Tongzhao Gong

Yun Chen
email: chenyun@imr.ac.cn

Shanshan Li

Yanfei Cao

Liyuan Hou

Dianzhong Li

Xing-Qiu Chen

Guillaume Reinhart

Henri Nguyen-Thi

Equiaxed dendritic growth in nearly isothermal conditions: A study combining in situ and real-time experiment with large-scale phase-field simulation

Keywords: equiaxed dendritic growth, polycrystalline solidification, phase-field method, large-scale simulation, synchrotron X-ray radiography

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Equiaxed dendritic growth in nearly isothermal

Introduction

The equiaxed dendrite is one of the most common microstructures formed during the solidification process of materials, and its morphology, size, and composition distribution are critical to mechanical properties of castings. A comprehensive and in-depth understanding of equiaxed dendritic growth is of vital importance to control the casting processing for optimizing the quality of final products [START_REF] Kurz | Progress in modelling solidification microstructures in metals and alloys: dendrites and cells from 1700 to 2000[END_REF][START_REF] Kurz | Progress in modelling solidification microstructures in metals and alloys. Part II: dendrites from 2001 to 2018[END_REF]. Many theoretical progresses have been gained on the steady-state growth of a dendrite growing freely during solidification of pure substances and alloys [START_REF] Pelce | Dendrites in the small undercooling limit[END_REF][START_REF] Ivantsov | Temperature field around a spherical, cylindrical, and needle-shaped crystal, growing in a pre-cooled melt Dok[END_REF][START_REF] Lipton | Dendritic growth into undercooled alloy melts[END_REF][START_REF] Lipton | Equiaxed dendrite growth in alloys at small supercooling[END_REF], even with considering the influence of melt convection [START_REF] Benamar | An exact solution for the shape of a crystal growing in a forced flow[END_REF][START_REF] Bouissou | Effect of a forced flow on dendritic growth[END_REF][START_REF] Wang | Equiaxed dendritic solidification with convection: Part I. Multiscale/multiphase modeling[END_REF][START_REF] Gandin | Boundary layer correlation for dendrite tip growth with fluid flow[END_REF][START_REF] Ananth | Self-consistent theory of dendritic growth with convection[END_REF]. Meanwhile, amounts of solidification experiments of transparent organic alloys [START_REF] Beckermann | Equiaxed dendritic solidification with convection: Part III. Comparisons with NH4Cl-H2O experiments[END_REF][START_REF] Li | Modeling of free dendritic growth of succinonitrileacetone alloys with thermosolutal melt convection[END_REF] were performed to validate these theories. However, exact prediction of the complex pattern formation during solidification is still a big challenge currently. This is because the transport of heat and mass, the melt convection, the crystal morphology, and the interaction among grains are too complex and highly nonlinear to be treated analytically.

The state-of-the-art numerical techniques, such as the quantitative phase-field (PF) method [START_REF] Karma | Atomistic to continuum modeling of solidification microstructures[END_REF], accompanied with the advanced in situ and real-time observation of solidification experiments by synchrotron X-ray radiography [START_REF] Akamatsu | In situ observation of solidification patterns in diffusive conditions[END_REF], have become powerful tools to investigate the microstructure evolution and the related nonlinear physics of metallic alloys during solidification. By means of synchrotron X-ray radiography, Bogno et al. [START_REF] Bogno | Growth and interaction of dendritic equiaxed grains: In situ characterization by synchrotron X-ray radiography[END_REF] and Becker et al. [START_REF] Becker | Free dendritic tip growth velocities measured in Al-Ge[END_REF] respectively studied the equiaxed dendritic growth of Al-10 wt.% Cu alloys and Al-24 at.% Ge alloys under the continuous cooling condition, and they found that equiaxed dendrite development was characterized by two successive growth regimes, namely the free growth and impinged growth, which were well reproduced in several recent quantitative PF simulations by Chen et al. [START_REF] Chen | Quantitative Phase-field Simulation of Dendritic Equiaxed Growth and Comparison with in Situ Observation on Al-4 wt.% Cu Alloy by Means of Synchrotron X-ray Radiography[END_REF], Qi et al. [START_REF] Qi | The Effect of Natural Convection on Equiaxed Dendritic Growth: Quantitative Phase-Field Simulation and Comparison with Synchrotron X-Ray Radiography Monitoring Data[END_REF] and Boukellal et al. [START_REF] Boukellal | Scaling laws governing the growth and interaction of equiaxed Al-Cu dendrites: A study combining experiments with phase-field simulations[END_REF]. However, those simulations were limited to a very small computational domain and consider only the cases of one or two grains because of the low computational efficiency of the quantitative PF simulations. It is thus time to unveil the collective behaviors of a large number of grains during solidification for metallic materials. Besides, the XRMON-SOL experiment [START_REF] Murphy | XRMON-SOL: Isothermal equiaxed solidification of a grain refined Al-20wt%Cu alloy[END_REF] was carried out by Murphy et al. to investigate the equiaxed dendrite growth under the influence of solutal interactions, then the XRMON-SOL experiment were modelled using a recently developed 3D mesoscopic envelope method [START_REF] Souhar | Three-dimensional mesoscopic modeling of equiaxed dendritic solidification of a binary alloy[END_REF][START_REF] Olmedilla | Quantitative 3D mesoscopic modeling of grain interactions during equiaxed dendritic solidification in a thin sample[END_REF], which provided a feasible scheme for large-scale simulation of solidification microstructure evolution. However, the branched dendritic structure could not be obtained by this method and therefore it is still necessary to perform quantitative PF simulations to accurately predict pattern formation and evolution of materials during solidification.

In the last few years, the quantitative large-scale PF simulations have been achieved [START_REF] Takaki | Primary arm array during directional solidification of a single-crystal binary alloy: Large-scale phase-field study[END_REF][START_REF] Gong | Fast simulations of a large number of crystals growth in centimeter-scale during alloy solidification via nonlinearly preconditioned quantitative phase-field formula[END_REF][START_REF] Sakane | Two-dimensional large-scale phase-field lattice Boltzmann simulation of polycrystalline equiaxed solidification with motion of a massive number of dendrites[END_REF], due to the rapid development of numerical methods and computer hardware. In this paper, the large-scale quantitative 2D PF simulations are carried out using the fast simulation schemes proposed in our previous study [START_REF] Gong | Fast simulations of a large number of crystals growth in centimeter-scale during alloy solidification via nonlinearly preconditioned quantitative phase-field formula[END_REF], to reproduce the in situ and real-time observed polycrystalline solidification experiments of Al-4 wt.% Cu alloys under the continuous cooling condition. The simulated equiaxed dendritic growth is directly compared with the synchrotron X-ray radiography monitoring data, and the effects of the cooling rate and the liquid gravity-driven natural convection on the equiaxed dendritic growth during solidification are also discussed. The aim of the present work is to realize the large-scale quantitative PF simulation with the number of grains and computational domain comparable to the real solidification experiment, and the fluid dynamics equations are extremely difficult to solve efficiently for such a large-scale quantitative 3D PF simulation.

Based on these two points, it is reasonable and necessary to carry out the 2D large-scale PF simulation that is relatively easier to implement as a first step before more realistic but challenging 3D simulations.

In situ and real-time observation of solidification experiments

The in situ and real-time observation of solidification experiments of Al-4 wt.% Cu alloys were implemented at ID19 beamline of the European Synchrotron Radiation Facility (ESRF) in Grenoble (France), and the detailed description of the experiment set-up and procedure was given in ref. [START_REF] Bogno | Growth and interaction of dendritic equiaxed grains: In situ characterization by synchrotron X-ray radiography[END_REF]. The thin samples (40 mm × 6 mm × 0.2 mm) of Al-4 wt.% Cu alloys were set vertically in a Bridgman furnace, and two heating elements were placed on the top and bottom of the sample to melt the sample in nearly isothermal conditions. During the experiment, the sample was first melted and maintained to a temperature above the liquidus temperature for several hours to homogenize the solute composition in liquid. Then solidification was triggered by simultaneously applying the same cooling rate on both heating elements. A 10-μm-camera optics was chosen to obtain a good compromise between a large field of view (5 mm × 12 mm) and a sufficient spatial resolution (pixel size 7.46 μm).

The time evolution of equiaxed dendritic microstructure of Al-4 wt.% Cu alloys in the solidification experiment at various cooling rates is shown in Fig. 1Fig. 1, in which the Al-rich solid crystals appear in white and the Cu-rich liquid is in darker grey after a special image processing [27] to improve the quality of images. As seen in Fig. 1Fig. 1, a great number of grains nucleated and grew from the melt during the continuous cooling, and then the equiaxed grains rapidly interacted with neighbors owing to the spatial constraint. A comprehensive study of the equiaxed dendritic growth with the same experimental procedure but on Al-10 wt.% Cu alloys had been reported [START_REF] Bogno | Growth and interaction of dendritic equiaxed grains: In situ characterization by synchrotron X-ray radiography[END_REF], and the main conclusion was that the equiaxed dendritic growth experienced two regimes: an accelerating regime corresponding to the stage when a growing dendrite arm was isolated from its neighbors, and a decelerating regime where the solute interaction had a suppression effect. Furthermore, from Fig. 1Fig. 1 it could also be seen that with the cooling rate increasing, the number of grains in the experiments will increase and the secondary dendritic arm spacing (SDAS) will decrease, thus leading to a refined equiaxed dendritic microstructure. in our previous study [START_REF] Chen | Quantitative Phase-field Simulation of Dendritic Equiaxed Growth and Comparison with in Situ Observation on Al-4 wt.% Cu Alloy by Means of Synchrotron X-ray Radiography[END_REF]. However, compared with ref. [START_REF] Chen | Quantitative Phase-field Simulation of Dendritic Equiaxed Growth and Comparison with in Situ Observation on Al-4 wt.% Cu Alloy by Means of Synchrotron X-ray Radiography[END_REF], the 2D simulations presented in this paper have several improvements. The computational domain size (4233 μm × 7408 μm) in ref. [START_REF] Chen | Quantitative Phase-field Simulation of Dendritic Equiaxed Growth and Comparison with in Situ Observation on Al-4 wt.% Cu Alloy by Means of Synchrotron X-ray Radiography[END_REF] was comparable to the sample size shown in Fig. 1Fig. 1(b), but only one single isolated equiaxed crystal and two grains growing opposite each other were considered, which was not able to unveil the collective behavior of multi-grain growth. Moreover, the gravity-driven natural convection was not considered in our previous PF simulation [START_REF] Chen | Quantitative Phase-field Simulation of Dendritic Equiaxed Growth and Comparison with in Situ Observation on Al-4 wt.% Cu Alloy by Means of Synchrotron X-ray Radiography[END_REF]. Therefore, the experiment (E2), accompanied by other two experiments (E1 and E3) with different cooling rates of Al-4 wt.% Cu alloy which is suitable for PF simulations owing to the low composition, is reused in the present work to give a clear picture of multi-grain growth during solidification. The reason why the PF model is only suitable for dilute alloys with a low solute concentration is that the Gibbs free energies of both liquid and solid phases are calculated using the regular solution model, in which the high order terms in the excess free energy are ignored [START_REF] Echebarria | Quantitative phase-field model of alloy solidification[END_REF].

Phase-field model and numerical implementations

Phase-field model

The quantitative PF model presented in ref. [START_REF] Echebarria | Quantitative phase-field model of alloy solidification[END_REF] is adopted to model solidification of dilute binary alloys under continuous cooling condition, and the quantitative capability of this model to reproduce the time evolution of microstructure during alloy solidification has been demonstrated in many previous studies [START_REF] Boukellal | Scaling laws governing the growth and interaction of equiaxed Al-Cu dendrites: A study combining experiments with phase-field simulations[END_REF][START_REF] Chen | Quantitatively comparing phase-field modeling with direct real time observation by synchrotron X-ray radiography of the initial transient during directional solidification of an Al-Cu alloy[END_REF][START_REF] Chen | Tip-splitting instability and transition to seaweed growth during alloy solidification in anisotropically preferred growth direction[END_REF][START_REF] Clarke | Microstructure selection in thin-sample directional solidification of an Al-Cu alloy: In situ X-ray imaging and phase-field simulations[END_REF]. The governing equations of the phase field ϕ and the rescaled solute concentration U are respectively given as

( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 c , 0 2 3 2 c 0 1 1 1 i i x y i W Rt WW m C t Rt U k m C          =      + =    +             + -- - -  -   n n n n (1)
( ) ( ) ( ) ( ) ( ) S L L 0 1 1 1 1 2 2 2 11 11 2 22 k k k D U DU tD W k U kU tt        + -- +  - =   +         +-  +-      +  +       (2) with a mis en form ( ) ( ) ( ) ( ) ( ) 0 2 / 1 1 1 1 1 C C k k U k k k   -+ --   = - + --   (3)
where C is the local solute concentration, C0 is the initial concentration, k is the solute partition coefficient, Rc is the cooling rate, and m is the liquidus slope. The solid and liquid phases are represented by ϕ = +1 and ϕ = -1, respectively. The spatial length and time are rescaled respectively by the interface width W0 and relaxation time τ0, and therefore the dimensionless value for the solute diffusion coefficient in liquid is

L 0 0 12 2 00 DW D a a Wd  == % (4)
where d0 = Γ/[|m|C0(1-k)] is the chemical capillary length with Γ the Gibbs-Thomson coefficient, a1 = 5 2 / 8 and a2 = 47/75 [START_REF] Echebarria | Quantitative phase-field model of alloy solidification[END_REF]. It should be noted that for Al-Cu alloys, although the solute diffusion in solid is much slower than that in liquid (the solid diffusivity is about 3 orders of magnitude smaller than the liquid diffusivity), a small solid diffusivity can be taken into account, as indicted in the original derivation of the model in ref. [START_REF] Karma | Phase-field formulation for quantitative modeling of alloy solidification[END_REF]. And this way to model the real solute diffusion in solid was also employed in our previous study [START_REF] Chen | Quantitatively comparing phase-field modeling with direct real time observation by synchrotron X-ray radiography of the initial transient during directional solidification of an Al-Cu alloy[END_REF], in which DS in Eq. (2)(2) is the solute diffusion coefficient in solid. The coupling coefficient λ in Eq. (1)(1) is given as λ = a1W0/d0.

The standard form of the fourfold symmetry of the surface energy anisotropy is considered, and thus

W(n) = W0as(n) and τ(n) = τ0as(n) 2 where ( ) ( ) 4 s 4 4 4 , 1 3
4 i i x y a    =  = - +   n (5)
In order to expedite quantitative large-scale PF simulations, a nonlinear preconditioning [START_REF] Glasner | Nonlinear Preconditioning for Diffuse Interfaces[END_REF] of the phase field ϕ is employed to allow coarser grids to be used in the interface layer. The transformed phase field ψ is given as

tanh 2    =   (6)
( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 c 0 s 0 s 0 2 s 2 0s , 2 c 0 1 2 tanh 2 2 tanh 2 2 1 tanh 1 2 i i x y i Rt a W a m C t a Wa Rt U k m C          =     + =  -             +   +         - - -    -     nn n n (7) ( )   2 S 0 2 L L (1 ) (1 ) tanh 2 2 1 tanh 1 11 2 1 (1 ) 2 2 2 2 1 tanh 1 (1 ) 2 2 2 kk U t kD D U W k U Dt kU t          + --   - + -   =   +  + + -  - + -  +                     (8)
The numerical accuracy and efficiency of the preconditioned PF formula has been examined previously [START_REF] Gong | Fast simulations of a large number of crystals growth in centimeter-scale during alloy solidification via nonlinearly preconditioned quantitative phase-field formula[END_REF]. In the PF simulations, as the nucleation undercoolings of grains are neglected, all the grains nucleated simultaneously at the beginning of solidification (at the liquidus temperature), and the number, site, and orientation of the grains in simulations are measured directly from the radiography images in Fig. 1Fig.

1 for various cooling rates.

Numerical implementations

The parallel adaptive finite element method, with multiple processors using distributed memory based on MPI protocol, is employed to solve the preconditioned PF model. Numerical implementations are realized by the open source deal.II library [START_REF] Bangerth | The deal.II Library, Version 8.4[END_REF], and the adaptive mesh refinement/coarsening is achieved through a simple but efficient approach [START_REF] Gong | Fast simulations of a large number of crystals growth in centimeter-scale during alloy solidification via nonlinearly preconditioned quantitative phase-field formula[END_REF]. The front-tracking method [START_REF] Gong | Fast simulations of a large number of crystals growth in centimeter-scale during alloy solidification via nonlinearly preconditioned quantitative phase-field formula[END_REF] is also used to capture crystallographic orientations to improve computing efficiency dramatically. The computational domain size is 5.0 mm × 5.6 mm (marked by the red boxes in Fig. 1Fig.

1), which is comparable with the spatial scale of the sample in the solidification experiment. Therefore, ξ = 49.96 was chosen in the present work to obtain a reasonable compromise between the quantitative simulated results and the computational efficiency. Thus, using this interface width parameter, the corresponding simulation domain size in dimensionless is 5000 × 5600. It should be noticed that for the other cases with lower cooling rates, the choice of ξ = 49.96 is small enough owing to the relatively lower crystal growth rate. The material and numerical parameters are listed in Table 1Table 1.

All the simulations were implemented on a supercomputer with two AMD ® EPYC 7452 (2.35 GHz) CPUs and 256 GB memory at each computing node. For the case with the largest number of grains (46 grains and Rc = 1.5 K/min), it took about 60 hours to finish the 2D PF simulation using 128 cores until the solid fraction got close to that in the last radiography image (fS = 0.6) recorded in the corresponding solidification experiment. For the case with Rc = 0.2 K/min, although the number of grains was reduced to 15, the computing time until the solid fraction reached 0.525 was about 150 hours using 128 cores owing to the much slower cooling rate.

Results and discussions

Equiaxed dendritic morphology

The equiaxed dendritic microstructure in a local domain selected from the solidification experiments at various cooling rates (E1 to E3) and the corresponding 2D PF simulations (S1 to S3) are shown in Fig. 3Fig. 3. Overall, the simulated equiaxed dendritic microstructures are in a reasonably good agreement with the radiography images, except that the primary dendritic arms in the simulations are generally much thicker than those in the experiments. The difference in the thickness of the primary dendritic arms could be attributed to the different equiaxed dendritic growth dynamics. As indicated in our previous study [START_REF] Gong | Quantitative comparison of dendritic growth under forced flow between 2D and 3D phase-field simulation[END_REF], the tip velocity of the primary dendritic arm in the 2D PF simulations is 1~2 orders of magnitude slower than that in the 3D PF simulations. In practice, the crystal growth in the experiments is in a 3D space despite of the thin-sample configuration. This can be understood by comparing the typical dendrite tip radius to the sample thickness. Consequently, the crystal growth rate in the 2D PF simulations is much slower than that in the experiments [START_REF] Chen | Quantitative Phase-field Simulation of Dendritic Equiaxed Growth and Comparison with in Situ Observation on Al-4 wt.% Cu Alloy by Means of Synchrotron X-ray Radiography[END_REF], thus leading to a relatively larger tip radius [START_REF] Gong | Quantitative comparison of dendritic growth under forced flow between 2D and 3D phase-field simulation[END_REF]. In addition, a low growth rate allows a long time for solute diffusion, which would make the primary arm stems grow laterally. The differences of dendrite growth between 2D and 3D can also be directly clarified by the recent work by Mirihanage et al. [START_REF] Mirihanage | Non-steady 3D dendrite tip growth under diffusive and weakly convective conditions[END_REF], in which a novel image processing algorithm was applied to extract data on dendrite tip curvatures and accompanying solute concentration fields, from a sequence of synchrotron radiogramsphy collected in situ during directional solidification of the Al-Cu-Si alloy.

Moreover, the secondary dendritic arms in the simulations are not so developed as those in the experiments, which also arises from the different dynamical behavior of crystal growth in 2D and 3D. Because of the confinement of solute diffusion, the solute boundary layer in 2D is thicker than in 3D [START_REF] Gong | Quantitative comparison of dendritic growth under forced flow between 2D and 3D phase-field simulation[END_REF], which leads to a stronger solutal interaction between neighboring grains in the 2D PF simulations. As a result, the development of secondary dendritic arms in 2D PF simulations is suppressed by the solute interaction more severely, resulting in commonly shorter secondary dendritic arms. In order to quantitatively evaluate the difference of the equiaxed dendritic morphology in the experiments and the 2D PF simulations, the SDAS, λ2, is measured at the same solid fraction (fS = 0.5) and the results are given in Fig. 4Fig. 4.

In both experiments and simulations, λ2 decreases as the cooling rate increases (Fig.

4Fig. 4(a)

), which is consistent with the analytical prediction by Kattamis and Flemings who indicated that the λ2 could be expressed as a power function of the cooling rate, namely λ2 ~ Rc n with n = -1/3 [START_REF] Flemings | Coarsening in solidification processing[END_REF]. Furthermore, with the cooling rate increasing, the distribution range of the λ2 will be also reduced, which could be easily found from the detailed distribution information of the λ2 as shown in The colors (b1 to b3) represent the crystal orientations.

Moreover, from the quantitative perspective, the λ2 in the 2D PF simulations are slightly larger than those in the solidification experiments. The quantitative differences in λ2 between the experiments and simulations may be attributed to that in simulations it took a relatively longer time to get the same solid fraction than that in experiments (Fig. 5Fig. 5). Consequently, longer duration enables the secondary dendritic arms in the 2D PF simulations to coarsen more significantly, thus leading a slightly larger λ2. However, the relative deviations of the λ2 between the experiments and simulations at Rc = 0.2, 0.5 and 1.5 K/min are about 2%, 12% and 3%, respectively, so it seems reasonable to consider that the λ2 in the 2D PF simulations shows a good quantitative agreement with the experimental data.

where k is the frequency factor, n is the Avrami exponent and t is the solidification time. The general equation for the Avrami exponent is given as

n qd B =+ (10)
where q equals 1 for linear growth (interface controlled) and 1/2 for parabolic growth (diffusion controlled), B is 0 for site saturation nucleation and 1 for continuous nucleation with a constant rate, and d stands for the dimensionality of the growth. The fitting parameters, k and n for both the experimental and simulated data, are plotted in And the Avrami exponent n could be considered approximately as a constant (Fig.

6Fig. 6(b)).

As mentioned above, for the 2D PF simulation (d = 2) where the solidification process is diffusion-controlled (q = 1/2) and the nucleation is achieved by artificially setting the number of the pre-existing nuclei (B = 0), the Avrami exponent is n = qd + B = 1 that is close to the fitted value (0.9783). However, for the experiments in 3D actually (d = 3) with continuous nucleation (B = 1), the expected Avrami exponent is n = qd + B = 2.5 but the fitted value is 0.7368 that is only about 1/3 of the prediction by Eq. ( 10) [START_REF] Gandin | Boundary layer correlation for dendrite tip growth with fluid flow[END_REF]. The significant difference of n between the experimental data and the predicted value could be attributed to that the solid fraction is measured from the 2D radiography images in the experiments, instead of the 3D bulk sample. On the contrary, the 2D PF simulations are able to well describe the transformation kinetics in 2D as the JMAK equation.

Microsegregation

Microsegregation during alloy solidification has always been a concern in suggests that the interface curvature effect could be neglected safely. Consequently, the maximum solute concentration depends on the applied undercooling, namely the product of the cooling rate and the solidification time Rct, rather than the cooling rate itself, which has also been clarified in our recent study [START_REF] Gong | Revisiting dynamics and models of microsegregation during polycrystalline solidification of binary alloy[END_REF]. Cmin is located at the a mis en form 12 pt center of the solid grain and increases with the undercooling (or the solidification time), but varies much slower than Cmax owing to the very small solid diffusivity. With a large cooling rate, the diffusion time to get the same undercooling is shortened, thus leading to a lower Cmin and a larger SR.

Effects of the natural convection on the equiaxed dendritic growth

The inevitable gravity-driven natural convection arising from density difference in the liquid usually plays an important role in the microstructure evolution during solidification, by dramatically modifying the transport of heat and solute. In order to study the effects of the gravity-driven natural convection, the equiaxed dendritic growth of the Al-4 wt.% Cu alloy is simulated using the PF model coupled with fluid dynamics equations as ref. [START_REF] Beckermann | Modeling melt convection in phase-field simulation of solidification[END_REF]. In order to expedite the PF simulation, the vector-valued method [START_REF] Gong | Quantitative comparison of dendritic growth under forced flow between 2D and 3D phase-field simulation[END_REF] is employed here to solve the fluid dynamics equations more efficiently. However, owing to the significant increase in computing amount to solve the polycrystalline PF formula coupled with fluid dynamics equations, only the case of the experiment E2 (Fig. 3Fig. 3(b)) is considered, in which the cooling rate (Rc = 0.5 K/min) is not very slow and the number of grains is also not too large. It should be noted that the large-scale 3D PF simulations of dendrite growth with liquid convection have been realized by Zhang et al. [START_REF] Zhang | Dendritic Growth Under Natural and Forced Convection in Al-Cu Alloys: From Equiaxed to Columnar Dendrites and from 2D to 3D Phase-Field Simulations[END_REF] and Takaki et al. [START_REF] Takaki | Large-scale phase-field lattice Boltzmann study on the effects of natural convection on dendrite morphology formed during directional solidification of a binary alloy[END_REF] recently.

However, for the present work the size for a 3D computational size is much larger than those studies, and thus here we only perform 2D simulations. Furthermore, the natural convection considered in the present PF simulation is caused only by the solutal effect, without considering the thermal effect. It took about 240 hours using the same computing conditionconfiguration as mentioned above until the solid fraction reached 0.5, which was much more time consuming than the corresponding simulation (S2) without convection. previous study [START_REF] Qi | Modeling of coupled motion and growth interaction of equiaxed dendritic crystals in a binary alloy during solidification[END_REF]. Furthermore, in the real solidification experiment E2, several dendrites settle down while they are growing. The settlement arises from the liquid flow and the density difference between solid and liquid. And this phenomenon has also been studied in ref. [START_REF] Qi | Modeling of coupled motion and growth interaction of equiaxed dendritic crystals in a binary alloy during solidification[END_REF], by modeling of coupled motion and growth of equiaxed dendrite using the PF method. Recently, Sakane et al. [START_REF] Sakane | Two-dimensional large-scale phase-field lattice Boltzmann simulation of polycrystalline equiaxed solidification with motion of a massive number of dendrites[END_REF] also performed 2D large-scale PF lattice Boltzmann simulations of polycrystalline equiaxed solidification with motion of a massive number of dendrites. However, in the present work, the motion of dendrites is not considered in order to expedite the PF simulation. By comparing the shape of each grain in Fig. 8Fig. 8(b1) (without convection) and Fig.

8Fig. 8(b2) (with convection), it could be seen that the natural convection has a significant influence on the equiaxed dendritic morphology when grains grow into big size. However, this influence depends on the location and orientation of the a grain. In detail, the growth of the upstream arms will be promoted, while that of the downstream arms will be suppressed, which has been indicated in many previous studies [START_REF] Bogno | Growth and interaction of dendritic equiaxed grains: In situ characterization by synchrotron X-ray radiography[END_REF][START_REF] Gong | Quantitative comparison of dendritic growth under forced flow between 2D and 3D phase-field simulation[END_REF][START_REF] Sakane | Multi-GPUs parallel computation of dendrite growth in forced convection using the phase-field-lattice Boltzmann model[END_REF]. The influence of dendrite arm orientation with respect to gravity on equiaxed grain growth has been studied in ref. [START_REF] Bogno | Growth and interaction of dendritic equiaxed grains: In situ characterization by synchrotron X-ray radiography[END_REF]. The authors explained that the different behaviors between the upstream and downstream arms is due to the poisoning effect of arm growth by the solute rejected by the other arms in the adjacent melt. The rejected solute sinks down by gravity as copper is heavier than aluminum.

Furthermore, the secondary arms are also affected by the liquid convection. Similar to the primary dendritic arms, the sidebranches at the upstream side are also more developed than those on the downstream side. In order to quantitatively estimate the effect of the liquid convection on the sidebranches, the SDAS in the solidification experiment (E2) and 2D PF simulations (S2 without convection and S4 with convection) are measured, and the results are listed in Table 2Table 2. As mentioned above, because a much longer time is taken in 2D PF simulations to reach the same solid fraction than that in the experiment (Fig. 5Fig. 5(a) and Fig. 9Fig. 9(a)), the coarsening of the sidebranches is more significant, thus leading to a relatively larger SDAS in simulations. The natural convection does not affect this trend apparently. However, it is worth noting that the average SDAS is increased about 6% by convection in the 2D PF simulations, which suggests that the solute convection may promote the overall coarsening of sidebranches during solidification.

The effect of the natural convection on the solid fraction, fS, is shown in Fig. 9Fig. 9. In the 2D PF simulations (S2 and S4), fS is much lower than the experimental data. Besides, Fig. 9Fig. 9(a) also shows that the natural convection has no obvious influence on fS at the beginning of solidification (less than 100 s during solidification), but then it slightly affects the solid fraction before fS = 0.25. During this period, fS is slightly larger when convection is taken into account (S4) than that in the case without convection (S2). Then, no apparent difference can be found between the simulations with and without convection. This evolution behavior denotes that natural convection only takes effect when the solute gradient in liquid at the most of solid-liquid interface is sufficiently high. High concentration gradient results in a relative strong convection, thus leading to a little higher growth rate. Subsequently, as solidification proceeds (fS > 0.25), the convection effect gradually decreases and both fS become almost identical in PF simulations with and without convection. The variation rate of solid fraction with time, ΔfS/Δt, is plotted in Fig. 9Fig. 9(b). Also, the influence of convection on ΔfS/Δt is not obvious, and fS increases faster in S4 than that in S2 after solidifying for a while. And then during the subsequent stage of solidification (fS > 0.25), the time change rates of the solid fraction in different simulations gradually get identical. In addition, apparent oscillations are observed for Vmax, which may be attributed to the complicated flow pattern caused by the developed dendrites. As impinged growth takes place among grains, the liquid channel between two grains becomes more and more narrow, and the enlarged flow pressure makes Vmax increase again from t2 to t3. Whereafter, Vmax starts to decrease since the liquid channel is too narrow and the liquid flow nearby are very weak, which could not render sufficient pressure to drive the liquid pass through the narrow channel. When the solid fraction becomes higher and higher, all grains are impinged with each other. Meanwhile, the solute gradient in liquid is reduced and the density contrast is not sharp sufficiently to drive intensive convection. As a result, the solute convection weakens as demonstrated by the reduced mean flow velocity. Until time t4, Vmax reappears outside the liquid channel between primary tips and again varies in the same manner as ΔfS/Δt.

Compared with the apparently oscillating maximum liquid flow velocity, the variation of the average liquid flow velocity is much more moderate and evolves similarly to the variation rate of solid fraction.

Conclusions and outlook

In the present paper, the equiaxed dendritic growth of Al-4 wt.% Cu alloys during polycrystalline solidification under continuous cooling conditions is studied Owing to the significant differences in mass and heat transport behavior between 2D PF simualtions and experimental data, large-scale quantitative 3D PF simulations of multi-dendrite growth during solidification will be considered in our future work.

Moreover, the effects of the nucleation undercooling on dendritic growth should be also concerned, so that the PF simulations can exactly reproduce the multi-dendrite growth in the solidification experiments, leading to quantitatively good agreement with the experiments as much as possible.

  conditions: A study combining in situ and real-time experiment with large-scale phase-field simulation Tongzhao Gong, Yun Chen, Shanshan Li, Yanfei Cao, Liyuan Hou, Dianzhong Li, Xing-Qiu Chen, Henri Nguyen-Thi, Guillaume Reinhart To cite this version: Tongzhao Gong, Yun Chen, Shanshan Li, Yanfei Cao, Liyuan Hou, et al.. Equiaxed dendritic growth in nearly isothermal conditions: A study combining in situ and real-time experiment with large-scale phase-field simulation. Materials Today Communications, 2021, 28, pp.102467. 10.1016/j.mtcomm.2021.102467. hal-03605914

Fig. 1 .

 1 Fig. 1. Sequence of in situ radiographs showing the equiaxed dendritic microstructure evolution in the solidification experiments of Al-4 wt.% Cu alloys at various cooling rates, Rc. (a) E1: Rc = 0.2 K/min; (b) E2: Rc = 0.5 K/min; (c) E3: Rc = 1.5 K/min.

  And hence the standard PF model (Eqs. (1)(1) and (2)(2)) is transformed into the nonlinearly preconditioned formula, as follows

Fig. 2 . 1 .

 21 Fig. 2. The convergence test of the interface width parameter, ξ = W0/d0, used in the quantitative 2D PF simulation of the Al-4 wt.% Cu alloy at the cooling rate Rc = 1.5 K/min. (a) Variation of the tip velocity with solidification time; (b) Variation of the maximum tip velocity with ξ.

Fig. 4Fig. 4 Fig. 3 .

 43 Fig. 3. Equiaxed dendritic morphology in the solidification experiments (E1 to E3 shown in a1 to a3) and the 2D PF simulations (S1 to S3 shown in b1 to b3) at the same solid fraction fS = 0.5 in local domains marked by the red boxes in Fig. 1Fig. 1.

Fig. 4 .

 4 Fig. 4. Comparison of the SDAS, λ2, between the solidification experiments (E1 to E3) and 2D PF simulations (S1 to S3) at the same solid fraction (fS = 0.5). (a) The average λ2 at various cooling rates (the error bars represent the standard deviation); (b) Box plot of the λ2 (the cross symbols are the maxima and minima, the filled circles are the means, the whiskers represent the 5 to 95 percentiles, the box top and bottom respectively represent the second and third quartiles, and the lines inside the box represent the medians).

4. 2 5 .

 25 Solid fractionThe solid fraction, fS, in the solidification experiments and the 2D PF simulations is plotted and compared quantitatively in Fig.5Fig. At the early solidification stage when grains are isolated from each other, the crystal growth rate increases with the undercooling, and therefore the fS increases gradually (Fig.5Fig. 5(a)) at an ever-increasing rate (Fig.5Fig. 5(b)) in both experiments and simulations. Then, the solute interaction between neighboring grains slows the crystal growth, but the solidification still proceeds (e. g., the coarsening of primary and secondary dendritic arms) to make the fS continue to increase (Fig.5Fig. 5(a)), despite of the ever-decreasing variation rate (Fig.5Fig. 5(b)). However, there indeed exist significant quantitative differences in the fS between the experimental data and the simulated results. At the same time, the fS and its time derivative, ΔfS/Δt, in the simulations are both lower than those in the experiments. The quantitative differences in the fS between the experiments and simulations could be mainly attributed to three factors. Firstly, the fS measured directly from the radiography images, which are 2D projections of the complicated 3D dendritic microstructure, usually overestimates the real solid fraction in a 3D sample. Secondly, the much lower crystal growth rate in 2D PF simulations compared with that in 3D cases[START_REF] Gong | Quantitative comparison of dendritic growth under forced flow between 2D and 3D phase-field simulation[END_REF] results in slower solidification kinetics than the experiments. More importantly, all the grains nucleated simultaneously at the beginning of solidification in PF simulations, and the nucleation undercooling was not taken into consideration. In the real solidification experiments, the grains nucleated at different temperatures due to different nucleation undercoolings. Therefore, the nucleation undercooling which was neglected in the simulations may also play an important role on the difference between the simulations and the experiments.

Fig. 5 .

 5 Fig. 5. Comparison of the solid fraction between the solidification experiments (E1 to E3) and the 2D PF simulations (S1 to S3). (a) Solid fraction, fS; (b) Variation rate of the solid fraction, ΔfS/Δt. t0 is the time when the first grain is observed by the naked eye in the radiography images in the experiments. The solid and dashed lines are the fitting curves by the JMAK function for the experimental data and simulated results, respectively.

FigFig. 6 .

 6 Fig. 6Fig. 6.

  materials and metallurgical sciences owing to its influence on the mechanical properties, corrosion resistance of cast products and formation of macrosegregation. Effects of cooling rates on solute microsegregation, including the maximum concentration Cmax in the interdendritic liquid, the minimum concentration Cmin in the core of the solid dendrite, and the solute segregation ratio SR = Cmax/Cmin, in the 2D PF simulations are shown in Fig. 7Fig. 7. It could be seen that the cooling rate has no obvious effect on Cmax, which is almost the same as the equilibrium concentration, CL E = C0 -ΔT/m. For Al-Cu alloys, the solute partition coefficient k < 1, which means that the solute is rejected into the liquid phase from the solid phase during solidification. As a result, Cmax is achieved in the interdendritic liquid around the root of sidebranches. The approximate equality relationship between Cmax and CL E

Fig. 7 .

 7 Fig. 7. Effects of cooling rates on solute microsegregation in the 2D PF simulations. (a) Maximum and minimum solute concentration, Cmax and Cmin; (b) Segregation ratio, SR = Cmax/Cmin.

Fig. 8 . 8 .

 88 Fig. 8. Equiaxed dendritic morphology in the experiment E2 (a) and the 2D PF simulations without (b1) and with (b2) convection at the same solid fraction fS = 0.3 in local domains marked by the red boxes in Fig. 1Fig. 1. In b1 and b2, the grayscale indicates the solute concentration, and the arrows and their colors indicate the liquid flow velocity and its magnitude. The red dots denote the location of the minimum solute concentration in the whole domain.

1 Fig. 9 .

 19 Fig. 9. Effects of gravity-driven natural convection on solid fraction. (a) Solid fraction, fS; (b) Variation rate of the solid fraction, ΔfS/Δt, and the maximum and mean flow velocity, Vmax and Vmean.

Fig. 10 .

 10 Fig. 10. Solute concentration and liquid flow velocity at the time t1 to t4 marked by

Fig. 11 .

 11 Fig. 11. Effects of liquid gravity-driven natural convection on solute segregation. (a) Maximum and minimum solute concentration, Cmax and Cmin; (b) Solute segregation ratio, SR = Cmax/Cmin.

  using the in situ and real-time observation of experiments by synchrotron X-ray radiography and large-scale quantitative 2D PF simulations. It is revealed that the equiaxed dendritic morphology and the SDAS in the 2D PF simulations are both in reasonably good quantitative agreement with those in the experiments. With the cooling rate increasing, the SDAS will be reduced as a power function of the cooling rate, which is consistent with the Kattamis-Flemings model. The gravity-driven natural convection will slightly increase the SDAS in the 2D PF simulation. Moreover, in both of the experiments and 2D PF simulations, the solid fraction evolves with time in the same manner as the JMAK equation, namely fS = 1 -exp[-(kt) n ]. The parameter k can be fitted well as a linear function of the cooling rate, and another parameter n is approximately a constant that relates to the parabolic growth controlled by diffusion for slow solidification, and the site saturation nucleation in the PF simulations or the continuous nucleation in the real solidification experiments. From the quantitative perspective, at the same solidification time (or undercooling), the solid fraction in the 2D PF simulations is much lower than that in the experiments, and the two fitting parameters, k and n, in the JMAK equation are respectively smaller and larger than the experimental values. Simulations with natural convection show that the flow strength increases with the solute segregation in melt, but has a slight effect on solid fraction only when the severe solute gradient is generated around most of the solid-liquid interface.Furthermore, the maximum solute concentration in the interdendritic liquid is nearly the same as the equilibrium concentration, which depends on the undercooling rather than the cooling rate. While the minimum solute concentration in the center of a grain is smaller with a larger cooling rate, owing to the shorter time for solute diffusion in the solid phase. The almost same Cmax in liquid and the smaller Cmin with a larger cooling rate in solid result in a larger segregation, SR = Cmax/Cmin. The liquid natural convection has no effect on Cmax, but slightly enhances Cmin due to the solute segregation induced by melt convection, thus leading to a smaller SR.

Table 1 .

 1 Material and numerical parameters used in the present 2D PF simulations.

	Symbol	Meaning	Value
	k	solute partition coefficient	0.14 [27]
	TL	liquids temperature	920 K [35]
	DL	solute diffusion coefficient in liquid	2.42×10-5 cm 2 /s [36, 37]
	DS	solute diffusion coefficient in solid	1.15×10 -8 cm 2 /s [36]
	Г	Gibbs-Thomson coefficient	2.41×10 -5 cm•K [38]
	m	liquidus slope	-3.5 K/wt.%
	ε4	surface energy anisotropy strength	0.0106 [39]
	Rc	cooling rate	0.2, 0.5, 1.5 K/min
	C0	sample global concentration	4 wt.% Cu
	ξ	interface width parameter	49.96
	(Δx)min	grid size in the interface layer	1.0 μm
	Δt	time step	1.14 × 10 -3 s

Table 2 .

 2 Comparison of the average SDAS in the solidification experiment and 2D PF

	a mis en form

a mis en form

Acknowledgement

This work was supported by the Youth Innovation Promotion Association CAS, the Science Challenge Project (Grant No. TZ2016004), the Special Scientific Projects of Inner Mongolia, and French National Space Center (CNES), Convention N°174800/00.

Data Availability

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.


Gaëlle Lebrun

Sanae Benaissa

Claude Le Men

Véronique Pimienta

Gilles Hébrard

Nicolas Dietrich
email: dietrich@insa-toulouse.fr

Effect of surfactant lengths on gas-liquid oxygen mass transfer from a single rising bubble

Keywords: Surfactants, Oxygen mass transfer, Bubble, PLIF-I, adsorption

This work is an experimental investigation of the effect of the nature of surfactants on oxygen mass transfer. The study focuses on three cationic surfactants with different hydrophobic chain lengths, and four nonionic surfactants with different hydrophilic chain lengths. Equilibrium adsorption isotherms are calculated for each surfactant from experimental values of surface tension in static conditions. Surfactant solutions at concentrations between 2.5×10 -8 and 5×10 -3 mol/L were prepared and oxygen transfer from millimetric air bubbles (between 0.82 and 1.08 mm) was measured by Planar Laser Induced Fluorescence with Inhibition (PLIF-I). When the bulk concentration of surfactant was increased, results showed a sharp decrease of bubble velocity, in the range of 283 -75 mm/s, and of liquid-side mass transfer coefficient, in the range of 5.6 ×10 -4 -0.4 ×10 -4 m/s. This effect was observed for all surfactants studied. However, the length of the hydrophilic chain did not appear to affect the hydrodynamics of the rising bubble or the oxygen transfer at the same bulk concentration.

Furthermore, for the same bulk concentration, increasing the hydrophobic chain length had an impact on the velocity and the mass transfer coefficient of oxygen. Finally, the Sherwood number was calculated in each medium and compared with classical correlations for gas-liquid mass transfer prediction. Those correlations seemed to reach a limit for a very concentrated medium.

Introduction

Gas/liquid systems involving mass transfer occur in numerous chemical, biochemical or environmental processes [START_REF] Roustan | Transferts gaz-liquide dans les procédés de traitement des eaux et des effluents gazeux[END_REF][START_REF] Charpentier | Considérations générales sur les contacteurs gaz-liquide Problèmes fondamentaux et appliqués de métrologie qui se posent. Place de nos recherches[END_REF][START_REF] Treybal | Mass-transfer Operations[END_REF][START_REF] Green | Perry's Chemical Engineers' Handbook[END_REF]. Among these, wastewater treatment plants using activated sludge tanks are at the core of process intensification for oxygen mass transfer. The oxygen mass transfer coefficient is a key parameter to optimize the process because oxygen is used to keep pollutantdegrading bacteria alive, so a good prediction of the mass transfer coefficient is required to enable the process to be adjusted toward a decrease of power consumption for the same performance. Seeing this strong industrial demand, intensive studies have been carried out to investigate the global mass transfer in gas-liquid contactors [START_REF] Bouaifi | A Comparative Study of Gas Hold-up, Bubble Size, Interfacial Area and Mass Transfer Coefficients in Stirred Gas-Liquid Reactors and Bubble Columns[END_REF][START_REF] Cockx | Modeling of Mass-Transfer Coefficient K-L in Bubble-Columns[END_REF][START_REF] Billet | Predicting Mass Transfer in Packed Columns[END_REF][START_REF] Albal | Mass Transfer in Multiphase Agitated Contactors[END_REF]. For two decades, studies have been oriented toward hydrodynamics and contactors at a more local scale but, for good comprehension of all the phenomena occurring in bubble columns, the molecular, bubble and laboratory scales have to be taken into account [START_REF] Besagni | The Effect of Liquid Phase Properties on Bubble Column Fluid Dynamics: Gas Holdup, Flow Regime Transition, Bubble Size Distributions and Shapes, Interfacial Areas and Foaming Phenomena[END_REF].

The physicochemical properties of the liquid phase can affect the hydrodynamics and mass transfer in a column and some studies have focused their research on viscous media [START_REF] Xu | Mass Transfer and Diffusion of a Single Bubble Rising in Polymer Solutions[END_REF][START_REF] Duran | Impact of the Activated Sludge Suspended Solids on Its Non-Newtonian Behavior and Oxygen Transfer in a Bubble Column[END_REF] or media close to the one found in aerated sludge tanks with the coupled effect of rheology and surface tension [START_REF] Lebrun | Gas-Liquid Mass Transfer around a Rising Bubble: Combined Effect of Rheology and Surfactant[END_REF][START_REF] Kováts | Influence of Viscosity and Surface Tension on Bubble Dynamics and Mass Transfer in a Model Bubble Column[END_REF][START_REF] García-Abuín | Oxygen Absorption in Polymer+surfactant Aqueous Solutions[END_REF]. The effect of the surface tension on mass transfer has also been largely studied. Surfactants come from all kinds of domestic detergents and, due to their amphiphilic properties, they adsorb on the gas/liquid interface, influencing hydrodynamics and mass transfer. They are considered as the most common contaminants hindering oxygen transfer [START_REF] Dumont | Mass Transfer Enhancement of Gas Absorption in Oil-in-Water Systems: A Review[END_REF][START_REF] Nekoeian | Effect of Surfactants on Mass Transfer Coefficients in Bubble Column Contactors: An Interpretative Critical Review Study[END_REF] but the mechanisms explaining this decrease are still under debate. Various authors have identified a decrease that occurs because of hydrodynamic modifications brought by surfactants, such as an increase of interfacial rigidity that decreases the renewal rates close to the bubble and decreases its terminal velocity [START_REF] Rosso | Effects of Interfacial Surfactant Contamination on Bubble Gas Transfer[END_REF][START_REF] Alves | Effect of Bubble Contamination on Rise Velocity and Mass Transfer[END_REF][START_REF] Griffith | The Effect of Surfactants on the Terminal Velocity of Drops and Bubbles[END_REF][START_REF] Weber | The Effect of Surface Active Agents on Mass Transfer from Spherical Cap Bubbles[END_REF][START_REF] Vasconcelos | Gas-Liquid Mass Transfer to Single Bubbles: Effect of Surface Contamination[END_REF] or a modification of the velocity field surrounding the bubble [START_REF] Weiner | Experimental and Numerical Investigation of Reactive Species Transport around a Small Rising Bubble[END_REF], and also physicochemical modifications [START_REF] Painmanakul | Effect of Different Contaminants on the α-Factor: Local Experimental Method and Modeling[END_REF][START_REF] Jimenez | Oxygen Mass Transfer and Hydrodynamic Behaviour in Wastewater: Determination of Local Impact of Surfactants by Visualization Techniques[END_REF][START_REF] Ahmia | Impact of Cellulose and Surfactants on Mass Transfer of Bubble Columns[END_REF]. Although the decrease of mass transfer and velocity is widely reported in the literature, Gómez-Díaz et al.

underlined the increase of mass transfer coefficient appearing with the addition of a very weak concentration of cationic surfactant. This increase was attributed to greater gas/liquid interfacial buoyancy due to the Marangoni effect [START_REF] Gómez-Díaz | Mass-Transfer Enhancement or Reduction by Surfactant Presence at a Gas-Liquid Interface[END_REF], an effect also identified later by Alvarez et al. with an anionic surfactant [START_REF] Álvarez | Continuous Removal of Carbon Dioxide by Absorption Employing a Bubble Column[END_REF]. In their studies, [START_REF] Painmanakul | Effect of Surfactants on Liquid-Side Mass Transfer Coefficients[END_REF] and [START_REF] Sardeing | Effect of Surfactants on Liquid-Side Mass Transfer Coefficients in Gas-Liquid Systems: A First Step to Modeling[END_REF] highlight the link between the mass transfer coefficient, the diameter of the bubbles and the surface coverage by surfactants [START_REF] Painmanakul | Effect of Surfactants on Liquid-Side Mass Transfer Coefficients[END_REF][START_REF] Sardeing | Effect of Surfactants on Liquid-Side Mass Transfer Coefficients in Gas-Liquid Systems: A First Step to Modeling[END_REF].

In addition, the surface properties of amphiphilic molecules depend on their structure, so they affect mass transfer in different ways. It has been observed that the charge of the head [START_REF] Painmanakul | Effect of Different Contaminants on the α-Factor: Local Experimental Method and Modeling[END_REF][START_REF] Moraveji | Effects of Surface Active Agents on Hydrodynamics and Mass Transfer Characteristics in a Split-Cylinder Airlift Bioreactor with Packed Bed[END_REF][START_REF] Caskey | A Study of the Effects of Soluble Surfactants on Gas Absorption Using Liquid Laminar Jets[END_REF][START_REF] Chen | Effects of Surfactant Contamination on Oxygen Mass Transfer in Fine Bubble Aeration Process[END_REF][START_REF] Jia | Effect of Surfactant Type on Interfacial Area and Liquid Mass Transfer for CO2 Absorption in a Bubble Column[END_REF][START_REF] Kouzbour | Experimental Analysis of the Effects of Liquid Phase Surface Tension on the Hydrodynamics and Mass Transfer in a Square Bubble Column[END_REF] or the length of the tail [START_REF] Vasconcelos | Gas-Liquid Mass Transfer to Single Bubbles: Effect of Surface Contamination[END_REF][START_REF] García-Abuín | Effect of Surfactant Nature upon Absorption in a Bubble Column[END_REF] largely affects mass transfer, since these parameters affect the affinity of surfactants with the interface. However, although these parameters have been studied, to the best of our knowledge, no general conclusion can be drawn about the influence of the surfactant's nature.

Mass transfer in presence of surfactants seems to involve very complex mechanisms and some research has focused on studying the numerical resolution of these complex systems. Some authors use direct numerical simulation (DNS) to predict the local distribution of surfactants around a bubble, and thus gain a better understanding of hydrodynamic and physicochemical phenomena close to the interface, which is promising for mass transfer models [START_REF] Pesci | Computational Analysis of Single Rising Bubbles Influenced by Soluble Surfactant[END_REF][START_REF] Deising | Direct Numerical Simulation of Mass Transfer in Bubbly Flows[END_REF] . In addition, for a better understanding of local phenomena, a few techniques for direct visualization of mass transfer have been developed. This is the case of Laser Induced Fluorescence methods (LIF) which use a pH sensitive dye to visualize gas/liquid transfer of CO 2 in clean or contaminated media [START_REF] Kováts | Influence of Viscosity and Surface Tension on Bubble Dynamics and Mass Transfer in a Model Bubble Column[END_REF][START_REF] Valiorgue | Concentration Measurement in the Wake of a Free Rising Bubble Using Planar Laser-Induced Fluorescence (PLIF) with a Calibration Taking into Account Fluorescence Extinction Variations[END_REF][START_REF] Stöhr | Visualization of Gas-Liquid Mass Transfer and Wake Structure of Rising Bubbles Using PH-Sensitive PLIF[END_REF][START_REF] Kong | Dual Emission LIF Technique for PH and Concentration Field Measurement around a Rising Bubble[END_REF][START_REF] Hiby | Eine Fluoreszenzmethode Zur Untersuchung Des Stoffübergangs Bei Der Gasabsorption Im Rieselfilm[END_REF][START_REF] Someya | DeLIF Measurement of PH Distribution around Dissolving CO2 Droplet in High Pressure Vessel[END_REF][START_REF] Huang | Influences of Gas-Liquid Interface Contamination on Bubble Motions, Bubble Wakes, and Instantaneous Mass Transfer[END_REF][START_REF] Lacassagne | Ratiometric, Single-Dye, PH-Sensitive Inhibited Laser-Induced Fluorescence for the Characterization of Mixing and Mass Transfer[END_REF]. Another visualization technique uses the properties of fluorescence inhibition by a quencher. This is the case of Planar Laser Induced Fluorescence with Inhibition (PLIF-I) to measure oxygen transfer, which uses a fluorescent dye that is inhibited in the presence of oxygen and can measure oxygen transfer in different geometries, such as plane interfaces [START_REF] Wolff | A Fluorescence Technique to Measure Concentration Gradients near an Interface[END_REF][START_REF] Wolff | Instantaneous Concentration Profiles of Oxygen Accompanying Absorption in a Stratified Flow[END_REF][START_REF] Woodrow | Laser-Induced Fluorescence Studies of Oxygen Transfer Across Unsheared Flat and Wavy Air-Water Interfaces[END_REF] Taylor bubbles [START_REF] Butler | Mass Transfer in Taylor Flow: Transfer Rate Modelling from Measurements at the Slug and Film Scale[END_REF][START_REF] Butler | Modelling of Mass Transfer in Taylor Flow: Investigation with the PLIF-I Technique[END_REF][START_REF] Roudet | Experimental Investigation of Interfacial Mass Transfer Mechanisms for a Confined High-Reynolds-Number Bubble Rising in a Thin Gap[END_REF][START_REF] Roudet | Hydrodynamic and Mass Transfer in Inertial Gas-Liquid Flow Regimes through Straight and Meandering Millimetric Square Channels[END_REF], and bubble columns [START_REF] Weiner | Experimental and Numerical Investigation of Reactive Species Transport around a Small Rising Bubble[END_REF][START_REF] Francois | Direct Measurement of Mass Transfer around a Single Bubble by Micro-PLIFI[END_REF][START_REF] Bork | The Impact of Local Phenomena on Mass Transfer in Gas-Liquid Systems[END_REF].

The present work focuses on the measurement of oxygen transfer from isolated millimetric air bubbles in media containing various surfactants of different, accurately-chosen natures. To achieve this goal, the technique of PLIF-I presented above was used. Cationic and nonionic surfactants were used and were chosen because of their similar global structures but different lengths of the hydrophobic or hydrophilic part. First, an in-depth study of interface colonization by these surfactants was performed.

Then mass transfer was studied for different concentrations of each surfactant, the objective being to highlight the link between the chemical structure of the surfactant, its interface colonization and gas/liquid oxygen transfer.

Material and methods

Oxygen transfer measurements

Planar Laser Induced Fluorescence with Inhibition (PLIF-I) was used in this study to quantify the oxygen transfer and hydrodynamic parameters of the bubble in different media. The experimental setup is represented in Figure 1. It is composed of a column (1) (10x10x30 cm 3 ) filled with the solution. A 75 µm internal diameter needle connected to a syringe pump, which can be activated remotely (Harvard apparatus PHD, 22/2000 Programmable, USA) (2), generates single air bubble. A CMOS camera (3) (12 bits, 1920x1200 pixel 2 , Basler ace, Germany) placed next to the column, 50 mm above the injection point, can record 250 images/s in a window of 1920x500 pixel 2 at 20.31±0.04 µm/pixel. This camera was positioned in such a way as to measure the surface area and velocity of the bubble; the viewing window was then around 39x10 mm 2 (the highest visualization axis on the vertical direction). The backlight used was a white LED (100x100 LLUB, PHLOX, France). The geometrical calibration of the camera was performed with a ruler and concerning the uncertainty of this method it is calculated as the thickness of one graduation bar. Then, the velocity is calculated with the time taken by the bubble to cross the 39 mm window, and the size of the bubble by a binarization of the bubble images and contour analysis (see figure 1).

The amount of oxygen transferred in the wake of the bubble was measured by fluorescence inhibition of a ruthenium complex (CAS: 20782-45-7, Sigma-Aldrich, USA), which was excited by a horizontal laser sheet (4) (Nd:YAG, laser: Dualpower 200-15, 15Hz, 2x200 mJ, Dantec Dynamics, Denmark) at intensity was recorded by a CCD camera (5) (FlowSense, 12 bits, 2048x2048 pixels, Dantec Dynamics, Denmark). This camera was placed at the bottom of the column with a sampling of 4.37±0.04 µm/pixel (viewing window about 9x9 mm 2 ), focused on the laser sheet. It was equipped with a 570 nm high pass filter in order to record only the fluorescence from the ruthenium complex, which has an emission wavelength of 582 nm. The camera speed was synchronized with the laser frequency and focused on the laser sheet. Before each experiment, a calibration curve was built by using an oxygen probe (multimeter: HQ40D, probe: Intellical LDO101, Hach, Germany) to associate each grey level with an oxygen concentration, according to the Stern Volmer [START_REF] Stern | Uber die Abklingzeit der Fluoreszenz[END_REF] equation ( 1)

1/G=1/G 0 +K sv /G 0 × [O 2 ] ( 1
)
where G is the grey level, G 0 is the grey level in absence of the quencher, [O 2 ] mg/L is the oxygen concentration and K sv is the Stern Volmer constant. Once the calibration curve was applied to the images obtained in the bubble wake, the oxygen concentration was displayed on each pixel. To calculate a mass transfer coefficient from experimental data, a threshold criterion was verified according to equation [START_REF] Charpentier | Considérations générales sur les contacteurs gaz-liquide Problèmes fondamentaux et appliqués de métrologie qui se posent. Place de nos recherches[END_REF], where I is the maximal intensity of the signal, I 0 the intensity of the background, and σ n the standard deviation of the background (square root of the expected value of the squared deviation from the mean). The signal was considered as significant if the signal-to-noise ratio was higher than 10.

(I-I 0 )/σ n >10 (2)

Figure 1: Experimental setup used to measure gas/liquid oxygen transfer by PLIF-I

The second criterion to be verified was the mass conservation in the wake plane in function of time if the absence of convection was to be assumed. Figure 2 displays the integral of the spot, which represents the total mass transferred in the plane, versus time. It is constant from 0.4 to 2.4 seconds, with a small standard deviation. The average of these values gave the total mass transferred in a plane perpendicular to the bubble wake. After 2.4 seconds, the noise was too great to allow an accurate integral to be calculated and, finally, the criterion described by equation ( 2) was no longer valid.

Figure 2: Integration of the oxygen field concentration in the wake of the bubble with time Once these criteria had been verified, it was considered that the mass transferred in a horizontal plane was constant and the change in the size of this spot structure was thus due solely to molecular diffusion of oxygen. In previous studies [START_REF] Francois | Direct Measurement of Mass Transfer around a Single Bubble by Micro-PLIFI[END_REF][START_REF] Jimenez | Mass Transfer in the Wake of Non-Spherical Air Bubbles Quantified by Quenching of Fluorescence[END_REF][START_REF] Dietrich | Fast Measurements of the Gas-Liquid Diffusion Coefficient in the Gaussian Wake of a Spherical Bubble[END_REF], it was observed that, for quasi-spherical bubbles, the diffusion spot was circular and presented a Gaussian profile. As a result, the diffusion spot was fitted by equation [START_REF] Treybal | Mass-transfer Operations[END_REF], where the concentration [O 2 ] in each pixel x p , y p was estimated.

[O 2 ](x,y)=A×exp((-(x p -X) 2 +(y p -Y) 2 )/B+C) (3)
Parameters were found by fitting the equation with the raw image using the "fminsearch" solver (Matlab R2017a). Note that A and B are parameters, and (X, Y) is the centre of the spot, the parameter C representing the background.

It was possible to calculate the flux of oxygen (mg/s) transferred by the rising bubble with equation ( 4), where V b is the velocity of the bubble (m/s) measured by the side camera.

F O2 =V b ×∫∫[O 2 ](x,y)dxdy (4)
The mass transfer coefficient was then deduced from equation [START_REF] Bouaifi | A Comparative Study of Gas Hold-up, Bubble Size, Interfacial Area and Mass Transfer Coefficients in Stirred Gas-Liquid Reactors and Bubble Columns[END_REF], where S b is the surface area of the bubble, deduced from the equivalent diameter measured with the side camera; [O 2 ] is considered as the concentration at the beginning of the experiment (measured with the optical oxygen probe, close to 0 mg/L); and [O 2 ] * is the saturation concentration of oxygen, measured by the optical oxygen probe.

k L =F O2 /(S b ×([O 2 ] * -[O 2 ])) (5)
Finally the diffusion coefficient of oxygen in the bulk was measured by a method developed by Xu et al. [START_REF] Xu | Fast Determination of Gas-Liquid Diffusion Coefficient by an Innovative Double Approach[END_REF] assuming that the surface area of the spot S spot increased with time following equation [START_REF] Cockx | Modeling of Mass-Transfer Coefficient K-L in Bubble-Columns[END_REF] S spot =2πD O2 tη D [START_REF] Cockx | Modeling of Mass-Transfer Coefficient K-L in Bubble-Columns[END_REF] where η D is a constant that can be fixed following the procedure described by Xu et al. [START_REF] Xu | Fast Determination of Gas-Liquid Diffusion Coefficient by an Innovative Double Approach[END_REF]. Then the surface area of the spot was plotted versus time and the diffusion coefficient was extracted from the slope.

Each condition was repeated 6 times in order to be sure of the repeatability of measurements. All measurements were performed at room temperature (294±1K).

Physicochemistry of systems studied

The surfactants used were cationic or nonionic. They are presented in Table 1. The three cationic mol/L and 5x10 -3 mol/L. The equilibrium surface tension was measured for each solution with the Nouÿ ring method (tensiometer: K6, Krüss, Germany), assuming that equilibrium surface tension was reached after 2 hours [START_REF] Caskey | A Study of the Effects of Soluble Surfactants on Gas Absorption Using Liquid Laminar Jets[END_REF]. To be sure of the repeatability of the measurements, each point was repeated 6 times and the deviation was lower than the error predicted by the supplier (1 mN/m). The surface tension was linked to the surface concentration; this effect is described by the Gibbs equation ( 7) [START_REF] Gibbs | On the Equilibrium of Heterogeneous Substances[END_REF].

Γ=-nRTln(dγ/dC) [START_REF] Billet | Predicting Mass Transfer in Packed Columns[END_REF] where Γ is the surface concentration (mol/m 2 ), n is equal to 1 for nonionic or 2 for ionic surfactants, R is the perfect gas constant (J.mol -1 K -1 ), T the temperature (K), γ the surface tension (N/m) and C the bulk concentration (mol/m 3 ). The Gibbs equation comes from a thermodynamic description of the system and, to access the surface concentration, adsorption models are more commonly used. In this study, the Langmuir model [START_REF] Langmuir | The Constitution and Fundamental Properties of Solids and Liquids[END_REF] was used for the nonionic surfactants, and is described by equation ( 8)

𝛤=𝛤 max ×K LG ×C/(1+ K LG ×C) ( 8
)
where Γ max is the maximal surface concentration (mol/m 2 ), and K LG is the Langmuir constant (m 3 /mol).

Concerning the cationic surfactants, the Langmuir model is not the most suitable since it assumes that there is no interaction between monomers. For this kind of surfactants, the Frumkin model [START_REF] Frumkin | Die Kapillarkurve Der Höheren Fettsäuren Und Die Zustandsgleichung Der Oberflächenschicht[END_REF] is more useful because it takes electrostatic interactions between molecules into account. The model is described by equations ( 9) and ( 10)

γ=γ 0 +(R×T×𝛤 max )×[ln(1-θ)+a×θ 2 ] (9) K F ×C=θ/(1-θ)×exp(-2×a×θ) (10)
where K F is the Frumkin constant (m 3 /mol), θ is the degree of coverage by the surfactant, given by θ=𝛤/𝛤 max , and a is a constant accounting for intermolecular interactions. Both models were solved numerically to fit the experimental points. The numerical resolution was performed by a free access software "Sa" (http://cinet.chim.pagesperso-orange.fr/tele_sa/install_Sa_Eng.html).

The critical micellar concentration (CMC) given by Table 1 was extracted from the curves of surface tension according to the bulk concentration. Details of this value will be given in the next section.

Concerning the Hydrophilic Lipophilic Balance (HLB), the values were provided by the supplier for nonionic surfactants and calculated from the Davies formula [START_REF] Davies | A Quantitative Kinetic Theory of Emulsion Type, I. Physical Chemistry of the Emulsifying Agent, Gas/Liquid and Liquid/Liquid Interface[END_REF] for cationic surfactants. The HLB is a scale representing the affinity of the surfactant with the different phases. The higher the HLB, the more hydrophilic the surfactant and, thus, the greater its affinity with water. That is why, in the table, HLB is observed to decrease when the hydrophobic part of a cationic surfactant increases, and HLB increases when the hydrophilic part of the nonionic surfactants increases.

In order to confirm that surfactants did not affect the rheology of the solution, viscosity was measured for shear rates between 1 and 500 s -1 with a rheometer (HAAKE MARS III, Germany). For all surfactants in the range of concentrations of the study, the viscosity was found to be the same as that of water (1.00±0.07 mPa.s).

Finally, mass transfer experiments were also tried with anionic surfactants. However, due to their negative charge, which is opposite to the positive charge of the ruthenium complex, interaction between the two created a complex that settled in the solution. This complex made measurement by PLIF-I impossible. As far as we know, no fluorescent dye has been used for measuring oxygen transfer in presence of an anionic molecule.

Results and discussion

Study of the interface colonization

Surfactants were selected for this study because they present different structures and, as a result, may have different ways of colonizing the interface. This part deals with the equilibrium state of interfaces with chosen surfactants. To characterize interfaces in presence of surfactants, a key parameter is the surface tension. Figure 3 A and B present the equilibrium surface tension according to the bulk concentration for all of the surfactants studied. Each curve is representative of a classical surfactant behavior. (i) At low concentration, the surface tension is equal to the surface tension of water, so noor a negligible amount ofsurfactant is adsorbed. (ii) A sharp decrease in surface tension occurs with increasing concentration, corresponding to a gradual adsorption of surfactant on the interface. (iii) At high concentration, from a value called the critical micellar concentration (CMC), the surface tension becomes constant: the system is no longer considered as diluted, the surfactant aggregates in the form of micelles, and the interface has reached its maximal surface concentration in the pressure and temperature conditions considered.

Even though the surfactants present the same classical curve shape, some differences can be noticed at this stage of the study. First, increasing the hydrophobic chain length of cationic surfactant leads to a large change in the value of the CMC, and the decrease of surface tension does not take place for the same bulk concentration. We expected these surfactants to reach approximately the same value of surface tension after the CMC, as is usually reported in the literature [START_REF] Tadmouri | Partition and Water/Oil Adsorption of Some Surfactants[END_REF][START_REF] Bergeron | Disjoining Pressures and Film Stability of Alkyltrimethylammonium Bromide Foam Films[END_REF]. This was the case for C16TAC and C8TAC, which reached a value close to 43 mN/m. However, for C12TAC, the measured value was around 10 mN/m weaker. This can be attributed to an impurity in the product, which was used as received from the supplier. The impurity may have been the corresponding long chain alcohol [START_REF] Gilányi | Adsorption of Alkyl Trimethylammonium Bromides at the Air/Water Interface[END_REF]. The behavior is different if the hydrophilic chains of nonionic surfactants are lengthened as the change in both the CMC value and the surface tension value above the CMC is very small. These 7 curves were used to calculate the equilibrium surface concentration since the systems were solved numerically, the dotted lines of Figure 3 representing the model that best fits the experimental points of surface tension. From these fitted parameters, values of equilibrium surface concentrations according to bulk concentrations are presented in Figure 4 A and B for each surfactant studied. From Table 2, Table 3, and Figure 4, regarding the maximal surface concentrations (𝛤 max ), the longer the chain is, the lower is the maximal surface concentration. This tendency was observed for nonionic surfactant and cationic surfactant, even though C8TAC and C12TAC reached a very close maximal surface concentration.

This can be attributed to the impurity discussed above in this section. However, although this effect is verified for each kind of surfactant, it does not hold true between the two different kinds of surfactants: even though the nonionic surfactants present a bigger molar mass than the cationic ones, their maximal interfacial concentrations are higher or very close. The parameter a is higher for the longest chains, but decreases and is almost negligible for C8TAC. This means that the molecular interactions are important for the longest chains but do not play an important role for the shortest chain and the Frumkin isotherm is very close to a Langmuir isotherm. Finally, the adsorption constants of cationic surfactants increase with the length of the tail and this is consistent with the values of HLB:

increasing the proportion of hydrophobic part decreases the solubility of the surfactants. Concerning the nonionic surfactants, their high adsorption gives them the same solubility properties.

In accordance with the constants K K and K LG , adsorption of nonionic surfactant took place for the same bulk concentrations for each chain length studied. In contrast, for cationic surfactants, the more hydrophilic they were, the more they adsorbed at high concentrations. To view this colonization from another standpoint, it is interesting to calculate the surface concentration, not in terms of number of moles, but in mass quantity. Figure 5 A and B represent the surface concentration in grams per square meter according to the molar concentration of the bulk. The first point highlighted by this figure is the difference between the maximal surface concentrations of nonionic surfactants and cationic surfactants. The mass of triton adsorbed on the interface was around 3 times the maximal surface concentration of cationic surfactants. This time, the curves show that the longer the chain of a nonionic surfactant, the higher its maximal mass concentration: even though the maximal surface concentration of TX165 reached a value between TX102 and TX100, the tendency was more marked with the higher value of TX305. This effect was less marked for cationic surfactants, for which no trend could be noted. Once the equilibrium surface colonization of surfactants had been highlighted, mass transfer measurements in the same kind of media could contribute information about the influence of the chain length, and also the affinity with the interface at different concentrations, on oxygen transfer.

Oxygen transfer in different media

The objective of this part is to highlight the effect of surfactants on oxygen transfer, comparatively to their affinity with the interface. First, the impact of the surfactant on the hydrodynamic parameters of the bubble needs to be considered since these parameters play an important role in mass transfer. The bubble diameter varied slightly between experiments, as the presence of surfactant tends to facilitate the detachment of bubbles and lead to smaller bubbles. Nevertheless, recorded diameters were between 0.82±0.02 mm and 1.08±0.02 mm. The mean diameter of each condition can be found in the supplementary material. The velocities of bubbles were strongly impacted by the presence of surfactants. Figure 6 A and B represent the velocity according to the bulk concentration. For all surfactants studied, increasing the concentration led to a decrease in the bubble velocity. For nonionic surfactants, the curve plotting this decrease has the same shape for any chain length and reaches a plateau around 0.1 m/s. In contrast, changing the tail length of cationic surfactants led to a translation of the curve. The impact on velocity appears at high concentrations with short tail surfactants and at low concentrations with long tail surfactants. These results can be explained by differences in the concentration of surfactants adsorbed at the interface. Indeed, while nonionic surfactant are adsorbed at the interface for the same bulk concentration, the cationic surfactants are adsorbed on the interface for very different concentration in the liquid phase. For example, for a concentration in the liquid phase of 10 -4 M, the maximal surface concentration of C16TAC is reached while C8TAC has not yet begun to colonize the interface. These differences in surface concentration lead to differences in the Marangoni force, which is opposite to the convection and responsible for the decrease in the velocity of bubbles when surfactants are present [START_REF] Palaparthi | Theory and Experiments on the Stagnant Cap Regime in the Motion of Spherical Surfactant-Laden Bubbles[END_REF]. It is important to keep these bubble velocities in presence of surfactants in mind because they are used to calculate the mass transfer coefficient according to equation (4).

Figure 6: Velocities of bubbles according to bulk surfactant concentrations. A. Cationic surfactants. B.

Nonionic surfactants

Concerning the study of oxygen transfer, Figure 7 gives examples of results obtained from PLIF-I for the TX100 series. Since the velocities of the bubbles are different, it represents different position in the tail of the bubble, however, the total mass transferred is conserved with time (see figure 2), as a results the images can be significantly compared. From these first results, it is obvious that the amount of mass transferred decreases with the increase of TX100 bulk concentration. However, the total mass transferred-in other words the integral of the spot-for the higher concentration of 3.7×10 -4 mol/L seems to be higher than for the lower concentration of 2.8×10 -5 mol/L. This effect does not follow the trend of the mass transfer to decrease with increasing concentrations of surfactant and stresses the importance of taking the velocity of the bubble into account. The velocity is higher for the lowest concentration and, as a result, the flux of oxygen transferred is higher even if the total mass transferred is lower. All information about important calculated parameters in each case is given in Tables 4,5,6,7,8,9 and 10, available in the supplementary material. This result was observed for each surfactant studied.

In order to compare all the results obtained for our sets of surfactants, the mass transfer coefficient was plotted against the bulk concentration of surfactants; the results are presented in Figure 8. The error bars represent the uncertainty of the measures. Since each measurement was made 6 times, the relative standard deviation of mass transfer coefficient is given in Tables 4,5,6,7,8,9 and 10, available in the supplementary material.

For each surfactant, a decrease of mass transfer coefficient was observed with increasing concentration of surfactant. This effect is described in the literature, where it is usually associated with hydrodynamic effects on the bubble [START_REF] Alves | Effect of Bubble Contamination on Rise Velocity and Mass Transfer[END_REF][START_REF] Weiner | Experimental and Numerical Investigation of Reactive Species Transport around a Small Rising Bubble[END_REF][START_REF] Huang | Influences of Gas-Liquid Interface Contamination on Bubble Motions, Bubble Wakes, and Instantaneous Mass Transfer[END_REF]: adsorption of surfactant on the bubble tends to make it more rigid and the renewal at the interface decreases, leading to a decrease of transferred flux. behaviors can be seen in Figure 8 for the two kinds of surfactant studied. Error bars are displayed for each point although, for very concentrated solutions, the bar is smaller than the symbol itself. First of all, it is interesting to compare the curve of mass transfer coefficient with those of velocity (see Figure 6). The decrease of mass transfer coefficient is closely linked with the velocity of the bubble. The slowdown due to the presence of surfactant leads to a decrease of the surface renewal and this seems to be the major effect leading the decrease of mass transfer. The effects of surfactants on the hydrodynamics leading to a decrease of mass transfer coefficient are described by current correlations and this point will be discussed in the next part. Concerning cationic surfactants, for the same concentration, an addition of 4 carbons on the carbonated chain leads to a different mass transfer coefficient. This decrease of mass transfer coefficient with the chain length of the same kind of surfactant has been observed by Garcia-Abuin et al. [START_REF] García-Abuín | Effect of Surfactant Nature upon Absorption in a Bubble Column[END_REF] in a bubble column. This information may be linked with results described part 3.1 (Figure 4 and Figure 5), because the adsorption of surfactant on the interface is translated from low concentrations for long chains to high concentration for short chains. These facts suggest that the decrease of mass transfer coefficient is linked to the equilibrium adsorption at interfaces. A similar analysis can be applied to nonionic surfactants but, this time, whatever the hydrophilic chain length, the mass transfer coefficient is the same for an identical molar concentration. Interestingly, this effect can be linked with Figure 4 and Figure 5 and, unlike the situation for cationic surfactants, the surface concentration increases for all hydrophilic chains when the bulk concentration is the same. These observations encourage us to assume that the adsorption isotherm and the surface concentration are very important parameters, and the hydrophobic chain length influences these parameters. any chain length at very high concentration. This means that the hydrophilic part of the surfactant has no effect on the mass transfer coefficient, even though it changes the interface density. It would have been interesting to have the same information on reaching a plateau for cationic surfactants in order to draw a conclusion about the effect of the hydrophobic chain length. Unfortunately, although the plateau is reached for the C16TAC, this is not the case for C12TAC and C8TAC because, for concentrations higher than 1.3×10 -3 and 5.1×10 -3 mol/L the signal received by the camera is too low and so not sufficient to satisfy the criterion determined by equation ( 2). However, we can still guess that their k L are different from the value of the plateau for C16TAC since the signals received are lower. We assume that, unlike the hydrophilic chain length of nonionic surfactant, the hydrophobic chain length has an impact on mass transfer. However, a method to capture lower flux would be needed to confirm this assumption.

Finally, although some conclusions can be drawn about the effect of the length of the surfactant, it is important to keep in mind that adsorption is a dynamic phenomenon and some studies have highlighted the importance of adsorption kinetics, especially for its effect on the velocity of rising bubbles [START_REF] Krzan | On the Influence of surfactant's polar group on the local and terminal velocities of bubbles[END_REF]. This means that surfactants are not immediately adsorbed on the bubble with the same concentration as described in Figure 4 and Figure 5. So a parameter that needs to be taken into consideration is the dynamics of the adsorption of surfactant on a rising bubble. From the short distance and with the window of visualization used here, this parameter is difficult to measure.

Oxygen transfer in presence of surfactants: limitation of the model

This article provides many data concerning mass transfer for bubbles that are clean, or partially or fully contaminated. These results can be compared with commonly used models for predicting mass transfer. We chose to use the Sherwood number according to equation [START_REF] Duran | Impact of the Activated Sludge Suspended Solids on Its Non-Newtonian Behavior and Oxygen Transfer in a Bubble Column[END_REF] in order to nondimensionalize the calculation.

Sh=k L ×d eq /D O2 [START_REF] Duran | Impact of the Activated Sludge Suspended Solids on Its Non-Newtonian Behavior and Oxygen Transfer in a Bubble Column[END_REF] Figure 9 represents the Sherwood number as a function of the Reynolds number. Error bars were calculated for each point but are not visible for low Sherwood and Reynolds numbers. In order to also represent the ratio of contamination of the bubble, every point is colored according to θ (=𝛤/𝛤 max ).

Experimental results are compared with predictions of the Higbie [67] and Frössling [68] models, which represent clean and fully contaminated bubbles, respectively, calculated with equations ( 12) and [START_REF] Kováts | Influence of Viscosity and Surface Tension on Bubble Dynamics and Mass Transfer in a Model Bubble Column[END_REF].

Sh Higbie =1.13×Re 0.5 ×Sc 0.5 [START_REF] Lebrun | Gas-Liquid Mass Transfer around a Rising Bubble: Combined Effect of Rheology and Surfactant[END_REF] Sh Frössling =2+0.66×Re 0.5 ×Sc 0.33 (13) Figure 9: For cationic and nonionic surfactants with different coverage ratios, Sherwood number as a function of Reynolds number.

Some interesting information can be obtained from this figure. First, the Reynolds number is strongly impacted by the increasing coverage ratio. As predicted in the literature [69], the presence of surfactants leads to a decrease of velocity and bubble diameter; these parameters are taken into account with the Reynolds number. Secondly, a sharp decrease of the Sherwood number is observed with a decrease of the Reynolds number and an increase of the coverage ratio. This decrease of Sherwood number can be described with two different slopes (represented by dotted lines on Figure 9). The first, for high Reynolds numbers, is the steepest. The equilibrium contamination of bubbles is weak and even a slight increase leads to a marked decrease in the Sherwood and Reynolds numbers.

As expected, the value of Sherwood numbers in this first slope are found to lie between the values predicted by Frössling and by Higbie. The same trend has been observed by Sardeing et al. for low surface coverage [START_REF] Sardeing | Effect of Surfactants on Liquid-Side Mass Transfer Coefficients in Gas-Liquid Systems: A First Step to Modeling[END_REF]. The second slope is for low Reynolds number and high level of contamination.

The slope is slight but the surprising result is that the Sherwood numbers are lower than those predicted by Frössling for all the bubbles with a coverage ratio between 95 and 100%, and for a large majority of bubbles with a coverage ratio higher than 90%. This result shows the limits of the Frössling model for highly contaminated bubbles. This overestimation can be as high as 50% for very concentrated solutions, which is not negligible and cannot be attributed to experimental errors. Similar results have already been observed by [START_REF] Jimenez | Mass Transfer in the Wake of Non-Spherical Air Bubbles Quantified by Quenching of Fluorescence[END_REF] [70] with monoglyceride caprylate as nonionic surfactant, and by Sardeing et al. for high surface coverage [START_REF] Sardeing | Effect of Surfactants on Liquid-Side Mass Transfer Coefficients in Gas-Liquid Systems: A First Step to Modeling[END_REF] with Lauryl dimethyl benzyl as cationic surfactant and a fatty alcohol as nonionic surfactant. These two previous studies measured the transfer for bigger bubbles, with diameters between 1.2 and 1.6 mm, which led to higher Reynolds numbers than the ones calculated in this study. The calculated Sherwood numbers are presented in

Conclusion

In this work, the impact of the nature of the surfactants used on oxygen mass transfer has been

experimentally measured with the PLIF-I technique. A description of the surface contamination has been given for each system studied and this reveals that the hydrophobic tail of cationic surfactant has an impact on the adsorption isotherm. The longer the chain, the more it adsorbs for low concentrations. In contrast, changing the length of a nonionic surfactant does not significantly affect the adsorption isotherm. Concerning oxygen transfer in liquids containing these surfactants, for every case studied, an increase of the bulk concentration led to a sharp decrease of velocity and mass transfer coefficient. The results suggest that these decreases with increasing concentration of surfactant are strongly related to the adsorption of surfactants on the interface. Concerning the nature of surfactants, if the hydrophobic chain of a cationic surfactant is lengthened, at the same bulk concentration, then the mass transfer coefficient is strongly impacted and is lower for the cases studied here. In contrast, increasing the length of the hydrophilic part of a nonionic surfactant does not have any effect on mass transfer at the same bulk concentration, even if the density of matter at the interface predicted by the Langmuir model is different. Finally, results have been compared with current correlations and show that, for a highly contaminated bubble, the Frössling model overestimates the mass transfer coefficient. These results underline the necessity to create an adapted model that takes account of the adsorption of surfactants at the interface and the resulting changes in physicochemical parameters.

[67] Higbie, R

Supplementary material

  surfactants were hexadecyltrimethylammonium chloride (C16TAC, Sigma Aldrich, CAS: 112-02-7, USA), dodecyltrimethylammonium chloride (C12TAC, Sigma Aldrich, CAS: 112-00-5, USA) and trimethyloctylammonium chloride (C8TAC, Sigma Aldrich, CAS: 10108-86-8, USA) with respectively 16, 12 and 8 carbons on their linear carbonated chain. They shared the same, positively charged, hydrophilic head: an ammonium chloride. Concerning nonionic surfactants, experiments were run with Triton X-100 (TX100, CAS: 9002-93-1, Sigma Aldrich, USA), Triton X-102 (TX102, CAS: 9036-19-5, Sigma Aldrich, USA) and Triton X-165 (TX165, CAS: 9636-19-5, Sigma Aldrich, USA) and TX305 (CAS: 9002-93-1, Sigma Aldrich, USA). These nonionic surfactants present the same hydrophobic structure and a hydrophilic polyethylene oxide chain with a different average number of repetitions; 9.5, 12, 16 and 30 for TX100, TX102, TX165 and TX305 respectively. All surfactant solutions were prepared with ultra-pure water (18 mΏ/cm).

Figure 3 :

 3 Figure 3: Equilibrium surface tension of surfactant solutions according to bulk concentration. A:

Figure 4 :

 4 Figure 4: Molar equilibrium surface concentration of surfactants in function of bulk concentration. A.

Figure 5 :

 5 Figure 5: Mass equilibrium surface concentration of surfactants versus bulk concentration. A. Cationic

Figure 7 :

 7 Figure 7: Oxygen concentration field in the wake of a bubble 0.33 seconds after the bubble's passage,

Figure 8 :

 8 Figure 8: Mass transfer coefficient of oxygen according to bulk concentration of surfactant. A: cationic

Figure 9

 9 Figure 9 and overestimation by the Frössling model is observed. These results highlight the necessity

Table 1 :

 1 Information on the surfactants used in the studyThe mass transfer coefficient was measured in solutions presenting concentrations between 2.5x10-8

					Molar
						CMC	HLB (without
	Surfactant	Type	Molecular formula	n	mass
						(mmol/L)	unit)
					(g/mol)
	C 8 TAC			7	208	320	12.7
			CH 3 (CH 2 ) n N(CH 3 ) 3 Cl
	C 12 TAC	cationic		11	264	6	10.8
	C 16 TAC			15	320	0.8	8.9

Table 2 and

 2 Table 3 present the parameters obtained for cationic and nonionic

	surfactants respectively.
		C8TAC	C12TAC	C16TAC
	K F (m 3 .mol)	0.02	2.58	3.38
	a (without unit)	0.04	0.63	2.75
	𝛤 max (mol.m -2 )	2.43×10 -6	2.48×10 -6	1.54×10 -6

Table 2 :

 2 Adsorption parameters obtained by using Frumkin model for cationic surfactants

		TX100	TX102	TX165	TX305
	K LG (m 3 .mol)	1053	1194	2440	3294
	𝛤 max (mol.m -2 )	2.67×10 -6	2.43×10 -6	1.90×10 -6	1.36×10 -6

Table 3 :

 3 Adsorption parameters obtained by using Langmuir model for nonionic surfactants

  ., 1935, "The Rate of Absorption of a Pure Gas into Still Liquid during Short Periods of Exposure", Transaction of the Institution of Chemical Engineers, 35(1), pp. 36-60. [68] Frössling, N., 1938, "Uber Die Verdunstung Fallender Tropfen," Beitr. Geophys. Gerlands, 52, pp. 170-216. [69] Clift, R., R Grace, J., and E Weber, M., 1978, Bubbles, Drops, and Particles. American Press, New York. [70] Jimenez M., 2013, Etude du transfert de matière gaz/liquid en milieu complexes : quantification du transfert d'oxygène par techniques optiques. INSA, Toulouse.

Table 4 :

 4 Details on calculated parameters of bubbles generated in C8TAC solutions

	C8TAC

of oxygen concentration field (10 -5 mg/m)

	Reynolds number	138±4	182±5	252±9	257±9	300±10
	Sherwood number	44±10	119±10	210±10	220±10	290±20
		1.69±0.2	3.51±0.07	4.68±0.1	4.41±0.03	6.10±0.29
	Oxygen diffusion coefficient (10 -9
		1.9±0.3	1.9±0.1	1.95±0.03	1.89±0.06	2.0±0.1
	m 2 /s)
	Mass transfer coefficient (10 -4 m/s) 0.85±0.05	2.3±0.1	4.2±0.3	4.2±0.3	5.4±0.3
	Relative standard deviation of mass
		0.12	0.02	0.05	0.01	0.06
	transfer coefficient
	Schmidt number	530±90	520±40	513±8	530±20	510±30

Table 5 :

 5 Details on calculated parameters of bubbles generated in C12TAC solution

	C12TAC

Table 6 :

 6 Details on calculated parameters of bubbles generated in C16TAC solutions

	C16TAC

Table 7 :

 7 Details on calculated parameters of bubbles generated in TX100 solutions

	TX100

Table 8 :

 8 Details on calculated parameters of bubbles generated in TX102 solutions

	TX102

Mass transfer coefficient (10 -4 m/s)

		0.63±0.04 0.61±0.04 0.84±0.04 1.00±0.05 1.9±0.1	3.3±0.2	5.5±0.3
	Relative standard
	deviation of mass	0.05	0.07	0.05	0.05	0.03	0.03	0.04
	transfer coefficient
	Schmidt number	530±30	530±40	530±50	520±20	550±50	540±10	500±30
	Reynolds number	83±2	103±2	103±2	103±2	105±2	192±5	266±9
	Sherwood number	28±2	30±3	42±5	48±3	98±9	170±9	270±20

Table 9 :

 9 Details on calculated parameters of bubbles generated in TX165 solutions

	TX165

Table 10 :

 10 Details on calculated parameters of bubbles generated in TX100 solutions

	TX305

Acknowledgments

The authors wish to thank the French National Research Agency (ANR) for the support from the project MAMOTHS ANR-17-CE06-001 and Bilal El Mokdad for his kind help.