals to 1.7 ps independently of the pump intensity. The confinement of the molecules (water, Et3N, TEA + ) in the BEA zeolite, or the zeolite framework itself, might favor the chemical interface damping. The UV-vis spectra of Ag 0 -BEA-2, exhibiting an absorption in the red wing of the plasmon band (Figure 3) support the existence of a charge transfer at the surface of the metal. The chemical interface damping has been recognized as an efficient mechanism to promote the photocatalytic activity of metal nanoparticles. The influence of the chemical reduction route on the hot electron relaxation mechanism of Ag-BEA samples is represented schematically in Figure 9.

The nanosized Ag NPs stabilized by the BEA zeolite, due to their high surface-volume ratio are in a favorable situation for application in plasmonic chemistry. The results show that by controlling the size of Ag NPs using the confinement effect of zeolites can be of great interest for designing efficient photoactive materials.

Conclusion

In this work, we present the preparation of silver nanoparticles in nanosized BEA type zeolite stabilized in colloidal suspensions. We demonstrated that after the introduction of silver cations (Ag + ) by ion exchange, nm-sized silver particle can be formed by instantaneous chemical reduction using an excess of trimethylamine. The photo-dynamics of the hot electrons in two BEA zeolite samples was characterized by femtosecond pump-probe transient absorption measurements in transmission, owing to the stability and transparency of the zeolite colloidal suspensions. The hot electron dynamics of Ag NPs embedded in the BEA zeolite matrix is strongly governed by the interactions at the surface of the metals. The capacity of the BEA zeolite to stabilize 1-2 nm-size Ag NPs is advantageous since the small sized Ag NPs increase the density of energy absorbed by the metals and it raises the surface interactions. However, the exact nature and mechanism of these chemical interface interactions is difficult to be predicted regarding the complexity of the zeolitite frameworks. More systematic time-resolved experiments in combination with advanced material characterizations will be beneficial to optimize these particular class of photoactive zeolites.

Experimental Section

Preparation of zeolite Beta colloidal suspensions

In this work two types of zeolite Beta samples, BEA-1 and BEA-2 of different Si/Al ratios have been investigated. Zeolite Beta nanocrystals were prepared under hydrothermal conditions from clear precursor suspensions with molar composition 9TEAOH: 0.5Al2O3: 25SiO2: 295H2O (BEA-1) and 9TEAOH: 0.25Al2O3: 25SiO2: 295H2O (BEA-2). Tetraethylammonium hydroxide (TEAOH) was used as an organic template for the preparation of BEA. Initially, the source of alumina (aluminium iso-butoxide, Al2O3) was dissolved in a solution containing TEAOH and distilled water, and then the silica source (colloidal silica 30 wt%) was added into the mixture under stirring. The precursor suspension was stirred for 60 minutes, and then subjected to hydrothermal treatment at 100 °C for 3 days. The obtained crystalline suspension was then purified by multistep high-speed centrifugation (20 000 rpm, 1 h) till a pH of the decanting solution of 9 was reached. The purified zeolite samples were redispersed in distilled water.

Preparation of Ag + containing zeolite Beta nanocrystals by ion-exchange process

The BEA zeolite colloidal suspensions were mixed with 0.5 mol.l -1 AgNO3 and continuously stirred for 3 hours at room temperature. The obtained suspensions were purified twice using high speed centrifugation (10 000 rpm, 60 mn) by adding water upon each centrifugation step to remove excess of free silver. The recovered nanocrystals were redispersed in water with a concentration of solid (zeolite) particles of 0.8 wt% and pH=7. The ionexchanged samples are denoted as Ag + -BEA-1 and Ag + -BEA-2.

Preparation of Ag NPs in colloidal zeolite BEA by chemical reduction method:

Figure 1 .

 1 Figure 1. A1) X-ray diffraction patterns recorded of (a) parent BEA-1 nanosized zeolite and (b) ion exchanged Ag + -BEA-1; B1) SEM picture, (C1) DLS curve, (D1) zeta potential curve recorded for sample Ag + -BEA-1. A2) X-ray diffraction pattern recorded of (a) parent BEA-2 nanosized zeolite and (b) ion exchanged Ag + -BEA-2, B2) SEM picture, (C2) DLS curve, (D2) zeta potential curve recorded for sample Ag + -BEA-2.

Figure 2 .

 2 Figure 2. TG/DTG curves of a) Ag + -BEA-1, b) Ag + -BEA-2, c) Ag 0 -BEA-1 and d) Ag 0 -BEA-2 samples

Figure 3 .

 3 Figure 3. UV-vis spectra recorded for samples Ag 0 -BEA-1 (A) and Ag 0 -BEA-2 (B) during the chemical reduction process using triethylamine (Et3N): (a) prior chemical reduction of Ag + -BEA sample, (b) immediately after chemical reduction (c) after 1 hour and (d) after 24 hours.The molar concentration of silver in the suspensions Ag 0 -BEA-1 and Ag 0 -BEA-2 are 7.1×10 -5 mol.l -1 and 2.25×10 -5 mol.l -1 , respectively; inset pictures (g) and (j) correspondingly). Inset pictures (e) and (f) show samples Ag + -BEA-1 (0.8wt%) and Ag 0 -BEA-1 (0.8wt%), correspondingly. Insets (h) and (i) show pictures of samples Ag + -BEA-2 (0.8wt%) and Ag 0 -BEA-2 (0.8wt%), correspondingly.

Figure 4 .

 4 Figure 4. A) HAADF images coupled with the point EDS technique of colloidal Ag 0 -BEA-2, performed at two selected zones A1 and A2. The contrast of the HAADF image is identified by the atomic number Z-contrast: Ag NPs are visualized as bright dots due to the high atomic number, and the zeolite matrix as a dark background due to its low Z-contrast. HAADF images with corresponding magnifications B) 20 nm and C) 10 nm.

Figure 5 .

 5 Figure 5. Transient absorption data recorded upon photo-excitation of sample Ag0-BEA-1 at pump =400 nm. 2D map of the transmission changes (bottom-left), transient spectra at representative pump-probe delays (top) and selected kinetics reconstructed from the transient spectra (bottom-right).

Figure 6 .

 6 Figure 6. Kinetics of the transmission change probed at probe = 380 nm, with different pump intensity (pump = 400 nm), a) I0 = 1.2 mJ/cm 2 , Ag 0 -BEA-1 and Ag 0 -BEA-2, and b) I0 = 1.2 mJ/cm 2, Ag 0 -BEA-2b.

Figure 7 .

 7 Figure 7. Electron-phonon decay time, t(1/e), deduced from the fit of the kinetic at probe=380 nm for different laser intensity (pump = 400 nm).

Figure 8 .

 8 Figure 8. Comparison of the kinetic recorded with probe = 380 nm for Ag 0 -BEA-1 and Ag-PVA samples (normalized transient absorbance).

Figure 9 .

 9 Figure 9. Influence of the chemical reduction route on the hot electron relaxation mechanism in Ag-BEA samples.

Table 1 :

 1 Chemical composition of ion exchanged Ag + -BEA-1 and Ag + -BEA-2 nanosized zeolite samples Si

			Al	TEA +	H2O	Ag +	Ag	[Ag + ] [b]
		/Al	/u.c	/u.c.	/u.c.	wt% [a]	+
							/u.
							c.
	Ag + -BEA-1	7.5	7.5	3.8	2.1	3.4	1.2	2.5
	Ag + -BEA-2	13.1	4.5	4	10	1.4	0.5	0.78

[a] weight percentage per gram of zeolite. [b] mmol/L in suspension with a solid (zeolite) concentration of 0.8 wt%

Acknowledgements

Financial support from the Chevreul institutes (FR 2638)

Entry for the Table of Contents

Silver nanoparticles with a size 1-2 nm have been prepared in colloidal zeolites. The plasmonic response was characterized by UVvis femtosecond absorption spectroscopy and it has revealed that the hot electron lifetime is governed by interactions and couplings at the interface of the silver nanoparticle. Thus, the as-prepared samples are promising in the perspective of plasmonic chemistry applications.

Institute and/or researcher Twitter usernames: @CNRS @LasireTeamPCS @LCPlab @labo_LCS @univ_lille

Wet chemical reduction procedure towards preparation of Ag NP in the zeolite suspension was applied. Triethylamine (denoted as Et3N) was dissolved in ethanol (EtOH) and stirred for few minutes in order to ensure its complete dissolution. Two chemical reduction procedures were implemented: Method A: Fast one-step chemical reduction procedure, in which Ag + -BEA-1,-2 (0.8wt%) samples were added directly to the Et3N mixture (Et3N : EtOH ratio 1 : 5) under continuous stirring at room temperature (Ag + -BEA-1, -2 : Et3N ratio 3.33 : 1). The suspensions were further stirred for one hour. The obtained samples will hereafter be denoted as Ag 0 -BEA-I and Ag 0 -BEA-2. Method B: Slow drop-by-drop chemical reduction procedure, in which Et3N mixture (Et3N : EtOH ratio 1:1) was added slowly to a diluted suspensions of ion-exchanged Ag + -BEA-1, -2 samples containing 0.032wt % of solid (zeolite) particles using micropipettes under vigorous stirring.

Preparation of pure Ag NP colloidal suspension stabilized by polyvinyl alcohol (PVA) Ag-NPs were prepared according to the procedure described in ref [46] . The solutions were made using ultra-pure water from a Millipore system (18 MΩ cm) and reagent-grade chemicals without further purification: AgClO4 and poly(vinyl alcohol) (PVA, 99% hydrolysed, MW=86000) from Aldrich, 2-propanol from Prolabo. The solutions were de-aerated by bubbling with nitrogen (N2, Air Liquide) before irradiation. All the aqueous samples were kept at natural pH. The γ-irradiation source was a 60 Co γ-facility of 7000 Curies with a dose rate of 1.75 Gy s -1 (6300 Gy h -1 ). The composition of the solution exposed to γ-rays was 2×10 -3 M AgClO4, 0.1 M PVA and 2 M 2-propanol. This procedure leaded to the formation of Ag-NPs with an average size of 20 nm characterized by a maximum of absorbance at 412 nm ( ε412=9200 dm 3 mol -1 cm -1 ).

Characterization

Powder X-ray diffraction (XRD) patterns were recorded in Debye-Scherrer geometry using a Siemens D5005 or PANalytical X'Pert Pro diffractometer with Cu Kα radiation (λ=1.5418 Å). Prior XRD experiments, the zeolite colloidal suspensions were freeze-dried, and the powders obtained were homogenized and spread on silicon holders. The samples were scanned in 2θ range of 3-50° with a step of 0.028° and 2.5 minute/degree step time.

The morphological features of zeolite nanocrystals were visualized by scanning electron microscopy (SEM) recorded with TESCAN-MIRA field-emission scanning electron microscope operating at 30 kV. Before carrying out the measurements, samples in powder form were coated with a thin layer of conducting material (platinum or gold) to improve the electrical conductivity using CRESSINGTON sputter machine. Additionally, the morphology, crystallinity and size of both zeolite BEA nanocrystals and Ag NPs were characterized using High Resolution Transmission Electron Microscopy (TEM). TEM micrographs and HADDF measurements were performed using a 200 kV JEOL 2010 FEG STEM electron microscope equipped with EDAX Energy Dispersive Spectrometer (EDS). In order to lower beam damages on samples low dose conditions were used for the images recording.

The particles size distribution and the stability of zeolite colloidal suspensions (0.12 wt% of solid particles) were measured by Dynamic Light scattering (DLS). DLS curves were recorded by Malvern Zetasizer Nano Series DLS instrument. The chemical composition of the samples was determined by Inductively Coupled Plasma-Atomic Emission Spectrometry (ICP) using a Varian ICP-OES system. Thermogravimetric (TG/DTG) analyses were performed to determine the amount of organic template and water in the zeolite samples with and without silver. TG/DTA analyses were measured using a SETSYS evolution instrument (SETARAM). The measurements were carried out on ~10 mg of sample under air with a flow rate of 40 ml.min -1 . The nitrogen adsorption-desorption measurements were recorded on samples calcined under vacuum at 500 °C. N2 sorption isotherms were measured using Micromeritics ASAP 2020 analyzer.

The UV-vis absorption spectra were recorded from the colloidal zeolite suspensions containing Ag NPs in quartz cuvettes of 1 cm path length using a Carry 4000 spectrometer working in transmission mode.

Femtosecond transient absorption measurements

UV-vis femtosecond transient absorption measurements were performed using the experimental set-up described in ref [54] . Briefly, the laser pump excitation (pump = 400 nm) was generated by frequency doubling of the fundamental emission at 800 nm from the Ti:Sa amplifier (Libra Coherent) using a BBO crystal. A white light super continuum probe covering the 320-800 nm spectral range was generated by focusing a few µJ of the fundamental laser pulses at 800 nm into a moving CaF2 plate. The transient spectra are recorded using a LN2-cooled CCD camera (Princeton Instrument).

The measurements were performed in quartz cuvettes with an optical path 1 cm or using a flow cell with an optical path of 1 mm. No photodegradation of the Ag-BEA suspensions during the experiments was observed; the measurements in the static cell were preferred as considered to be less invasive. The electron-phonon dynamics is determined by the initial quantity of light absorbed by the sample. Thus, special attention was paid to keep the pumpprobe overlap conditions unchanged when comparing the photodynamic at different pump intensity or between different samples. At least two samples were measured per day for cross-referencing of the data.


G Hurel

A Ture Savadkoohi

C.-H Lamarque

Design of a nonlinear absorber for a two degrees of freedom pendulum and experimental validation

Keywords: passive control, nonlinear energy sink, pendulum, experimental validation

A nonlinear energy sink is designed to control the oscillations of a two degreesof-freedom pendulum under the following types of excitations: horizontal and vertical base displacements and initial energy. Coupled nonlinear governing system equations are treated with the multiple scale methods with different assumptions depending on applied loads. Fast and slow system dynamics are revealed leading to detection of different possible dynamical regimes. The choice of the design parameters of the nonlinear energy sink is discussed and a tuning method is proposed. The efficiency of the control is validated by an experimental study: a physical model of the pendulum is excited by the shake table. The response of the system shows that the nonlinear energy sink is able to control the main system.

Introduction

In transportation systems, vibrations can be an issue for mechanical structures but also for passengers comfort. In order to control the vibrations of such systems, many types of devices are proposed. These control devices cover large categories of systems ranging between active, passive and hybrid ones [START_REF] Housner | Structural control: past, present, and future[END_REF]. In passive control systems, the most well-known device the tuned mass damper [START_REF] Frahm | Device for damping vibrations of bodies[END_REF], which is a spring-mass system with the same natural frequency than the frequency of the targeted mode to be controlled. It has been shown that, via including a cubic non-linearity in the restoring forcing function of the spring, the efficiency of this device is improved [START_REF] Roberson | Synthesis of a nonlinear dynamic vibration absorber[END_REF]. In the nonlinear energy sink (NES) [START_REF] Vakakis | Energy Pumping in Nonlinear Mechanical Oscillators: Part II. Resonance Capture[END_REF][START_REF] Gendelman | Energy Pumping in Nonlinear Mechanical Oscillators: Part I-Dynamics of the Underlying Hamiltonian Systems[END_REF][START_REF] Lee | Suppression of limit cycle oscillations in the van der Pol oscillator by means of passive non-linear energy sinks[END_REF], the restoring force function is purely nonlinear, i.e. there is no linear component. This function can be polynomial, like the cubic function, or non-polynomial [START_REF] Gendelman | Targeted energy transfer in systems with non-polynomial nonlinearity[END_REF]. It is shown that the nonsmooth nonlinearity can be considered for the absorbers [START_REF] Gendelman | Targeted energy transfer in systems with non-polynomial nonlinearity[END_REF][START_REF] Lamarque | Targeted energy transfer in mechanical systems by means of non-smooth nonlinear energy sink[END_REF]. Such systems are very efficient when the system possesses pre-stressing terms [START_REF] Weiss | Multi-scale energy exchanges between an elasto-plastic oscillator and a light nonsmooth system with external pre-stress[END_REF]. An example of such systems is the piece-wise linear one which can be built with a mass moving in a clearance between two linear springs or elastic barriers. The NES can be used in various domains such as civil engineering for buildings [START_REF] Lu | Optimization design and experimental verification of track nonlinear energy sink for vibration control under seismic excitation[END_REF] or stay-cable [START_REF] Izzi | Investigating the use of Targeted-Energy-Transfer devices for stay-cable vibration mitigation[END_REF][START_REF] Weiss | Control of vertical oscillations of a cable by a piecewise linear absorber[END_REF]. The passive control of a single degree-of-freedom (dof) pendulum by a tuned mass damper is studied by Matsuhisa et al. [START_REF] Matsuhisa | Vibration Control of a Ropeway Carrier by Passive Dynamic Vibration Absorbers[END_REF] The application of a NES with cubic and nonsmooth nonlinearities on controlling planar oscillations of a two dof pendulum under external excitations is studied by Hurel et al. [START_REF] Hurel | Nonlinear vibratory energy exchanges between a two degrees-of-freedom pendulum and a nonlinear absorber[END_REF][START_REF] Hurel | Passive control of a two degrees-of-freedom pendulum by a non-smooth absorber[END_REF]. In addition, the same authors studied the control process of a single dof pendulum under base and external excitations with cubic NES [START_REF] Hurel | Nonlinear passive control of a pendulum submitted to base excitations[END_REF]. For the systems in gravitational field, the cubic NES looses its pure nonlinear nature due to the prestressing terms induced by the gravity. To overcome this fact, a nonsmooth NES can be used [START_REF] Alireza | Vibratory energy exchange between a linear and a nonsmooth system in the presence of the gravity[END_REF].

In this article, we propose a design procedure for controlling a two dof pendulum under two-dimensional base excitation (horizontal and vertical) and initial energies.

The paper is organized as it follows: In section 2, we present the system and its governing equations accompanied by some explanations about the principle of the NES. Then, in section 3, we tune the parameters to have a good and reasonable control. In section 4, an experimental validation is presented. At last, in section 5, the paper is concluded.

2 Principle of the nonlinear absorber 2.1 The main system: a two degrees-of-freedom pendulum

The mechanical system under consideration is a two dof pendulum of mass M in a gravity field g = -g k shown in Figure 1. It is hung from a moving base O. The pendulum can rotates around i and j axes with angles ψ and ϕ respectively, transforming global reference frame ( i, j, k) in local reference frame ( i , j , k ). It is excited by the imposed displacement at its base O represented as x O (t), y O (t) and z O (t). The center of mass of the pendulum is located at the distance L from the base O. Viscous damping coefficients relative to the rotation of angles ϕ and ψ are represented as c ϕ and c ψ , respectively.

g = -g k α k j i ψ ϕ O k j i Figure 1:
The main system: a two-degrees-of freedom pendulum under gravitational effects with g = -g k.

The nonlinear absorber

The control of the oscillations of the main system is made by a nonsmooth NES [START_REF] Lamarque | Targeted energy transfer in mechanical systems by means of non-smooth nonlinear energy sink[END_REF] coupled to the pendulum. The nonlinear absorber is composed of a mass m, moving in a guide between two linear springs of stiffness k, with a clearance of 2d (see Figure 2a). When its displacement u = 0, the mass is located at a distance a from the base O. The mass of the absorber is very small compared to the mass of the main system:

m = εM, ε 1 (1)
In order to control the system in both directions, the NES is set to have an angle α with the axis i as seen on Figure 2a. The restoring force function s of the NES is depicted in Figure 2b as a function of the displacement u of the mass m relative to the pendulum.

Equations of the system

The kinetic (K) and potential (U) energies of the system read [START_REF] Hurel | Passive control of a two degrees-of-freedom pendulum by a non-smooth absorber[END_REF]:

K = 1 2 M ẋ2 G + ẏ2 G + ż2 G + 1 2 I xx φ2 + 1 2 I yy ψ2 + 1 2 m ẋ2 m + ẏ2 m + ż2 m (2) U =g(M z G + mz m ) + S(u) (3)
where • stands for the time derivation of the variable •, (x G , y G , z G ) are the coordinates of the center of mass G of the main system and (x m , y m , z m ) are the coordinates of the NES in the main reference of frame, I yy and I xx are the product of inertia of the main system around i and j, respectively, and ∇S(u) = s(u), where ∇ stands for the gradient operator. The non-conservatives forces applied on the system are:

F ϕ = -c ϕ φ (4)
F ψ = -c ψ ψ (5)
F u = -c u u (6)
The equations of the system are obtained with the Euler-Lagrange equations. They are provided in appendix A.

Assumptions

The main system is a two-dof system with generalized coordinates represented as ϕ and ψ.

If we linearize the system around its equilibrium state (ϕ = ψ = 0), we can calculate two eigenmodes with the natural frequencies represented as ω ϕ and ω ψ . Because of the geometry and the inertia of the pendulum, we assume that the natural frequencies are close to each other:

ω ϕ = ω ψ + σ ω ε (7)
We assume also that the displacement of the base O is periodic and its fundamental frequency is close to the natural frequencies of the main system:

x O (t) = ∞ j=0
x j e ijΩt (8)

y O (t) = ∞ j=0 y j e ijΩt (9)
z O (t) = ∞ j=0 z j e ijΩt (10)
where i 2 = -1 and Ω = ω ϕ + σ ϕ ε = ω ψ + σ ψ ε. Finally, we assume that the angles and the displacements are small: ϕ = √ εϕ, ψ = √ εψ, u = √ εu, x j = ε √ εx j , y j = ε √ εy j and z j = √ εz j . This type of scaling of system variables with respect to the ε parameter is not unique and it can vary depending on the nature of the problem (e.g. the type of excitation) [START_REF] Hurel | Nonlinear vibratory energy exchanges between a two degrees-of-freedom pendulum and a nonlinear absorber[END_REF][START_REF] Hurel | Passive control of a two degrees-of-freedom pendulum by a non-smooth absorber[END_REF][START_REF] Hurel | Nonlinear passive control of a pendulum submitted to base excitations[END_REF].

Table 1 collects numerical values of system parameters which will be used in all numerical examples of this paper.

c ϕ 0.05 N m rad -1 s -1 c ψ 0.05 N m rad -1 s -1 NES m 0.08 kg a 0.207 m α π 4 k 120 N m -1 d 0.02 m c u 2.5 N m -1 s -1
2.5 Detection of fast/slow dynamics: complexification and multiple scale method

We introduce the complex variables of Manevitch [START_REF] Manevitch | The Description of Localized Normal Modes in a Chain of Nonlinear Coupled Oscillators Using Complex Variables[END_REF]:

Φe iΩt = φ + iΩϕ (11)
Ψe iΩt = ψ + iΩψ (12)
Ue iΩt = u + iΩu

In the multiple time scale method [START_REF] Hasan | Nonlinear oscillations. Wiley classics library ed[END_REF], we consider that the time can be decomposed into fast (τ 0 ) and slow (τ j , j = 1, 2, ...) time scales.

τ 0 = ε 0 t, τ 1 = ε 1 t, τ 2 = ε 2 t... (14)
Then, the derivation operator is redefined as:

d dt = ∂ ∂τ 0 + ε 1 ∂ ∂τ 1 + ε 2 ∂ ∂τ 2 + ... (15)
In this analysis, we keep only the first harmonics of the response of the system. This is done for an arbitrary function of the system h(τ 0 , τ 1 , τ 2 ...) by following operator:

H = Ω 2π 2π Ω 0 h(τ 0 , τ 1 , τ 2 ...)e -iΩτ0 dτ 0 (16)
Applying Equation 16, we assume that Φ, Ψ and U are independent of fast time scale τ 0 . This will be verified during multiple scale method or we will look at an asymptotic system behavior where fast time (τ 0 ) tends to infinity. After applying equations 11-13 and 16 on governing system equations, following system is obtained:

Lg dΦ dτ 0 + i Lg(Ω 2 -ω 2 ϕ ) 2Ω Φ + εω 2 ϕ a 2 dΦ dτ 0 + Lg ω 2 ϕ dΦ dτ 1 + a cos(α) dU dτ 0 + c ϕ + ia 2 Ω 2 -i ag 2Ω Φ + i Lg 16Ω 3 |Φ| Φ + i aΩ 2 -g 2Ω cos(α)U + iL L 8Ω + g 16Ω 3 Ψ 2 Φ * -i L 2 4Ω Φ |Ψ| + i Lg 8Ω 3 Φ |Ψ| -LΩ 2 x 1 -2iLΩz 2 Φ * + O(ε 2 ) = 0 ( 17
)
Lg dΨ dτ 0 + i Lg(Ω 2 -ω 2 ψ ) 2Ω Ψ + εω 2 ψ a 2 dΨ dτ 0 + Lg ω 2 ψ dΨ dτ 1 + a sin(α) dU dτ 0 + c ψ + ia 2 Ω 2 -i ag 2Ω Ψ + i Lg 16Ω 3 |Ψ| Ψ + i aΩ 2 -g 2Ω sin(α)U + iL L 8Ω + g 16Ω 3 Φ 2 Ψ * -i L 2 4Ω Ψ |Φ| + i Lg 8Ω 3 Ψ |Φ| -LΩ 2 y 1 -2iLΩz 2 Ψ * + O(ε 2 ) = 0 ( 18
)
ε dU dτ 0 + a cos(α) dΦ dτ 0 - gi 2aΩ Φ + Ωi 2 Φ + a sin(α) dΨ dτ 0 - gi 2aΩ Ψ + Ωi 2 Ψ + λ + Ωi 2 U -i K 2Ω US(|U|) + O(ε 2 ) (19) with c ϕ = c ϕ m , c ψ = c ψ m and S(|U|) = 2 π      0 if |U| ≤ D arccos D |U| - D |U| 2 -D 2 |U| 2 if |U| > D (20)
where

λ = c u m , D = Ωd √ ε and K = k m [7].

Analysis of the system

At the first order O(ε 0 ), corresponding to the fast time scale τ 0 , the equations of the system read:

∂Φ ∂τ 0 = 0 (21) ∂Ψ ∂τ 0 = 0 (22) ∂U ∂τ 0 + a cos(α) ∂Φ ∂τ 0 + a sin(α) ∂Ψ ∂τ 0 + i A 2Ω (cos(α)Φ + sin(α)Ψ) + iΩ + λ 2 U -i K 2Ω US(|U|) = 0 (23)
with A = (aΩ 2 -g). We express the complex variables in polar form: Φ = N ϕ e iδϕ , Ψ = N ψ e iδ ψ and U = N u e iδu . We seek for an asymptotic state when τ 0 tends to infinity, i.e., ∂U ∂τ 0 -→ 0. Equation 23 gives the expression of the slow invariant manifold (SIM):

cos 2 (α)N 2 ϕ + sin 2 (α)N 2 ψ + 2 sin(α) cos(α) cos(δ)N ϕ N ψ = KS(N u ) -Ω 2 2 + λ 2 Ω 2 A 2 N 2 u (24)
where δ = δ ϕ -δ ψ . The SIM is function of four real variables N ϕ , N ψ , N u and δ. It can be represented in the space (N ϕ , N ψ , N u ) for several values of δ. The SIM is composed of stable and unstable zones delimited by singular points as demonstrated in [START_REF] Hurel | Passive control of a two degrees-of-freedom pendulum by a non-smooth absorber[END_REF]. Figure 3 shows the SIM with stable and unstable zones for an arbitrary value of δ and a particular value δ = π.

The ε 1 order of equations corresponding to the slow time scale τ 1 , yields to:

Lg ∂Φ ∂τ 1 =x 1 LΩ 4 + 2iz 2 LΩ 3 Φ * -i 2σ ϕ Lg 2 + c ϕ Ω 2 2 Φ -i AΩ 2 (aΦ + cos(α)U) -i Lg 16Ω |Φ| 2 Φ + 2 - 4LΩ 2 g Φ |Ψ| 2 + Lg + 2LΩ 2 g Φ * Ψ 2 (25)
Lg ∂Ψ ∂τ 1 =y 1 LΩ 4 + 2iz 2 LΩ 3 Ψ * -i 2σ ψ Lg 2 + c ψ Ω 2 2 Ψ -i AΩ 2 (aΨ + sin(α)U) -i Lg 16Ω |Ψ| 2 Ψ + 2 - 4LΩ 2 g Ψ |Φ| 2 + Lg + 2LΩ 2 g Ψ * Φ 2 (26)

Different cases of excitations

We identify three cases of excitations for which the NES can control the pendulum oscillations:

Horizontal base excitation

Initial energy

Vertical base excitation

These different excitation types are considered separately in following sections in order to simplify and also to solve Equations 25 and 26.

Horizontal base excitation

In this case, there is a horizontal periodic motion of the base O. The NES is designed to keep the amplitude of the main system below a given threshold. For design, it is necessary to know either the maximal amplitude of excitation to find the lowest threshold, or the maximal amplitude of the main system allowed to be effective at the highest excitation amplitude. In this study, we suppose that we know the maximum amplitude of excitation.

The effect of the vertical displacement of the base O is no more visible at this order. The equilibrium points are obtained via setting ∂Φ ∂τ 1 = 0 and ∂Ψ ∂τ 1 = 0 in Equation 25 and 26 accompanied by consideration of the Equation of the SIM (24):

X 1 LΩ 4 -i 2σ ϕ Lg 2 + c ϕ Ω 2 2 Φ -i AΩ 2 (aΦ + cos(α)U) = 0 (27) Y 1 LΩ 4 -i 2σ ψ Lg 2 + c ψ Ω 2 2 Ψ -i AΩ 2 (aΨ + sin(α)U) = 0 (28)
In order to determinate the efficiency of the NES for a given excitation, we calculate the equilibrium points of the system for a range of frequency in the interval Ω ∈ [ω min , ω max ]. Figure 4 shows that N ϕ , N ψ and N u of the equilibrium points of the system with NES are lower than the system without NES. We can also see that for some values of the frequency Ω there is no stable equilibrium point. These zones can correspond to different non periodic responses such as strongly modulated response (SMR) [START_REF] Hurel | Nonlinear vibratory energy exchanges between a two degrees-of-freedom pendulum and a nonlinear absorber[END_REF][START_REF] Starosvetsky | Strongly modulated response in forced 2DOF oscillatory system with essential mass and potential asymmetry[END_REF] which is due to existence of fold singularities in the system. Figure 5 shows that the system oscillates between both stable zone of the SIM. The amplitudes N ϕ and N ψ oscillate and stay below a threshold.

However, as the SIM has dimension four, which creates difficulty in its visualization. That's why analytical predictions should be compared carefully with results obtained from direct numerical integration of the system equations.

The system under initial energy

In this case, the base O is motionless but an initial energy is applied to the mass M of the pendulum. This energy can be due to a sudden displacement of O. The role of the NES is to absorb this energy rapidly. Here, the time t c to reach a given amplitude from an initial amplitude of pendulum is a criterion to be minimized for tuning the NES.

Without excitation, we can observe from Equations 25 and 26 that the only equilibrium point of the system is Φ = Ψ = 0. We have to evaluate the decreasing speed of the amplitude. We introduce the new complex variable: Θ = cos(α)Φ + sin(α)Ψ. By considering almost equal damping scenarios for the system as: c θ = c ϕ + εσ cϕ = c ψ + εσ c ψ and very close frequency σ θ = σ ϕ + εσ σϕ = σ ψ + εσ σ ψ , the Equations 27 and 28 give:

∂Θ ∂τ 1 = -iσ θ + c θ Ω 2 + iΩaA 2Lg Θ -i ΩA 2Lg U ( 29
)
Equation 23 becomes:

AΘ = iλΩ -Ω 2 + KS(|U | 2 ) U (30)
By replacing expression of Θ from Equation 30 in equation 29, we obtain:

∂N u ∂τ 1 (N u ) = - Ω 2 2Lg A 2 λ + c θ λ 2 Ω 2 + KS(N 2 u ) -Ω 2 2 λ 2 Ω 2 + 2KS (N 2 u ) (KS(N 2 u ) -Ω 2 ) N 2 u + (KS(N 2 u ) -Ω 2 ) 2 N u (31)
From Equation 30, we write the expressions of

∂N θ ∂N u and ∂N u ∂τ 1 to calculate ∂N θ ∂τ 1 : ∂N θ ∂τ 1 (τ 1 ) = ∂N θ ∂N u ∂N u ∂τ 1 (τ 1 ) (32)
Figure 6a illustrates the variation of ∂N θ ∂τ 1 as a function of N θ . It is seen that N θ decreases abruptly to a considerable level and then it reduces its energy in an exponential manner. A numerical integration of the Equation 32 gives the theoretical evolution of N θ as a function of time τ 1 . One can see two distinct behaviours: the first one for N θ > 0.2 with an abrupt decrease in amplitude and the second one (N θ < 0.2), with an exponential manner. This evolution is compared in Figure 6b with results obtained from the numerical integration of the system governing equations (see Equations 35, 36 and 37). The main difference is due to the approximation σ = σ ϕ + εσ σϕ = σ ψ + εσ σ ψ . In reality the frequencies ω ϕ and ω ψ are different and a phase angle δ increases over time. Since the SIM depends on δ, the behaviour of the NES and the variation of N θ vary with δ.

Vertical excitation

In this case, there is only a vertical periodic motion of the base O. Here, a validation of the parameters of the NES is necessary to verify the possibility of existence of high amplitudes of the oscillations.

To understand the behaviour of the system, we consider Ψ = 0, y 1 = 0, x 1 = 0 and α = 0, i.e. we consider a planar system. Here, the hypothesis of small Φ (Φ = √ εΦ) would lead to irrelevant results. The Equation 25 leads to:

2iz 2 LΩ 3 Φ * -i 2σ ϕ Lg 2 + c ϕ Ω 2 2 Φ -i AΩ 2 (aΦ + U) -i Lg 16Ω |Φ| 2 Φ = 0 (33)
The equilibrium points of the system can be evaluated from Equations 23 and 33. They are shown in Figure 7. In previous study [START_REF] Hurel | Nonlinear passive control of a pendulum submitted to base excitations[END_REF], we showed that the NES prevents the pendulum to reach high amplitude equilibrium point from a state with low amplitude. It is worth to mention that in the design of the NES, the isolas should be treated and identified carefully as they could correspond to very high energy level of the main dofs and be counter-productive.

Tuning of parameters of the nonlinear absorber

To design the NES, we have to determine six parameters: the mass m, the position a, the angle α, the stiffness k, the clearance d and the damping coefficient c u . These parameters have to be tuned to find good efficiency of the NES. These six parameters should be tuned all together. However, due to technical constraints, we suppose that the mass m and the position a of the NES are determined independently. The orientation angle α is also tuned independently because it depends on excitation and on the demand of the industry about the direction of the control.

The mass of the NES

The mass m does not appear directly in Equations 23, 25 and 26 but its effect is hidden behind the slow time scale τ 1 = m M t. Hence, the higher value of the mass m implies that the infinity of slow time scale τ 1 starts faster, i.e. the system can reach its equilibrium points faster [START_REF] Lamarque | Multi-scale dynamics of two coupled nonsmooth systems[END_REF]. However, due to the main assumptions in analytical developments (ε 1) and due to technical constraints, m cannot be too high.

The position of the NES

The position of the NES is also determined by technical constraints, but several options can exist. In Equations 25 and 26, the decreasing of Φ and Ψ relative to slow time τ 1 due to the absorber is proportional to A = aΩ 2 -g. In order to maximize A, the absorber has to be placed as far as possible from G, the equivalent mass center of the pendulum:

OG = - g Ω 2 k (34)

Orientation angle of the NES

In order to understand the effect of the orientation of the NES, we fix α the orientation angle of the NES and we study the response of the system for variable angles of horizontal base excitation, β = tan y 1 x 1 . Figure 8 shows the results for angle α ∈ [0,

π 2 ].
Here are some remarks:

1. It is confirmed that with α = 0 + j π 2 , j ∈ Z, the NES can be efficient and can control both ϕ and ψ angles [START_REF] Hurel | Nonlinear vibratory energy exchanges between a two degrees-of-freedom pendulum and a nonlinear absorber[END_REF].

2. If α → 0, then the NES is not efficient in controlling ϕ. If α → π 2 , then the NES is not efficient in controlling ψ.

3. If the direction of excitation β is known, then setting α = β provides a good control of the pendulum since both angles ϕ and ψ have relatively low amplitude.

Parameters maps

Horizontal excitation

Let us suppose that parameters m, a, α are already determined and we would like to find the good combination of parameters k, d and c. To do this, we build parameter maps in the plan (k, d) for several values of c with the amplitudes of angles ϕ and ψ. This is performed by numerical integration of the equations of the system (Equations 35, 36 and 37). We simulate the system with horizontal base excitation x 1 = 17 mm from initial condition with no energy (ϕ(0) = ψ(0) = u(0) = φ(0) = ψ(0) = u(0) = 0). The time of simulation is long enough to reach the asymptotic state of fast time: τ 0 = 100.

Figure 9 shows the parameters maps for the amplitudes N ϕ and N ψ . The zone where the NES is more efficient is for low values of λ and d with the frequency k m being close to the frequencies ω ϕ and ω ψ . This fact is similar to the design logic of a tuned mass damper. However, the zone of efficiency is very narrow and the absorber is not robust. In fact, as the main system is nonlinear, the frequencies of oscillations vary with the amplitude. Furthermore, it is technologically difficult to ensure a precise and low value of damping because of different phenomena such as friction, interactions, etc... So we prefer to take higher values of c u and d to provide more robustness to the NES. Color scale corresponds to the maximal amplitude the system can reach for angles ϕ and ψ.

Initial energy

In this case, we would like to minimize the time t c for which the system decreases its amplitude from an initial state to half of it. To evaluate this time, we consider Equation 29.

For each combination of parameters (k, d, c u ), a numerical integration of the equation 32 is performed with an initial energy N θ = 100. Figure 10 shows the time t c needed for the system to reach 50 % of its initial amplitude. We can observe following aspects:

when the damping λ increases, the clearance d decreases;

the NES is still efficient for high values of stiffness k.

To design a robust NES we choose a value of clearance d for which high values of stiffness k provide a good control. 4 Experimental validation

The prototype

A pendulum is made of a threaded rod. A parallelepiped of steel gives inertia in both directions. The hinge is made of two pivot linkages allowing the pendulum to rotate around i and j axes (see Figure 11b). The system is excited by the shake table. The values of x 1 and y 1 are set with the amplitude of acceleration of the shaker and the angle β between the table and the pendulum. It is possible to change the configuration of the shaker to excite the pendulum vertically and provide vertical displacement of the base z O (t) (see Figure 12). The mass of the NES consists of a translating rod (via two linear ball bearings for minimizing friction), between two linear springs (see Figure 11a). For simulating the tests without NES, the mobile mass is blocked.

The parameters of the pendulum and the NES are those displayed on Table 1.

The test procedure

The tests consist of sinusoidal excitations with sweeping frequencies. The sweeping is carried out with increasing and decreasing frequencies. An uni-axial accelerometer measures the acceleration of the base O. The amplitude of excitation is ensured with a control loop. A triaxial accelerometer is glued on the pendulum which provides the values of angles ϕ and ψ. We excite the system with constant velocity amplitude of the base O for two reasons: compared to the test with constant acceleration, constant velocity reduces the displacement of the table at low frequencies ;

the shake table provides the same power to the system independently of the frequency.

For these reasons, the amplitude of acceleration of the base is proportional to the frequencies.

Preliminary tests

Some preliminary tests are performed to characterize the system. As far as the system is nonlinear, several tests with different amplitudes of excitation are performed. The results are presented in Figure 13. It can be seen that:

The frequency at the peak response decreases when the excitation increases. The main system presents a nonlinear softening behaviour.

The shape of the curve with increasing and decreasing frequencies are very different.

The system follows the stable branch of the response curve. When it meets an unstable state, it jumps on the other stable branch.

Horizontal base excitation

A test with and without the NES is performed with α = β = 0. As shown in Figure 14, the NES reduces the amplitude of the pendulum with a factor of 2. To prove the efficiency of the NES on both angles ϕ and ψ, another test is performed with α = π 4 . The results shown in Figure 15 show the NES is still efficient.

Vertical base excitation

The shaker of the table is oriented vertically to give a vertical displacement at the base O.

It is necessary to give an initial energy at the pendulum in order to create oscillations of the system. This is done by an initial angle θ. Figure 16 shows that without NES, the pendulum reaches a state with high amplitude of oscillations. With the NES, the system goes rapidly to the equilibrium point N θ = 0. This confirms that the NES is efficient for the vertical base excitation. Frequency (Hz)

Conclusion

We propose a method to design a nonsmooth nonlinear energy sink to control a two degreesof-freedom pendulum for three types of excitations: horizontal base excitation, vertical base excitation and the initial energies. First we analysed the governing equations of a mathematical model of the system with a time multiple scale method. Thanks to this analysis, we can evaluate the efficiency of the control process by the NES under each of three types of excitations. The design process consists of tuning six parameters of the nonsmooth absorber. The mass, the position and the orientation angle are first fixed due to the technical constraints. Then, the good combinations of the stiffness, clearance and damping of the nonlinear energy sink are found with parameters maps taking account the robustness of the device. Finally, the proposed design tools are tested on a prototype pendulum mounted on the shake table. The tests show that the nonlinear energy sink is efficient to control the oscillation of the pendulum. The explained procedure in this paper provides design tools for passive control of pendulum type systems, such as gondola lifts, by nonsmooth absorber. Obtained results in this paper (theoretical and experimental) will be used to design a nonlinear system for controlling pendulum movements of full-scale gondola lift.

Figure 2 :

 2 Figure 2: Nonsmooth NES: a) schematic of the piece-wise linear absorber; b) The restoring forcing function of the absorber.

Figure 3 :

 3 Figure 3: The three-dimensional views of the SIM for two values of δ.

Figure 4 :

 4 Figure 4: Equilibrium points of the system for a horizontal excitation x 1 = 17 mm.

Figure 5 :

 5 Figure 5: Numerical results showing SMR type oscillations of the system x 1 = 13.5 mm and Ω = 8.5.

  Comparison of analytic result to numerical integration.

Figure 6 :

 6 Figure 6: Evolution of N θ as a function of slow time τ 1 .

Figure 7 :

 7 Figure 7: Equilibrium points of the system for a vertical excitation z 2 = 0.05.

Figure 8 :

 8 Figure 8: Amplitude of ϕ and ψ as a function of angle of applied horizontal base excitation (β) for several values of NES orientation (α).

  Maximal value of N ψ

Figure 9 :

 9 Figure 9: Maximal value of N ϕ and N ψ for the system with parameters k, d and u and excited by horizontal base displacement (x 1 = 17 mm). Color scale corresponds to the maximal amplitude the system can reach for angles ϕ and ψ.

Figure 10 :

 10 Figure 10: Time t c (s) to reach 50 % of the initial amplitude as a function of parameters k, d and c u .

( a )

 a Experimental piece-wise linear NES. (b) Ball joint made of two pin joints.

Figure 11 :

 11 Figure 11: Details of the experimental setup: (a) the NES is made by a rod moving between two springs; (b) the ball joint is made with two pin joints.

Figure 12 :

 12 Figure 12: Pictures of the experimental setup: a pendulum hung to a jib crane and excited by an electrodynamical shaker in horizontal (a) or vertical (b) directions.

Figure 13 :

 13 Figure 13: Response of the pendulum to several amplitudes of excitation with α = β = 0. Velocity amplitude V are given in mm s -1 .

Figure 14 :

 14 Figure 14: Test result with α = β = 0 with and without NES.

Figure 15 :

 15 Figure 15: Test result with α = π/4 and β = 0 with and without NES for amplitudes of angles ϕ and ψ.

( a )

 a Initial energy given to the pendulum.

  Evolution of θ amplitude.

Figure 16 :

 16 Figure 16: Comparison of the response of the pendulum with and without NES, with vertical excitation and initial energy.

Table 1 :

 1 Parameters of the considered system for all presented examples.

	Part	Parameter Value
			g	9.81 m s -2
			M	1.68 kg
			L	0.10 m
	Pendulum	ω ϕ	8.29 rad s -1
			ω ψ	8.42 rad s -1

We assume that excitation and complex variables are small. A second rescaling is performed:x 1 = √ εX 1 , y 1 = √ εY 1 , z 2 = √ εZ 2 , Φ = √ εΦ, Ψ = √ εΨ, U = √ εU and D = √ εD.

Acknowledgement This research was funded by "La Région Auvergne Rhone Alpes" in the frame of the CALIPSO project.

Author contributions All authors of this paper have contributed equally in its different scientific aspects (analytical and numerical developments, discussion of results, etc.) and also in writing the article.

Data availability statement Research data are not shared.

A Equations of the system