Jason Devaux

Mélanie Mignot

Florent Rouvière

Isabelle François

Carlos Afonso

Sabine Heinisch
email: sabine.heinisch@univ-lyon1.fr

On-line reversed-phase liquid chromatography x supercritical fluid chromatography coupled to high-resolution mass spectrometry: a powerful tool for the characterization of advanced biofuels

Keywords: On-line RPLC x SFC, Comprehensive two-dimensional chromatography, advanced biofuels, microalgae bio-oil, HRMS

Bio-oils obtained by thermochemical or biochemical conversion of biomass represent a promising source of energy to complement fossil fuels, in particular for maritime or air transport for which the use of hydrogen or electricity appears complicated. As these bio-oils are very rich in water and heteroatoms, additional treatments are necessary before they can be used as biofuel. In order to improve the efficiency of these treatments, it is important to have a thorough knowledge of the composition of the bio-oil. The characterization of bio-oils is difficult because they are very complex mixtures with thousands of compounds covering a very wide range of molecular weight and polarity. Due to the high degree of orthogonality between the two chromatographic dimensions, the on-line combination of reversed-phase liquid chromatography and supercritical fluid chromatography (on-line RPLC x SFC) can significantly improve the characterization of such complex matrices. The hyphenation was optimized by selecting, in SFC, the stationary phase, the co-solvent, the make-up solvent prior to high resolution mass spectrometry (HRMS), and the injection solvent. Additionally, a new interface configuration is described. Quality descriptors such as the occupation of the separation space, the peak shapes and the signal intensity were considered to determine the optimal conditions. The best results were obtained with bare silica, a co-solvent composed of acetonitrile and methanol (50/50, v/v), a make-up solvent composed of methanol (90%) and water (10%) with formic acid (0.1%), an addition of co-solvent through an additional pump for SFC separation in a 2.1 mm column, and an hydro-organic solvent as injection solvent. The optimized setup was used to analyse two microalgae bio-oils: the full bio-oil coming from hydrothermal liquefaction and Soxhlet extraction of microalgae, and the gasoline cut obtained after distillation of the full bio-oil. Results in on-line RPLC x SFC-qTOF were particularly interesting, with very good peak shapes and high reproducibility. Moreover, the high degree of orthogonality for microalgae bio-oils of RPLC and SFC was highlighted by the very large occupation of the separation space. Isomeric profiles of compound families could be obtained in RPLC x SFC-qTOF and many isomers not separated in SFC alone were separated in RPLC and vice versa, thus showing the complementarity of the two chromatographic techniques.

Introduction

With the depletion of fossil fuels and the increasing global energy demand, the development of sustainable energy sources is of paramount importance. Microalgal biomass is a promising resource to produce biofuels and value-added products because of its high photosynthetic efficiency and fast growth rate. Moreover, microalgae only need water, light, nutrients, and carbon to grow [START_REF] Li | Characteristics and Evolution of Nitrogen in the Heavy Components of Algae Pyrolysis Bio-Oil[END_REF][START_REF] Oliveira | An overview on microalgae as renewable resources for meeting sustainable development goals[END_REF]. Microalgae can then be converted into bio-oils by thermochemical processes. Although some studies used pyrolysis to convert the microalgae [START_REF] Adamczyk | Pyrolysis Behaviours of Microalgae Nannochloropsis gaditana[END_REF][START_REF] Hu | A study on experimental characteristic of microwave-assisted pyrolysis of microalgae[END_REF][START_REF] Rahman | Valorizing of weeds algae through the solar assisted pyrolysis: Effects of dependable parameters on yields and characterization of products[END_REF], they are converted most often by hydrothermal liquefaction [START_REF] Neveux | Biocrude yield and productivity from the hydrothermal liquefaction of marine and freshwater green macroalgae[END_REF][START_REF] Xu | Characterization of biocrudes recovered with and without solvent after hydrothermal liquefaction of algae[END_REF][START_REF] Xu | Experimental investigation and comparison of bio-oil from hybrid microalgae via super/subcritical liquefaction[END_REF][START_REF] Zhang | Effect of acidic, neutral and alkaline conditions on product distribution and biocrude oil chemistry from hydrothermal liquefaction of microalgae[END_REF] in order to avoid the drying step of the raw material and thus additional costs [START_REF] Fan | Algal biorefinery to value-added products by using combined processes based on thermochemical conversion: A review[END_REF][START_REF] Sharma | Effect of catalyst and temperature on the quality and productivity of HTL bio-oil from microalgae: A review[END_REF]. The main components of microalgae include proteins, carbohydrates, and lipids. During the thermochemical conversion, they are transformed into smaller oxygenated and nitrogenous molecules [START_REF] Barreiro | Hydrothermal liquefaction (HTL) of microalgae for biofuel production: State of the art review and future prospects[END_REF]. The high levels of water and heteroatomic compounds make the bio-oils chemically unstable and corrosive. Upgrading treatments are therefore necessary to reduce the heteroatomic content and increase the calorific value of the bio-oils before use [START_REF] Staš | Petroleomic Characterization of Pyrolysis Bio-oils: A Review[END_REF][START_REF] Tang | Preparation of low-nitrogen and high-quality bio-oil from microalgae catalytic pyrolysis with zeolites and activated carbon[END_REF]. To optimize these treatments, a deep understanding of bio-oils composition is required. However, their characterization is a challenging task as those are complex mixtures including thousands of compounds covering a wide range of molecular weight and polarity. Most of the time complex mixtures are analyzed by direct introduction (DI) into a Fourier transform ion cyclotron resonance mass spectrometer (FTICR-MS) [START_REF] Smith | 21 Tesla FT-ICR Mass Spectrometer for Ultrahigh-Resolution Analysis of Complex Organic Mixtures[END_REF][START_REF] Mikhaylova | Molecular analysis of nitrogen-containing compounds in vacuum gas oils hydrodenitrogenation by (ESI+/-)-FTICR-MS[END_REF], allowing the assignment of a molecular formula to thousands of m/z ratios with a high level of confidence. However, two issues arise with direct introduction: (1) the impossibility to differentiate the numerous isomers present in bio-oil samples and (2) the possible matrix effects that prevent the detection of some compounds due to ion suppression in the ionization source.

To overcome these issues, chromatography is often hyphenated to mass spectrometry. Gas chromatography (GC) has been used to separate bio-oils from lignocellulosic biomass [START_REF] Silva | Quantification of real thermal, catalytic, and hydrodeoxygenated biooils via comprehensive two-dimensional gas chromatography with mass spectrometry[END_REF][START_REF] Schena | Influence of acquisition rate on performance of fast comprehensive two-dimensional gas chromatography coupled with time-offlight mass spectrometry for coconut fiber bio-oil characterization[END_REF]. However, GC is limited to volatile and high-temperature stable compounds. As a result, many compounds cannot be analysed by GC unless a derivatization step is carried out. Liquid chromatography (LC) and supercritical fluid chromatography (SFC) are promising techniques to separate polar and neutral compounds. Crepier et al. [START_REF] Crepier | Ultra-high performance supercritical fluid chromatography hyphenated to atmospheric pressure chemical ionization high resolution mass spectrometry for the characterization of fast pyrolysis bio-oils[END_REF] used SFC hyphenated to an ion trap time-of-flight mass spectrometer with an atmospheric pressure chemical ionization source in negative mode (APCI(-)-IT-TOF) for the characterization of fast pyrolysis bio-oil from conifer sawdust, and they compared the results obtained with DI-APCI(-)-FTICR. The interest of SFC for isomer separation was demonstrated for several molecular formulae, with sometimes more than three peaks observed at different retention times. Moreover, although the number of assigned molecular formulae was lower in SFC-HRMS (1379 vs. 3949 in DI-FTICR), differences in the oxygen distribution were highlighted. The most intense compounds in SFC-HRMS were less oxygenated compared to DI-FTICR, suggesting that the matrix impacted the ionization of some compounds [START_REF] Crepier | Ultra-high performance supercritical fluid chromatography hyphenated to atmospheric pressure chemical ionization high resolution mass spectrometry for the characterization of fast pyrolysis bio-oils[END_REF].

For complex mixtures, comprehensive two-dimensional liquid chromatography (LC x LC) is a technique of choice, leading to an impressive separation power when the separation mechanisms are different. Although the mechanisms are similar using two reversed-phase liquid chromatography systems (RPLC x RPLC), on-line RPLC x RPLC was found to be very attractive for ionizable compounds (peptides or pharmaceuticals) by just changing the mobile phase pH and possibly the stationary phase between the two dimensions [START_REF] Guillarme | Theoretical and practical comparison of RPLC and RPLC × RPLC: how to consider dilution effects and sensitivity in addition to separation power?[END_REF]. As bio-oil compounds are essentially neutral, RPLC x RPLC is much less attractive, the stationary phase change alone being usually insufficient. However, RPLC x RPLC was applied to the analysis of the aqueous phase of lignocellulosic biomass pyrolysis bio-oil. A porous graphitic carbon column and a bonded silica-based column were used in first ( 1 D) and second ( 2 D) dimension respectively, leading to a fairly good occupation of the retention space [START_REF] Masle | Development of on-line comprehensive two-dimensional liquid chromatography method for the separation of biomass compounds[END_REF]. Two different bonded silica-based columns were also used [START_REF] Lazzari | Quantitative analysis of aqueous phases of bio-oils resulting from pyrolysis of different biomasses by two-dimensional comprehensive liquid chromatography[END_REF][START_REF] Lazzari | Comprehensive two-dimensional liquid chromatography-based quali-quantitative screening of aqueous phases from pyrolysis bio-oils[END_REF], but despite different stationary and mobile phases, the retention mechanisms were related, leading to peaks distributed along the 2D-separation space diagonal and hence to a low retention space coverage. The occupation of the retention space is a key parameter and the combination of RPLC and SFC is a much more promising technique for neutral compounds. Off-line RPLC x SFC has been applied to pharmaceuticals [START_REF] Crépier | Characterization of positional isomers of drug intermediates by off-line RPLC x SFC hyphenated to high resolution MS[END_REF], natural products such as Chinese plants [START_REF] Xin | Construction of an off-line two dimensional reversed-phase liquid chromatography/ultra-high performance supercritical fluid chromatography method for rapid and comprehensive analysis of Piper kadsura[END_REF][START_REF] Wei | A high-efficiency strategy integrating offline two-dimensional separation and data postprocessing with dereplication: Characterization of bufadienolides in Venenum Bufonis as a case study[END_REF], and fragrant oil [START_REF] Stevenson | Comprehensive two-dimensional chromatography with coupling of reversed phase high performance liquid chromatography and supercritical fluid chromatography[END_REF]. The two main advantages of the off-line approach are (i) the possibility to treat the collected fraction and hence to avoid detrimental injection effects arising from the injection of large hydro-organic solvent volumes in SFC and (ii) the possibility to optimize the two dimensions separately, therefore, the performance in off-line mode is expected to be higher but at the cost of a longer analysis time. However, the off-line approach has many drawbacks that do not exist with the on-line one: (i) the fractions are diluted during the fraction collection, (ii) some compounds may be lost during the fraction treatment, (iii) contamination can occur all along the process and (iv) data processing is not straightforward at all. On-line SFC x RPLC was first set up by François et al. [START_REF] François | Construction of a new interface for comprehensive supercritical fluid chromatography×reversed phase liquid chromatography (SFC×RPLC)[END_REF][START_REF] François | Comprehensive supercritical fluid chromatography×reversed phase liquid chromatography for the analysis of the fatty acids in fish oil[END_REF] but due to the nature of the first dimension mobile phase, the authors had to use two two-positions/ten-port switching valve with two loops packed with octadecyl (C18) silica to avoid signal interferences due to the decompressed CO2 which became apparent in the UV trace. The packed loops allowed CO2 depressurization and the trapping of the analytes before their injection into the RPLC system by adding water after the backpressure regulator (BPR) to enhance focusing. The water was also used to serve solvent displacement and hence efficiently remove the residual CO2 gas prior to introduction of the fractions in the 2 D. In contrast, in RPLC x SFC, since the first dimension eluent is liquid, a conventional LC x LC valve with two empty loops can be used. Sarrut et al. compared on-line RPLC x RPLC with a Hypercarb column in 1 D and RPLC x SFC for the analysis of an aqueous phase of lignocellulosic bio-oil [START_REF] Sarrut | Potential and limitations of on-line comprehensive reversed phase liquid chromatography×supercritical fluid chromatography for the separation of neutral compounds: An approach to separate an aqueous extract of bio-oil[END_REF]. They showed that the degree of orthogonality was much higher in RPLC x SFC, leading to a small increase in peak capacity.

The objective of this work was to demonstrate the benefits of on-line RPLC x SFC hyphenated to a quadrupole time-of-flight (qTOF) mass spectrometer for the characterization of microalgae based bio-oils. Key parameters were optimized for this analytical technique. Those parameters included the stationary and mobile phases in SFC, the make-up solvent prior to the qTOF mass spectrometer and the interface between the two dimensions of separation. An illustrative application is given by comparing two different samples coming from two different stages of the conversion process.

Material and methods

Chemicals and sample preparation

Acetonitrile (ACN), methanol (MeOH), and formic acid (LC-MS grade) were purchased from Sigma-Aldrich (Steinheim, Germany). HPLC grade toluene was from Biosolve Chimie (Dieuze, France). Water was provided by an Elga Purelab Classic UV purification system (Veolia water STI, Le Plessis Robinson, France). High purity CO2 (99.99%) was obtained from Air Liquide (Paris, France) Two bio-oil samples coming from the VASCO2 project were supplied by TotalEnergies TRTG (Total Research and Technology Gonfreville). The first sample was a bio-oil obtained by hydrothermal liquefaction of a wild microalgae consortia and the second one was the gasoline cut of the full bio-oil after distillation. A detailed description of bio-oil production was made by Barrère-Mangote et al. [START_REF] Barrère-Mangote | Study of Biocrudes Obtained via Hydrothermal Liquefaction (HTL) of Wild Alga Consortium under Different Conditions[END_REF]. The full bio-oil and the gasoline cut were first dissolved in toluene at 22 mg/mL and in methanol at 178 mg/mL respectively. Stock solutions were further diluted at 5 mg/mL for the full bio-oil and 16 mg/mL for the gasoline cut in methanol and filtered through PVDF membrane (0.22 µm) before injection.

Columns

RPLC separations were carried out using a XBridge BEH C18 column (50x1.0 mm; 3.5 µm) and three columns were evaluated for the unidimensional SFC separation: Viridis BEH 2-EP (100x3.0 mm; 1.7 µm), Viridis BEH (100x3.0 mm; 1.7 µm), and Torus 2-PIC (50x3.0 mm; 1.7 µm) all from Waters. For on-line RPLC x SFC analyses, an additional column from Waters was used in the second dimension: Acquity BEH HILIC (50x2.1 mm; 1.7 µm).

Instruments and chromatographic methods:

1D-SFC

SFC separations were carried out using a Waters UPC² (Milford, MA, USA) equipped with a binary solvent delivery pump, an autosampler with a 10 µL loop, an isocratic pump for the make-up solvent, a column oven compatible with temperatures up to 90°C, a diode array detector (DAD) with a 8.4 µL flow-cell and a backpressure regulator (BPR). The dwell volume was 250 µL, it was measured according to the procedure described by Sarrut et al. [START_REF] Sarrut | Potential and limitations of on-line comprehensive reversed phase liquid chromatography×supercritical fluid chromatography for the separation of neutral compounds: An approach to separate an aqueous extract of bio-oil[END_REF]. Three different columns, all from Waters were assessed: Viridis BEH 2-EP (100x3.0 mm; 1.7 µm), Viridis BEH (100x3.0 mm; 1.7 µm), and Torus 2-PIC (50x3.0 mm; 1.7 µm) and three solvents B were compared: methanol, acetonitrile/water (98/2, v/v), and methanol/acetonitrile (50/50, v/v). The gradient times were adapted to the column dead times (t0): 1%B for 4t0, 1-60%B in 59t0 with acetonitrile/water or 1-40%B in 40t0 with methanol and methanol/acetonitrile. For all experiments, the injected volume, the flow rate, the oven temperature, and the BPR pressure were set at 1 µL, 1400 µL/min, 30°C, and 137 bar respectively. The DAD outlet was connected to the mass spectrometer via a double T-union allowing pressure regulation and the addition of a make-up solvent with a flow rate of 500 µL/min. The instrument was controlled using Empower software from Waters.

On-line RPLC x SFC

RPLC analyses were carried out using the first dimension of a 2D I-Class system from Waters (Milford, MA, USA) equipped with a binary solvent delivery pump, an autosampler with a 20 µL flow-through needle, two column ovens with a maximum temperature of 90°C, a DAD equipped with a 0.5 µL cell. The Dwell volume was 110 µL. The 2D I-Class and the UPC² were connected via a 10-port/2-positions valve equipped with two identical loops (geometrically estimated volume of 6 µL). Two valve configurations were compared, the standard (Fig. 1a) and the new (Fig. 1b) one. This latter was selected for the bio-oil analyses as discussed in section 4. Both dimensions were controlled using Empower 3 and were synchronized thanks to external events. On-line RPLC x SFC conditions are given in Table 1.

Mass spectrometry parameters

Mass spectra were acquired using an Agilent 6560 IM-qTOF from Agilent Technologies (Waldbronn, Germany) equipped with a JetStream electrospray ionization (ESI) source in positive mode. Data acquisition was performed using MassHunter software from Agilent Technologies over the m/z 90-900 with an acquisition rate of 5 Hz. The ionization parameters were set as follows: drying gas temperature, 300°C; drying gas flow rate, 11 L/min; sheath gas temperature, 350°C; sheath gas flow rate, 11 L/min; nebulizer pressure, 40 psi; fragmentor, 185 V and capillary voltage, 3500 V. After acquisition, data were sent to an Excel sheet and rearranged in 2D-contour plots using an in-house program developed with Matlab (V7.12.0635).

Calculations

The mobile phase composition at elution was calculated according to:

𝐶𝑒 = 𝐶 𝑖 + 𝐶 𝑓 -𝐶 𝑖 𝑡 𝑔 × (𝑡 𝑟 -𝑡 0 -𝑡 𝑑 -𝑡 𝑖𝑠𝑜 ) (1)
With Ci and Cf being the initial and final gradient composition and tr, tg, t0, td, and tiso being the solute retention time, the gradient time, the column dead time, the instrument dwell time, and the duration of the initial isocratic hold respectively.

The peak capacity for 1D separations is given by:

𝑛 1𝐷 = 1 + 𝑡 𝑛 -𝑡 1 𝑤 (2)
with tn and t1 corresponding to the retention time of the most and the least retained peak respectively and w to the average 4σ peak width.

In RPLC x SFC, the effective peak capacity can be calculated according to [START_REF] D'attoma | On-line comprehensive two-dimensional separations of charged compounds using reversed-phase high performance liquid chromatography and hydrophilic interaction chromatography. Part I: Orthogonality and practical peak capacity considerations[END_REF]:

𝑛 𝑅𝑃𝐿𝐶 × 𝑆𝐹𝐶 = 𝑛 𝑅𝑃𝐿𝐶 × α × (1 -γ) + n RPLC × n SFC × αγ ( 3
)
where γ is the fraction of the separation space occupied by the peaks and α is the undersampling correction factor. γ is calculated according to the procedure described by D'Attoma et al. [START_REF] D'attoma | On-line comprehensive two-dimensional separations of charged compounds using reversed-phase high performance liquid chromatography and hydrophilic interaction chromatography. Part I: Orthogonality and practical peak capacity considerations[END_REF].

α = 1 √1+0.21×( 4𝑡 𝑠 𝑤 )² (4)
With ts being the sampling time.

The injection volume in the second dimension is given by:

2 𝑉 𝑖 = 𝑡 𝑠 × 1 𝐹 (5)
With 1 F being the flow rate of the first dimension.

Results and discussion

1D-SFC method development

Stationary and mobile phases

Three different stationary phases were assessed in 1D-SFC for the separation of a gasoline cut. Those include: Viridis BEH 2-EP, Viridis BEH, and Torus 2-PIC. The three corresponding base peak chromatograms (BPC) (MeOH as co-solvent) are presented in Fig. 2. Viridis BEH 2-EP and Torus 2-PIC show similar space coverage with a composition range from 1%B to 15%B and 1%B to 16%B respectively (Eq. 1). The composition range with Viridis BEH was almost twice as wide (27%) suggesting that this stationary phase was more suitable for this study. The peak capacity was calculated for the three columns using Eq. 2, because the column lengths were different, peak capacities were normalized to a 10cm-length. It represents the number of peaks that can theoretically be separated side by side with a resolution of 1, which makes it a reliable descriptor of the separation power. Due to the complexity of the sample, leading to a large number of coelutions, the peak capacities were calculated using the same eight extracted ion chromatograms (EICs). The 4σ peak width values and the resulting peak capacities are listed in Table S1. The peak capacity was more than twice as high (i.e. 185) using the Viridis BEH column than with the other two columns (about 80). It is important to note that peak widths with MS detection are much larger than with UV detection due to additional peak broadening (tubing, introduction capillary, source, CO2 depressurization [33]…), resulting in lower peak capacities than those that could be expected. Given these results, the Viridis BEH column was selected in SFC for the rest of the study.

Co-solvents are used to increase the polarity of CO2-based mobile phase and to modify the stationary phase (by adsorption onto its surface). Their concentration in the mobile phase impacts the critical point (pressure and temperature above which the mixture is in a supercritical state) and the density of the mobile phase. Co-solvents also affect sample solubility, eluent strength and selectivity [START_REF] West | Effects of mobile phase composition on retention and selectivity in achiral supercritical fluid chromatography[END_REF]. Three co-solvents were investigated including methanol, a mixture of acetonitrile and water (98/2, v/v), and a mixture of methanol and acetonitrile (50/50, v/v). BPCs for the separation of the gasoline cut are presented in Fig. S1. The selected criterion values are listed in Table 2. As it can be shown, the peak capacity is similar for the three co-solvents (about 190), with a larger occupation of the retention space using the mixture of acetonitrile and water. This is related to the lower eluent strength of this co-solvent compared to the other two. While the large composition range can be attractive for one-dimensional separations, it is much less so for the second dimension of on-line comprehensive 2D-LC, which must be rapid while maintaining a sufficiently high peak capacity. The upper pressure is also important for selecting the co-solvent. With the same flow rate of 1.4 mL/min, the upper pressure was slightly lower (Table 2) with the mixture of methanol and acetonitrile (50/50), which allowed us to increase the flow rate and hence to decrease the normalized gradient slope. Considering the above, our selected SFC system consisted of a Viridis BEH column and a mixture of methanol and acetonitrile (50/50, v/v) as cosolvent.

SFC make-up optimization

In SFC-MS the addition of a make-up solvent is often recommended to avoid solute precipitation when CO2 depressurization occurs, especially at the beginning of the gradient elution. Since this solvent is supplied by an additional pump, it can be individually optimized to improve the ionization yield without affecting the chromatographic separation. Plachká et al. showed that the addition of water in the make-up solvent could increase the MS response up to 50% [START_REF] Plachká | Fast Optimization of Supercritical Fluid Chromatography-Mass Spectrometry Interfacing Using Prediction Equations[END_REF]. Thus, the gasoline cut was injected under the above selected conditions by varying the percentage of water (0 to 30%) in the make-up solvent composed of MeOH, 0.1% formic acid and water. The results are presented in Fig. 3a. For easier interpretation, 5 EICs (m/z 223.181, 270.280, 278.248, 284.295 and 317.102) were chosen along the chromatogram as representative compounds. For all the observed peaks, the intensity increased sharply (by a factor of up to 5) by adding 10% of water, which could be explained by the acidic properties of the SFC mobile phase and make-up. Indeed, CO2 reacts with MeOH and water to form methoxylcarbonic acid and carbonic acid respectively. The apparent pH is therefore acidic and the MS response is increased in ESI(+) [START_REF] Plachká | Fast Optimization of Supercritical Fluid Chromatography-Mass Spectrometry Interfacing Using Prediction Equations[END_REF][START_REF] Gazárková | Current state of supercritical fluid chromatography-mass spectrometry[END_REF]. For microalgae bio-oils, this increase is significant due to the high proportion of nitrogenous compounds. The greater the amount of water, the higher the signal is obtained. However, as illustrated in Fig. 3b, the intensity seems to reach a plateau at high water percentage. Indeed, when the mixture entering the ionization source is too rich in water, the solvent desolvation is more difficult and the signal intensity no longer increases but may even decrease. This phenomenon and the variation in signal intensity were found to be compound-dependent as illustrated by the curves in Fig. 3b. For the compound represented by the blue square, the curve increases sharply from 0 to 10% but is constant thereafter. The trend is different for the compound eluted at 2.641 min (green triangles), the increase in intensity from 10 to 30% is significant, suggesting that the plateau has not yet been reached. Although the highest intensity could be obtained with 30%H2O or more, the RPLC x SFC analyses were performed with 10%H2O for several reasons: (i) at 30%H2O, the water amount is too high and phase separation can occur between the gaseous and the liquid phase, leading to a heterogeneous fluid and pressure instability. (ii) the increase in intensity being compound-dependent, a further increase in the water content could have led to too high signal for the compounds that responded the most and therefore to a too wide dynamic range.

On-line RPLC x SFC

Interface development

In comprehensive 2D-LC, although easier to implement, the off-line mode has major drawbacks compared to the on-line mode. Those include (i) additional dilution during fraction collection, (ii) possible loss of compounds or contamination during fraction treatment, and (iii) time-consuming data processing. We therefore opted for the on-line mode that has become usual in LC x LC but is much less straightforward to implement in LC x SFC. First, unlike on-line LC x LC, there is no commercial instrument in LC x SFC. The two devices have to be hyphenated using a home-built interface design. The current research uses a via a 10-port/2-positions valve equipped with two identical loops. Second, flow splitting after LC and before the valve cannot be applied due to CO2 depressurization. Third, for the same reason, the loops have to be completely filled which is not the case in LC x LC. Additionally, due to the absence of commercially available instrumentation and software, the operator needs to take care of data acquisition and data interpretation.

With a view to limit detrimental injection effects while injecting larger volumes in the SFC second dimension, we studied a new configuration for the interface (Fig. 1b) that we compared to the conventional one (Fig. 1a). With the conventional interface configuration, the two components of the mobile phase are mixed by the mixer device of the SFC instrument. The loop is thus flushed by the mobile phase (co-solvent and CO2). With the new configuration, the co-solvent is supplied after the valve via a tee-union. The loop content is therefore flushed with CO2 only. To compare the performance of the two interface configurations, the LC mobile phase was replaced by a mixture of acetonitrile and water (50/50) spiked with 50 ppm of caffeine, and the LC column was replaced by a zero dead volume union. The LC flow rate and the sampling time were set at 10 µL/min and 1.2 min respectively. With sample loops of 6 µL, 6 µL of the mixture was expected to be injected in SFC. The obtained caffeine peaks using the classical valve configuration are presented in Fig. 4a andc, it is clearly noticeable that the peaks are large and distorted in this case. However, with the second configuration (Fig. 4b andd) the peak shapes are clearly improved, leading to thinner peaks and therefore to a higher peak capacity. With the conventional configuration and partial loop mode, droplets of co-solvent cover the walls of the loop during valve switching due to CO2 depressurization [START_REF] Sarrut | Potential and limitations of on-line comprehensive reversed phase liquid chromatography×supercritical fluid chromatography for the separation of neutral compounds: An approach to separate an aqueous extract of bio-oil[END_REF]. Using this new configuration, the effluent is composed of pure CO2. The fraction coming from the first dimension enters a loop completely empty of liquid and co-solvent droplets cannot exist, suggesting that a partial loop injection becomes possible. This new interface configuration was therefore chosen for our on-line RPLC x SFC analyses.

Injection effects in 2 D-SFC

In SFC, severe injection issues can occur with too large injection volumes, especially with solvents that contain water [START_REF] Desfontaine | A systematic investigation of sample diluents in modern supercritical fluid chromatography[END_REF][START_REF] Batteau | Effect of the injection of water-containing diluents on band broadening in analytical supercritical fluid chromatography[END_REF]. In comprehensive LC x LC, the injected volume in the second dimension depends on both the first dimension flow rate and the sampling time (Eq. 5). Due to the nature of the SFC mobile phase, the flow cannot be split before the valve without depressurizing CO2. The first dimension flow rate was therefore reduced to the minimum possible according to the gradient system used (i.e. 10 µL/min). With a sampling time of 0.6 min and a sample loop of 6 µL, the injection volume was therefore expected to be 6 µL. Injection effects in on-line LC x SFC were studied following the same approach as previously for the interface study. Two different column internal diameters (3.0 and 2.1 mm) were considered. The resulting peaks of caffeine are shown in Fig. 5. Surprisingly, the peak shapes are as good with 2.1 mm (Fig. 5a) as with 3.0 mm (Fig. 5b) i.d., the obtained peak widths being quite similar with ACN/water (50/50, v/v) as injection solvent. Based on these results, the 2.1 mm i.d. column was preferred because (i) sensitivity is expected to be higher for a given injection volume and (ii) the normalized gradient slope is lower for a given gradient time thus, increasing the peak capacity. The composition of the injection solvent was also studied since it has often been reported as a critical factor [START_REF] Pauw | Understanding and diminishing the extra-column band broadening effects in supercritical fluid chromatography[END_REF][START_REF] Enmark | A closer study of peak distortions in supercritical fluid chromatography as generated by the injection[END_REF]. Fig. 5 c) shows the caffeine peaks obtained with three ACN/water (v/v) compositions (10/90, 50/50 and 90/10) and the 2.1 mm i.d. column. It was once again surprising to see that the injection solvent seemed to have no significant effects on both the peak shapes and the peak widths. The average peak width was about 0.47 s, leading to a peak capacity of about 20. These results showed that despite a high volume (6 µL) injected in different hydro-organic solvents, the caffeine peak kept a good shape, suggesting that the on-line RPLC x SFC coupling could provide promising results under such injection conditions. It should be underlined that such a result is in disagreement with studies highlighting the detrimental effect of hydro-organic solvents on peak shapes [START_REF] Batteau | Effect of the injection of water-containing diluents on band broadening in analytical supercritical fluid chromatography[END_REF], but it should also be noted that this latter study is undertaken using traditional injection techniques and hence a mixed flow of CO2 and co-solvent is passing through the injection loop at the time of injection. Therefore, the fact that sharp peak shapes are obtained in RPLC x SFC should be ascribed to the performance of the new interface configuration and the fact that the loop is flushed with CO2, reducing diffusion in the tubing.

When using CO2 only for the transfer and hence the injection of the fraction onto the SFC dimension, immiscibility undeniably occurs, certainly when the transfer solvent contains high concentrations of water. The extremely sharp peak shapes in the SFC dimension when injecting relatively large volumes of "strong" solvent might therefore be explained as a positive effect of that immiscibility. Similar behaviour has been observed by François et al. [START_REF] François | Considerations on the possibilities and limitations of comprehensive normal phase-reversed phase liquid chromatography (NPLC×RPLC)[END_REF] when combining NPLC and RPLC in an on-line comprehensive setup. The authors carried out a gradient of 90/10 n-hexane/ethylacetate to 34/66 n-hexane/ethylacetate in the NPLC 1 D separation. Compounds present in fractions eluting at the end of the 1 D gradient (hence containing higher ethylacetate content) showed severe peak distortion in the 2 D RPLC separation, when the 2 D gradient was started using 40/60 H2O/ACN. While n-hexane is immiscible in both H2O as well as in ACN, ethylacetate is miscible in ACN, but not in H2O. When starting the 2 D gradient with 100% H2O, and hence creating full immiscibility of the fraction solvent at the start of the 2 D gradient, significant peak sharpening was observed.

As the injection of larger volumes of sample dissolved in relatively polar solvent is problematic in regular SFC analyses, the application of a similar injection approach as applied in the current RPLC x SFC interface could be considered advantageous as well.

Full bio-oil in on-line RPLC x SFC

The conditions in on-line RPLC x SFC, selected according to the above study, are listed in Table 1. Due to connection issues between the two instruments, it was impossible to apply a gradient time higher than 0.25 min with a sampling time of 0.6 min. The final composition in the SFC dimension was therefore set at 30%B so that the flow rate could be raised and the the normalized gradient slope reduced as much as possible (here 5.8%). The linear retention model is widely used in HPLC, especially in RPLC. It was recently found that it could be applied in SFC for bio-oil compounds [START_REF] Crepier | Development of a supercritical fluid chromatography method with ultraviolet and mass spectrometry detection for the characterization of biomass fast pyrolysis bio oils[END_REF]. The coefficients (logk0 and S) are usually calculated according to the linear solvent strength (LSS) theory from two preliminary runs. These coefficients were calculated for nine compounds by Osiris software (Euradif, France) from the retention data obtained with two normalized gradient slopes (i.e. 0.5 and 1.5%). Two different co-solvents (ACN/H2O 98/2, v/v and ACN/MeOH 50/50, v/v) were considered for these calculations. Predicted retention times with extrapolated normalized gradient slopes (i.e. 3 and 5%) were found to be close enough to experimental retention times (errors of less than 5% and 10% with normalized gradient slopes of 3% and 5% respectively), thus validating the LSS model in the studied range. Fig. S2 shows the resulting variation of S with logk0. As can be observed, for both co-solvents, S decreases as the retention increases (logk0 increases). The S values become very low with the ACN/H2O co-solvent for the most retained compounds, theoretically indicating that the final gradient composition must be very high to elute them unless using a very low normalized gradient slope, which is impossible in the second dimension. The ACN/MeOH co-solvent was therefore found to be more appropriate in on-line RPLC x SFC. However, even with this co-solvent and due to the reduction of S with the retention, some highly retained compounds could not be eluted under RPLC x SFC conditions as highlighted in Fig. S3, showing the same 3 EICs in 1D-SFC (normalized gradient slope of 1%) and in on-line RPLC x SFC (normalized gradient slope of 5.8%).

The contour plot obtained for 44 EICs from the full bio-oil is presented in Fig. 6a. The complementarity of the two techniques is clearly demonstrated as several compounds not separated in RPLC alone appear separated in SFC and conversely. The peaks are well distributed over the separation space, leading to a distribution across the retention space close to 100% and a calculated effective peak capacity of about 700 (Eq. 3, 4 and Table S2). The SFC chromatograms of three successive injections are presented in Fig. 6b. The obtained excellent peak shapes are in accordance with the observations made in section 4.2.2 for caffeine, although, in both cases, the injection conditions were unfavourable. Moreover, the SFC retention times proved to be highly repeatable between several consecutive injections as illustrated by the vertical dashed lines. Isomeric separations in both dimensions are illustrated in Fig. 7 by three extracted ion chromatograms (EICs with m/z 242.248, 277.216 and 284.295). The high separation power enabled to separate the different compounds in both RPLC and SFC. RPLC mainly separates compounds according to their m/z ratios but also a few isomers such as EIC 277.216. The two spots at approximately 33 and 35 minutes are separated in RPLC only. SFC well separates the isomers. This is clearly demonstrated with EIC 284.295 as three spots well separated in SFC, are coeluted in RPLC. Finally, for the three considered EICs, five peaks were separated in RPLC and four in SFC, leading to 8 well separated spots in on-line RPLC x SFC, showing the great interest of using these two dimensions simultaneously.

1D-SFC vs. on-line RPLC x SFC

One of the many compelling examples of the value of on-line RPLC x SFC over 1D-SFC is shown in Fig. 8 with the comparison of the separations (EIC 172.170 and BPC) between 1D-SFC-HRMS (Fig. 8a) and on-line RPLC x SFC-HRMS (Fig. 8b). The peak intensity for the EIC 172.170 is much higher in on-line RPLC x SFC compared to 1D-SFC (up to 10 times higher) while the quantity injected was only 3 times higher. This large difference can probably be attributed to the upstream RPLC separation. In 1D-SFC, the peaks of EIC 172.170 are coeluted with several other compounds as illustrated by the overlapped chromatograms in Fig. 8a (EIC in red and BPC in black). In on-line RPLC x SFC (Fig. 8b), the RPLC separation allows to separate the red peaks from the main peaks of the matrix, resulting in less ionization competition and an increased sensitivity.

Comparison of two bio-oil samples

Two microalgae bio-oil samples were analyzed in on-line RPLC x SFC-qTOF. The full bio-oil (sample #1) was obtained after hydrothermal liquefaction and Soxhlet extraction. The gasoline cut (sample #2) was obtained after distillation of the full bio-oil. The two samples were therefore related and thus, expected to have many compounds in common. The contour plots (BPCs) are presented in Fig. 9a for sample #1 and 9b for sample #2. As expected, the separations are very similar for both samples. The most intense compounds are strongly retained in RPLC (elution between 40 and 55 minutes). Some differences can nevertheless be observed between the two BPCs, especially in the dashed black squares, showing the presence of many spots in Fig. 9a that do not exist in Fig. 9b. This suggests that several compounds were removed or converted during the additional processing step (i.e. distillation). This is also illustrated in Figs. 9c and9d that show the EICs (m/z 200.201) for sample #1 and #2 respectively. Two spots can be observed in Fig. 9d while three spots in Fig. 9c, the additional spot being circled in red. Considering the distribution of the spots in the two contour plots, this additional compound (at 16 s in SFC and 33 min in RPLC) would have been eluted with another one in both 1D-SFC and 1D-RPLC, thus showing the interest of using both techniques simultaneously.

Conclusions

The characterization of microalgae bio-oils is difficult because these are very complex mixtures with thousands of compounds covering a very wide range of polarity and molecular weight. Two microalgae bio-oils were analysed for the first time in on-line RPLC x SFC-qTOF to illustrate the potential of this technique for such complex matrices. These included a full bio-oil coming from hydrothermal liquefaction and Soxhlet extraction of microalgae, and the gasoline cut obtained after distillation of the full bio-oil. Considering quality descriptors such as peak capacity, peak shape, and peak intensity, key parameters were optimized in 1D-SFC and then in on-line RPLC x SFC.

The best results in 1D-SFC were obtained with bare silica as stationary phase, acetonitrile/methanol (50/50, v/v) as co-solvent, and a make-up solvent composed of methanol (90%), water (10%) and formic acid (0.1%).

In on-line RPLC x SFC, the peak capacity was found to be similar, for the same injection column between 3.0 and 2.1 mm i.d. columns. Consequently, the 2.1 mm i.d. column was selected, which increased the sensitivity while improving the peak capacity by decreasing the normalized gradient slope. Injection solvent effects were also assessed by injecting caffeine in various hydro-organic solvents, under on-line RPLC x SFC conditions. Interestingly, acetonitrile/water solvents with up to 90% water still provided good peak shapes. Such good results could be ascribed to a new interface configuration, with the co-solvent supplied directly after the valve. It is suggested in this study that creating full immiscibility between CO2 and the injection solvent at the beginning of the 2 D gradient could make the peaks more focused.

The large retention space coverage provided by the two orthogonal dimensions and the resulting high separation power allowed us to discriminate the two samples studied despite their high degree of similarity. Very good time repeatability was shown between the analyses of successive fractions in the second dimension, demonstrating the reliability of our on-line RPLC x SFC setup. However, incomplete elution in SFC of the most retained compounds was reported due to the current limitations of the equipment used.

Isomeric profiles of compound families could be obtained in RPLC x SFC-qTOF. Many isomers that could not be separated in SFC alone could be in RPLC and vice versa, showing the great complementarity of the two chromatographic techniques. c,d). Main differences are highlighted by the black dashed squares and the red dotted circles. Analytical conditions in Table 1. ) and (b) the RPLC x SFC conditions (steep gradient, see Table 1). The dashed line corresponds to the determined RPLC x SFC limit showing the missing compounds beyond. ) and (b) RPLC x SFC conditions (steep gradient, see Table 1). The dashed line corresponds to the determined RPLC x SFC limit showing the missing compounds beyond.

CRediT authorship contribution statement

Fig. 1 :Fig. 2 :Fig. 3 :

 123 Fig. 1: Schematic representation of the on-line RPLC x SFC setup. (a) standard configuration and (b) new configuration.Fig. 2: 1D-SFC separation of gasoline cut with three different columns: (a) Viridis BEH, (b) Torus 2-PIC and (c) Viridis BEH 2-EP. The dashed blue lines correspond to gradient profiles at elution. Analytical conditions in section 2.3.1. Fig. 3: Effect of the addition of water in the make-up solvent. (a) Sum of 5 EICs (m/z 223.181, 270.280, 278.248, 284.295, and 317.102) with different amount of water in the make-up solvent. (b) Peak intensity (retention times of 1.536, 2.246 and 2.641 min in Fig.3 (a) versus percentage of water in the make-up solvent.

Fig. 4 :Fig. 5 :

 45 Fig. 4: Three successive separations of caffeine in on-line LC x SFC with (a) the standard configuration and (b) the new configuration. Corresponding contour plots with (c) the standard configuration and (d) the new configuration. The two configurations are shown in Fig. 1. 1 D conditions: no column, isocratic infusion of caffeine (50 ppm in acetonitrile/water 50/50) at 10 µL/min, T = 30°C. 2 D conditions: BEH HILIC (50x2.1 mm; 1.7 µm) at 1.8 mL/min, 30°C., PBPR = 150 bar, gradient from 5% to 30%ACN/MeOH (50/50) in 0.25 min, ts = 1.2 min, 6 µL injected, make-up solvent: MeOH/H2O (90/10) + 0.1%FA.Fig. 5: Illustration of injection solvent effects in 2 D-SFC under on-line LC x SFC conditions. Two successive separations of caffeine in SFC with two column internal diameters: (a) 2.1 mm and (b)

Fig. 6 :

 6 Fig. 6: (a) On-line RPLC x SFC separation of the full bio-oil sample: contour plot representing the sum of 44 EICs (b) 2 D separation of three successive fractions. Analytical conditions in Table1.

Fig. 9 :

 9 Fig. 9: On-line RPLC x SFC-qTOF separation of (a, c) the full bio-oil and (b, d) the gasoline cut. Contour plots: BPC: (a, b); EIC 200.201(c, d). Main differences are highlighted by the black dashed squares and the red dotted circles. Analytical conditions in Table1.

Fig. S1 .

 S1 Fig. S1. 1D-SFC base peak chromatogram of the gasoline cut using the Viridis BEH column (100x3.0 mm; 1.7 µm) and (a) methanol, (b) a mixture of acetonitrile and water (98/2) and (c) a mixture of acetonitrile and methanol (50/50) as co-solvent. Other analytical conditions: see section 2.3.1.

Fig. S2 .

 S2 Fig. S2. Variation of the S as a function of log(k0) for nine compounds in 1D-SFC with the ACN/H2O (98/2, v/v) (orange) or ACN/MeOH (50/50) (blue) co-solvent.

Fig. S3 .

 S3 Fig.S3. Three EICs in 1D-SFC using the BEH HILIC column (50x2.1 mm; 1.7 µm) and a mixture of acetonitrile and methanol (50/50) in (a) optimized 1D conditions (smooth gradient, see section 4.1.1.) and (b) the RPLC x SFC conditions (steep gradient, see Table1). The dashed line corresponds to the determined RPLC x SFC limit showing the missing compounds beyond.

Figure 8 26 Figure 9

 269 Figure 8

Fig. S2 .

 S2 Fig. S2. Variation of S as a function of log(k0) for nine compounds in 1D-SFC with the ACN/H2O (98/2, v/v) (orange) or ACN/MeOH (50/50) (blue) co-solvent.

  in Table 1. Fig. 7: On-line RPLC x SFC-qTOF separation of the full bio-oil sample: EIC 242.248 (green dotted line), 277.216 (red dotted line) and 284.295 (black dotted line). Analytical conditions in Table 1. Fig. 8: Separation of a full bio-oil sample in (a) 1D-SFC with ACN/MeOH (50/50, v/v) as co-solvent (1 µL injected) and (b) on-line RPLC x SFC (3 µL injected): BPC (black) and EIC 172.170 (red). Other analytical conditions given in section 2.3.1. for 1D-SFC and in Table 1 for on-line RPLC x SFC.

Table 2 :

 2 Comparison of elution composition range, peak capacity, and total pressure for three cosolvents with the Viridis BEH column (100x3.0mm, 1.7µm). See analytical conditions in section 2.3.1.

	Co-solvent	Elution composition	Peak capacity	System pressure
		range (%)		(bar) at 1.4mL/min
	MeOH	27	186	355 at 40%B
	ACN/H2O (98/2, v/v)	41	195	360 at 60%B
	ACN/MeOH (50/50, v/v)	29	193	350 at 40%B

Acknowledgements

The authors would like to thank TotalEnergies TRTG, CEA LITEN Grenoble and COLDEP for giving the two bio-oil samples coming from the VASCO2 project. This work has been partially supported by University of Rouen Normandy, INSA Rouen Normandy, Institute of Analytical Science (Labex iMust, University of Lyon) the Centre National de la Recherche Scientifique (CNRS), European Regional Development Fund (ERDF) Nº HN0001343, Labex SynOrg (ANR-11-LABX-0029) for financing Jason Devaux's PhD, Carnot Institute I2C, the graduate school for research XL-Chem (ANR-18-EURE-0020 XL CHEM), Region Normandie.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Paolo Visco

Frédéric Van Wijland

Emmanuel Trizac

Non Poissonian statistics in a low density fluid

(Dated: November 19, 2018)

Our interest goes to the collisional statistics in an arbitrary interacting fluid. We show that even in the low density limit and contrary to naive expectation, the number of collisions experienced by a tagged particle in a given time does not obey Poisson law, and that conversely, the free flight time distribution is not a simple exponential. As an illustration, the hard sphere fluid case is worked out in detail. For this model, we quantify analytically those deviations and successfully compare our predictions against molecular dynamics simulations.

The purpose of the present paper is to address, for an arbitrary low density fluid in a stationary state, basic properties that have been under-appreciated or overlooked, which bear upon the collisional statistics: What is the probability distribution P (N , t) of the number of collisions N suffered in equilibrium by a tagged particle over a given duration t? Conversely, what is the probability distribution function of the free flight time, P FFT (τ ), again for a tagged particle? Such fundamental questions, relevant in their own right, have also consequences on the evaluation of transport coefficients and when transposed to the related realm of granular gases [START_REF] Brilliantov | Kinetic Theory of Granular Gases[END_REF][START_REF] Barrat | [END_REF], directly quantify dissipation. Our message is that intuitive expectation fails -except in highly untypical cases-as far as the collisional statistics is concerned, for an interacting fluid in and also, arguably less surprisingly, out of equilibrium. All results reported are new, together with the kinetic theory techniques used.

Whereas in a dense fluid, velocity correlations and hydrodynamic effects are responsible for a non trivial collisional statistics, one could naively expect that in the dilute limit where collisions become uncorrelated and molecular chaos is enforced [START_REF] Résibois | Classical Kinetic Theory of Fluids[END_REF][START_REF] Cercignani | Rarefied Gas Dynamics[END_REF], collisional events define a Poisson process P(ω) so that P (N , t) = exp(-ωt)(ωt) N /N !, where ω is the mean collision frequency (i.e. N /t → ω at long times, where the brackets denote an ensemble average). The corresponding free flight time distribution would then be

P FFT (τ ) = - d dτ P (N = 0, τ ) = ω e -ωτ . (1)
However, as we shall see below -and this seems to have been ignored in the literature [START_REF]The fact that N is not a Poisson variable has already been re[END_REF]-such a point of view is flawed. In essence, the collision frequency for a particle with velocity v depends on v (it generically increases with v = |v|), which in turn induces correlations between successive collision times. After general considerations that encompass equilibrium and non equilibrium steady states, we will show that the collisional statistics is generically non Poissonian. This is what prompted us to focus on the simplest -analytically tractableinteractions and consider the equilibrium hard sphere fluid for illustrative purposes. Such a model is one of the most useful paradigms in statistical mechanics and has played an essential role in the development of the theory of liquids [START_REF] Hansen | Theory of Simple Liquids[END_REF]. It is one of the simplest system exhibiting a phase transition. Remarkably, it does not only provide a valuable theoretical starting point, but also enjoys direct experimental realizations [START_REF] Pusey | [END_REF]. It is therefore surprising that such a well studied system yields non trivial properties in a limit where little would have been expected. We will see that at late times, when ωt ≫ N , the effect of such correlations is that P (N , t) is of Poissonian form but with a renormalized frequency ω/ √ 2 instead of ω. This result holds irrespective of space dimension. In addition, explicit and accurate results will be reported for the number of collision cumulants N p c . Our analytical predictions will be compared to numerical simulations.

For the sake of simplicity, we begin the analysis by the free flight time distribution P FFT (τ ). The evolution of a tagged particle in a large stationary homogeneous fluid defines, in the low density limit, a Markov process where the transition rates can be computed from a linearized Boltzmann equation, see e.g [8]. In simplified situations, the velocity dependent collision rate r(v) can be computed analytically; the inset of Fig. 1 shows an illustrative example for hard spheres in equilibrium. Velocities are expressed here in rescaled units, and for equilibrium situations, the velocity distribution function reads, with d the space dimension

φ(v) = 1 √ 2π d e -v 2 /2 . (2)
Out of equilibrium, φ is a stationary measure. In any case, the mean collision frequency ω follows from φ:

ω = φ(v) r(v) dv = r(v) . (3)
From the Markovian property, it follows that the conditional probability of having a free flight time τ given a velocity v reads P FFT (τ |v) = r(v) exp(-r(v)τ ). To proceed further and obtain P FFT (τ ) from some average of P FFT (τ |v), attention must be paid to the fact that the relevant weight to use is not the velocity distribution φ(v) itself, but the velocity on collision, r(v)φ(v)/ω. The prefactor r(v) biases the distribution toward more energetic events and accounts for the fact that in a given time interval, a particle with a larger than typical velocity will experience more collisions. We therefore obtain

P FFT (τ ) = dv r 2 (v) ω e -r(v)τ φ(v). (4)
This expression explicitly differs from the result reported in [9], where r 2 /ω is replaced by r (in other words, the weight used in [9] is φ and not rφ/ω). To see why such an approach is incorrect, one can compute the mean collision time τ = τ P FFT (τ ) dτ , that should be equal to 1/ω. This is indeed the case with the distribution given in (4), whereas the formula of Ref. [9] gives τ = 1/r , which differs from 1/ r = 1/ω [START_REF]This problem also pertains to the definition of time in Direct Simulation Monte Carlo algorithms, as discussed[END_REF]. More importantly, upon neglecting the v dependence of the rate r (i.e. assuming r = ω), the integral in ( 4) is readily integrated and yields expression (1) for P FFT . It has been shown that a v-independent collision rate corresponds to particles interacting via an inverse power law pair potential with exponent 2d -2 [START_REF] Cercignani | Rarefied Gas Dynamics[END_REF], which defines the socalled Maxwell model [11], a particularly convenient framework in kinetic theory. Maxwell molecules are nevertheless highly untypical and for any other fluid, r depends on v so that (4) cannot be exponential. We therefore conclude here that the collisional statistics is in general non Poissonian, except for Maxwell molecules where successive collisions turn out to be uncorrelated. We will clarify below the conditions for the occurrence of correlations, and show that while the v dependence of r is a necessary condition for non Poissonian behavior, it is in general not sufficient.

After the previous qualitative remarks, our goal is to quantify the deviations for Poissonian behavior, and to this end, we hereafter consider the prototypical hard sphere model where the frequency r(v) takes the form [8]:

r(v) = ω √ 2 v 2 d 1 F 1 1 2 , 1 + d 2 , - v 2 2 + e -v 2 2 1 F 1 d -1 2 , d 2 , v 2 2 , (5)
where 1 F 1 denotes a confluent hypergeometric function of the first kind. Although a closed-form expression cannot be obtained for P FFT (τ ) due to the lack of simplicity of the collision rate r(v), finding the large τ behavior calls for a saddle point approximation for the integral appearing in Eq. ( 4), which yields

P FFT (τ ) ωτ ≫1 ∼ exp - ωτ √ 2 ω 2 1 - 2 d + ωτ √ 2d -d/2
.

(6) Interestingly, to leading order, τ is distributed exponentially, as naively expected [see Eq. ( 1)], but with a renormalized rate ω/ √ 2. The validity of expression ( 6) is illustrated in Fig. 1, which displays results of numerical simulations. We have checked that the molecular dynamics data in Fig. 1 precisely coincide with the numerical integration of Eq. ( 4) for all velocities (not shown).

We also note here that very similar considerations hold for the distribution of path lengths: for a particle with velocity v, the free flight distance (FFD) ℓ travelled in a time t is ℓ = vt, so that P FFD (ℓ|v) = v -1 P FFT (ℓv -1 |v). P FFD then follows from the counterpart of Eq. ( 4) ). The circles correspond to the results of molecular dynamics simulations at density ρ = 0.04σ -2 where σ is the discs diameter, with N = 1000 particles. The dotted line shows the prediction of Eq. ( 6) while Eq. ( 1) is shown with a dashed line. The inset shows the tagged collision frequency r(v)/ω versus v, also for d = 2. The value 1/ √ 2 at v = 0, indicated with an arrow, is d-independent.

P FFD (ℓ) = dv r 2 (v) ω v exp - r(v) v ℓ φ(v). (7)
The resulting probability density is not a simple exponential, at variance with a claim sometimes found in the literature (see e.g. [START_REF] Hecht | Statistical Thermodynamics and Kinetic Theory[END_REF][START_REF] Blair | [END_REF]). However, a saddle point computation akin to that leading to (6) provides here the long distance behavior of P FFD , which is governed by the minimum of the function r(v)/v [START_REF] Conversely | the minimum of r(v) is the relevant quantity for the long time behavior of Eq. (4); it is reached at v = 0 ([END_REF]. This leads a) to P FFD (ℓ) ∝ exp(-ω v -1 ℓ/ √ 2) for ℓ much larger than the mean free path, and b) to the remark that for the Maxwell model with a v independent rate r, the minimum of r(v)/v is reached for v → ∞ and vanishes, which leads to a non-exponential large ℓ limiting behavior [START_REF] Visco | [END_REF].

We now turn to the related but more complex question of the number of collisions. We introduce the joint probability f (v, N , t) of having velocity v and having suffered N collisions in a time window t, for our tagged particle. The corresponding time evolution, again in the dilute limit, follows from the linear Boltzmann-like equation

∂ t f (v 1 , N , t) = dv 2 d σ(v 12 • σ)θ(v 12 • σ) [f (v * * 1 , N -1, t)φ(v * * 2 ) -f (v 1 , N , t)φ(v 2 )] , ( 8
)
where θ is the Heaviside function, v 12 is the relative velocity, σ is a unit vector and the * * superscript refers to precollisional velocities:

v * * 1 = v 1 -(v 12 • σ) σ and v * * 2 = v 2 + (v 12 • σ) σ.
The latter equation encodes a full description of the collisional statistics for the tagged particle, in the low density limit. Here again we stress that such an analytical approach can be extended to other interaction potentials, leading to the already mentioned remark that, apart from Maxwell molecules, it does not admit a Poisson solution. Moreover, the above equation does not admit a stationary solution in the long time limit, due to the time dependent behavior of the collision number N . More precisely, we expect the large time dependence to be exponential, as a consequence of the linear character of the equation. For analytical progress, it turns convenient to introduce the generating function f through

f (v, λ, t) = ∞ N =0 e -λN f (v, N , t) . (9)
Of course, upon summing ( 8) over all possible values of N , in the equilibrium state one recovers [START_REF] Barrat | [END_REF]. It then appears that the cumulant generating function µ (λ), such that in the large time limit

∞ N =0 f (v, N ) = f (v, 0, t) = φ(v), the Maxwell-Boltzmann distribution
N p c ωt≫1 ∼ t(-1) p ∂ p µ ∂λ p λ=0 , (10)
is the largest eigenvalue of an evolution operator that straightforwardly follows from (8) [START_REF] Visco | [END_REF]. Furthermore, µ(λ) is directly related to the large time behavior of P (N , t) through its large deviation function π, defined as

P (N , t) ωt≫1 ∼ e tπ(n) (11)
where n = N /t. Indeed, π is the Legendre transform of µ:

π(n) = min λ (µ(λ) + λn) . (12)
The quantity µ(λ) therefore bears an important physical information, and has been the technical focus of our study.

From perturbation theory, we have obtained the behavior of µ at large λ in the form

µ(λ) ∼ ω √ 2 (e -λ -1) + O(e -2λ ) , (13)
which implies that

P (N , t) ∼ e -ωt √ 2 N ! ωt √ 2 N , for N ≪ ωt . (14)
For this Poissonian behavior P(ω/ √ 2), the large deviation function easily follows:

π(n) = n -n log(n √ 2/ω) -ω/ √ 2 . (15)
We note that Eq. ( 14) with N = 0 is compatible with the time integral of the leading exponential order of free flight time distribution given in [START_REF] Hansen | Theory of Simple Liquids[END_REF], as it should. However, the dependence embodied in Eqs. ( 13) and ( 14) follows from a large λ expansion and only holds for N ≪ ωt (hence n/ω → 0). It is therefore not indicative of the typical behavior, for which it is more useful to consider the low λ limit. We then dwell on the remark made after (9) that for λ = 0, we have f (v, 0, t) = φ(v), which leads to the approximation scheme where f (v, λ, t) is taken as a Gaussian with temperature T (λ) and the requirement that T (0) = 1 [START_REF]More precisely, it can be shown that at late times[END_REF]. This approximation is the lowest order of a more systematic expansion (see [START_REF] Visco | [END_REF]), but it provides a useful and reasonably accurate information given its simplicity. The idea is to project the Boltzmann equation ( 8) governing the evolution of f (v, λ, t) onto the first two velocity moments. This leads to

0 1 2 3 n = N / t -2 -1 0 π(n) / ω t=10,
µ(λ) = - ω √ 2 (1 -e -λ ) 1 + T (λ) T 0 , (16a)
with

T (λ) = √ 2T 0 √ 1 + e λ . ( 16b
)
It can be checked that µ(∞) = -ω/ √ 2, as implied by Eq. ( 13). The corresponding first three cumulants follow from [START_REF]This problem also pertains to the definition of time in Direct Simulation Monte Carlo algorithms, as discussed[END_REF] and read, irrespective of dimension d

N c ωt = 1 (17a) N 2 c ωt = 9 8 (17b)
N 3 c ωt = 289 256 . (17c)
These values are compared to molecular dynamics simulation data in Table I. Moreover, the large deviation function π(n) of collisions, which follows from ( 12), appears to be in excellent agreement with its molecular dynamics counterpart, see Fig. 2 At this point, it proves instructive to consider the Lorentz gas with only one mobile particle and a collection of spherical fixed scatterers. One readily gets r(v) ∝ v, but the collisional statistics is nevertheless Poissonian. The reason is that the velocity modulus of the mobile particle is constant along the trajectory: in other words, there is no thermalization. It therefore appears that non Poissonian behavior arises from two key properties, that induce collisional correlations: first, the collision rate depends on the velocities, and, second, the particle thermalizes to some stationary non singular measure. These criteria also apply out of equilibrium.

In conclusion, we have shown that the statistics of the number of collisions N experienced by a tagged particle in a low density homogeneous and stationary fluid, in or out of equilibrium, is a subtler quantity than it might seem. Our general statements have been illustrated with the hard sphere fluid. For the distributions of both N and the related free flight time -for which several incorrect results may be found in the literature-, we have quantified the corresponding non Poissonian behavior which follows from the simple physical ingredient that a particle with a high velocity statistically collides more often than a typical particle. A key quantity in the theoretical analysis is the cumulant generating function µ(λ), which can be computed explicitly for large λ and approximately for small λ. The Gaussian ansatz worked out here can be considered as the lowest order of a systematic expansion. The resulting analytical predictions have been confronted against molecular dynamics numerical simulations, with a very good agreement. These numerical results show that the deviations from Poisson behavior P(ω), although not dramatic -which may be the reason why they are underdocumented in the literature-are nevertheless clearly observable. In particular, we have obtained the a priori surprising result that for long times, the distribution of N is Poissonian, but with a "dressed" rate ω/ √ 2. Conversely, the distribution of free flight time τ is exponential, with a behavior ∝ exp(-ωτ / √ 2).

Figure 1

 1 Figure1Free flight time distribution as a function of ωτ , on a linearlog scale, for a two dimensional hard disc gas (d = 2). The circles correspond to the results of molecular dynamics simulations at density ρ = 0.04σ -2 where σ is the discs diameter, with N = 1000 particles. The dotted line shows the prediction of Eq. (6) while Eq. (1) is shown with a dashed line. The inset shows the tagged collision frequency r(v)/ω versus v, also for d = 2. The value 1/ √ 2 at v = 0, indicated with an arrow, is d-independent.

  Large deviation function π(n) characteristic of the long time behavior of the probability P (N , t) of suffering N collisions in a time t (and defined by π(N /t) ∼ t -1 log P (N , t) for ωt ≫ 1). The symbols correspond to molecular dynamics measures for a hard disc system with N = 1000 particles at two different but low densities. The solid line shows the Gaussian result following from Eqs. (16) while the dashed line is the large deviation function π(n) = nn log(n/ω)ω associated to the Poisson law P(ω). On the graph, the time scale has been set by the choice ω = 1.

	ρ=0.001 t=50, ρ=0.001 t=10, ρ=0.01 t=50, ρ=0.01
	Gaussian
	approximation
	Poisson
	Figure 2

TABLE I

 I Cumulants for the number of collisions N from molecular dynamics simulations (performed on a two dimensional system with reduced density ρσ 2 = 0.04), and comparison with both the Poisson P(ω) result and Gaussian approximation (17).

	ωt = 10	N c/t 1.	N 2 1.123 c/t	c/t 1.129 N 3
	ωt = 50	1.	1.135	1.105
	Poisson	1	1	1
	Gaussian	1	1.125	1.129

Acknowledgements We would like to thank J. Piasecki, J.M.J. van Leeuwen, M.H. Ernst, D. Frenkel and H. van Beijeren for useful discussions. This work was supported by the French Ministry of Education through a JCJC ANR grant.


Jean-Jacques Sinou
email: jean-jacques.sinou@ec-lyon.fr

Nicolas Coudeyras

Samuel Nacivet

Study of the nonlinear stationary dynamic of single and multi instabilities for disc brake squeal

Keywords: squeal, self-excited vibrations, periodic and quasi-periodic oscillations, nonlinear and harmonic components

The focus of this article is to show and discuss the nonlinear dynamical behaviors of brake system subjected to friction-induced vibration that can be generated due to the co-existence of multi-unstable modes. A finite element model with friction coupling is used to analyze the stability of the brake system and the stationary nonlinear oscillations for squeal noise prediction. The mechanism of squeal instability considers a mode coupling phenomenon that is classically referred to as coalescence.

Introduction

One of the major challenges for vehicle manufacturers is the detection and prediction of squeal noise arising due to friction-induced vibrations in brake systems. To study the squeal phenomenon in brake, the determination of the mechanism of the self-excited vibration is one of the first and most important phases to be discussed. As explained by Ibrahim [START_REF] Ibrahim | Friction-induced vibration, chatter, squeal and chaos: Part Imechanics of contact and friction[END_REF][START_REF] Ibrahim | Friction-induced vibration, chatter, squeal and chaos: Part IIdynamics and modeling[END_REF], the first mechanisms that are based on friction characteristics and consider the variation of the friction coefficient are not sufficient to explain the occurrence of squeal. The second hypotheses in explaining the squeal phenomenon state that friction-induced vibrations are based on the structural dynamics properties and geometrical mode coupling. One of the first studies was presented by Spurr [START_REF] Spurr | A theory of brake squeal[END_REF] with the sprag-slip model. Then, Earles and co-workers [START_REF] Earles | Instabilities arising from the frictional interaction of a pin-disc system resulting in noise generation[END_REF][START_REF] Earles | Disc brake squeal noise generation: predicting its dependency on system parameters including damping[END_REF], and North [START_REF] North | A mechanism of disc brake squeal[END_REF] proposed various analytical models with two or more degrees-of-freedom that illustrated the importance of structural dynamics and demonstrated that the mode coupling can lead to onset of squeal instability. We refer the interested reader to the reviews [START_REF] Ibrahim | Friction-induced vibration, chatter, squeal and chaos: Part Imechanics of contact and friction[END_REF][START_REF] Ibrahim | Friction-induced vibration, chatter, squeal and chaos: Part IIdynamics and modeling[END_REF][START_REF] Kinkaid | Automotive disc brake squeal[END_REF][START_REF] Ouyang | Numerical analysis of automotive disc brake squeal : a review[END_REF] for an extensive overview of the different models and mechanisms of automotive disc brake squeal that are not the subject of this study.

Even if the first models to study the squeal phenomenon were based on analytical models with few degrees-of-freedom, being generally sufficient to understand the problem of friction-induced vibrations, extensive studies using finite element method are actually performed to analyze brake squeal propensity [START_REF] Massi | Brake squeal: Linear and nonlinear numerical approaches[END_REF][START_REF] Ouyang | Complex eigenvalue analysis and dynamic transient analysis in predicting disc brake squeal[END_REF][START_REF] Ouyang | A methodology for the determination of dynamic instabilities in a car disc brake[END_REF][START_REF] Fritz | Investigation of the relationship between damping and mode-coupling patterns in case of brake squeal[END_REF]. So the aim of this paper is to investigate the squeal prediction and to clarify the nonlinear behaviors that might be observed by using a classical finite element model of brake system including the disc and the pad. The nonlinear dynamical behaviors of squeal noise are discussed in order to obtain some indications that might be useful in detecting and understanding the nonlinear friction-induced vibrations in brake system. The study is set up as follows: first, the nonlinear finite element model of modecoupling instability in brake system is presented. Second, a brief review of the stability analysis and squeal prediction are given. Then, the nonlinear behaviors of squeal phenomenon due to a single mode coupling or multi mode couplings are presented and discussed. Different analyses are considered to provide information on the effects of multi-instabilities and the importance of the nonlinear harmonic components of the nonlinear vibrations in squeal prediction.

Numerical model

The system under study represents a simplified brake. It is composed by a pad and a disc that are the main components contributing to squeal. The pad and disc system without frictional interfaces are discretized by using a finite element approximation via Abaqus software, as shown in Figure 1. The caliper-piston assembly is not defined in this simplified brake. Hence the hydraulic pressure is directly applied to the backplate of the pad. The system is reduced by using a Craig and Bampton technique and keeping certain contact nodes (i.e. at the disc/pad interface) and retaining the first fifty eigenmodes of each component of the brake system. The set of equation describing the dynamic system can be written by where M and C , respectively, the mass and damping matrices. K represents the structural stiffness matrix. x is the displacement vector and dot denotes derivative with respect to time. ext F is the vector force due to brake pressure applied on the pad. NL F contains the linear and nonlinear terms due to the friction contact between the pad and the disc. The friction interface is modelled by using dedicated software in Matlab and by introducing in the finite element model the nonlinear contact force vectors that are defined by

    NL ext Mx Cx Kx F F   (
      3 if otherwise 0 0 p l p d nl p d p d contact ,X k x -x k x -x x -x F               3 if otherwise 0 0 d l p d nl p d p d contact ,X -k x -x -k x -x x -x F          p p contact ,Y contact ,X r F F s i g n v     d d contact ,Y contact ,X r F F s i g n v  
where "d" denotes the disc and "p" the pad. The displacements d

x and p x are defined in the direction normal to the contact surface; then contact ,Y F and contact ,X F are the contact forces along the tangential and normal directions respectively.  is the friction coefficient that is assumed to be constant and the classical Coulomb's law is applied. v r defines the relative velocity between the pad and disc. l k and nl k correspond to the linear and nonlinear stiffnesses contact respectively. This formulation has been chosen to fit the first and the third order of pad compression curves obtained from experimental tests, as showed in Figure 2. As indicated in the previous equations, contact and nocontact configurations at the friction interface are possible so that the pad and disc can separate locally or can open and close repeatedly at some local nodes during the vibration.

Stability analysis

The complex eigenvalue analysis is the first tool to provide the physical parameters that lead to instabilities of the brake system and the squeal occurrences. The stability of the nonlinear system is calculated by considering the linearized system at the equilibrium point. This static equilibrium position 0

x is obtained by solving the nonlinear static equations that satisfy the following conditions:

A   0 N L e x t

Kx F F

Stability of the static position 0

x is then investigated for a small perturbation x around the equilibrium point:

x x x 0   After calculation, the linearized system is given by

        L 0 Mx Cx K K x x 0  
where L K , which is asymmetric, represents the linearized expression of the nonlinear frictional contact NL F at the equilibrium position

      i i i x x      0 NL NL L 0 x F x F x K x x
So the local stability around the equilibrium point 0

x is investigated by calculating the eigenvalues of the Jacobian matrix

J     1 1             L 0 0 I J M K K x M C
The complex eigenvalues  can be expressed as

a ib   
where the real part a indicates the stability of the equilibrium point and b represents the pulsation of the mode. As long as the real part of all the eigenvalues remains negative, the equilibrium point of the system is stable. When at least one of the eigenvalues has a positive real part, the equilibrium point is unstable and squeal noise occurs. Figures 3 display the evolution of real parts and frequencies as a function of the friction coefficient. For a friction coefficient equal to zero, the real parts of eigenvalues are negative due to the structural damping introduced in the system; so, the system features stable separate modes in the frequency range. As the friction coefficient increases, Figure 3(b) illustrates that the real parts increase slowly. The associated frequencies around 1510Hz tend to get closer, as indicated in Figure 3(a). The Hopf bifurcation point occurs for a friction value of 0.26 (see Figure 3(b)). Showing Figure 3(a), it can be noted that the two modes do not reach exactly the same frequency at the bifurcation point due to a non-equally damped configuration between the two modes (see the paper of Fritz et al. [START_REF] Fritz | Investigation of the relationship between damping and mode-coupling patterns in case of brake squeal[END_REF] for more details). For the reader comprehension, the red lines in Figures 3 correspond to the unstable mode. This phenomenon is classified as a non-perfect coalescence. This non-perfect coalescence can also be seen by considering the gap in real part between the two modes at =0. As explained by Hoffmann and Gaul [START_REF] Hoffmann | Effects of damping on mode-coupling instability in friction induced oscillations[END_REF] and Fritz et al. [START_REF] Fritz | Effects of damping on brake squeal coalescence patterns-application on a finite element model[END_REF], a ''smoothing effect'' of the curves with respect to the friction coefficient is observed, both for real parts and frequencies in the case of non-equally damped configuration. After the first Hopf bifurcation point at =0.26, a second instability appears when the value of the friction coefficient is greater than 0.28 (see Figure 3(b) where the continues and dotted lines are associated with the frequencies at 920Hz and 1510Hz, respectively). The upper Figure 3(a) illustrates the coalescence phenomena around 920Hz. Finally, Figures 4 show the evolutions of both real parts and frequencies with increasing the friction coefficient for the two instabilities (at 920Hz and 1510Hz). It can be observed that one of the two modes that coalesce is stable whereas the other one is unstable for the two instabilities. However, for the first instability (at 1510Hz), the unstable mode corresponds to the higher mode, whereas for the second instability (at 920Hz), the unstable mode corresponds to the lower mode. In these two cases, the less damped eigenvalue becomes unstable.

Nonlinear self-excited vibrations

Even if the stability analysis is the first step for designing mechanical systems subjected to friction-induced vibration, the determination of the vibrational amplitudes is essential for a robust design of brake systems. It can be noted that a brake system may be considered to be efficient if the self-excited vibrations do not become excessive and so no offensive noises are generated. In other words, a brake system with small selfexcited oscillations but performing properly can be validated in a design process. So not only the stability analysis but also the amplitudes and the vibrational behaviour of the nonlinear oscillations are key factors. The time-history responses of the nonlinear dynamic system can be obtained by solving the original full nonlinear equation via a numerical integration. Here, the nonlinear system is rewritten in state variables and variable order Adams-Bashforth-Moulton PECE solver is used. In this section, specific cases (for various friction coefficients) will be given to illustrate the different nonlinear behaviour of the brake system that can be observed during squeal vibrations. The first case corresponds to self-excited oscillations of the brake system with a "single instability" participation (with only one unstable mode). It represents one of the most classical nonlinear behavior observed during squeal phenomenon. The second one illustrates the "multi-instabilities" case when the nonlinear self-excited vibrations appear due to the presence of two unstable modes.

Single instability

Firstly, Figures 5(a) illustrate the nonlinear oscillations of the pad (Y-component) in the time domain for the friction coefficient =0.26 when only one instability is present (as previously seen in Section 2). The displacement and velocity growth leading to periodic oscillations is one of the most typical nonlinear features of the squeal phenomenon. Figure 5(b) shows the associated limit cycle amplitudes. Moreover, Figures 6 illustrate the limit cycle amplitudes of the pad in the X and Z directions.

Friction coefficient

Frequency The spectrum components of the nonlinear self-excited amplitudes are analyzed by using the Fast Fourier Transformation method to derive the Power Spectrum Density of the displacement. Results for a node of the pad (in the X,Y and Z directions) are plotted in Figure 7: the fundamental frequency that corresponds to the unstable frequency of the mode around 1512Hz (as previously showed by the eigenvalues analysis) and its second and third harmonics with decreasing amplitudes are present. The contribution of the 1 and 2 harmonic components are also clearly shown on the limit cycles with the small "double-inside loops" that characterizes the presence of the second order for the complete nonlinear response of mechanical systems (see Figure 5(b)). The contribution of the third order in the nonlinear response can be identified in the spectrum (see Figure 7). However the contribution of the 3 harmonic components is less significance than the 1 and 2 harmonic components. In view of the above, it can be concluded that the nonlinear self-excited oscillations during the "single squeal instability" is composed by harmonic oscillations.

The nonlinear time simulation agrees with the instability predictions of the possible unstable modal coupling. In fact, not only the fundamental component of the unstable mode but also contributions of the second and third orders are detected in periodic oscillations of the squeal signal. So the nonlinear contributions and the associated harmonic frequency components must be taken into account not only for predicting the complete nonlinear periodic vibration of the "single squeal" instability but also for avoiding worse designs that could neglect the presence of extra harmonic resonant peaks of the fundamental unstable mode. Then, it is observed that the fundamental frequency of the stationary self-excited oscillations is not exactly those of the unstable mode obtained by using eigenvalues analysis, as indicated in Table 1.

Finally, it can be noted that the unstable conditions, the limit cycle amplitudes and the associated self-excitation mechanism have been obtained even if a constant friction coefficient is used in the numerical simulation. As reported above, the present work deals with squeal phenomenon as a modal instability.

Multi-instabilities

In this section, the nonlinear behaviour of the brake system subjected to multi squeal instabilities is investigated. Figure 8(a) shows the quasi-periodic oscillations of the pad (Y-component) for the friction coefficient =0.28. This value of the friction coefficient corresponds to the appearance of the second instability, as previously indicated in Section 3. The associated limit cycle is given in Figure 8(b). Moreover, the limit cycles of the pad in the X and Z-directions are plotted in Figures 9. It clearly appears that the nonlinear behaviour of the brake system is complex due to the contributions of the two instable modes.

Figure 10 shows the Power Spectrum Density of the three previous components of the pad (X,Y and Z components): in this case, it is observed that the fundamental frequencies in the spectrum are the frequency f 2 of the first squeal instability (at 1497Hz) and the frequency f 1 of the second squeal instability (at 897Hz). The second and third harmonics of the second instability (i.e. 2f 1 , 3f 1 ) and the second harmonics of the first instability (i.e. 2f 2 ) are also identified. Moreover, it clearly appears that the two unstable modes f 1 and f 2 interact together to product sum and difference frequencies f 2f 1 and f 2 +f 1 . The component f 2 -f 1 is prominently seen, being equal to f 2 . The spectrum hence indicates the combination harmonics such as 2f 1 -f 2 , 3f 1 -f 2 , 2f 2 -f 1 , f 2 +f 1 , 2f 1 +f 2 , etc. Even if these combination of frequency components are less important than the fundamental frequencies (f 1 and f 2 ), its harmonics (2f 1 , 3f 1 and 2f 2 ), and the sum and difference frequencies (f 2 -f 1 and f 2 +f 1 ) , they are prominently seen in the spectrum. The presence of the combination harmonics of the two instabilities is indicative of strong participation and "coupling" of the two unstable modes that leads to quasi-periodic selfexcited oscillations of the brake system. Finally, other combinations (4f 1 , 3f 2 , f 2 +f 1 , 2f 2 +2f 1 and 3f 1 +f 2 ) of less significance could be detectable, as indicated in Figure 10. cycles of the pad in the X and Z directions. As previously shown (for =0.28), a complex nonlinear vibrational behavior is observed. It can be noted that the amplitudes of the pad in this case are smaller than the amplitudes for =0.28. So increasing the friction coefficient does not imply an increase of the limit cycles amplitudes of the brake system. Figure 13 shows the Power Spectrum Density for the friction coefficient =0.35 for the X,Y and Z components of the pad. The fundamental frequencies in the spectrum of this case are the frequency f 2 of the first squeal instability (at 1495Hz) and the frequency f 1 of the second squeal instability (at 904Hz). In comparison with the previous case (=0.28), it is observed that the frequency of the first squeal instability decreases whereas the frequency of the second instability increases (see Table 1). The second and third harmonics of the second instability (i.e. 2f 1 , 3f 1 ) are important whereas the second harmonics of the first instability (i.e. 2f 2 ) are less significant. Then, interactions between the two unstable modes f 1 and f 2 are clearly identified: the combination harmonics such as 2f 1 -f 2 and f 2 -f 1 are prominently observed, being equal to 2f 1 or 3f 1 . Other combination harmonics of less significance that have been previously observed for =0.28 are also present for =0.35, such as 3f 1 -f 2 , -f 1 +f 2 , 2f 1 +f 2 , 2f 2 -f 1 , 4f 1 , 3f 1 +f 2 , 2f 2 +f 1 , 2f 1 +2f 2 . It is noted that the participation of these combination harmonics are practically identical for the two cases. Finally, the presences of new combination harmonics of the two instabilities are shown (5f 1 -3f 2 , -3f 1 +2f 2 , -2f 1 +2f 2 , 5f 1 -2f 2 , 4f 1 -f 2 , 5f 1 -f 2 , 6f 1 -f 2 , 5f 1 and 7f 1 -f 2 ). They are of less significance but they are clearly seen in the spectrum and so they indicate a strong participation and coexistence of the two unstable modes of fundamental frequencies f 1 and f 2 that leads to bi-periodic self-excited oscillations. Then, even if some combination harmonics appear to be very close (see for example -3f 1 +2f 2 and 2f 1 -f 2 , -2f 1 +2f 2 and 3f 1f 2 , or 2f 2 and 5f 1 -f 2 ), they are clearly distinguished. It can be noted that these new combination harmonics correspond to the upper orders of the sum and difference frequencies  nf1  mf 2 (with n and m integers). In this last case, the fundamental frequency f 1 of the first unstable mode and its second and third harmonics components 2f 1 and 3f 1 are predominant. The typical "inside loop" of the limit cycle is shown in Figures 14(b) and 15(a). Contributions of less significance that corresponds to the n harmonic components (for n=4 to 11) are also detected in the spectrum, as indicated in Figure 16.

Here, the presence of the first instability and the sum and difference frequencies or harmonics combinations of the two unstable modes are found to be absent in the spectrum. Hence, the nonlinear behavior of the self-excited system is similar to periodic oscillations and so appears more simple than the two previous cases (for the friction coefficient =0.28 or =0.35). So, it may be concluded that even if the eigenvalues analysis indicates the presence of two instabilities, the second unstable mode of frequency f 2 does not participate in the nonlinear oscillations of the self-excited vibrations. One general explanation can be that the first instability of frequency f 1 governs the nonlinear oscillations of the brake system hiding or "removing" the contributions of the second instability of frequency f 2 .

As reported in the previous section, the fundamental frequencies of the self-excited oscillations are not exactly those obtained by using eigenvalues analysis, as indicated in Table 1.

Conclusion

In this study, we investigated numerically the nonlinear behaviour of disc brake squeal by using a simplified finite element model. Firstly, two classical results are observed:

-when only one instability occurs in the brake system, the fundamental frequency and its harmonic are present in the nonlinear self-excited vibrations. The fundamental frequency of the nonlinear oscillations is not exactly the same than those obtained by applying eigenvalues analysis. -The resonance peaks due to the fundamental frequencies component are usually greater than the resonance peaks due to the harmonics components or combinations of frequency components. However, the amplitudes of the harmonics components, sum and difference frequencies can not be neglected and clearly contribute to the self-excited oscillations of the brake system. Secondly, two additional observations are obtained:

-The resonances of the nonlinear self-excited vibrations that satisfy the relationships  nf1  mf 2 (with n and m integers) occur due the coexistence of two instabilities of fundamental frequencies f 1 and f 2 , respectively. So the amplitudes of peaks that do not correspond to the fundamental frequencies of unstable modes do not indicate the presence of new instability but show the harmonics of these frequencies that can contribute to the overall vibration. -Even if the stability analysis indicates the existence of two instabilities, the nonlinear behaviour can be governed by only one unstable mode without a significant contribution of the other instability. As a result, the nonlinear selfexcited oscillations are only composed by the fundamental frequency f 1 or f 2 and its harmonics. So a method for the suppression of brake squeal should be to introduce energy to an other frequency that is out of the audible range (see for example the application of the Dither mechanism [START_REF] Cunefare | Experimental active control of automotive disc brake rotor squeal using dither[END_REF]). In conclusion, the nonlinear self-excited vibrations of disc brake squeal can be very complex due to the presence of multi resonance peaks. Therefore, it becomes easier to avoid a worse design by utilizing not only the fundamental frequencies but also the harmonic components and the combination of frequencies components.

Figure 1 :Figure 2 :

 12 Figure 2: Experimental tests -pad compression

Figure 3 :

 3 coalescences patterns (a) frequency (b) real parts (red lines = unstable modes) (a) (b) Figure 4 : 3Dplot of the coalescence patterns (a) at 920Hz (b) at 1510Hz

Figure 5 :

 5 stationary response of the Y-component of the pad for =0.26 (a) zoom on the displacement and velocity (b) limit cycles (a) (b) Figure 6 : Limit cycles of the pad for =0.26 (a) X-direction (b) Z-direction

Figure 7 :

 7 Figure 7 : Power Spectral Density of the stationary response of the pad for =0.26 ( X-direction, Y direction, Z-direction)

Figure 8 :

 8 Figure 10 : Power Spectral Density of the stationary response of the pad for =0.28 ( X-direction, Y direction, Z-direction)

  Figure 11 : stationary response of the pad (Y-component) for =0.35 (a) zoom on the displacement and velocity (b) limit cycles (a) (b) Figure 12 : limit cycles of the pad for =0.35 (a) X-direction (b) Z-direction Then Figures 11 illustrate the nonlinear oscillations of the pad (Y-component) and the associated limit cycle for the friction coefficient =0.35. Figures 12 show the

  limit

Figure 13 :

 13 Figure 13 : Power Spectral Density of the stationary response of the pad for =0.35 ( X-direction, Y direction, Z-direction)

Figure 14 :

 14 Figure 16 : Power Spectral Density of the stationary response of the pad for =0.4 ( X-direction, Y direction, Z-direction)

Table 1 :

 1 Fundamental frequencies of the nonlinear responses

		f 1 (Hz)	Frequency f 2 (Hz)
	0.26	-	1512
	0.28	897	1497
	0.35	904	1495
	0.4	907	-

Acknowledge

The first author gratefully acknowledges the financial support of the french National Research Agency through the programme Young researcher ANR-07-JCJC-0059-01-CSD 2.


Samuel Rufat

Karsten Uhing

Maike Vollmer

Alexander Fekete

Giacomo Bianchi

Christian Kuhlicke

Conclusions of the Third European Conference on Risk Perception, Behaviour, Management and Response -ECRP22

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

In an hybrid format, the Third ECRP conference has gathered 40 researchers, experts and practitioners from 14 countries (Austria, Belgium, Estonia, France, Germany, Italy, the Netherlands, Norway, Poland, Romania, Spain, Sweden, Switzerland, and the United Kingdom) at the Fraunhofer Forum in Berlin, Germany. The panels and workshops crossed disciplines (complexity science, economics, engineering, geography, history, life science, psychology, sociology, among others), including fields from all the Disaster Risk Reduction cycle phases (from early warning to insurance through perception, vulnerability, behaviour, management, resilience, etc.), and using diverse case studies from ongoing European projects to bring together a panoramic European discussion of the on-going research and practice.

CONCLUSION OF THE EUROPEAN CONFERENCE ON RISK PERCEPTION, BEHAVIOUR, MANAGEMENT AND RESPONSE

BERLIN ACTION POINTS

CHALLENGES IN USING THEORIES IN DISASTER RISK RESEARCH

All research is based on theoretical assumptions. Yet, such assumptions are often not clearly stated. In research on risk perception and adaptive behaviour a rather small proportion of studies explicate their theoretical basis and how it informed their empirical analysis. However, explicating theoretical assumption is beneficial to various reasons:

A greater reflexivity about and appreciation of the role of theories in this field of research, including tolerance for new ideas and for younger researchers An awareness of the relevance of institutional capacity and long-term commitment to certain topics, themes and frameworks A greater encouragement to engage with theories in applied research projects to ensure outcomes that are of relevance not just for specific settings and contexts but contribute to a larger body of empirical evidence

  The Third ECRP conference in June 2022 in Berlin, Germany, has gathered again our two communities, the Risk Perception and Behaviour Survey of Surveyors (Risk-SoS) and the H2020-DRS-01 Cluster on risk perception and adaptive behaviour (a grouping of several Horizon Europe -Disaster Resilient Societies projects, most notably RESILOC, BUILDERS, CORE, ENGAGE, LINKS, RiskPACC). One of the key challenges for risk, vulnerability and resilience research is how to address the role of risk perceptions and how perceptions influence behaviour. It remains unclear why people fail to act adaptively to reduce future losses, even when there is ever-richer information available on natural and humanmade hazards (flood, drought, etc.). The current fragmentation of the field makes it an uphill battle to cross-validate the results of existing independent case studies. This, in turn, hinders comparability and transferability across scales and contexts and hampers recommendations for policy and risk management. The ECRP conference cycle aims to contribute to improve the ability of researchers in the field to work together and build cumulative knowledge.

Explicit engagement is challenged by different reasons: What is needed?

		Theories are usually grounded in disciplinary traditions and
		"languages" which often hamper cooperation in an interdisciplinary
		research environment
	Increasing transparency
	i.e. other researchers can assess underlying
	assumptions and rationales	A lack of awareness of and/or knowledge (incl. training) about the
		existence and relevance of theories in research on risk perception and
		adaptive behaviour
	Stimulating incremental or revelatory advancements
	i.e. building on existing theoretical framework or developing new frameworks allowing counter-intuitive insights	Responding to the demands of practitioners are often favoured over contributing to an established theoretical reference body
	Ensuring comparability of findings
	i.e. comparing studies conducted in different geographical
	settings and, by doing so, drawing more general conclusions
	Informing and influencing decision-making
	processes on the policy-level
	i.e. distilling a clear-cut message from science to stakeholders,
	especially for decision-making processes, based on a robust
	evidence base

Further reading: Kuhlicke, C., de Brito, M. M.,

Bartkowski, B., Botzen, W., Doğulu, C., Han, S., Hudson, P., Karanci, A. N., Klassert, C., Otto, D., Scolobig, A., Moreno Soares, T., & Rufat, S. (2023)

. Spinning in circles? A systematic review on the role of theory in social vulnerability, resilience and adaptation research. Global Environmental Change, 80, 102672.

RECOMMENDATIONS ON SHARING AND TESTING QUESTIONNAIRES OPEN QUESTIONS FROM THE ECRP