Laureanne Parizot

Hugo Dutilleul

Maria-Elena Galvez

Tony Chave

Patrick Da Costa
email: patrick.da_costa@sorbonne-universite.fr

Sergey I Nikitenko
email: serguei.nikitenko@cea.fr

Physical and chemical characterization of shock-induced cavitation

Keywords: Shock induced cavitation, advanced oxidation processes, sonochemistry

A strong impact on a water surface induces a shock wave propagation with a significant pressure variation leading to cavitation bubble formation. A new shock induced cavitation reactor described in this work was characterized by physical and chemical techniques. Water hammer model verification with Joukowsky approach allowed to determine the wave speed propagation and gas fraction in water submitted to shock. These values were used for frequency analysis and compared with direct bubble visualization in order to estimate the influence of the experimental parameters on the shock-induced cavitation. Thereby, the shock wave contains a broad spectrum as decomposed into frequencies. This multi-frequency nature induces heterogeneous bubbles with calculated radii of 0.01 to 3.5 mm and observed radii of 0.01 to 2.8 mm depending on experimental conditions (initial pressure, impact height, gas atmosphere). For the first time, the formation of hydroxyl radicals was proven under impact-induced cavitation. The concentration of radicals increases with increasing number of successive impacts, reaching ca. 1.3 µmol.L -1 after 500 impacts in the presence of 20% O2-Ar as saturating gas. Radical generation seems to be relatively independent of the impact height but strongly depend on the type of gas saturating water, being substantially lower in the presence of air. Moreover, radical generation increases when decreasing the initial pressure and depends on the frequency at which water is impacted by the piston. Nevertheless, yield of OH radicals during shock-induced cavitation remains much lower than that produced by power ultrasound.

Introduction

Pressure gradient in fluids may produce cavitation, which is generally described as a formation, growth and transient implosive collapse of the gas and vapor filled microbubbles [START_REF] Mason | Applied sonochemistry: the uses of power ultrasound in chemistry and processing[END_REF]. The physical and chemical effects of cavitation are interesting for multiple applications [START_REF] Shah | Cavitation Reaction Engineering[END_REF]. Shock waves and micro jets produced by bubble collapse lead to disaggregation of powdered solids, surface erosion and micro-deformation, and depassivation of metals as well [START_REF] Hihn | Respective contribution of cavitation and convective flow to local stirring in sonoreactors[END_REF]. Violent collapse yields nonequilibrium plasma inside the bubble [START_REF] Nikitenko | Toward a new paradigm for sonochemistry: Short review on nonequilibrium plasma observations by means of MBSL spectroscopy in aqueous solutions[END_REF] which emits the light, called sonoluminescence [START_REF] Suslick | On the origin of sonoluminescence and sonochemistry[END_REF], and generates chemically reactive species [START_REF] Riesz | Free radical generation by ultrasound in aqueous and nonaqueous solutions[END_REF][START_REF] Pflieger | Effect of operational conditions on sonoluminescence and kinetics of H2O2 formation during the sonolysis of water in the presence of Ar/O2 gas mixture[END_REF].

The latter phenomenon is directly linked to the sonolysis of water molecules inside the cavitation bubbles. In this sense, cavitation is promising for the development of advanced oxidation processes for liquid waste management without the addition of any chemical product [START_REF] Dukkanci | Ultrasonic degradation of oxalic acid in aqueous solutions[END_REF][START_REF] Bremner | The Chemistry of Ultrasonic Degradation of Organic Compounds[END_REF][START_REF] Chave | Sonocatalytic degradation of oxalic acid in the presence of oxygen and Pt/TiO2[END_REF][START_REF] Qiu | A review on heterogeneous sonocatalyst for treatment of organic pollutants in aqueous phase based on catalytic mechanism[END_REF][START_REF] Parizot | Sonocatalytic oxidation of EDTA in aqueous solutions over noble metal-free Co3O4/TiO2 catalyst[END_REF]. Cavitation and its consequences have been widely studied in the last decades. Though hydrodynamic cavitation was discovered first, cavitation bubbles are also produced commonly by power ultrasound, generated by piezoelectric transducers or magnetostrictors. Ultrasonically induced cavitation has found practical application in many industrial processes [START_REF] Gągol | Effective method of treatment of industrial effluents under basic pH conditions using acoustic cavitation -A comprehensive comparison with hydrodynamic cavitation processes[END_REF][START_REF] Covinich | Advanced Oxidation Processes for Wastewater Treatment in the Pulp and Paper Industry: A Review[END_REF][START_REF] Gogate | Large-scale sonochemical reactors for process intensification: design and experimental validation[END_REF][START_REF] Shrikant | Hydrodynamic Cavitation: A Novel Treatment Approach[END_REF][START_REF] Yi | Degradation of organic wastewater by hydrodynamic cavitation combined with acoustic cavitation[END_REF]. Cavitation can be moreover induced upon high intensity light irradiation or upon electrolysis, what has been very useful for studying single bubble phenomena and for understanding the fundamentals of bubble dynamics [START_REF] Brennen | Cavitation and bubble dynamics[END_REF][START_REF] Lauterborn | Optic cavitation[END_REF][START_REF] Fuster | Influence of the accommodation coefficient on nonlinear bubble oscillations[END_REF].

This paper focuses on much less studied cavitation produced by shock waves in solutions. Indeed, shock-induced cavitation is a relatively recent innovation. To the best of our knowledge only few studies of shock-induced cavitation have been reported so far [START_REF] Driels | Estimation of the dynamic cavitation tension of water by a shock tube method[END_REF][START_REF] Randolph | Do Not Drop: Mechanical Shock in Vials Causes Cavitation, Protein Aggregation, and Particle Formation[END_REF]. Bubble formation and collapse upon the impact of a piston on a liquid surface were recently evidenced, as well as its contribution to the acceleration of oxidation reactions [START_REF] Dutilleul | Shock-induced cavitation as a way of accelerating phenol oxidation in aqueous media[END_REF][START_REF] Daou | Investigation of the collapse of bubbles after the impact of a piston on a liquid free surface[END_REF]. The impact of the piston generates a trail of shockwaves that propagates within the bulk of the liquid, inducing pressure variations that result in the formation of cavitation bubbles. The pressure profile is similar to the one obtained by lithotripsy, with a high pressure followed by a rarefaction [START_REF] Johnsen | Shock-induced collapse of a gas bubble in shockwave lithotripsy[END_REF]. Then, both phenomenon might have some properties in common [START_REF] Henglein | Oxidation of Iodide by the Intense Acoustic Bursts of an Extracorporeal Lithotripter[END_REF][START_REF] Gambihler | Influence of dissolved and free gases on iodine release and cell killing by shock waves in vitro[END_REF][START_REF] Matula | In vitro sonoluminescence and sonochemistry studies with an electrohydraulic shock-wave lithotripter[END_REF]. It was reported recently that shock-induced cavitation can be accurately described by the water hammer model [START_REF] Bergant | Water hammer with column separation: A historical review[END_REF][START_REF] Beuthe | Review of two-phase water hammer[END_REF] (i.e. the Joukowsky equation [START_REF] Joukowsky | Über den hydraulischen Stoss in Wasserleitungsrohren[END_REF]) and, contrary to ultrasonic cavitation, the shock wave propagated in liquid phase has multiple frequencies [START_REF] Dutilleul | Shock-induced cavitation as a way of accelerating phenol oxidation in aqueous media[END_REF]. On the other hand, chemical effects of shock-induced cavitation have never been studied up to date. The occurrence and intensity of shock-induced cavitation seemed so far to strongly depend on many experimental parameters, but no quantitative analysis was performed either on the physical or on the chemical consequences of this type of cavitation. In order to potentially apply this type of cavitation to wastewater remediation, similarly than when using ultrasounds, further insight needs to be gained, particularly concerning the formation of radical or oxidant species in water. Moreover, there is a lack of information regarding the amount of energy provided in shock-induced cavitation, this being of key importance when comparing this impactinduced process to ultrasonic sonochemistry. Then, more detailed physico-chemical characterizations of this phenomenon are required. This work focuses on the multiparametric study of shock-wave induced cavitation by the impact of a piston on the water contained in a cylindrical reactor. The impact and resulting shock wave features were further studied from a physical point of view, considering a modified water hammer model as well as through the visualization and determination of bubble occurrence and size. The quantification of the chemical species was performed using analytical dosimetry techniques currently used in ultrasound sonochemistry.

Materials and methods

Shock-induced cavitation experimental setup

Designed shock-induced cavitation reactor used in this study was composed of a stainless-steel piston (7,8 cm 2 surface, 0,6 kg weight) operated by an electrical engine that delivers a rotation movement, raising and letting it fall on the liquid surface (Fig. 1). Then, the piston movement into the created space is accelerated by gravity and by outside air pressure. In these conditions, the impact piston is highly strong and induces a shock wave which creates cavitation bubble during its propagation. Piston bounces can be observed depending on the impact force due to water being nearly incompressible and the resulting non-elastic mechanical shock force resultant [START_REF] Sisemore | Classical Shock Theory[END_REF]. The frequency (0 to 1.5 Hz) and the number of impacts was settled via a Matlab code piloting a Red Pitaya interface. The height of impact (8 to 20 mm) depends on the volume of liquid contained in a stainless-steel cylindrical reactor (75 mm diameter, 13 mm thickness, 200 mm height). A PMMA reactor, having the same dimensions, was used for bubble visualization experiments. This visualization was performed with the aid of a CDD camera (Phantom V711). The initial pressure was fixed (20-80 mbar) using a vacuum pump and a regulation valve. Pressure target was established either at the beginning of each experimental run or maintained all through the experiment (continuous vacuum conditions, when indicated). Static pressure measurements were performed by a piezoelectric sensor (Kistler 601H, connected to a 5018A amplifier), placed at the bottom of the reactor. Piston position was monitored using a linear transducer (Gefran, type PC). All signals were acquired and recorded using the Red Pitaya interface (16 386 data acquisition) compiled and triggered by a Matlab code or by an oscilloscope (MSO7054A of Keysight-Agilent, 500 000 data acquisition). Gas atmosphere was controlled by sparging at 100 mL.min -1 , inside the liquid contained in the reactor from the bottom or from the middle part of the reactor, during 15 min before each experiment. During impact treatment, dissolved gas content was self-adjusting as impact series couldn't be interrupted to maintained the settled impact frequency. When a continuous vacuum was applied, gas atmosphere was modified. In order to adjusted gas atmosphere, a 5 min sparging was achieved every 100 impacts. Initial conditions were checked thank to impact pressure measurement and physical characterization. Liquid samples were taken at positions of gas inlet. No gas sparging was achieved in the experiments performed under air atmosphere. Overall scheme of the reactor is shown in Fig 2.

Reagents

2-Hydroxyterephthalic acid (97%, Sigma Aldrich), terephthalic acid (99%, Acros Organics), NaOH (98% Alfa Aesar), KH2PO4 (99%, Sigma Aldrich), Na2HPO4 (99%, Sigma Aldrich) were used as received without further purification. All solutions were prepared with commercial demineralized water (<10 μS.cm -1 at 20°C). Ar/O2 gas mixture with 20 vol.% of O2, Ar of 99.999% purity and O2 99.999% purity were supplied by Air Liquide.

Data treatment

Static pressure and piston position data treatment was achieved using different Matlab codes. The signals were smoothed using a moving average algorithm. The parameters of this algorithm were chosen according to the type of analysis performed: an average step was set at 10 points for frequency analysis

Gas

Liquid

and at 200 points for maximum pressure and position study. The images of water submitted to the impacts were obtained with Phantom V711 camera and processed using as well a Matlab code yielding bubble size histograms for each image treated. This Matlab code runs a binarization with background subtraction and a gain for each frame, before using a circle detection function to save radius and position for each detected bubble. Bubble's formation was visually verified by drawing a circle on the original pictures representing bubbles with each obtained radii and position. Then, this method was used to determine the best binarization threshold and sensitivity setting detecting all bubbles in the obtained images.

Chemical dosimetry

The concentration of OH radicals formed during impact cavitation was determined using terephtalic acid dosimetry, as describe by Iida et al. [START_REF] Iida | Sonochemistry and its dosimetry[END_REF]. The initial solution was prepared as follows: 332 mg of terephtalic acid was dissolved under mechanical stirring at ambient temperature in a 500 mL buffer solution prepared with 200 mg of NaOH, 590 mg of KH2PO4 and 980 mg of Na2HPO4 dissolved in demineralized water. Once the solution is homogenized, dilution was completed with demineralized water until 1 L. This solution was directly introduced into the shock-induced cavitation reactor and subsequently submitted to a series of impacts. It is well known that terephtalic acid reacts selectively with OH radicals producing fluorescing 2-hydroxyterephtalic acid [START_REF] Gągol | Effective method of treatment of industrial effluents under basic pH conditions using acoustic cavitation -A comprehensive comparison with hydrodynamic cavitation processes[END_REF]. A Fluoromax-4 with a Horiba Nanoled laser (Jobin Yvon) was used with a 3 mm slit width and a 315 nm excitation wavelength to observe an emission at 425 nm for each sample analyzed. The detection limit of 2-hydroxyterephtalic acid (OH radicals) was 10 -10 mol.L -1 , whereas the measurement error was calculated at 6.10 -9 mol.L -1 of terephthalic acid, after a comparative study considering several calibration plots in our conditions.

Results and discussion

Parameters study and water hammer model

Several modifications were introduced to the studied shock-induced reactor compared to the previously studied setups [START_REF] Dutilleul | Shock-induced cavitation as a way of accelerating phenol oxidation in aqueous media[END_REF][START_REF] Daou | Investigation of the collapse of bubbles after the impact of a piston on a liquid free surface[END_REF]. Therefore, the influence of the different operating parameters and water hammer model was first checked in order to validate that both apparatus have a similar response to the experimental conditions and to anticipate the impact effects. Indeed, water hammer model is a simple way to define the proprieties of this system for a better comprehension. The verification that the impact was adequately represented by the water hammer model consist in the evaluation of the initial pressure (20, 40, 60 and 80 mbar) and impact height (from 8mm to 19mm). Each experimental condition was tested for 15 successive impacts in order to average pressure and velocity results. Measurement error was estimated at 20% by this method. The piston velocity was obtained from the position derivative of its position at each time.

Plotting the maximal pressure at the impact versus the impact height for different initial pressures (Fig. 3 (a)) evidences a linear relationship between these parameters depending on the initial pressure in the cavitation reaction. At low initial pressures, i.e. 20 mbar, a stronger influence of piston height on the impact pressure can be observed, whereas at higher pressures, i.e. 80 bar, the impact pressure is not as importantly influenced by the piston height. Indeed, the slope of each plot in Fig. 3(a) varies also linearly with the initial pressure in the reactor (Fig. 3(b)). In this way, the peak pressure at the impact can be easily predicted using these linear trends. This is clear that initial pressure and impact height have a big influence on impact intensity. A light variation of initial pressure could decrease drastically impact pressure meaning probably that the rest of gas between piston and water surface limit the energy transfer from the piston to the water column by its resistance. Otherwise, the piston velocity is principally influenced by impact height, as shown in Fig. S1.

The initial pressure variation does not seem to importantly influence the piston velocity, considering the experimental error. Since the initial pressure inside the reactor varies from 20 mbar to 80 mbar here, the pressure force on the outer surface of the piston, exposed to atmospheric pressure, is always more important than the one on the inner surface (with a pressure variation of maximum 8%for each experimental condition). Some friction force depending on the piston position may occurs in the reactor as a simple free fall fit without friction force can't match with experimental results neither if a constant force of 340 N based on experimental results is considered (Fig. S1.).

As the maximal pressure attained at the instant of the impact depends linearly on the impact height (Fig. 3), plotting the impact pressure against the piston speed (Fig. S2) further evidences a linear relationship between these two parameters. This is in complete agreement with the predictions based on the water hammer model, as observed previously for other similar shock-induced cavitation setups [START_REF] Dutilleul | Shock-induced cavitation as a way of accelerating phenol oxidation in aqueous media[END_REF][START_REF] Daou | Investigation of the collapse of bubbles after the impact of a piston on a liquid free surface[END_REF]. The water hammer phenomenon is indeed the hydraulic shock that occurs when a fluid in motion is suddenly forced either to stop or to modify its direction. This shock results in a pressure surge, noise and, sometimes, in the formation of cavitation bubbles. In industry, this shock can be provoked by opening or closing a valve, by starting/stopping a pump, or in the feeder pipe elbows [START_REF] Beuthe | Review of two-phase water hammer[END_REF]. The force balance corresponding to the impact of a solid piston on the surface of a static fluid is analogous to the case of a moving fluid suffering an hydraulic shock, and can be therefore described using the Joukowsky equation [START_REF] Joukowsky | Über den hydraulischen Stoss in Wasserleitungsrohren[END_REF]:

∆𝑃 = 𝜌 𝑐 ∆𝑣 (1)
where ∆𝑃 = 𝑃 𝑖𝑚𝑝 -𝑃 𝑖 ≈ 𝑃 𝑖𝑚𝑝 , since 𝑃 𝑖 (initial pressure) can be neglected in front of 𝑃 𝑖𝑚𝑝 (impact pressure, in Pa), 𝜌 is fluid density (kg/m 3 ), c is the shockwave celerity in the fluid (m/s), and ∆𝑣 = 𝑣 -0 = 𝑣, the piston velocity (m/s).

This simple model highlights the possible influence of initial parameters on the shock wave celerity at the impact time. Indeed, the linear relation, in Fig. S2, means that the slope 𝜌 𝑐 could be a constant for each initial pressure indicating that the variation of liquid/gas ratio with initial pressure should be taken into account. Considering a constant density 𝜌 = 1000 kg/m 3 for water, as the gas amount in our conditions is very low, the shock-wave celerity 𝑐 can be calculated from the value of each slope.

The celerity values calculated are 1177 m/s for 20 mbar, 913 m/s for 40 mbar, 603 m/s for 60 mbar and 374 m/s for 80 mbar. These estimations are consistent with speed of sound propagation in water (1500 m/s) and in air (340 m/s). Shock wave celerity should indeed be highly influenced by the gas layer between piston and water during impacts. This means that at higher initial pressures, i.e. 80 mbar, the volume of air/vapor in contact with the liquid surface is more important that at lower initial pressures, i.e. 20 mbar. At the impact, this volume is violently compressed and mixed with the liquid water inside the reactor, generating a gas-liquid mixture that contains more and more gas as the initial pressure increases. Therefore, as this initial pressure increases, the celerity of the shockwave generated upon the impact gets closer to sound speed in air. Moreover, sound celerity is similarly influenced by gas fraction in a liquid/gas system as shown in Fig. S3 reproduced from Werner Kieffer et. al. [START_REF] Kieffer | Sound speed in liquide-gas mixtures : water-air ad water-steam[END_REF] and plots sound celerity in an air/water mixture depending on air fraction for various ambient pressure. Each estimated wave celerity shown in Fig. S3 corresponds to an air fraction from 0.05% to 1% depending on impact pressure. These results are consistent since an impact pressure increase corresponds to an impact height increase for a same initial pressure. As impact height determines the introduced gas volume, it settles also initial void fraction.

Related density for each impact gas fraction was calculated from Fig. S3 and indexed in Table S1.

As expected, density decreases with impact pressure increase and hence with impact height. Density adjustments in 𝜌 𝑐 does not significantly influence shock wave celerity. It is worth noting that a minimal variation of gas fraction could change significantly shock wave propagation due to the wave reflections on the gas/liquid interface [START_REF] Brennen | Cavitation and bubble dynamics[END_REF]. It means that impact intensity is probably not the only factor which could influence shock wave propagation effect as cavitation. In this work, we considered water hammer model that reveals a possible shock wave celerity influence but a better understanding should be possible with a complete Rayleigh equation study.

Pressure signals and bubble size relation

Hypothesis that shock-induced wave has multiple frequencies was already verified with a wavelet transformation of pressure signals in previous studies [START_REF] Dutilleul | Shock-induced cavitation as a way of accelerating phenol oxidation in aqueous media[END_REF][START_REF] Daou | Investigation of the collapse of bubbles after the impact of a piston on a liquid free surface[END_REF]. In this work, a diagram of frequency distribution was obtained and reveals frequency from 0.1 to 100 kHz limited by used pressure sensor. Multiple frequency means the shock contains a broad spectrum decomposed into frequencies representing various excitation level in water which could eventually produce cavitation bubble according to Minnaert equation (a simplification from Rayleigh-Plesset equation for the dynamic of cavitation bubble [START_REF] Hilgenfeldt | Analysis of Rayleigh-Plesset dynamics for sonoluminescing bubbles[END_REF]):

𝜔 𝑁 = [ 3𝑘(𝑃 ∞ ̅̅̅̅ -𝑃 𝑣 ) 𝜌 𝐿 𝑅 𝐸 2 ] 1 2 (2)
where 𝜔 𝑁 is the natural frequency (kHz), 𝑘 is the polytropic exponent (dimensionless), 𝑃 ∞ (𝑡) represents fluid pressure at infinity distance from bubble (Pa), 𝑃 𝑣 the steam pressure (Pa), 𝜌 𝐿 is liquid density near bubble (kg.m -3 ) and 𝑅 𝐸 the excitation bubble radius (m).

Wavelet transformation was also studied for the new shock-induced motorized reactor and Minnaert equation was applied on the obtained frequency distribution to estimate probable excitation bubble radii [START_REF] Minnaert | On musical air-bubbles and the sounds of running water[END_REF]. The Haar wavelet, which transform signals in a square fit, was used to do the wavelet transformation here [START_REF] Haar | Zur Theorie der orthogonalen Funktionensysteme: Erste Mitteilung[END_REF]. An example of this transformation was given in Fig. S4. Motorized reactor frequency range was larger than the previous reactor operated by electromagnet due to a different smoothing treatment and sensor resonance frequency involves a signal saturation at 250 kHz. Yellow color indicates a strong intensity of frequencies and reveals that shock wave delivers mostly low frequencies less than 10 kHz. High frequency apparition is not homogeneous during shock wave propagation but all signals, regardless initial parameter setting, show high frequency during impact and 2 ms after impact when piston bounces.

Minnaert equation allow us to simply estimate the possible bubble radii: greater probability of 1 to 3.5 mm radii and non-homogenous apparition of bubbles below 1 mm radii. These represent bubble resonance radii which could be formed, growth and collapse after shock wave propagation.

Nevertheless, theoretical Minnaert estimation and experimental results often present important difference as it was demonstrated by Brotchie et al. [START_REF] Iida | Sonochemistry and its dosimetry[END_REF]. Theoretical radii were overestimated compare to reality (for example: at 213 kHz, resonance radius was calculated at 14 µm and experimental radius determined at 3.9 µm) [START_REF] Brotchie | Effect of Power and Frequency on Bubble-Size Distributions in Acoustic Cavitation[END_REF]. Multiple frequencies produced by shock wave increases resonance probabilities but are not constant over time and formed bubbles have various size. However, wavelet study does not permit precise analysis of all parameters since all pressure signals are within the similar frequency range. High intensity of low frequency occurs all along impact cycles and high frequency only appears during impact and bounce for any initial parameters. No relevant parameter influence was found among our signal pressure analysis as this method can't be considered as a reliable model compare to a Fourier transform study at each time, for example. Direct bubble visualization was therefore necessary to identify bubble radii during shock wave propagation depending on operational conditions.

Bubble visualization

Bubble apparition seems random due to multiple frequency excitation and instantaneous propagation of shock wave. The aim of this part of study was to evaluate bubble size distribution and the number of bubbles created at each impact in a specific volume depending on various operational condition. Several parameters were studied, such as the time of bubble apparition, radius range (compare to Minnaert estimation), and impact gas fraction (compare to water hammer model). Some typical pictures during initial transient phase are shown in Fig. 4 for the impact at 12 mm, 20 mbar under air.

Before the impact at t= -0.1 ms, stable bubbles are already formed due to piston rise (pressure of water is initially close to vapor pressure). These stable bubbles constitute nuclei sites which help bubble formation after impact. At the impact, t = 0 ms, gas is dissolved in water and pressure is maximal. Next, pressure decreases by shock wave propagation for forming cavitation bubbles which will growth and collapse from t = 0.7 ms to t = 1.3 ms after impact and from t = 2 ms to t = 3.5 ms after piston bounce. was observed in previous work on phenol degradation [START_REF] Dutilleul | Shock-induced cavitation as a way of accelerating phenol oxidation in aqueous media[END_REF]. Bubble radii are observed here from 0.01 mm (lowest detectable size by the camera and numerical treatment) to 2.8 mm depending on pressure wave propagation. First, biggest appear, at the impact, then bubbles divide themselves after the first collapse given new nuclei sites. At piston bounce, bubbles were more abundant but smaller. More detailed results of bubble image treatment summarized in Table S2 underline some challenges in the determination of bubble behavior in shock-induced cavitation reactor. Indeed, each setting parameter used in Matlab code for image treatment influences the well detection of bubble. Furthermore, bubbles in a same video often need a different treatment which necessitates to adjust these parameters by hand:

-Binarization threshold setting determines the position of bubble border and finally bubble radii.

-Sensitivity setting could permit to detected large bubbles, but it hinders smaller bubble detection.

(a) (b)

-Completely dark bubbles and bubble with only dark border which are observed in the same picture need a different treatment.

-Gas fraction calculation was achieved at maximum detected gas volume considering perfect spherical bubbles in camera field volume which is closed to reactor side where bubbles are easily formed.

All these factors induce a numerical treatment error difficult to estimate. Furthermore, circle detection limit of 0.01mm inhibits bubble visualization corresponding to frequencies beyond sensor resonance at 250 kHz (Fig. S4.(b)).

Maximum detected bubble number is considered while volume percentage calculation normalized by camera field which is adjusted for each video due to some shadow. The time of bubble apparition was considered from the first formed cavitation bubbles after impact until the last formed cavitation bubbles after piston bounce. Number of maximum impact bubbles was considered after impact rarefaction and bubble size ranking allows us to compare them at the same time for various experimental parameters. Then, maximum detected bubbles number is different than maximum impact bubble number. Indeed, maximum bubble number appears after piston bounce due to bubble division after first impact collapse. Particularly, at 8 mm and 80 mbar, maximum detected bubbles corresponding to the piston rise before impact and not to shock wave propagation.

Heterogeneous bubble sizes complicate the linking of experimental data and experimental conditions, but a few observations could be noticed:

-At 8 ms, bubbles are smaller than 0.5mm of radius.

-At 12 and 16 mm, bubble ranges are wider (smallest bubbles were detected at these heights).

-Time of bubble apparition increase with impact height.

-Under Ar/20%O2, more bubbles were observed and this profusion of bubble could obstruct the picture which is constituted of mostly black shade such as compromising the circle detection.

-At 8mm and 80 mbar, no cavitation was observed. The too important gas layer may stop shock wave propagation.

-Gas fraction is lower than 0.65% that is consistent with sound wave propagation in liquid/gas mixture (Fig. S3) and it decreases with initial pressure increase. When initial pressure increases, the gas layer increases also which induced the decrease of shock wave propagation velocity.

Then, the shock wave is less strong to produce cavitation bubble which decreases gas fraction in liquid media.

These results confirm the conclusions of water hammer and Minnaert approaches [START_REF] Dutilleul | Shock-induced cavitation as a way of accelerating phenol oxidation in aqueous media[END_REF][START_REF] Brotchie | Effect of Power and Frequency on Bubble-Size Distributions in Acoustic Cavitation[END_REF]. Indeed, initial pressure and height conditions are essential to define impact pressure intensity and shock wave propagation speed. For a chosen initial pressure, propagation speed is approximatively the same for each impact height, and then for each impact pressure. However, gas fraction at the impact increase with impact pressure for each shock wave propagation velocity, probably meaning that a stronger impact induced more cavitation bubble formation in liquid (Fig. S3). For example, at 80 mbar and 8 mm, wave propagation is too weak to obtain an efficient rarefaction and cavitation. At 40 mbar for a same 8 mm height, cavitation bubble was well produced. Thus, initial pressure is essential to create bubbles and a maximum of 40 mbar seems to be necessary. Bubble sizes observed by camera are similar to the estimated ones by Minnaert study (Fig. S4). Whereas speed camera resolution and numerical picture treatment don't permit to check bubble radii less than 0.01mm corresponding to high frequency beyond 250 kHz, the sensor resonance frequency.

Influence of successive impact at 1.1 Hz was studied with a camera frame on all reactor height. Four impacts have been saved on video. The previous bubble detection method is unable on these pictures due to abundant bubble formation from the second impact and especially under Ar/20% O2 atmosphere.

An example of impact at 16 mm, 20 mbar and under Ar/20%O2, is shown in Fig. S5. These pictures have to be compared to the associated pressure signals in Fig. 6. and the beginning of a third one [START_REF] Pflieger | Effect of operational conditions on sonoluminescence and kinetics of H2O2 formation during the sonolysis of water in the presence of Ar/O2 gas mixture[END_REF]. First impact shows same pictures than Fig. 4: during piston rising

(1) at t = -0.5 s, stable bubbles are produced due to the decreasing pressure until chosen initial pressure setting. Bubbles are dissolved (2) at the impact of the piston (at t = 0 s) and then wave propagation induces formation, growth and collapse of cavitation bubbles until t = 2 ms. Next, few stable bubbles appear at the reactor top at t = 5 ms due to piston contact on the water surface. From t = 0.2 s and 0.3 s, the motor cam collides with piston (3) before raising it gradually which induces a small pressure decrease and then a pressure increase when piston come back on the water surface. Stable bubbles appear and are dissolved again at this moment. The motorized piston is then progressively risen during 0.5 s (4), which produces numerous stable bubbles coalescing to water surface. At the end of this piston rising at t = 0.7 s, an abundance of bubbles has appeared. Indeed, for the first piston rising, liquid has just been degassed by initial pressure setting. Thus, there is less nucleation sites than for the second piston raise due to the first cavitation bubble division when they collapse which create more site. Second impact [START_REF] Suslick | On the origin of sonoluminescence and sonochemistry[END_REF] occurs at t = 0.95 s. The multiple bubbles are first dissolved before anew be formed, growth and collapse. Cavitation bubbles are smaller and not as much numerous than stable bubbles formed by piston raise. It is noteworthy that some bubbles are not dissolved and stay stable during wave propagation. Again, from t=1.2 s to 1.3 s, the cam collides with piston (6), even more stable bubbles are formed that coalesce and became undistinguishable anymore since the picture contains mostly black shade. Finally, from t = 1.5 s to t=1.8 s, piston raises and forms again a great number of stable bubble [START_REF] Pflieger | Effect of operational conditions on sonoluminescence and kinetics of H2O2 formation during the sonolysis of water in the presence of Ar/O2 gas mixture[END_REF] which dissolve themselves before their formation, growth and collapse at t=1.8s. Last cycle until the fourth impact is similar.

At each impact, the number of stable bubbles formed by piston raise increases. The gas coming from the dissolution of bubble at the impact constitutes new nucleation sites for cavitation bubble.

Thereby, reactional medium is stirred and degassed gradually which could modify the pressure between piston and liquid surface after several impacts. Then, during high number of impacts treatment, impacts series were limited to 100 impacts before adjusting atmosphere and initial pressure. This profusion of bubble could increase probabilities to obtain interesting chemical effects despite there are mostly big bubbles with probable associated physical effect. Indeed, acoustic cavitation studies revealed that higher frequency induced smaller bubble formation [START_REF] Brotchie | Effect of Power and Frequency on Bubble-Size Distributions in Acoustic Cavitation[END_REF] and a better chemical activity [START_REF] Pflieger | Effect of operational conditions on sonoluminescence and kinetics of H2O2 formation during the sonolysis of water in the presence of Ar/O2 gas mixture[END_REF]. Whereas, low frequency with bigger bubbles is more efficient for mechanical effects, which is used for emulsification or cleaning in industry [START_REF] Shah | Cavitation Reaction Engineering[END_REF]. Chemical effects produced by shocked-induced cavitation need to be experimentally measured to compare the formation yield of oxidant species to these physical observations.

Free radical dosimetry

Terephtalic acid dosimetry is the most appropriated to detect hydroxyl radicals with a detection limit estimated as 10 -10 mol.L -1 [START_REF] Iida | Sonochemistry and its dosimetry[END_REF][START_REF] Mason | Dosimetry in sonochemistry: the use of aqueous terephthalate ion as a fluorescence monitor[END_REF]. Then the influence of frequency, impact height, initial pressure and gas atmosphere was evaluated with a continuous vacuum setting to keep a stable intern pressure during no more than 100 impacts before adjusting gas atmosphere. Vacuum application on the reactor side involve a 12 mm impact height minimum. First, the study of the impact height influence (Fig. 7) shows a linear tendency for hydroxyl radical formation depending on impact number. Maximum concentration after 500 impacts was only 1.3x10 -7 mol.L -1 which is very low compared to ultrasonic radical formation yield [START_REF] Pflieger | Effect of operational conditions on sonoluminescence and kinetics of H2O2 formation during the sonolysis of water in the presence of Ar/O2 gas mixture[END_REF]. Apparently, height variation has no specific influence on the radical production. As observed with high speed camera study, height increase induces stronger impacts and a wider bubble size distribution and chemical effects appear with smaller bubbles [START_REF] Mason | Applied sonochemistry: the uses of power ultrasound in chemistry and processing[END_REF]. Thereby, height increase allows bubble size distribution to increase but not the number of small bubbles presumably producing free radicals. Neither impact intensity seems to be decisive on chemical effects. Initial pressure influence was tested at 20 mbar 40 mbar (Fig. 8). At 20 mbar, the atmosphere contains less oxygen representing oxidant species, compare to a 40 mbar pressure condition but intern reactor conditions near water vapor pressure (20 mbar at 20 °C) induces a stronger impact on the water surface and probably a higher shock wave propagation speed. Thus, both pressure conditions could be favorable to radical species formation. Nevertheless, Fig. 8 reveals that 20 mbar is most efficient to produce hydroxyl radicals. Indeed, impact force is stronger, then rarefaction too, and bubble formation is easier near water vapor pressure. The system is maintained to evaporation limit and wave propagation, facilitated as there is less introduced gas, causes vapor bubble formation. An initial pressure reduction 0,00E+00 Impact frequency is a new parameter for this shock induced cavitation reactor, compare to the previous studied device operated by an electromagnet [START_REF] Dutilleul | Shock-induced cavitation as a way of accelerating phenol oxidation in aqueous media[END_REF][START_REF] Daou | Investigation of the collapse of bubbles after the impact of a piston on a liquid free surface[END_REF], and it is adjustable from 0 to 1.5 Hz.

Terephtalic acid analysis was achieved at 0.5 Hz, 0.8 Hz and 1.1 Hz. Results in Fig. 9 do not show an important frequency influence. Considering the experimental error, only last dots present a difference following this order: 0.5<1.1<0.8 Hz. In this way, frequency at 0.8 Hz represent an interesting ratio experiment time/oxidant species yield. Furthermore, a 1.1 Hz frequency increases the reactor wear risk.

Finally, various gas atmospheres were tested in comparison to ultrasonic experiments [START_REF] Pflieger | Effect of operational conditions on sonoluminescence and kinetics of H2O2 formation during the sonolysis of water in the presence of Ar/O2 gas mixture[END_REF].

Results in Fig. 10 reveals the importance of introduced gas on the formation yield of OH radicals which is consistent with the results observed during ultrasonic irradiation. Indeed, gas mixture influences radical formation rate according to following order: Ar/20%O2 > Ar > air. These results are quite similar to the sonochemical effects produced by ultrasound [START_REF] Dukkanci | Ultrasonic degradation of oxalic acid in aqueous solutions[END_REF]. Ar/20%O2 atmosphere permit to obtain maximum radical formation in this study. Argon atmosphere is more efficient than air due to the 0,00E+00 The yields of hydroxyl radical obtained for impact-induced cavitation and ultrasonic cavitation in the presence of different gases are summarized in Table 1. A factor of almost 100 is observed between the two different techniques. Much higher chemical efficiency of ultrasonic cavitation can probably be attributed to the fact that shock induced reactor produces cavitation bubbles only at impact and bounce time (that being so approximately every second depending impact frequency) whereas ultrasound produces a continuous cavitation bubble cloud. Nevertheless, detection of OH radicals during impactinduced cavitation is important for better understanding of the cavitation mechanisms in general. Calculated delivered power for impact induced cavitation required for yield determination is indicated in Table .2. It was calculated following the delivered kinetic energy ∆𝐸𝑐 at each impact which only depends on impact height as describe in Fig. S1 where a linear relation was observed between the piston speed and the impact height:

∆𝐸𝑐 = 1 2 𝑚 𝑝 𝑣 2 = 0.00867 ℎ 2 (3)
where ℎ is the impact height in mm.

Table 2 contains the results of the power (in Wh) delivered through 1 h impact treatment at different heights from 10 to 16 mm and at an impact frequency of 0.8 Hz. The results are compared to an ultrasonic treatment at 345 kHz and 50 W acoustic power (determined through calorimetric measurements using ultrasonic device reported recently [START_REF] Bremner | The Chemistry of Ultrasonic Degradation of Organic Compounds[END_REF]). These data allow to compare the yields of OH radicals produced by shock wave cavitation and acoustic cavitation. The values evidence that the power delivered upon 1h impact treatment is much lower than the power transferred using ultrasounds at 345 kHz. Contrary to ultrasounds, energy is transmitted in shock-induced cavitation is not provided in a continuous way, but only at the instant the piston impacts the liquid contained in the reactor.

Furthermore, all the kinetic energy of the piston is not directly transmitted to the liquid. In fact, the piston rebounds as it touches the liquid surface due to non-elastic propriety of the piston shock on the incompressible water surface [START_REF] Sisemore | Classical Shock Theory[END_REF]. Therefore, a part of its kinetic energy is used in this movement and subsequently transmitted during a second impact caused by this rebound. In any case, at the sight of our energy calculations we can conclude that the amount of energy involved in continuous ultrasound treatment and through shock-induced cavitation are not comparable, and that, moreover, this energy is provided in a very different way in both cases.

Conclusions

In this work, new shock-induced cavitation reactor was characterized using physical and chemical techniques. Physical characterization confirmed the possible application of water hammer model to describe shock propagation at studied conditions. Wavelet transformation and Minnaert equation applied for the frequency analysis of pressure signal revealed multiple frequency wave propagation within the reactor and possible various level of bubble excitation (from 0.01 to 3.5 mm radii). Indeed, high-speed bubble imaging allowed to observed bubble size distribution from 0.01 to 2.8 mm radii, consistent with theoretical study from signals pressure, over time of shock wave propagation. It was shown that the initial gas pressure and impact height are the most influent parameters determining the pressure wave propagation in the liquid. While impact height settled impact pressure intensity and piston velocity, initial pressure influence drastically impact energy transfer but probably also shock wave celerity. It is interesting that these results provided some new insights about the influence of shock wave propagation on bubble cloud formation in the reactor.

For the first time, hydroxyl radical formation has been proven during shock-induced cavitation that rank it among new advanced oxidation processes. A lower initial pressure increases the amount of hydroxyl radical, certainly due to a stronger impact inducing more active bubbles. Furthermore, gas atmosphere is determinant for the yield of OH radicals, similar to conventional sonochemistry, during shock-induced cavitation and follows a general trend: Ar/20% O2 > Ar > air.

Fig. 1 .

 1 Fig. 1. Motorized piston system used in the shocked-induced cavitation experiments.

Fig. 2 .

 2 Fig. 2. Diagram of shock-induced cavitation reactor.

Fig. 3 .

 3 Fig. 3. (a) Dependence of impact pressure 𝑃 𝑖𝑚𝑝 on impact height ℎ and (b) linear relationship between the slope of the pressure vs. height curves and initial pressure 𝑃 𝑖 .

Fig. 5 .

 5 Fig. 5. (a) shows bubble size distribution over time of shock wave propagation obtained from

Fig. 4 .Fig. 5 .

 45 Fig. 4. Typical pictures of obtained bubbles after an impact at 12mm, 20mbar, under air.

Fig. 6 .

 6 Fig. 6. Pressure signal during several successive impacts at 1.1Hz, 16 mm, 20 mbar after Ar/20% O2 gas paddling (step number description in the text).

Fig. 7 .

 7 Fig. 7. Impact height influence on hydroxyl radical formation during impact treatment at 20 mbar, 0.8 Hz, under Ar/20%O2 at ambient temperature.

  previous paddling which produces nucleation sites. Moreover, argon is a gas favoring plasma formation with species ionization and chemical reaction initiation, and oxygen is a supplementary oxidant species.

	HO -1 ) • radical concentration (µmol.L HO -1 ) • radical concentration (µmol.L	2,00E-08 4,00E-08 6,00E-08 8,00E-08 1,00E-07 1,20E-07 1,40E-07 1,60E-07 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 2,00E-08 4,00E-08 6,00E-08 8,00E-08 1,00E-07 1,20E-07 1,40E-07 0.16 1,60E-07 0.14 0.12 0.1 0.08 0.06 0.04 0.02		20 mbar 40 mbar 1.1 Hz 0.8 Hz 0.5 Hz
	0 0,00E+00 0
		0	0	100 100	200	200	300	300 400	400 500	500
				Impact number Impact number
	-1 ) • radical concentration (µmol .L	4,00E-08 6,00E-08 8,00E-08 1,00E-07 1,20E-07 1,40E-07 1,60E-07 0.16 0.14 0.12 0.1 0.08 0.06 0.04		Ar/20%O2 Argon Air 2
	HO	2,00E-08 0.02
		0,00E+00 0
			0	100		200	300	400	500
						Impact number

Fig. 9. Influence of the impact frequency on hydroxyl radical formation during impact treatment at 20 mbar, 12 mm, under Ar/20%O2 at ambient temperature.

Fig. 10. Gas atmosphere influence on hydroxyl radical formation during impact treatment at 12 mm, 20 mbar, 0.8 Hz at ambient temperature.

Table 1 .

 1 Hydroxyl radical formation yield: comparison between ultrasonic and impact treatment at

	ambient temperature.
	Gas	US 207 kHz (45 Wh)	US 345 kHz (50 Wh)	Impact 12 mm (1.2 Wh)
	Ar/20%O2	3.16 µmol.kJ -1	5.68 µmol.kJ -1	0.063 µmol.kJ -1
	Argon	0.56 µmol.kJ -1	1.02 µmol.kJ -1	0.045 µmol.kJ -1
	Air	0.60 µmol.kJ -1	0.46 µmol.kJ -1	0.027 µmol.kJ -1

Table 2 .

 2 Calculated power delivered during 1 hour at various impact heights at 0.8 Hz impact frequency compared to 1 hour 345 kHz ultrasonic treatment[START_REF] Parizot | Sonocatalytic oxidation of EDTA in aqueous solutions over noble metal-free Co3O4/TiO2 catalyst[END_REF].

	US 345 kHz		Impact height
	(Pac = 50 W)	10 mm (1 J)	12 mm (1.5 J)	14 mm (2 J)	16 mm (2.5 J)
	50 Wh	0.8 Wh	1.2 Wh	1.6 Wh	2.0 Wh

Acknowledgements

This work was performed within the Investments for the future program of the French Government and operated by the French National Radioactive Waste Management Agency (ANDRA), Project CADET. The authors would like to thank Jerome Bonnety, Jean-Marie Citerne, Renaud Jalain, Aemilius Gitantra, Maria-Alejandra Castrillon Ramirez, Juan Buceta for participating in experiments.


Dragan Doder
email: d.doder@uu.nl

Srdjan Vesic
email: vesic@cril.fr

Madalina Croitoru
email: croitoru@lirmm.fr

Ranking Semantics for Argumentation Systems With Necessities

Bipolar argumentation studies argumentation graphs where attacks are combined with another relation between arguments. Many kind of relations (e.g. deductive support, evidential support, necessities etc.) have been defined and investigated from a Dung semantics perspective. We place ourselves in the context of argumentation systems with necessities and provide the first study to investigate ranking semantics in this setting. To this end, we (1) provide a set of postulates specifically designed for necessities and (2) propose the first ranking-based semantics in the literature to be shown to respect these postulates.

Introduction

Argumentation framework is a simple yet powerful knowledge representation and reasoning paradigm [Dung, 1995]. The underlying representation structure is a directed graph where the nodes represent arguments and the directed edges represent attacks between the arguments. Over this structure, two main classes of semantics were proposed for reasoning. Extension-based semantics rely on selecting subsets of arguments (called extensions) based on specific properties within the graph [Dung, 1995]. Ranking-based semantics provide an order between the arguments of the graph [START_REF] Matt | [END_REF]Leite and Martins, 2011;[START_REF] Amgoud | Ranking-based semantics for argumentation frameworks[END_REF][START_REF] Bonzon | [END_REF][START_REF] Amgoud | Leila Amgoud and Dragan Doder. Gradual semantics for weighted graphs: An unifying approach[END_REF][START_REF] Amgoud | Leila Amgoud and Dragan Doder. Gradual semantics accounting for varied-strength attacks[END_REF]. Roughly speaking, an intuitive difference between the two classes of semantics is that, in extension based semantics, the attack relation is mostly used to destroy its target (two arguments attacking each other will not be in the same extension) whereas in ranking based semantics it is often used to only weaken its target.

Bipolar argumentation frameworks are a generalization of argumentation graphs in which we consider as underlying representation a bi-colored graph: the nodes represent arguments and the bi-colored edges represent, respectively, support and attack relations between the arguments [Cayrol and Lagasquie-Schiex, 2005b;[START_REF] Amgoud | [END_REF][START_REF] Boella | [END_REF][START_REF] Cayrol | Claudette Cayrol and Marie-Christine Lagasquie-Schiex. Bipolarity in argumentation graphs: Towards a better understanding[END_REF]. * Contact Author Bipolar argumentation systems have received much attention in the literature due to the versatility of their representation power. The various relations defined in complement to attacks can express several kinds of endorsements: deductive [START_REF] Villata | [END_REF], evidential [START_REF][END_REF], necessities [START_REF] Nouioua | Argumentation frameworks with necessities[END_REF][START_REF] Nouioua | AFs with necessities: further semantics and labelling characterization[END_REF]Cohen et al., 2015;Gottifredi et al., 2018] etc.

In this paper we turn our attention to the study of ranking semantics over bipolar argumentation frameworks with necessities (i.e. argument A is necessary for argument B) that have been proposed in order to model knowledge such as "if argument B is accepted then necessarily argument A is accepted". Please note that these argumentation frameworks have only been investigated from an extension-based semantics point of view [START_REF] Nouioua | Argumentation frameworks with necessities[END_REF][START_REF] Nouioua | AFs with necessities: further semantics and labelling characterization[END_REF]Gottifredi et al., 2018].

To illustrate the significance of our contribution, let us consider a motivating example. In certain countries (France, Italy, Germany etc) a diploma of habilitation is required for the candidate in order to apply for a professorship. The selection is then based on the significance of the research and teaching activities. The necessity of having a habilitation cannot be discussed, it is a requirement. Thus, the argument of having the habilitation is not reinforcing the argument of being a professor, but, on the contrary, it can destroy it.

To illustrate the different nature of the two relations, observe the argumentation graph from Figure 1. Let argument a be "Tom should be hired as a professor". Since the habilitation is a necessary condition, argument b, standing for "Tom has a habilitation diploma", is necessary for a. This is depicted by a dotted blue arrow from b to a. Let c stand for "Tom has only a few publications", and d for "Tom does not have a lot of teaching experience". The facts that c and d attack a are represented by the solid red arrows.

What we need to model in this example is the following: On one hand, each of the attacks from c and d weakens the argument a, but none of those attacks is sufficient to completely reject a, hence it is convenient to use a ranking-based semantics to model this behaviour. On the other hand, if b is rejected, a should be rejected as well. This necessity link can not be modeled by an attack relation in ranking based semantics. Namely, suppose that one introduces an argument b with meaning that "Tom does not have a habilitation", and an attack from b to a. This modeling choice is not appropriate, Hence, to model this situation, we will use ranking based semantics with necessities. To this end, the contribution of this paper is two-fold as follows:

• A modelling contribution unravelling the particular role that the necessity relation plays in the context of ranking based semantics for bipolar argumentation frameworks. The rest of the paper is organized as follows: Section 2 introduces the basic notions necessary for this work. In Section 3 we propose the set of principles for ranking based semantics for argumentation frameworks with necessities. Section 4 presents the formal analysis of the principles, studies the links between them and shows the impact of the principles to the behavior of the framework. In Section 5 we define the first gradual semantics for argumentation frameworks with necessities, we prove a characterization result and we verify that the introduced semantics satisfies all the principles. We conclude in Section 6.

Formal Setting

In this paper we follow the standard framework for weighted argumentation graphs [Amgoud et al., 2017], and extend it with a necessity relation between arguments. The resulting framework, that we named weighted argumentation graph with necessities is a graph that consists of a finite set of arguments and two binary relations between them: a necessity relation and an attack relation. In addition 1 , an initial weight from the unit interval of reals [0, 1] is attached to each argument.

Definition 1 (WGN) A weighted argumentation graph with necessities (WGN) is a tuple G = A, w, R, N , where

• A is a non-empty finite set of arguments,

• w : A → [0, 1],
• R ⊆ A × A,

1 Please note that, for generality reasons, we will consider weighted argumentation frameworks in the remainder of the paper.

• N ⊆ A × A, where N is acyclic, and

• R ∩ N = ∅.
Intuitively, w(a) is the initial strength of argument a, i.e. the intrinsic weight attached to it independently of the attacks between the arguments. For example, an argument coming from a more trustworthy source will be attached a higher initial weight. Notation (a, b) ∈ R (or aRb) means that argument a attacks argument b and (a, b) ∈ N (or aN b) means argument a is necessary for argument b. Att G (a) denotes the set of all attackers of a in G, and Nec G (a) denotes the set of all arguments necessary for a in G. As in the previous work on argumentation frameworks with necessities [Gottifredi et al., 2018;[START_REF] Boudhar | [END_REF], we assume that N is acyclic and that R ∩ N = ∅.

Definition 2 (Semantics) A semantics is a function S which maps any WGN G = A, w, R, N into a function Deg S G : A → [0, 1].
For any a ∈ A, Deg S G (a) represents the overall strength of argument a in the graph G.

Principles for Weighted Argumentation With Necessities

In this section, we propose the first set of principles for ranking-based semantics for argumentation frameworks with necessities. Those principles can be categorized in three classes. Some of them are direct adaptations of the existing principles for weighted argumentation frameworks [Amgoud et al., 2017] (e.g. independence states that two disjoint graphs do not influence the degrees of each other's arguments). Other principles extend the existing principles (e.g. weakening soundness states that there are two reasons for the loss of strength of an argument, one of them being the necessity relation).

In addition, we propose five novel principles that describe the impact of necessity relation to the semantics: n-neutrality, n-weakening, n-counting, n-reinforcement and trimming. Some of them have counterparts amongst the principles for attack relations (e.g. n-reinforcement states that increasing the strength of a necessary argument increases the strength of its target, in contrast to a-reinforcement which states that increasing the strength of an attacker decreases the strength of the target), while some of them do not (e.g. trimming states that an argument cannot be more acceptable than any argument that is necessary for it).

We point out that a-reinforcement corresponds to the principe bi-variate reinforcement designed by [START_REF] Amgoud | Weighted bipolar argumentation graphs: Axioms and semantics[END_REF] for the support relation. The other four novel principles are essentially different from the existing principles of Amgoud and Ben-Naim due to the different natures of necessity and support relations.

Let us first define the notion of isomorphism.

Definition 3 (Isomorphism) Let G = A, w, R, N and G = A , w , R , N be two WGNs. An isomorphism from G to G is a bijective function f from A to A such that: • ∀ a ∈ A, w(a) = w (f (a)), • ∀ a, b ∈ A, aRb iff f (a)R f (b). • ∀ a, b ∈ A, aN b iff f (a)N f (b).
We can now formulate the anonymity principle.

Principle 1 (Anonymity) A semantics S satisfies anonymity iff, for any two WGNs G = A, w, R, N and G = A , w , R , N , for any isomorphism f from G to G , the following property holds: ∀ a ∈ A,

Deg S G (a) = Deg S G (f (a)
). According to the independence principle, unconnected parts of a graph do not impact each other.

Principle 2 (Independence) A semantics S satisfies independence iff, for any two WGNs G = A, w, R, N and

G = A , w , R , N s.t. A ∩ A = ∅, the following holds: ∀ a ∈ A, Deg S G (a) = Deg S G⊕G (a), where G ⊕ G = A ∪ A , w , R ∪ R , N ∪ N with w such that for ev- ery a ∈ A (resp. a ∈ A ) we have w (a) = w(a) (resp. w (a ) = w (a )). A path from a to b in a WGN G = A, w, R, N is any sequence a = a 0 , a 1 , . . . , a n = b of arguments such that (a i-1 , a i ) ∈ R ∪ N for every i ∈ {1, . . . , n}.
Principle 3 (Directionality) A semantics S satisfies directionality iff, for any two WGNs G = A, w, R, N and

G = A, w, R , N s.t. either • R = R ∪ {(a, b)} and N = N , or • N = N ∪ {(a, b)} and R = R , for every c ∈ A such that there is no path from b to c, we have Deg S G (c) = Deg S G (c).
According to the a-neutrality principle, the attackers of degree 0 do not impact their targets.

Principle 4 (A-neutrality) A semantics S satisfies Aneutrality iff, for any WGN

G = A, w, R, N , a, b ∈ A, if • w(a) = w(b), • Att G (b) = Att G (a) ∪ {c} with c ∈ A \ Att G (a), Deg S G (c) = 0, and • Nec G (b) = Nec G (a), then Deg S G (a) = Deg S G (b).
According to the N-neutrality principle, the necessary arguments of degree 1 do not impact their targets.

Principle 5 (N-neutrality) A semantics S satisfies N- neutrality iff, for any WGN G = A, w, R, N , a, b ∈ A, if • w(a) = w(b), • Att G (b) = Att G (a), and • Nec G (b) = Nec G (a) ∪ {c} with c ∈ A \ Nec G (a), Deg S G (c) = 1, then Deg S G (a) = Deg S G (b).
The equivalence principle states that two arguments having the attackers and the necessary arguments with the same degrees also have the same degrees, provided that their initial weights are same. Recall that by n-neutrality, perfect necessary arguments (of degree 1) do not impact the target. On the contrary, if a necessary argument is imperfect (i.e has a degree strictly smaller than 1), that will lead to some loss of the initial weight of the target. This is formalized by the following principle.

Principle 9 (N-weakening) A semantics S satisfies nweakening iff, for any WGN G = A, w, R, N , for any argument a ∈ A, if

• w(a) > 0, and

• there exists b ∈ Nec G (a) such that Deg S G (b) < 1, then Deg S G (a) < w(a)
. The a-counting principle states that every attacker will decrease the degree of the target, provided that the attacker's acceptability degree is strictly positive.

Principle 10 (A-counting) A semantics S satisfies acounting iff, for any WGN G = A, w, R, N and a, b ∈ A, if

• w(a) = w(b), • Deg S G (a) > 0, • Nec G (a) = Nec G (b), and • Att G (b) = Att G (a) ∪ {c} with c / ∈ Att G (a) and Deg S G (c) > 0, then Deg S G (a) > Deg S G (b).
The next principle, called n-counting, formalizes the intuition that adding a necessary argument having a degree non equal to one will decrease the acceptability degree of the target (provided that the target does not already have the zero degree).

A, if • w(a) = w(b), • Deg S G (a) > 0 or Deg S G (b) > 0, • Nec G (a) = Nec G (b), • Att G (a) \ Att G (b) = {x}, • Att G (b) \ Att G (a) = {y}, and • Deg S G (y) > Deg S G (x), then Deg S G (a) > Deg S G (b).
On the contrary to a-reinforcement, n-reinforcement claims that a necessary argument with a stronger degree weakens its target less than a necessary argument with a weaker degree.

Principle 14 (N-reinforcement) A semantics S satisfies reinforcement iff, for any WGN G = A, w, R, N , and The introduced principles emphasize that the necessity relation is essentially different from both attack and support relations. Indeed, according to n-weakening, necessities and attacks share a crucial property, which clearly distinguishes them from support relations -they both decrease the degrees of their targets. In addition, according to n-reinforcement, increasing the strength of a necessary argument increases the strength of the target, which obviously does not hold for the attack relation (see a-reinforcement).

a, b ∈ A, if • w(a) = w(b), • Deg S G (a) > 0 or Deg S G (b) > 0, • Att G (a) = Att G (b), • Nec G (a) \ Nec G (b) = {x}, • Nec G (b) \ Nec G (a) = {y}, and • Deg S G (y) > Deg S G (x), then Deg S G (b) > Deg S G (a).
The next principle says that if an initial weight of an argument decreases, its acceptability degree decreases as well (provided that it was not already zero).

Principle 15 (Proportionality) A semantics S satisfies proportionality iff, for any WGN G = A, w, R, N and a, b ∈ A such that

• Att G (a) = Att G (b), • Nec G (a) = Nec G (b),
• w(a) > w(b), and

• Deg S G (a) > 0 or Deg S G (b) > 0, then Deg S G (a) > Deg S G (b).
The following principle captures the main intuition behind the necessity relation: if the argument b is necessary for a, then a cannot be accepted if b is not accepted as well. Therefore, the acceptability degree of a is at most the acceptability degree of b. Another principle for ranking semantics is resilience [Amgoud et al., 2017], which says that an argument cannot lose all the intrinsic weight, i.e. that w(a) > 0 implies Deg S G (a) > 0. This principle is not desirable in presence of necessities. Indeed, the main intuition behind the necessity relation is that an argument cannot be accepted if any of its necessary arguments is not accepted. This is formalized by the trimming principle, which implies that a necessary argument of degree 0 will always reduce the strength of its target to 0, regardless of the initial weight of the target.

Formal Analysis of the Principles

In this proposition we show that if an argument is only attacked by arguments with a 0 acceptability value and the necessary arguments for it have an 1 acceptability value then its acceptability equals its basic strength for semantics satisfying independence, directionality, maximality, a-neutrality and nneutrality.

Proposition 1 Let a semantics S satisfy independence, directionality, maximality, a-neutrality and n-neutrality. Let G = A, w, R, N be a WGN and a ∈ A an argument such that for every x ∈ Att G (a), Deg S G (x) = 0, and for every y ∈ Nec G (a), Deg S G (y) = 1. Then, Deg S G (a) = w(a). In the next proposition we show that if a semantics satisfy independence, directionality, maximality, a-weakening, n-weakening, a-neutrality and n-neutrality then its basic strength is an upper bound for its acceptability.

Proposition 2 Let a semantics S satisfy independence, directionality, maximality, a-weakening, n-weakening, a-neutrality and n-neutrality. Then, for every WGN G = A, w, R, N and every a ∈ A we have Deg S G (a) ∈ [0, w(a)]. Arguments that have a basic strength equal to 0, or more generally an acceptability degree 0, cannot change their degrees if they become targets of additional attacks or necessities. This follows from directionality, independence, maximality, a-neutrality, n-neutrality, a-reinforcement and nreinforcement.

Proposition 3 Let a semantics S satisfy directionality, independence, maximality, a-neutrality, n-neutrality, areinforcement and n-reinforcement. Then, for every WGN

G = A, w, R, N and every a, b, c ∈ A such that w(a) = w(b), c / ∈ Att G (a) ∪ Nec G (a) and • Nec G (a) = Nec G (b) and Att G (b) = Att G (a) ∪ {c}, or • Att G (a) = Att G (b) and Nec G (b) = Nec G (a) ∪ {c},
we have that Deg S G (a) = 0 implies Deg S G (b) = 0. The monotony principle by [START_REF] Amgoud | Leila Amgoud and Jonathan Ben-Naim. Axiomatic foundations of acceptability semantics[END_REF] is stated for standard argumentation graphs (i.e. the graphs with attacks only) and says that a set S of attackers weakens its target at least as much as any subset of S. The following results states that a generalization of that principle, that properly takes into account necessity relation of our framework, follows from a set of principles from Section 3.

Proposition 4 Let a semantics S satisfy directionality, independence, maximality, a-neutrality, n-neutrality, areinforcement and n-reinforcement. Then, for every WGN

G = A, w, R, N and every a, b ∈ A if • w(a) = w(b) • Att G (a) ⊆ Att G (b), and • Nec G (a) ⊆ Nec G (b) then Deg S G (a) ≥ Deg S G (b).
The following result shows that a set of our principles implies a property that reminds of counter-transitivity [START_REF] Amgoud | Ranking-based semantics for argumentation frameworks[END_REF]. The following two results give some dependencies between principles. By the following proposition, both a-counting and n-counting follow from some other principles.

Proposition 6 Let S be a semantics which satisfies directionality, independence and maximality. Then

• If S satisfies a-neutrality and a-reinforcement, then S satisfies a-counting. • If S satisfies n-neutrality and n-reinforcement, then S satisfies n-counting.

Gradual Semantics for Argumentation Framework With Necessities

In this section we provide the first gradual semantics for argumentation frameworks with necessities. As in case of any bipolar framework, it is hard to obtain a semantics that is well-defined for all graphs of the framework. The usual approach in the literature is to restrict the attention to the class of acyclic graphs [START_REF] Amgoud | Weighted bipolar argumentation graphs: Axioms and semantics[END_REF][START_REF] Rago | [END_REF]. The only exception is the work of [START_REF] Mossakowski | Modular semantics and characteristics for bipolar weighted argumentation graphs[END_REF]. However, that semantics is very simple and particular one, since it only takes into account the strongest attacker and the strongest supporter, while ignoring the values of other attackers and supporters. We will follow the standard approach and define semantics for several subclasses of graphs, including acyclic graphs. We consider the following three classes of graphs.

Notation: We denote by G ac the set of all acyclic graphs (meaning that R ∪ N is aciclyc), by G att the set of graphs with no necessities (i.e. where N = ∅) and by G nec the set of graphs with no attacks (i.e. R = ∅).

We use the probabilistic sum to aggregate the attackers, i.e. the commutative and associative function x⊕y = x+y -xy. We define the semantics using the iterative method.

Definition 4 For a WGN G = A, w, R, N and i ∈ {1, 2, 3, . . .} we define the function f i G inductively as follows: .

f i G (a) = w(a) x∈Nec(a) f i-1 G (x) e y∈Att(a) f i-1 G (y)
(1)

The following theorem shows that for any graph from any of the considered three classes (G ac , G att and G nec ), the set (one for each argument) of equations (1) has a unique solution.

Theorem 3 Equation (1) uniquely characterizes Npe.

Theorem 3 allows us to check the principle compliance. Our principles are properties of semantics that should hold for every graph. However, Npe is defined on three classes of graphs: G ac , G att and G nec , therefore it is only sensible to check the properties of Npe by restricting the principles to those classes. We next show that Npe satisfies all the principles.

Theorem 4 The Npe semantics satisfies all the principles on each of the three classes of graphs G ac , G att and G nec .

Finally, let us note that by applying the semantics Npe to the class G att , we obtain a novel semantics for all weighted argumentation graphs (not only for the acyclic ones). Similarly, restricting the Npe semantics to G nec leads to the first ranking-based semantics for weighted graphs with necessities only.

Conclusions and Related Work

In this paper, we study for the first time how to model necessities in ranking based semantics. This allows us to combine two things both needed in order to model certain practical scenarios:

• First, the ability of ranking based semantics to model the attacks that only weaken their targets (and not completely destroy them).

• Second, the use of necessity relation in order to model the fact that an argument a is necessary for the argument b (i.e. the rejection of a should lead to complete rejection of b).

As explained in the paper and made intuitively obvious by our motivating example in the introduction, necessity does not behave as a type of support, nor a type of attack. The study of its behaviour for ranking semantics allowed us to basically identify a new type of relation. To this end necessity shares the property of attack relations in weakening their targets, but also an important property of support in the fact that the stronger the source, the stronger the target.

Placing our work within the relevant literature is thus a difficult exercise because none of the approaches identified the specificities of the necessity relations as explained above. However, we would like to highlight in chronological order five important papers that should be mentioned at this point. First we mention the noteworthy work of Cayrol and Lagasquie-Schiex [2005a], who were the first to propose to rank arguments in bipolar argumentation frameworks by aggregating the values of all parents of arguments. [START_REF] Rago | [END_REF] introduce DF-QuAD to deal with quantitative argumentation debates in order to allow for automated decision support, whereas [START_REF] Amgoud | Weighted bipolar argumentation graphs: Axioms and semantics[END_REF] provide a comprehensive and inspiring study of ranking semantics and their principles for bipolar argumentation systems when considering a general approach to support. The work of [START_REF] Mossakowski | Modular semantics and characteristics for bipolar weighted argumentation graphs[END_REF] is an important indepth analysis of the technicalities behind convergence results in bipolar argumentation frameworks with a particular emphasis on the underlying graph structure of the argumentation framework. Finally, let us also mention the work by Potyka [2019], which generalizes the convergence conditions introduced by Mossakowski and Neuhaus.

Figure 1 :

 1 Figure 1: Bipolar argumentation graph with necessities.

  Principle 6 (Equivalence) A semantics S satisfies equivalence iff, for any WGN G = A, w, R, N and a, b ∈ A, if• w(a) = w(b),• there exists a bijection f :Att G (a) → Att G (b) such that Deg S G (c) = Deg S G (f (c)) for every c ∈ Att G (a), • there exists a bijection g : Nec G (a) → Nec G (b) such that Deg S G (c) = Deg S G (g(c)) for every c ∈ Nec G (a), then Deg S G (a) = Deg S G (b).According to the maximality principle, the degree of an argument which does not have any attackers nor necessary arguments is exactly its initial weight. Principle 7 (Maximality) A semantics S satisfies maximality iff, for any WGN G = A, w, R, N , and every a ∈ A such that Att G (a) = Nec G (a) = ∅, we have Deg S G (a) = w(a). The a-weakening principle states that any attack from an argument having a positive degree necessarily leads to the loss of the initial weight of the target. Principle 8 (A-weakening) A semantics S satisfies aweakening iff, for any WGN G = A, w, R, N , for any argument a ∈ A, if • w(a) > 0, and • there exists b ∈ Att G (a) such that Deg S G (b) > 0, then Deg S G (a) < w(a).

  Principle 11 (N-counting) A semantics S satisfies ncounting iff, for any WGNG = A, w, R, N and a, b ∈ A, if • w(a) = w(b), • Deg S G (a) > 0, • Att G (a) = Att G (b), and • Nec G (b) = Nec G (a) ∪ {c} with c / ∈ Nec G (a) and Deg S G (c) < 1, then Deg S G (a) > Deg S G (b).Weakening soundness identifies the only two possible reasons for losing some initial weight, namely, being attacked by non zero degree arguments and having necessary arguments with degree different from 1. Principle 12 (Weakening soundness) A semantics S satisfies weakening soundness iff, for any WGN G = A, w, R, N , and a ∈ A, if Deg S G (a) < w(a), then there exists either b ∈ Att G (a) such that Deg S G (b) > 0, or b ∈ Nec G (a) such that Deg S G (b) < 1. A-reinforcement states that a stronger attacker has a bigger impact on a target. Principle 13 (A-reinforcement) A semantics S satisfies areinforcement iff, for any WGN G = A, w, R, N , and a, b ∈

Figure 2 :

 2 Figure 2: Graph G from Example 1.

Principle 16 (

 16 Trimming) A semantics S satisfies trimming iff, for any WGN G = A, w, R, N and a, b ∈ A such that b ∈ Nec G (a), Deg S G (a) ≤ Deg S G (b). Example 2 (Ex. 1 continued) Since Deg S G (x) = 0.6 and Deg S G (y) = 0.8, by trimming we have that Deg S G (a) ≤ 0.6 and Deg S G (b) ≤ 0.8.

Proposition 5

 5 Let a semantics S satisfy directionality, independence, maximality, equivalence, a-neutrality, n-neutrality, a-reinforcement and n-reinforcement. Then, for every WGN G = A, w, R, N and every a, b ∈ A if • w(a) = w(b) and • there exist two injective functions f : Att(a) → Att(b) and g : Nec(a) → Nec(b) such that: for all c ∈ Att(a), Deg S G (c) ≤ Deg S G (f (c)) for all d ∈ Nec(a), Deg S G (d) ≥ Deg S G (g(d)) then Deg S G (a) ≥ Deg S G (b).

  with f 0 G (a) = w(a). In the above definition we follow the usual convention that α∈∅ α = 1 and α∈∅ α = 0. The next result shows that f i G converges whenever G is acyclic, without attacks or without necessities. Theorem 1 Let α ∈ {ac, att, nec}. Then, for every G = A, w, R, N ∈ G α , for every a ∈ A, f i G (a) converges when i → +∞. This result allows us to define the semantics for the classes G ac , G att and G nec . Definition 5 (Npe) The necessity product exponential semantics is a function Npe transforming any WGN G = A, w, R, N ∈ G α , where α ∈ {ac, att, nec}, into the function Deg Npe G : A → [0, 1], such that for any a ∈ A, Ex. 1 continued) Let G be the WGN from Example 1. By applying Npe, we obtain Deg Npe G ) = 0.72 √ e ≈ 0.44. Under Npe, the degree of every argument is bounded by its initial weight. Proposition 7 For any WGN G = A, w, R, N ∈ G α , where α ∈ {ac, att, nec}, for any a ∈ A, Deg Npe G (a) ∈ [0, w(a)]. We now prove that the Npe semantics can be represented by a set of equations relating the degree of an argument with the degrees of its attackers and necessities. Theorem 2 Let α ∈ {ac, att, nec}. Then, for every G = A, w, R, N ∈ G α , for every a ∈ A Deg Npe G (a) = w(a) x∈Nec(a) Deg Npe G (x) e y∈Att(a) Deg Npe G (y)