Dylan Le Maux

Vincent Klapczynski

Mickaël Courtois
email: mickael.courtois@univ-ubs.fr

Thomas Pierre

Philippe Le Masson

Surface tension of liquid Fe, Nb and 304L SS and effect of drop mass in aerodynamic levitation

Keywords: surface tension, liquid metals, aerodynamic levitation, effect of drop mass, numerical modelling

The oscillating drop method has been widely used for surface tension measurements of liquids at high temperatures such as molten metals. An experimental device that permits such measurements has been developed at IRDL. This device relies on aerodynamic levitation for the sample to be contactless and on acoustic excitation to make it oscillate at its resonance frequency, which is related to the surface tension with Rayleigh's equation. Beforehand, experimentally, a lateral gas jet blows and controls the rotation of the drop due to its unpredictable behavior. Moreover, the effect of drop mass, usually studied for electromagnetic levitation, remains unknown for aerodynamic levitation. To fill this knowledge gap, levitated drops of different masses were numerically modelled to observe the influence of gravity or additional oscillation modes. It resulted that for light mass drops, such as those used for our surface tension measurement, it is not significant. Finally, our results on iron, niobium, and 304L stainless steel have been compared with literature. For pure metals, we observed a similar decreasing behavior versus temperature. For steel, results highlighted the impact of experimental parameters on the surface tension (length of experiments and composition of the atmosphere). These measurements thus provide original results that will be useful for the numerical modeling of innovative industrial processes such as additive manufacturing or welding.

Introduction

This bibliography only focuses on characterization methods for liquid metals. Surface tension is a parameter of interest regarding numerical modeling of industrial processes including molten metal, such as welding or additive manufacturing. Some methods where the sample is in a container have been developed such as the sessile drop method [START_REF] Dal | Laser based method for surface tension and density measurements for liquid refractory metals (Nb, Ta, and W)[END_REF] or the maximum bubble pressure method [START_REF] Takeda | Development of Maximum Bubble Pressure Method for Surface Tension Measurement of High Viscosity Molten Silicate[END_REF]. But chemical reactions between the container and the sample can occur at high temperatures (above the melting point of common metals), resulting in the sample pollution. Therefore, the surface tension value is hampered by the modification of its composition. With a global uncertainty around ± 1 %, the quasi-containerless pendant drop [START_REF] Wu | Surface Tension of High Temperature Liquids Evaluation with a Thermal Imaging Furnace[END_REF] helps preventing this pollution but cannot permit to reach high temperatures, the drop being attached to the cold rod of the same material. The most used method for surface tension measurement is the oscillating drop associated with containerless methods such as the free-fall drop or multiple levitation methods (electromagnetic, electrostatic, aerodynamic, etc.). The mentioned uncertainty is generally higher, around ± 5 %.

In situ measurement has been performed in arc welding by observing the oscillations of the projected drops of steel [START_REF] Subramaniam | Effect of shield gas composition on surface tension of steel droplets in a gas-metal-arc welding arc[END_REF], [START_REF] Bachmann | In situ droplet surface tension and viscosity measurements in gas metal arc welding[END_REF]. The mentioned surface tension uncertainty is approximately ± 20 %, but it permits to consider the influence of all phenomena implied during the welding process like arcplasma. Moreover, due to the very short time of the experiment, it is possible to perform measurement above 2 000 K [START_REF] Monier | Etude expérimentale du comportement dynamique des phases liquides en soudage par court-circuit contrôlé[END_REF].

The oscillating drop method, which is chosen for this work, is based on the Rayleigh's equation, that gives the surface tension as a function of the resonance frequency of an oscillating drop [START_REF] Rayleigh | On the Capillary Phenomena of Jets[END_REF]. It has been widely used associated with levitation that has already been briefly reviewed in a previous paper [START_REF] Le Maux | Density measurement of liquid 22MnB5 by aerodynamic levitation[END_REF]. Aerodynamic levitation is used in this work for its (proposition:) capacity to measure various physical properties within a large temperature extent. Some levitation devices (only electromagnetic ones to our knowledge) have been used under micro-gravity conditions by sending it in space [START_REF] Egry | Surface Tension of Liquid Metals: Results from Measurements on Ground and in Space[END_REF], in parabolic flight [START_REF] Mohr | Surface tension and viscosity of liquid Pd43Cu27Ni10P20 measured in a levitation device under microgravity[END_REF], or in drop-shaft [START_REF] Fujii | Surface tension of molten silicon measured by the electromagnetic levitation method under microgravity[END_REF]. The levitation tends to complicate the drop oscillations by adding an extra force with the surface tension. Micro-gravity helps to quasi-overcome levitation force and thus simplify the oscillations, making easier the estimation of the resonance frequency in spite of an important cost for each experiment. The free-fall drop method also permits to perform measurements in micro-gravity conditions in a more simple way but with a very short time of experiment, depending on the height of the fall [START_REF] Matsumoto | Measurement of surface tension of molten copper using the free-fall oscillating drop method[END_REF].

The oscillating drop is generally chosen to reach high temperature (above 2 000 K). However, the effect of levitation forces and gravity might be considered and the frequency spectrum has to be analyzed to determine the surface tension. Section 2 of this publication presents the oscillating drop method to measure surface tension of liquid metals with an innovative approach to determine the resonance frequency. Then, the effects of gravity on the surface tension measurement are studied in Section 3. Finally, Section 4 presents and discusses the results of surface tension after measurements on iron, niobium, and 304L stainless steel.

Materials and methods

The oscillating drop method

The oscillating levitated drop is a convenient method for the determination of the surface tension.

Rayleigh first described the relation between the surface tension and the resonance frequency of an oscillating drop in Eq. (1) [START_REF] Rayleigh | On the Capillary Phenomena of Jets[END_REF]:

Eq. (1)
where is the surface tension, is the mass, the resonance frequency, and describes the oscillation mode. In this study, we place ourselves in the case of the mode (Eq. ( 2)), the other ones being insignificant.

Eq. ( 2)

In many cases, static deformation due to both the levitation force and the rotation occurs during the levitation of the sample, leading to a splitting of the mode 2 into five characteristic modes. Each characteristic mode has its own resonance frequency, , , and the shape of the oscillating drop under these modes can be predicted using Legendre functions [START_REF] Egry | Surface tension measurements of liquid metals by the oscillating drop technique[END_REF] (not presented here). Fig. 1 presents a side view corresponding to our experimental device. If we look at that radius variation with a 54.7° angle (considering the vertical to be 0°), the pure mode 2 is not visible from the side view.

Theoretically, it is possible, thanks to a fast Fourier transform (FFT) on the variation of the radii at 0°, 54.7° and 90° over time, to find by discrimination the characteristic frequencies related to the different oscillation modes [START_REF] Sauerland | Surface tension measurements on levitated liquid metal drops[END_REF]. Pure mode 2 should be visible at 0° and 90°, mode 2±1 only at 54.7° and mode 2±2 at 54.7° and 90°. Finally, a sum rule is usually used to calculate the surface tension [START_REF] Dubberstein | Surface tension and density data for Fe-Cr-Mo, Fe-Cr-Ni, and Fe-Cr-Mn-Ni steels[END_REF], [START_REF] Schneider | Measurement of the Surface Tension of Undercooled Liquid Ti 90Al 6V 4 by the Oscillating Drop Technique[END_REF]. The oscillations amplitude is an important point for the oscillating drop method, indeed, Rayleigh's equation being linear is adapted for small drop deformations. Greater deformations would induce a frequency shift that has been previously numerically studied [START_REF] Watanabe | Numerical simulation of oscillations and rotations of a free liquid droplet using the level set method[END_REF]. For a 30 mg drop of 1.9 N•m -1 surface tension, a ± 10 % shift of amplitude oscillations would imply ± 2 % error on the calculated surface tension. This error is limited to ± 0.3 % with ± 5 % of amplitude oscillations. We will thus limit the amplitude to ± 5 % for all the experiments. This agrees with other works mentioning their experimental parameters [START_REF] Langstaff | Aerodynamic levitator furnace for measuring thermophysical properties of refractory liquids[END_REF].

The experimental device

The experimental device is presented in Fig. 2 (see also [START_REF] Le Maux | Density measurement of liquid 22MnB5 by aerodynamic levitation[END_REF]). An argon hydrogen mix Ar + 2.5 %vol.

H 2 is blown through a 60° convergent-divergent nozzle to levitate an approximately 30 mg sample (about 2 mm of diameter for iron alloys) (capital A in Fig. 2). Other atmospheres such as pure argon have recently been studied with this experimental device [START_REF] Klapczynski | Surface tension measurements of liquid pure iron and 304L stainless steel under different gas mixtures[END_REF]. A first numerical flowmeter Brooks SLA5850S controls the lift gas (B). The levitation part is set in a vacuum chamber equipped with borosilicate viewports (C). The upper window is dedicated to the heating laser (IPG Photonics YLR 300/3000 QCW at 1 070 nm) (D) and the pyrometric measurement (E). The heating laser has a 280 W maximal power in continuous mode. It is equipped with a 500 mm focalization optic slightly defocused to make the aimed spot at about 2 mm. The sample temperature measurement is performed by a commercial bichromatic pyrometer (Optris CT Ratio 2MH1) with 1.45 µm and 1.75 µm wavelengths. For this pyrometer, a ratio of the emissivities at the two wavelengths is set to match the solidification plateau as a known temperature of the tested metal. Contrary to the solid state, at liquid state it is assumed that the possible emissivities ratio variation implies temperature variation inferior to the uncertainty measurements [START_REF] Pierre | Simultaneous estimation of temperature and emissivity of metals around their melting points by deterministic and Bayesian techniques[END_REF]. Moreover, considering the temperature reached by the metallic samples, the influence of surrounding environment is insignificant.

A high-speed camera (F) Phantom VEO-E 310L is set in front of a side viewport and the backlighting laser (300 mW, 532 nm diode laser with a beam expander) (G) in front of the opposite window. The sample is heated up by the laser until it becomes liquid. Then an acoustic solicitation is used to make oscillate the liquid sample: a sinusoidal signal is generated, amplified (H), and transformed into pressure waves thanks to a speaker (I). The second flowmeter (B) controls a lateral gas jet to force the drop rotation along the vertical axis (see Section 2.3). Finally, an oxygen analyzer measures the residual oxygen in the pipe of the levitation gas (J).

Rotation of the sample

One of the difficulties of this work is the determination of the characteristic frequencies with the method previously described in Section 2.1. Indeed, it appears that the drop is subject to random rotations. These rotations have been observed thanks to small corundum particles used as tracers thanks to both their high emissivity and the self-illumination of the sample. These rotations make difficult the identification of the peaks on the spectra obtained with the FFT. For example, Fig. 3 presents a normalization of the sample radii variations versus the frequency for three angles. One peak is clearly visible around 200 Hz on the three spectra although none of the oscillation mode should appear. Moreover, among all the peaks on the left, it is not possible de distinguish which ones are related to an oscillation mode or to background noise. Finally, even for measurements attempted with constant experimental parameters, the spectra obtained are different making the experiments unrepeatable and uncertainty important. In electromagnetic levitation, the sample naturally rotates along the vertical symmetry axis of the coil [START_REF] Fraser | Surface tension measurements on pure liquid iron and nickel by an oscillating drop technique[END_REF]. With electrostatic levitation, supplementary small coils impose a rotation along the vertical axis, making possible to control the rotation speed [START_REF] Rhim | Noncontact electrical resistivity measurement technique for molten metals[END_REF]. With aerodynamic levitation, the effect of an imposed rotation has been studied by Millot et al. [START_REF] Millot | Analysis of Surface Tension from Aerodynamic Levitation of Liquids[END_REF], but no measurement is reported. Nevertheless, the authors implicitly consider that the drop naturally rotates along the vertical axis at the speed of about 30 rad•s -1 . Since no such rotation is observed, our experimental device is modified to control the drop rotation: a thin gas jet is blown through a needle on the side of the levitated drop, as seen in Fig.

4.

The needle is positioned at the equator of the drop. The variation of the size of the sample with temperature makes it difficult to have an optimal position, resulting in a rotation axis slightly shifted from the vertical. The main point here is to have a fixed rotation axis instead of an unpredictable one as described before. Once again corundum particles are used to visualize the drop rotation. The speed is estimated around 150 rad•s -1 , which is higher than the value previously stated by [START_REF] Rhim | Noncontact electrical resistivity measurement technique for molten metals[END_REF]. This is the slowest speed our system can achieve, a smaller gas flow through the needle being unable to induce a rotation of the drop. However, it is important to keep the rotation speed as slow as possible. Indeed, some experiments with different gas flows have been performed on iron. Fig. 5 shows the evolution of the surface tension versus the shape ratio rx/ry, meaning the radii ratio. The values close to a unity shape ratio are the ones obtained with the smallest gas flow allowing a control of the rotation. It appears that a higher rotation speed induces a deformation of the drop due to centrifugal force, modifies its resonance frequency, and leads to overestimate the surface tension. Numerous experiences allow to set up a method that does not deform the drops by adjusting the gas flow. Then, this method has been used for the measurements presented in this paper. It appears that not only the experiments are now repeatable thanks to the controlled rotation, but also that split modes are suppressed, leaving alone the pure mode 2. Fig. 6 present the spectra, in the same manner as Fig. 3, obtained for an experiment with controlled rotation. One single peak is clearly visible at 0° and 90°, but nothing appears at 54.7°, which is characteristic of the pure mode 2 (see Fig. 1). The calculus of surface tension is then straightforward as the frequency read on the spectrum that we attribute to the pure mode 2 is considered as the resonance frequency.

Fig. 6 Spectra obtained with a FFT on the variation of the radii over the frequency with a controlled rotation of the drop. The amplitude of the spectra is normalized to 1 and shifted for clarity.

Post-treatment

The video captured by the high-speed camera is automatically post-treated with Matlab ® frame by frame, as already described in [START_REF] Le Maux | Density measurement of liquid 22MnB5 by aerodynamic levitation[END_REF]. Each frame is cropped to only keep the drop in the frame. Then the image is binarized and an edge detection is applied. The coordinates of each point of the edge are used as input data for an ellipse fitting function that returns the long radius, the short radius, and the shift angle. These parameters permit us to calculate the radii at 0°, 54.7°, and 90°. A FFT is then applied on the radii variation signals, returning a spectrum where the characteristic frequencies can theoretically be read (see Fig. 6).

Numerical analysis of the effect of drop mass

Numerical modeling of the experiment

The effect of gravity has been studied by previous researchers for electromagnetic levitation, leading to a correction law to consider it [START_REF] Cummings | Oscillations of magnetically levitated aspherical droplets[END_REF], then mostly used in many following works for surface tension measurements. Some works has been performed in microgravity by sending the experimental device in a plane for a parabolic flight [START_REF] Mohr | Surface tension and viscosity of liquid Pd43Cu27Ni10P20 measured in a levitation device under microgravity[END_REF] or in the International Spatial Station [START_REF] Brillo | Thermophysical property measurements of liquid metals by electromagnetic levitation[END_REF], thus to suppress the effect of gravity. To our knowledge, this kind of study has never been done for aerodynamic levitation. We assume that gravity is usually neglected because of the small size of the drops (generally 2 to 3 mm with a mass of 30 to 150 mg for metals [START_REF] Langstaff | Aerodynamic levitator furnace for measuring thermophysical properties of refractory liquids[END_REF]). The absence of static deformation of the levitated drop, which profile appears to be experimentally perfectly circular, leads to the hypothesis that gravity can be neglected. Significantly larger drops are flattened due to their own mass affected by gravity, which is the case for the sessile drop method [START_REF] Dal | Laser based method for surface tension and density measurements for liquid refractory metals (Nb, Ta, and W)[END_REF]. Moreover, results obtained on pure metals such as nickel with aerodynamic levitation [START_REF] Glorieux | Aerodynamic levitation: An approach to microgravity[END_REF] agree well with the ones obtained with electromagnetic levitation once applied the correction law.

However, we propose to verify the impact of gravity (potentially leading to significant static error on surface tension the measurement) on a levitated drop of metal with a numerical model. Measurements with the same material but with a varying sample mass from 20 mg to 160 mg have been carried out [START_REF] Glorieux | Surface Tension of Liquid Alumina from Contactless Techniques[END_REF], but the authors found it difficult to draw conclusions from these results. We prefer a numerical approach over an experimental one because of the difficulty to stabilize large levitated drop with our device due to strong surface oscillations appearing with 100 mg drops in our former experiments.

We chose to modelized a drop of iron for two reasons. Firstly, the surface tension of iron very wellknown. Secondly, this is the closest pure material to steel the experimental device is mainly designed for. A 2D axisymmetric geometry is preferred for faster computing reasons (see Fig. 7), supposing that there is no fluid movement around the z-axis (no rotation of the drop for example). The density of liquid iron drop has been measured previously with this experimental device [START_REF] Le Maux | Density measurement of liquid 22MnB5 by aerodynamic levitation[END_REF], and has been chosen around its solidification point ( 1 = 7 000 kg•m -3 ). For a 2.2 mm diameter, the mass of the drop is around 30 mg, which matches with the aimed mass for the experiments. The value of surface tension is chosen to 1.9 N•m -1 •K -1 , corresponding to the data found in the literature around the solidification point. The model solves only fluid flow equations in liquid metal and gas. The free surface deformation is modeled through a mobile mesh method A.L.E [START_REF] Donea | An arbitrary lagrangian-eulerian finite element method for transient dynamic fluid-structure interactions[END_REF].

In order to describe fluid movements in both gas and liquid metal, two identical equations of momentum conservation are solved (Eq. ( 3)). Additionally, a source term is added to consider the effect of gravity on the gas and on the drop. Regarding the very low expected velocities, both fluids (gas and liquid metal) are considered incompressible with mass conservation equation (Eq. ( 4)).

Eq. (

Eq. ( 4)

where ρ is the density, p the pressure, is the velocity field and the subscript i is either the metal (1) or the gas (2). To simulate the acoustic excitation, a sinusoidal gas inlet boundary conditions is imposed on the fluid velocity (Eq. ( 5)):

Eq. ( 5)

where Ve is the average fluid velocity (or the fluid velocity without acoustic excitation) that has been calculated from the experimental gas flow and diameter of the nozzle, A is set so that the amplitude of the oscillation response of the drop is around ± 5 % of the original radius of the drop (as the same manner as the acoustic volume of the speaker is set), and f is a frequency sweep of 60 Hz centered on the theoretical resonance frequency (obtained with Rayleigh's equation and the surface tension data from the literature).

At the interface between both fluids, surface tension (Eq. 6) and velocities continuity (Eq. 7) are taken into account:

Eq. ( 6)

Eq. ( 7)

where is tensor of viscous stresses, κ is the curvature, and μ is the dynamic viscosity. Since the drop is subject to deformation under the influence of two types of oscillations, a mobile mesh is used (Eq. ( 8)). Firstly, the initial height of the drop set for the model is generally slightly higher than the optimal one, so the drop oscillates on a mode 1 (translation) before stabilizing. Secondly, the acoustic excitation generates mode 2 oscillations.

Eq. ( 8) The laser heating and the radiative and convective heat losses are not modelized, the drop remains at the same temperature. A part of the nozzle is modelized to simulate the gas flow around the drop, which is used to simulate the acoustic excitation: a sweep of 60 Hz around the theoretical resonance frequency over few seconds. Three probes are set on the drop to measure its variations: on the top, on the bottom and on the equator. The difference between the first two gives the vertical diameter and the third directly gives the horizontal radius. A fourth probe is used to observe the vertical translation of the drop. A FFT is applied on the spectrum representing the evolution of the drop diameter or radius variation over frequency, allowing identify the resonance frequency. Then the surface tension is calculated with Rayleigh's equation and compared with the initial one set for the model.

Comparison with experiment

To validate the model, a simulation is performed with the same parameters as an experiment chosen as representative of the surface tension measurements. The mass of drop for this experiment is 33 mg and the estimated surface tension is 1.84 N•m -1 . Fig. 8 presents the comparison between experimental and simulated normalized amplitude of the drop diameter versus the frequency. In both cases, a single peak is visible, centered at 218 Hz. The peak from the model is much larger, which is probably due to the acoustic excitation, in reality a pressure variation modelled here as a variable velocity for numerical reasons. With this method, the drop is excited during the whole frequency sweep but the resonance frequency can still be read on the spectrum. Fig. 8 shows that consistent results are obtained between the experiment and the numerical simulation.

Results of the model and discussion

To exploit this model, a parametric analysis is performed with a varying mass from 30 mg (as in our experiment) to approximately 500 mg. Another nozzle has been resized to stabilized drops massing more than 100 mg.

In the case of drops of more than 100 mg, we observe non-linear effects on the oscillations. This is visible on the diameter variation over time as displayed in Fig. 9. Strong oscillations appear between 0.75 s and 1.2 s, but the frequency of these oscillations, being out of the frequency range swept by the excitation, is not related to the resonance frequency. Actually, it is related to mode 1 oscillations (translational), and induces mode 2 oscillations as the drop bounces in the nozzle. The oscillations related to the resonance frequency are barely visible around 1.75 s. This shows the limit of the observation of only mode 2 oscillations. For comparison, the diameters variation for a 30 mg is displayed on Fig. 10. In this case, only large amplitude oscillations are seen when the resonance frequency is reached around 1.2 s. The diameter variations for several drops of different masses are post-treated with FFT, and the spectra are displayed in Fig. 11. For a sake of clarity, the possible mode 1 oscillations (described before) have been removed. As the drop mass increases, the resonance frequency decreases, which is predictable with Rayleigh's equation. A frequency shift is calculated with the ratio of the numerically obtained frequency to the theoretical resonance frequency, the results are summarized in Fig. 12. For light mass drops, for which the gravity should not have significant effect, the frequency shift is expected to be equal to 1. However, a shift is observed for the 30 mg drop (0.995) that would induce a 0.9 % error (not negligible but rather low). This shift is attributed to error related to numerical calculus (mesh and time steps), and therefore is not considered. For heavier drops, the frequency shift is significant from 234 mg (1.01 inducing a 2.5 % error) and increases with the mass: at 515 mg, the shift is 1.11 inducing a 23.8 % error. It is important to state that it is not the mass of sample itself that induces the frequency shift, but the deformation due to gravity. For example, no frequency shift should be observed in microgravity where gravity will not deform heavy drops. In conclusion, for samples used with aerodynamic levitation weighing between 30 mg and around 200 mg, a correction law of the type used for electromagnetic levitation should not improve significantly the accuracy of surface tension measurement, but could be considered for heavier drops. performed. Errors bars are displayed for each measurement points with the uncertainty expressed previously. It appears that, generally, the dispersion is greater than the error bars. We explain this mostly by the pollution of the samples, which is a source of uncertainty that we cannot quantify and highlights the need of extreme care for cleanliness of both samples and atmosphere.

The surface tension results on iron (solidification temperature T f = 1 808 K) are compared with two previous works representative of the literature [START_REF] Morohoshi | Surface Tension of Liquid Iron as Functions of Oxygen Activity and Temperature[END_REF], [START_REF] Wille | Thermophysical Properties of Containerless Liquid Iron up to 2500 K[END_REF] and displayed in Fig. 13. Each work finds out that results can be approximated with a decreasing linear law over the temperature ( for our work with T the temperature in kelvin). The values globally overlap but the slopes are different: our one is -1.54 N•m -1 •K -1 while other works reported -2.85 N•m -1 •K -1 and -4.55 N•m -1 •K -1 .

However, it is important to precise that the purity of our samples is 99.95 % against 99.99 % and 99.997 % for the works mentioned here, and that the surface tension is very sensitive to pollution. The difference in the measurements is assumed coming mainly from the difference of purity.

Fig. 13 Surface tension of iron. Dotted line is the solidification temperature (1 808 K). Literature [START_REF] Morohoshi | Surface Tension of Liquid Iron as Functions of Oxygen Activity and Temperature[END_REF], [START_REF] Wille | Thermophysical Properties of Containerless Liquid Iron up to 2500 K[END_REF].

For niobium (solidification temperature T f = 2 750 K), our results are compared with results from other works (see Fig. 14), that we supposed have mostly been obtained thank to methods with contact because of the presence of only one value at the melting point [START_REF] Allen | The surface tension of liquid transition metals at their melting points[END_REF]- [START_REF] Arkhipkin | Installation for measuring the surface tension of liquid refractory metals[END_REF]. Only one publication reporting values as function of the temperature has been found [START_REF] Paradis | Non-Contact Measurements of Surface Tension and Viscosity of Niobium, Zirconium, and Titanium Using an Electrostatic Levitation Furnace[END_REF]. Again, our results can be approximated with a decreasing linear law ( for our work). Our slope is -3.59 N•m -1 •K -1 while the other work reported -1.99 N•m -1 •K -1 . Our sample purity is 99.9 % (this detail is not mentioned by [START_REF] Wille | Thermophysical Properties of Containerless Liquid Iron up to 2500 K[END_REF]- [START_REF] Paradis | Non-Contact Measurements of Surface Tension and Viscosity of Niobium, Zirconium, and Titanium Using an Electrostatic Levitation Furnace[END_REF]). In any case, our measurements can complete niobium surface tension literature, especially above 3 000 K. Only that last part agrees with literature [START_REF] Ozawa | Influence of Oxygen Partial Pressure on Surface Tension of Molten Type 304 and 316 Stainless Steels Measured by Oscillating Droplet Method Using Electromagnetic Levitation[END_REF] showing a linear decreasing behavior. One assumption to explain this strong difference is the duration of experiments under a mixture of argon and dihydrogen atmosphere. Dihydrogen is used for its ability to capture oxygen present in the atmosphere and at the surface of the liquid naturally present in the sample. Our experiments last less than 10 s to keep the chemical composition of the alloy. So results presented are close to the welding and additive manufacturing processes in terms of time and gas environment. This behavior, increasing up to 1 900 K and then decreasing is classically seen for Fe-O and Fe-S couples [START_REF] Sahoo | Surface tension of binary metal-surface active solute systems under conditions relevant to welding metallurgy[END_REF] and commonly used for not deoxidized steels like in [START_REF] Ozawa | Influence of Oxygen Partial Pressure on Surface Tension of Molten Type 304 and 316 Stainless Steels Measured by Oscillating Droplet Method Using Electromagnetic Levitation[END_REF].

Conclusion

This paper presented the measurement of the surface tension of iron, niobium, and 304L stainless steel using Rayleigh's equation. This equation relies the surface tension with the resonance frequency. This latter parameter has been experimentally measured with the help of the oscillating drop method. One of the challenges that has been faced is the difficulty to determine it on the spectrum obtained from the oscillating drop because of its unpredictable rotation. The proposed solution has been to blow a gas jet on the side of the drop to control its rotation along the vertical axis. In result, the spectrum displays only one peak considered representative of the resonance frequency, which makes the analysis straightforward. This raises questions about the effect of deformation of the drop induced by the rotation, since an overestimation of surface tension with strong deformations has been observed.

Awaiting further analysis, we performed surface tension measurements with the slowest rotation speed reachable.

A numerical study of levitated drops has been performed to analyze the effect of drop mass on the calculus of the surface tension. It appears that for light mass drops (less than 200 mg), it does not have a significant impact, but for heavier ones, both mode 3 oscillations and gravity should be considered for a more precise surface tension calculus.

Our results on iron show quite good agreement with literature. The difference (concerning mostly the slope) stems mostly from the initial purity of the sample. The results on niobium can complete the literature, which lacks of data on this material. Finally, the results on 304L stainless steel show an increasing-decreasing behavior, unlike literature which is linearly decreasing. Indeed, our time experiments are much shorter and thus, closer to industrial processes conditions using this material.

Fig. 1

 1 Fig. 1 Side view of spherical drop with exhibiting oscillations at the characteristic frequencies of mode 2.

Fig. 2

 2 Fig. 2 Scheme and photography of the experimental device. Only elements A, C, D, E, and G are visible on the photography.

Fig. 3

 3 Fig.3Spectra obtained with a FFT on the variation of the radii over frequency. The amplitude of the spectra is normalized to 1 and shifted for clarity.

Fig. 4

 4 Fig. 4 Needle positioned to control the rotation of the levitated drop.

Fig. 5

 5 Fig. 5 Calculated surface tension as function of the shape ratio. Radius rx is the horizontal one and radius ry the vertical one.

Fig. 7

 7 Fig. 7 Geometry and mesh of the model with the example of a 30 mg drop and the nozzle corresponding to the one used experimentally. Homogeneous elements of 60 µm.

Fig. 8

 8 Fig. 8 Spectra comparison between an experiment and the corresponding model (same mass and surface tension).

Fig. 9

 9 Fig. 9 Diameters and height of a 515 mg drop subject to oscillations.

Fig. 10

 10 Fig. 10 Diameters and height of a 30 mg drop subject to oscillations.

Fig. 11 FFT

 11 Fig. 11 FFT spectra obtained for different drop masses.

Fig. 12

 12 Fig. 12 Frequency shift as function of the drop mass. The frequency shift is defined as the ratio of numerically obtained frequency to the theoretical frequency.

Fig. 14 15 .

 1415 Fig. 14 Surface tension of niobium. Dotted line is the solidification temperature (2 750 K). Literature [31]-[34].

Fig. 15

 15 Fig.[START_REF] Dubberstein | Surface tension and density data for Fe-Cr-Mo, Fe-Cr-Ni, and Fe-Cr-Mn-Ni steels[END_REF] Surface tension of 304L stainless steel. Dotted line is the solidification temperature (1 694 K). Literature[START_REF] Ozawa | Influence of Oxygen Partial Pressure on Surface Tension of Molten Type 304 and 316 Stainless Steels Measured by Oscillating Droplet Method Using Electromagnetic Levitation[END_REF].

Results

Uncertainties

According to Rayleigh's equation, the surface tension measurement depends on two parameters, the resonance frequency and the mass of the drop. The uncertainty is thus given by Eq. ( 9): Eq. ( 9)

Main uncertainty sources for the determination of these two parameters are the pollution of the sample, the effect of external forces such as gravity, the presence of other oscillation modes than mode 2, the mass measurement, the determination of the resonance frequency on the spectrum, and the resonance frequency shift due to oscillations amplitude.

A special attention has been paid to avoid the pollution of the samples. The oxygen level has been lowered to less than 1 ppm (sealing of the enclosure, use of hydrogen to capture oxygen). However, pollution can still come from the impurities initially present in the sample. The nature of these impurities being unknown, the impact on the measurement of the surface tension are not quantifiable, especially for steels, which composition already includes many elements in variable quantities depending on the supplier and the casting quality. As explained in Section 3.3, the gravity and the presence of other oscillation modes do not seem to have significant impact. The uncertainty on the measurement of the mass is given by the resolution of the balance (± 5 µg): the average mass of a sample (30 mg) giving Δm/m = ± 0.02 %. The uncertainty on the resonance frequency of the spectrum is due to the sharpness of the peak: its precision holds within ± 1 Hz in the worst case, giving Δv r /v r = ± 0.45 % for a 220 Hz resonance frequency. These quantifiable uncertainties give Δγ/γ = ± 0.9 %.

The uncertainty on the temperature measurement is difficult to estimate considering the lack of data in literature concerning the evolution of emissivity of liquid metal with temperature. Therefore, the temperature uncertainty is not evaluated here.

Surface tension results

The method for surface tension measurement has been applied on two pure metals (iron and niobium) and on an alloy (304L stainless steel). Iron has been widely studied and thus is an appropriate material for comparison with literature. On the opposite, niobium and 304L steel are not very documented, especially at high temperature. Our measurements will therefore complete the database on those materials. The highest temperature able to be reached with our experimental device is limited by heat losses mostly due to evaporation. Also, it is difficult to go down to low temperatures (close or below solidification point) because of the lack of temperature regulation when these measurements were

Conflict of interest

The authors declare no conflicts of interest.


Pedro Ramaciotti Morales
email: pedro.ramaciottimorales@sciencespo.fr

Multidimensional online American politics: Mining emergent social cleavages in social graphs

Keywords: social graphs, graph embedding, network homophily, ideological scaling, ideologies, polarization

Dysfunctions in online social networks (e.g., echo chambers or filter bubbles) are studied by characterizing the opinion of users, for example, as Democrat-or Republican-leaning, or in continuous scales ranging from most liberal to most conservative. Recent studies have stressed the need for studying these phenomena in complex social networks in additional dimensions of social cleavage, including anti-elite polarization and attitudes towards changing cultural issues. The study of social networks in high-dimensional opinion spaces remains challenging in settings such as that of the US, both because of the dominance of a principal liberal-conservative cleavage, and because two-party political systems structure preferences of users and the tools to measure them. This article builds on embedding of social graphs in multi-dimensional ideological spaces and NLP methods to identify additional cleavage dimensions linked to cultural, policy, social, and ideological groups and preferences. Using Twitter social graph data I infer the political stance of nearly 2 million users connected to the political debate in the US for several issue dimensions of public debate. The proposed method shows that it is possible to identify several dimensions structuring social graphs, non-aligned to liberal-conservative divides and related to new emergent social cleavages. These results also shed a new light on ideological scaling methods gaining attention in many disciplines, allowing to identify and test the nature of spatial dimensions mined on social graphs.

Introduction

The study of socio-political dysfunctions or disorders unfolding in digital social media and social networks [START_REF] Benkler | Network propaganda: Manipulation, disinformation, and radicalization in American politics[END_REF] has raised to prominence in the past decade, including studies of algorithmic bias [START_REF] Bakshy | Exposure to ideologically diverse news and opinion on facebook[END_REF], extremism [START_REF] O'callaghan | Down the (white) rabbit hole: The extreme right and online recommender systems[END_REF], or echo chambers [START_REF] Barberá | Tweeting from left to right: Is online political communication more than an echo chamber?[END_REF]. These studies hinge on ontologies for the political positions or stances of online users. Bakshy et al. [START_REF] Bakshy | Exposure to ideologically diverse news and opinion on facebook[END_REF], for example, classified users and content on Facebook as Democrat-or Republican-leaning to analyze cross-cutting recommendations, and Barbera et al. [START_REF] Barberá | Tweeting from left to right: Is online political communication more than an echo chamber?[END_REF] positioned Twitter users on liberal-to-conservative continuous scales to investigate the so-called echo chambers. These categorical (e.g., user classification) or geometrical (e.g., opinion scales) have been leveraged in several studies. In many settings, including that of the US, categorical approaches are limited to binary classifications, while geometrical ones are often limited to onedimensional opinion scales. This stems from the fact that the US is in essence a two-party system, and that the US has undergone several decades of issue alignment [START_REF] Poole | Congress: A political-economic history of roll call voting[END_REF]: the alignment of issue positions on several matters, resulting in highly-correlated views on gun control, abortion, racial issues, among several others [START_REF] Mason | i disrespectfully agree": The differential effects of partisan sorting on social and issue polarization[END_REF]. Methods for embedding social networks in higher dimensional opinion spaces (with dimensions standing for opinion indicators for different issues of social cleavage) have been developed only recently. These methods allow to tackle, for example, several European socio-political settings, known to require higher dimensionality [START_REF] Bakker | Complexity in the european party space: Exploring dimensionality with experts[END_REF] to account for observed social choice data, from roll call voting [START_REF] Cointet | What colours are the yellow vests? an ideological scaling of facebook groups[END_REF] to online social network activity [START_REF] Morales | Unfolding the dimensionality structure of social networks in ideological embeddings[END_REF]. Accordingly, recent methods have proposed to embed online social graphs in empirical geometrical spaces of several dimensions, where dimensions stand for indicators of opinions on immigration, income redistribution, or ideological stances such as left-and right-wing positions [START_REF] Morales | Inferring attitudinal spaces in social networks[END_REF]. In settings such as that of the US and that display a main dominant dimension of social cleavage, there is comparatively less development on methods to mine additional dimensions. In this context, cleavages refers to historically-determined political, social or cultural dimensions dividing society in the sense that individuals into groups that structure political debate [START_REF] Lipset | Cleavage structures, party systems, and voter alignments: an introduction[END_REF]. The study of online social networks in higher opinion dimensions calls nonetheless for renewed efforts, as recent research highlights the importance of new additional cleavages, in particular relating to anti-elite sentiment and populism, as driving forces in many relevant political and informational phenomena in the US [START_REF] Uscinski | American politics in two dimensions: Partisan and ideological identities versus anti-establishment orientations[END_REF]. This article builds on recent ideological scaling [START_REF] Imai | Fast estimation of ideal points with massive data[END_REF] and aforementioned graph embedding methods for spatializing social graphs in multi-dimensional ideological spaces [START_REF] Morales | Inferring attitudinal spaces in social networks[END_REF]. Exploiting graph embedding and NLP methods, it proposes a method to mine cleavage dimensions linked to cultural, policy, social, and ideological groups and preferences in social graphs. The method is applied to Twitter social graph data of nearly 2 million users strongly connected to the online political debate in the US. The proposed method shows that several cleavage dimensions traditionally considered in social network analysis (e.g., conservatism, gun control, patriotism, religion) are indeed strongly related, as most studies find, but further allows to quantitatively measure the relative alignment of these issues. More importantly, the method identifies and computes positions of large numbers of users in emerging, non-aligned dimensions identified in political science research as emerging cleavages, and that several works in social science have hypothesized and put forward in recent years. These results extend previous research works showing that it is possible to spatialize users in the US in several space dimensions that are not highly correlated. These results also shed light on the ubiquitous practice of analyzing social networks with ideological scaling methods, by showing ways to assess its validity and improving existing methods. While traditional ideological scaling methods yield a scale that is highly correlated with Democrat-Republican cleavages (resulting from an issue align-ment process), party cleavage in specifically can be more accurately determined among many other aligned cleavages.

Related Work

Binary categorical classification of social media and network users counts numerous works, from exploitation of self-reporting and surveys [START_REF] Bakshy | Exposure to ideologically diverse news and opinion on facebook[END_REF] to sophisticated methods using neural networks on heterogeneous graphs [START_REF] Xiao | Timme: Twitter ideology-detection via multi-task multi-relational embedding[END_REF]. More related to this article, ideological spatialization of relational behavioral data can be traced back at least to the Nominal Three-Step Estimation (NOMINATE) method [START_REF] Poole | A spatial model for legislative roll call analysis[END_REF]. The NOMINATE method is used to position parliamentarians and bills in one-dimensional liberal-conservative scales. It assumes 1) the existence of unobservable geometrical ideological parameters or positions, and 2) an underlying homophilic generative process (i.e., parliamentarians vote to approve bills that are ideologically close to them in the unobservable space), and uses roll call data in Bayesian inference of the unobservable parameters. This process, called ideological scaling, has been used in numerous settings involving social choice: court rulings, campaign donations, parliamentary vote, and social network activity among others [START_REF] Imai | Fast estimation of ideal points with massive data[END_REF]. In its most wide-spread form, the homophilic probabilistic generative process is modeled as [START_REF] Barberá | Birds of the same feather tweet together: Bayesian ideal point estimation using twitter data[END_REF]:

P (i ← j|α, β, φ i , φ j ) = logit -1 α -β φ i -φ j 2 , (1)
where the probability of observing user j interacting with user i (i.e., i ← j) depends on position and scale parameters α and β, and, most importantly, on the distance between the unobservable positions φ i and φ j of users i and j. Social choice data (i.e., pairs i ← j) can then be used to infer positions φ i for all users i. A large number of works assumes that one dimension is enough to retrieve the main social cleavage in the US, implicitly assuming it is the liberal-conservative one, and uses social network choice data (e.g., Facebook likes [START_REF] Bond | Quantifying social media's political space: Estimating ideology from publicly revealed preferences on facebook[END_REF] or Twitter following link [START_REF] Barberá | Birds of the same feather tweet together: Bayesian ideal point estimation using twitter data[END_REF]) to compute the position of users in this dimension. These works often rely on ex post validation using text cues to argue that the latent dimension indeed represents the political positions of users. Multi-dimensional inference for φ i can be achieved in a computationally-tractable manner with Correspondence Analysis [START_REF] Greenacre | Correspondence analysis in practice[END_REF] ,as it has been shown to approximate the inference of unobservable parameters of (1), both theoretically [START_REF] Lowe | Understanding wordscores[END_REF] and empirically [START_REF] Barberá | Tweeting from left to right: Is online political communication more than an echo chamber?[END_REF].

Because (1) depends on unobservable parameters φ through pairwise distances, their inference is invariant to isometric transformations. In particular rotation transformations mean that retrieved dimensions cannot be assured to be aligned with social cleavages that might be structuring social choice. In European settings, using the position of referential users such as politicians of known political parties, and party positions in reference issue spaces (provided, e.g., by political polls or surveys), it has been shown how inferred dimensions display different levels of alignment with issues of public political debate [START_REF] Morales | Unfolding the dimensionality structure of social networks in ideological embeddings[END_REF]. This means that, in general, it cannot be assured that a single-dimensional ideological scaling model will yield a political opinion scale completely aligned with some presumed main left-right or liberal-conservative cleavage. Using the position of several political parties, this fact has been leveraged in embedding large numbers of users in multi-dimensional space where dimensions stand for identifiable and separate political issues, not requiring ex post interpretation or validation [START_REF] Morales | Inferring attitudinal spaces in social networks[END_REF]. Because the US is in essence a two-party system, and because of the difficulty in obtaining reference users that have both 1) known positions φ in latent space and 2) positions known through political surveys (in particular for issues that might not be highly correlated with liberal-conservative cleavages), similar methods have proved to be more challenging in this setting. This article thus proposes a method for constructing such referential positions for large numbers of users, relating to hypothesized cleavages, based on text descriptions written by users in their online profiles on social network sites. Using both, multidimensional scaling and embedding based on (1), and NLP methods for constructing groups of referential users, this article proposes a method to mine several spatial opinion dimensions, and most importantly, emergent opinions that are not highly-correlated with liberal-conservative cleavages.

Social Network Data

Following multidimensional ideological scaling works in Europe [START_REF] Morales | Unfolding the dimensionality structure of social networks in ideological embeddings[END_REF] and in the US [START_REF] Barberá | Tweeting from left to right: Is online political communication more than an echo chamber?[END_REF], we select a bipartite sub-graph of the Twitter social graph. To capture online social choices that might be revealing of several social and political preferences we take as reference users members of the US Congress. We manually annotate the Twitter accounts of 550 members of the 116th United States Congress (looking for verified accounts corresponding to each congressperson), and collected their 17 952 824 followers (collection performed using Twitter's API in October 27th, 2020, see the Acknowledgements section for privacy-compliance information and references). To minimize the probability of followers being bots, we follow criteria adopted by several studies [START_REF] Morales | Unfolding the dimensionality structure of social networks in ideological embeddings[END_REF][START_REF] Morales | Measuring the accuracy of social network ideological embeddings using language models[END_REF][START_REF] Morales | Your most telling friends: Propagating latent ideological features on twitter using neighborhood coherence[END_REF][START_REF] Morales | Auditing the effect of social network recommendations on polarization in geometrical ideological spaces[END_REF] and further identified followers with more than 25 followers (7 325 940), and users that have posted more at least 100 tweets (7 471 365). See [START_REF] Barberá | Birds of the same feather tweet together: Bayesian ideal point estimation using twitter data[END_REF] for further details behind the rationale for these parameters. To identify users that are strongly connected to political debate and that follow spatial preference models we identify followers that follow at least three members of congress (3 846 925) [START_REF] Barberá | Birds of the same feather tweet together: Bayesian ideal point estimation using twitter data[END_REF]. We select the 1 821 272 unique followers that satisfy all three conditions.

To establish reference points in latent space, we collect the text self-descriptions made by users in their Twitter profiles (also on October 27th, 2020). Out of 1 821 272 users, 1 442 716 had written any text entry in their Twitter profiles. This collection, performed in the days leading to the 2020 US Presidential Election allows us to investigate cleavages in candidate preferences. The strategy presented in this article consists in finding space dimensions capable of dichotomizing pairs of groups of users identified with keywords used in their profiles, and revealing of social cleavages. We follow several studies to identify several possible cleavages of interest, including [START_REF] Poole | A spatial model for legislative roll call analysis[END_REF][START_REF] Baumann | Modeling echo chambers and polarization dynamics in social networks[END_REF][START_REF] Morales | Inferring attitudinal spaces in social networks[END_REF][START_REF] Uscinski | American politics in two dimensions: Partisan and ideological identities versus anti-establishment orientations[END_REF]: party, candidate, racial cleavages, cleavages in regional politics (urban vs rural, or state vs federal po-sitions), religious cleavages, cleavages relating to gun control, the long-standing issue of communism in the US, cleavages related to liberal "life-styles" [START_REF] Bakker | chapel hill expert survey[END_REF] (e.g., homosexuality, feminism), welfare, military, patriotism, globalization, conspirationism and mistrust (including mistrust of the establishment and experts), and cleavages around attitudes towards elites. Table 1 summarizes proposed issues of partition according to the surveyed literature, with the keywords identified for the classification of users in binary partitions. User classification relies on a keyword-based strategy of minimalist choice: keywords do not intend to portrait the diverse forms of expression of position on each issue, but rather to identify groups of users with low ambiguity of position. In addition to keywords, we rely on sentiment analysis of profile text to distinguish positive and negative mentions of keywords (using a pre-trained BERT base model for uncased words [START_REF] Devlin | Bert: Pre-training of deep bidirectional transformers for language understanding[END_REF]), assigning to each profile text a sentiment from 1 (very negative) to 5 (very positive). We label text profiles as negative [-] if sentiment is equal to 1, and as positive [+] if sentiment is equal to 5. In Table 1 we also distinguish users whose profiles are not negative [ -]. This is needed, for example, to identify users that might use the word "republican" in their profiles, but in order to utter critique (e.g., "I hate republicans!").

Homophily Network Embedding

To identify dimensions that might be revealing of cleavages, we first produce a multi-dimensional space embedding in which these dimensions might emerge as directions. For this, we take the bipartite social subgraph of the 550 members of congress and their 1 821 272 followers to produce an homophily embedding using Correspondence Analysis on the adjacency matrix to compute values φ of (1) following the procedure in [START_REF] Morales | Unfolding the dimensionality structure of social networks in ideological embeddings[END_REF]Section IV]. In this multi-dimensional space, dimensions δ i (i = 1, 2, ...) are ranked according to the information they contain about choices represented in the bipartite social graph, as measured by the inertia. Fig. 1 shows the inertia of each dimension and their relative gain, showing that at most the three first dimensions are relatively more informative than the rest. Fig. 1 also shows the embedding positions of both, congressional members and followers, and the marginal density on these first three dimensions, estimated for the purposes of visualization with kernel density estimation. We can compute party positions as the mean position of congressional members. As expected, the first -most explicative-dimension, δ 1 , stands as a good candidate of dimension for attitudes towards parties (or related cleavages, such as liberal-conservative). However, the question remains whether isometric transformations can improve the distinction made by any dimension between Democrat-and Republicanleaning followers. In particular, we do not know if a rotation might improve the ability of a classifier to distinguish between Democrat and Republicans on the first dimension. We know that δ 1 stands for a latent cleavage, and we know that it is highly aligned with Democrat-Republican cleavages, but we do not know if it is the best spatial direction for distinguishing these two groups. More broadly, Anti-elitism Elite "ceo" OR "founder" OR "representative" OR "expert" OR "phd" OR "professor" 47 032

Anti-Elite "people" OR "elite" OR "bankers" OR "politicians" OR "millionaires" OR "experts" 38 086

Table 1. Summary of the proposed issue partitions of users into minimal groups for mining spatial cleavage dimensions capable of classifying them. For each issue we identify two disjoint groups defined by queries of the Twitter profile text descriptions, including keywords (case insensitive, here all written in lowercase), and sentiment:

positive [+], negative [-], and non-negative [ & -].
it is not trivial to attribute an inductive meaning to what δ 2 and δ 3 might stand for, or to any other space direction for that matter.

Mining Cleavage Dimensions

To extract cleavage dimensions as directions in this latent ideological space, we leverage our proposed set of binary labels from Section 3. We first illustrate this on party cleavages. Among users our users, 7 895 use the word "republican" and 14 481 the word "democrat" in their profile without negative sentiment. To measure the degree to which δ 1 , δ 2 or δ 3 might be good candidate directions for distinguishing these two groups, we fit a logistic regression model on each dimension. With this model, we can compute users that might be rightly or wrongly classified with it, and thus compute a precision, recall and F1-score metric. Fig. 2 shows these values and the distribution of these two groups along δ 1 , δ 2 and δ 3 , showing that δ 1 indeed is the only dimension among the three to produce a meaningful distinction. Alternatively, instead of using a given dimension, we can fit a multivariate logistic regression model, and identify the direction perpendicular to the decision hyper-plane boundary. In the case of our three-dimensional model, the decision boundary will be a plane and the direction a three-dimensional vector (see in Fig. 2). This discovered direction separating these two groups of users is well aligned with δ 1 , but it does not produce an improvement in the F1-score. Still, ideological scaling cannot rely on the a priori assumption that this will always be the case as it is standard practice in many works in several disciplines, especially in light of research suggesting a decline in left-right cleavages structuring social choice [START_REF] Grossman | Economic internationalization and the decline of the left-right dimension[END_REF][START_REF] Morales | Unfolding the dimensionality structure of social networks in ideological embeddings[END_REF][START_REF] Uscinski | American politics in two dimensions: Partisan and ideological identities versus anti-establishment orientations[END_REF]. This method can be further used to test all proposed binary groups of Table 1, hypothesized to represent potentially relevant cleavages. We fit a multivariate logistic regression model for each pair of groups, and measure the classification accuracy of the model, reported in Table 2, highlighting the cases with F1-score accuracy equal or greater to 0.6. As some pairs are highly imbalanced (e.g., for religious cleavages), we systematically sub-sample the majority group with a Near-Miss strategy [START_REF] Mani | knn approach to unbalanced data distributions: a case study involving information extraction[END_REF]. Fig. 3 shows a selection of groups of pairs of labeled users, with the decision boundary and discovered orthogonal direction of the fitted multivariate decision model. This selection highlights the different qualities in the accuracy of the multivariate logistic regression classifier, corresponding to different strengths of cleavages for the pairs in each labeled group, under the assumption that the chosen criteria identify a relevant group of users. The identification of different plausible dimensions of social cleavage presents us with several interesting possibilities. First, the coordinate of each user along these new identified directions can be used to disentangle issue alignment: e.g., we can produce separate and explicit dimensions for party cleavages and attitudes towards gun control. In contrast, δ 1 (usually reported in works using ideological scaling) is a proxy for party cleavages, but also for other positions on correlated issues (e.g., racial or religious issues, see Fig. 3) in non-explicit ways. Similarly, by inspecting the alignment between different cleavage dimensions we can identify and quantify this issue polarization. Fig. 4 shows the mined cleavage dimensions (i.e., with F1-score ≥ 0.6) and their pairwise angular distance, with metrical clusters computed with Un-weighted Pair Group Method with Arithmetic (UPGMA) mean [START_REF] Sokal | A statistical method for evaluating systematic relationships[END_REF]. The mined cleavages can be organized in four ideologies (in the sense of issue alignment): 1) a dominant ideology comprising party, candidate, and other stances correlated with δ 1 , 2) an ideology separating people defining themselves using the words "local" and "global", 3) an ideology separating those defining themselves using the words "welfare" and "libertarian", and 4) an ideology separating those with positive and negative mentions of issues relating to sexual diversity and feminism, and the use of the word "communism". Four directions cannot be perfectly orthogonal in three-dimensions, but any two cleavages belonging to two different identified ideological groups will display enough angular distance, so as to not be considered as highly correlated. Being able to disentangle issues in separate dimensions, allows us to conduct different investigations against the map positions of actors in now identifiable axes. Because we can also measure the position of reference users (politicians) in mined cleavage dimensions, we can investigate intra-party diversity on separate issues: e.g., of support for their presidential candidate, or attitudes towards welfare, religious diversity, or diversity of views on racial issues. Fig. 5 shows, for example, that Republicans are more heterogeneous in their support for Donald Trump than the Democrats in their support for Joseph Biden, both the members of congress (in crosses in Fig. 5) and the followers (density shown in light blue in Fig. 5).

Discussion and conclusions

This article argued for the importance of re-examining the assumptions implicitly leveraged by studies that use ideological scaling, showing that the dominant latent dimensions cannot be assumed to be the direction that is most aligned with traditional Democrat-Republican cleavages in the US. The degree of alignment between latent dimension and different cleavages can and must be determined and measured. This article further presented a way of mining dimensions of social cleavage with explicit meaning using both network embedding and NLP methods. Furthermore, using this combination of methods, this article analyzed the case of a political Twitter friend network in the US, identifying the main dominant cleavage, but also additional ones hypothesized as relevant by recent studies in social sciences [START_REF] Uscinski | American politics in two dimensions: Partisan and ideological identities versus anti-establishment orientations[END_REF]. In particular, four ideologies, or bundled groups of cleavage dimensions were identified. These groups of dimensions are not highly aligned among them, and represent new cleavage dimensions that can be used in further studies. This method also offers the possibility of developing new applications for explicitly measuring issue polarization as the alignment of bundled social cleavages, as well as a method for projecting large numbers of users onto space dimensions with explicit meaning in terms of the issues to which it measures positive and negative views. This new possibility, it was argued, opens interesting paths for research, which was illustrated with a brief example. By measuring positions of Democrat and Republican congressional members on both, party cleavage and candidate cleavage dimensions with data collected in the days leading to the 2020 US presidential election, this article showed that, when compared with Democrats, it may be proved that Republicans display higher heterogeneity in their support for their candidate. Beyond this example, many others could leverage these results and methods. In particular, having multidimensional distributions of political attitudes could be leveraged in the study of social mobilization by analyzing the determinants and characteristics of political movements (see for example [START_REF] Cointet | What colours are the yellow vests? an ideological scaling of facebook groups[END_REF][START_REF] Morales | Atlas multi-plateformes d'un mouvement social: Le cas des gilets jaunes[END_REF][START_REF] Morales | Posters and protesters[END_REF]). Additionally, by leveraging information consumption practices and media diets, attitudinal positions could be attributed to news media articles and outlets, allowing for the study of diversity, or lack thereof, in information consumption patterns [START_REF] Morales | Role of the website structure in the diversity of browsing behaviors[END_REF][START_REF] Morales | Measuring diversity in heterogeneous information networks[END_REF]. This, in turn, presents interesting possibilities for large-scale analysis of wide news and informational ecosystems [START_REF] Cointet | Uncovering the structure of the french media ecosystem[END_REF].

Fig. 1 .

 1 Fig. 1. Multi-dimensional homophily embedding of the collected Twitter network. Dimensions ranked by inertia, and relative gain of including each dimension (top left). Scatter plot and estimated density of the conditional probabilities for the position of users in first three dimensions (top right). Density of followers and positions of members of congress colored by party, and party positions as mean of groups shown in first three dimensions (bottom).

Fig. 2 .

 2 Fig. 2. Distribution Republican-or Democrat-leaning according to their Twitter text profile description, their distribution along the first 3 latent space dimensions, and the accuracy of logistic regression models fitted on each dimension (left). Conditional distributions and positions of labeled users in three-dimensions, and distributions along the direction perpendicular to the boundary of a multivariate logistic regression (right).

Fig. 3 .

 3 Fig. 3. Selection of groups of pairs of labeled users, with the decision boundary and orthogonal direction of the fitted multivariate decision model.

Fig. 4 .

 4 Fig. 4. Mined cleavage dimensions (left) can be organized in four ideologies (in the sense of issue polarization), shown in four groups in blue in the angular distance matrix (right).

Fig. 5 .

 5 Fig. 5. Density of Twitter users and positions of members of congress along mined dimensions. The distribution of members of congress shows the intra-party diversity of stances towards presidential candidates and welfare.

Table 2 .

 2 Groups of pairs of labeled users (according to criteria of Table1), naming of the mined dimension perpendicular to the decision boundary of a multivariate logistic regression classification model, and the accuracy of the fitted model.

	Label 1	Label 2	Dimension	Precision	Recall	F1-score
	Democrat	Republican	d Dem-Rep	0.814	0.823	0.819
	Trump sup.	Biden sup.	d T rump-Biden	0.944	0.874	0.908
	Pro-BLM	Anti-BLM	d BLM	0.488	0.205	0.289
	BlackLivesMatter	BlueLivesMatter	d P olice	0.687	0.925	0.789
	Urban	Rural	d U rban	0.529	0.577	0.552
	State	Federal	d St-F ed	0.626	0.536	0.578
	Christian	Atheist	d Christian	0.706	0.559	0.624
	Pro-Gun	Anti-Gun	d P ro-Gun	0.713	0.544	0.617
	Pro-Communism	Anti-Communism	d Comm.	0.647	0.650	0.648
	Anti-Liberal LifeStyle Pro-Liberal LifeStyle d Lif eStyle	0.537	0.867	0.664
	Libertarian	Welfare	d W elf are	0.695	0.720	0.707
	Pro-Police	Anti-Police	d P olice	0.617	0.624	0.620
	Pro-Military	Anti-Military	d M ilitary	0.599	0.474	0.529
	Patriot	Anti-imperialist	d P atriot	0.709	0.692	0.700
	Global focus	Local focus	d Local	0.547	0.676	0.605
	DeepState	Rationalist	d DeepState	0.569	0.667	0.614
	Elite	Anti-Elite	d AntiElite	0.549	0.439	0.488

Acknowledgments

This work has been been funded by the "European Polarisation Observatory" (EPO) of the CIVICA Consortium, and by the French National Agency for Re-