B Betting

E Varea

C Gobin

G Godard

B Lecordier

B Patte-Rouland

Experimental and numerical studies of smoke dynamics in a compartment fire

Keywords: Compartments fire, PIV large scale, enclosure fire, FDS, smoke dynamics

Fires in confined spaces are of major concern in fire safety engineering. Indeed, fires with ensuing fatalities, generally occur in apartment or room fires. The decision and response time of rescue teams mainly results from empirical understanding. However, since compartment fires are multi-physical and multi-scale problems, a clear fundamental approach is needed. One of the main issues concerns the transition from localized to generalized fire. The most important vector leading to generalized fires is smoke. Indeed, smoke temperature is very high and many species i.e. burned and unburned gases are already mix or might mix. In this study, an experimental facility composed of a maritime container is set-up. It allows enclosure fires up to 1 MW of power. This power is representative of moderate room fires. The global behavior of the smoke is investigated through the analysis of mean experimental temperature fields, smoke dynamics by large scale PIV as well as numerical simulations. For numerical simulations, the Fire Dynamics Simulator (FDS) software is used. The experimental measurements are used to evaluate the validity of FDS in under-ventilated conditions. It is observed that FDS code is able to reproduce both temperature and velocity data of enclosure fires with accuracy depending on the power to volume ratio. A criterion capable to discriminate the ventilation status of confined fire is observed and discussed.

Introduction

Every year in France, compartment fires are responsible for more than 88,000 interventions involving more than 15,000 people, including several hundred deaths and serious injuries ( [START_REF] Betting | Etudes Expérimentales et lois prédictives des foyers d'incendies[END_REF], [START_REF] Betting | Experimental and numerical investigations of the flow characteristics in confined fires[END_REF], ANR Firediag). Nowadays, the decisionmaking process of emergency services during fire interventions is mainly based on a rapid disaster analysis. This analysis comes from the experience gained over the years. However, this type of analysis does not permit a complete understanding of the disaster since it is mainly based on empirical knowledge. The evolution of a fire, the appearance of a secondary focus or of thermal phenomena, i.e. back-draft [START_REF] Gottuk | The development and mitigation of backdraft : a real-scale shipboard study[END_REF][START_REF] Bepoix | Les phénomènes thermiques[END_REF] or flash-over [START_REF] Francis | Observable characteristics of flashover[END_REF][START_REF] Thomas | Flashover and instabilities in fire behavior[END_REF][START_REF] Bishop | Nonlinear dynamics of flashover in compartment fires[END_REF], are difficult to predict with accuracy.

In order to optimize and secure the intervention of rescue services safety engineering, including knowledge of fire dynamics and fire-related hazards is crucial [START_REF] Johansson | Review of the use of fire dynamics theory in fire service activities[END_REF]. More precisely, one of the main critical aspects that remains to be determined is the smoke dynamics at openings where fresh air and hot fumes mix. Indeed, this particular phenomenon, typical of under-ventilated fires, may be encountered in many enclosures fires. This phenomenon is called pulsing fire and has been extensively studied by Merci [START_REF] Merci | Fluid mechanics aspects of fire and smoke dynamics in enclosures[END_REF], Bengtsson [START_REF] Bengtsson | Enclosure fire[END_REF] and Pretrel [START_REF] Prétrel | Pressure variations induced by a pool fire in a well-confined and force-ventilated compartment[END_REF]. Pulsating fires occur in oxygen-limited conditions. These conditions may be related to natural (opening) or mechanical ventilation (nuclear power plants, High-rise buildings) [START_REF] Merci | Fluid mechanics aspects of fire and smoke dynamics in enclosures[END_REF].

Study of smoke dynamics is made possible using different devices or techniques. These include, among others Pitot probes and bidirectional probes or McCaffrey probes [START_REF] Mccaffrey | A robust bidirectional low-velocity probe for flame and fire application[END_REF]. However, these probes are intrusive and potentially impacting the smoke dynamics. Moreover only one point data is evaluated. For this reason, laser techniques can be used. The smoke velocities can be studied using the PIV technique [START_REF] Tieszen | Experimental study of the flow field in and around a one meter diameter methane fire[END_REF][START_REF] Hou | Measurements and prediction of air entrainment rates of pool fires[END_REF][START_REF] Koched | Application de la piv sur un écoulement de fumée à un passage de porte induit par une source incendie, 13ième Congrès de Techniques Laser[END_REF]. This technique has already been used in the study of fires [START_REF] Bryant | A comparison of gas velocity measurements in a full-scale enclosure fire[END_REF][START_REF] Bryant | A guide to characterizing heat release rate measurement uncertainty for full-scale fire tests[END_REF][START_REF] Koched | Utilisation de la stereo piv pour l'optimisation des mesures de vitesse de fumee d'incendie a un passage de porte par des sondes bidirectionnelles, 14eme Congrés Francophone de Techniques Laser[END_REF] for well-ventilated and underventilated conditions. A natural extension of this technique is to use the PIV to highlight exchanges between hot exhaust fumes and fresh incoming air. These large-scale measurements highlight areas of interest for a better understanding of fires.

In order to gain further insights into smoke dynamics in a confined environment, we propose a dual approach based on experimental and numerical data. The scalar quantity (temperature) and the vector quantity (velocity) are evaluated and compared considering both experimental and numerical data. The objectives of the paper are threefold :

-To design an experimental set-up to ensure PIV measurements inside the fire volume. -To observe the specific behavior of confined fires for ventilated or under-ventilated conditions for both temperature and velocity data.

The capability of FDS code to reproduce confined fires with accuracy is discussed. -To obtain more information about the transition from ventilated to under ventilated fire conditions. This information will be use to design a helpful decision-making tool for fire officers.

Experimental set-up

To investigate the smoke dynamic characteristics of enclosure fires, an experimental test bench is set-up. The thermal phenomena studied using the test bench are likely to be encountered by firefighters during their missions. This cell is composed of two maritime containers mounted perpendicularly. One is the measure cell and the other is the test cell. In this configuration, only one 1m 2 outlet (0.8m 2 of clear aperture due to insulation) in the opposite direction to the burner), is open, Fig. 1.

The standard size of a maritime container is 6 m × 2.59 m × 2.45 m. These structures have emerged as training tools in rescue services. Moreover, the volume of these containers is close to that of a room of a standard flat, which is approximately 35 m 3 [START_REF] Bepoix | Les phénomènes thermiques[END_REF].

The fuel source consisted of 36 propane burners with a surface area of 1 m 3). Insulation is provided using :

-40 cm of cellular concrete blocks (floor) ; -10 cm of rock wool (0.034 W/m.K) and 5 cm mineral wool (0.034 W/m.K) (roof) ; -10 cm of mineral wool (0.034 W/m.K) and 3cm of vermiculite plates (0.21 W/m.K). A 1 m 2 access enables both the visualization and setup of advanced laser diagnostics such as Particle Image Velocimetry (PIV) [START_REF] Koched | Utilisation de la stereo piv pour l'optimisation des mesures de vitesse de fumee d'incendie a un passage de porte par des sondes bidirectionnelles, 14eme Congrés Francophone de Techniques Laser[END_REF].

For temperature measurements, 20 type-K thermocouples (diameter of 1.5 mm) are used. Two racks of 10 thermocouples are mounted vertically and horizontally along the center line of the chamber. Position is adjusted to be below the outlet which is opposite side of the burner, Large scale PIV is a complex task because it is a compromise between spatial resolution and field of view. To overcome this issue, a special configuration is set-up. Instantaneous, two-dimensional velocity measurements relied on Particle Image Velocimetry (PIV) in which alumina particles are used (50 µm diameter). These particles are injected through an in-house air supply corona which is positioned below the burner. Particles are injected in such a way that the fresh air injection dynamics are not modified. As low as 10 % of air to reach stoichiometric condition is used to feed the burner with particles seeded air. The optical arrangement is : Nd : Yag laser (Quantel QSmart, 2*400 mJ/pulse) 1, mirrors 2 and 3 and a -20mm cylindrical lens which pro- Then PIV processing is performed using a cross-correlation technique between pairs of successive images. The size of the PIV interrogation windows is 64 pix 2 with a 50 % overlap. The time interval between two consecutive images is in the range of 1 to 2.5 ms, depending on the powers and therefore flow velocity.

Numerical simulations

Simulations are performed using FDS code version 5 [START_REF] Desanghere | Détermination des conditions d'échauffement de structure extérieure à un bâtiment en situation d'incendie[END_REF] which was developed by the National Institute of Standards and Technology (NIST). Due to its extensive validation [START_REF] Mcgrattan | Fire dynamics simulator user's guide[END_REF][START_REF] Mcgrattan | Fire dynamics simulator (version 5) technical reference guide[END_REF], FDS is commonly used in engineering studies of fire safety.

The numerical domain is represented in Fig. 4.

A mesh sensitivity is shown in Fig 6 . Temperature profiles from simulations are plotted against time (600 s) with cell sizes of 2.5 cm, 5 cm, 10 cm and 20 cm. Results indicate no significant differences at long time scales. Note that with the smaller mesh, a slightly higher temperature is observed in the ramp-up phase.

Table 1 shows for grid resolutions of 2.5, 5, 10 and 20 cm together with the cell number and computation time.

From the experiments, spatial resolution for the PIV is approximately 2.5 cm. However, a computation time of 9 days for 2.5 cm mesh is not acceptable. Therefore, the 5 cm cell size is chosen to optimize computational time and

Results

In this section, data from temperature profile and velocity fields are reported and discussed. Conditions are defined in the previous section (see Table 2) and are briefly given at the beginning.

Temperature data 4.1.1. Experimental and numerical profiles

Simulations are performed for three different heat release rates : 200 kW, 500 kW and 800 kW. The results of numerical and experimental temperature profiles plotted against time are compared in Fig. 7. The temperatures correspond to the evolution of the average values of the ten thermocouples below the ceiling.

In Fig. 7(a), 200 kW, it is observed that the temperature evolution against time for experimental and numerical results are very similar. The HRR increases are almost identical, as well as the temperature values. In addition, the steady-state is not reached. In Fig. 7(b), the temperature evolution against time is plotted for a HRR of 500 kW. Numerical and experimental results exhibit similar profiles. However, it is observed that a slight shift appears between the two temperature evolution. This is due to the beginning of under-ventilated conditions which are difficult to model in FDS. In Fig. 7(c), 800 kW an significant discrepancy between the experimental and numerical results exists from the beginning. This difference is resolved after 100 seconds. This deviation of the numerical data from the experimental results might be due to FDS code which uses to overestimate the heat release rate in under-ventilated conditions. This was discussed in [START_REF] Magnognou | Etudes numériques et expérimentales sur le risque d'inflammation des gaz imbrûlés au cours d'un incendie en milieu sous-ventilé[END_REF]. In order to verify this assumption, 2D temperature distributions are evaluated via simulations.

Numerical results of 2D fields

In this section, the development of the fire are studied for the complete configuration considering 2D fields. 200, 500 and 800kW cases are studied.

The burner is located on the right side of the cell while the open outlet is on the left side. In Fig. 8, the instantaneous at t=300s (right) and the average (left) temperature fields are presented.

In Fig. 8(a) the instantaneous temperature field at 200 kW is reported. The smoke take place in the upper part of the domain. Re-circulation of fresh air is observed at the outlet. This fresh air transports oxygen to the fire. At this time, the volume is considered to be properly ventilated. This observation can be confirmed by the long time averaged field, see Fig. 8(b). On the averaged field, clear plume and stratification are observed. The re-circulation zones are also present.

The 2D fields for 500 kW configuration are given in Figs. 8(c) and 8(d). In this case, the fire dynamics are completely different. The temperature distribution is above 350 • C. The initial fire focus is no longer localized at the burner. It is observed that the flame moved towards the outlet. During this simulation it is observed that the flame position is highly unstable (RMS not shown here). This phenomenon represents the alternation of fresh air intake and evacuation of hot gases.

The pressure in the enclosure decreases and allows fresh air to enter. This fresh air entrance restarts combustion which causes the pressure to increase and a release of hot gases. This phenomenon, characteristic of under ventilated fire, is repeated until there is no more fuel.

Figures 8(e) and 8(f) show the 800 kW configuration. The flames and hot gases are concentrated at the outlets and below the burner. These locations correspond with the presence of oxygen in the cell.

From the temperature profiles, it is observed that the quality of the simulations degrades with the increase in HRR. This might be due to the appearance of under-ventilated conditions which are difficult to model in FDS, [START_REF] Magnognou | Etudes numériques et expérimentales sur le risque d'inflammation des gaz imbrûlés au cours d'un incendie en milieu sous-ventilé[END_REF]. This is supported by the 2D temperature distributions where under-ventilated behaviors are observed for higher HRR. Under ventilated conditions dominate this type of configuration and strongly influence fire dynamics including temperature and velocity. The limitations of FDS with regard to under-ventilated fires is know.

In order to evaluate if the flame is quenched with increased powers, the numerical Heat Release Rate (HRR) is reported for powers ranging from 200 to 800kW. In Figure 9(a) HRR at 200 kW is plotted. The HRR remains constant which is typical for well-ventilated conditions. At 500 kW, Fig. lation conditions become less favorable. The HRR is less stable. However, the value remains close to the specified value, making experimental and numerical temperature profiles fairly similar. Finally, for 800 kW, Fig. 9(c), the HRR exhibits large fluctuations. This is typical for under-ventilated conditions. The simulations are less accurate since FDS is less valid for these configurations due to both high HRR and under-ventilated conditions. This confirmation of under-ventilated conditions explains the differences observed in Fig 7(c).

Velocity fields

In this section, both experimental 2D fields from large scale PIV measurements and simulations are reported and discussed.

Experimental fields

This section present the results obtained from the PIV measurements. Smoke velocities are measured for HRR ranging from 100 kW to 1 MW. This makes it possible to study will-ventilated to severely under-ventilated cases. The position of the stagnation plane, the main location of mixing, can be seen to be relatively insensitive to the power evolution..

To check this hypothesis, Figures 12 and13 shows the velocity fields from 500 kW to 1000 kW.

For these powers, the smoke layer seems to drop slightly before it stabilizes. The smoke layer position seems almost identical in Figures 12 and13. In addition, the mixing zone intensified, especially at 700 kW, 12(c). This is confirmed for HRR of 900 kW and 1 MW, Figs 13(a)) and 13(b), where the stagnation plane is stretched and became longer.

For powers of 800 kW, 12(d), 900 kW and 1 MW, a PIV signal loss is observed at the top right of the image. This can be linked to strong exchanges with the outside, particularly because of the high entrance of fresh air. This air supply leads to strong mixing resulting in a dilution of particles, hence less signal for PIV calculation. In addition, the velocity of hot gases increases for powers from 100 kW to 500 kW but then appears to decrease. To confirm this hypothesis, Figure 14 is plotted. The latter represents the evolution of the maximum average velocity present in the volume as a function of the input power. In this figure, a strong increase in the maximum velocity is observed for HRR from 100 kW to 500 kW. For the 600 kW case, the maximum velocity drops sharply, before rising slightly again for increased powers.

Under-ventilated conditions might explain the lack of oxidizer which limits the development of the fire and thereby the smoke dynamics. Indeed, for the power of 500 kW, 10 g/s of propane are fueling the burner. Considering that the combustion occurs at stoichiometric conditions i.e. for an adiabatic flame temperature of 2267K, there is around 1 m 3 /s of hot gases which is produced. Considering a negligible pressure rise in the enclosure fire and the outlet area of 0.8 m 2 (clear aperture), the balance equations impose exit velocities of 1.25 m/s. This calculation resulting from first principals is in total agreement with the data of Figure 14, where the inflection point is between 1.42 and 1.12 m/s. Therefore, the analysis of Figure 14 shows that the velocity drop against power is clearly related to pressure rise in the chamber. Pressure rise induces accumulation of hot gases in the enclosure, which limits the volume of inlet fresh gases, hence under-ventilated conditions are reached. As a conclusion of the section, we demonstrated that from temperature profiles, HRR analysis and maximum velocity distribution seem to reveal conditions of well-ventilated and under-ventilated fires.

Comparison of numerical and experimental fields

To complete the analysis of the velocity fields, the results obtained numerically and experimentally are compared. In order to study cases in wellventilated, poorly-ventilated and under-ventilated conditions, the HRR of 200 kW, 500 kW and 800 kW are considered. As a reminder, the studied configuration is available in the Figure 10. For each image presented below, the numerical field is located on the left while the experimentally measured field is on the right. Figure 15 shows the velocity fields for a power of 200 kW. On these two images, one can see that the numerical data overestimates the velocity in the upper part of the image. Moreover, the layer of hot fumes is more significant in the numerical image. However, the velocities remain on the same order of magnitude. Although overestimated, the overall behaviour seems consistent with the experimental results. Figure 16 shows results for 500 kW. The overall gas distribution is quite well simulated on the upper part of the image. The re-circulation of fresh gas towards the burner is slightly underestimated but maintains a coherent aspect with that observed experimentally by PIV. To complete the analysis of the numerical and experimental velocities, the results obtained for a power of 800 kW are presented in Figure 17.

On the upper part of these images, a similarity in the velocity distribution and in the hot gas distribution can be observed. However, the second part of the numerical image deviates from what is observed experimentally. Indeed, no fresh air return is present in the numerical results. One can incriminate the validity of the model in under-ventilated conditions.

Vertical velocities -Experimental measurements

Figure 18 shows the vertical component of the velocity profiles for 200 kW, 500 kW and 800 kW power ratings from experiments. In order to get as close as possible to the outlet, the velocities are measured in the upper part of the laser sheet. Velocities are plotted along the x axis.

The colours red, blue and black represent the HRR of 200 kW, 500 kW and 800 kW, respectively.

For the three cases, the first part of the figure (0 to 500 mm) indicates positive velocities and therefore a hot smoke outlet. Higher HRR corresponded to higher velocities. This can linked to the increase of hot fumes which are released with increased powers. In the second part of the image, negative velocities are observed and therefore a fresh air return for 200 kW and 500 kW HRR exists. For the 800 kW power, no fresh air return is observed. Following this observation, oit can be concluded that hot fumes leaving the volume interfere with the entry of fresh air, resulting in only positive velocities.

Conclusion

In this study, the flow topology of a large scale confined fire is evaluated. The test cell is representative of a standard room and the power of the burner went up to 1 MW. A dual approach is proposed. Experimental data for the scalar quantity temperature and 2D velocity fields are recorded and analyzed. Simulations using FDS are performed. The results are listed below.

First, the analysis of temperature evolution against time shows very good agreement between experimental data and simulation results since wellventilated conditions are respected. For under-ventilated conditions, i.e. power of 800kW, discrepancies exist between the experimental and simulation results. FDS is shown to be inadequate for predicting under-ventilated conditions.

Second, large scale PIV data are recorded below the outlet. Beyond the inherent difficulty in managing large scale PIV in confined environment, the position of the smoke layer is shown to be in sensitive to the power inside the test cell. However, an extension of the interface between hot gases and fresh gases is noticed. Third, the comparison of PIV data with 2D fields extracted from the simulations showed that good agreement exists for well-ventilated conditions. Fourth, the evaluation of the maximum velocity against power revealed the clear existence of two behaviors. One for powers below 500 kW, where the maximum velocity is growing with the HRR and one for powers above 500 kW where a strong drop in the maximum velocity is observed before growing again. This illustrates the existence of a quantity which qualifies the ventilated status of the cell. Fifth, the velocity profiles of the vertical component along the x-axis are evaluated for both well-ventilated and under-ventilated conditions. For ventilated conditions, it is shown that a well-established structure where fresh air and hot fumes enter and exit the cell exists. However, for under-ventilated conditions there is no significant fresh air inlet and only the air provided by the injection system facilitates combustion.

In conclusion, the clear existence of well-ventilated and under-ventilated conditions from temperatures and velocity fields is demonstrated. The experimental data is in agreement with the numerical data under well-ventilated conditions. Moreover, we pointed out the criteria based on flow dynamics that might make it easier the determination of ventilation status considering large scale confined fires.

Figure 1 :

 1 Figure 1: Experimental set-up configuration, top view.

Fig 4 .

 4 Particle Image Velocimetry (PIV) configuration is represented in Fig 5.

Figure 2 :

 2 Figure 2: Burner view.

Figure 3 :

 3 Figure 3: Inside view of the test cell.

Figure 4 :

 4 Figure 4: FDS view of the cell.

Figure 5 :

 5 Figure 5: PIV set-up.

Figure 6 :

 6 Figure 6: Grid resolution for 2.5, 5, 10 and 20 cm.

  Grid resolution (cm) Simulation time Cell number 2

Figure 7 :

 7 Figure 7: Temperature profiles at 200 kW (a), 500 kW (b) and 800 kW (c) ; Experimental : red line, Simulation : blue line.

Figure 8 :

 8 Figure 8: (300s)Instantaneous(t=300s), left, and average, right, temperature fields at 200 kW (a, b), 500 kW (c, d) and 800 kW (e,f).

Figure 9 :

 9 Figure 9: HRR evolution against time for 200 kW (a), 500 kW (b) and 800kW (c).

Figure 10

 10 Figure 10 shows the position of the experimental field of view. The blue frame represents the 1 m 2 measuring area. Figure 11 shows velocity fields for HRR ranging from 100 kW, Fig 11(a), 200 kW, Fig 11(b), 300 kW, Fig 11(c) and 400 kW, Fig 11(d). The horizontal velocity U along the x axis is reported. The black frame in Fig 11(a) represents the outlet position and is not reported for other.

Figure 10 :Figure 11 :

 1011 Figure 10: Position of the measuring area.

Figure 12 :

 12 Figure 12: Average velocity fields at 500 kW (a), 600 kW (b), 700 kW (c), 800 kW (d).

Figure 13 :

 13 Figure 13: Average velocity fields at 900 kW (a), 1000 kW (b).

Figure 14 :

 14 Figure 14: Exit velocity evolution as a function of power.

Figure 15 :

 15 Figure 15: Comparison of numerical and experimental velocities, 200 kW.

Figure 16 :

 16 Figure 16: Comparison of numerical and experimental velocities, 500 kW.

Figure 17 :

 17 Figure 17: Comparison of numerical and experimental velocities, 800 kW.

Figure 18 :

 18 Figure 18: Comparison of vertical velocities

Table 1 :

 1 Summary of simulation times.

	spacial resolution. The domain is discretized into more than 365000 cells.
	To ensure reliability between experimental and numerical results, the follo-
	wing points are verified.
	The boundary conditions used in this work are :
	-Total opening of the domain to simulate the outside ;
	-Ramp fire very fast because the fuel is propane ;
	-Consideration of insulation for heat losses.
	To this end, we propose a test matrix which summarizes the test cases for
	both experimental and numerical data, Table 2.

Table 2 :

 2 Summary of simulation.

Acknowledgements

Financial support from the French National Research Agency, under the project 'Firediag' is gratefully acknowledged. This project has been funded with the support from the European Union with the European Regional Development Fund (ERDF) and from the Regional Council of Normandie. The authors thank the computing center CRIANN for the computation time.


Luc Girard

Bappaditya Naskar

Jean-François Dufrêche

Jonathan Lai

Olivier Diat

Pierre Bauduin

A Thermodynamic Model of Non-Ionic Surfactants' Micellization in the Presence of Polyoxometalates

Polyoxometalates (POMs) are nanometric metal-oxide anions with an unmatched range of chemical and physical properties. During the past decade, significant efforts have been made to study POM surface activity and self-assembly properties that are essential for catalysis applications and for producing organic-inorganic hybrid materials. A previous work highlighted the tungstosilicate (SiW12O40 4-) and tungstophosphate (PW12O40 3-) Keggin POM anions' spontaneous and noncovalent adsorption at the micellar surface of nonionic surfactants. In this study, the critical micelle concentrations (CMC) of two non-ionic surfactants, the n-octyl-β-glucoside (C8G1) and the tetraethylene glycol monooctyl ether (C8E4), were measured in the presence of POMs, and we propose herein thermodynamic models to explain an increase or a decrease of the CMC depending on the choice of the POM/surfactant couple.

INTRODUCTION

Polyoxometalates (POMs) are molecular oxo-clusters of the early transition metals in their highest oxidation states. They can be considered either small water soluble oxides or nanoions (1-4 nm). [START_REF] Pope | Polyoxometatlate Chemistry -An old field with new dimensions in several disciplines[END_REF] The variety of their chemical and physical structures at the atomic scale makes them a key nano-particle used in numerous applications including in the medical, analytical and material science fields [START_REF] Long | Polyoxometalate clusters, nanostructures and materials: From self assembly to designer materials and devices[END_REF][START_REF] Long | Polyoxometalates: Building Blocks for Functional Nanoscale Systems[END_REF][START_REF] Blazevic | The Anderson-Evans polyoxometalate: From inorganic building blocks via hybrid organic-inorganic structures to tomorrows "Bio-POM[END_REF][START_REF] Chen | Mo polyoxometalate nanoclusters capable of inhibiting the aggregation of A beta-peptide associated with Alzheimer's disease[END_REF][START_REF] Secheresse | Polyoxometalate Chemistry: Some Recent Trends[END_REF] . Among these properties, the catalytic properties are well known and noteworthy. [START_REF] Hill | HOMOGENEOUS CATALYSIS BY TRANSITION-METAL OXYGEN ANION CLUSTERS[END_REF][START_REF] Zhao | Porous Metal-Organic Frameworks for Heterogeneous Biomimetic Catalysis[END_REF] Their significant development in the last twenty years is also related to their ability to self-assemble in large structures [START_REF] Liu | Self-assembly in aqueous solution of wheel-shaped Mo-154 oxide clusters into vesicles[END_REF] with enhanced physical properties but also to make hybrid organic-POM building blocks for designing smart complex [START_REF] Yang | A Photo-driven Polyoxometalate Complex Shuttle and Its Homogeneous Catalysis and Heterogeneous Separation[END_REF][START_REF] Jallet | Covalent amphiphilic polyoxometalates for the design of biphasic microemulsion systems[END_REF][START_REF] Leclercq | Pickering Emulsion Stabilized by Catalytic Polyoxometalate Nanoparticles: A New Effective Medium for Oxidation Reactions[END_REF] or advanced functional materials and devices. [START_REF] Zhang | Self-Assembly of Organic-Inorganic Hybrid Amphiphilic Surfactants with Large Polyoxometalates as Polar Head Groups[END_REF] Nanometer-sized POM clusters, such as the ones of Kegging type, can be classified as super-chaotropic anions [START_REF] Assaf | Water Structure Recovery in Chaotropic Anion Recognition: High-Affinity Binding of Dodecaborate Clusters to -Cyclodextrin[END_REF] and adsorb at neutral 2 polar interfaces, [START_REF] Naskar | Nanometer-Size Polyoxometalate Anions Adsorb Strongly on Neutral Soft Surfaces[END_REF][START_REF] Chaumont | Polyoxometalate Keggin Anions at Aqueous Interfaces with Organic Solvents, Ionic Liquids, and Graphite: a Molecular Dynamics Study[END_REF] a property that was exploited to organize them into a lyotropic liquid crystal structure. [START_REF] Constantin | Lamellar L-alpha Mesophases Doped with Inorganic Nanoparticles[END_REF][START_REF] Poulos | A PGSE-NMR Study of Molecular Self-Diffusion in Lamellar Phases Doped with Polyoxometalates[END_REF][START_REF] Poulos | Photochromic hybrid organic-inorganic liquid-crystalline materials built from nonionic surfactants and polyoxometalates: Elaboration and structural study[END_REF] This latter property is particularly noteworthy, considering that the weak interactions taking place between the negatively charged POMs and neutral interfaces are of non-electrostatic origin and are sufficiently strong to promote adsorption. Indeed, most of the previous studies aiming at adsorbing POMs on interfaces used the more classical approach based on the electrostatic coupling between cationic interfaces, for instance covered by cationic surfactants, and the negatively charged POMs. Langmuir films were for example made at the water-air interface by using an electrostatic coupling method. [START_REF] Musumeci | Smart High-kappa Nanodielectrics Using Solid Supported Polyoxometalate-Rich Nanostructures[END_REF][START_REF] De Viguerie | Surface pressure induced 2D-crystallization of POM-based surfactants: preparation of nanostructured thin films[END_REF][START_REF] Liu | Polyoxometalate-based functional nanostructured films: Current progress and future prospects[END_REF][START_REF] Kurth | In situ analysis of metallosupramolecular coordination polyelectrolyte films by surface plasmon resonance spectroscopy[END_REF][START_REF] Kurth | Ultrathin composite films incorporating the nanoporous isopolyoxomolybdate[END_REF][START_REF] Liu | Functional polyoxometalate thin films via electrostatic layer-by-layer self-assembly[END_REF] The electrostatic coupling approach between cationic surfactants and POMs was also used many times with other goals, for example to produce catalytic nano-particles [START_REF] Leclercq | Pickering Emulsion Stabilized by Catalytic Polyoxometalate Nanoparticles: A New Effective Medium for Oxidation Reactions[END_REF] or POM selfassembly in non-aqueous solvents. [START_REF] Li | Designed Assembly of Heterometallic Cluster Organic Frameworks Based on Anderson-Type Polyoxometalate Clusters[END_REF] An alternative approach was proposed by using surfactants with POMs as polar heads covalently bound to alkyl chains. [START_REF] Jallet | Covalent amphiphilic polyoxometalates for the design of biphasic microemulsion systems[END_REF][START_REF] Zhang | Self-Assembly of Organic-Inorganic Hybrid Amphiphilic Surfactants with Large Polyoxometalates as Polar Head Groups[END_REF][START_REF] Dufaud | Inorganic Hybrid Materials with Encapsulated Polyoxometalates[END_REF][START_REF] Landsmann | Smart Inorganic Surfactants: More than Surface Tension[END_REF] POMsurfactants proved to be efficient in structuring POM building blocks in bulk and at interfaces by spontaneous self-assembly in 2D or 3D, playing with the wealth of amphiphilic structures and organization in liquid crystals or fluid phases, e.g. micelles or microemulsions.

It appears that the non-covalent approach has many advantages over the electrostatic and covalent ones: (i) it produces more flexible and liquid-like structures (micelles, lyotropic phases) compared to the electrostatic approach, and (ii) it is much less costly and timeconsuming than the covalent approach that requires multiple-step synthesis. However, a key issue for the control of the adsorption and self-assembly properties of POMs in aqueous media is to understand their non-electrostatic interactions with polar non-ionic moieties at interfaces.

In the present study, we investigate how the POMs' adsorption on micelles influences the micellization process in order to go further in the understanding of the non-electrostatic interactions between POMs and non-ionic surfactants. The critical micellar concentration (CMC) values of the octyl-beta-glucoside (C8G1), and the tetraethylene glycol monooctyl ether (C8E4), two non-ionic surfactant systems with identical alkyl chain length and thus volume, were determined by the surface tension measurement at different POM concentrations with PW12O40 3-(or [PW12] 3-) and SiW12O40 4-(or [SiW12] 4-), two Keggin's POMs for which we already know that they adsorb on these micellar surfaces. [START_REF] Naskar | Nanometer-Size Polyoxometalate Anions Adsorb Strongly on Neutral Soft Surfaces[END_REF] The determination of the CMC is therefore used here as a thermodynamic probe of the POM/surfactant interactions [START_REF] Wennerstrom | Micelles -Physical-Chemistry of Surfactant Association[END_REF] and we propose a thermodynamic model of the micellization process to explain our observation.

EXPERIMENTAL SECTION

Materials. H4SiW12O40.12H2O 99.9% and H3PW12O40.12H2O 99.995% were obtained from Aldrich. n-octyl-β-D-monoglucoside (C8G1) 99% was obtained from Anatrace. Tetraethylene glycol monooctyl ether (C8E4) 95,5% was synthesized following the steps described in Naskar et al.. [START_REF] Naskar | Nanometer-Size Polyoxometalate Anions Adsorb Strongly on Neutral Soft Surfaces[END_REF] All the chemicals were used as received unless otherwise stated. Doubly distilled water (κ= 5-6 μS.cm -1 at 25°C) was employed for solution preparation. pH values of the solutions were measured and were always below 4.0, i.e. in the range of stability of the POMs in water.

Methods. The surface tension (γ) was measured as a function of the surfactant concentration using a drop shape analyzer (Krüss DSA 100). The surface tension values are the average values of 5 measurements at least for each point. The absolute precision is typically 1%. The CMC was determined at the apparent break in the surface tension. An error bar is estimated from the error made in the higher γ-slope determination. Various solutions of POMs at fixed concentrations were prepared either in water or in brine (100 mM of NaCl). Stock solutions of concentrated surfactants were also prepared in the same media. The density of each solution was measured at 23°C using a vibrating tube densitometer (Anton Paar DSA 5000).

The liquid/air surface tension was measured after injecting a fixed number of mother solution drops into the cuvette containing the POMs solution. The γ values were accurate within ± 0.1 mN.m -1 .

RESULTS AND DISCUSSION

The micellization of surfactants in water is a thermodynamic mechanism to minimize the free energy of the solution via the aggregation of the surfactants above a critical concentration (CMC). 1B shows the surface tension of C8G1 solutions in the presence of 2, 5, 10 and 25 mM of [PW12] 3-. The critical micelle concentrations were determined from the point of intersection of two straight lines of surface tension vs logarithm bulk concentration curves above and below CMC. The averaged surface tension observed at and above the CMC is called γCMC. CMC and γCMC values were reported in Table 1 for both surfactants. The CMC variations as a function of POM concentration were also plotted in Fig. 2 to highlight the differences between the different surfactant/POM couples.

It was not possible to study the C8E4 / [PW12] 3-system, as the solution turns to turbid at lower concentration of C8E4 and becomes clear at higher concentration of C8E4 (observation for 2 mM [PW12] 3-). It seems to be due to precipitation or coacervate formation at low concentration. [START_REF] Jing | Organic-inorganic macroion coacervate complexation[END_REF] This indicates that [PW12] 3-interacts more strongly with C8E4 than [SiW12] 4-.

The CMC values of C8G1 and C8E4 in water without POMs are in line with literature data, i.e. between 19 and 25 mM for C8G1 [START_REF] Kjellin | Studies of Ndodecyllactobionamide, maltose 6'-O-dodecanoate, and octyl-beta-glucoside with surface tension, surface force, and wetting techniques[END_REF][START_REF] Shinoda | Surface Chemical Properties in Aqueous Solutions of Nonionic Surfactants -Octyl Glycol Ether, Alpha-Octyl Glyceryl Ether and Octyl Glucoside[END_REF][START_REF] Ericsson | Effects of temperature, salt, and deuterium oxide on the self-aggregation of alkylglycosides in dilute solution. 1. n-nonyl-beta-D-glucoside[END_REF][START_REF] Bauer | Hydration of sugar based surfactants under osmotic stress: A SAXS study[END_REF] (titration calorimetry measurement which is known to be very accurate gave 27.1 mM) [START_REF] Király | Adsorption and Aggregation of C8E4 and C8G1 Nonionic Surfactants on Hydrophilic Silica Studied by Calorimetry[END_REF] and around 8 mM for C8E4 [START_REF] Naskar | Nanometer-Size Polyoxometalate Anions Adsorb Strongly on Neutral Soft Surfaces[END_REF][START_REF] Le Maire | Interaction of membrane proteins and lipids with solubilizing detergents[END_REF][START_REF] Vulliezlenormand | Determination of Detergent Critical Micellar Concentration by Solubilization of a Colored Dye[END_REF] (the titration calorimetry value is 8.4 mM). In the presence of POMs, we determined a shift of the CMC to lower values in the case of [SiW12] 4-with C8E4 or [PW12] 3-with C8G1 as expected for an electrolyte effect with non-ionic surfactant. However this effect is significant at very low concentration of POM (below 10 mM) whereas it is observed with such amplitude above 100 mM for usual electrolytes. [START_REF] Ericsson | Effects of temperature, salt, and deuterium oxide on the self-aggregation of alkylglycosides in dilute solution. 1. n-nonyl-beta-D-glucoside[END_REF][START_REF] Molina-Bolivar | Self-assembly, hydration, and structures in N-decanoyl-N-methylglucamide aqueous solutions: Effect of salt addition and temperature[END_REF][START_REF] Miyagishi | Salt effect on critical micelle concentrations of nonionic surfactants, N-acyl-N-methylglucamides (MEGA-n)[END_REF][START_REF] Carale | Salt Effects on Intramicellar Interactions and Micellization of Nonionic Surfactants in Aqueous-Solutions[END_REF] On the other hand we determined a slight increase of the CMC when [SiW12] 4-is added to a solution of C8G1. The CMC were determined from the point of intersection between the highest slope when the surface tension decreases before the CMC and the regime where the surface tension is constant.

We have also observed that the higher the POM concentration, the higher the surface tension at the CMC, , whatever the surfactant/POM couple. This effect has already been observed if we refer to other published works. [START_REF] Matsubara | Interaction between ionic and nonionic surfactants in the adsorbed film and micelle. Dodecylammonium chloride and tetraethylene glycol monooctyl ether[END_REF][START_REF] Micheau | Specific Salt and pH Effects on Foam Film of a pH Sensitive Surfactant[END_REF][START_REF] Bagheri | Synergism between non-ionic and cationic surfactants in a concentration range of mixed monolayers at an air-water interface[END_REF] It was indeed shown that increases for mixed non-ionic and ionic surfactants, increasing the concentration of ionic surfactants or when using a pH sensitive surfactant for which the ratio of charged and non charged polar heads can be tuned. In all these cases an increase of the surface tension at the CMC can be correlated to the concentration of charge at the water/air interface enhancing the surface energy.

The surface tension without surfactant and for different concentrations of POM in solution was measured (see Table 1). Small variations of surface tension were observed in the presence of POMs, which indicates a very weak adsorption at the bare water/air surface, which likely arises from the slight surface activity of protons. [START_REF] Kunz | Osmotic coefficients and surface tensions of aqueous electrolyte solutions: Role of dispersion forces[END_REF] The effect of background salt was also studied for comparison between the C8G1 system and C8E4, with 100 mM of NaCl and at 10 mM of [SiW12] 4-with 100 mM of NaCl. The background salt has a slight effect on the CMC of C8G1 and C8E4 without POM. Indeed for C8G1 the CMC in 100 mM of NaCl (25.7 mM) is slightly lower than the CMC in pure water, i.e. 27.6 mM (see Table 1). As mentioned previously, this is not surprising for a non-ionic surfactant for which CMC is usually not or only slightly affected by the presence of salt at low concentrations. Ion effect on the non-ionic surfactant CMC becomes indeed significant only at high salt concentrations, i.e. in the molar range or when they show specific effect, with the CMC's decrease much more pronounced for "salting-out" ions than for "salting-in" ions. [START_REF] Miyagishi | Salt effect on critical micelle concentrations of nonionic surfactants, N-acyl-N-methylglucamides (MEGA-n)[END_REF][START_REF] Carale | Salt Effects on Intramicellar Interactions and Micellization of Nonionic Surfactants in Aqueous-Solutions[END_REF][START_REF] Palladino | Ionic Strength Effects on the Critical Mice liar Concentration of Ionic and Nonionic Surfactants: The Binding Model[END_REF] On the other hand, a significant CMC decrease is observed for the C8G1/[SiW12] 4-(10 mM) system from 30.1 down to 23.5 mM when adding 100 mM of NaCl whereas an increase was determined adding only 10 mM of [SiW12] 4-POM to the pure system (from 27.6 up to 30.1 mM). This behavior is similar to the effect of the addition of salt to an anionic surfactant solution, which leads to a decrease of the CMC by a screening effect. [START_REF] Palladino | Ionic Strength Effects on the Critical Mice liar Concentration of Ionic and Nonionic Surfactants: The Binding Model[END_REF] The same observation can be done for the C8E4//[SiW12] 4-(10 mM) system. This observation could be qualitatively explained by a screening effect of the charges of an "ionic surfactant" formed by the association of POMs and the non-ionic surfactants.

The affinity of the POMs for surfactants was previously shown by studying the POMs' adsorption onto micellar surfaces using the same surfactants at higher concentration. [START_REF] Naskar | Nanometer-Size Polyoxometalate Anions Adsorb Strongly on Neutral Soft Surfaces[END_REF] SAXS data were indeed analyzed taking into account a strong adsorption of the Keggin-type POMs,

[PW12] 3-and [SiW12] 4-, at the surface of non-ionic micelles made either with C8G1 or C8E4. It was also shown that depending on the chemical nature of the surfactant polar head (glucose or PEG), the POMs were more or less embedded in the polar heads corona and without a direct bond involved between POMs and surfactants. The consequences of the adsorption of POMs on the micelles were a huge increase of the stability of the micellar solution and a change in the micelle shape from elongated to spherical. It was also shown that for C8E4 solutions, the cloud point (CP), i.e. the temperature above which a liquid-liquid phase separation takes place, was increased from 40°C to 90°C by the addition of Keggin POMs. This is a tremendous effect compared to the one of thiocyanate, SCN -, one of the most salting-in anions in the Hofmeister series, for which the CP only increases by a few degrees.

The salting-in effect and the adsorption propensity of POMs onto micelles were shown to be stronger for the more polarizable POMs, [PW12] 3-compared to [SiW12] 4-or other larger POM nanoions such as DAWSON type of polyoxometalates i.e. following the POM's charge density. [START_REF] Buchecker | Polyoxometalates in the Hofmeister series[END_REF] It was proposed that the driving force of the adsorption of POMs mainly originates from the gain of entropy brought about by the release of several hydration water molecules in the bulk. Indeed, a partial dehydration of both the POMs and the surface polar heads is expected along with the adsorption of POMs at the micelle surface. The effect of the addition of POMs on the CMC is significant compared to the effect of classical salts. [START_REF] Molina-Bolivar | Effect of NaCl on the selfaggregation of n-octyl beta-D-thioglucopyranoside in aqueous medium[END_REF][START_REF] Zhang | Electrolyte effects on the surface tension and micellization of n-dodecyl beta-D-maltoside solutions[END_REF] Moreover, the CMC either increases with POM concentration, as for the C8G1/[SiW12] 4-couple or decreases for C8G1/[PW12] 3-and C8E4/[SiW12] 4-couples respectively, as observed in Fig. 2. This significant effect of POMs could be related to their super-chaotropic character or super "salting-in" property.

In the following section an explanation is proposed for the different effects of the addition of

[SiW12] 4-and [PW12] 3-on the CMC of C8G1 and C8E4 using a thermodynamic approach based on the pseudo-phase model and thanks to a statistical mechanic calculation to determine the ∆G of POM adsorption onto the micelles.

B) Phase equilibrium model in micellar solutions of non-ionic surfactants in the presence of POMs.

The well-known pseudo-phase (see SI. 1) is used to describe the system as follows. [START_REF] Shinoda | Pseudo-phase separation model for thermodynamic calculation on micellar solutions[END_REF] This model is generally considered to be valid at high aggregation numbers. In our case, the latter is between 25 to 45, depending on the adsorbed POMs. Thus pseudo-phase is a valid approximation as in that case, straightforward calculations show that the error on the Gibbs energy is around 1.3 % when the aggregation number is 30. Moreover, it has the big advantage that it does not depend on the aggregation number so that only one fitted parameter by equilibria is required.

As shown in Fig. 2, we have observed two different behaviors for the CMC variation versus POMs concentration. We already know that two POMs studied here are adsorbed at the micelle surface of both surfactants. [START_REF] Naskar | Nanometer-Size Polyoxometalate Anions Adsorb Strongly on Neutral Soft Surfaces[END_REF] A CMC increase indicates that the monomer form is stabilized. In order to explain this stabilization, the assumption is made that a fraction of surfactant molecules is not free but associated with POMs despite the fact that we have no direct or indirect evidence of the existence of these POM/monomer association and that the formation of this association is the predominant effect compared to the adsorption on POMs on the micelles. On the opposite, a decrease of the CMC can be explained by a stabilization of the micelles, which suggests that POMs' adsorption onto the micelles becomes the predominant effect compared to the POM/monomer interaction.

Thus, when POMs are added to the system, different species could be considered and the simplest ones are: the monomer m, the POM p, the micelle M, the POM associated with the monomer ma and the micelle associated with the POM Ma, with respective fractions according the three following equilibria and as sketched in Fig. 3:

m ( ) ⇔ m in M
the basic equilibrium as already sketched and described in SI. 1.

m + p ( ) ⇔ ma and m ( ) ⇔ m in Ma
The equilibrium constant K of the Equilibrium (1) can be expressed as (See SI. 1):

= = e _ = e ∆ (1)
Likewise for the Equilibrium (2), we can write the chemical potential equilibrium as:

+ ! " #ln& + ' + ! " #ln& ' = ( + k " #ln& ( (2)
where & ' and & ( are the POM concentration and the concentration of monomers associated with POM.

This new equilibrium constant K can thus be expressed as:

= * ( * * + = e ( , , + = e ∆ ( (3)
and depends on the energy of POM/monomer association, ∆-. .

For the Equilibrium (3), the equality of the chemical potentials can be expressed as:

+ ! " #ln& = ( (4)
with the defined equilibrium constant as:

= = e ( _ = e ∆ /01,3 (5)
Equations ( 1) and ( 5) are not compatible, as the Equilibrium (1) cannot be present together with Equilibrium (3). In other words, only the most stable micellar aggregates, either the micelles without POMs (Equilibrium 1) or the one with POMs (Equilibrium 3) can be present as a pseudo-phase. Thus, on one hand, if 4 ( , the micelle without POMs is the most stable state for aggregation and Equilibrium (1) and ( 2) take place according to Eq. ( 1) and

(3). On the other hand, if 5 ( , the micelle with POMs is the most stable pseudo-phase and Equilibrium (3) take place according to Eq. ( 5).

First case, the monomers are stabilized by the POMs (C8G1/[SiW12] 4-system).

In this first case, the association POM-monomer is non negligible and results in an increase of the surfactant solubility and thus of the CMC. From Eq. ( 1) and ( 3), we get the concentration of monomers associated with POM:

& ( = 676 & ' e ,∆ ( = c ' e ,∆ ( 9∆ (6)
with the micellar critical concentration 676 without POM into the system:

676 = e ∆ (7)
The total concentration of monomers not involved in a micelle reads:

676 = & : + & ( = 676 + & ( (8)
Then, using Eq. ( 6), the CMC can be written as:

676 = 676 ;1 + & ' e ,∆ ( = = 676 >1 + & ' ? (9)
Thus, the model predicts a linear increase of the CMC versus the POM concentration in solution. This expression (Eq. 9) can be compared with the experiments if we express CMC as a function of the total POM concentration:

[POM] = & ' + & ( (10)
From Eq. ( 6) we obtain:

[POM] = & ' E1 + e ,∆ ( 9∆ F (11)
and finally

676 = 676 G1 + [POM] H I JH I K L ( 12
)
By fitting in Fig. 2 the CMC variation versus [SiW12] 4-concentration for C8G1 system, we obtain: for the equilibrium m + p = ma and we find (for details see SI. 2.): ∆-QR ≈ -9 kJ. mol , ∆-. ≈ -5 kJ. mol and = 8.8 L . mol -1 , values that indicate a lower Gibbs energy of micellization compared to the basic system without POM [START_REF] Garrido | STAND: Surface Tension for Aggregation Number Determination[END_REF] but a non negligeable POM/surfactant energy of complex formation.

CMC = 0.
The association constant can be simply expressed as a function of the proportion α of association of surfactant with POM and the total initial concentration of POM and monomers,

c init : = 8,8 L. mol = ] * 0^0_ ( ]) * 0^0_ ( ]) * 0^0_ (13)
If we consider c init = 25 mM, then α = 16 %, which means that about 1/6 of surfactants would be associated with POM for concentrations close to the CMC. Even if the fraction of POM/monomer complexes is weak, it is sufficient to slightly increase the CMC.

Second case, the micelles are stabilized by the POMs (C8G1/[PW12] 3-and C8E4/[SiW12] 4- systems).

As mentioned above, a plausible alternative to explain a decrease of the CMC is that POMs adsorption onto the micelle becomes the predominant effect in comparison to the POM/monomer dimeric complex, which is negligible here. K 3 is now lower than K . Thus only Equilibria (3) has to be considered:

m ( ) ⇔ m in Ma
and characterized by the equilibrium constant that depends on the free energy of micellization (∆-QR,. ) with POMs. Due to the charges of the POMs adsorbed onto the micelles, the previous method based on the equilibrium between the species cannot be solely applied and an approach based on the specific calculation of the free energy of micellization (∆-QR,. ) with POMs has to be used by establishing a lateral equation of state (EOS). [START_REF] Benshaul | Theory of Chain Packing in Amphiphilic Aggregates[END_REF][START_REF] Israelachivili | Intermolecular and Surface Forces[END_REF][START_REF] Bauduin | Perpendicular and lateral equations of state in layered systems of amphiphiles[END_REF][START_REF] Ennis | Spontaneous Curvature of Surfactant Films[END_REF][START_REF] Leontidis | Effects of monovalent anions of the Hofmeister series on DPPC lipid Bilayers part II: Modeling the perpendicular and lateral equation-of-state[END_REF][START_REF] Marcelja | Chain Ordering in Liquid-Crystals .2. Structure of Bilayer Membranes[END_REF][START_REF] Muller | How Do Anions Affect Self-Assembly and Solubility of Cetylpyridinium Surfactants in Water[END_REF] This equation quantifies the different interactions between the surfactant heads within the aggregates (whatever their shape) and also the interactions between the charged adsorbed species (POMs) onto the micelles. For a non-ionic micellar system without POMs, the total free energy of micellization (∆-QR ) is considered to be the sum of two terms that represents i) the hydrocarbon-water free energy, proportional to the water/alkane surface tension and ii) the short-range steric repulsion energy of the surfactant head groups, inversely proportional to the variation of the polar head surfaces (`-` ).

Expressed here per mole of surfactant, the free energy of micellization without POM is usually described by the eq. ( 14) as followed:

∆-QR = a b c d + a ef.g hf' = (`-` ) + . . ( 14
)
with a0 is the minimal head group area and which is fixed, from pure steric considerations, at 0.36 nm 2 for C8G1 and 0.23 nm 2 for C8E4. We assume that the aggregation numbers, the size parameter and the surface tension do not depend on the POMs. Experimentally, the dependence is actually weak and steric effects only slightly depend on the surrounded associations.

The factor c is calculated in the absence of POMs when the function is minimal thus with & = (a -a ) with a the surface per polar head within the micelles taken from literature and fixed at 0.49 nm 2 for C8G1 and 0.45 nm 2 for C8E4. [START_REF] Bauer | Hydration of sugar based surfactants under osmotic stress: A SAXS study[END_REF][START_REF] Matsubara | Surface adsorption and micelle formation of aqueous solutions of polyethyleneglycol and sugar surfactants[END_REF] In the presence of POMs, an additional contribution to the free energy is considered that can be split into two terms and accounts for (i) the adsorption process of the POM, j

The first contribution related to the POM adsorption onto the micelle can be written as a .gs = t .gs (∆-.gs -! " #ln6 uv * )

where ∆-.gs denotes the variation of the Gibbs energy for one POM during the adsorption.

6 uv *
is the POM concentration in the solvent.

For the second term, Felec explicitly stands for the electrostatic energy of the ions adsorbed to the surface and can be expressed as: [START_REF] Muller | How Do Anions Affect Self-Assembly and Solubility of Cetylpyridinium Surfactants in Water[END_REF] a fxfR = (yf) I n (lm

I z{| | } ~ •rr (17)
with z = 3 or 4 the charge number of the POM, e the elementary charge, € and € h the vacuum and relative permittivities of the solvent, respectively. • R'xx is the radius of the micelle (4π• R'xx = at .…… ) and t .gs is the average number of adsorbed POMs onto the micelles.

For the electrostatic formula, eq. ( 17), we should consider that the charges onto the micelle are uniformly distributed and that no correlation between the adsorbed ions is explicitly taken into account. This assumption remains valid when the ion adsorption is strong enough and when their distribution can be considered as a uniform shell of charges around the aggregates.

However, in our case the number of POMs adsorbed is rather low (typically between 4 and 7 [START_REF] Naskar | Nanometer-Size Polyoxometalate Anions Adsorb Strongly on Neutral Soft Surfaces[END_REF] ) so that charge correlations have to be taken into account for ε r (for details see SI. 3).

Finally the total free energy of micellization with POM adsorbed onto micelles can be written as:

∆-QR,. = (`-` ) +

This total free energy is minimized versus t P"" and a in two steps (for details see SI. 4).

From previous fitting of SAXS spectra, [START_REF] Naskar | Nanometer-Size Polyoxometalate Anions Adsorb Strongly on Neutral Soft Surfaces[END_REF]

Taking into account the results of the first step minimization, ∆G QR,. can be expressed now as follows:

∆-QR,. = (a -a ) +

. .

-š√a

with š =

The second step of ∆-QR,. minimization versus a was obtained by the Newton Raphson method (for details see SI. 4) and makes it possible to get its minimum value at each POM's concentration as well as the micellar characteristics i.e the Nagg and a, the surface per polar head.

We found ∆-.gs to be -16 kBT and -22 kBT for C8G1/[PW12] 3-and C8E4/[SiW12] [START_REF] Blazevic | The Anderson-Evans polyoxometalate: From inorganic building blocks via hybrid organic-inorganic structures to tomorrows "Bio-POM[END_REF]-respectively (see dotted lines in Fig. S5 in SI. 5). These values cannot be compared directly as they concern two different POMs adsorbed on different surfactant molecules. Nevertheless, these values confirm the strong interaction of the POM with the polar interface of the non-ionic micelles, whether the sugar heads or the EO chains. The ∆-.gs values may appear rather high but are not so surprising when we consider some recent results concerning the POM crystal formation in aqueous phase using EO oligomers. [START_REF] Buchecker | Polyoxometalate/polyethylene glycol interactions in water: From nano-assemblies in water to crystal formation by electrostatic screening[END_REF] Indeed, EO oligomers/POM crystals can be formed spontaneously at low concentration and increasing the ionic strength, which imply strong attractive interactions between both species in solution. The adsorption of a polarizable anion on a polar surface implies that a few water molecules are released from the hydration shells of both the surface and the ion. [START_REF] Collins | Charge density-dependent strength of hydration and biological structure[END_REF] This partial dehydration process has an enthalpy cost which is higher for small salting-out anions compared to large salting-in or chaotropic anions of low charge density. Moreover, the entropy gain associated to the release of many water molecules into the bulk phase is supposed to be prominent over the enthalpy cost for POMs, owing to their large size and large water shell. [PW12] 3-anion is then expected to interact more strongly than [SiW12] 4-with the surface of non-ionic micelles that are highly hydrated. This adsorption mechanism stabilizes the micelles and results in a decrease of the CMC. This effect is stronger when the POM is adsorbed into the hydrated micellar shell with the release of a large number of water molecules, as for the C8E4/[SiW12] 4- system with a ∆Gads of -22 kBT (as the polyethoxylated chains are more hydrated that the sugar heads). This interaction can even be so strong that POM may become a "sticky anchorage" between two micelles leading to the formation of a coacervate [START_REF] Jing | Organic-inorganic macroion coacervate complexation[END_REF] or a demixion as observed in the case of C8E4/[PW12] 3-system. On the other hand, the lower ∆Gads value obtained for the C8G1/[PW12] 3-system can be explained by the (rigid) sugar head that prevents the penetration of [PW12] 3-in the polar corona of the C8G1 micelle. This model does not take into account that the aggregation number can vary as a function of number of adsorbed POM. However, to check this effect, it will require synchrotron radiation to exploit weak scattering data at the CMC.

CONCLUSION

In this study, we have shown that the variation of the non-ionic surfactant CMC in presence of Keggin POM can be positive or negative as a function of the choice of the polar head/POM couple. Indeed, we observe either a stabilization of the micelle for C8G1/[PW12] 3- and C8E4/[SiW12] 4-system, a stabilization of the monomeric form for C8G1/[SiW12] 4-and even a coacervate for C8E4/[PW12] 3-. Thermodynamic approaches were used to quantitatively describe the CMC increase and decrease. A classical pseudo-phase model was used when the surfactant monomer form was stabilized (CMC increase) whereas a lateral equation of state was applied when the POM adsorption onto the micelles was the dominant effect (CMC decrease). This work contributes to the general understanding of the non-electrostatic interactions taking place in water between POMs and non-ionic amphiphilic molecules and can be extended to different kinds of hydrophilic surfaces or interfaces. It represents a keystone for the future development of innovative materials with original nanostructures and functional properties. Considering also that polyoxometalates (POMs) have been previously investigated for their antimicrobial, anti-tumoral properties, [START_REF] Nabika | Activity of Keggin and Dawson polyoxometalates toward model cell membrane[END_REF][START_REF] Rhule | Polyoxometalates in medicine[END_REF][START_REF] Yang | Polyoxometalate-biomolecule conjugates: A new approach to create hybrid drugs for cancer therapeutics[END_REF][START_REF] Yong | Gadolinium polytungstate nanoclusters: a new theranostic with ultrasmall size and versatile properties for dual-modal MR/CT imaging and photothermal therapy/radiotherapy of cancer[END_REF] to prevent amyloid plaque formation [START_REF] Chen | Mo polyoxometalate nanoclusters capable of inhibiting the aggregation of A beta-peptide associated with Alzheimer's disease[END_REF][START_REF] Geng | Polyoxometalates as Inhibitors of the Aggregation of Amyloid beta Peptides Associated with Alzheimer's Disease[END_REF] and more generally to adsorb on phospholipid membranes, [START_REF] Jing | Polyoxometalate macroion induced phase and morphology instability of lipid membrane[END_REF][START_REF] Kobayashi | Interplay of Hydrophobic and Electrostatic Interactions between Polyoxometalates and Lipid Molecules[END_REF][START_REF] Nabika | Imaging Characterization of Cluster-Induced Morphological Changes of a Model Cell Membrane[END_REF] this work will help to understand their physicochemical properties in a biological medium.

29

 29

A

  ) CMC as a function of POM concentration. CMC values of the two surfactant systems (C8E4 and C8G1) in water or in brine were determined via surface tension measurement as a function of the surfactant concentration and for different POM concentrations. Fig. 1A and 1C show the surface tension of C8E4 and C8G1 solutions respectively in the presence of 0, 5, 10 and 25 mM of [SiW12] 4-. Fig.

Figure 1 .

 1 Figure 1. Surface tension as a function of logarithm of the surfactant concentration at various POM concentrations: A) C8E4 with [SiW12] 4-, B) C8G1 with [PW12] 3-, C) C8G1 with [SiW12] 4-.

Figure 2 .

 2 Figure 2. CMCs' variation as a function of POM concentration in water for the three couples C8G1/[SiW12] 4-, C8G1/[PW12] 3-and C8E4/[SiW12] 4-. Dashed lines correspond to the thermodynamic model presented later in the document.

Figure 3 .

 3 Figure 3. Sketch of the three equilibria of the surfactants taking into account the interaction of POM anions with the non-ionic surfactant in an aqueous medium: the surfactant monomers /micellar aggregation (equilibrium 1), the POM/surfactant association (equilibrium 2) and the surfactant monomers/micellar aggregation taking into account the adsorption of POM ontonon-ionic micelles as already observed in reference (equilibrium 3).[START_REF] Naskar | Nanometer-Size Polyoxometalate Anions Adsorb Strongly on Neutral Soft Surfaces[END_REF]

  196 [MN7]+27.706 (in mmol.L -1 ). Then, we deduce ∆-:O* , ∆-P and the constant =

  repulsions between the POMs adsorbed onto the micelles, j k qrq n (oo p. Nagg is the aggregation number which have to be taken into account to model the curvature of the system (Nagg C8G1 = 25 to 45 and Nagg C8E4 =33 as obtained for a spherical model in presence of POMs 15 ) : ∆-QR,. = a b c d + a ef.g hf' + (k (lm Jk qrq ) n (oo

  the ratio [surfactant]/[POM] on the micelle was estimated to be 4.3 (3 < Nads <6 roughly depending on Nagg and the POMs concentration) for a solution of C8G1 with of [PW12] 3-and 5.3 ( 4< Nagg <6 ) for a solution of C8E4 with [SiW12] 4-. ∆-.gs has been fitted to find a value for Nads close to the experimental value for the same POM concentrations (10 mM). see Fig. S4 in SI. 4) leads to: t .gs = >Ž •x• '' * ∆• (lm ?-{| | } ~ •rr (-˜) I

Table 1 . Micellization parameters at 23°C of C8G1 and C8E4 in water or in brine with various [SiW12] 4-or [PW12] 3-concentrations: CMC and surface tension at CMC.

 1

	Solvent (medium)			C8G1		C8E4
		γsolvent	CMC	γCMC	CMC	γCMC
		/mN . m -1	/mM	/mN . m -1	/mM	/mN . m -1
	H2O	72,6	27.6	30.5	9.2	29.9
	100 mM NaCl	72.3	25.7	32.0	7.5	28.0
	100 mM NaCl +10 mM		23.5	34.5	3.2	32.6
	[SiW12] 4-
	5 mM [SiW12] 4-	72.5	29.4	32.3	5.9	32.1
	10 mM [SiW12] 4-	72.6	30.1	34.1	4.5	34.3
	25 mM [SiW12] 4-	69.3	31.5	35.4	4.7	35.2

mM [PW12]

3- 72.2 20.2 36.3 5 mM [PW12] 3- 72.0 14.8 37.9 10 mM [PW12] 3- 71.2 11.0 39.7 25 mM [PW12] 3- 70.1 11.0 39.0

ACKNOWLEDGEMENTS

The authors would also like to thank Professor Epameinondas Leontidis for his fruitful and helpful discussions. This work was financially supported by the Agence Nationale de la Recherche (ANR CATASURF/ ANR-10-CD21-001 and ANR CELADYCT/ANR-12-BS08-0017 projects).

AUTHOR INFORMATION

Corresponding Author *Fax: (+33) 466 339 288. E-mail: Luc.girard@enscm.fr

Present Address

ICSM, UMR 5257 (CEA, CNRS, UM, ENSCM) CEA Marcoule, BP17171, 30207 Bagnolssur-Céze, France

Notes

The authors declare no competing financial interest.

SUPPORTING INFORMATION Supporting Information Available:

The following information is given in the supplementary file: description of the monomer micelle equilibrium using the pseudo phase model; determination of ∆G QR , ∆G . and the constant for the C8G1/[SiW12] 4-system; determination of ε r and electrostatic energy for a number of POMs adsorbed on the micelle; first minimization of ∆-QR,. as a function of t .gs ; second minimization of ∆-QR,. as a function of `.