Amit Raj Dhawan

Michel Nasilowski

Zhiming Wang

Benoît Dubertret

Agnès Maître
email: agnes.maitre@insp.upmc.fr

Efficient single-emitter plasmonic patch antenna fabrication by deterministic in situ optical lithography using spatially modulated light

Single-emitter plasmonic patch antennas are room-temperature deterministic single photon sources, which exhibit highly accelerated and directed single photon emission. However, for efficient operation these structures require three-dimensional nanoscale deterministic control of emitter positioning within the device, which is a demanding task, esp. when emitter damage during fabrication is a major concern. To overcome this limitation, our deterministic room-temperature in situ optical lithography protocol uses spatially modulated light to position a plasmonic structure non-destructively on any selected single-emitter with three-dimensional nanoscale control.

In this paper we analyze the emission statistics of such plasmonic antennas that embed a deterministically positioned single colloidal CdSe/CdS quantum dot that highlight acceleration and brightness of emission. We demonstrate that the antenna induces a 1000-fold increase of the emitter absorption cross-section, and under high pumping, these antennas show nonlinearly enhanced emission.

T he possibility of deterministically controlling the local environment of an emitter to make its emission faster, the capability of collecting more emitted photons, and the effectiveness of exciting a system with only one photon will benefit several optical quantum technologies [START_REF] O'brien | Photonic quantum technologies[END_REF]. The highly optimized light-matter interactions in devices with these features will highlight the involved physics [START_REF] Haroche | Exploring the Quantum: Atoms, Cavities, and Photons[END_REF] and find application in fields like single photon generation [START_REF] Lounis | Single-photon sources[END_REF][START_REF] Pelton | Efficient source of single photons: A single quantum dot in a micropost microcavity[END_REF][START_REF] Lodahl | Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals[END_REF], quantum plasmonics [START_REF] Chang | A single-photon transistor using nanoscale surface plasmons[END_REF][START_REF] Tame | Quantum plasmonics[END_REF], and optical antennas [START_REF] Bharadwaj | Optical antennas[END_REF][START_REF] Knight | Photodetection with active optical antennas[END_REF][START_REF] Belacel | Controlling spontaneous emission with plasmonic optical patch antennas[END_REF]. Advances in microlithography [START_REF] Smith | Microlithography: Science and Technology, Second Edition, ser. Optical science and engineering[END_REF] produced superior electronics and helped attain the first quantum revolution, and the developments in nanofabrication [START_REF] Cui | Nanofabrication: Principles, Capabilities and Limits[END_REF] are paving the way for the second quantum revolution [START_REF] Jones | The second quantum revolution[END_REF]. By means of Purcell effect [START_REF] Purcell | Spontaneous emission probabilities at radio frequencies[END_REF], structures such as optical cavities [START_REF] Vahala | Optical microcavities[END_REF] and plasmonic antennas [START_REF] Giannini | Plasmonic nanoantennas: Fundamentals and their use in controlling the radiative properties of nanoemitters[END_REF][START_REF] Esteban | Optical patch antennas for single photon emission using surface plasmon resonances[END_REF], modify light emission. Their optimal operation necessitates precise spatial positioning of the emitter with respect to its environment, which is a challenging task. Although electron-beam lithography obtains higher resolution than optical lithography, it requires more expensive and specialized equipment and working conditions. Moreover, the beam electrons can damage the emitter. Optical lithography is a cheaper and convenient solution that achieves good writing resolution without electron exposure. However, the intense light required to perform lithography locally above the emitter can photodegrade it. Our fabrication protocol [START_REF] Dhawan | Optical lithography process adapted for a sample comprising at least one fragile light emitter[END_REF] resolves this problem by selectively not exposing the emitter to intense light during the lithography as it uses a custom-designed donut laser beam. At room-temperature and under ambient working conditions, we have employed this protocol to position a plasmonic patch on any selected CdSe/CdS QD with nanometric vertical and horizontal precision, and fabricate plasmonic patch antennas with remarkable emission properties. Core/shell QDs [START_REF] Reiss | Core/shell semiconductor nanocrystals[END_REF], vacancy centers in nanodiamonds [START_REF] Aharonovich | Diamond photonics[END_REF], and defects in two-dimensional materials [START_REF] Tran | Quantum emission from hexagonal boron nitride monolayers[END_REF][START_REF] Toth | Single photon sources in atomically thin materials[END_REF] are deterministic room-temperature single photon sources. Coupling them to optical or plasmonic cavities can make their emission brighter and faster. Optical cavities require very precise spectral matching, which makes them unsuitable for broadband em-itters like colloidal QDs at room-temperature. However, such broadband emitters can couple very efficiently to plasmonic antennas due to their wide spectral resonance and low volume, thus making them promising room-temperature single photon sources. This has been discussed [START_REF] Esteban | Optical patch antennas for single photon emission using surface plasmon resonances[END_REF][START_REF] Akselrod | Probing the mechanisms of large Purcell enhancement in plasmonic nanoantennas[END_REF][START_REF] Marquier | Revisiting quantum optics with surface plasmons and plasmonic resonators[END_REF] and demonstrated [START_REF] Dhawan | Extreme multiexciton emission from deterministically assembled single-emitter subwavelength plasmonic patch antennas[END_REF][START_REF] Hoang | Ultrafast room-temperature single photon emission from quantum dots coupled to plasmonic nanocavities[END_REF] recently. Embedding the QD between the nano-spaced metal-dielectric interfaces couples its radiation to generate surface plasmon polaritons (SPPs) at the interfaces [START_REF] Esteban | Optical patch antennas for single photon emission using surface plasmon resonances[END_REF], which creates very high electromagnetic field around the QD and accelerates its emission. As the patch thickness is less than the skin depth of the field, the SPP-coupled interaction in the antenna emits photons from the patch as shown in Figure 1a. Here we describe the deterministic fabrication of single-emitter patch nanoantennas and demonstrate that the antenna can dramatically modify QD emission, which is illustrated by high recombination rate and fluorescence enhancement, increased absorption cross-section, and the nonlinear emission of the antenna. The metallic losses in plasmonic antennas that reduces their radiative efficiency [START_REF] Bigourdan | Design of highly efficient metallo-dielectric patch antennas for single-photon emission[END_REF] can be mitigated by their acceleration and directionality of emission, thus achieving faster and brighter emission. We show that a plasmonic antenna can increase the absorption cross-section of a QD by more than three orders, and lead to highly nonlinear emission under low power excitation.

Results

Deterministic in situ spatially modulated laser lithography

Conventional photolithography [START_REF] Levinson | Principles of Lithography[END_REF] utilizes light of appropriate wavelength and intensity to locally expose the photoresist, which is then chemically etched. Using in situ photolithography, Belacel et al. [START_REF] Belacel | Controlling spontaneous emission with plasmonic optical patch antennas[END_REF] demonstrated deterministic fabrication of plasmonic patch antennas that contain an aggregate of colloidal QDs. This method cannot be extended straightforwardly to room-temperature deterministic fabrication of single-emitter antennas because the fluorescence of a single QD is considerably lower than that of an aggregate, and therefore it is masked by the luminescence of the photoresist. The high light intensity required to observe a QD leads to unintended exposure (chemical modificat-ion due to light) of the photoresist above it and makes localized lithography impossible. Our protocol avoids this problem by using a resist bi-layer which has low luminescence and is not exposed like conventional photoresists during the process. The bi-layer resist stack consisting of a lift-off resist (LOR® [START_REF] Corp | LOR® lift-off resists[END_REF]) and polymethyl methacrylate (PMMA) does not rely upon conventional exposure because it is evaporated by intense laser light during the laser etching process [START_REF] Forsén | Laser lithography on resist bi-layer for nanoelectromechanical systems prototyping[END_REF]. However, the intense light of the fundamental laser mode required to remove the resist bi-layer above a single QD usually results in photobleaching it. We resolve this problem by using spatially modulated laser light with a donut profile (Figure 1d), which does not expose the QD to light during the lithography, and the carefully designed intensity profile of the laser mode leads to a successful positioning a plasmonic patch centered/off-centered above the QD with a lateral precision of ±50 nm. The QD can be positioned vertically in the antenna with a precision of ±3 nm using spin-coated thin films. This lithography protocol can work with a variety of emitters-single or aggregates-such as QDs, vacancy centers in nanodiamond, molecules, and defects in two-dimensional materials. This article describes the use of this protocol to deterministically select and position a single colloidal CdSe/CdS QD (Figure 1b) inside a plasmonic patch antenna as illustrated in Figure 1a. Light-matter interactions in nanophotonic devices can be optimized by deterministic control over emitter selection and device fabrication, and the described lithography protocol is capable of achieving this. The antennas discussed in this paper were fabricated at room-temperature and under atmospheric pressure, but the fabrication protocol can be used at low temperature and in vacuum conditions.

Compared to our other protocol [START_REF] Dhawan | Extreme multiexciton emission from deterministically assembled single-emitter subwavelength plasmonic patch antennas[END_REF] that utilizes a laser with wavelength tunability to maximize absorption of light by the resist and minimize QD photobleaching, this method excels at preventing emitter damage as no light is seen by the QD during the laser etching step. Moreover, by modifying the phase map on the spatial light modulator (SLM) screen, a variety of high-resolution shapes and patterns can be created. The confocal scan and the laser etching pattern of Figures 2a,b were produced by a Laguerre-Gaussian (LG) donut mode (here LG 40 ) that was focused through a microscope objective. The mode was created by reflecting a fundamental mode laser beam using an optimally tuned liquid crystal SLM displaying a phase mask, and then filtering out the ±1 diffraction order as shown in Figure 1d. On a 280 μm thick Si wafer, a 10 nm thick adhesive layer of Cr is deposited, above which a 200 nm thick optically opaque layer of Au is evaporated. Then a 10 nm thin layer of PMMA is spin-coated and baked, on which a QD dispersion of adequate concentration is spincoated to obtain well-separated single QDs. Then another layer of PMMA (35 nm) is spin-coated and baked, thus embedding individual QDs in a dielectric layer. Then a lithography resist bi-layer is created by firstly spin-coating and baking a 450 nm thick layer of LOR®5A, and then a 50 nm thick layer of PMMA.

The sample (Figure 2a) is confocally scanned using low intensity laser light (473 nm continuous wave laser). After positioning the emitter, high intensity laser light is used to remove the resist bi-layer locally above the emitter as shown in the atomic force image (AFM) above Figure 2b. As the light intensity of the laser mode is almost zero at the center as seen in Figure 2f, the resist at the center is not removed and the QD below it does not photodegrade during the lithography. Then by selective chemical etching of LOR®5A, the remaining central cylinder of the resist bi-layer is removed, and an undercut, which facilitates the final lift-off, is created in the upper PMMA (Figure 2c). The size and shape of the etched feature can be controlled precisely by modifying the laser beam profile and the etching time or chemicals. After evaporating 20 nm of gold (Figure 2d) on the sample, a chemical lift-off is performed to obtain single-emitter patch antennas (Figure 2e).

The performance of these antennas depends on the dipolar orientation of the QD, the dielectric spacer thickness around the QD, the position of the QD with respect to the patch, and the size of the patch. This protocol allows optimal control over all these parameters. In this study, we controlled all these parameters except QD orientation [START_REF] Lethiec | Measurement of three-dimensional dipole orientation of a single fluorescent nanoemitter by emission polarization analysis[END_REF].

Figure 3 Antenna photoluminescence decay and photon correlation

Fitted time-resolved photoluminescence responses and decay times of three antennas fabricated using in situ optical lithography with spatially modulated light a-c. The inset plots show the corresponding normalized coincidences, which signify g 2 (0). d shows the complete decay of an antenna (blue curve) and its post-processed decay after removing events < 3.3 ns (orange curve), and e shows the corresponding second order photon correlation curves-complete decay (blue) and filtered decay (orange). The inset in d is a simulated illustration of multiexciton (MX) and exciton (X) recombination in a biexponential decay. f Energy level diagram illustrating the transitions in QD decay-solid and dashed lines represent radiative and non-radiative transitions, resp.

Single photon emission from antennas under low excitation

In homogeneous PMMA medium and under low intensity off-resonance excitation (405 nm laser pulsed at 2.5 MHz), a QD from the studied batch typically emitted single photons with a monoexponential decay of characteristic lifetime of about τ = 90 ns (Figure 1c). Coupling a QD to the antenna changes its decay rate (Figure 3) and its single photon emission characteristics. Most of the spontaneous decay response curves of these antennas can be fitted with a biexponential function with decay constants τ fast (multiexciton) and τ slow (exciton). Removing photon events corresponding to the fast decay through post-processing leaves mainly exciton recombination events, which are single photon emission events and therefore show lower g 2 (0) . This is demonstrated in Figures 3d,e, where filtering out the fast component of the antenna decay (blue trace), leaves mostly exciton events (orange trace), which exhibits a negligible zero delay peak in the photon coincidence curve (Figure 3e, low g 2 (0) curve), thus validating the above hypothesis about the studied systems.

An excited QD decays in a cascade from the multiexciton states to the exciton state and finally to the ground state as depicted in Figure 3f Auger processes are inherently less efficient in these relatively large QDs due their size [START_REF] Klimov | Quantization of multiparticle Auger rates in semiconductor quantum dots[END_REF]. Purcell effect changes QD decay characteristics when it is coupled to the antenna. As the Purcell effect acts only on non-Auger electromagnetic transitions, Auger processes become even more ineffective in an antenna [START_REF] Dhawan | Extreme multiexciton emission from deterministically assembled single-emitter subwavelength plasmonic patch antennas[END_REF][START_REF] Coolen | Handbook of Self Assembled Semiconductor Nanostructures for Novel Devices in Photonics and Electronics[END_REF]. This makes these antennas prone to multiexciton emission as demonstrated by the increased g 2 (0) (>0.25) in Figures 3b,c. Purcell factor F P can be calculated by comparing the QD exciton decay rate in a homogeneous medium Γ X to its exciton decay rate inside an antenna Γ Antenna X , where Γ Antenna X = F P Γ X . The values of Γ Antenna X and Γ Antenna X are estimated respectively from τ slow of the fitting curves. Using the typical emission lifetime of a QD in homogeneous medium, the F P of the three single photon source antennas of Figures 3a,b,c was noted as 38, 10, and 19, resp. Figure 3a is the emission response of the antenna of Figure 2e (AFM image), and Figure 4c shows its directive radiation pattern.

The distinct rapidity of multiexciton emission allows for temporal filtering via post processing as shown in Figure 3d, e. The possibility of temporal filtering can be exploited to make such antennas better single photon sources [START_REF] Abudayyeh | Purification of single photons by temporal heralding of quantum dot sources[END_REF]. As long as the exciton quantum yield of the QD is stable, the ant-enna emission can be switched very rapidly between single photon and multiphoton emission by varying the excitation intensity-this effect can find application in optoelectronic technologies [START_REF] Kambhampati | Multiexcitons in semiconductor nanocrystals: A platform for optoelectronics at high carrier concentration[END_REF].

Emission enhancement under low excitation

The widefield fluorescence images of Figure 4a demonstrate the enhancement in brightness due to a patch antenna. Both the images were acquired under similar conditions using continuous wave mercury lamp illumination at 438±12 nm. The top image of Figure 4a was collected after removing the resist bilayer above the QD (at step c of Figure 2). We note that the QD fluorescence signal collected with this acquisition time is quite low. The luminescing edges are a result of laser and chemical etching. Positioning the plasmonic patch on this QD, and carrying out the required lift-off, made the fluorescence signal 6 times brighter (Figure 4a, bottom image). Under low intensity excitation, this bright antenna emitted at 640±33 nm (Figure 4b). The enhancement in fluorescence can be attributed to mainly two effects: 1) Plasmonic acceleration of recombination rates (generally, smaller patch antennas cause higher acceleration), and 2) increase in excitation and collection efficiency due to the antenna (generally, larger patch antennas exhibit more directional emission).

It is noteworthy that under very low pulsed laser excitation intensity of only 15 mW/cm 2 , we detected 1.7 × 10 4 photons/s with g 2 (0) = 0.2. The remarkably high brightness of the antenna despite a moderate F P ≈ 3 can be attributed to the high collection of the photons emitted by it due to the directivity of the antenna and to the efficient excitation of the QD. This latter will be discussed in detail in the next section. The antenna patch directs emission very effectively-this is evident from highly directional radiation patterns of two antennas (Figures 4c,d) measured in the far field by Fourier plane imaging, which is in agreement with the simulation results for similar patch sizes [START_REF] Belacel | Controlling spontaneous emission with plasmonic optical patch antennas[END_REF]. The symmetry of the lobes in the polar plot depicts that the QD was positioned at the center with respect to the circular patch within 5 nm, which demonstrates the lithographic precision of this technique.

Antenna emission under increasing excitation

Time-correlated single photon counting on a Hanbury Brown-Twiss optical measurement setup reveals that the emission characteristics of these antennas change considerably as the excitation intensity is varied. As these antennas involve plasmonically coupled large QDs, they are prone to multiexciton emission [START_REF] Dhawan | Extreme multiexciton emission from deterministically assembled single-emitter subwavelength plasmonic patch antennas[END_REF]. Exciting the bright antenna of Figure 4a more intensely resulted in a larger contribution of multiexcitons to its emission. The low average power of the pulsed laser (405 nm laser at 2.5 MHz in fundamental mode) required to excite this antenna demonstrate its efficiency (Figure 5a) and its sensitivity to nonlinear emission transition. At low power excitation (< 0.5 nW), the antenna emission is mainly due to excitons and its g 2 (0) remains around 0.3 as shown by in the graph corresponding to P1 (Figure 5a, b), and curve follows a typical 1e -βP trend, where P is the average power of the laser and β is some coefficient (here β = 9.2 nW -1 ). However, around point P2 and beyond, the emission becomes highly nonlinear and the trend was noted as P α , where α = 5.3 in this case. Analyzing QD emission as a function of pump intensity, Zhang et al. have reported a very similar trend [START_REF] Zhang | New insights into the multiexciton dynamics in phase-pure thick-shell CdSe/CdS quantum dots[END_REF]; here we observe higher nonlinearity and a larger absorption cross-section due to the antenna. Plasmonic interactions provide efficient ways of trapping and propagating light [START_REF] Atwater | Plasmonics for improved photovoltaic devices[END_REF][START_REF] Ferry | Plasmonic nanostructure design for efficient light coupling into solar cells[END_REF][START_REF] Esparza | Differential reflectivity spectroscopy on single patch nanoantennas[END_REF], which increases the effective absorption cross-section of these antennas [START_REF] Geddes | Surface Plasmon Enhanced, Coupled and Controlled Fluorescence[END_REF][START_REF] Bohren | Absorption and Scattering of Light by Small Particles[END_REF]. Our measurements reveal a 1000-fold increase in the absorption cross-section due to the antenna. We calculated the absorption cross-section of the antenna using the fitting parameter β as σ abs = A spot hc λ f rep β, where A spot is the area of the laser spot focused by the objective, hc/λ is the excitation photon energy (h: Planck's constant, c: speed of light in vacuum, λ: laser wavelength), f rep is the repetition rate of the laser. Interestingly, the antenna above the QD increases its effective absorption cross-section to 4 × 10 -11 cm 2 . When compared to σ abs of a QD (typically measured about 1 -6 × 10 -14 cm 2 , see Supplementary Information), this is a very significant increase, and it manifests in the very low power required to excite the antenna in the linear regime and to switch it to the nonlinear emission regime. This huge increase in the absorption cross-section, which leads to efficient excitation of the antenna, explains its high brightness (Figure 4a) despite its moderate F 3 ≈ 3 (Figure 5b).

From P2 to P4, g 2 (0) increases from 0.5 to 1 showing stronger multiexciton emission, and the lifetime reduces significantly because multiexciton recombination is considerably faster. This nonlinear emission suggests a superradiance effect [START_REF] Bacher | Biexciton versus exciton lifetime in a single semiconductor quantum dot[END_REF][START_REF] Zhang | New insights into the multiexciton dynamics in phase-pure thick-shell CdSe/CdS quantum dots[END_REF], where localized dipolar emitters emit collectively. Although the antenna contains a single QD, the multiple states created in the QD under higher excitation can be thought of as interacting dipolar emitters localized in it similar to coupled emitters in Dicke superradiance [START_REF] Dicke | Coherence in spontaneous radiation processes[END_REF]. Another way to view this superradiance effect is as a plasmonic Dicke effect [START_REF] Pustovit | Cooperative emission of light by an ensemble of dipoles near a metal nanoparticle: The plasmonic dicke effect[END_REF], where the cooperative emission from the antenna is due to the resonant energy transfer between the emitter and the plasmons rather than between multiple emitters. Such nonlinearity in emission introduced at a slightly higher power results in a substantial increase in brightness and recombination rate, thus creating an efficient and fast photon source.

Conclusion

The described deterministic nanolithography protocol helped us realize single-emitter plasmonic patch antennas with very high control over emitter positioning inside the antenna. The fabricated antennas showed directional and rapid single photon emission, and demonstrated nonlinear increase in brightness due to the high optical confinement inside the antenna and the high electron-hole pair confinement inside the nano-emitter. The dramatic increase in the absorption cross-section noted in plasmonic antennas opens opportunities for engineering these effects to create efficient light nanosources. Further investigation into the factors behind this effect, and controlling these factors by precise lithography will allow ways to overcome plasmonic losses, and create bright and fast emission sources. Moreover, a high absorption cross section is boon to applications that rely on light absorption such as solar energy conversion. In our future studies, we will investigate if the nonlinear emission of such patch antennas present opportunities that will benefit quantum technologies.

Figure 1

 1 Figure1Single-emitter plasmonic patch antenna with a QD A single QD is selected and embedded in a dielectric layer sandwiched between a plasmonic metal film and a patch to obtain a single-emitter plasmonic patch antenna, as illustrated in a. A transmission electron microscopy image b of the colloidal CdSe/CdS QDs used in this work, and c, monoexponentially fitted emission decays of four QDs in a homogeneous medium. d Optical setup used for the generation of spatially modulated light using a reflective phase-only spatial light modulator. PBS is a polarizing beam splitter.

Figure 2

 2 Figure 2 Deterministic in situ lithography on single-emitters using spatially modulated light Figures a to e illustrate the steps of the lithography protocol. The image above a is a confocal scan of a QD. The AFM image and the corresponding height profile above b depict a cylinder etched into the resist bi-layer, which is removed by chemical etching at step c as illustrated by the AFM image above it. The AFM image on the left of e shows a 20 nm high circular antenna patch of diameter 1.6 μm obtained after lift-off. f Cross-section intensity maps of LG 20 and LG 40 laser beams imaged by a camera, and the corresponding intensity profiles.

  Photon coincidences

  . Auger processes, which are non-radiative and qu-ench multiexciton emission, only influence multiexciton transitions such as biexciton to exciton, and are included in the multiexciton to exciton rate Γ and non-Auger multiexciton rates.

Figure 4

 4 Figure 4 Fluorescence enhancement due to plasmonic nanoantennas a Widefield fluorescence images of the same QD before and after plasmonic patch deposition. Both images were acquired in grey scale by a charge coupled device camera in 200 ms and under similar excitation and collection conditions. The scale line represents 10 μm. b Emission spectrum of this antenna. Radiation patterns and the corresponding (θ, φ) polar plots of two patch antennas showing their directional emission c, d.

Figure 5

 5 Figure[START_REF] Lodahl | Controlling the dynamics of spontaneous emission from quantum dots by photonic crystals[END_REF] Excition and multiexciton emission in plasmonic antenna a Detected photon intensity vs excitation power displays the transition from exciton emission to multiexciton emission b, is further highlighted by the emission decay and the normalized photon correlation curves for points P1-P4.


Thi-Phuong-Huyen Tran

Sy-Hung Nguyen
email: sihung.nguyen@hcmute.edu.vn

Stéphane Bouissou
email: bouissou@geoazur.unice.fr

Experimental Study of the Strain Localization in a Rock Analogue Material at Brittle-Ductile Transition

Keywords: Deformation bands, brittle-ductile transition, digital image correlation. I

Many studies on strain localization of geomaterials have been carried out over the past few decades. Theoretical analysis and numerical models show that the dilatancy property strongly affects the onset of the deformation localization bands. The evolution of deformation bands, however, remains poorly known. In this work, we analyzed the initiation and evolution of strain localization in a Granular Rock Analogue Material (GRAM1) performed under axisymmetric compression at brittle-ductile transition by using Digital Image Correlation (DIC) technique. DIC provides a full-field measurement of displacement and thus allows the analysis of non-homogeneous deformation. The onset of band/fracture localization is well detected and located on the stress-strain curve. During the deformation process, the deformation bands as well as sets of sub-parallel strands which contribute the final thickness of the main shear band form sequentially. The analyses of DIC data and post-mortem sample observation enhance the understanding of the strain localization phenomena at brittle-ductile transition.

INTRODUCTION

Rock failure is generally classified in three regimes: brittle, ductile and transitional brittle-ductile regimes. A large amount of experimental tests of rocks performed under axisymmetric compression, covering the whole range of material behavior from brittle to brittle-ductile transition, showed the important role of porosity change induced by the deviatoric stresses over the failures modes. In the brittle regime, the deviatoric stress can cause dilatancy which ultimately leads to failure by shear dilatancy band localization and brittle faulting. During cataclastic flow, the deviatoric stress provides significant contribution on the compaction strain which induces "shear-enhanced compaction" phenomenon [START_REF] Wong | The transition from brittle faulting to cataclastic flow in porous sandstones: Mechanical deformation[END_REF]. The reduction of porosity leads the rock to become harder, thus inhibiting the development of shear localization. Byerlee [START_REF] Byerlee | Brittle-ductile transition in rocks[END_REF] studied the frictional characteristics of rocks and found that the brittle-ductile transition pressure bdt is the pressure at which the stress required to form a fault surface is equal to the stress required to cause sliding on the fault. At m < bdt (mmean stress), the failure envelope is rather linear, the failure corresponds to an abrupt stress drop for axisymmetric compression conditions. At m > bdt, its slope progressively reduces to zero and then to the negative values, there is a smooth stress reduction with an axial strain increase [START_REF] Wong | The transition from brittle faulting to cataclastic flow in porous sandstones: Mechanical deformation[END_REF]. The transition between two regimes, which respectively lead to a localized strain with dilatancy inside a shear band and a quasi-homogeneous strain with compactancy, is still not well understood. In the brittle-ductile transition, a set of conjugated shear bands is typically formed, and becomes more and more numerous and closely spaced [START_REF] Bésuelle | Experimental characterisation of the localisation phenomenon inside a Vosges sandstone in a triaxial cell[END_REF]. The evolution of this process, including the moment and place of the onset of the first band and then the others, however remains poorly known. The detection of the onset of deformation banding is crucial for the theoretical analysis of the underlying mechanism, which is believed to be a constitutive instability resulting from the deformation bifurcation [START_REF] Rudnicki | Conditions for the localization of deformation in pressure-sensitive dilatant materials[END_REF]. One of the major criteria of applicability of the theory is a correspondence the critical hardening modulus predicted theoretically to that measured in the experiments. To make a correct measurement, one has to precisely define the onset of band localization and locate it on the stress-strain curve.

In conventional triaxial compression tests, the sample is within the steel pressure cell, which makes it impossible to measure directly the deformation field unless using heavy and expensive X-ray tomography techniques that can be applied in certain cases [START_REF] Lenoir | Volumetric digital image correlation applied to x-ray microtomography images from triaxial compression tests on argillaceous rock[END_REF], [START_REF] Bésuelle | X-ray micro C for studying strain localization in clay rocks under triaxial compression[END_REF]. In most cases only the postmortem sample observation is possible. Over the past few decades, extensive experimental investigations on strain localization of geomaterials have been carried out. In such tests, full-field measurements are being used to study the strain localization phenomenon which provides information concerning deformation localization pattern, onset of band/fracture localization and its propagation to the ultimate state. One of the most widespread full-field measurement techniques is the method of displacement field measurement using Digital Image Correlation (DIC). DIC technique provides qualitative and quantitative descriptions of local responses, which allow evaluating the strain distribution and localization [START_REF] Doumalin | Characterisation of the strain distribution in heterogeneous materials[END_REF], [START_REF] Bésuelle | A new true triaxial cell for field measurements on rock specimens and its use in the characterization of strain localization on a vosges sandstone during a plane strain compression test[END_REF]. Recently, DIC has also been extended to 3D volumetric image correlation [START_REF] Bornert | Mesure tridimensionnelle de champs cinématiques par imagerie volumique pour l'analyse des matériaux et des structures, Instrumentation[END_REF], [START_REF] Lenoir | Volumetric digital image correlation applied to x-ray microtomography images from triaxial compression tests on argillaceous rock[END_REF] by 3D imaging techniques such as X-Ray Computed Tomography (CT) or X-Ray Micro-Computed Tomography (MCT) [START_REF] Hild | Three-dimensional analysis of a compression test on stone wool[END_REF].

In the present work, we present the results on the studying the initiation and evolution of strain localization in a rock analogue (granular, frictional, dilatant, and cohesive) material GRAM1 under triaxial axisymmetric compression at a confining pressure Pc corresponding to the transition from brittle faulting to ductile deformation Pbdt. This material has the strength two orders of magnitude smaller than that of the most rocks, so it provides considerable technical/ methodological advantages compared to the rock tests. In the context of the present work, this is particularly important as for Pc not exceeding a few mega-Pascals, the pressure cell can be made of transparent plastic allowing thus a direct observation of the sample surface during loading. From the images capturing the sample during loading, we apply the DIC technique to obtain the displacement and strain fields during the sample deformation.

II. MATERIAL AND EXPERIMENTAL PROCEDURE

A. GRAM1 Material characterization

The experiments presented in this article were carried out on GRAM1 (Granular Rock Analogue Material 1) which are fabricated from a finely ground powder of TiO2 with the average grain size of ~ 0.3 μm. The powder is subjected to the hydrostatic pressure of Pfab = 2 MPa at which the grains are bonded one to another. An extensive program of GRAM1 stress-strain measurements in axisymmetric compression and extension tests at different Pc below Pfab shows that this material has frictional, cohesive, dilatant properties and mechanical behavior very similar to those of hard rocks (Fig. 1) [12], [START_REF] Mas | An experimentally constrained constitutive model for geomaterials with simple friction-dilatancy relation in brittle to ductile domains[END_REF].

GRAM1 produces the same macro failure features as rocks, from very brittle splitting to shear and compaction bands with Pc increase, but is about 2 orders of magnitude less strong and less rigid than rocks. Both the friction coefficient  and the dilatancy factor  reduce with Pc from positive to negative. GRAM1 presents a brittle-ductile transition behavior at Pc = Pbdt ≈ 0.3 MPa. The main physical and mechanical properties of GRAM1 are presented in Table 1.

TABLE I.

MAIN PROPERTIES OF GRAM1 [START_REF] Saillet | Evolution of cataclastic faulting in highporosity sandstone, Bassin du Sud-Est[END_REF] Properties GRAM1

Porosity (n) 0.

B. Sample and Experimental setup 1) Sample

The testing was carried out on cylindrical test samples of about 80 mm in height and 40 mm in diameter. For DIC analysis, a speckle-like pattern is sprayed on a very thin (~300 µm) jacket to provide the surface with characteristic features. The droplets should be of an almost uniform size (of a few pixels, in our case about of 3-10 pixels), and sufficient density (Fig. 2). The latter helps to increase the accuracy of DIC measurements [START_REF] Dautriat | Localized deformation induced by heterogeneities in porous carbonate analysed by multi-scale digital image correlation[END_REF].

2) Experimental setup

Fig. 3 shows the experimental setup using the new pressure cell developed in Géoazur laboratory [START_REF] Tran | Initiation and Evolution of a Network of Deformation Bands in a Rock Analogue Material at Brittle-Ductile Transition[END_REF]. This new pressure cell with two transparent plane rectangular windows allow direct observations of the sample surfaces during loading. The sample and two rigid platens at the ends are jacketed and placed into a pressure cell of 2 MPa capacity filled with clean water and subjected to the confining pressure P c of 0.3 MPa by a 3 MPa capacity-pressure generator (the accuracy is of 4.5×10 3 Pa). The sample is then loaded in the axial direction by a piston with a velocity of 10 -6 m.s -1 , the axial stress induced is measured by an internal force sensor of 8 kN capacity with 0.1% precision. The deviatoric stress is computed as the major principal stress minus the minor principal stress q = .

The sample deformation was recorded throughout each test using two digital cameras with high resolution of 2752×2204 pixels (with a pixel size of about 32×32 m  ) set up on two sides of plane windows of the pressure cell. The lighting system consists of four LED bulbs; two bulbs for each side are fixed on the top and bottom of window with a 45° inclination. To minimize spatial and temporal differences in the DIC uncertainties, it is important to align the axes of the cameras perpendicularly to these plane windows and keep a diffuse, homogeneous and unchanging illumination of the sample during the exposure time. During loading, images covering on the whole sample height are taken simultaneously at a frequency of 10 Hz; this allows investigating deformation bands from the initiation to their evolutions occurred in 0.1 s.

3) Digital Image Correlation technique

Digital Image Correlation (DIC) is a method for noncontact full field kinematics measurement of planar or nonplanar surfaces undergoing deformation. This method is based on the comparison of a reference image with an image at a deformed stage. Numerous DIC softwares have been developed, either academically (7D, Correla, CorrelManuV, KelKins, COSI-Corr…) or commercially (Aramis 2D, VIC-2D…). In this work, we use a software package 7D, which was developed at the Université de Savoie [START_REF] Vacher | Bidimensional strain measurement using digital images[END_REF], for sub-pixel correlation.

A square grid is firstly defined in the analyzed area on a reference image of a sample and one or a set of sample images at later deformation stages. Around each grid node, a square correlation window (or subset) is then defined. In our analysis, the grid element size and correlation window size are both 10×10 pixels (Fig. 4). The DIC algorithm is designed to find the most similar subset between the reference and deformed images by optimizing a correlation coefficient C defined as (1):

                              D i D i i i i D i i g x g f X f g x g f X f C 2 2 1 ( 1
)
Where D is the subset over which C is computed. Xi and   The difference in the positions of grid nodes (centers of correlation windows) between the reference and deformed images yields a displacement vector, from which strain tensors might be calculated. Strains are calculated, in this case, using large strain Green Lagrange formulation.

III. RESULTS AND DISCUSSION

In this section, the analysis of two representative tests conducted under the same condition, but different total shortening 1.9% and 8.1% respectively, will be presented.

A. Band initiation and propagation

Fig. 5 presents the stress-strain curve for the presented test C0.3-12T which have clear peak succeeded by a smooth stress reduction and then, by a stress plateau. The nominal axial strain 𝜀̃𝑎 𝑥 calculated from the DIC strains corresponds very well to the internal LVDT measurements [START_REF] Tran | Initiation and Evolution of a Network of Deformation Bands in a Rock Analogue Material at Brittle-Ductile Transition[END_REF] where the LVDT sensors were glued directly to the jacket of sample [12]. From 17000 photographs, 13 photographs corresponding to key moments in the test were selected for the DIC analysis. In order to study the evolution of the band network, we analyze ax, maximum shear m = axc)/2, and the incremental shear m strains (ax and c are the axial and circumferential/horizontal strain).

From the evolution of accumulated shear strains m in the test C0.3-12T presented in the Fig. 6 for 13 representative points indicated on the stress-strain curve, the onset of strain localization can be seen to appear on the lower right corner of the left hand side at point/stage 3. The initiated deformation band I form early, near the stress peak (q = 1.06 Mpa), and then propagates rapidly with the average rate Vb of 5×10 -1 mm/s (the rate of the sample shortening Vc is 10 -3 mm/s). Two other deformation bands (II) and (III) are initiated at stages 4 and 5 and then propagate with the rate of about 3×10 -1 mm/s, which is much lower than the velocity of the first band propagation. These master bands are the most developed and remain active throughout the whole deformation history. Along with the master bands, secondary bands (bands a, b, c, d, g, e, f, h, i) form respectively during the deformation history (for more detail in [START_REF] Tran | Initiation and Evolution of a Network of Deformation Bands in a Rock Analogue Material at Brittle-Ductile Transition[END_REF]). The orientation of the shear bands with respect to the loading axis remains stable over the whole loading stage. This angle is of about 34±1°.

B. Contribution of set of sub-parallel strands to band's thickness

The Fig. 8 clearly shows that the evolution of an individual band includes its along-strike propagation and thickening. The thickness of the band increases progressively during the test and is contributed by a serie of small bands, known as strands. Photos of the surface of a master band show the contribution of strands to final thickness of the master band (limited between two dotted lines) (Fig. 8). These strands are quasi-sub parallel and formed sequentially, which is well seen on the incremental strain maps in Fig. 9. Fig. 9a presents the evolution of the band (I)'s formation through the cumulative strain fields at six different levels of loading in the small area marked in red rectangle in Fig. 8c. These cumulative strain fields are computed in comparison to the reference image at the initial stage (0). Two dotted lines present the final thickness of the band (I) at the end of the test, they therefore reveal the increasing of the band's thickening over time. Thus, by comparison of evolution of incremental strain fields in Fig. 9b, it shows the process of forming of subsequent strands in this shear band zone. At step 2-3, the first strand initiates on the upper zone near the dotted line on the right side, and then the other strands in turn appear and shift to the bottom dotted line (steps 3-4, 6-7, 9-10, 12-13). This allows identifying the evolution of the strands in deformation band in the analyzed zone on post-mortem sample (Fig. 8c).

The thickness of the individual band/stand measured from DIC data dD is thicker than that measured by zooming on the high-resolution photos of the post-mortem sample dR. The actual thickness dR of the incipient band is around 0.1 mm while dD is about two grid elements (~ 0.64 mm) since the size of DIC grid element is 10 pixels ( (Fig. 10b). When dD is bigger than the grid element size, dD approaches dR. Fig. 10 shows that both dR and dD of master band are about 2.3 mm.

As shown in Fig. 11, in the case of 8.1% axial shortening, we can observe a network of dense conjugate bands in which each shear band zone (master bands as well as secondary bands) is contributed by a set of sub-parallel strands to the shear direction. The contribution of these strands to the main shear bands makes their orientation lightly decrease to of about 32±2° with respect to the axial principal stress. The thickness of these shear bands dR (master bands as well as secondary bands) is about 2-3 mm, corresponds well to one measured from DIC data dD.

C. Discussion

The results obtained from DIC analyses and post-mortem sample observation show the sequential formation of deformation bands. The evolution of each band includes its along-strike propagation and the forming of sets of subparallel strands to the shear direction contributing the thickness of the main shear band.

One explanation for deformation localization and the formation of band network at brittle-ductile transition is based on the theory of strain hardening materials, where the material failure is defined not only through the stress-strain evolution, but also through inelastic straining. That means considering the evolution the normalized hardening modulus h and dilatancy factor with mean stress m and equivalent inelastic shear strain 𝛾̅ 𝑝 . From analysis of extended experimental data set for GRAM1 and data for Tavel limestone and Solnhofen limestone, Mas and Chemenda [START_REF] Mas | An experimentally constrained constitutive model for geomaterials with simple friction-dilatancy relation in brittle to ductile domains[END_REF] showed for the first time the yield surfaces in the space (𝜏̅ , 𝜎 𝑚 , ~𝛾̅ 𝑝 ), where 𝜏̅ is the von Mises stress and 𝛾̅ 𝑝 is the equivalent inelastic shear strain obtained by separation of elastic and inelastic strain. Their analysis confirmed that the deformation localization and rupture usually occurs when the normalized hardening modulus ℎ(𝜎 𝑚 , 𝛾̅ 𝑝 ) = [𝜕 𝜏̅ (𝜎 𝑚 , 𝛾̅ 𝑝 ) 𝜕𝛾̅ 𝑝 ⁄ ]/𝐺 is close to zero (G is the shear modulus). The inelastic deformation starts well before the peak stresses and is characterized by positive hardening h > 0. Softening h < 0 and dilatancy are correlated after the peak stresses, then correlation between hardening and compacting is observed at high mean stress m, and h become positive (hardening).

At brittle-ductile transition, the mean stress at deformation localisation is located between the hardening and softening deformation regimes, m = 0.653 MPa in the reference test [START_REF] Tran | Initiation and Evolution of a Network of Deformation Bands in a Rock Analogue Material at Brittle-Ductile Transition[END_REF]. At this m, h changes from positive to negative value and then again increases close to 0. Therefore, one can assume that the first band initiates h < 0 and evolved along the same band until h became positive.  respectively first increases from negative to positive values with 𝛾̅ 𝑝 and then reduces to negative values again. As mentioned, the compaction is associated with the hardening, whereas the dilatancy is associated with the softening. The formation of band network occur thus at the transition from positive (softening) to negative (hardening)  with 𝛾̅ 𝑝 growth. In consideration of  p (𝛾̅ 𝑝 ) function, the formation of the first band (band I) in the reference test occurs between points 2 and 3 (on the stressstrain) where 𝛾̅ 𝑝 ≈ 0.016. The corresponding  is positive (close to the maximum value) and h is negative. After this point, the 𝛾̅ 𝑝 increases rapidly, but then slows down again when  becomes negative and h is positive at 𝛾̅ 𝑝 = 0.02 (Fig. 12). The material within the band therefore becomes stronger with deformation, it thus makes continued localization at the same location difficult. In contrast, stress is locally enhanced in the surrounding regions which have been mechanically weakened and at a certain point it leads to form a new band in this weakened region.

IV. CONCLUSION

The experimental methodology, combining both DIC analyses, post-mortem sample observation, allows us to detect the initiation of localisation deformation bands and their evolution during the loading and to understand their sequential activity at the macroscopic sample scale. The deformation bands do not form simultaneously but sequentially while they are not able to be identified by only post-mortem sample observation. Furthermore, DIC analysis allows giving some constraints on the velocity propagation of the deformation bands and showing the forming of sets of sub-parallel strands which contributes to the final thickness of the band. Both the dilatancy factor  and hardening modulus h evolve with 𝛾̅ 𝑝 . The strain localisation occurs in the dilatant ( deformation regime. During the band evolution, the deformation within it becomes compactive (which leads to widen the band and the formation of new bands. These experimental findings are close to that documented in the natural (geology) cataclastic bands [START_REF] Aydin | Analysis of faulting in porous sandstones[END_REF], [START_REF] Saillet | Evolution of cataclastic faulting in highporosity sandstone, Bassin du Sud-Est[END_REF]. The process of strand network's formation contributing to the total band thickness may propose an explanation to the evolution of cluster network in field [START_REF] Philit | Cluster of cataclastic deformation bands in porous sandstones[END_REF].

Fig. 1 .

 1 Fig. 1. Comparison of 𝑞(𝜀 𝑎𝑥 ) and 𝜀(𝜀 𝑎𝑥 ) curves for GRAM1 and rocks for compression tests. The plots for the rocks are from: Solnhofen limestone, Adamswiller sandstone [12].

Fig. 2 .

 2 Fig. 2. Cylindrical GRAM1 sample jacketed with a very thin transparent latex film in axisymmetric compression tests Fig. 3. Photo of the experimental setup

  coordinates (in pixels) of homologous points in the reference and deformed images. 0  is the transformation function. 𝑓(𝑋 𝑖 ) and levels of the point i in the reference image and in deformed images. 𝑓 ̅ and g are respectively the averages of the grey levels of all the pixels in subset of the reference and deformed images. C varies from 0 for perfect correlation, to the value 1 for no match at all.

Fig. 4 .

 4 Fig. 4. Sample and the analyzed areas defined for computing the strains at different scales; Correlation parameters used in this study

Fig. 7

 7 Fig.7 shows the evolution of accumulated axial strains ax in the case of large shortening (8.1%) C0.3-8T. Network of conjugate bands is formed by two families of deformation bands. Each family consists of both master bands and secondary bands. The master bands form quite early (from point/stage 3) and remain active during the deformation process. Along with these master bands, secondary bands form sequentially and network of conjugate bands therefore becomes denser.

Fig. 6 .

 6 Fig. 6. Shear strains fields 𝛾 𝑚 from DIC analyses. For each point, two images corresponding to the two opposite sides present strains 𝛾 𝑚 accumulated between point 1 and points n, where n=2, 3…13. The numbers on color palettes are the strain values multiplied by 10 3[START_REF] Tran | Initiation and Evolution of a Network of Deformation Bands in a Rock Analogue Material at Brittle-Ductile Transition[END_REF]

Fig. 7 .

 7 Fig. 7. Evolution of the εax obtained from DIC in the test C0.3-8T. The numbers on the stress-strain curve from this test (in the upper left part corner of the figure) indicate the points for which the strains are shown. The number on color palettes are the strain values multiplied by 10 3

Fig. 8 .

 8 Fig. 8. Photos of post-mortem sample surface from test C0.3-12T zoomed at different scales: (a) the whole sample from the side 1, (b) in the black rectangle, (c) in the red rectangle.

Fig. 9 .

 9 Fig. 9. Evolution of (a) the εax, γm fields and (b) incremental axial strains ∆𝜀 𝑎𝑥 , incremental shear strains ∆𝛾 𝑚 obtained from DIC in the test C0.3-12T for the analyzed area defined in Fig.4 and Fig.8c. The number on color palettes are the strain values multiplied by 10 3

Fig. 10 .

 10 Fig. 10. (a) The thickness of master band dR measured by zooming on the photo of post-mortem sample surface from test C0.3-12T, (b) (c) axial strain fields εax obtained from DIC at the deformation stages 3 and 13, respectively. dD is the thickness of master band measured from DIC data.

Fig. 11 .

 11 Fig. 11. Photo taken the whole sample from the test C0.3-8T and zoomed on the deformation band network. Different types of deformation band: major deformation band (MB), secondary deformation band (SB), strands contributing to the principle band (S), band saturation zone (SZ), open fissure (F)

5 th International Conference on Green Technology and Sustainable Development (GTSD)978-1-7281-9982-5/20/$31.00 ©2020 IEEE

ACKNOWLEDGMENT

This work was supported by the Côte d'Azur Observatory, the Region Provence Alpes Côte d'Azur and GeoFrac-Net Consortium.


Maxime Mauduy

Fabien Gierski

Virginie Bagneux

Nicolas Cabe

Denis Jacquet

Pascale Leconte

Cécile Sénémeaud

Nicolas Margas

Nicolas Mauny

Ludivine Ritz

Nicolas Cabé

Hélène Beaunieux

Jessica Mange

A Dominance Analysis Approach Applied to Psychological Factors Associated with Binge Drinking among University Students

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

Introduction

Binge Drinking (BD), generally defined as a heavy alcohol consumption over a short period (NIAAA, 2004), is considered as a major public health issue, and especially among university students as it concerns 28% of European student samples (European Union: Directorate General Communication, 2010). While previous studies have evidenced the role of several psychological factors related to BD (e.g., drinking motives 1 , drinker identity 2 , alcohol subjective norms 3 , emotional states [START_REF] Hussong | The stress-negative affect model of adolescent alcohol use: disaggregating negative affect[END_REF][START_REF] Schuler | Age-varying associations between substance use behaviors and depressive symptoms during adolescence and young adulthood[END_REF] , metacognitions beliefs [START_REF] Gierski | Positive and negative metacognitions about alcohol use among university students: Psychometric properties of the PAMS and NAMS French versions[END_REF] ), little is known about the strength of each of these factors as determinants of BD, mostly because they were often considered separately. The present study therefore proposes to examine the strength of their relative contribution and to identify the most associated factors to BD.

Material and Methods

A community sample of 2851 university students (Mage = 20.50; SD = 2.00; 62.9% of women) participated in an internet survey-based study that was included in the larger research ADUC project ("Alcool et Drogues à l'Université de Caen"). The participants provided information on alcohol related variables (AUDIT, BD score), demographics (sex, age), as well as subjective norm, social identity, external motivations, internal motivations, meta-cognitions, impulsivity and personality traits.

The data were first processed via a backward linear regression analysis with repeated K-fold crossvalidation [START_REF] Bruce | Practical Statistics for Data Scientists: 50+ Essential Concepts Using R and Python[END_REF] to identify the factors that are significantly associated to BD. Second, to counteract the limitations of traditional statistical methods assessing the strength of factors in models, the relative importance of significant psychological factors associated with BD was assessed by performing a bootstrapping dominance analysis [START_REF] Grömping | Variable importance in regression models[END_REF] .

Results

First, the backward linear regression analysis evidenced a best 14-variable model (R² = .512, F(14,1893) = 142, p <.001; RMSE = 14.659; MAE = 10.172), excluding positive metacognitions, negative metacognition (i.e., prejudice), some dimensions of impulsivity (i.e., perseverance and sensation seeking), anxiety trait, and depression. Second, the bootstrapping dominance analysis showed that the strongest variables associated with BD were enhancement motives and drinking identity (average ΔR² = 21.81%), followed by alcohol subjective norm and social motives (average ΔR² = 13.99%). Other associated variables (average ΔR² = 2,84%) were negative metacognition on uncontrollability, sex, coping motives, lack of premeditation, positive metacognition on cognitive self-regulation, positive urgency, lack of perseverance, age, conformity motives and loneliness.

Conclusion

The comparison of the relative importance of variables associated to BD has allowed to identify four major psychological factors, namely enhancement motives, drinking identity, alcohol subjective norm and social motives, suggesting that BD is largely motivated by inter-individual factors. These results have major empirical and clinical implications. Indeed, they highlight specific psychological factors which, not only, should be explored more in depth in the understanding of BD, but also, should be targeted as a priority in new prevention strategies to solve the BD issue.

Keywords: binge drinking, dominance analysis, identity, enhancement motives, subjective norm, social motives


Bencsura

Tibor Eric Henon

B Ga Bor Vasva Ri,A Istva

Szila Gyi

A Š Kos Sa Ndor Do

Sebastien Canneauxb

Reaction and complex formation between OH radical and acetone

Recent Ðeld measurements have revealed that acetone is present in surprisingly high concentration in the atmosphere.1 It has been shown1ah3 that atmospheric degradation of acetone can be the dominant source of (OH and HO

x HO 2 ) radicals in the upper troposphere resulting in increased ozone production.4 Sources of acetone in the atmosphere include oxidation of non-methane hydrocarbons, biogenic and anthropogenic emissions and biomass burning. 5 Major sinks are photodecomposition6 (the main loss process) and reaction with OH. 7 Kinetic studies of the reaction of OH radical with acetone have been reviewed.8,9 Very recently, numerous new investigations have been performed and reported in close succession.7,10h12 In most of the kinetic experiments, the overall reaction (1) was studied at and above room temperature where the rate coefficients determined were found to obey Arrhenius law.8,9

OH ] CH 3 C(O)CH 3 ] products (1)
The reaction was assumed to proceed via H-atom abstraction, (1a), producing acetonyl radical, CH

C(O)CH 3 : OH ] CH 3 C(O)CH 3 ] CH 2 C(O)CH 3 ] H 2 O (1a) 2
Wollenhaupt et al.7 was the Ðrst who extended the temperature-range of the investigations down to that typical of the upper troposphere (T B 200 K). The authors observed strong non-Arrhenius behaviour for the overall reaction of OH with acetone. The rate coefficient, was found to k 1 , decrease with decreasing temperature, Ðrst fast, but only slowly below room temperature, reaching a minimum at around 240 K, then increasing again slightly below this temperature. The temperature dependence was explained7 by the existence of two reaction routes. Accordingly, the reaction proceeds via H atom abstraction (1a) at higher temperatures, while below room temperature methyl elimination (1b) dominates :

OH ] CH 3 C(O)CH 3 ] CH 3 ] CH 3 COOH (1b)
In a subsequent publication, Wollenhaupt and Crowley11 presented experimental evidence for the formation of in the CH 3 reaction of OH with acetone. Based on the temperature dependence of and the measured yields, reaction (1b) k 1 CH 3 was proposed7,11 to occur via an additionÈelimination mechanism, where addition of OH to the carbonyl C atom is followed by elimination of from the vibrationally excited CH 3 addition radical (CH

3 C(OH)(O)CH 3
)*. The very recent experimental studies7,11,12 clearly indicate a complex mechanism for the reaction of OH radical with acetone. However, despite a wealth of information supplied by these new studies, important features of the reaction are not yet known. Such are, for example, the e †ect of pressure on the reaction, the branching ratio for the H abstraction channel and the question of how the assumed reaction mechanism conforms to theoretical models.

In this paper we present experimental results obtained at room temperature for and the branching ratio

k 1 C 1a \ k 1a
/k 1 , and report parts of the results from ab initio molecular orbital computations. [START_REF] Bencsura | [END_REF] The discharge-Ñow (DF) method was applied to carry out the experiments. Two kinetic set-ups were used. One of them served for the determination of the rate coefficient of the overall reaction using resonance Ñuorescence monitoring of the OH radical (DF/RF). The other apparatus was equipped with laser induced Ñuorescence detection (DF/LIF) to determine the reaction branching ratio. OH radicals were generated inside of a moveable injector by reacting H atoms with and F atoms with in the DF/RF and DF/LIF NO 2 H 2 O experiments, respectively. OH and acetonyl (1-methylvinoxy) radicals were detected in the DF/LIF experiments following excitation in the (1,0) band of the AÈX transition at 282 nm14 and in the electronic transition at 340 nm,15 respectively. Laser radiation was provided by a Nd : YAG pumped frequency-doubled tunable dye laser. Helium (Messer-Griesheim, 99.9990%) was the carrier gas. High purity acetone (Aldrich, 99.9]%) was used in the experiments which was degassed by repeated freezeÈpumpÈthaw cycles prior to use. 2 bath gas under conditions where probable systematic errors were virtually eliminated. The very good agreement of the rate coefficients gives credit to them, particularly because we used the non-photolytic discharge-Ñow method while in all previous determinations the OH radicals were produced by UVphotolysis.7h12 Our measurements were carried out at low pressures (P B 3 mbar He). Comparison with published results7h12 reveals no systematic variation of with reaction k 1 pressure. This is actually what is anticipated for a direct H atom abstraction reaction. The observation of pressure dependence would have provided strong evidence for the formation of an energized adduct (an addition radical or radicalÈ molecule complex) that could be stabilized by collisions. Obviously, the reverse statement is not true : the absence of pressure dependence does not exclude the formation of adduct intermediates. That is, the pressure independence observed for neither proves nor disproves the existence of an additionÈ k 1 elimination mechanism for the reaction of OH with acetone.

In the DF/LIF kinetic experiments, the consumption of OH was accompanied by a parallel build-up of CH 2 C(O)CH 3 , providing evidence for reaction (1a). Representative kinetic curves for OH and are shown in Fig. 2. The CH 2 C(O)CH 3 branching ratio of reaction (1a) has been determined by computer simulations of the measured concentration vs. time proÐles13 and a comparative method, similar to the one we used in a previous work. [START_REF] Temps | Do be[END_REF] Results obtained by the comparative method are presented here. In this case, the reaction of F atom with acetone served as a reference, the acetonyl-yield for which was assumed to be 100%. Experiments were carried out in the following way : F atoms were introduced through the moveable injector into the reactor at a Ðxed distance (constant reaction time), were mixed up with acetone in high excess to assure complete conversion, and the H abstraction reaction channel :

([CH 3 C(O)CH 3 ] A [F] 0 ) the LIF
C 1a \ k 1a k 1 \ 0.50 ^0.04 (T \ 298 K)
The assumption that the reference reaction F ] CH 3 C(O)CH 3 proceeds exclusively via H-atom abstraction was tested by comparison with the reaction in an Cl ] CH 3 C(O)CH 3 experimental arrangement similar to that described above (conversion of F to Cl was achieved by the reaction of F with HCl). The average of the ratio of the respective signalstrengths was found to be S CH2C(O)CH3

(Cl`acetone)/S CH2C(O)CH3 (F`acetone) \ 1.03 ^0.07 which is believed to be a strong indication for practically 100% acetonyl yields in the reactions of acetone with both F and Cl. The branching ratio determined by us for reaction channel (1a) is signiÐcantly less than one, indicating that a simple H atom abstraction can not be the only reaction mechanism for the reaction of OH with acetone. This is in line with the Ðndings of Wollenhaupt and Crowly11 who reported the value of plexes preceding transition states. Ab initio calculations were performed using the GAUSSIAN 94 software package.17 Reactants, complexes, transition-state structures and products were fully optimized using the analytical gradients at the MP2 level of theory,18 with the 6-31G(d,p) basis set. The minimum energy path (MEP) was examined by the IRC procedure ;19 the transition states were found to connect proper reactants and products. Single-point calculations applying the CCSD(T) method20 with the 6-311G(d,p) basis set were carried out to obtain energy di †erences for stationary structures. The "" frozen-core ÏÏ approximation was used both in MÔller-Plesset and CCSD(T) calculations. Some of the structural features and the zero-point energy corrected relative energies computed for the reaction are summarized OH ] CH 3 C(O)CH 3 in Fig. 3. Detailed geometries and harmonic vibrational frequencies will be given in a later publication. [START_REF] Bencsura | [END_REF] Our theoretical investigations have revealed a pathway for the H atom abstraction reaction which can be summarized as

C 1b \ k 1b k 1 \ 0.50 ^0.15
OH ] CH 3 C(O)CH 3 ] MC1 ] TS1 ] CH 2 C(O)CH 3 ] H 2 O
where MC1 and TS1 designate hydrogen-bridged complex and reaction transition state, respectively. MC1 was found as local minimum on the potential energy surface with 26. reaction (1a), have been thought of as direct metathesis reactions with a single barrier and transition state along the reaction coordinate. There are more and more indications, however, from theoretical studies (e.g. ref. [START_REF] Jodkowski | Be rces Do be[END_REF]) and also from sophisticated experimental works (e.g. ref. 23) that the kinetic and dynamic properties of many of the H atom abstraction reactions are connected to the formation of lose radicalÈ molecule complexes. In certain cases, the direct H atom transfer is "" assisted ÏÏ24 by intermolecular noncovalent bonding interactions. This is what is believed to be the case for reaction (1a) suggested by our theoretical results as well.

In experimental works of the reactions of atoms and free radicals with carbonyl compounds (aldehydes and ketons), the addition to the C2O bond was occasionally suggested to explain the observed product yields, but unambigous experimental evidence has been scarce (see e.g. ref. 25 and references therein). We are aware of only two theoretical studies which dealt with the addition reaction of OH to the carbonyl double bond 26,27. In our CCSD(T) computations, addition to the carbon atom of the C2O group was found to occur through a loosely bound complex, MC2, with similar structure to that of the transition state, TS2, that leads to the formation of the addition radical a-hydroxy-i-propoxy in the reaction :

OH ] CH 3 C(O)CH 3 ] MC2 ] TS2 ] CH 3 C(OH)(O)CH 3
The incoming OH radical [O(2)H( 7)] approaches the C(1) site from a direction nearly perpendicular to the molecular plane containing the C(2)C(1)C(3) skeleton of the acetone molecule (see Fig. 3). In MC2, a four-membered ring is formed by the hydroxyl radical and the C(1) and O(1) atoms of the acetone molecule. As in the case of the abstraction complex, the intermolecular hydrogen bond is formed between the hydrogen atom of the OH radical, H (7), and the oxygen atom of the acetone molecule, O(1). The binding energy for MC2 was calculated to be 16.3 kJ mol~1, that is, the addition complex is less stable by 10 kJ mol~1 than its abstraction counterpart. The four-membered ring structure is preserved in the transition state, TS2, but with a much smaller O( 2 Atmospheric implications of our kinetic results are in accordance with recent conclusions. 7,11,[START_REF] Gierczak | 16th International Symposium on Gas Kinetics[END_REF] The pressure independent rate coefficient measured in the present study supports the view7,12 that reaction (1) is a more important sink for acetone in the upper troposphere than it was thought before. On the basis of product studies performed by Wollenhaupt and Crowley11 and in the current work, it appears well established that the branching ratios are B0.5 for both the (1a) and the (1b

) CH 2 C(O)CH 3 ] H 2 O C H 3 ] CH 3
COOH reaction channels at room temperature. A consistent mechanism is still lacking, however, which would explain the observed product yields and the temperature dependence of and would allow reliable predictions to be made for k 1 ,7,12 atmospheric modeling studies. Experimental determination of the temperature dependencies of product yields and a search for further stationary structures on the potential energy surface would be particularly useful in elucidating the mechanism of the reaction of OH with acetone.

  Kinetics and mechanism of the reaction of OH with have been studied by discharge-Ñow CH 3 C(O)CH 3 experiments and CCSD(T) quantum chemical computations. In the experiments, the rate coefficient for the overall reaction, (1), and the branching ratio for the speciÐc reaction channel OH ] 04 ^0.03) ] 1011 mol~1 s~1 and respectively (T \ 298 K). Two di †erent reaction pathways have C 1a \ k 1a /k 1 \ 0.50 ^0.04, been characterized by ab initio calculations. Both H atom abstraction and OH addition to the C2O group have been found to occur through hydrogen bonded complexes. Most of our results OHÉ É ÉCH 3 C(O)CH 3 support recent Ðndings (M. Wollenhaupt, S. A. Carl, A. Horowitz and J. N. Crowley, J. Phys. Chem. A, 2000, 104, 2695 ; M. Wollenhaupt and J. N. Crowley, J. Phys. Chem. A, 2000, 104, 6429) but contradictions remain concerning the mechanism of this atmospherically important reaction.

Fig. 1

 1 Fig. 1 Plots used for the determination of the rate coefficient for the overall reaction of OH with acetone. The kinetic data plotted were obtained from DF/RF experiments at T \ 298 K. The inset shows representative pseudo-Ðrst-order decay plots, where and S OH on S OH off designate OH resonance Ñuorescence signal-strengths in the presence and absence of acetone, respectively. The numbers given in square brackets refer to concentrations in 10~10 mol cm~3. CH 3 C(O)CH 3 The slopes of the straight lines of the semilogarithmic decay plots give the pseudo-Ðrst-order rate coefficient, The bimolecular rate coeffik 1 @ . cient, is obtained from the plot of The k 1 , k 1 @ \ k 1 [acetone] ] const. straight lines are linear least-squares Ðts to the data.

  signal-strengths of acetonyl radicals formed, were recorded. Under the very same conditions, S CH2C(O)CH3 (F`acetone) , with the exception that Ñuorine atoms were converted to OH by the reaction with inside the injector, the LIF signal-H 2 O strengths of the acetonyl radicals for the studied reaction, were recorded. The branching ratio for reaction S CH2C(O)CH3 (OH`acetone), (1a) was obtained from the ratio A S CH2C(O)CH3 (OH`acetone)/S CH2C(O)CH3 (F`acetone) . small correction was made to take into account the di †erent wall-losses of F and OH. Multiple determinations were performed with an approximately 10-fold variation in The [F] 0 . average of the results is the recommended branching ratio for

Fig. 2

 2 Fig. 2 Consumption of OH and formation of acetonyl in a typical DF/LIF experiment of the reaction. Experimen-OH ] CH 3 C(O)CH 3 tal conditions were as follows :T \ 300 K, P \ 2.83 mbar, [F] o \ mol cm~3, mol cm~3, [OH] o \ 3 ] 10~13 [H 2 O] \ 7 ] 10~12 [acetone] \ 9.7 ] 10~10 mol cm~3.and designate the S OH S CH2C(O)CH3 concentration-proportional LIF signal strengths for the OH and acetonyl radicals, respectively. The solid curves were obtained from computer simulations of a multi-reaction mechanism, where the only Ðtted parameter was the branching ratio of the acetonyl-forming reaction channel, i.e. C 1a \ k 1a /k 1 .13

  Ñash photolysis experiments ; was C 1b obtained by Ðtting of an assumed reaction mechanism to the measured proÐles.11 The two branching ratios, deter-CH 3 O mined by Wollenhaupt and Crowly and by us, complement each other, which may be taken as an indirect C 1a ] C 1b B 1, support for their reliability. Collaterally with the experimental work, we have performed ab initio molecular orbital computations to get an insight into the mechanism of the reaction of OH with acetone. Reaction pathways have been explored on the potential energy surface for the hydrogen abstraction reaction and the OH addition to the carbon atom of the C2O group. Both reactions were found to occur through loosely bound com-OHÉ É ÉCH 3 C(O)CH 3

  3 kJ mol~1 binding energy relative to the reactants. It has a (D 0 ) six-membered ring-like structure formed by the hydroxy and H(1), C(2), C(1) and O(1) atoms of the acetone molecule (see Fig. 3.). The calculations predicted a nearly planar structure for OH approach to the C(2)C(1)C(3) frame of acetone. In a very recent work, Aloisio and Francisco21 have found an complex, very similar to MC1, using OHÉ É ÉCH 3 C(O)CH 3 density functional theory method. Applying the best level of theory [B3LYP/6-31&&G(3df,3pd)] the authors reported 19.3 kJ mol~1 binding energy. Besides the OHÉ É ÉCH 3 C(O)CH 3 complex, no other portion of the potential energy surface was investigated by Aloisio and Francisco.21 The structure of TS1 has a close resemblance to that of MC1, the transition state being practically "" preformed ÏÏ by the hydrogen-bonded complex. The computed barrier is 16.7 kJ mol~1 above the reactants. Traditionally, hydrogen-transfer reactions, such as

Fig. 3

 3 Fig. 3 Schematic molecular structures and energetics for obtained from ab initio computations at the CCSD(T)/6-311G(d, OH ] CH 3 C(O)CH 3 p)//MP2/6-31G (d,p) level of theory. MC1 and MC2 designate complexes formed on along the H atom abstraction reaction OHÉ É ÉCH 3 C(O)CH 3 path and the C2O addition reaction path, respectively. TS1 and TS2 are transition states, the structures of which are similar to those of the respective complexes, MC1 and MC2. OH-attack takes place approximately in the plane and perpendicularly to the plane OHÉ É ÉCH 3 C(O)CH 3 deÐned by the C(2)C(1)C(3) skeleton of the acetone molecule in case of the abstraction reaction and addition reaction, respectively. The energy data are given in kJ mol~1 and include zero-point energy corrections.

  Representative pseudo-Ðrst-order plots are presented in Fig. 1. Experimental conditions and results are summarized in Table 1. Most of the experiments were carried out by the DF/RF method resulting in the recommended rate coefficient of Wollenhaupt et al.7 applied the laser Ñash photolysis method combined with laser induced Ñuorescence detection of OH radicals at 27, 67, and 133 mbar of Ar or N

	cm~3 ; acetone was present in large excess ([CH 3 C(O)CH 3 ] A [OH] ). 0 The usual pseudo-Ðrst-order experiments and evaluation
	procedure were used to determine the rate coefficient of the
	overall reaction (1). k 1 cm3 mol~1 s~1.	(298
				The initial concentration of OH was	[OH] 0	O 7 ] 10~13	mol
	DOI : 10.1039/b009601f			Phys. Chem. Chem. Phys., 2001, 3, 551È555	551
		This journal is	(	The Owner Societies 2001

k 1 (298 K) \ (1.04 ^0.03) ] 1011 cm3 mol~1 s~1, with 1p statistical error quoted. In a few experiments, the DF/LIF method with di †erent OH source was applied giving essentially the same result (see Table 1). Our rate coefficient value agrees well with the recent IUPAC recommendation,9 K) \ (1.14 ^0.09) ] 1011 cm3 mol~1 s~1, and is in k 1 (298 excellent agreement with the very recent determination by Wollenhaupt et al.7 which is K) \ (1.04 ^0.05) ] 1011

Table 1

 1

		Experimental conditions and kinetic results for the reaction	OH ] CH 3 C(O)CH 3	] products	(1)a
	Method	OH source	No. of expts.	T / K	P/ mbar	v/ cm s~1	1 0 11 [CH 3 C(O)CH 3 ]/ mol cm~3	k w s~1 b/	k s~1 1 @ /	/ 1 0 ~10 k 1 cm3 mol~1 s~1
	DR/RF DF/LIF	H ] NO 2 F ] H 2 O	19 5	298 ^2 300 ^2	2.63 2.82	779 345		2.10È85.5 97.0È160		9 ^6c 3 ^2d	17È103 91È150	10.4 ^0.30 9.97 ^0.65

a Quoted errors represent 1p (precision only). b Heterogeneous loss of OH radicals. c PTFE wall coating. d Halocarbon wax wall coating.

  room temperature.7,11 Computations at even higher level of theory and with the application of more sophisticated basis sets are not expected to bring down the addition-channel barrier height close to the value of the experimental activation energy. Therefore, given the high yields observed experi-CH 3 mentally,11 we speculate that there must be a hitherto unexplored mechanism that does not involve the formation of the

					barrier computed for the addition reaction of OH to the car-
					bonyl C2O bond of acetone (33.9 kJ mol~1) is clearly incon-
					sistent with the negative ([3.5 kJ mol~1) experimental
					activation energy and therefore with the assumption of a fast
					additionÈelimination reaction through the energized
					(CH 3 C(OH)(O)CH 3 values and high CH )* 3	radical that might explain the large yields in the experiments at and below k 1
					addition radical assume, for example, the occurrence of a concerted mechanism It is not unreasonable to CH 3 C(OH)(O)CH . 3
					for the reaction of OH radical with acetone that directly leads
					to energy process. and CH 3	CH 3	COOH	formation via a low activation
				)ÈC(1) bond dis-
	tance [O(2)ÈC(1) \ 1.836	A ].	This bond distance is very close
	to that calculated by Soto and Page26 at the CASSCF(3,3)
	/DZP level of theory [O(2)ÈC(1) \ 1.834	A 26]	in their study of
	the potential energy surface for addition of OH to the carbon
	atom of formaldehyde. The barrier height computed for the
	addition channel is 33.9 kJ mol~1 which agrees favourably
	with the barrier of 29.2 kJ mol~1 predicted by Soto and
	Page26 at the MRCI/DZP level for OH addition to formalde-
	hyde. The exit channels on the potential energy surface for
	(e.g. investigated in the current work. We note, moreover, that have not been CH 3 C(OH)(O)CH 3 CH 3 ] CH COOH) 3
	addition of OH to the O atom of the carbonyl group is
	strongly endothermic26 and does not play any role under
	ambient conditions.
	The experimentally measured temperature dependence of k 1 has been described by Wollenhaupt et al.7 by a double expo-
	nential equation :
	k 1 (202È395 K)
	\ (5.3 ^2.2) ] 1012exp[([11.0 ^1.4 kJ mol~1)/RT ]
	] (1.0 ^0.5) ] 1010exp[(3.5 ^0.9 kJ mol~1)/RT ]
	] cm3 mol~1 s~1
	attributing the Ðrst term to direct H abstraction and the
	second one to addition. The zero-point energy corrected
	barrier computed in the current work for the abstraction
	pathway at the CCSD(T) level of theory is about 6 kJ mol~1
	larger (16.7 kJ mol~1) than the experimental activation energy
	(11.0 kJ mol~1), but this is a reasonable agreement and within
	the likely uncertainties (the agreement between the A-factors is
	also satisfactory13). On the other hand, the high activation

Phys. Chem. Chem. Phys., 2001, 3, 551È555

Phys. Chem. Chem. Phys., 2001, 3, 551È555

Acknowledgements

This work has been supported by the FrenchÈHungarian Intergovernmental Program (contract No. F-8/98) and the GermanÈHungarian Intergovernmental Program (contract No. D-8/98). The I.D.R.I.S. and C.R.I.H.A.N. computing centers are acknowledged for the CPU time granted. The authors appreciate the assistance of Mr T. in some of Ko cher the experiments.