Keywords: particles: an intricate behavior

Ces trois années à l'institut d'Alembert ont été extrêmement enrichissantes, tant d'un point de vue scientifique que personnel, et je ne pourrais imaginer y mettre un terme sans remercier chaleureusement les innombrables personnes qui ont fait de cette aventure une véritable épopée humaine, que je garderai longtemps en mémoire. Alors que je pars pour de nouveaux horizons, je ne peux retenir l'émotion de m'envahir à l'idée de tourner la page de ces formidables rencontres. La vie est faite d'avancées, nous en sommes tous conscients, mais le savoir ne rend pas les séparations qui la jonchent moins douloureuses.

Je ne peux commencer ces remerciements sans exprimer ma plus profonde gratitude envers Suzie, sans qui rien n'aurait été possible. Depuis ce jour où je suis venu visiter la salle Savart, comme le petit étudiant de Master que j'étais alors, jusqu'à ma soutenance, tu as su m'accompagner dans mon travail, parler de sciences mais aussi de tout et de rien, me soutenir, te rendre disponible, répondre présente quand j'avais besoin. C'est sans nul doute grâce à toi que jamais l'enthousiasme des manips ne m'a quitté au cours de ma thèse, et pour cela je t'en suis reconnaissant.

Merci également grandement à Christophe, avec qui j'ai pris un grand plaisir à discuter de l'ensemble de mon travail tout au long de ma thèse. Tu t'es rendu disponible sans faillir pendant 3 ans, et tu as toujours su m'apporter le regard de théoricien qui me manquait souvent dans l'étude de mes résultats. Nombre d'analyses seraient sans doute moins abouties si nous n'avions pas confronté nos points de vue au cours du doctorat.

Je souhaite également remercier l'ensemble des membres de mon jury de m'avoir accordé un peu de leur temps. Un merci tout particulier à Anne-Laure Biance et Sébastien Michelin d'avoir accepté le rôle de rapporteur, et d'avoir lu en détail mon manuscrit. Vos rapports m'ont permis d'avoir un regard plus clair sur ce que j'avais fait, et je ne peux en demander plus. Merci également à Étienne Reyssat de m'avoir suivi tout au long de ma thèse en tant que membre de mon comité, nos discussions ont toujours été enrichissantes. Merci enfin à Elizabeth Charlaix, Pierre Jop et Manouk Abkarian, que je ne connaissais pas avant ma soutenance, mais avec qui j'ai pris un vif plaisir à parler de sciences.

Vient désormais le tour du laboratoire, des thésards, ATER, et postdoctorants. Et sur ce point, je pourrais étendre mes remerciements à l'infini, tant j'ai l'impression d'être redevable envers vous tous. Si on m'avait dit avant que je ne commence ma thèse que l'ambiance serait aussi formidable, et que je découvrirais tant de gens incroyables, j'aurais signé encore plus vite. J'ai adoré travailler à vos côtés, discuter de sciences mais aussi et surtout de la vie, faire durer les pauses café, débattre de tout et de rien, déjeuner à vos côtés, reprendre une tisane dans l'après-midi, déguster une bière le soir. En un mot, j'ai adoré avancer avec vous pendant ces trois années. Nous nous sommes connus collègues, nous nous quittons amis, et je ne doute pas que nos routes continueront à se croiser encore longtemps.

Je ne peux me priver de citer plus spécifiquement certains d'entre vous. À commencer par Jeanne, merci d'avoir éclairé ce labo de ta présence, de ta joie, de ton enthousiasme. Je suis vraiment content d'avoir commencé ma thèse en même temps que toi, parce que ça m'a permis de te côtoyer pendant trois ans, et j'espère que ça continuera ! Tu n'es pas la seule aux côtés de qui j'ai eu le plaisir de débuter ma thèse : merci à Alexis, mon co-bureau, que j'ai rencontré pendant mon stage, et avec qui j'ai pu déclamer à l'infini d'innombrables citations de films, toutes plus cultes les unes que les autres ; à Alverède, que j'ai vu sans discontinuer pendant trois ans débarquer dans mon bureau à 11h30 pour partir à la cantine ; à Mathis, fidèle partenaire des pichets de bière.

Une thèse, c'est aussi de nouvelles fournées de doctorants / postdoctorants qui arrivent chaque année. Et avec chacune de ces fournées, l'occasion de découvrir de nouvelles personnes incroyables, et de former de nouvelles amitiés tout aussi fortes que les précédentes. Merci Antoine pour ta bonne humeur, tous ces bons moments partagés en salle de manips

et ailleurs, et pour l'ambiance musicale en Savart (je le sais, le rock celtique te manquera). Merci Cécile pour ta gentillesse, pour cette école d'été qui aurait eu moins de saveur sans toi. J'ai hâte de continuer à vous voir en dehors. Merci aussi Anaïs, c'est toujours un vrai plaisir de discuter avec toi, et j'espère que ça durera. Merci Franck et Adrien, notamment pour ces parties en plein confinement (et pour ces calculs de maths Franck ;) ).

Merci Alice, Antoine, Aidan, Nelson, Serena, Sagar, Hugo, Valentin, Virgile, Paul, Raphael, Alexandre, Quentin. Merci à tous. Vous êtes tous des personnes incroyables, et je profite de cette thèse pour le redire. Sans vous, ces trois années ne m'auraient sans doute laissé qu'un goût d'amertume. Avec vous, elles ont pétillé de bonheur. J'espère que vous le savez déjà, mais je le réécris ici, car j'en ai l'occasion, et car on n'ouvre jamais assez notre coeur : je suis extrêmement heureux d'être votre ami.

Mes remerciements vont aussi au reste du labo, à tous les chercheurs avec qui j'ai eu le plaisir d'interagir pendant ces trois années, et au service administratif et financier qui épaule quotidiennement les doctorants.

Je remercie également grandement mes amis de prépa, que je revois à chaque fois avec un vif plaisir, et mes amis de l'X, de la section bad, que je connais depuis maintenant six ans, mais que j'ai hâte de voir encore pour toute une vie. Merci à Rémi, Émeline et Nico, mes colocs de naguère, avec qui j'ai tant partagé, et que je suis toujours heureux de retrouver. Merci à la team Jullouville, Mog, Paul, et Nico à nouveau, avec qui j'ai toujours autant de plaisir à discuter malgré la distance. Hâte qu'on voyage tous ensemble chez les uns et les autres. Merci Tiphaine, merci Kevin, vous m'avez apporté et vous continuez à m'apporter beaucoup, j'espère continuer à partir en vacances avec vous (le Rhône à vélo n'est-ce pas ?), et à vous retrouver en soirée ou ailleurs. Tiphaine, j'attends maintenant ta soutenance avec impatience ! Merci Élodie et Claire, ces déjeuners entre thésards me manqueront, j'espère qu'on perpétuera d'une manière ou d'une autre cette tradition une fois qu'on travaillera tous ! Je finis par ma famille, parce que tout a commencé par eux, et que leur soutien n'a jamais cessé depuis vingt-six ans. Papa, Maman, merci d'avoir toujours été derrière moi, d'avoir toujours défendu mes choix, d'avoir toujours apporté votre aide de bon coeur, de m'avoir fait grandir tout au long de ma vie. Nombreux sans doute sont les enfants à le dire, mais vous êtes les meilleurs parents possible. Toi aussi Julien, tu es le meilleur grand frère possible. Je sais pouvoir me reposer sur toi quel que soit mon problème, et lorsque je cherche conseil, c'est vers toi que je me tourne. Tu sais comme personne me pousser vers l'avant, quel que soit le sujet. Merci à toi aussi Marine, pour ta bonne humeur et ton enthousiasme. Merci à tous mes cousins, à Charlotte d'être venue assister à ma soutenance, et plus largement à toute ma famille.

À vous tous dont j'ai un jour croisé la route, je vous dois tout. Alors je ne me perdrai pas plus longtemps en palabres, car il suffit d'un mot : MERCI.

Introduction

Capillarity: a daily experience

Our journey in the fabulous world of capillarity starts like most journeys do: with a small coffee to dissipate the remaining mists of sleep. And as we play with a piece of sugar that we place negligently at the surface of our coffee, an amazing phenomenon strikes us. Against all odds, the liquid ascends the sugar cube, defying the almighty gravity itself [START_REF] Jurin | An account of some experiments shown before the royal society; with an enquiry into the cause of some of the ascent and suspension of water in capillary tubes[END_REF]. Disturbed by this counter-intuitive miracle, we try to regain our senses with a cold shower. But as we stare blankly at the water flow ejected by the shower head, a new question pops into our head: how come the water jet destabilizes into a myriad of small droplets (figure 1.a) instead of preserving its integrity [2]?

Besieged by more and more interrogations, and fewer and fewer answers, we flee our house in search of a peaceful haven where our mind will not be disturbed anymore. A quiet lake attracts our attention, and we decide to rest at its edge, letting our gaze wander at the surface of water. A small bug with long legs suddenly appears in our field of view, magically sliding on water without efforts [START_REF] David | The hydrodynamics of water-walking arthropods[END_REF] (figure 1.b). Conscious to be ourselves unable of such a wonder, we are stunned again by this umpteenth twist.

Not really sure if our heart can resist any new capillary marvels, we end up at the restaurant, where we hope to drown our astonishment with a few glasses of wine. This naive attempt fails even more than expected, as we witness long tears of wine flowing down the surface of the glass after the first sip [4] (figure 1 [START_REF] David | The hydrodynamics of water-walking arthropods[END_REF]. c) Tears of wine. The photo is taken from the french Wikipedia article "Larmes de vin", and is in the public domain.

INTRODUCTION

liquids flow, their surface cannot be deformed at will. They can be seen as membrane characterized by a surface tension. This surface tension gives their spherical shape to droplets and bubbles. We will not explain here the origin of surface tension, but a complete review on capillarity and wetting phenomena can be found in [START_REF] De Gennes | Capillarity and wetting phenomena: drops, bubbles, pearls, waves[END_REF].

Capillary forces between two particles

In the present work, we are interested in a particular aspect of capillarity: the interaction between particles floating at an interface. Capillary forces are usually divided into two categories [START_REF] Peter | Capillary forces and structuring in layers of colloid particles[END_REF]: normal forces, as illustrated in figure 2, due to capillary bridges between two small objects, and lateral capillary forces. Normal forces are the consequence of either liquid bridges in gas, or gas bridges in liquid. This leads to an attractive interaction between two particles, or between a particle and a wall. The force is produced by the pressure drop across the curved interface and by the surface tension exerted along the contact line, leading to an attractive force directed normally to the plane of the contact line. As children, we all learn to master normal capillary forces in order to erect the biggest possible sand castle [START_REF] Dj Hornbaker | What keeps sandcastles standing?[END_REF], just as the one of figure 2. But their importance is not limited to the amusement of children: they are responsible for the dispersion of pigments or the adhesion of powders on surfaces, amongst many other things [START_REF] Peter | Capillary bridges and capillary-bridge forces[END_REF].

Normal forces

In this thesis, we will only focus on lateral capillary forces, which arise when the shape of an interface is perturbed by several objects. Here again, two main categories can be drawn: flotation forces, where the particles deform the interface because of their weight, and immersion forces, where the deformation is due to the wetting properties of the bodies. Several situations can lead to immersion forces [START_REF] Peter | Capillary forces between colloidal particles[END_REF]: as illustrated in figure 3, the perturbation of the contact line can appear when particles are confined inside a liquid layer that decays either at infinity or at a finite distance, or when they float at the surface of a liquid. In that case, the force is produced by an irregular contact line due to surface roughness or chemical inhomogeneity [START_REF] Stamou | Long-range attraction between colloidal spheres at the air-water interface: The consequence of an irregular meniscus[END_REF].

Some animals are able to use this effect in order to generate a horizontal force: the larva of the waterlily leaf beetle is a great example (figure 3.e). Because it is a terrestrial animal, it needs to be able to escape a water puddle in order to survive. By curving its own body, this larva manage to interact with a meniscus and propel itself [START_REF] David | Meniscus-climbing insects[END_REF].

But capillary forces are not limited to millimeter-sized objects and insects. Immersion forces exist even for colloidal particles for which gravity can be neglected. Even for Figure 3: Lateral immersion forces between particles. The liquid is represented in blue, while the color white stands for the surrounding fluid. Immersion forces can arise in various situations: if particles are confined in a liquid film that decays at infinity, whether it is in a fluid (a) or against a wall (b), if the meniscus decays at a finite distance (c), or finally if the contact line undulates at the surface of the particles (d). e) Waterlily leaf beetle larva curving its own body to propel at the surface of water. Scale bar: 3 mm. Image taken from [START_REF] David | Meniscus-climbing insects[END_REF]. f) Polystyrene particles of 95 nm and 144 nm in diameter aggregated into a two-dimensional array thanks to lateral capillary forces. Scale bar: 500 nm. Image taken from [START_REF] Yamaki | Size-dependent separation of colloidal particles in two-dimensional convective self-assembly[END_REF].

particles of a few nanometers, capillary forces can overcome brownian motion and lead to aggregation, as demonstrated in figure 3.f where polystyrene particles of approximately 100 nm in diameter are aggregated into a two-dimensional array.

For that type of colloidal particles floating at an interface, the capillary interaction created is not axisymmetric, and a multipolar expansion is needed to account for the undulation of the contact line [START_REF] Krassimir D Danov | Interactions between particles with an undulated contact line at a fluid interface: Capillary multipoles of arbitrary order[END_REF][START_REF] Krassimir | Capillary forces between particles at a liquid interface: General theoretical approach and interactions between capillary multipoles[END_REF] and describe the behavior of monolayers composed by such particles [START_REF] Peter A Kralchevsky | Particles with an undulated contact line at a fluid interface: interaction between capillary quadrupoles and rheology of particulate monolayers[END_REF].

Contrarily to colloidal particles, large objects deform the interface because of their weight: this is what is called flotation forces, as illustrated in figure 4. The deformation can be directed downwards if the particle is denser than water, or upwards if the object considered is lighter (an air bubble for instance). For spherical particles, the deformation is axisymmetric: the bead acts as a monopole.

Flotation forces

Figure 4: Flotation forces between particles. The liquid is represented in blue, while the color white stands for the surrounding fluid. Flotation forces are caused by the weight of the object, leading to lateral forces. If the two objects deflect the interface in the same direction, the force is attractive (two heavy particles for instance), otherwise it is repulsive (a bubble and a dense particle).

When two particles are placed at the interface, their respective menisci overlap, leading INTRODUCTION to:

• an attractive motion if both objects deflect the interface in the same direction;

• a repulsive motion if the two objects deflect the interface in opposite directions.

These flotation forces are going to be at the heart of our work.

But whatever the nature of the lateral capillary forces is, particles at an interface interact, and this interaction may lead to collective dynamics and aggregation.

Assembly of particles

As already seen in figure 3.f, nanometer particles at a liquid interface can aggregate and form two-dimensional arrays. Such a self-assembly is in fact very common in nature and technology [START_REF] George | Self-assembly at all scales[END_REF]. Who has never stared at their cereal bowl during a lethargic breakfast, trying to find the courage to go on with the day, without realizing the intricate physical beauty lying behind the aggregated cereals [START_REF] Walker | The Flying Circus of Physics 2nd edition (with answers)[END_REF]? Who has never cheerfully drunk a glass of champagne to celebrate New Year's Eve, without wondering how come the small bubbles gather before bursting, as illustrated in figure 5.b? Aggregation can also be found at larger scales in nature. Isolated fire ants struggle when placed at the surface of water, but collectively they are able to survive floods by aggregating into a strong waterproof raft, allowing them to remain afloat [START_REF] Nathan J Mlot | Fire ants self-assemble into waterproof rafts to survive floods[END_REF]. A layer of stationary ants constitutes the base of the raft, while the others stay dry on top of them. The raft can then resist high deformations, as demonstrated by figure 5.a. Quite amazingly, the spreading of a sphere of ants deposited at a liquid interface is similar to the spreading of a drop. The raft is constructed quickly, and even more importantly is self-assembling. [START_REF] Nathan J Mlot | Fire ants self-assemble into waterproof rafts to survive floods[END_REF]. b) Aggregated bubbles floating at the surface of water in a petri dish. Photo taken from [START_REF] Vella | The "cheerios effect[END_REF]. c) Cristalline aggregate generated by the assembly of objects of various shapes. Photo taken from [START_REF] Bowden | Self-assembly of mesoscale objects into ordered two-dimensional arrays[END_REF].

Capillary self-clustering is not the prerogative of small animals. A controlled aggregation process can be used to manufacture complex structures, and lead to the fabrication of macroscopic objects with a well-defined microstructure [START_REF] Srinivasan | Microstructure to substrate self-assembly using capillary forces[END_REF], whether in a two-dimensional array [START_REF] Bowden | Self-assembly of mesoscale objects into ordered two-dimensional arrays[END_REF] as illustrated by figure 5.c, or in a three-dimensional configuration [START_REF] Thomas D Clark | Self-assembly of 10-µm-sized objects into ordered three-dimensional arrays[END_REF]. All these techniques rely on a thorough understanding of the capillary forces between particles [START_REF] Peter | Capillary interactions between particles bound to interfaces, liquid films and biomembranes[END_REF] that are generated by the deformation of the interface around each object. When the different menisci overlap, the system is no longer at rest, and the minimization of the total energy leads to a motion in the plane of the interface.

As emphasized by the two examples of figure 5.c, the shape of the interacting objects is a key aspect for a controlled aggregation [START_REF] Botto | Capillary interactions between anisotropic particles[END_REF]. In that specific example, small objects aggregate into two-dimensional self-assembled arrays because of their individual geometry. The shape and wettability of each object govern the direction of the interacting forces, and therefore the structure of the overall system.

Micrometer-sized anisotropic particles, for instance ellipsoids as in figure 6.b, exhibit interactions far greater than thermal agitation or capillary forces generated by spherical particles of a similar size and surface properties [START_REF] Loudet | Capillary interactions between anisotropic colloidal particles[END_REF], and aggregate in specifically-orientated assemblies [START_REF] Lewandowski | Oriented assembly of anisotropic particles by capillary interactions[END_REF].

a) b)

Figure 6: Shape and chemistry. a) Janus particles, with colour-encoded molecules emitting at different wavelengths to show the biphasic nature of their surface. Scale bar: 4 µm. Photo taken from [START_REF] Roh | Biphasic janus particles with nanoscale anisotropy[END_REF]. b) Silicon ellipsoidal particles at an oil-water interface, aggregating side to side. Scale bar: 21 µm. Photo taken from [START_REF] Loudet | Capillary interactions between anisotropic colloidal particles[END_REF].

The aggregation can also be monitored using the surface chemistry of the particles in a controlled way, as introduced by Casagrande and Veyssié in 1989 with Janus particles [START_REF] Casagrande | janus beads": realization and behaviour at water/oil interfaces[END_REF]: nano or microparticles with different chemical properties along their surface (see figure 6.a). Their pioneering work, quickly brought into light by P.-G. de Gennes lecture for the 1991 Nobel Prize, has given rise to a fast improvement of the fabrication techniques [START_REF] Roh | Biphasic janus particles with nanoscale anisotropy[END_REF] as well as numerous studies of their assembly, whether in a single medium [START_REF] Amar | Fabrication, assembly, and application of patchy particles[END_REF] or at a fluid-fluid interface [START_REF] Bp Binks | Particles adsorbed at the oil-water interface: A theoretical comparison between spheres of uniform wettability and "janus" particles[END_REF][START_REF] Bum | Janus particles at an oil-water interface[END_REF].

A precise understanding of the forces at play is necessary to control the aggregation process. In the rest of this thesis, we will only focus on flotation lateral capillary forces, for which the attractive force results from the weight of the particles. We will focus on a specific system called a granular raft, that we will now describe.

Granular rafts

All the experiments will be conducted at an oil-water interface. A fluid-liquid interface is commonly characterized by its capillary length, defined as follows:

INTRODUCTION globally spherical: their surface tension energy that tends to minimize their surface exceeds their gravitational energy. This effect is perfectly illustrated by figure 7, where four droplets of increasing size are aligned on a superhydrophobic substrate. The smaller ones clearly adopt a spherical shape, while the biggest one is flattened by gravity, and adopts the shape of a pancake. From left to right, the size of the droplets increases. Image taken from [START_REF] Séon | Les Lois d'échelle: La physique du petit et du grand[END_REF], gracefully given by T. Séon.

Between oil and water, this very same surface tension is greatly enhanced by buoyancy effects, leading to c ≈ 1 cm. This high capillary length allows greater deformations of the interface, and therefore higher capillary forces (this relation between the deformation and the force will be explored in chapter 1).

For all our experiments throughout this work, a thick layer of silicone oil is poured into a tank filled with pure water, as represented in figure 8. We then sprinkle particles denser than oil and water from above. They cross the first air-oil interface, go through the layer of oil, and finally reach the oil-water interface. There, two behaviors can be observed:

• if the bead is too large or too dense, it crosses again the interface and settles at the bottom of the water tank;

• otherwise, the bead remains stuck at the interface, because of the combined effects of buoyancy and surface tension coming from the deformation of the interface.

The possibility for an object to float at an interface depends on its size, its density, on the fluid properties, the contact angle, but also its shape, its mechanical properties (flexibility for instance) or its surface geometry and chemical properties [START_REF] Vella | Floating versus sinking[END_REF].

Here, we will use spherical particles small enough so that they individually remain afloat. When several particles are deposited at the interface, they interact and start to aggregate, until they form a closely packed monolayer of beads that we call a granular raft, as visualized in figure 8.a. The range of the attractive capillary force is of the order of the capillary length c .

Other aggregating systems at a fluid interface have already been presented in this introduction. The main difference between all these examples and our system is that we use far heavier particles (denser or larger). Gravity is therefore the dominant effect leading to the deformation of our interfaces.

When more and more particles are added, two main behaviors can again be distinguished as the granular raft grows in size. The first possibility is the coverage of the whole interface by the particles, as studied in [START_REF] Jambon-Puillet | Folds in floating membranes: from elastic sheets to granular rafts[END_REF]. In that specific situation, the particle-covered interface can be partially modeled as a heavy elastic sheet [START_REF] Jambon-Puillet | Wrinkles, folds, and plasticity in granular rafts[END_REF] that wrinkles and buckles when compressed, exactly as would do a compressed floating elastic film [START_REF] Jambon-Puillet | The compression of a heavy floating elastic film[END_REF]. This models however fails to account for example for a hysteretic behavior during several cycles of compression, probably due to the granular nature of the raft.

The second possibility is the sinking of the granular raft. Indeed, the interface can be completely covered by particles only if the raft remains stable. But as it grows in size with more and more particles, the raft can become unstable, and the particles can sink collectively even though they were floating individually [START_REF] Vella | Equilibrium conditions for the floating of multiple interfacial objects[END_REF]. This collective destabilization has been studied in [START_REF] Protière | Sinking a granular raft[END_REF], where a critical size at which the granular raft sinks has been derived. Two dimensionless numbers control this behavior:

D 1 = ρ part ρ w -ρ o 2R part c D 2 = (ρ w + ρ o )/2 ρ w -ρ o 2R part c (2)
with ρ part the density of the beads, ρ w the water density, ρ o the oil density, R part the radius of the beads. D 1 and D 2 both come from a modeling of the granular raft as an axisymmetric continuous elastic sheet partially composed of a material with the same density as the beads, and partially of another material with a density of (ρ w + ρ o )/2. D 1 and D 2 compare the weight of the raft (D 1 for the part of the raft of density ρ part , D 2 for the rest of the elastic sheet) with its buoyancy.

For low enough values of D 1 (meaning small enough and light enough particles), the critical size of the raft diverges: it becomes possible to cover the whole interface with particles, as described earlier. In our study, we limit ourselves to higher D 1 , for which INTRODUCTION we always witness the sinking of the granular raft, as visualized in figure 8.b. During this sinking process, the raft detaches from the upper oil phase, creating a stretched oil filament which breaks up into millimeter-scale armored droplets, as for example in figure 8.c. These particle-covered droplets are similar to what is called liquid marbles [START_REF] Aussillous | Liquid marbles[END_REF]: water droplets covered by a hydrophobic powder. The main difference concerns the size of the particles: up to 1 mm in diameter for our armored droplets, while liquid marbles are made of particles with a diameter in the range 0.1-100 µm [START_REF] Aussillous | Properties of liquid marbles[END_REF].

The destabilization can also be triggered by mechanically pushing the raft down with a stick. But whatever the origin of the destabilization, the upper oil phase is encapsulated inside a shell of particles which sinks and remains stable at the bottom of the water tank [START_REF] Abkarian | Gravityinduced encapsulation of liquids by destabilization of granular rafts[END_REF] (see figure 8 Figure 9: Encapsulated droplets. Photos of an encapsulated oil droplet at the bottom of the water tank. Its volume is increased by injecting oil from a syringe from a to d. Images taken from [START_REF] Abkarian | Gravityinduced encapsulation of liquids by destabilization of granular rafts[END_REF].

These encapsulated oil droplets are very stable, as demonstrated by figure 9: they can trap several times the initial volume of oil when injected with oil thanks to a syringe. After a critical volume, the shell of particles will of course no longer be able to contain the oil droplet, leading either to the floating of the whole armored droplet or the pinch-off of the upper portion of the droplet. In that latter case, the particles remain at the bottom of the water tank.

This technique constitutes a non-chemical method to achieve encapsulation of viscous fluids in water, to the difference of all the current treatments used to remedy oil spills for examples. Indeed, burning the oil releases a lot of fumes toxic for the environment and the human health, and using detergent is just a way of hiding the oil spill by dispersing it into micro droplets that can be dangerous for marine life.

Thesis organization

Throughout this thesis, we wish to understand several aspects of the life of a granular raft at an oil-water interface, starting with its birth when two single particles meet, followed by the simultaneous interaction of multiples beads. Once the granular raft is formed, we explore its cohesion and resistance to erosion. Finally, we study its sinking, and more precisely the dynamics of pinch-off of the oil thread formed.

To understand the formation of a granular raft, we start in chapter 1 by the analysis of the attraction between two identical spherical beads floating at the interface. Their respective motion is satisfactorily described by the existing model of the Cheerios effect, that we derive again. These initial experiments are followed by the experimental measurement of the velocity profiles of two non-identical interacting granular rafts. We explore the link between their motion and their respective number of particles, and find out that both the capillary and the drag forces strongly depend on the sizes of the two interacting rafts. This dependence is explained experimentally, numerically and theoretically by looking at the morphology of a granular raft, and especially at the interfacial deformation surrounding it. Once the interaction between two rafts is understood, we briefly explore the attraction between a single raft and a static cylindrical object deforming the interface. Here again, the capillary force experienced by the raft strongly depends on its size and on the external interfacial deformation, but the two-raft model developed earlier fails to account for the motion of the raft. We discuss possible explanations for this disagreement between the experiments and the theory.

In chapter 2, we explore the interaction between numerous granular rafts of various sizes. In that situation, the collective aggregation prevents any analytical solution. However, a statistical description of the aggregation can be undertaken, inspired by the numerous studies on collective motion and self-clustering. Driven by this large literature, we explore the aggregation of many particles initially randomly distributed at the interface, and highlight the role of the initial surface concentration in beads on the time scale of the clustering. The cluster-mass distribution is also studied, again both with experiments and numerical simulations. The distribution of sizes appears to depend on the number of particles and on time, a dependence that we extensively characterize. But despite the very good agreement between the experimental and numerical results, it is still unclear if the clustering follows a self-similar process and if scaling laws can be extracted to describe the aggregation dynamics.

After having studied the formation of a granular raft, we investigate in chapter 3 how a given isolated raft manage to preserve its integrity. By deflecting the interface, we are able to impose a controlled motion on the raft, and examine the conditions leading to its erosion by loss of particles. A cohesion force is deduced from the experiments, but surprisingly, it exceeds by several orders of magnitude the capillary force expected. In order to understand this unexpectedly high cohesion, we develop a model experiment where the attraction between two beads in contact is measured. Here again, their capillary interaction appears to be far stronger than expected by the classical linear theory. A precise description of the contact line around the spheres brings us some insight on this high cohesion.

Finally in chapter 4, we focus on the sinking of the granular raft after its destabilization, and on the dynamics of pinching of the oil thread formed during the process. The measurement of the minimum radius of the thread is compared to classical results on pinch-off dynamics for droplets extruded from a nozzle. The self-similar regimes expected are found for granular rafts, but the shrinking thread seems to oscillate between the two self-similar regimes. To explore this intriguing behavior, we perform a new experiment, where we mimic the sinking of a granular raft by stretching the interface until pinch-off. We show that the oscillation is a function of both the pulling velocity and the initial radial expansion of the liquid thread. We also perform a numerical simulation to confirm these experimental observations. The pinching of a liquid thread covered by particles is also briefly tackled, but because the particles interact with the pinching neck, no clear results have been extracted.

INTRODUCTION

Chapter 1 1.4.2 Interaction between a single particle and a static object . . . . . 1.4.3 Interaction between a raft and a static object . . . . . . . . . . .

Attraction between two granular rafts

Objectives

As presented during the introduction, one of our goal in this thesis is to describe the aggregation of multiples heavy particles at an interface. This is a substantial task that we need to approach step by step. As a consequence, before tackling the clustering of many particles, we need to understand the interaction between two rafts. This is already not an easy problem. Models have been developed for two identical rigid particles, but as soon as we consider either more than two particles [START_REF] Dalbe | Aggregation of frictional particles due to capillary attraction[END_REF] or deformable nonidentical bodies, the global dynamics become more tricky. Here, we want to describe how two granular objects interact, knowing that their shape depends on the number of particles and is not know a priori. This situation is far more intricate than the capillary interaction between two spherical identical beads. With granular rafts, the morphology has also to be solved. And to complicate things even further, the two granular rafts considered will not necessarily have the same number of particles.

After having looked at a first experiment where two rafts are attracted towards one another, we will describe experimentally and theoretically the interaction between two identical beads. The model used for two particles will then be generalized for two granular rafts, whose capillary interaction will be quantitatively described, and then explained by looking at the morphology of a raft experimentally, numerically, and theoretically. Finally, the question of the interaction between a single granular raft and a static object deforming the interface will be addressed.

Some results of this chapter can be found in [START_REF] Lagarde | The capillary interaction between pairs of granular rafts[END_REF].

Granular rafts dynamics

In this section, we want to quantify the forces exerted on a granular raft during the interaction with another raft, and to see how they depend on the number of particles.

The experiment: first results

We begin by looking at the raw measurements one can extract from a typical experiment.

Experimental procedure

The experimental setup is the one presented in the introduction. In a typical experiment, a thick layer of silicone oil (kinematic viscosity ν o = 50.10 -6 m 2 .s -1 , density ρ o = 960 kg.m -3 , oil-water surface tension γ = 38 mN.m -1 ) is carefully poured into a tank of dimensions 0.2 x 0.2 x 0.25 m filled with pure water. The surface tension is measured using the pendant drop technique. In this chapter, we vary neither the type of oil nor its viscosity. The only control parameters are the number of particles, as well as their density and radius (see table 1.1).

A precise number of particles n A are sprinkled from above. We deposit beads far away from one another, from every direction around the raft. Due to the very strong gravitational interaction between such objects (which we aim to describe quantitatively), they automatically aggregate into a compact axisymmetric raft, as visualized from the top view in figure 1.1.b. In this chapter, we use six different types of beads, whose main characteristics are summarized in table 1.1. The particles rearrange themselves quickly after contact, and as a consequence we neither observe a loosely packed assembly such as the one described in [START_REF] Loudet | Capillary interactions between anisotropic colloidal particles[END_REF][START_REF] Xue | Strongly metastable assemblies of particles at liquid interfaces[END_REF], nor an elongated one. The raft radius can be controlled by adding beads progressively.

We reproduce the same procedure elsewhere in the tank, with n B particles. The motion of the two rafts thus formed (respectively identified by the letters A and B) is recorded either from above or from the side by a camera at 250 frames per second, as illustrated in figure 1.1. All velocity measurements are performed using top views, while the side view is only used to visualize the deformation of the interface. The properties of the various beads used in the experiments are summarized in table 1.1, along with the approximate maximum number of particles n sink the corresponding raft can reach before sinking [START_REF] Protière | Sinking a granular raft[END_REF]. At the oil-water interface, the capillary length c = γ/((ρ w -ρ o )g) is greatly increased by buoyancy effects, so that c ≈ 10 mm (with ρ w the density of water, and g the gravitational acceleration). The maximal possible deformation is therefore far more important than at an air-water interface. This leads to unusually high long-range capillary forces.

A first typical experiment: interaction between two rafts of different sizes

As illustrated by the visualization of a typical experiment in figure 1.1, the two rafts move towards one another until they come into contact, at which point they rearrange to form a bigger raft or sink. Here, we focus on the interaction of two rafts before they reach one another. The fluid interface obviously undergoes strong deformations due to the weight of each raft. This induces a vertical displacement which cannot be neglected in our measurement of the raft speed. For example, in figure 1.1.a, the amplitude of the vertical motion of raft B between the last two images is more than half the length of its horizontal motion. In our study, all the speed measurements are derived from top views of the granular raft motion. Yet, a movie taken from above only gives us information on the horizontal projection of the speed. The vertical component is not directly accessible. To overcome this problem, we deduce the vertical displacement from the radial motion, via measurements of the interface deformation.

The equation for the interface is obtained by a classic equilibrium between the hydrostatic pressure and the pressure jump due to the curvature. For a cylindrical coordinate system centered in the middle of the raft, the height of the interface beyond the raft B is obtained as the solution of the following system of equations:

         γ h 1 + h 2 + h r -(ρ w -ρ o )gh 1 + h 2 = 0 h(r → ∞) = 0 h(r = R raf t B ) = h raf t B (1.1a) (1.1b) (1.1c)
with h the height of the interface (h=0 for a flat interface), r the radial coordinate, h the derivative of h with respect to r, R raf t B the radius of the raft and h raf t B the height of the interface at the edge of the raft (see figure 1.2). We neglect the irregularities at the edge of the raft due to the presence of the particles, making the assumption that the shape of the interface around the raft is isotropic. We can thus write equation (1.1) in an axisymmetric configuration.

Oil

Water After having measured R raf t B and h raf t B from a side view of raft B without the presence of A (more details on the morphology of a raft will be given in section 1.3), we solve equation (1.1) numerically, and obtain the expression of h(r) imposed by the presence of the raft B. Then, from the knowledge of the radial position L of the center of A with respect to the center of B, which is measured directly from the movie, we can deduce the expected vertical position of the granular raft A along its motion. Following this procedure, we can calculate the total speed V of a granular raft from its radial displacement. Implicitly here, we use the Nicolson linear approximation [START_REF] Mm Nicolson | The interaction between floating particles[END_REF] by saying that the derivative of the vertical position of raft A only depends on the interface depth imposed by B.

Velocity measurements

The dimensionless distance between the two centers L/R raf t as well as the total speed V of the rafts are plotted in figure 1.3, for two different experiments: the aggregation of two particles (blue curve), and the aggregation of two rafts, made respectively of 30 and 50 particles (red and black curves). The velocity curves have to be read from right to left, with an increase of V as the two rafts get closer, until it reaches a maximum speed. Then, the two rafts briefly slow down just before contact, because of a hydrodynamic coupling in the drag force that will be discussed in the next section: the liquid that separates the two rafts before collision has to be expelled, causing an increase of the drag.

Qualitatively, we can already see that the sizes of the two interacting rafts have a strong effect on their motion. This can be deduced either from figure 1.3.a, where the distance between the two particles (blue curve) decreases much slower than between the two rafts, or from figure 1.3.b, where we can see that the velocity of the 30-particle raft, when attracted by a 50-particle raft, exceeds by more than one order of magnitude the velocity of a single In blue, the two rafts are made of one particle each, while in red and black, one is made of 30 particles, the other of 50 (see figure 1.1 for the experimental visualization). b) Velocity of the two rafts in the same two experiments, as a function of the dimensionless distance between the centers of the two rafts. Red circles: velocity of the 30-particle raft; black squares: velocity of the 50-particle raft, blue squares : velocity of a single particle attracted by another isolated particle of same size. For both figures, the error bars are of the order of the thickness of the curves, and as a consequence are not represented.

particle attracted by another single one. Moreover, the raft made of 30 particles moves at a larger speed than the raft made of 50 particles during their interaction.

To describe more quantitatively these preliminary observations, we need an expression for the interaction of objects at an interface. Luckily, a lot of efforts have been made in the past decades to develop such a model.

Dynamics of two particles

Now that we have a better view of the typical velocity profiles of two interacting rafts, we need to understand the capillary interaction between two particles at an interface, in order to later generalize the theory to two rafts.

The Cheerios effect: a short history

In 1949, Nicolson studied the attractive force between two bubbles floating on a fluid [START_REF] Mm Nicolson | The interaction between floating particles[END_REF]. This pioneering work was used as a cornerstone for most of the work that came after, especially for the assumptions made to conduct the calculation. After the interaction between two bubbles, the capillary attraction between parallel infinite cylinders floating on a liquid was tackled [START_REF] Wa | On the attraction of floating particles[END_REF][START_REF] Allain | Interaction between particles trapped at fluid interfaces: I. exact and asymptotic solutions for the force between two horizontal cylinders[END_REF]. In parallel, the capillary forces between floating particles [START_REF] Pa Kralchevsky | Capillary meniscus interaction between colloidal particles attached to a liquid-fluid interface[END_REF][START_REF] Vn Paunov | Lateral capillary forces between floating submillimeter particles[END_REF][START_REF] Nikolina | Capillary forces between spherical particles floating at a liquid-liquid interface[END_REF] or between a particle and a vertical wall [START_REF] Peter A Kralchevsky | Capillary image forces[END_REF][START_REF] Orlin | Capillary image forces: Ii. experiment[END_REF] were extensively studied, leading to an analytical solution for the capillary force between two particles. Yet, this expression cannot be used directly because it depends on the position of the contact line around the particle, a physical quantity unknown a priori.

Finally in 2005, a calculation conducted by Vella and Mahadevan [START_REF] Vella | The "cheerios effect[END_REF] led to an analytical expression depending only on the fluid and bead parameters. We detail this calculation here.

The Cheerios effect: vertical position of a single particle

The key aspect of this calculation is to first consider the vertical equilibrium of a single particle at an interface [START_REF] Bernard | Colloidal particles at liquid interfaces[END_REF], and then add the second particle to calculate an interaction energy. Let's first consider an isolated particle floating at a fluid-fluid interface, as represented in figure 1 Let's first notice an obvious geometrical relation between the various angles:

ψ s = arctan(ξ s ) = ϕ s + θ -π (1.2)
with ξ s the slope of the interface at the contact line. Then, we just write the vertical equilibrium of forces exerted on the particle: the weight -→ F g , the surface tension around the contact line -→ F γ , and the generalized Archimedes' principle ---→ F archi . This last one is the trickiest one, and needs some explanations.

---→ F archi corresponds to the weight of fluid that would have occupied the volume between the undisturbed interface and the region wetted by the lower liquid, designated by B in figure 1.4.b. A more rigorous justification is given in [START_REF] Joseph | Surface tension force on a partly submerged body[END_REF]. To simplify the calculation, we will neglect the effect of the upper fluid (as if it was air), but a similar calculation can be performed if we take into account the two fluids. This volume Ω B is then the sum of a spherical cap and a cylinder:

Ω B = πR 2 part sin 2 ϕ s ξ s cylinder + π 3 (R part (1 -cos ϕ s )) 2 (3R part -R part (1 -cos ϕ s )) spherical cap (1.3)
where all the parameters are defined in figure 1.4. The three vertical forces, projected onto an upward vertical axis, give the following equations:

         F g = - 4 3 πρ part R 3 part g F γ = 2πγR part sin ϕ s sin(arctan(ξ s )) F archi = ρ w gΩ B (1.4a) (1.4b) (1.4c)
with g the acceleration of gravity, ρ part the density of the particle, γ the surface tension, and ρ w the density of the lower fluid. The equilibrium of forces gives:

4 3 πρ part R 3 part g =2πγR part sin ϕ s ξ s 1 + ξ 2 s + ρ w gπR 2 part ξ s sin 2 ϕ s + R part 3 (1 -cos ϕ s ) 2 (2 + cos ϕ s ) (1.5)
where we used a classical trigonometric relation for sin(arctan(ξ s )). Then, we use equation (1.2) to substitue ϕ s in the buoyancy term of equation (1.5), and we linearize in ξ s by assuming a small slope of the interface:

cos ϕ s = cos(π -θ + arctan(ξ s )) = -cos θ -ξ s sin θ sin ϕ s = cos(π -θ + arctan(ξ s )) = sin θ -ξ s cos θ (1.6a) (1.6b)
Using equation (1.5), we then look for an expression of ξ s sin ϕ s :

ξ s sin ϕ s = 2 3 BD -B 1 3 + 1 2 cos θ - 1 6 cos 3 θ -B 1 2 ξ s sin θ(1 -3 cos 2 θ) + ξ s 2 c B 1/2 sin 2 θ - ξ s ξ s R part sin θ cos θ (1.7)
where D = ρ part /ρ w is the particle-liquid density ratio, B = R 2 part / 2 c is the Bond number of the system, and c = γ/(ρ w g) the capillary length.

Then, we simplify again equation (1.7) for small Bond numbers, small deflections of the interface, and small particles (basically we get rid of the second line of equation (1.7)), leading to the following expression:

ξ s sin ϕ s = BΣ (1.8)
where Σ = 2D-1

3

-1 2 cos θ + 1 6 cos 3 θ is a dimensionless number. We also need to compute the depth h of the interface at a distance r from the particle. To do so, we assume the interface to be axisymmetric, and we linearize equation (1.1) as follows:

     ∇ 2 h = h/ 2 c h(r → ∞) = 0 h (r = R part sin ϕ s ) = ξ s (1.9a) (1.9b) (1.9c)
The solution of equation (1.9) is very well known:

h(r) = - ξ s c K 1 (R part sin ϕ s / c ) K 0 (r/ c ) ≈ -BΣR part K 0 (r/ c ) (1.10)
with K i the modified Bessel function of the second kind of order i. We used the asymptotic expansion of K 1 (x) for x 1:

K 1 (x) ∼ x→0 1/x.
The Cheerios effect: the capillary force between two particles

We now have everything we need to compute the interaction energy between two particles, with a few more assumptions. First, we are going to assume that the only horizontal contribution of the forces comes from surface tension. Doing so, we neglect the hydrostatic pressure contribution that comes from the tilting of the contact line around the particle (the pressure is then different depending on the azimuthal position around the particle).

Using again the Nicolson approximation [START_REF] Mm Nicolson | The interaction between floating particles[END_REF], the energy of interaction can be expressed as the product between the effective weight of the particle F g + F archi = 2πγR part BΣ with the vertical deflection of the interface generated by the other particle [START_REF] Allain | Interaction between particles trapped at fluid interfaces: I. exact and asymptotic solutions for the force between two horizontal cylinders[END_REF][START_REF] Vella | The "cheerios effect[END_REF][START_REF] Dalbe | Aggregation of frictional particles due to capillary attraction[END_REF]:

E(r) = 2πγR part BΣ(-BΣR part K 0 (r/ c )) (1.11)
By deriving equation (1.11) with respect to r, we end up with the final expression for the horizontal force between two identical particles at a fluid-air interface:

F cap 1→1 (r) = 2πγR part B 5/2 Σ 2 K 1 (r/ c ) (1.12)
where the subscript of F cap 1→1 indicates that we are looking at the force exerted by one particle on another one. In our situation, the contribution of the upper fluid in the vertical equilibrium needs to be taken into account. A similar calculation leads to the same expression [START_REF] Dani | Hydrodynamics of particles at an oil-water interface[END_REF], with a modified dimensionless number Σ:

         Σ = χ o + χ w 3 - 1 2 cos θ + 1 6 cos 3 θ χ o = (ρ part -ρ o )/(ρ w -ρ o ) χ w = (ρ part -ρ w )/(ρ w -ρ o ) (1.13a) (1.13b) (1.13c)
with ρ part , ρ o and ρ w the solid, oil and water densities (from now on, we will always consider an oil-water interface). A more accurate calculation, performed by Cooray et al [START_REF] Cooray | Floating and sinking of a pair of spheres at a liquid-fluid interface[END_REF], will be presented in chapter 3, where fewer assumptions are made, especially on the axisymmetry of the interface. Both surface tension and hydrostatic pressure will be taken into account in the calculation of the attractive force.

Balance of forces for particles at an oil-water interface

F cap 1→1 is counterbalanced by a drag around each particle F drag 1 , as illustrated by figure 3.18. Here again, we need a few assumptions. First, we consider the beads to be only slowed down by the oil phase. For two identical beads of one millimeter attracted by one another, the typical velocity V is equal to a few mm/s. We can therefore estimate the Reynolds number

Re = ρ o R part V /µ o ≈ 0.02 1.
We also estimate the Capillary number Ca = µ o V /γ ≈ 0.002 1, meaning we can neglect the motion of the contact line.

Oil

Water Figure 1.5: Forces acting on a particle attracted by another. Schematic representation of the interaction between two beads at an oil-water interface. L is the distance between the centers of the two particles. They are both subjected to a capillary attraction F cap 1→1 and a viscous drag F drag 1 .

The drag force acting on a particle F drag 1 can therefore be expressed as a Stokes drag corrected by the mobility function G for two spheres in a single phase, in order to take into account the drainage of the liquid between the particles as they get closer:

F drag 1 = 6πµ o R part kV G -1 L R part (1.14)
where L is the distance between the centers of the two particles, and k accounts for the fact that the particles move along an interface and as a consequence are immersed in two phases [START_REF] Dani | Hydrodynamics of particles at an oil-water interface[END_REF]. The mobility function G was tabulated by Batchelor [START_REF] Batchelor | Brownian diffusion of particles with hydrodynamic interaction[END_REF][START_REF] Batchelor | The hydrodynamic interaction of two small freelymoving spheres in a linear flow field[END_REF], and can be approximated by various interpolation formulae [START_REF] Nikolina | Capillary forces between spherical particles floating at a liquid-liquid interface[END_REF][START_REF] Boneva | Effect of electric-field-induced capillary attraction on the motion of particles at an oil-water interface[END_REF]. We will use the following one:

G(x) = 1 - 3 2 x -1 + x -3 - 15 4 x -4 - 4.46 1000 (x -1.7) -2.867 (1.15)
At infinity, there is no coupling and G(+∞) = 1, whereas when the two particles are in contact, G(2) = 0, which ensure a contact between the two particles with a zero velocity.

Because Re 1, there is no inertia in our system, so the equilibrium of forces directly gives us an expression of the velocity of one particle separated by a distance L from the other:

V = γB 5/2 Σ 2 3µ o k G L R part K 1 L c (1.16)
The only unknown parameter is k, but we expect its value to be in the range [0.4 ; 1] [56], since at first order, the particle is half surrounded by oil, and half by water.

Attraction between two beads: experimental results

This model can be checked experimentally very easily, by simply recording the aggregation of two identical particles and measuring their velocity, as illustrated in figure 1.6. As predicted by the model, we do recover an increasing velocity (figure 1.6.b) until a maximum, at which point the velocity decreases because of the drainage of the liquid separating the two beads. The parameter k of equation (1.16) is taken as a fitting parameter, since we do not know a priori its value. As emphasized by figure 1.6, we perfectly describe the aggregation dynamics of the two particles. Moreover, as expected, we end up with a value of k between 0.5 and 1.

The interaction between two particles is therefore satisfactorily explained. Both the drag and the capillary forces have been expressed, and their balance leads to the expected velocity profiles. The goal is now to look at the aggregation of two granular rafts, and to quantitatively understand how this velocity profile is modified by the number of particles in each raft.

Kinematics of two granular rafts

The model derived for two particles needs to be generalized in order to account for the motion of two granular rafts.

CHAPTER 1. ATTRACTION BETWEEN TWO GRANULAR RAFTS

Forces acting on a raft

To describe the dynamics of a granular raft, we adapt the previous model, that was developed for only two small identical spherical particles. What we want to know is how the capillary force and the drag force between two rafts A and B (see figure 1.7) depend on the number of particles in each raft. To that end, we generalize equation (1.12) to the attraction of two granular rafts, as schematically represented in figure 1.7:

F cap A→B = f (n A , n B )a(R part , ρ part , ...)K 1 L c (1.17)
where f is the function we want to determine experimentally, F cap A→B the force exerted by the raft A on B, n A and n B the number of particles in each raft, and a(R part , ρ part , ...) = 2πγRB 5/2 Σ 2 .

To make such a generalization, we assume here that the wavelength and amplitude of the undulation of the edge of the raft are small enough so that they can be neglected at long range and the interface around the raft can be described as isotropic. We neglect here the granular nature of the raft, the edge of which may be roughened by individual particles. Similarly, we generalize equation (1.14) to the motion of a granular raft:

Oil

F drag A = g(n A )bV A G -1 l + 2R part R part (1.18)
where g is the function we want to determine experimentally, b = 6πµ o R part k the Stokes coefficient, and l designates the distance between the edges of the two closest particles of the two rafts, as defined in figure 1.7. We approximate the hydrodynamic coupling between the two rafts by considering the drainage of the liquid only between the two closest particles of each interacting raft. Such a formulation of the drag is only valid for a sphere, but here we use it for a granular raft, which has the shape of a curved disk. As a consequence, the scaling law for the drag should also differ from the Stokes drag of a sphere, although we expect the general scaling laws for the Stokes drag to be valid. Keeping in mind all the assumptions lying under the scaling law of equation (1.18), we combine it with equation (1.17) and finally deduce an expression for the speed of a raft made of n A particles attracted by a second raft constituted by n B beads:

V A = f (n A , n B ) g(n A ) a b G l + 2R part R part K 1 L c (1.19)
The ratio a/b is measured once and for all for each type of particle (given radius and given density, see table 1.1) thanks to a two-bead experiment, as for instance represented in figure 1.6, for which f (1, 1) = g(1) = 1 by definition. Then, f /g can be experimentally determined as a function of n A and n B , giving us information on the ratio of the two forces.

Influence of the number of beads on the velocity profiles

By varying the density and size of the beads, we explore different ratios a/b, and for each one, we can change n A and n B . Some results are displayed in figure 1.8, where we plot the speed of the raft A as a function of the distance between the centers of the two rafts L, first with n A = 1 and an increasing n B , and then with n B = 60 and an increasing n A , for a given type of particle. The speed of the raft appears to increase both with n A and n B . This is due to the deformation of the interface that increases with n B . Similarly, F cap B→A is related to the weight and the size of the raft A, an increasing quantity with n A . Focusing on a single curve of figure 1.8, we recover the behavior described previously: an increasing speed as the two rafts are attracted towards one another, until a maximum where the speed starts to decrease due to the drainage of the liquid between the two rafts.

As n A or n B increases, the curves are shifted to the right since L min = R raf t A + R raf t B , a value which increases as we add particles. The black dotted curves of figure 1.8 represent the fit of equation (1.19) for each couple of rafts (n A , n B ). There is a good agreement between the experimental data and the theory for n A = 1 (figure 1.8.a), the fitting line being within the experimental noise. As we increase n A , the fit starts to drift from the measured speed, in particular regarding the position of its maximum. This is not so surprising regarding the numerous assumptions made in our model. These limitations will be discussed extensively in section 1.3.6.

Influence of the number of beads on the forces

Keeping in mind that for high n A , our model does not explain the entire dynamics, we observe the evolution of the fitting coefficient f /g as a function of n A and n B for different types of beads (see table 1.1), and a large variety of combinations of n A and n B (figure 1.9).

CHAPTER 1. ATTRACTION BETWEEN TWO GRANULAR RAFTS

All the data collapse onto the same straight line when represented as a function of n B √ n A , on approximately three decades, giving the following experimental result: Equation (1.20) gives us an empirical relation between the ratio of the two forces and the number of particles in each raft. Even though both forces are functions of n A , n B only intervenes in the capillary force. This immediately proves that F cap A→B ∝ n B . Moreover, applying Newton's third law [START_REF] Peter | Capillary forces between colloidal particles[END_REF][START_REF] Vn Paunov | Capillary meniscus interaction between a microparticle and a wall[END_REF][START_REF] Orlin | Direct measurement of lateral capillary forces[END_REF], we obtain that this force has to be symmetrical in n A and n B and thus F cap A→B ∝ n A n B , implying also for the drag force g(n A ) ∝ √ n A . The far-field behavior of the capillary force F cap A→B ∝ n A n B follows a Coulomb or gravitational-like law, replacing the electric charge or the mass by the number of grains; since this force only depends on the number of grains, it indicates that the organization of the particles within such compact granular rafts plays no role in the interaction. This is consistent with the superposition approximation, where interface deformation at large distances can be considered as the sum of the individual menisci of each particle. However, we lack a physical understanding of these scalings and need to go back to the definition of the capillary force, and to morphological measurements of a granular raft.

f (n A , n B ) g(n A ) = n B √ n A (1.

Granular raft morphology

By looking at the velocity profiles of two interacting rafts of various sizes, we have been able to quantify the dependence of the forces at play with the number of particles:

F cap A→B ∝ n A n B and F drag A ∝ √ n A .
Nevertheless, these scaling laws still appear obscure, which is why we need to link them to the actual deformation of the interface generated by the rafts.

Modified aspect ratio of a raft

As explained in section 1.2.2, the capillary force exerted between two particles, assuming an isotropic shape and small deformations of the interface, is proportional to the slope of the interface deformed by the presence of one sphere multiplied by the weight of the other. Using the same approach, we can assume the capillary force exerted by a raft is also proportional to the slope of the surrounding interface. In the limit of small deformations, we can derive an analytical formula for this slope by solving a linearized version of equation (1.1), as we did for the calculation of the capillary force between two beads:

         ∇ 2 h = h 2 c h(r → ∞) = 0 h(r = R raf t ) = h raf t (1.21a) (1.21b) (1.21c)
which gives us the following expression: .22) with K 0 the modified Bessel function of the second kind of order zero, R raf t the radius of the raft, and h raf t the depth of the edge of the raft. Equation (1.22) can be reconciled with the scaling F cap A→B ∝ n A if we show that the modified aspect ratio of a raft

dh dL = h raf t / c K 0 (R raf t / c ) K 1 L c . ( 1
h raf t / c K 0 (R raf t / c)
is linear with its number of particles n. Experimentally, we can easily have access to the value of this aspect ratio, as demonstrated in the following section. In section 1.3.3, the geometry of a raft will be described with numerical simulations, and finally, a theoretical model will be developed in section 1.3.4.

Morphology of a raft: experiments

We start by looking experimentally at a measurement of the modified aspect ratio

h raf t / c K 0 (R raf t / c) .
We measure h raf t from side views of the granular raft (blue arrows, figure 1.10.a). We then measure the area of the raft A raf t from top views and calculate the equivalent radius by R raf t = A raf t /π. The experimental modified aspect ratio is plotted in figure 1.10.b for different types of beads and raft sizes, as a function of the effective weight of a raft F vert , that we define as the difference between the weight of the particles and their buoyancy:

F vert (n) = nF vert (1) = n 4 3 πR 3 part g(ρ part - ρ o + ρ w 2 ) (1.23)
with n the number of particles in the raft. In equation (1.23), both the weight and the buoyancy of the particles are considered linear with n, with the supplementary assumption that a bead is equally immersed in oil and water. However, we are aware that the generalized buoyancy of the raft, which takes into account surface tension effect, is not strictly additive, as discussed in previous collective sinking experiments [START_REF] Vella | Floating versus sinking[END_REF].

All the data collapse onto a single curve, and the modified aspect ratio appears linear with F vert (a power law regression on the experimental data gives an exponent of 1.01), which is by definition proportional to n. As a consequence, the modified aspect ratio appears to be linear with the number of particles.

In order to consolidate this experimental result, we turn to numerical simulations.

Morphology of a raft: numerics

We want to calculate the shape of a granular raft, in order to add new evidence to the previous experimental measurement of the modified aspect ratio. As a consequence, we need to be able to model a granular raft floating at an interface.

Model for the numerics

We use the code described in [START_REF] Protière | Sinking a granular raft[END_REF]. The raft is simulated as an axisymmetric continuous membrane of thickness b = 2R part , with a given volume fraction φ part of a material of density ρ part , a complementary volume fraction of a material of density (ρ o + ρ w )/2, and a tension -→ T directed tangentially to the raft surface. First, let us express the two vectors -→ t and -→ n , as well as the elementary distance ds, as function of the local slope of the granular raft h :

- → t = 1 √ 1 + h 2   1 0 h   , - → n = 1 √ 1 + h 2   -h 0 1   , ds = dr 1 + h 2 (1.24)
By using these three formulas, we can now express the different forces acting on the elementary volume of figure 1.11.b in the local coordinate system ( -→ t , -→ n ): the weight -→ F g , the balance of tension ∆ -→ T , and the hydrostatic forces --→ F hyd . For the calculation of --→ F hyd , we need to compute the depth of the upper and lower surface of the elementary volume, a calculation that is a little tricky and may need a few comments. For a point localized in (r, θ, h), the center line of the membrane has a depth of h ∞ -h, with h ∞ defined in figure 1.11. The depth of the lower surface can be expressed as h ∞ -h + x, with x defined in figure 1.11.c. Simple geometrical relations lead to:

x = b 2 cos α = b 2 dr ds = b 2 1 √ 1 + h 2 (1.25)
with α the inclination angle, as defined in figure 1.11.c. The three forces can now be written as follows:

                         -→ F g = -φ part ρ part + (1 -φ part ) ρ o + ρ w 2 bdsgrdθ× h √ 1 + h 2 - → t + 1 √ 1 + h 2 - → n ∆ - → T = (T (r + dr) - → t (r + dr)(r + dr) -T (r) - → t (r)r)dθ + T (r)ds( ---→ e θ+dθ -- → e θ ) --→ F hyd = ρ w g h ∞ -h + b/2 √ 1 + h 2 -ρ o g h ∞ -h - b/2 √ 1 + h 2 dsrdθ - → n (1.26a) (1.26b) (1.26c)
Let us simplify the equation of equilibrium of the tension (1.26b):

∆ - → T = (r dT dr - → t + rT d - → t dr + T - → t )drdθ -T (r) 1 + h 2 drdθ - → e r = (r dT dr - → t + rT d - → t dr + T - → t )drdθ- T (r) 1 + h 2 drdθ( 1 √ 1 + h 2 - → t + -h √ 1 + h 2 - → n ) = ( dT dr - → t + T ( d - → t dr + h r - → n ))rdrdθ (1.27a) (1.27b) (1.27c) d -→ t
dr can be easily calculated:

d - → t dr = - h h (1 + h 2 ) 3/2   1 0 h   + 1 (1 + h 2 ) 1/2   0 0 h   = h (1 + h 2 ) 3/2   -h 0 1   = h 1 + h 2 - → n (1.28) (1.29) (1.30)
We end up with a set of two equations, corresponding to the normal and tangential force balances:

                   T (r) -φ part ρ part + (1 -φ part ) ρ o + ρ w 2 bgh (r) = 0 T (r) h 1 + h 2 + h r + g(h ∞ -h)(ρ w -ρ o ) 1 + h 2 + bgφ part ρ o + ρ w 2 -ρ part = 0 (1.31a) (1.31b)
These two equations are only valid where the raft is present. Beyond, the interface will be described by the classical equation for a fluid-fluid interface that we already used a couple of times (equation (1.1)). For the boundary conditions, we take h (0) = 0 (the raft is flat in its center), h(r → ∞) = h ∞ (undisturbed interface at infinity), and T (r = R raf t ) = γ (continuity of the tension at the edge of the raft).

A dimensionless version of the system of equation (1.31) is then used. As explained in [START_REF] Protière | Sinking a granular raft[END_REF], the shape of the raft h(r) is computed with a relaxation scheme, and as long as the tension remains positive, a static solution is found. This can be used to predict the stability of a granular raft, or to compute its morphology, and deduce its modified aspect ratio, as defined in equation (1.22).

Geometry of the simulated interface

The simulated interface is plotted on top of experimental visualizations of the corresponding granular raft in figure 1.12.a, with a visually good agreement for the top three photos. For the last two granular rafts shown in figure 1.12.a, no numerical result is available: the simulations predict the sinking of the raft for such a deformation. This is probably due to the simplicity of the model used for the simulations, the limitations of which are discussed in [START_REF] Protière | Sinking a granular raft[END_REF]. As a consequence, the numerics can only give us information for part of the sizes accessible experimentally. However, they allow us to explore a far broader range of densities and radii of particles, and appear in that sense complementary to the experiments.

The numerical measurements of the modified aspect ratio are represented in colored circles in figure 1.12. Here again, the data collapse onto the same curve when represented as a function of F vert , over more than four decades, with an exponent of 1.08 for a power fitting law. In conclusion, we recover the following result both experimentally and numerically:

h raf t / c K 0 (R raf t / c ) ∝ n (1.32)
Interestingly, in figure 1.12.b, we find that beyond a critical vertical force, the linear relation between the modified aspect ratio and F vert no longer holds (see the right portion of the blue curve). Such a curve corresponds to a situation where the raft reaches a maximal depth which corresponds to a balance between the hydrostatic pressure and the raft weight, and then extends indefinitely as more particles are added [START_REF] Protière | Sinking a granular raft[END_REF]. As a consequence, h raf t is bounded, leading to the saturation of the capillary force. Since K 0 (R raf t / c ) ∼ π 2R raf t e -R raf t and F vert ∝ R 2 raf t , it explains the vertical deviation in figure 1.12.b for large rafts.

Here again, the modified aspect ratio appears to be linear with the number of particles, in total agreement with the experiments. But we can try to do even better, by finding again this result with a minimal theoretical model.

Morphology of a raft: theory

To the previous experimental and numerical evidence, we propose to add a theoretical justification to this linear relation between the modified aspect ratio

h raf t / c K 0 (R raf t / c
) and the number of particles.

To account for the vertical force the raft is imposing on the interface, we distribute F vert along the perimeter of the raft, through the introduction of the pressure P at the point -→ r :

P ( - → r ) = F vert 2πR raf t δ( - → r -R raf t ) (1.33)
where δ is a Dirac function. We then calculate the pressure jump at the oil-water interface, for -→ r ≥ R raf t , in the limit of small deformations. The classic differential equation that describes the height of the interface h is modified by the presence of P :

∇ 2 - 1 2 c h = P ( - → r ) γ . (1.34)
The solution of equation (1.34) is given by the convolution of P and the green function

G( - → r , - → s ) = -1 2π K 0 ( -→ r --→ s c
), associated with the linear operator (∇ 2 -1/ 2 c ). K 0 is the modified Bessel function of the second kind of order zero.

h( - → r ) = 2 G( - → r , - → s ) P ( - → s ) γ sdsdα h( - → r ) = - F vert 4π 2 γ 2π 0 K 0   - → r 2 + R 2 raf t -2R raf t - → r cos α c   dα (1.35a) (1.35b)
with α the azimuthal angle. Because we are only interested in the depth of the interface at the edge of the raft, we apply the previous equation for -→ r = R raf t :

h raf t = F vert π 2 γ π/2 0 K 0 2R raf t c
sin α dα.

(1.36)

In the limit R raf t c , we can go further with the calculation, using K 0 (x) ∼ x→0 -ln(x):

h raf t ∼ - F vert π 2 γ π/2 0 ln 2R raf t c sin α dα h raf t ∼ - F vert 2πγ ln R raf t c . (1.37a) (1.37b)
Our minimal model therefore gives the following result (in the limit of small deformations and small granular rafts):

-

h raf t / c ln(R raf t / c ) ∼ F vert 2πγ c ∝ n, (1.38)
which corresponds to the previous experimental and numerical results observed in the same limits. All these results validate that:

F cap B→A = n A n B F cap 1→1 . (1.39)
where F cap 1→1 stands for the capillary force between two isolated identical particles.

The drag force

Finally, we need to explain how the drag force depends on the raft geometry:

F drag A = √ n A F drag 1 (1.40)
with F drag 1 the drag force on a single sphere at the interface. One could understand the drag dependency with n A by considering the equivalent sphere of radius R raf t . For such a sphere, the Stokes drag is proportional to R raf t , which for a large range of granular raft sizes evolves as √ n, as demonstrated by figure 1.13. This argument holds even for the largest granular rafts (see for instance the last photo of figure 1.12), for which we could have expected the high curvature of the raft to make R raf t deviate from this law. However at first order, R raf t ∝ √ n, leading to the expected drag on the granular raft.

But even with this argument, the proportionality between the drag force and the raft radius is still quite surprising, since the shape of a granular raft is rather a curved disk than a perfect sphere.

To be perfectly convinced of our result, we need to explore the limitations of the model we used in order to describe the dynamics of two interacting rafts.

Limitations of the model

A number of hypotheses underlying the approach described in this paper are no longer valid for a granular raft. They can give us some clues to understand the limits of our model. In particular, we observe an increasing discrepancy between the fit and the experimental data measured for the velocities when the rafts become large (the two highest velocities of figure 1 The two-sphere canonic example we used to derive equations (1.17) and (1.18) strongly relies on the assumption of small deformations around spherical beads (small with respect to the capillary length). With granular rafts, we can reach a vertical deformation of the interface up to almost half the capillary length.

Moreover, the morphology of a raft, a two-dimensional circular monolayer of beads, cannot be approximated as a spherical shape, and one could therefore expect the drag to be drastically different.

Finally, the viscous hypothesis used for the calculation of the drag can be questioned for the bigger rafts. In this case, the maximum speed reached during their motion is almost a hundred times higher than for two single beads, leading to a Reynolds number of order one. All these differences may change the amplitude of the forces through f and g and even the form of these functions.

We also assumed that the capillary force is proportional to K 1 (L/ c ), a result that comes from the linearized solution of equation (1.1). However, for some experiments, when two dense granular rafts interact, as in figure 1.14, equation (1.1) becomes highly non-linear, and the approximate linear solution is no longer sufficient to account for the curvature of the interface. Figure 1.14 illustrates this limitation of our model: on top of the experimental visualization of the interface when two granular rafts are almost in contact, we plot the interfacial shape we would get if the deformations could be superimposed, deduced from the experimental measurement of the shape of the interface around a single granular raft of the same size. The superposition principle seems to account rather well for the curvature on the left and right side of each raft, but clearly overestimates the depth of the interface between the two rafts, leading to an error on the calculation of the capillary force. This could be part of the explanation for the disagreement between the fit and the measured speed in figure 1.8 for the motion of the two biggest rafts.

Another important feature that we did not take into account is the lateral extension of a granular raft. As already emphasized, the derivation of equation (1.12) for the capillary force between two spheres at an interface relies on various hypotheses, including the size of the particles: R part c . However, the largest rafts, as illustrated in figure 1.14 for instance, can exhibit a diameter of the order of the capillary length. In this situation, equation (1.12) should not be valid anymore. Here again, this could explain the discrepancy between our experimental data and the corresponding fitting curve for large rafts.

Despite all these limits, our model still accounts for the major part of our data, proving that in a first approximation, a granular raft does behave as a heavy membrane which is not deformed by the interaction with other rafts. Its motion is the result of the equilibrium between a capillary attractive force and a drag force, whose dependence with the number of particles can be expressed as follows:

F cap B→A = n A n B F cap 1→1 F drag A = √ n A F drag 1 (1.41a) (1.41b)
The interaction between two granular rafts have been described. We can now wonder how a granular raft will interact with other objects deforming the interface.

Motion of a raft along a curved interface

Up to now, we have only considered capillary interactions between two moving granular rafts. The interfacial deformation was produced by the particles themselves. Nevertheless, a particle will also move if the interface itself is curved [START_REF] Liu | Capillary assembly of colloids: Interactions on planar and curved interfaces[END_REF]. This can be an issue for insects trying to move beyond a meniscus for instance, which need to develop techniques to climb menisci [START_REF] David | Meniscus-climbing insects[END_REF].

The curvature gradient of the interface acts as an external field [START_REF] Blanc | Capillary force on a micrometric sphere trapped at a fluid interface exhibiting arbitrary curvature gradients[END_REF][START_REF] Sharifi-Mood | Curvature capillary migration of microspheres[END_REF], and can allow a precise control of the trajectories and orientations of the moving objects [START_REF] Cavallaro | Curvature-driven capillary migration and assembly of rod-like particles[END_REF]. We can naturally wonder if the laws of motion we ended up with for two granular rafts remain valid when a single raft moves along a curved interface.

Deflecting the interface

The experimental setup is presented in figure 1.15. A circular metal cylinder of radius R cyl is placed at the oil-water interface, and then moved downwards until the desired depth H cyl . Because the interface remains pinned at the edge of the cylinder, we end up with a very well-controlled curved interface (see figure 1.15 for an experimental visualization of the interface), with a depth that can be described by the solution of equation (1.42). In comparison with the two-raft experiment, here we decouple the effect of H cyl and R cyl , while the radius and depth of a granular raft cannot be changed separately. Moreover, we can reach far greater vertical deflections of the interface, and as a consequence far larger capillary forces.

         γ h 1 + h 2 + h r -(ρ w -ρ o )gh 1 + h 2 = 0 h(r → ∞) = 0 h(r = R cyl ) = H cyl (1.42a) (1.42b) (1.42c)
As previously, we need to address the issue of the difference between the horizontal velocity, measured from top views of the experiment, and the total velocity along the interface. In section 1.2.1, we justified the need of computing the total velocity by noticing that the vertical displacement of two interacting rafts could be of the same order as their horizontal displacement when close enough. Nevertheless, we did not measure the total velocity from side views, and therefore we have never checked quantitatively the agreement between the computed curved velocity and the experiment. This new experiment gives us the opportunity to do that very easily, because the visualization from the side is less challenging than in a two-raft experiment (since the cylinder is not moving, we know the trajectory followed by the raft as well as its final position).

As explained before, we can compute the total velocity of the particle by numerically solving the system (1.42) and then deduce from its horizontal motion the corresponding vertical displacement. The results are displayed in figure 1.15.c. There is a very good agreement between the computed velocity (black curve) and its experimental value measured on a side view (green curve). This justifies a posteriori our previous measurements for the aggregation of two rafts, that were done using top views of the process. In the following, we will again perform all our velocity measurements using photos taken from above.

The last point we need to discuss here is the need to solve equation (1.42) compared to the possibility of solving a linearized version of the very same equation, for which there is an analytical solution (see equation (1.9) for the linearized equation, and equation (1.22) for its solution). Here again, the side view is going to bring us the answer. In figure 1.15.b, we plot the interface calculated either from the full equation or from the linearized one, in a situation where the deformation is important. The full solution perfectly accounts for the deflection of the interface, while the linear solution overestimates its depth. As a consequence, we will always solve the full equation (1.42) in order to compute the total velocity of an object along a curved interface.

Interaction between a single particle and a static object

Before looking at the motion of a granular raft along the interface, we first use this experimental device to characterize the motion of a single particle along a curved interface. In this section, we will only use one type of ceramic bead (density ρ part = 3,800 kg.m -3 , radius R part = 0.45 mm).

Influence of the size and depth of the cylinder on the velocity profiles

A given cylinder (fixed radius R cyl ) is placed at the oil-water interface at rest, and is then pulled down until the desired depth H cyl . The particle is placed far from the cylinder, and we record its motion from above. We then start again the experiment for a different couple of parameters (R cyl , H cyl ). Some results are displayed in figure 1. [START_REF] Peter A Kralchevsky | Particles with an undulated contact line at a fluid interface: interaction between capillary quadrupoles and rheology of particulate monolayers[END_REF].

Let us look at each variable one at a time:

• the velocity of the particle for a given set of parameters behaves exactly as expected, with a first increase as it gets closer to the cylinder until it reaches a maximum. The particle accelerates as the curvature of the interface increases. Just before contact, the bead slows down because of the hydrodynamic coupling of the drag extensively described previously: the liquid separating the bead from the edge of the cylinder needs to be ejected;

• the velocity of the particle clearly increases with the depth of the cylinder. This was expected, since the capillary force is linked to the local slope of the interface, which increases when the cylinder goes deeper. The larger the cylinder is, the deeper the cylinder can go without destabilizing the meniscus. Hence, larger cylinders allow us to access larger deformations and therefore larger attractive forces;

• the effect of the radius of the cylinder is less clear. Indeed, the velocity curves in figure 1. [START_REF] Peter A Kralchevsky | Particles with an undulated contact line at a fluid interface: interaction between capillary quadrupoles and rheology of particulate monolayers[END_REF].c and 1.16.d do not appear to be fundamentally different, except for the shift in abscissa due to the fact that L corresponds to the distance from the center of the cylinder. Now that the velocity profiles of a single particle has been understood, we can use them to quantify the force exerted by the cylinder on the particle.

Influence of the size and depth on the forces experienced

Similarly to the relative motion of two granular rafts, we can try to fit the experimental data with a theoretical velocity. Here again, we base our fit on the model developed for two small identical spherical beads, but instead of assuming a fitting coefficient that depends on the number of particles in each interacting rafts, we consider it will depend on the number of particles inside the raft and on the parameters of the cylinder (R cyl , H cyl ):

V A = f (n A , R cyl , H cyl ) g(n A ) a b G l + 2R part R part K 1 L c (1.43)
with n A the number of particles inside the raft (n A = 1 in the current section), l the horizontal distance between the edge of the cylinder and the edge of the interacting raft (l = 0 at the end of the motion), and L the horizontal distance between the center of the cylinder and the center of the raft. Again, we approximate the hydrodynamic coupling between the particle and the cylinder through the function G, and consider that the drainage of the liquid is the same as the one between two beads. Moreover, we use a formulation of the force calculated in the limit of small deformations, even though we have proven in figure 1.15.b that this assumption is not valid when the cylinder deflects a lot the interface.

H cyl / c K 0 (R cyl / c
) , for different radii and depths of the cylinder (colored markers). In black diamonds, we plot the fitting parameter for the motion of a single particle attracted by a granular raft of various sizes, as a function of the modified aspect ratio of the granular raft

h raf t / c K 0 (R raf t / c
) . In a and b, the solid lines have a slope of 1.

Keeping in mind these approximations, we plot the fitting coefficient f /g in figure 1.17.a, as a function of the depth of the cylinder, for different radii. We end up with what appears to be a linear relation with H cyl , but as expected, we do not capture the dependence of the dynamics with R cyl .

In order to collapse all our data, we can use the results of the previous section. Hence, according to equation (1.32), we can test if the capillary force exerted by the cylinder is proportional to what we called the modified aspect ratio:

F cyl ∝ H cyl / c K 0 (R cyl / c ) (1.44)
This collapse is displayed in figure 1.17.b, in which we can see that all the colored markers are aligned in a log-log plot, on a line of slope 1. Furthermore, we can compare the attraction the cylinder exerts on a single particle to the attraction a granular raft exerts on the same isolated particle. The fitting coefficient for the latter is plotted in figure 1.9.b, as a function of n B √ n A . For the ceramic bead of density ρ part = 3,800 kg.m -3 and radius R part = 0.45 mm, we select the data for which n A = 1, and plot them as a function of the modified aspect ratio of raft B, deduced from the experimental measurement of its radius and depth. The corresponding points are displayed in black diamonds in figure 1.17.b, and are perfectly consistent with the fitting coefficient for the motion of a bead attracted by a cylinder.

From the point of view of a single sphere, the forces exerted by a granular raft or a cylinder through the deformation of the interface are identical, and depend on the modified aspect ratio

H cyl / c K 0 (R cyl / c)
of either the cylinder or the raft. The next natural step is therefore to study the motion of a whole raft along the same curved interface.

Interaction between a raft and a static object

We can now look at how a granular raft moves along a curved interface, as well as how its motion depends on its number of particles and the cylinder depth and radius.

A typical experiment a) b) The experimental protocol is quite similar: here again, a given cylinder (fixed radius R cyl ) is placed at the oil-water interface at rest. n particles are sprinkled far from the cylinder and aggregate into an axisymmetric monolayer. When the raft is formed, the cylinder is pulled down until the desired depth H cyl , and the motion of the granular raft is recorded from above, as illustrated by figure 1.18. As expected, the raft is attracted by the cylinder and accelerates along its motion.

Influence of the number of particles

For a given radius and depth of the cylinder, we can now vary the number of particles inside the raft. As in the previous section, we will only use one type of ceramic bead (ρ part = 3,800 kg.m -3 , R part = 0.45 mm). Some experimental results are presented in figure 1. [START_REF] Nathan J Mlot | Fire ants self-assemble into waterproof rafts to survive floods[END_REF], where we plot the velocity of the raft for a given set of parameters (R cyl , H cyl ), and different numbers of particles. Here again, several observations can be drawn:

• qualitatively at least, all the granular rafts behave as expected: their velocity increases as they get closer to the cylinder, until a maximum where a short decrease can be observed just before contact;

• the velocity of the raft increases with its number of particles. This could have been expected, because according to our previous results, V ∝ √ n for a granular raft (see equation (1.41)).

We can try to fit our data according to equation (1.43), as we did for single particles, and see how the fitting coefficient depends on the number of particles n, the depth of the cylinder, and its radius f (n, R cyl , H cyl )/g(n). Yet, as emphasized in figure 1. [START_REF] Nathan J Mlot | Fire ants self-assemble into waterproof rafts to survive floods[END_REF].b and 1.19.c, our model is only able to describe the motion of a single particle, but fails to fit the data for large rafts (n = 50 in figure 1.19.c). The discrepancy between the fit and the velocity measured experimentally increases with the number of particles in the raft. Hence, our model does not allow us to quantitatively explain the motion of a granular raft attracted by a cylinder that deflects the interface: we are only able to predict the motion of two interacting rafts on a flat interface (equation (1.41)), or of a single particle on an interface deformed by a cylinder.

This failure can come from a large variety of reasons: first, the numerous assumptions made in the model, which have already been extensively discussed. They are even more critical for the cylinder experiment since we can explore greater interfacial deformations. A second important difference between the cylinder-raft interaction and the raft-raft interaction is the fact that the interface is pinned at a fixed position for the cylinder, while for two interacting rafts, the vertical position of both rafts evolves throughout the experiment. This can lead to fundamentally different behaviors. For instance, if the depth of the cylinder is lower than the depth of the raft (H cyl < h raf t ), then the raft will never reach the edge of the cylinder: it will stop before touching the cylinder, at an equilibrium distance that increases with h raf t -H cyl .

The hydrodynamic coupling in the expression of the drag force is probably also a source of error: we consider the drainage of the liquid between the raft and the cylinder to be the same as the drainage of the liquid between two identical spheres, whatever the sizes of the cylinder and the raft are.

Comparison between the interactions with a cylinder or a granular raft

To be completely convinced the motion of a raft towards a cylinder is not the same as the motion towards another raft, we can perform the following comparison: we measure the velocity of a raft A (made of n A particles) attracted by a raft B, and then we choose a cylinder with similar parameters (H cyl = h raf t B , R cyl = R raf t B ), and measure the velocity of the raft A when attracted by the cylinder. Such results are displayed in figure 1.20. , radius Rpart = 0.45 mm), either attracted by a cylinder with a radius R cyl = 2.5 mm at a depth H cyl = 1.9. mm (red circles), or attracted by another granular raft (blue squares) with parameters similar to the cylinder (R raf t = 2.7 mm, h raf t = 2.1. mm).

The differences between the two curves are quite clear: even if the maxima appear to be superimposed, the path to reach them is not the same. The blue curve would be perfectly well fitted by the model, while the fit would fail to describe the red curve, exactly like in figure 1. [START_REF] Nathan J Mlot | Fire ants self-assemble into waterproof rafts to survive floods[END_REF].

More experiments would be needed to understand the specificities of the cylinder-raft interaction, and to perfectly well describe the motion of the raft. For instance, we could explore other types of interfacial deformations: instead of an axisymmetric cylinder that deforms the interface both radially and azimuthally, we can deflect the interface downwards using an infinitely long rectangle, so that the vertical deflection only depends on one space variable in a cartesian grid.

Conclusion

In this chapter, we have investigated the interaction dynamics of a two-body system formed after the clustering of beads into granular rafts at an oil-water interface. In particular, we described the capillary forces generated by each granular raft, as well as the drag due to their individual motion. Because the deformation around a granular raft strongly depends on its size (depth and width), the capillary interaction can surpass by several orders of magnitude the forces created by individual beads, with a very strong dependence on the number of particles.

A generalization of the two-particle model is found to account very satisfactorily for the motion of the granular rafts, and only starts to deviate from the experimental data for objects of dimensions close to the capillary length. Despite the clear non-linearity of the equations describing the shape of the interface for such large rafts, the capillary force appears linear both with the size of the attracting and the attracted raft. The drag, on the other hand, increases linearly with the radius of the considered raft (or in other terms, as the square root of the number of particles). The morphology of a granular raft (lateral and vertical extensions) was also studied experimentally, numerically, and theoretically.

If the interaction between two granular rafts was satisfactorily explained, the motion of a raft on an interface curved by the presence of a cylinder, on the contrary, still lacks a complete understanding. A refined model needs to be developed in order to better describe the force exerted by the curved interface on a granular raft, and more experiments need to be conducted.

Up to now, we have completely neglected the intrinsic discrete nature of a granular raft by modeling it as a two-dimensional membrane. However, in some cases the granular aspect of the raft may play an important role. We have observed that rafts made of smaller particles (diameter < 400 µm) show a weaker cohesion between the grains within the raft, which may lead to erosion and internal motions for a fast enough raft displacement. This phenomenon will be extensively investigated in chapter 3.

The aggregation between two granular rafts of random sizes was studied. Quite naturally, the next step is now to explore the aggregation of a random number of granular rafts, each one made by a random number of particles. In other words, after the two-body problem, it is time to tackle the n-body problem.

Chapter 2

Statistics of aggregation

Collective motion and clustering dynamics

What do a planetary system, a bunch of cereals floating at the surface of a milk bowl, a fog slowly invading the peaceful streets of Paris in the morning, a water spray, or a long chain of polymers on the verge of gelling have in common? This could appear as the beginning of a sick joke, but also of an endless field of study for any physicist interested in aggregating systems [START_REF] Leyvraz | Scaling theory and exactly solved models in the kinetics of irreversible aggregation[END_REF].

Planets are produced by the collision of blocks of various sizes [START_REF] Hoi | On the validity of the coagulation equation and the nature of runaway growth[END_REF]. These collisions are quite frequent in the Asteroid Belt (on astronomical time scales), where countless objects orbit around the sun, as artistically represented in figure 2.2.a. Depending on the relative speed of the two colliding objects, it can lead either to the fragmentation of the asteroids, or to their merging.

Similarly, fogs result from the suspension of micrometer-sized water droplets that can coalesce and break apart [START_REF] Sheldon K Friedlander | Smoke, dust, and haze[END_REF], while polymers are formed through the assembling of elementary units called monomers [START_REF] Walter | Theory of molecular size distribution and gel formation in branched-chain polymers[END_REF].

The same process is found in water sprays, as illustrated in figure 2.2.b, where we can see the destabilization of a liquid sheet into droplets of various volumes [START_REF] Gd Crapper | A note on the growth of kelvin-helmholtz waves on thin liquid sheets[END_REF]. The distribution of sizes of the resulting droplets depends on the destabilization process itself, but also on the merging and breaking events occurring beyond the liquid sheet. In all these physical systems, a lot of objects interact with one another. In that very simple sentence lies all the difficulties of multi-body systems. Perhaps the most wellknown is the n-body gravitational problem, first introduced by Isaac Newton in 1687 [72] for n = 3. The aim is to solve the motion of three objects interacting through gravity, given their initial positions and velocities. And quite astonishingly, no general solution exists as long as n 3.

Similarly to the classic 3-body problem, most of the time, no exact solution can be found when too many bodies interact. As stated by Philip Warren Anderson, "More is different" [START_REF] Philip W Anderson | More is different[END_REF]. Hopefully, this does not mean their collective behavior cannot be described. Statistical tools can be used to account for the global dynamics, and describe phase ordering and coarsening kinetics [START_REF] Bray | Theory of phase-ordering kinetics[END_REF]. Statistical physics is now widely used for numerous interdisciplinary fields, from biology [START_REF] Sella | The application of statistical physics to evolutionary biology[END_REF] to computer science [START_REF] Pastor | Evolution and structure of the Internet: A statistical physics approach[END_REF] or even social science [START_REF] Castellano | Statistical physics of social dynamics[END_REF]. This large range of application is due to the fact that even though the specific interactions between the objects considered differ drastically, the tools used to study the whole system remain the same.

This approach is particularly useful to describe the behavior of active matter, a generic term to describe systems able to consume and dissipate energy [START_REF] Ramaswamy | The mechanics and statistics of active matter[END_REF]. It can consist of non-living moving self-propelled objects [START_REF] Deseigne | Collective motion of vibrated polar disks[END_REF][START_REF] Bricard | Emergence of macroscopic directed motion in populations of motile colloids[END_REF], as in figure 2.2.a [START_REF] Narayan | Long-lived giant number fluctuations in a swarming granular nematic[END_REF]. In that specific example, a monolayer of rodlike particles confined in a circular cell is simply agitated at a frequency of 200 Hz. Even in such a simple system, giant long-lived density fluctuations are observed, at a scale far larger than an individual object. This is surprising since the particles communicate with one another only by collisions and contact with their direct neighbors. Very simple interactions can lead to global order and large inhomogeneities.

The same behavior can be found with living matter, such as birds flocks or bacteria in suspension [START_REF] Zhang | Collective motion and density fluctuations in bacterial colonies[END_REF] that can arrange in large clusters [START_REF] Peruani | Collective motion and nonequilibrium cluster formation in colonies of gliding bacteria[END_REF], as illustrated in figure 2.2.b. In this experiment, a monolayer of gliding bacteria is observed as a function of time. Collisions between the different cells lead to the emergence of large clusters, inside which all the cells move in the same direction. Clusters grow in size by colliding with other aggregates, while simultaneously, bacteria on the border of a cluster can escape. This leads to a transient cluster-size distribution that reaches a stationary state after a certain time, as the one of figure 2.2.b. This steady state mainly depends on the initial concentration of bacteria. In all these situations, a global collective order arises from the individual motion of each object. This collective motion can lead to aggregation, similarly to our system of interest. The statistical behavior of active particles has been studied numerically [START_REF] Levis | Clustering and heterogeneous dynamics in a kinetic monte carlo model of self-propelled hard disks[END_REF], experimentally [START_REF] Ginot | Aggregation-fragmentation and individual dynamics of active clusters[END_REF] and theoretically [START_REF] Gabriel S Redner | Classical nucleation theory description of active colloid assembly[END_REF], always with the aim of extracting scaling laws able to describe the growth of the clusters and the distribution of their sizes throughout the whole dynamics [START_REF] Michael | Motility-induced phase separation[END_REF].

Aggregation kinetics are not the prerogative of active particles. As described in the previous chapter, passive objects placed at a fluid interface may attract one another through capillarity, without any energy input. Up to now, we have only considered the interaction between two granular rafts, but nothing prevents us to place numerous rafts at an interface. Such a system has already been extensively studied for far smaller particles, such as nanometer-sized gold particles whose motion is completely controlled by Brownian thermal agitation. And even though the individual motion of each colloid is completely random, the aggregation process appears to be amazingly universal [START_REF] Da Weitz | Dynamic scaling of cluster-mass distributions in kinetic colloid aggregation[END_REF][START_REF] My Lin | Universality in colloid aggregation[END_REF].

Two main regimes of aggregation have been identified for colloidal particles, as illustrated in figure 2.25, depending on the amplitude of the repulsive forces that resist clustering:

• diffusion-limited aggregation [START_REF] Ta Witten | Diffusion-limited aggregation, a kinetic critical phenomenon[END_REF][START_REF] Thomas | Diffusion-limited aggregation[END_REF], when the repulsive forces are negligible. The aggregation is then limited by the time needed by the clusters to bump into each other because of diffusion. A typical cluster obtained is presented in figure 2.25.a;

• reaction-limited aggregation [92,[START_REF] Meakin | Structure and dynamics of reaction-limited aggregation[END_REF], when the repulsive forces are strong, but can still be overcome. The aggregation is then limited by the time needed by the clusters to overcome this repulsion through their brownian motion. A typical cluster obtained is presented in figure 2.25.b.

In both cases, the cluster-mass distribution evolves according to specific dynamical scalings, with a shape that becomes universal after a certain time. The resulting clusters exhibit fractal dimensions that can be measured, as emphasized by figure 2.25. The fractal dimension clearly depends on the rate of aggregation, and equals approximately 1.8 for diffusion-limited colloid aggregation, and 2.1 for reaction-limited colloid aggregation. The theoretical framework needed to analyze these aggregation processes will be presented later. The two photos are taken from [START_REF] My Lin | Universality in colloid aggregation[END_REF], where more details on the experiment are available.

The cluster-mass distribution is what we wish to characterize and understand in our specific situation, in which instead of colloids, large heavy particles interact. This difference is fundamental. Indeed, contrarily to colloids for which the random motion prevents any direct calculation, here the force of interaction is very well-known. We characterized it in the previous chapter. But since the number of interacting objects is very high, we still need these statistical tools to describe the aggregation, even if all the motions are perfectly deterministic.

We will start by looking at experimental and numerical measurements of the aggregation, before looking deeply into the distribution of sizes of our granular rafts.

Experimental aggregation

We first study the aggregation of particles at an interface from an experimental point of view, by simply sprinkling the beads at the interface, and looking at their collective clustering.

Setting up the experiment

Experimental device and control parameters

The experimental setup is identical to the one used in the previous chapter. A thick layer of silicone oil (kinematic viscosity ν o = 100.10 -6 m 2 .s -1 , density ρ o = 966 kg.m -3 , oil-water surface tension γ = 42 mN.m -1 ) is poured into a tank of dimensions 0.3 x 0.3 x 0.25 m filled with pure water. We use a silicone oil twice more viscous than in the previous chapter, in order to slowdown the aggregation process and have more time to record the dynamics. Moreover, we use a larger water tank, because we want to make sure that the particles do not interact with the walls.

Only one type of particles will be used in the whole chapter (at the exception of the very last section): ceramic particles of density ρ part = 3,800 kg.m -3 and radius R part = 125 µm ± 25 µm. For this type of beads, the rafts can gather several thousand particles before sinking (we did not look at the critical size at which it destabilizes), giving us the opportunity to study a broad range of total interacting particles.

Instead of counting the beads, a task too challenging regarding the number of beads used, we weigh them. The corresponding number will be determined later with a photo of the experiment. In a typical experiment, we first add particles in a homemade salt shaker until the desired mass. A funnel of radius R domain is then positioned a few millimeters above the oil-air interface, and we sprinkle the particles inside, as schematically represented in Two control parameters drive our study: the total number of particles n tot and the radius of the domain in which we sprinkle the particles R domain . Nevertheless, as explained before, we do not have a direct access to the number of particles, but only to their total mass. Moreover, the particles are polydisperse (R part = 0.125 mm ± 25 µm), which means that simply dividing the total mass by the mass of one particle could lead to a large error on n tot .

To avoid this pitfall, we determine the number of particles by counting them on the first photo of each experiment, as the one presented in figure 2.4. As expected, for a given mass, n tot varies, but by no more than ten percent. We consider that for less than ten percent of variation, the results can be compared with one another.

The different values we use for n tot and R domain are summarized in table 2.1. As emphasized in figure 2.4, the initial distribution at the interface is random. However, we do not want the initial positions of the particles to have an impact on the statistical results we will derive later. As a consequence, for each couple of parameter (n tot , R domain ), we do the experiment five times, in order to average the measurements between the five clustering processes. By doing so, the results after averaging will be relatively independent of the initial configuration.

For the lowest surface densities, the experiments can last three hours. Because of this very long time scale, and also because the capillary forces are quite low for particles of the size we use, the smallest perturbation of the system disturbs a lot the clustering of the particles. To avoid any perturbation, a plastic cover is placed on top of the water tank to limit convection above the oil interface. The water tank is moreover placed on foams to absorb any vibration of the environment.

Visualization of an experiment

A typical experiment is displayed in figure 2.5. For each experiment, we take one photo every ten seconds. The aggregation starts by a scattered configuration where all particles are isolated, but after 40 minutes, bigger clusters appear, and continue to grow as they attract all the remaining isolated particles. These clusters will finally merge with one another, until all particles are collected into a single granular raft. We can see on these series of photos that quite quickly, the distribution of sizes becomes somehow bimodal, with two or three large rafts and a collection of isolated particles. The clustering does not go through a period during which a continuous range of sizes is present. We think this is due to the fact that the capillary forces, as demonstrated in the previous chapter, strongly depend on the size of the interacting rafts. Thereby, as soon as a large raft is created, it is going to exert the largest force on all other particles, leading to a collapse of all the remaining beads towards it.

Total number of particles during the experiment

We want to know the number of particles in each cluster as a function of time during the whole process. Counting them came out to be an unrealistic method, since it is difficult with the image resolution to distinguish particles once they are aggregated. Instead of that, we measure on each photo the area of every cluster. By performing this measurement on the first photo, we deduce the average area of one particle, since they are mainly isolated at the beginning of the experiment. After that, we just need to divide the area of each cluster by the area of one particle, corrected by a coefficient that takes into account that the packing of spheres on a plan presents holes. This coefficient also depends on the lighting of the experiment. As a consequence, we choose this coefficient for each experiment in order to make sure that the measured number of particles stays constant during the whole aggregation, as illustrated by figure 2.6. n tot varies by no more than 2%, meaning that our measurement technique is reliable to perform the analysis of the cluster sizes. Figure 2.6 also allows us to confirm the polydispersity that we discussed earlier. Indeed, the five curves correspond to the exact same mass of particles, but the corresponding number of beads varies from 900 to 1000. Now that we have some confidence in our experimental setup, it is time to look at the quantities we are able to measure.

Preliminary measurements

Reproducibility of the results

The two easiest variables we can measure are the number of clusters n clusters as a function of time, and the size of the biggest cluster n max . But before that, let us define some notations. Small letters, as i or n, will describe the size of a given cluster, which is defined by the number of particles inside it, while N (i) will stand for the number of clusters of size i. As a consequence, if we sum all the different number of clusters N (i), we end up with the total number of cluster n clusters . Similarly, if we weight each N (i) by the size of each cluster, we end up with the total number of particles n tot . Hence, these two variables can be defined as follows:

n tot = iN (i) n clusters = N (i) (2.1)
n max and n clusters are represented in figure 2.7 for a given domain and a given mass of particles. Quite naturally, the size of the biggest cluster n max increases during the aggregation, and at the same time, the number of clusters decreases until only one remains. n clusters (t = 0) ≤ n tot because when we sprinkle the particles, some of them fall really close to one another. As a consequence, at the beginning of the experiment, some pairs or even triplets are already constituted.

In figure 2.7, five experiments with the same set of parameters are plotted. The five curves are rather well superimposed, even though as explained earlier, n tot is not strictly constant between consecutive aggregations. Our experiment is therefore reproducible. From now on, for a given set of parameters, we will only plot the average value of our measurements for the five experiments we perform each time. One may notice that the curves are truncated before the end of the aggregation: at the end of all five curves, the biggest cluster has not collected all the particles. This is due to the nature of the interaction between the rafts. We know the capillary force decreases exponentially with the distance. As a consequence, when two rafts are too far apart, the amplitude of the interacting force is going to be so low that it may become comparable to the inherent noise of the experiment. The aggregation would then happen after a very large time, or maybe even an infinite time. Indeed, it is really not clear if a force lower than the noise still has an influence on the motion of the particles.

Each experiment is stopped when most particles are already aggregated, but we do not wait for all the particles to be inside the same cluster.

We can now begin to play with our two input parameters n tot and R domain , and see how they both affect the aggregation dynamics.

Role of the number of particles

In figure 2.8, we look at the influence of the total number of particles. A first comment concerns the shape of the three curves: whatever the number of particles, n max and n clusters displays a common behavior: n max increases while n clusters decreases, as already described in the previous figure.

The effect of n tot is also rather clear. When the total number of particles is increased, the clustering dynamics are much faster. This result is not surprising at all. Indeed, when we increase n tot for a given R domain , the initial distance between the particles decreases. We have seen in the previous chapter that the capillary force between two particles evolves as K 1 (l/ c ), with l the distance between the two particles, c the capillary length, and K 1 the modified Bessel function of the second kind of order 1. As a consequence, when l decreases, the capillary force is much higher, leading to a much faster aggregation.

The clustering is faster, but up to a time rescaling, the curves seem to evolve in the same manner: n clusters decreases slower and slower as the aggregation proceeds for the three curves, and similarly, n max increases with a sigmoidal shape for the three different numbers of particles. This tends to indicate that the number of particles only influences the rate of aggregation, but without changing the nature of the process.

Role of the size of the domain

After having analyzed the influence of the number of particles, we can focus on how the size of the domain changes the aggregation kinetics. The results are presented in figure 2.9.

Here again, we recover an increase of n max and a decrease of n clusters . The effect of R domain is limpid. When the size of the domain is decreased, the aggregation is faster. This is not surprising, since a decrease of R domain for a given n tot has the same effect as an increase of n tot for a given R domain : a reduction of the initial distance between the particles.

Figures 2.8 and 2.9 indicate that both n tot and R domain act on the time scale of the clustering by changing the inter-particle distance. It would be nice to be able to gather these two parameters into a single variable.

Role of the surface density

Changing n tot and R domain is similar as changing the initial concentration of particles at the interface. Their combined effect may therefore be taken into account with a single parameter: the initial surface density of particles φ surf , defined according to equation (2.2):

φ surf = n tot πR 2 domain (2.2)
The different values of φ surf are summarized in table 2.1. We can try to gather all our measurements using this parameter φ surf , as it is done in figure 2.10. As intuited, a higher surface density results in a faster aggregation. Moreover, if we consider two different curves with two very different sets of parameters but a similar value of φ surf , as for instance the green and turquoise curves, we can see they are almost superimposed. This tends to prove that indeed φ surf is the correct rescaling parameter. Yet, this argument does not hold for all the curves. For the two lowest values of φ surf , the size of the biggest cluster follows different trends. We think this disagreement is due to finite-size effects. For an infinite system, the surface density would be the only important parameter, but here, in a real system, boundary and initial conditions still have an impact: n tot and R domain are not strictly equivalent. Nevertheless, in a first approximation, φ surf appears as a reasonable control parameter. All the curves can be superimposed when the time along the x-axis is multiplied by φ surf , as demonstrated by the insets of figure 2.10. This rescaling is completely phenomenological, in the sense that it is not based on any theoretical argument but only on experimental observations.

Distribution and probabilities

Up to now, we have characterized two rather simple quantities: the number of clusters n clusters and the size of the biggest cluster n max . As it has been done in many studies on various aggregating systems [START_REF] Da Weitz | Dynamic scaling of cluster-mass distributions in kinetic colloid aggregation[END_REF][START_REF] My Lin | Universality in colloid aggregation[END_REF], but even on fragmentation dynamics that can be studied using the same tools [START_REF] Vledouts | Fragmentation as an aggregation process[END_REF], the only way to fully describe the aggregation is by characterizing the distribution of all different sizes as a function of time.

Distribution of sizes

In figure 2.11, we plot the distribution of sizes at four successive times separated by 200 s. The x-axis corresponds to the size of a cluster, while in ordinate we can read the corresponding number of clusters. Several comments can be made. First, here again, the distribution results from the averaging between the five experiments we performed for this specific set of parameters. As a consequence, the lowest value of N (n) is not 1 but 1/5. This is fundamental, because the more experiments we have, the more relevant the distribution will be. In figure 2.11, all points located on the dotted lines are wrong estimates of the real number of clusters of the corresponding sizes. We detected them once, meaning that in the five experiments, at t = 620 s for instance (orange curve), only one experiment presented a cluster of size n = 616. But if hundreds of experiments were performed (which is totally unrealistic), this specific size would be observed only a few times, and N (616) would be far lower than our measurement.

Keeping in mind this inherent limitation, we can now focus on the evolution of the distribution. The blue circles represent the initial distribution. As expected, the particles are mostly isolated, but we also start with a bunch of pairs and triplets. During the clustering process, the number of isolated particles appears to be strictly decreasing (from blue to orange at n = 1). The behavior is more complicated for other sizes. Let's focus for instance on n = 4. We can see on the four points that N (4) is first increasing with time (dark blue to light blue), then remains almost constant for 200 s, and finally starts to decrease.

Concerning the global trend, we can see that N (n) transits from a sharp left-centered distribution to a more scattered one with larger and larger clusters as the aggregation proceeds.

We also need to be careful with the normalization of our data. The integral of each curve in figure 2.11 is not normalized, and changes with time. What we represented is not the probability of existence of a given raft, but only the number of rafts observed, which is not the same information. This choice will be justified later when we will try to rescale the distributions. Indeed, in the literature, most studies have rescaled the distributions rather than the probabilities.

Yet, it is interesting to look at this other way of plotting our data.

Probabilities

Two different probabilities can be calculated:

• the probability P part (n) that a given particle belongs to a raft of size n:

P part (n) = nN (n) n tot (2.3)
• the probability P raf t (n) that a given raft has a size n:

P raf t (n) = N (n) n clusters (2.4)
To the difference of the distribution of sizes N (n), these two quantities are normalized, meaning that P part = P raf t = 1. These two probability density functions, or PDF, are plotted in figure 2.12. In figure 2.12.a, we find again the same low boundary for P part (n) as we did with N (n), coming from the fact that we performed only five experiments for each parameter. But contrarily to figure 2.11, the bimodal aspect of the distribution is more obvious: at large times (t = 620 s in figure 2.12.a), a particle has a huge probability to be either in the biggest raft or to be isolated. The probability to find a raft of size n (figure 2.12.b) evolves very differently of course, because except at the very end, there are always a lot of isolated particles for only one large cluster.

We now face two problems. First, we are limited by the low number of experiments we can perform. Our truncated distributions can only produce frustration to anyone that really wants to understand the physical mechanisms leading to the collapse of all beads into a single raft. Secondly, we managed to plot various quantities that depend on time, but without the right tools, these data do not bring us any insight on the clustering dynamics.

To solve this problem we will call on numerical simulations to make up for the lack of statistics.

Numerical aggregation

We now want to study the aggregation of particles at an interface numerically, by mimicking the experiment: we want to start by a random initial distribution of the particles, followed by their aggregation through capillarity. We will then perform measurements of the sizes of the different clusters so that we can plot the distribution of sizes at each time. Because many simulations can be performed, we hope to obtain satisfying statistics.

Setting up the numerics

The algorithm

The idea is the following: we want to simulate the aggregation of n tot particles randomly distributed inside a circle of radius R domain . Quite conveniently, the previous chapter was devoted to the calculation of the capillary interaction between two granular rafts. We recall here one of the main results, since all this section will be based on it. The motion of a raft A attracted by another raft B is controlled by the equilibrium between the capillary force and the drag, expressed as follows:

     F cap B→ A = n A n B 2πγR part B 5/2 Σ 2 K 1 (L/ c ) F drag A = √ n A 6πµ o R part kV A G -1 l + 2R part R part (2.5a) (2.5b)
with k the coefficient taking into account that the raft moves at an interface. In this chapter, we will always take k = 0.5 (see previous chapter for details on the value of k).

All the parameters are defined in the previous chapter. This leads to the following velocity for a raft A attracted by a raft B:

V A = √ n A n B γB 5/2 Σ 2 3µ o k G l + 2R part R part K 1 L c (2.6)
In the previous chapter, we have discussed the importance of the vertical component of the velocity, due to the deformation of the interface. All our fits were performed on the total velocity, which means that equation (2.6) gives us the total velocity. Yet, we are going to perform two-dimensional simulations. As a consequence, we need to project equation (2.6) on the horizontal plane. This means that we need to know the geometry of the interface.

In the linear limit, the slope of the interface at a distance L from the raft is given by equation (1.22), that we mention again here:

dh dL = h raf t / c K 0 (R raf t / c ) K 1 L c .
(2.7)

Moreover, we have already proven the following relation, again in the limit of small deformations and small granular rafts (see equation (1.38)):

h raf t / c K 0 (R raf t / c ) ≈ F vert 2πγ c = 2nR 3 part g(ρ part -ρo+ρw 2 ) 3γ c (2.8)
It is now possible for us to compute the value of dh/dL for any raft simply by knowing its size. The horizontal velocity of a raft made of n i particles, attracted by a raft of size n j , is then expressed simply as:

V (x) i (i, j) = √ n i n j γB 5/2 Σ 2 3µ o k G l ij + 2R part R part K 1 L ij c × cos tan -1 2n j R 3 part g(ρ part -ρo+ρw 2 ) 3γ c K 1 L ij c (2.9)
with L ij the distance between raft i and j, and l ij the distance between their edge. From equation (2.9), we can compute the respective velocities of two interacting rafts. But we aim to simulate many rafts, each one interacting with all the others. This is where we make our major hypothesis: we assume that all forces can be superimposed. This means that we can calculate V (x)

i (i, j) for each couple of rafts, and then sum all the different velocities. The velocity vector of a raft of size i is therefore:

--→ V (x) i (i) = n clusters j=1 V (x) i (i, j) -→ e ij (2.10)
with -→ e ij the unitary vector along the direction connecting the rafts i and j, directed towards j.

This calculation is performed for all the different rafts at each time step, and then the position of each raft is moved accordingly. We use an adaptive time step defined as follows:

∆t = max   R part max i V (x) i , min i,j L ij 10max i V (x) i   (2.11)
The first term of the max in equation (2.11) ensure that even the fastest raft will never travel by more than the radius of one particle, while the second one allows bigger time steps when all particles are very far from one another, otherwise the simulation goes on almost forever.

Testing the algorithm

In order to be sure our adaptive time step is relevant, we compare the data obtained using it with simulations performed with various constant time steps. The results are presented in figure 2.13. For the four different simulations (three with a constant time step, one with the adaptive one as defined in equation (2.11)), we start with exactly the same initial configuration. All the different curves are superimposed, meaning that all the calculations are sufficiently refined.

We are not interested in the specific arrangements of the particles inside the clusters, but only in the aggregation dynamics. Therefore, when two rafts of sizes n i and n j touch one another, we merge them into a bigger raft of size n i + n j and radius √ n i + n j R part (we neglect the fact that particles inside a raft do not cover 100% of the area of the raft).

First numerical result, compared with the experiment

We use the same parameters as in the experiments: same oil and water properties, same particles. Each simulation starts with a random distribution of n tot particles inside a circle of radius R domain . A typical simulation is displayed in figure 4.28. At least qualitatively, the clustering process is very similar between the numerics and the experiments, with a collapse of individual particles into a few large rafts that later merge with one another, until one remains at the end.

The simulations will give us ideal results: the forces of interactions are imposed by our model. Yet, we have seen in the previous chapter that this model partially fails to describe the complete motion of the larger rafts. Furthermore, no polydispersity is included in the simulations, and we choose the coefficient of the drag k = 0.5 while its value should be measured. Finally, we start the simulations with only isolated particles, but in the experiments, as it can be seen in figure 2.8.b, only 90% of the particles are isolated, while the others are already in pairs or triplets. For all these reasons, we do not expect that for similar parameters, experiments and numerics behave strictly in the same way, and that their respective dynamics can be superimposed, as illustrated in figure 2 Concerning the evolution of the number of clusters and the size of the biggest cluster, the two curves are globally superimposed for the first portion of the aggregation, and start to differ at large times. This disagreement between the dynamics of aggregation is not fundamental. First, because studying an ideal system where the forces are perfectly controlled is the best way to understand the physical real system. Secondly, because even though the rate at which rafts encounter each other may differ, the distribution of sizes along the dynamics can still be identical, up to a time rescaling (this assumption will be explored later). And finally, because we are particularly interested in the first half of the clustering process, when a large variety of sizes can be found. After a certain time, the distribution of sizes is due to the fact that only one raft attracts all the remaining particles.

Another main difference between the experiment and the numerics concerns the end of the aggregation: as explained before, we end the experiments before all particles are aggregated. Indeed, the total time of aggregation may be infinite if the amplitude of the interaction becomes lower than the noise of the experiment. Nevertheless, in the simulations, there is no noise, and even for particles very far apart, we can compute a positive force that will always lead to aggregation. As a consequence, as demonstrated by figure 2.15, all simulations are performed until only one raft remains.

Focusing on the first part of the dynamics, we can also compare the distributions, as it is done in figures 2.15.c, 2.15.d and 2.15.e. We can see that for small sizes, the distributions are perfectly well superimposed, which proves that our simulations do reproduce the real clustering dynamics of the experiments. These three figures also perfectly demonstrate how the simulations are able to make up for the poor statistics of the experiments: the lower boundary for N (n) is decreased by two orders of magnitude for the simulation. The shape of the distribution is not truncated anymore: we have access to the whole function, instead of just a portion. Now that we have faith in our numerical results, we can start to play with the two input parameters n tot and R domain .

Preliminary results: influence of the surface density

The main reason we decided to perform numerical simulations was to be able to reproduce the same experiment with different random initial positions of the particles a lot of times, in order to measure statistically relevant distributions. If we just want to check the evolution of n max or n clusters , 40 simulations is enough. But if we want to look at the distributions of sizes, more is needed.

To that end, we first perform 40 simulations for several sets of parameters, and then for a few chosen ones, we perform several hundreds of simulations. Here again, the results are quite reproducible, so we average all quantities between the simulations, as it was done for the experiments. The different parameters used are summarized in table 2.2. Before looking at the distributions, we can try to recover one of the previous experimental results: the influence of the initial surface density on the clustering dynamics. To do so, we plot for six different simulations the evolution of the size of the biggest cluster and the number of clusters in figure 2. [START_REF] Peter A Kralchevsky | Particles with an undulated contact line at a fluid interface: interaction between capillary quadrupoles and rheology of particulate monolayers[END_REF]. The results are quite similar to the experiments: when φ surf decreases, the particles are farther apart, which results in a slower clustering. As a consequence, for a larger initial surface density, n max increases faster while n clusters decreases more sharply.

Characterizing the numerical aggregation

Distribution of sizes and probabilites

For now on, we will only use the numerical results for which a lot of simulations have been performed. Similarly to what we did with the experiments, we plot in figure 2.17 for one set of parameters the distribution of sizes at different times of the aggregation, along with the two different types of probabilities defined earlier: the probability that a given particle belongs to a raft of size n (figure 2.17.b) and the probability that a given raft has a size n (figure 2.17.c). As explained before, the distributions at the end of the aggregation do not interest us, because for large times, only one single raft attracts all the other particles. The finite aspect of our system starts to play a role, a situation which will not be taken into account by the theoretical tools we will present later. This is why we look at the distributions only up to the time t 60 at which the biggest cluster has collected 60% of all particles. t 60 is defined in figure 2. [START_REF] Krassimir | Capillary forces between particles at a liquid interface: General theoretical approach and interactions between capillary multipoles[END_REF].

These three figures are very similar to their experimental counterparts, but because they result from the averaging of 980 simulations, they bring us far more insight on the aggregation process. We recover the behavior for the smallest rafts: a decrease of the number of isolated particles, and a non-monotonic time evolution for all the other sizes, with a first increase followed by a decrease.

Since we can perform a large number of simulations, we now have a clear idea of the aspect of the distribution for a given time. First, at a fixed t, N (n) is a decreasing function of n, a behavior which is not obvious at all. In fact, this decrease will not remain true throughout the whole clustering, since we know for sure that at the end, all probabilities of existence will equal zero except for the biggest raft containing all the particles. But for t < t 60 , N (n) is a decreasing function of n.

This decrease appears to be subdivided in two regimes: for low values of n, N (n) decreases with a slope close to n -2 , and for large values of n, the decrease steepens. Nevertheless, because the values do not extend on enough decades, it is impossible to differentiate a power law decrease from an exponential one.

The position (n, N (n)) at which the transition between these two regimes appears seems also to be a function of time. When t increases, the transition happens for larger n at a lower value of N (n). These observations can also be made by looking at the probabilities instead of the distributions of sizes. There again, we recover the existence of two regimes with a time-dependent transition between them. The numerous observations we have just made are valid for all the parameters of table 2.2: for the four sets of parameters where enough simulations were performed, the distributions present the same characteristics, with a transition between two regimes.

Other statistical quantities: Cumulative distribution function, standard deviation, and mean size

Other probabilistic quantities can be used to represent our data. Instead of the distribution of sizes, we can for instance look at the cumulative distribution function [95], as represented in figure 2.18.a, or at more global quantities such as the mean size of the clusters < n >, or their standard deviation σ (see figure 2.18.b). The cumulative distribution function is defined as the probability for a particle to be in a raft of size less than or equal to n. Here, we will rather plot P (X ≥ n), the complementary probability for a particle to be in a raft of size more than or equal to n.

This probability is quite obviously a decreasing function of n, from P (X ≥ 1) = 1 to 0 when n ≥ n tot + 1. The distribution of sizes transits from a situation where particles are isolated to a situation where more and more rafts are formed: P (X ≥ n) is an increasing function of t for any given n. When the aggregation is over, the cumulative distribution function is constant at 1. The mean size of the clusters < n > increases with time, as emphasized by figure 2.18.b, while the standard deviation has a non-monotonic behavior, with first an increase of its value towards a maximum value at around n tot /2, followed by a slower decrease. Interestingly, we can see that during the first part of the clustering, σ is higher than the mean value < n >. This means that the data is widely spread around the mean, as qualitatively intuited with the visualizations of the aggregation, both in the experiments and the simulations: the particles are globally distributed between a larger raft and isolated particles.

After a certain amount of time, the curve of the standard deviation intersects the mean size < n >. At this time, the biggest cluster has already collected almost all the particles, since < n >≈ n tot /2, meaning that approximately only two rafts remain. The following slow decrease of σ comes from the averaging of the 980 simulations that do not end at the same time.

We have spoken a lot about the most undeniable advantage of performing numerical simulations: being able to average a very large number of results in order to derive statistically relevant distributions. These distributions remain to be explained, of course. But the simulations have another important interest compared to the experiments: the interacting forces can be chosen as desired.

Changing the forces

Removing the influence of the number of particles

Up to now, we have performed numerical simulations by choosing the system (2.5) to model the interaction between two rafts. This choice was motivated by the results of chapter 1, where the attraction between two rafts was extensively studied. In the previous section, we have proven that the balance between the forces of system (2.5) were able to satisfactorily reproduce the experiments, and a specific distribution of sizes was deduced.

Yet, we can also choose to change the interaction between the rafts. By doing so, the clustering process will not mimic a physical system anymore. The dynamics will certainly differ from what we have observed in the experiments and the "physical" simulations. And precisely because it will differ, it will give us some information on the real system.

We choose to simplify the interaction by removing any dependence with the sizes of the clusters. The velocity of a given raft will therefore only depends on its distance from the other rafts, but neither on its size nor the size of the other rafts. Equation (2.9) now becomes:

V (x) i = γB 5/2 Σ 2 3µ o k G l ij + 2R part R part K 1 L ij c × cos tan -1 2n j R 3 part g(ρ part -ρo+ρw 2 ) 3γ c K 1 L ij c
(2.12)

By doing so, we hope to decouple the influence of the number of particles in each raft from the role played by the interparticle distance L ij . A visualization of the process is given in figure 2. [START_REF] Nathan J Mlot | Fire ants self-assemble into waterproof rafts to survive floods[END_REF].a, along with the measurements of the size of the biggest cluster and the number of clusters. andc) Evolution of the size of the biggest cluster and the number of clusters, as a function of time, for two types of simulations: in red, the simulations are performed with a non-physical interaction between the rafts (see equation (2.12)), while in black, we use the real interaction of equation (2.9). For both curves, R domain = 5.8 cm and ntot = 200.

Even qualitatively, the aggregation dynamics appear really different from what we have seen: instead of a single large cluster collecting all the remaining particles, we find here several rafts of intermediate sizes. This is not surprising, since now the number of particles inside a raft no longer has any influence on its dynamics. Big rafts do not attract particles more efficiently than small ones.

This leads to a second feature: the clustering becomes far slower. Here again, this was expected, since removing the factor √ n i n j decreases mechanically the norm of the velocity.

This is why we choose to represent the evolution of n max and n clusters with a logarithmic scale for the abscissa. Otherwise, since the time scales between the two simulations are far too different, we would not be able to compare them. The two figures confirm this observation: there is more than one decade between the two simulations for the total time of aggregation. Moreover, the slope of both n max and n clusters is far larger when forces depend on the sizes of the rafts.

Influence of the initial surface density

This delay in the clustering being established, we can now try to see if we recover in this new situation the same dependence of the aggregation dynamics with respect to the surface density φ surf . To that end, as we did before, we perform several simulations with various numbers of particles n tot and different radii of the domain R domain . The results are presented in figure 2.20. At first sight, we seem to find here again the same type of dependency, with a faster clustering when particles are more concentrated. Yet, the order of the curves does not strictly follow the φ surf -scale. For instance, the yellow curve has a higher concentration of particles than the orange one, but its evolution is slower. The same observation can be made between the blue and the light blue curves. In both cases, the faster dynamics are found in the simulations for which the particles are distributed in the smaller domain. This can be understood at least qualitatively: there is no more effect of the number of particles on the forces experienced. As a consequence, when the particles occupy a larger domain, even though their number is higher, the time taken for a particle from one side of the domain to reach the center will be higher than in a smaller domain. Boundary effects prevent φ surf from being the only relevant parameter: an increase in n tot is not counterbalanced by a parallel increase of R domain .

Distribution of sizes and probabilities

Following the same path as before, we can now have a look at the quantities that really describe the clustering in figure 2.21: the distribution of sizes at different times, and the two types of probabilities defined previously: the probability that a given particle belongs to a raft of size n (figure 2.21.b), and the probability for a given raft to be of size n (figure 2.21.c). Here again, we focus on the first part of the aggregation, since at the end, the distribution is trivial: only one raft remains. The analysis of this figure has to be made in comparison with figure 2.17, where the aggregation was simulated using real capillary forces. As before, we recover a decrease of the number of isolated particles with time. But for the other sizes, the behavior is less clear. A first important difference concerns the behavior of the curves at a given time: we no longer have a monotonic decrease of N (n) for large times (or equivalently a decrease of the probabilities), whereas it was the case in figure 2.17. Here, N is neither monotonic in n nor in t.

Here again, N (n) seems to be subdivided in two regimes: for large values of n, N (n) decreases very quickly, a behavior which was already observed for real forces. Nevertheless, for low values of n, the evolution of N (n) completely differs from what we had previously: instead of a decrease with a slope close to n -2 , we now have either a plateau or a slow increase. This is fundamental, since the goal of this section is to bring insight on the real dynamics by removing the influence of the size of the clusters on the velocities. As a consequence, it is clear that with realistic capillary forces, the evolution of N (n) for low values of n, and more precisely of its slope, is directly linked to the dependence of the capillary forces with the number of particles.

As before, the position (n, N (n)) of the transition between the two regimes is a function of time. As time increases, the transitions happens for larger n and lower N (n).

We have seen how a modification of the interactions between the different clusters can directly influence the shape of the distribution, as well as the time scale of the aggregation. But having seen it does not mean having explained it, we just know the two pieces of information are correlated. Moreover, whether it is for non-physical forces or for realistic capillary forces, we have only described the distributions of sizes and the various probabilities, without explaining neither their time-dependency nor their behavior at a fixed time. The task is not simple, but whether we manage at the end to understand the behavior of the distributions or not, we need theoretical tools to at least try.

Smoluchowski coagulation equation

Theoretical framework

Assumptions on the system

As briefly mentioned in the introduction of the current chapter, aggregation is a general phenomenon: particles at an interface may aggregate, but as do dust in the universe to form planets and droplets in a water spray. And even though the interaction between the different constituents of the system undergoing clustering has nothing in common, general tools can be used. From a probabilistic point of view, the goal is to model the distribution of sizes thanks to the probabilities of encounter between clusters. These probabilities will be system-dependent, of course, but once they are known, the general equations describing the evolution of the distribution of sizes will be the same.

The theoretical framework was established by Smoluchowski in 1916 [START_REF]Drei vortrage uber diffusion, brownsche bewegung und koagulation von kolloidteilchen[END_REF] for a diffusive process by Brownian motion, but due to the extraordinary general aspect of his work, it has since been applied for countless systems, even the development of structures in the expanding universe [START_REF] Silk | The development of structure in the expanding universe[END_REF]. Very good reviews exist on the mathematical dimension of these coagulation equations [START_REF] David | Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists[END_REF][START_REF] Leyvraz | Scaling theory and exactly solved models in the kinetics of irreversible aggregation[END_REF]. We will present here the basic ingredients, in a general situation (and not in the specific case of particles at an interface).

We consider two clusters of mass x and y (for particles, the mass corresponds to the number of particles). When they are close, they can aggregate into a cluster of mass x + y. In principle, the probability of merging should depend on their respective masses, positions and velocities. Such a model would be very complicated to implement, so we are going to perform probably the most severe assumption we could have made: we make the hypothesis that this probability only depends on the masses. In this mean-field approach, we consider that no spatial correlation exists during the aggregation. This is obviously wrong: indeed it is clear that the respective positions of the clusters in a system of finite size will strongly affect the aggregation. This assumption can sometimes be justified if the transport mechanism is faster than the reaction of merging. This is not the case in our situation.

Nevertheless, trying to apply the Smoluchowski coagulation equation to our system is still really interesting, even if we do not manage to account for all our results. First, it will give us some tools to study our data. Secondly, we can still learn information from theories that do not match the experiments by analyzing what is working and what is not.

We also neglect the cluster morphology on the aggregation rates. Furthermore, we consider that the reaction is binary, meaning that no more than two clusters can merge simultaneously.

Equation of evolution

Keeping in mind all these assumptions, and especially the mean-field hypothesis, we can now focus on an elementary merging event. If we designate by C x a cluster of mass x, then a merging event can be written as follows:

C x + C y K(x,y) ----→ C x+y (2.13)
with K(x, y) the Kernel of the reaction, representing the rate of reaction between two clusters of mass x and y. x and y can be either continuous or discrete variables. We will keep the letters x and y for the continuous case, and the letters i and j for the discrete one. We can assume that K is symmetric:

K(x, y) = K(y, x)
The number of clusters of a given mass x can increase because two clusters of smaller size merge into a new cluster of size x, and can decrease because a cluster of size x has merged with another cluster. Here again, a new assumption is made: we consider that the mass of a given cluster can only grow: no erosion exists in our model. We can therefore write the Smoluchowski coagulation equation, either in its continuous form or discrete one:

                                 ∂N av (i, t) ∂t = 1 2 i-1 j=1 K(j, i -j)N av (j, t)N av (i -j, t)- ∞ j=1 K(i, j)N av (j, t)N av (i, t) ∂N av (x, t) ∂t = 1 2 x 0 K(y, x -y)N av (y, t)N av (x -y, t)dy- ∞ 0 K(x, y)N av (y, t)N av (x, t)dy (2.14a) (2.14b)
where N av (x, t) is the average number of clusters of size x at time t. We need to be careful here: N av is not defined as N in the previous sections. Since the system can a priori contain an infinite number of particles, we cannot use directly the number of clusters as in our experiments and simulations. But if the system is finite, the two quantities are proportional by a factor n tot .

Because they constitute an infinite system of coupled non-linear differential equations, these equations are extremely challenging. The existence and uniqueness of a solution is not guaranteed at all without further assumptions on the kernel.

Exact solutions exist for three specific Kernels [START_REF] Deaconu | Smoluchowski's coagulation equation: probabilistic interpretation of solutions for constant, additive and multiplicative kernels[END_REF], the constant one [START_REF]Drei vortrage uber diffusion, brownsche bewegung und koagulation von kolloidteilchen[END_REF], the additive one [START_REF] Am Golovin | The solution of the coagulation equation for cloud droplets in a rising air current[END_REF] and the multiplicative one [START_REF] Mcleod | On an infinite set of non-linear differential equations[END_REF]:

K(x, y) = 1 K(x, y) = x + y K(x, y) = xy (2.15)
The multiplicative kernel belongs to the class of gelling systems, and therefore deserves a few words. The solution for this kernel is an example of a system exhibiting a phase transition called gelation: a portion of the mass belongs to a cluster of infinite size in a finite amount of time. The solution is therefore defined only up to a critical time T gel at which gelation occurs. Some criteria exist on the kernel for the existence of a non-infinite gelling time. Usually, kernels are assumed to be homogeneous, meaning that:

K(xc, cy) = c γ K(x, y) (2.16)
A common criterion is the following: a system is non-gelling if γ ≤ 1. More elaborate criteria exist to determine if a kernel is gelling, but this is not the subject of the present study.

Self-similarity

Formulation of the self-similar solution

The Holy Grail of anyone trying to study the distribution of an aggregation process (and therefore our Holy Grail) is to find a self-similar solution of the Smoluchowski coagulation equation [START_REF] Sk Friedlander | The self-preserving particle size distribution for coagulation by brownian motion[END_REF][START_REF] Van Dongen | Dynamic scaling in the kinetics of clustering[END_REF], in order to explain the dependency of N av (n, t) both with n and t. We can look for a solution of the form:

N av (n, t) = s(t) -τ ψ n s(t)
(2.17)

with ψ the self similar function, s(t) a time rescaling, and τ an unknown exponent. τ can be easily determined using the conservation of mass (for this calculation, we will use the continuous case):

∞ 0 xN av (x, t)dx = 1 = ∞ 0 xs(t) -τ ψ x s(t) dx = s(t) -τ ∞ 0 s(t)xψ(x)s(t)dx = s(t) -τ +2 ∞ 0 xψ(x)dx (2.18a) (2.18b) (2.18c)
This directly indicates that τ = 2. The self-similar solution can now be written as follows:

N av (n, t) = s(t) -2 ψ n s(t) (2.19)
But the work is not over: we still have no clue on both the self-similar function ψ and the time function s(t). To make further progress, we define the k th moment of the distribution as:

S k (t) = ∞ j=1 j k N av (j, t) or S k (t) = ∞ 0 x k N av (x, t)dx (2.20)
and we define P k as the ratio between two successive moments:

P k = S k /S k-1 .
Then if we inject the self-similar expression of N av in P k , we can manage to express the function s(t):

P k = ∞ 0 x k N av (x, t)dx ∞ 0 x k-1 N av (x, t)dx = ∞ 0 x k s(t) -2 ψ x s(t) dx ∞ 0 x k-1 s(t) -2 ψ x s(t) dx = ∞ 0 s(t) k x k ψ(x)s(t)dx ∞ 0 s(t) k-1 x k-1 ψ(x)s(t)dx = s(t) ∞ 0 x k ψ(x)dx ∞ 0 x k-1 ψ(x)dx ∼ s(t) (2.21a) (2.21b) (2.21c) (2.21d)
Since s(t) is only a function of time, we have proven that it is proportional to any ratio P k between two successive moments. Since s(t) is only supposed to capture the time dependency, the scaling law s(t) ∼ P k should be sufficient to rescale our data. As it is done in [START_REF] Da Weitz | Dynamic scaling of cluster-mass distributions in kinetic colloid aggregation[END_REF], we will use the moment of order 2 P 2 = S 2 /S 1 , that can be easily calculated for any experiment or simulation.

Self-similarity for the aggregation of granular rafts

If we plot N av P 2 2 = N P 2 2 /n tot as a function of n/P 2 , we expect all our data to be superimposed onto a single curve representing the self-similar function ψ. This is what we have done in figure 2.22 for a given experiment and a numerical simulation with a similar surface density (with real forces, and not the modified forces studied in the previous section). This figure constitutes one of the main result of this whole chapter.

First, let us focus on the rescaling of the experiment. We can see that a large majority of the points seems to have collapsed onto the same curve. Nevertheless, the points located inside the dotted circle appear to be completely out of this master curve. This is completely logical, since they correspond to sizes of rafts that were detected only once in the five experiments. Their probability of existence is therefore completely overestimated, meaning that these points are not relevant at all. But since we measure them, we chose to include them anyway in the figure .  We find here again the same limitation of the experiments that we encountered earlier: we do not have access to the whole distribution of sizes. As soon as n is larger than 10, the measurement is not relevant. As a consequence, it is not surprising that part of the curve does not rescale well.

If we focus on the left portion of figure 2.22.a however, we find that the points all align onto a same line of slope approximately -2, no matter what their corresponding time t is. This seems to indicate that indeed the aggregation proceeds according to a self-similar evolution described by equation (2. [START_REF] Nathan J Mlot | Fire ants self-assemble into waterproof rafts to survive floods[END_REF]). Yet, we need to remain careful here, because even though the points are globally aligned, they are still scattered on approximately one decade on the vertical axis for each abscissa. The rescaling is not completely clear.

The experiments do not allow us to study the whole distribution, but no similar limitation exists for the simulations. In figure 2.22.b, we look at the same rescaling but for a simulation. Again, all the data seem to collapse relatively well onto the same curve. We find again a decrease of slope close to -2 for the left side with still some kind of vertical scatter of the points which tends to indicate that the rescaling is not perfect. But contrarily to the experiment, we can also look at the right side of the curve, since a sufficient number of simulations has been averaged. Here also, all the data collapse onto a same curve of higher slope. Because there is not even one decade in abscissa, it is however impossible to discriminate between a power law and an exponential decrease. The transition between these two regimes is located at n/P 2 ≈ 1.

All these features were already found with the raw distributions before rescaling (see (ntot = 300, R domain = 2.9 cm). The input parameters between the experiments are different, but the surface density is similar. The color bars represent the time. The points encircled in the dotted circle correspond to statistically non-relevant data, as explained section 2.2.3. In c, we superimpose the two curves of a and b, in red for the experiment and black for the simulation. In the three figures, the solid lines have a slope -2, and are only a guideline to the eye.

figure 2.17): the existence of two regimes and of a transition whose position was evolving with time. Here, the rescaling seems to have captured this dependence with time, leading to a relatively satisfying superimposition of the two regimes for all times and the localization in one point of the transition. This rescaling also allows us to compare more easily the numerics with the experiments, as it is done in figure 2.22.c. The two curves are perfectly superimposed, giving us once again faith in the expressions of the forces we used for the simulations. We can also see that the regime with the highest negative slope is located at the position where the experimental points are not relevant for statistical reasons. This is why we can observe it only in the simulations. If we go a bit into details, this regime corresponds to values of n/P 2 larger than one. A quick look at the definition of P 2 is enough to see that it is an increasing function of time (more details on P 2 can be found in the next section). This means that the highly-negative slope regime corresponds to times smaller than a transition time t c , t c being an increasing function of n.

Influence of the initial surface density

Keeping in mind that the rescaling is not completely satisfying, we can now explore the dependence of the self-similar function with the surface density of particles. To that end, we plot in figure 2.23 the rescaled distributions for all the different surface densities we explored, both for the experiments in a and for the numerics in b. All the curves are clearly superimposed whatever the corresponding surface density is. This tends to indicate that the self-similar function ψ of equation (2. [START_REF] Nathan J Mlot | Fire ants self-assemble into waterproof rafts to survive floods[END_REF]) is independent of φ surf . In fact, this result is not really surprising. As explained before, the surface density influences the time scale of the aggregation. Yet, the goal of this rescaling is precisely to remove the time-dependency thanks to the moment P 2 . This function depends on the surface density, as demonstrated in section 2.4.4, and appears to capture all the effects of the surface density. This comment is not anecdotal at all: it means that in the range of φ surf explored, the distributions of sizes are identical up to a time rescaling. φ surf has no effect on the shapes of the distributions, but only on their time scale.

Comparison with other aggregating systems

Simulations with modified forces

Before trying to quantify in details this rescaling function P 2 , we can have a look at the rescaled distributions for the simulations with modified forces of section 2.3.4, where the interacting force no longer depends on the number of particles in each raft. We use the same type of rescaling function. The results are displayed in figure 2. [START_REF] Peter | Capillary interactions between particles bound to interfaces, liquid films and biomembranes[END_REF].

We recover what was described with the raw distributions: the existence here again of two regimes of evolution, with a transition for n/P 2 ≈ 1. The regime with the highest negative slope seems to be quite similar with the one obtained for the real forces. This is not at all the case for the portion of the curve with n/P 2 ≤ 1: the rescaled distributions there appear to be slightly increasing functions of n/P 2 . Moreover, in that area, the rescaling is not completely convincing, which was also the case for the simulations with the real capillary forces. This could indicate that the distributions do not really evolve according to the self-similar function of equation (2.17). We may have different rescalings for the two regimes identified.

But even though the rescaling is not perfect, the differences between figure 2.23 and figure 2.24 are more than clear. The self-similar functions are completely different, meaning that the detail of the specific interaction between two rafts has a fundamental effect on the evolution of the distribution of sizes during the aggregation.

The two previous situations (real and modified forces) can also be compared to what has already been done in the colloid literature, where dynamical scalings have been found to describe the cluster size distribution.

Distributions of sizes for DLA (diffusion-limited aggregation) and RLA (reactionlimited aggregation)

In the introduction, we have presented two very famous clustering processes: the diffusion and reaction limited aggregations, that can be observed for colloids moving at an interface because of Brownian motion. The difference between DLA and RLA comes from the comparison between the time scale needed for two clusters to encounter one another and the time scale needed for them to overcome their repulsion.

Just as we did with our data, we can look for a self-similar solution for DLA and RLA. This is what is done in figure 2. [START_REF] Botto | Capillary interactions between anisotropic particles[END_REF], where both the raw and rescaled distributions are presented [START_REF] Da Weitz | Dynamic scaling of cluster-mass distributions in kinetic colloid aggregation[END_REF]. Both processes appear to exhibit dynamical scalings of the cluster mass distribution, as emphasized by the two insets. But the similarities stop right there: the rescaled distributions are very different between the RLA and DLA. A power law decrease is found for the RLA, while a different scaling is observed for DLA, with a sharp decrease of the rescaled distribution for m/P 1 > 1, and some kind of asymptote for m/P 1 < 1.

The rescaled distributions of the DLA are a bit similar to the one we obtained by removing the dependence of the capillary forces with the number of particles. By some aspects, these two situations are similar: in both cases, the velocity of the particles does not depend on the size of the other objects at the interface, and the aggregation is limited by The main figures represent the raw distributions, while in the inset, the data have been rescaled according to equation (2.17). The figures are taken from [START_REF] Da Weitz | Dynamic scaling of cluster-mass distributions in kinetic colloid aggregation[END_REF], where more details and results can be found.

the time needed for two clusters to bump into one another. Yet, for DLA, the individual motions of each particle is random, while with our modified forces, the trajectories are completely deterministic.

For the real capillary forces, the rescaled distributions completely differ from DLA, RLA, or simulations with modified forces. The self-similar function does seem to account for the aggregation process, since the data are superimposed satisfactorily, but its evolution is quite different: for n/P 2 < 1, it behaves as some kind of power-law decrease, while for n/P 2 > 1, the decrease becomes sharper.

To understand a bit deeper this rescaling, we need to focus more on the moment of order 2, which was used to take into account the time evolution of the aggregation.

Moment of order 2

Expected power-law Up to now, we still have not used the Smoluchowski coagulation equation. Indeed, we have only postulated a certain form of the self-similar function, and see if it was coherent with our data. But by injecting the self-similar expression inside the Smoluchowski coagulation equation, we can also try to derive some equations for ψ and s(t). First, let us write again the two equations we are going to use:

                 N av (x, t) =s(t) -2 ψ x s(t) ∂N av (x, t) ∂t = 1 2 x 0 K(y, x -y)N av (y, t)N av (x -y, t)dy- ∞ 0 K(x, y)N av (y, t)N av (x, t)dy (2.22a) (2.22b)
We inject equation (2.22a) in equation (2.22b):

-2 ṡ(t)s(t) -3 ψ x s(t) -s(t) -2 ṡ(t)s(t) -2 xψ x s(t) = 1 2 x 0 K(y, x -y)s(t) -4 ψ y s(t) ψ x -y s(t) dy- ∞ 0 K(x, y)s(t) -4 ψ y s(t) ψ x s(t) dy (2.23)
We simplify by s(t) -4 in each term, and use a change of variables to simplify the variables inside the functions ψ.

-

ṡ(t)s(t) 2ψ(x) + xψ (x) = 1 2 x 0 s(t)K(s(t)y, s(t)(x -y))ψ(y)ψ(x -y)dy- ∞ 0 s(t)K(s(t)x, s(t)y)ψ(y)ψ(x)dy (2.24)
Then, we use the assumption of homogeneity of the kernel : K(cx, cy) = c γ K(x, y).

-

ṡ(t)s(t) -γ 2ψ(x) + xψ (x) = 1 2 x 0 K(y, x -y)ψ(y)ψ(x -y)dy- ∞ 0 K(x, y)ψ(y)ψ(x)dy (2.25)
We now have a function of t which is equal to a function of x, meaning they are both equal to the same constant W . We end up with a differential equation for s(t) and another one for ψ(x):

             ṡ(t) =W s(t) γ -W 2ψ(x) + xψ (x) = 1 2 x 0 K(y, x -y)ψ(y)ψ(x -y)dy- ∞ 0 K(x, y)ψ(y)ψ(x)dy (2.26a) (2.26b)
What is really exciting in equation (2.26a) is that it does not depend on the kernel, and can be solved explicitly. The solution is the following:

s(t) = (C + (1 -γ)W t) 1/(1-γ) ∼ t→∞ t z (2.27)
with C a constant of integration, and z = 1/(1 -γ). This is really interesting for us, because the time-rescaling function s(t) appears to evolve as a power-law. And determining the exponent of this power-law would give us insight on the kernel of the aggregation process through the value of the homogeneity exponent γ. Luckily, we know that s(t) ∼ P 2 , so we have a direct experimental and numerical access to the evolution of the rescaling function s(t).

Experimental measurement of P 2

To explore this idea, we plot the evolution of P 2 for various surface densities, both for the experiments and the simulations, in figure 2.26. The expression of P 2 is reminded in equation (2.28)

P 2 = S 2 S 1 = ∞ j=1 j 2 N av (j, t) ∞ j=1 jN av (j, t) = ∞ j=1 j 2 N (j, t) n tot (2.28)
As illustrated by the figure, P 2 is an increasing function of time, from P 2 (t = 0) = 1 to P 2 (t → ∞) = n tot . With this graph, we recover an observation that was made a lot in this chapter: when the initial surface density of the particles increases, the aggregation is going faster.

Sadly, our data do not allow us to fit a power-law with enough confidence, since for most of the curves, the portion that could appear as a straight line in a logarithmic plot is extended in less than one decade in abscissa.

Moreover, for most of the curves, it seems the slope is still increasing when P 2 sharply slowdowns because of the finite aspect of our system: we are not working with an infinite number of rafts distributed all over the plane, but with a limited number of particles inside a finite domain. Since the power-law scaling of P 2 is only valid at large times, we may not be able to reach these large times with our experimental setup and our numerical methods. Furthermore, a fit based on the exact expression of s(t) as defined in equation (2.28) does not manage to capture the behavior of P 2 . Several interpretations can be drawn from the failing of the fit in figure 2.26. Basically, it can have two origins:

• first, it can be due to the finite aspect of our experiments and simulations, that can prevent us from having access to the long-time dynamics;

• secondly, it can simply be explained by the fact that the aggregation is not governed by the Smoluchowski coagulation equation, or at least that some of the numerous assumptions we have made have to be thought again. We can specifically evoke the mean-field hypothesis, where we considered that no spatial correlation exists during the aggregation.

Some studies have tried to describe systems where the Kernel depends both on the masses and the velocities of the interacting clusters: the ballistic agglomeration [START_REF] Gf Carnevale | Statistics of ballistic agglomeration[END_REF]. The probability density function will depend on mass and impulsion, and a collision between two clusters conserves the total mass and total impulsion. Some scalings can be derived [START_REF] Ben-Naim | Decay kinetics of ballistic annihilation[END_REF], and it has been proved that they significantly deviate from a mean-field scaling analysis, in particular in their dependence with surface density [START_REF] Trizac | Dynamic scaling behavior of ballistic coalescence[END_REF].

But whether it is because the model is too simple or because our system is too reduced, what is quite clear is that we cannot go really further in the analysis of our data by using the Smoluchowski coagulation equation. A solution could be to perform simulations with many more particles, inside a much larger domain. Yet, in our simulations, we calculate the interactions between each particle, so increasing a lot their number would result in a huge computational cost. Of course, a cutoff length in the calculation of the capillary force could be implemented, but we did not perform such a study. Despite these numerous limitations, the rescaling of the distributions is still very good. To understand the different regimes observed a bit deeper, we can analyze even further the time dependency of the distributions. We recover a behavior that was already described. For the isolated particles, N (1, t) is a strictly decreasing function of time, as expected, from N (1, t = 0) = n tot to zero. The other specific curve is the curve n = n tot , which is strictly increasing, from 0 to

A thorough analysis of the distributions

N (n tot , t = ∞) = 1.
In between these two specific behaviors, N (n, t) has a non-monotonic evolution, with first an increase of its value up to a maximum, followed by a decrease. This non-monotonic behavior could already be seen in all the previous representations by following the different colors. But now we can quantify it, for example by measuring the position of the maximum of N (n, t) for each value of n. This will give us three quantities:

• the value n of the curve;

• the maximum value N max (n) reached during the clustering (or the maximal probability P part max = nN max (n) to find a raft of size n during the whole aggregation);

• the time t max (n) at which N (n, t) is maximum.

Maxima of the distributions

We can therefore plot three figures in order to represent these three quantities as a function of one another, as it is done in figure 2. [START_REF] Roh | Biphasic janus particles with nanoscale anisotropy[END_REF]. In a, we plot directly the position of the maxima as a function of time, as it can be read in the previous figure 2. [START_REF] Lewandowski | Oriented assembly of anisotropic particles by capillary interactions[END_REF]. We recover the behavior described above: first a decrease of N max with time, followed by an increase for large times that also corresponds to large values of n. This can also be observed in figure 2.28.c, where the non-monotonic evolution of N max with n is quite clear. The time t max at which this maximum arises is strictly increasing with n, as demonstrated by figure 2.28.b. This is far from being obvious.

When looking at the evolution of the moment of order 2 P 2 , we were not able to measure scaling laws. Here, for the first time, we seem to have found a quantity that can be described by a scaling law: N max as a function of n, at least for the portion where the curve is decreasing. The second part of the curve corresponds to finite-size effects (a finite number of rafts in a finite domain). Therefore, we will only focus on the first part, where the decrease has roughly a slope -2.

If we prefer to think in terms of probabilities rather than number of clusters, this means that the maximum probability P part max = max(P part ) that a given particle belongs to a raft of size n throughout the whole aggregation evolves as 1/n:

   N max = max t (N (n, t)) ∼ n -2 P part max = max t (P part (n, t)) ∼ n -1 (2.29a) (2.29b)

Maximum number of rafts and surface density

Comparison between the experiments and a simulation

This new approach for the representation of the distributions as a function of time can also be conducted for the experiments, even though the poor statistics will limit what we can do. Indeed, as soon as n is too large, the corresponding probability is either zero (no such raft in all five experiments) or at the minimum value. In both cases, the probability will not be relevant, and neither will be the time at which the maximum is measured.

Yet, we can still compare the experiments with the simulation, as it is done in figure 2.29. The simulation is plotted in black squares, while the experiments are in colored circles. Exactly as expected, the values of the experiments are not relevant for n ≥ 15. Nevertheless, we can focus on the curve for n ≤ 15, where we recover the decrease of N max with n. Moreover, we find a very good agreement between the experiments and the numerics, which once again gives us faith in all the results we derive from the simulations. ). The color bar displays the surface density φ surf of each experiment. To distinguish the different curves even for two almost identical φ surf , the scale of the color bar is not linear.

The dependence of N max with the surface density is not clear on this figure. It seems that the slope of N max becomes slightly more negative when φ surf increases (the yellow curve decreases faster than the blue one), but since the data is only available on one decade, this observation is mainly qualitative.

Influence of the initial surface density

Figure 2.28 can be drawn again for our four sets of simulations with different surface densities, in order to see how the three variables depend on the initial concentration of particles. The results are displayed in figure 2.30. We recover some of the previous observations, for instance with t max : when the surface density decreases, the aggregation slows down, so quite naturally, t max is shifted to higher values. Concerning the evolution of N max with time, besides this shift, it seems that the slope also slowly decreases with φ surf . The evolution of N max as a function of n seems to answer a scaling law whatever the surface density is. The slope of this scaling law seems to vaguely depend on the surface density: the yellow curve decreases slightly faster than the blue curve. All slopes are close to -2, which means that the maximum numbers of clusters of size n approximately decreases as n -2 , or equivalently, that the maximal probability of finding a particle inside a raft of size n throughout the whole aggregation decreases as n -1 . This seems to remain true no matter how concentrated are the particles, in the range of surface densities explored: in that sense, it is a rather strong result. The scaling law for N max as a function of n seems to depend only on the aggregation process and not on the specificities of a given set of parameters.

Going back to the modified forces

Finally, in order to be exhaustive, we can also apply this same procedure to measure the maxima of the distributions for the modified forces presented in section 2.3.4. The curve of N max as a function of n is presented in figure 2.31 in red, compared with a simulation performed with the same input parameters but a real capillary interaction in black. Comparison between two types of simulations. Maxima of the distributions as a function of n, for two types of simulations: in red, the simulations are performed with a non-physical interaction between the rafts (see equation (2.12)), while in black, we use the real interaction of equation (2.9). For both curves, R domain = 5.8 cm, ntot = 200. The solid line has a slope -2, and is only a guideline to the eye.

The curves are almost superimposed up to n ≈ 150. Only for very large values of n, N max continues to decrease for the non-physical interaction. This behavior can be understood, and has in fact already been explained in section 2.3.4: since all the rafts interact with the same magnitude, whatever their sizes are, then compared to the real case, there is no formation of one single large raft collecting all the remaining particles. On the contrary, at the end we usually have several rafts of intermediate size collapsing into the final one.

But if we focus on the first portion of the curve, we can see that N max globally decreases as n -2 for both curves. This reinforces again the importance of this scaling law: we knew already that it was rather independent of the surface density; now, it appears this behavior is also independent of the magnitude of the forces of interactions, and on how the individual velocities depend on the different sizes. It is a more intrinsic feature.

Transition between the two regimes

Transition and maxima of the distributions

We now have two results which are not connected a priori:

• in section 2.4.2, we managed to somehow rescale the distributions of sizes, by using the moment of order 2 P 2 . This procedure highlighted two different regimes of evolution of N (n), with a transition that varies with time. This time-dependency of the transition is precisely what we captured in the rescaling. The two regimes are separated by a dotted line in figure 2.32;

• in section 2.5, we analyzed the distributions for fixed values of n, and measured their maxima throughout the whole aggregation. These maxima are characterized by a triplet (n, t max , N max ): for each value of n, we measure the time t max at which the number of rafts N (n, t) reaches its maximum N max .

One can legitimately wonder how these two results are connected to one another. To elucidate this mystery, we plot on top of the rescaled distributions of figure 2.22 the points defined by the coordinates (n, t max , N max ). Their rescaled abscissa becomes n/P 2 (t max ) and there rescaled ordinate P 2 2 (t max )N max /n tot . These specific points are drawn in black dots in figure 2.32. As one can see, they are astonishingly all located right at the transition between the two regimes of evolution of N (n). Therefore, this transition corresponds to the maxima of the distributions for each size. Because P 2 is an increasing function of time, we can deduce that the sharply decaying regime on the right of the transition (regime 2) corresponds to the short-time portion of each curve of figure 2.27, where the distribution is increasing with time. Similarly, the slowly decaying regime (regime 1), on the left of the transition in figure 2.32, corresponds to the large-time dynamics of each curve of figure 2.27, where N is decreasing with time for any given n.

This can help us understand what these two regimes are: on the right of the transition (regime 2), the points correspond to data where N (n, t) is increasing with time. This means that the rafts of size n are created faster than they disappear by merging with other particles. Similarly, the points on the left of the transition (regime 1) correspond to the portion of the distributions where the rafts are disappearing faster than they are created. As a consequence, the transition corresponds to the equilibrium between the creation and the disappearance of the rafts for all sizes.

But these two regimes have even more meaning than that: indeed, even for a curve taken at a given time, as it was done for example in figure 2.17, we recover the existence of two different decaying regimes for N (n), without any variation in time. Therefore, these regimes not only correspond to a faster or slower creation of rafts with respect to their disappearance, but also to the intrinsic variation of N (n) with n at any given time. The transition concerns both t and n.

Another interesting feature concern the abscissa of this transition, which is located at approximately n/P 2 ≈ 1. This means that the size of the clusters that reach their maximum number should verify n(t max ) ≈ P 2 .

Moment of order 2 and maxima of the distributions

The previous observation can be checked quite easily by plotting these two quantities onto the same figure, as it is done in figure 2.33. As expected, the two curves are almost superimposed. The main difference concerns the number of points: P 2 can be calculated for all the time steps, while n(t max ) has only n tot values, since it has a single value for each size of raft. The equivalence between P 2 and n(t max ) is far from being obvious, since the definitions of these quantities are completely different. We do not know if this is a mere coincidence, or if there is a more profound explanation.

We can summarize a bit our different findings here. We have studied the aggregation of granular rafts, and managed to rescale the distributions of sizes into a self-similar solution exhibiting two regimes. The transition between these two regimes of evolution of the distributions has been made clearer: for each size of raft, it corresponds to the moment at which there is an equilibrium between the creation and the disappearance of the rafts.

We still lack a complete understanding of the overall aggregation process. More theoretical work would be needed to understand our data, as well as simulations and experiments including far more particles in a larger domain.

Conclusion and discussion

In this chapter, we have explored the aggregation of many particles at an oil-water interface, from an experimental point of view as well as a numerical one. The individual dynamics cannot be solved analytically, but a statistical approach can provide a complete description of the clustering process. Because the capillary force strongly depends on the distance between particles, the aggregation dynamics are greatly enhanced when the particles are more concentrated at the interface. The number of clusters decreases faster, and similarly, the size of the biggest raft increases faster. Quite quickly, a large raft is formed, and collects all the remaining particles.

The clustering process is described by the distribution of sizes at each time, or equivalently by the probability to find a particle inside a raft of a given size. This distribution appears to depend both on the time and the size of the clusters, and can be subdivided into two regimes of evolution with different slopes in a logarithmic plot. The transition between them appears to be a function of time, a dependence that can be captured thanks to a rescaling inspired by the Smoluchowski coagulation equation.

Yet, it is still not completely clear if the self-similar solution obtained thanks to this procedure perfectly captures the complete behavior of the distributions of sizes. It is however clearer that the transition between the two regimes corresponds to the maxima of the distributions of sizes as a function of time, for any given number of particles. The maximal probability to find a particle in a raft of a given size seems to answer a scaling law, but here again, no theoretical argument backs up this observation.

A model would be needed to better grasp the evolution of the distributions of sizes, and understand their variations both with time and number of particles. The existence of scaling laws to describe the aggregation is also not that clear, and needs further exploration.

From the beginning of this section, we have only focused on the aggregation of particles initially randomly dispersed at the interface. But we can also be interested in the clustering when the particles are placed at specific initial positions. By doing so, we can end up with specific trajectories, as well as a very controlled aggregation. We will present here only very preliminary qualitative results.

The initial distance between a pair of particles only influences the time taken for the two beads to aggregate, but as soon as more than two particles are present, their respective trajectories are going to be affected by the initial positions. This is very well illustrated by figure 2 In figure 2.34.a, the particles are initially equally separated. As a result, they all display a rectilinear motion, and they aggregate all three at the same time, approximately at the position of the center of gravity of the triangle formed by their initial positions. But as soon as we break the symmetry between the three particles, the trajectories start to curve, as emphasized by figure 2.34.b. This result is in fact quite natural, because a given particle will be more attracted by the closest bead. We will not say much more on the subject. But just for the beauty of it, let's look at more complex initial configuration, involving more particles. We place 18 particles on the sides of an equilateral triangle, with a constant initial spacing between the beads. Three clusters grow from each summit of the triangle before collapsing together in the center, leading to quite a hypnotic shape when we superimpose the successive positions.

Erosion and cohesion of a granular raft

Structural changes of a monolayer

Up to now, we have studied the birth of a granular raft, from the attraction between two spheres and between two already formed rafts, to the collective aggregation of many particles. Once aggregated, the particles form a monolayer at the oil-water interface. This raft will then live its life, encounter other rafts, move with the flow, be subjected to capillary forces. All these events can lead to structural changes that we wish to describe.

The mechanisms responsible for the resistance of an aggregate to erosion are still far from being understood, even though it is fundamental in many applications involving disperse systems, where the size of the flocks created greatly conditions the efficiency of the desired process. Flocculation, namely the aggregation of dispersed particles inside a suspension [START_REF] Thomas | Flocculation modelling: a review[END_REF], is for instance essential for water and wastewater treatments [START_REF] Bratby | Coagulation and flocculation in water and wastewater treatment[END_REF], where it is used to clarify the liquid by agglomerating dispersed particles into large assemblies that settle down. It is also crucial for the most noble cause any french researcher can think about, that is the cheese industry [START_REF] Patrick F Fox | [END_REF].

Depending on the application, the goal can be to prevent the formation of the aggregates, to facilitate their formation [START_REF] Chai Siah | A review on application of flocculants in wastewater treatment[END_REF], or to disperse them once they are formed [START_REF] Powell | Dispersion by laminar flow[END_REF], as illustrated in figure 3.1.a and 3.1.b. In that example, a three-dimensional spherical flock made of polystyrene spheres of radius greater than 20 µm is placed in a simple shear. At short times, the cluster behaves as a solid body, and rotates with the flow. But as the experiment proceeds, particles start to detach, first by remaining in the vicinity of the sphere, and finally, as visualized in figure 3 Whatever the application, the size of the flocks results from the competition between the external constraints that tend to break the aggregates [START_REF] Zaccone | Breakup of dense colloidal aggregates under hydrodynamic stresses[END_REF] and the cohesive forces preserving the integrity of the flocks [START_REF] Powell | Dispersion by laminar flow[END_REF], that can lead to a specific flock-size distribution [START_REF] Patrick | Coagulation and fragmentation: Universal steady-state particle-size distribution[END_REF]. But without a good understanding of the cohesive forces, characterizing this distribution is beyond reach. Various techniques have been developed to measure the strength of a flock [START_REF] Anthony | Micromechanics: a new approach to studying the strength and breakup of flocs[END_REF][START_REF] Jarvis | A review of floc strength and breakage[END_REF].

One example is presented in figure 3.1.c, 3.1.d, 3.1.e, where the cohesion of a threedimensional calcium carbonate fractal aggregate, made of particles of an average size of 1.34 µm, is explored [START_REF] Anthony | Micromechanics: a new approach to studying the strength and breakup of flocs[END_REF]. A micromechanical technique is developed, where the flock is stretched by two micro pipettes, one of them being linked to a cantilever whose deflection provides a measurement of the rupture strength. In that study, no relation between the flock size and its strength was found, probably because the rupture happens at the weakest point of the structure.

But despite these inventive methods, three-dimensional aggregates remain challenging to study, because a direct visualization of the rupture is hindered by the surrounding particles.

Since the work of Camoin et al [START_REF] Camoin | Aggregation in a sheared 2d dispersion of spheres with attractive interactions[END_REF], some studies have focused on the behavior of 2D aggregates stuck at a liquid interface to understand their erosion and breakup. Indeed, in this case, reconfigurations are effortlessly observed as the object remains on the same plane throughout the experiment and the hydrodynamic forces at play may then be investigated more easily. The interest in 2D aggregates has moreover been renewed by an increasing enthusiasm towards Pickering emulsions [START_REF] Spencer Umfreville | Cxcvi.-emulsions[END_REF], where particles are arranged in a two-dimensional configuration at an interface to stabilize droplets [START_REF] Aveyard | Emulsions stabilised solely by colloidal particles[END_REF], as demonstrated by figure 3.2, where a water droplet covered by latex particles remains stable in a cyclohexane solution. The particles are well adsorbed at the interface, and are packed hexagonally with some gaps, due to the curvature of the interface. Pickering emulsions allow the development of processes to stabilize emulsions in an environmentally-friendly way, without the use of surfactants [START_REF] Bernard | Particles as surfactants-similarities and differences[END_REF], whether it is for drug delivery [START_REF] Frelichowska | Pickering w/o emulsions: drug release and topical delivery[END_REF] to encapsulate bioactive products [START_REF] Timgren | Starch particles for food based pickering emulsions[END_REF] or for food applications [START_REF] Claire C Berton-Carabin | Pickering emulsions for food applications: background, trends, and challenges[END_REF].

With different external stresses, a monolayer of particles can undergo several types of structural changes, as illustrated in figure 3.3:

• first, a simple reorganization of its internal structure for low stresses [START_REF] Edward | Structure and dynamics of particle monolayers at a liquid-liquid interface subjected to extensional flow[END_REF][START_REF] Edward | Structure and dynamics of particle monolayers at a liquid-liquid interface subjected to shear flow[END_REF]. Such a rearrangement can be produced by shear flows that force the particles to slide along one another;

• the formation of cracks or folds [START_REF] Vella | Elasticity of an interfacial particle raft[END_REF][START_REF] Jambon-Puillet | Wrinkles, folds, and plasticity in granular rafts[END_REF]: a monolayer can wrinkle and buckle when compressed, while cracks can appear when the monolayer is stretched;

• erosion, during which the aggregate looses single particles. Here again, erosion events can be produced by an external flow tearing off particles located at the border of the aggregate;

• fragmentation, during which the 2D structure breaks into several large pieces. The breakup of the aggregate can happen at weak points of the structure.

Numerical simulations [START_REF] Higashitani | Two-dimensional simulation of the breakup process of aggregates in shear and elongational flows[END_REF] along with experiments have tried to characterize the relative importance of both breaking phenomena (erosion and fragmentation) and to describe the structure of the aggregates, whether for large particles aggregating because of their weight [START_REF] Nikolina | Restructuring and break-up of two-dimensional aggregates in shear flow[END_REF][START_REF] Nikolina | Fragmentation and erosion of two-dimensional aggregates in shear flow[END_REF] or for micrometer-sized particles attracted to one another because of the undulation of the contact line at their surface [START_REF] Kasper Masschaele | Flow-induced structure in colloidal gels: Direct visualization of model 2d suspensions[END_REF][START_REF] Barman | Simultaneous interfacial rheology and microstructure measurement of densely aggregated particle laden interfaces using a modified double wall ring interfacial rheometer[END_REF]. In this chapter, we wish to quantify the cohesion of a granular raft, as well as the possible conditions leading to its breakup or at least to its erosion. We will start by studying the erosion a raft can be subjected to when moving along an interface, and then move to a model-experiment in order two understand a far simpler system composed of only two beads. A lot of the results of this chapter can be found in [START_REF] Lagarde | Probing the erosion and cohesion of a granular raft in motion[END_REF].

Erosion of a granular raft

First, let us focus on the conditions for which a granular raft can be subjected to erosion. To that end, we need to be able to exert a stress on the raft, with an increasing amplitude until particles start to detach. We will first present the experiment, and then we will quantify the cohesion of the raft.

A typical experiment

Experimental device

To study the erosion of a granular raft, several methods can be chosen. As described in various articles (see for instance [START_REF] Nikolina | Restructuring and break-up of two-dimensional aggregates in shear flow[END_REF]), an external flow can be imposed around the raft, whose amplitude can be tuned up until erosion is observed. Here, we choose to leave the fluid at rest, and to deflect the interface using a cylinder, leading to an experiment very close to the one described in section 1.4 (for clarity reasons, we will present again the experimental procedure in the next paragraph). The erosion of the raft will therefore come from its own motion rather than from an externally imposed flow.

In a typical experiment, as represented in figure 3.4, a circular metal cylinder (radius R cyl = 15 mm) is placed at an oil-water interface. In this chapter, the oil parameters are fixed: kinematic viscosity of the silicone oil ν o = 50×10 -3 m 2 .s -1 , density ρ o = 960 kg.m -3 , oil-water surface tension γ = 38 mN.m -1 . Elsewhere in the water tank, particles are sprinkled from above, and start to aggregate once they reach the oil-water interface due to capillarity until they form a large axisymmetric monolayer of particles, as described in the two previous chapters. Two types of ceramic beads are used in this experiment, with different densities and sizes ({ρ part ; R part }={ 3,800 kg.m -3 ; 0.125 mm } or { 6,000 kg.m -3 ; 0.1 mm }). Once the granular raft is formed, the cylinder is moved downwards until the desired depth H cyl , and because the curvature of the interface acts as an external field [START_REF] Cavallaro | Curvature-driven capillary migration and assembly of rod-like particles[END_REF][START_REF] Liu | Capillary assembly of colloids: Interactions on planar and curved interfaces[END_REF], the granular raft naturally follows the slope until it reaches the cylinder. The experiment is recorded from above at 100 frames per second, for different numbers of particles and different cylinder depths. A cylinder of radius R cyl deflects downwards an oil-water interface at a depth H cyl , the interface being pinned at the perimeter of the cylinder. A granular raft of radius R raf t is formed far away from the cylinder, and is attracted by the cylinder through the deflection of the interface.

Visualization of the erosion

As schematically represented in figure 3.3, several situations can arise depending on the external stress imposed on the raft. For low vertical deflections, the granular raft moves towards the cylinder without any internal motion of its constitutive elements. As H cyl is increased, slight rearrangements of the particles inside the raft may be observed, leading in some cases to erosion, as illustrated in figure 3.5. A first particle detaches in the second photo, quickly followed by other beads as the raft gets closer to the cylinder. As already explained several times throughout this thesis, the visualizations from above only give us access to the horizontal component of the speed of the raft. However, by numerically solving the equation of the interface (1.42), we can compute the vertical projection and deduce the total speed V , as illustrated in the inset of figure 3.5. As demonstrated in the inset, it is crucial to take into account the vertical speed of the raft in order to avoid an error that can go up to fifty percent.

The arrow in the inset of figure 3.5 indicates the position at which the first detachment of a particle from the edge of the raft occurs. We are interested in the threshold at which such an event is triggered. To that end, we need to play with the depth of the cylinder and the number of particles.

First results: existence of erosion

Our aim in this section is to determine the depth of the cylinder at which erosion occurs, and to study the dependence of this depth with the radius of the raft.

Non-reproducibility of the experiments

One particular feature of this experiment is its non-reproducibility, meaning that for a given position of the cylinder (R cyl and H cyl fixed) and a given size of the raft, erosion is not always observed. This is illustrated in figure 3.6, where the same experiment is reproduced twice, with the same number of particles and the same position of the cylinder. Erosion is present in b, while no particle detaches from the raft in a, even though the velocity of the raft is the same. We think this is due to the specific geometrical arrangement of the particles inside the raft that cannot be controlled, and changes from one experiment to the other. To circumvent this obvious limitation, the same experiment is reproduced several times for each couple of parameters (H cyl , n). If no motion of the particles is observed, not even the slightest reorganization or sliding, then we do the experiment only twice. We consider erosion to be present in one experiment as soon as one particle detaches from the edge of the raft.

Phase diagram of erosion

Each experiment is reproduced two or three times. Therefore, we can establish a phase diagram of erosion, as presented in figure 3.7, in the parameter space (R raf t , H cyl ), in which we describe three different situations:

• erosion is never observed: red circles;

• erosion is observed during all three experiments with the same parameters: green triangles;

• for the same control parameters (R raf t , H cyl ), erosion is sometimes observed, sometimes not: blue squares.

For deflections of the interface lower than 5 mm in depth, erosion is never observed, whatever the size of the raft is. This observation appears to be quite obvious, since the maximum velocity of the raft (and therefore the drag experienced by the particles) increases with the depth of the cylinder. The meniscus breaks for H cyl > 12 mm, preventing us from exploring larger stresses. Yet, it is hard to go beyond this first result with figure 3.7. Indeed, the depth of a raft increases with its number of particles n, so we can assume the attraction it exerts on a bead at its edge is also enhanced. But at the same time, we have previously shown that the velocity of a granular raft evolves as √ n. As a consequence, the drag force experienced also increases with the size of the raft. In brief, modifying R raf t influences both the cohesion of the raft and the force responsible for erosion. Moreover, even though H cyl plays a role through the deformation of the interface, it is not clear how it is directly linked to the eroding force.

To go beyond these limitations, we need to think again at the mechanisms responsible for the erosion.

Experimental cohesive forces of a raft

Velocity of detachment

As emphasized by figure 3.5, for each experiment, we can identify the time of detachment of the first particle (designated by the arrow in the inset of figure 3.5), and measure the velocity of the raft at the same instant V crit . The results are displayed in figure 3.8, which gives us information on the cohesion forces of a raft, through the knowledge of the velocity needed to detach a particle (V crit ≈ 0.01 -0.05 m/s for the denser ceramic beads, 0.005 -0.04 m/s for the lighter ones). Indeed, this critical velocity is directly linked to the force needed to extract a particle from the raft.

We get a lot of points for any given x-axis coordinate, since we explore a large range of values for H cyl , and because the same experiment is reproduced several times for each given depth. For each set of parameters (R raf t , H cyl ) the critical speed at which erosion occurs varies a lot from one experiment to the other, as emphasized by the great dispersion of the data of a given color at a fixed x-axis coordinate. Moreover, the speed of a raft increases with its size, as demonstrated in chapter 1. As a consequence, a bead belonging to the smallest raft will never experience a drag force as high as a particle inside a larger raft, leading to a smaller scattering of the data. However, no clear trend can be extracted from this data, especially knowing that for each x-axis coordinate, erosion was sometimes not observed, meaning the raft speed was not high enough to generate a drag force greater than the cohesion force. In other words, these experiments correspond to data points in figure 3.8 which may go beyond the limit of the y-axis and that we cannot observe with the experimental setup described here. The arrangements of the particles inside each raft may differ greatly from one experiment to the other and thus lead to strong differences in the global cohesion force. In order to go beyond in the analysis of the erosion, we need to describe the balance of forces experienced by the particles of the edge of the raft.

Balance of forces

Particles of the edge of a raft are subjected to three forces:

• the force exerted by the raft itself F raf t , that we wish to determine;

• the force exerted by the cylinder F cyl , through the slope of the interface;

• the hydrodynamic forces resulting from the motion of the raft F drag , that we can deduce from the experiment.

Experimentally, we observe that erosion mostly takes place at the rear of the raft. This can be understood thanks to the schematic representation of figure 3.9, where we can see that from a purely geometrical point of view, a particle strictly at the rear of the raft (particle B) is free to detach very easily, whereas a particle on either side of the raft (particle A) is blocked by the presence of the neighboring particles, and is more likely to slide and rearrange along the raft than to detach.

As a consequence, we will write the equilibrium of forces for the farthest particle from the cylinder (see the particle encircled in red in figure 3.4), for which all the forces are aligned.

A B Figure 3.9: Geometry constraints on the particles of the edge of a raft. Schematic representation of the forces acting on two particles of the edge of the raft. The forces considered are the drag force F drag , in the direction opposite to the motion, and the force exerted by the raft F raf t , directed towards the center of the raft. We do not consider the force exerted by the cylinder to keep this explanation as simple as possible, but it would not change anything. The resulting force is plotted in red.

First, following the linear approach described and used in chapter 1, and assuming small deformations and small particles with respect to the capillary length c = γ/((ρ w -ρ o )g), we can roughly estimate F cyl as the product of the weight of the particle with the slope of the interface:

F cyl = 4 3 πR 3 part (ρ part - ρ w + ρ o 2 )g dh dr (r = L + R raf t ) (3.1)
where h stands for the depth of the interface perturbed by the cylinder, but without the presence of the raft. The hydrodynamic forces F drag need a careful treatment. By considering the flow around the red particle in figure 3.4, we can calculate a Reynolds number based on the particle radius Re part = R part V raf t /ν o 1 for all our experiment, and a second one based on the raft radius Re raf t = R raf t V raf t /ν o ≈ 1 when the speed is maximum (for instance, Re raf t ≈ 3 for the maximum speed of figure 3.5). We thus typically expect the flow field to be completely laminar and the Stokes law to be valid.

This leads to a Stokes drag that can be expressed as follows:

F drag = 6πµ o kR part V raf t (3.2)
with µ o the dynamic viscosity of the oil phase, and k a coefficient that accounts both for the fact that the particle moves along an interface, and as a consequence is immersed in two phases, and for the screening of the drag experienced by the bead because of the rest of the granular raft. We assume the coefficient k is of the order of 0.5. In reality, the combined effects of the screening and the presence of two phases will lead to a lower coefficient, but the screening being complex to evaluate, we choose to keep k = 0.5 for all our experiments. In [START_REF] Nikolina | Fragmentation and erosion of two-dimensional aggregates in shear flow[END_REF], where a similar approach is developed, the coefficient k is taken as a fitting coefficient and is found to remain in the interval [0.2, 0.3] for the oil-water interface.

Possible role of other forces

Other forces could play a role in the detachment of a particle, and their respective importance needs to be discussed before carrying on with the analysis of the experimental results. As illustrated by figure 3.5, the raft does not move at a constant speed. The transient nature of its motion could influence the erosion by producing an inertial force. Yet, by measuring the acceleration of the raft, we can estimate the amplitude of the inertial forces F inertial = (4/3)πR 3 part ρ part (dv/dt), and compare it to the Stokes drag of equation (3.2), as it is done in figure 3.10. As expected, the drag force increases as the raft gets closer to the cylinder. Its evolution is exponential, an expected behavior: we know from chapter 1 that the velocity of a raft

V ∝ K 1 (L/ c ) ∼ L c exp(-L/ c ).
Because the acceleration is the derivative of the velocity, the same exponential behavior is found for the inertial forces. Just before contact, the raft briefly slows down, leading to a negative acceleration, which explains the sharp decrease of the inertial forces for low L.

For all our experiments, the inertial forces appear to be approximately two orders of magnitude lower than the Stokes drag, and as a consequence can be neglected.

One last assumption made here is that the velocity remains the same for every particle forming the raft. However, since the local curvature of the interface increases as the raft gets closer to the cylinder, we could expect the front of the raft to go faster than its rear, leading to an additional tangential stress that could stretch the raft. To address this issue, we measure the length of a raft L raf t as a function of its horizontal distance from the center of the cylinder, L raf t being defined as the distance between the particle closest to the cylinder and the farthest one. Here again, we solve numerically equation (1.42) to deduce from top views of the raft its arclength along the curved interface.

We measured L raf t for several sizes of rafts and several depths of the cylinder (for situations where erosion was observed). Two examples of such a measurement are presented in figure 3.11.

For low deflections of the interface, L raf t is globally constant. For H cyl = 10 mm, L raf t changes by no more than the size of two particles, even for the longest raft. As a consequence, considering the axisymmetric rafts studied in the present chapter we believe we can neglect the differential velocity across the area of the raft.

Cohesion of a raft

Keeping in mind that the drag force may be slightly overestimated, we write the balance of forces at the exact moment of detachment of the particle t crit , when the three forces are at equilibrium:

F raf t = F drag (t = t crit ) -F cyl (t = t crit ) F raf t ≈ F drag (t = t crit ) (3.3) (3.4)
where we simplify equation (3.3) into (3.4) by using the fact that experimentally, we always have F drag (t = t crit ) F cyl (t = t crit ). According to our model, F raf t equals the drag force, which is directly proportional to the velocity. By multiplying all the points of figure 3.8 by 6πµ o kR part , we end up with an estimation of the drag force experienced by the particle that is eroded, at the exact moment of its detachment. Here again, the scattering of the data is very important. And because for some experiments, erosion may not have occurred because the velocity of the raft was not high enough, it would make no sense to average the critical force for each raft radius. But by taking the minimum value for each x-axis coordinate, we can deduce a lower boundary for the cohesion force of a granular raft F cohesion as a function of its radius, as represented in figure 3.12 for two types of particles.

For the two types of ceramic beads, we find a slight increase of F cohesion as a function of R raf t . Moreover, the cohesion appears to be greater for the high-density beads, for a similar raft radius. This result is not surprising at all, since we know the capillary forces between two particles increase with their respective density.

To account for these experimental points, we can estimate the capillary force a raft exerts on a particle at its edge. This force can be estimated similarly to equation (3.1), as the product between the weight of a particle and the slope of the interface at the edge of a raft. This slope can either be measured on side views of the rafts or by solving equation (1.9), as already done in chapter 1, leading to the following result:

dh raf t dr (r = R raf t ) = h raf t / c K 0 (R raf t / c ) K 1 R raf t c (3.5)
where

dh raf t dr (r = R raf t
) is the slope of the interface at the edge of the raft (without the deflection caused by the cylinder), h raf t the depth of the edge of the raft, and K i the modified Bessel function of the second kind of order i. The slope remaining rather small for the sizes explored experimentally, the theoretical slope predicted by equation (3.5) and its experimental value differ by less than 10 percent. The corresponding force is represented by red and black stars in figure 3.12.

We find a strong disagreement between the experiments and the theoretical results for the measured cohesion force. As pointed out previously, the force is probably overestimated but the screening is not sufficient to account for the two orders of magnitude of difference between the theory and the experiment for the smallest raft. Indeed, the screening derived in [START_REF] Nikolina | Fragmentation and erosion of two-dimensional aggregates in shear flow[END_REF] would only lower our data by a factor 2 or 3, which would be far from sufficient to conciliate the experiment with the theory. Unfortunately, our experimental setup does not allow us neither to go further in the understanding of what seems to be an unexpectedly high cohesion, nor to undoubtedly exclude the possibility of a screening higher than expected. Moreover, friction due to the solid-solid contact between the spheres, as well as jamming phenomena preventing particles from moving can be observed and should be included to completely describe the cohesion of a raft.

A new experiment is needed to explore the very core of this interaction, if possible without having to make so many assumptions.

Cohesion between two aggregated particles

To circumvent the issues described in the previous section, we now focus on the characterization of the cohesion between two spheres at an oil-water interface: in this case, the capillary interaction can be described more finely, the screening will be less important a priori, and no jamming due to the specific arrangement of the beads is possible. Moreover, for two particles, the linear approximation is completely valid, giving more credibility to the capillary force calculated.

In search of an experimental setup

The two-bead aggregate With the previous experimental device, the speed a two-bead aggregate can reach during its interaction with a cylinder is way too low to separate the particles. Our aim is therefore to be able to apply a strong enough force on one particle belonging to a pair of aggregated beads. To that end, we developed two experimental setups. Only one proved out to be reliable enough to draw conclusions and understand deeper the capillary interaction between two particles in contact at a liquid interface. Nevertheless, we will briefly present both of them, because even the one that failed participated in the conception of the second one and to our global understanding.

They both rely on the same principle: a metallic wire with a diameter of 0.3 mm is glued on a bead, and connected to a translation mechanism. The particle is then placed at an oil-water interface, at the depth of a freely floating pair of particles h 2part (which is determined experimentally), and another identical bead is added not far and aggregates with the wire-connected particle. Thanks to this procedure, the couple of particles has the same vertical position it would have had without the presence of the wire, as illustrated by figure 3 What we now want to measure is the force needed to separate the two beads, hence the following question: how can we generate such a force?

First setup

A first idea could be to apply a known external force on the pair of particles while keeping the attached bead at a fixed position, until the second bead detaches. This is what is done in figure 3.14, where the external field in created thanks to the deflection of the oil-water interface by pulling down a cylinder. Because we can solve the equation of the interface, as already described many times in this manuscript, we can determine the local curvature and thereby the force applied on the bead. This gave us some results, but because of an experimental difficulty, we do not have enough confidence in these results to draw conclusions from them (or in other words, the parameters of the experiment are not controlled well enough). Indeed, when we move down the cylinder, the interface also moves down, but because one particle is attached to a wire, the pair of beads does not follow the descent of the interface. A solution could be to move vertically the wire so that the bead stays at the interface, but it turned out it is not possible to ensure the particle is at the right depth, or even to define what this right depth is, since the contact line around the bead is no longer horizontal. A metallic wire is glued on a bead which is then placed at the interface, and attracts another particle. The force applied on the second particle is produced by the local curvature of an interface deflected by a cylinder of radius R cyl placed at a depth H cyl .

Second setup

Because of these issues, a new experimental setup is used, as presented in figure 3 In such a configuration, we can play directly on the drag experienced instead of applying an external force field to change indirectly the speed. The pair of beads is moved horizontally at a given constant speed V . If no detachment is observed, the speed is increased gradually from one experiment to the next until the two beads separate. The horizontality of the translation stage being fundamental to keep the particles at the same level along their motion, a highly sensitive spirit level with a precision of 0.1 mrad is used.

Several types of particles are used in this experiment: plastic beads (ρ part = 1, 420 kg.m -3 , R part ∈ [2.5, 2, 1.5] mm), ceramic beads (ρ part = 3, 800 kg.m -3 , R part ∈ [0.45, 0.35, 0.25] mm), high density ceramic beads (ρ part = 6, 000 kg.m -3 , R part ∈ [0.5, 0.25, 0.2, 0.15, 0.1] mm), and finally glass beads (ρ part = 2, 500 kg.m -3 , R part = 0.17 mm). The three photos of figure 3.15 show the bonding between the beads and the wires.

Experimental result for the cohesion between two beads

Visualization of an experiment

A typical experiment is shown in figure 3.16, in which two plastic beads are moved horizontally at the critical detachment speed. The freely floating particle slowly detaches from the wire-connected one, a behavior which was not observed for a lower speed. No detachment was observed for the biggest plastic bead (ρ part = 1, 420 kg.m -3 , R part = 2.5 mm), for which the experimental device could not reach high enough velocities. In order to reach the desired speed V , a transient acceleration regime is needed. No clear influence of this acceleration ramp was noted: for any given experiment, the detachment velocity is the same whatever the acceleration needed to reach this velocity is, in the range [0 15] mm/s 2 . This observation is quite obvious, since we have already stated that the inertial forces could be neglected for the motion of a raft along a curved interface, even though the acceleration in that case could go up to 50 mm/s 2 .

An unexpectedly high cohesion force

Similarly to the erosion experiment of the previous section, the drag force experienced by the detaching particle is estimated by a Stokes drag corrected by a coefficient k (here again, we take k = 0.5):

F drag = 6πµ o kR part V crit (3.6)
with V crit the lowest speed for which detachment is observed. At detachment, since only the drag and the attractive cohesive forces are present, this estimate directly gives us the cohesion force between the two beads. These experimental results are displayed in figure 3.17 in colored stars. The error bars of our data are determined by reproducing several times the experiment first in the same conditions, then with slightly different depths of the beads (the depth was changed by no more than 10 %), in order to determine lower and upper boundaries of this critical speed. These experimental points need to be compared to the theoretical prediction of the capillary force between two particles. In the limit of small deformations, small spherical particles, and isotropy of the meniscus around a sphere, the capillary force between two identical spheres in contact at an interface can be expressed as follows [START_REF] Vella | The "cheerios effect[END_REF][START_REF] Dani | Hydrodynamics of particles at an oil-water interface[END_REF] (see chapter 1 for the complete calculation):

F cap = 2πγR part B 5/2 Σ 2 K 1 2R part c (3.7) with B = (R part / c ) 2 the Bond number, Σ = 2D-1 3
-1 2 cos θ + 1 6 cos 3 θ, θ the oil-waterparticle contact angle, and D = ρpart-ρo ρw-ρo . This theoretical cohesion force is represented in dotted lines in figure 3.17. For the biggest particles, this linear theory perfectly accounts for our results, and the experimental cohesion force strictly corresponds to the capillary attraction between the two beads. However, as the radius of the beads decreases, the disagreement greatly increases, up to two orders of magnitude for the smallest beads used.

This result is consistent with the previous experiment with a granular raft, where the same small particles were used (red points of figure 3.12), and a similar disagreement was found between the classic linear capillary theory and the experimental cohesion force measured. For this previous experiment, a lot of parameters could not be controlled, and the theoretical framework used relied on a number of assumptions that were not verified, leading to a difficult interpretation of the results.

However, in the situation presented here, the assumption of small deformations needed to derive equation (3.7) is valid. Furthermore, contrarily to the erosion experiment, no jamming between particles is possible and the screening of the drag created by the other particle cannot account for two orders of magnitude.

The last source of error in the interpretation of the experiment could come from the presence of the metallic wire, which could have an influence on the flow. A priori, the larger the wire is with respect to the particle, the more it will affect the drag experienced. In order to probe its influence, we used two wires of different diameters for one of the experiment (ρ part = 3, 800 kg.m -3 , R part = 0.45 mm): the usual wire with a diameter of 0.3 mm, and a larger one with a diameter of 1.1 mm, whose size is comparable to the particle. There is only a factor 2 of difference between the two cohesion forces deduced from these two experiments (black star and black diamond at R part = 0.45 mm). This tends to prove that even though the wire does affect the detachment, it cannot explain the two orders of magnitude found for the smallest bead.

This result indicates that a more fundamental explanation has to be found. The answer could lie within the assumptions needed for the calculation of equation (3.7), and especially under the hypothesis of isotropy which is not clear for two particles in contact.

Calculation of the capillary forces

In figure 3.17, we compared the experimental cohesive force with a theoretical prediction derived from a rather simple model, where the deformation around a particle is supposed to be isotropic, and where all equations are linearized. Yet, here we consider two particles in contact. As a consequence, it is rather clear that the contact line around each particle is going to be tilted: its depth will be higher in between the two particles than on the other side. Therefore, we need to perform a calculation for a tilted contact line.

Forces in presence

In 2017, Cooray et al. [START_REF] Cooray | Floating and sinking of a pair of spheres at a liquid-fluid interface[END_REF] precisely calculated the capillary forces for a pair of particles without assuming axisymmetry around a particle, by taking into account the tilting of the contact line around each sphere. By doing so, two components of the force arise:

• a hydrostatic force --→ F hyd , due to the difference of depth of the two-phases around the sphere, when integrating the pressure;

• a surface tension force -→ F st , resulting from the integration of surface tension along the non-planar contact line.

Both forces have vertical and horizontal components, but here we will only focus on the horizontal ones. The calculation is detailed in [START_REF] Liyanaralage | The capillary interaction between objects at liquid interfaces[END_REF], but since we find a slightly different result for homogeneity reasons, we are going to do again the whole calculation. All the important quantities are defined in figure 3.18. The subscript s stands for the situation of a single isolated particle, while the subscript p describes the situation of a pair of particles. α stands for the azimuth angle in a referential centered in the middle of a particle, ϕ p describes the position of the contact line around a particle (for a pair of particles), and as a consequence is a function of α. z p is the vertical position of the center of a particle (z p < 0 if the center of the particle is below the oil-water level at infinity, z p > 0 otherwise), and ξ p the vertical depth of the contact line (ξ p and ξ s are always negative, ξ p is a function of α).

First of all, let's try to express the two forces --→ F hyd .

-→ e x and -→ F st .

-→ e x as a function of ϕ p , ξ p and z p , -→ e x being the unitary horizontal vector, directed along the direction linking the centers of the two particles. We consider a point -→ P on the contact line of a sphere of radius R part , whose position with respect to the center of the sphere can be written as follows:

- → P = R part (sin ϕ p cos α - → e x + sin ϕ p sin α - → e y -cos ϕ p - → e z ) (3.8)
We can also define a local coordinate system along the contact line (

- → t , - → n , - → b )
, with -→ t the tangential vector along the contact line, -→ n the vector normal to the particle along the contact line, and

- → b = - → t ∧ - → n .

Surface tension force

The contact angle being define in the plane ( -→ n , -→ b ), the surface tension vector can be written as

- → γ = γ(cos θ - → b + sin θ - → n ),
with γ the oil-water surface tension, and θ the contact angle.

-→ F st .

-→ e x is then simply the projection of the surface tension -→ γ along the horizontal axis -→ e x integrated around the particle:

-→ F st . - → e x = γ 2π 0 (cos θ - → b . - → e x + sin θ - → n . - → e x )dl (3.9)
dl needs a careful treatment, since it represents the infinitesimal length along the contact line, and therefore depends on ϕ p (α). It can be calculated as follows:

dl = ∂ - → P ∂α dα = R part dϕ p dα cos ϕ p cos α -sin ϕ p sin α 2 + dϕ p dα cos ϕ p sin α + sin ϕ p cos α 2 + dϕ p dα sin ϕ p 2 1/2 dα = R part dϕ p dα 2 + sin 2 ϕ p λ dα = R part λdα (3.10a) (3.10b) (3.10c)
The last step is to express the vectors -→ b and -→ n to compute their horizontal component.

To that end, we use the following definitions:

- → n = ∂ - → P ∂R - → t = ∂ - → P ∂α - → b = - → t ∧ - → n (3.11)
The calculation leads to: This expression is exactly the same as the one obtained in [START_REF] Liyanaralage | The capillary interaction between objects at liquid interfaces[END_REF].

- → n =   sin ϕ p cos α sin ϕ p sin α -cos ϕ p   , - → t = 1 λ   

Hydrostatic force

For --→ F hyd .

-→ e x , our result differs from the expression of [START_REF] Liyanaralage | The capillary interaction between objects at liquid interfaces[END_REF].

The hydrostatic component can be obtained by integrating the hydrostatic pressure over the surface of the particle, and projecting it along the x-axis. In a spherical system of coordinates (r,α,φ), with φ the inclination angle and α the angle in the (x,y) plane, the elementary area is given by dA = R 2 part sin φdφdα. Moreover, the depth of a point at the surface of the bead is given by z p + R part cos φ, (z p and R part cos φ can be either positive or negative). This leads to:

--→ F hyd . - → e x = ρg(-z p -R part cos φ)( -→ -n. - → e x )dA (3.14)
The outgoing normal vector of the surface si given by -→ n = cos α sin φ -→ e x + sin α sin φ -→ e ycos φ -→ e z . Therefore, equation (3.14) can be rewritten:

--→ F hyd . - → e x = -R 2 part ρg(-z p -R part cos φ) cos α sin 2 φdφdα = R 2 part ρ o g 2π 0 π-ϕp 0 (z p + R part cos φ) cos α sin 2 φdφdα+ R 2 part ρ w g 2π 0 π π-ϕp (z p + R part cos φ) cos α sin 2 φdφdα (3.15a) (3.15b)
with ρ o (resp. ρ w ) the density of the oil phase (resp. water). Let's calculate the integer over φ:

π-ϕp 0 (z p + R part cos φ) sin 2 φdφ = π-ϕp 0 z p 1 2 (1 -cos 2φ)dφ+ π-ϕp 0 R part cos φ sin 2 φdφ = 1 2 z p (π -ϕ p + sin ϕ p cos ϕ p ) + R part 3 sin 3 ϕ p (3.16a) (3.16b) Similarly, π π-ϕp (z p + R part cos φ) sin 2 φdφ = 1 2 z p (ϕ p -sin ϕ p cos ϕ p ) - R part 3 sin 3 ϕ p (3.17)
Coming back to equation (3.15), we get:

--→ F hyd . - → e x = R 2 part g 2π 0 ρ o 1 2 z p (π -ϕ p + sin ϕ p cos ϕ p ) + R part 3 sin 3 ϕ p + ρ w 1 2 z p (ϕ p -sin ϕ p cos ϕ p ) - R part 3 sin 3 ϕ p cos αdα = R 2 part g(ρ w -ρ o ) 6 2π 0 (-3z p (sin ϕ p cos ϕ p -ϕ p ) - 2R part sin 3 ϕ p cos αdα (3.18a) (3.18b)
To summarize, we have now calculated the expression of the two components of the horizontal force: with λ given by equation (3.10). The total horizontal force is then simply obtained by:

                         --→ F hyd . - → e x = R 2 part g(ρ w -ρ o ) 6 2π 0 (-3z p (sin ϕ p cos ϕ p -ϕ p )- 2R part sin 3 ϕ p cos αdα -→ F st . - → e x =γR part
F cohez = --→ F hyd . - → e x + -→ F st . - → e x (3.20)
Thanks to equation (3.19), we now have a new expression for the attractive force F cohez that does not assume axisymmetry around the particle. We would now like to perform the calculation for our beads. Yet, some work is still needed to numerically calculate the two forces of the system (3.19): indeed, ϕ p , z p (and λ, since it is a function of ϕ p ) are not known a priori. But with a few hypotheses, they can be expressed as functions of ϕ s , θ and R (the quantities describing the position of a single isolated particle).

Vertical position of the contact line

Vertical position of the contact line around a pair of beads

For the moment, let us assume we know ϕ s (the position of the contact line for an isolated sphere with respect to the center of the particle). The first step will be to calculate the vertical position of this very same contact line ξ s , still for an isolated particle (with respect to the oil-water level at infinity). In the limit of small deformations, the equation we need to solve is the following (an equation we have already encountered a lot in this manuscript):

         ∇ 2 h = h 2 c h(r = R part sin ϕ s ) = ξ s h(+∞) = 0 (3.21a) (3.21b) (3.21c)
Applying the result in r = R part sin ϕ s (where we know the slope h (r = R part sin ϕ s ) = tan(ψ s ) = tan(ϕ s + θ)), we end up with equation (3.22):

ξ s = -tan(ϕ s + θ) c K 0 (R part sin ϕ s / c ) K 1 (R part sin ϕ s / c ) (3.22)
From equation (3.22), we can directly deduce the expression of z s from geometrical considerations:

z s = ξ s + R part cos ϕ s (3.23)
The vertical position of the pair of particles is then obtained by simply assuming we can superimpose the deformation of the two individual particles, and by using the solution of the system (3.21):

z p = z s -tan(ϕ s + θ) c K 0 (2R part / c ) K 1 (R part sin ϕ s / c ) (3.24)
Similarly to the center of the particles, we assume that the contact line moves vertically because of the other sphere, through the following displacement:

ξ p = ξ s -tan(ϕ s + θ) c K 0 (r(α)/ c ) K 1 (R part sin ϕ s / c ) (3.25)
with r(α) the horizontal distance between a point along the contact line and the center of the other sphere. r(α) can be calculated using Pythagore's theorem:

r(α) = R part (2 -sin ϕ s cos α) 2 + (sin ϕ s sin α) 2 = R part 4 + sin 2 ϕ s -4 sin ϕ s cos α (3.26a) (3.26b)
We can now calculate the angle ϕ p describing the position of the contact line with respect to the center of a particle, because it follows the same relation as in equation (3.23):

z p = ξ p + R part cos ϕ p . ϕ p = arccos z p -ξ p R part = arccos cos ϕ s + tan(ϕ s + θ) c R part K 1 (R part sin ϕ s / c ) × K 0 r(α) c -K 0 2R part c (3.27a) (3.27b)
with r(α) defined in equation (3.26b). To summarize, we managed to find the expression of both ϕ p and z p as a function of ϕ s , θ and R part , which can be measured experimentally. Therefore, we are now able to calculate the cohesion force F cohez .

Vertical position of the contact line of a single isolated sphere Nevertheless, we can go even further with the theory, by looking for an expression of ϕ s as a function of the density ratio, the Bond number, the contact angle θ, and the radius of the particle R part . This angle ϕ s can be determined through the vertical balance of the forces exerted on a sphere at an interface, as done in [START_REF] Cooray | Floating and sinking of a pair of spheres at a liquid-fluid interface[END_REF][START_REF] Vella | The "cheerios effect[END_REF]. The calculation was done in chapter 1 in order to compute the expression for the horizontal force between two identical particles at a fluid-fluid interface:

F cap (r) = 2πγR part B 5/2 Σ 2 K 1 (r/ c ) (3.28)
To end up with this result, we linearized all the expressions assuming a small slope of the interface. Here, we are going to follow basically the same steps but without linearizing the equations, and without introducing the slope of the interface (we will work only using the different angles of the problem). Since we already performed the calculation, we will skip most of the different steps here.

The forces acting on a particle at an interface are:

• its weight F w = 4 3 πR 3 part ρ part g;

• the vertical component of the surface tension

F γ = 2πγR part sin ϕ s sin ψ s = -2πγR part sin ϕ s sin(ϕ s + θ) (3.29a) (3.29b)
• the generalized buoyancy, that can be calculated following [START_REF] Vella | The "cheerios effect[END_REF] as the sum of two contributions, as illustrated in figure 1.4:

the weight of water that would have occupied the volume Ω B between the undisturbed interface and the water-wetted area, exerted upwards (volume B)

Ω B = -πR 2 part sin 2 ϕ s ξ s + π 3 R 3 part (2 + cos ϕ s )(1 -cos ϕ s ) 2 = πR 3 part 2 3 -cos ϕ s + 1 3 cos 3 ϕ s - ξ s R part sin 2 ϕ s (3.30a) (3.30b)
the weight of oil corresponding to the complementary volume 4 3 πR 3 part -Ω B , exerted downwards (area A).

We end up with the following vertical force for the generalized buoyancy:

F archi = πR 3 part g 4 3 ρ o + (ρ w -ρ o ) 2 3 -cos ϕ s + 1 3 cos 3 ϕ s - ξ s R part sin 2 ϕ s (3.31)
The balance between these three forces leads to (same equation as in [START_REF] Cooray | Floating and sinking of a pair of spheres at a liquid-fluid interface[END_REF]):

4 3 ξ o = 2 3 -cos ϕ s + 1 3 cos 3 ϕ s - ξ s B 1/2 l c sin 2 ϕ s -2B -1 sin ϕ s sin(ϕ s + θ) (3.32)
with l c = γ/((ρ w -ρ o )g) the capillary length, B = R 2 part /l 2 c the Bond number, and ξ o = (ρ part -ρ o )/(ρ w -ρ o ) the density ratio.

Injecting equation (3.22) (the expression of ξ s as a function of ϕ s ) in (3.32) leads to an equation that can be solved numerically in order to deduce the value of ϕ s only as a function of the liquids and beads properties.

Comparison between experiments and theory for the position of the contact line around a single sphere

There is a good agreement between the experimental angles ξ s exp , measured on side views of isolated particles as illustrated by figure 3.19, and the theoretical ones ξ s theo calculated from equation (3.32), as demonstrated by figure 3.19.e. For all our measurements, the mean difference between the experimental value of ϕ s and the theoretical one is ∆ϕ s =< ϕ s exp -ϕ s theo >≈ 3 • .

Cohesion between two beads: the explanation

Cohesive force as a function of ϕ s theo

After all these efforts, it is now possible to compute numerically the value of F cohez as a function of only (R part , θ), and to compare this theoretical value to our experimental measurements of F cohesion . The corresponding results are plotted in figure 3.20 in black squares, along with the previous experimental results in colored stars, and the first theoretical model of equation (3.7) in plain line. The rescaling of the x-axis comes from the asymptotic analysis of equation (3.7) for small radii (see equation (3.33)), which indicates that B 5/2 Σ 2 is the good rescaling parameter. Because there is a strong hysteresis of the contact angle (up to 40 • between its minimal and maximal value), we measure θ with different beads for each radius, and calculate as a consequence several cohesion forces for each radius, depending on θ. Yet, the values obtained are almost not affected by this hysteresis (this is why for a given abscissa, all the squares appear to be superimposed). The force obtained differs only slightly from the simple calculation of equation (3.7) (not more than 10 percent of difference which is consistent with the results reported in [START_REF] Cooray | Floating and sinking of a pair of spheres at a liquid-fluid interface[END_REF]). As a consequence, this approach does not explain our experimental result. Have we really done all these calculations for nothing?

F cap ∼ Rpart→0 γ c B 5/2 Σ 2 (3.33) 10 -7 10 -5 10 -3 10
Cohesive force as a function of ϕ s exp A last hope remains. As pointed out before, ϕ s exp differs from its theoretical value by ∆ϕ s ≈ 3 • . We can therefore shift all depth angles by a constant value ∆ϕ s and calculate F cohez (R part , θ mean , ϕ s theo + ∆ϕ s ). The corresponding result is plotted in dotted lines in figure 3.20 (we also use one single value of θ mean for each material, since we have already established that the hysteresis of the contact angle has no effect on the calculation of F cohez ).

Surprisingly, considering the small value of ∆ϕ s , both the general trend and the order of magnitude are now consistent with the experiments: for small radii, we recover an enhanced capillary force, whereas when the radius of the particle increases, both calculations lead to the same result. This suggests that the small variations of ϕ s have a strong effect on F cohez for small particles, an interpretation that needs some backup.

Up to now, we have considered F cohez for fixed values of ϕ s . As represented in figure 3.21, we can also investigate how F cohez varies with the angle ψ s = ϕ s + θ -π, with ψ s defined in figure 3.18.a (simple geometrical considerations lead to this relation), for a given radius. One needs to be careful in the analysis of this representation of F cohez . Indeed, ψ s , ϕ s and θ are angles that depends on the particle properties, but they are fixed for a given particle (with the exception of possible hysteresis). As a consequence, there is no real physical meaning in considering a random value of ψ s for a fixed radius R part . But by doing so, we can better understand the sensitivity of F cohez to its different input variables. The three curves in figure 3.21 correspond to three different radii of high density ceramic beads (ρ part = 6, 000 kg.m -3 ), and the dotted lines indicate the theoretical angles ψ s theo = ϕ s theo + θ -π for each radius. As illustrated, F cohez ∝ ψ 2 s for small angles. As a consequence, a variation of ∆ϕ s in x-axis for the bigger radius in pink would only result in a factor two or three for the force, while the very same variation on the x-axis for the smallest particle in green would result in more than one order of magnitude in ordinate. Thus, as ψ s gets smaller, the tiniest variation of its value has an increasing effect on F cohez . This is in perfect agreement with the result of figure 3.20: for the largest beads, the deflection of the interface ψ s is large, and so even a variation of ∆ϕ s ≈ 3 • has almost no effect: the dotted line, the black squares, and the plain line are similar. But as we travel to the left of figure 3.20, the particles become smaller, and so does ψ s . There, the increase of ϕ s by approximately 3 • can lead to an increase in the cohesive force F cohez of two orders of magnitude. Thanks to this effect, the dotted line in figure 3.20 follows the experimental points, while both the solid line and the squares underestimate greatly the capillary force for small radii.

Difference between ϕ s exp and ϕ s theo

The last remaining question concerns the differences between ϕ s exp and ϕ s theo . The approximations made in the calculation of ϕ s theo are perfectly valid for the smallest particles considered. Yet, it is well known that for small particles, capillary forces between objects are produced via geometrical details such as the object's shape or surface roughness [START_REF] Krassimir D Danov | Interactions between particles with an undulated contact line at a fluid interface: Capillary multipoles of arbitrary order[END_REF][START_REF] Botto | Capillary interactions between anisotropic particles[END_REF]. Such undulations of the contact line may also be present for larger particles, leading to an effective depth angle ϕ s exp different from the theoretical one by only a few degrees. However, as demonstrated earlier, these few degrees can lead to very strong variations of the capillary force between two spheres in contact. The tilting of the contact line as the two particles come into contact also plays an essential role in the observed large increase of the capillary forces. It is the combination of these two effects that leads to an enhanced cohesion force for small particles.

Using this result for the cohesive force between two beads, we can now go back to the erosion of a whole granular raft, where the cohesive force was found to be far higher than expected by the classical linear theory. The capillary force acting on a particle at the edge of a raft has probably a double origin: on the one hand, the global slope of the interface, which is precisely what we had already taken into account; on the other hand, the specific geometry of the contact line and of the interface around the particle, which now appears to be determinant for the cohesion by playing a dominant role.

Conclusion

In this chapter, we have investigated the cohesion inside an assembly of large dense particles at an interface, and between two aggregated beads at a liquid interface. The erosion a granular raft experiments during its motion occurs for unexpectedly high speeds. This enhanced cohesion appears to result from the precise description of the meniscus around each particle, as demonstrated by the two bead-experiment. For small particles, the attractive force between two beads in contact exceeds by two orders of magnitude the force expected by the classic theory, whereas no significative effect is observed for large particles. The tilting of the contact line as well as small variations in the pinning of the contact line seem to explain this enhanced force.

The small differences between the theoretical depth angle of a freely floating particle and the experimental one would require some further investigations. In particular, the role of roughness, non-spherical shapes, and surface inhomogeneity on the effective depth angle deserves to be investigated further. The present study could also be extended to the capillary forces between smaller particles, or to the erosion of particle-covered droplets under flow. A theoretical description of the geometry of the contact line when more than two beads are in contact would also help to understand this phenomenon, as well as when a couple of particles evolves along a curved interface.

As briefly presented in the introduction, when too many particles are added, the granular raft may destabilize and sink. The threshold of destabilization has already been studied in a previous work [START_REF] Protière | Sinking a granular raft[END_REF], but the near future of a destabilized raft remains to be explored. It is therefore natural to wonder about the dynamics of this sinking.

Pinch-off: from granular rafts to inkjet printing

Up to now, we have studied the birth of a granular raft, from the attraction between two single particles to the interaction between a vast number of aggregates of random sizes. This process leads to the formation of a granular raft that deflects the interface. This raft can then live its life by moving freely at the surface of the liquid and collecting more particles, and may experience erosion during its motion. But as stated before, the interface is not able to deform indefinitely. At some point, the deformation will be too large, and the raft will sink. We have studied its birth and its life; it may now be the time to address the issue of its "death".

Our experiment

Experimental procedure

The best way to start is simply to look at an actual experiment where a granular raft is sinking. The experimental procedure is the same as before. We fill a glass tank (0.2 x 0.2 x 0.25 m) with deionized water, and pour a layer of silicone oil on top of it, as explained in the introduction. In this chapter, we do not study the impact of the oil viscosity on the dynamics of sinking, and therefore we will only use silicone oil with a kinematic viscosity ν o = 1.10 -4 mm.s -2 and a density ρ o = 966 kg.m -3 . Particles are added progressively in order to form an axisymmetric monolayer. The deformation of the interface increases with the size of the raft, until a maximum size as visualized in figure 4.1, from which the raft becomes unstable and sinks. A high-speed camera (Phantom v2511) records the dynamics with a recording speed up to 25,000 frames per second. To resolve perfectly the contour of the oil filament, the lightning is very important. We insert a mask between the backlight and the sample, in order to increase the contrast with the ambient fluid [START_REF] Cohen | Testing for scaling behavior dependence on geometrical and fluid parameters in the two fluid drop snap-off problem[END_REF]. A typical sinking event is visualized in figure 4.2. In the situation presented here, the particles are encapsulating a droplet of oil connected to the upper layer by an oil thread, which elongates and thins as the sinking proceeds.

Sinking of a granular raft

The oil filament is perfectly axisymmetric. At the end of the sinking, the thinning of the oil filament begins to be localized near the top layer. This is where a first pinch-off occurs. Later, a second pinch-off will happen near the encapsulated droplet. After that, as described in the introduction, the armored droplet will reach the bottom of the water tank, where it will remain stable.

We have described qualitatively the sinking of a granular raft. But we still have no clue on the procedure we may follow to analyze such dynamics, or on the relevant variables we may want to measure in order to characterize the pinching dynamics. Luckily, we are not alone in this. Pinching dynamics have gathered considerable attention in the past decades, both because of countless applications and challenging fundamental questions.

Pinch-off in daily life

Countless applications

Cell division, ink-jet printing, dripping faucets, raindrop fragmentation, air bubbles in a fish tank, sparkling water: this long list of daily situations involving pinch-off dynamics could go on forever. Because most of these phenomena are commonly experienced, one can easily forget all the physical stunning wonders lying behind them. Who would think that behind a vulgar water stream dripping from a faucet [START_REF] Bala Ambravaneswaran | Dripping-jetting transitions in a dripping faucet[END_REF][START_REF] Clanet | Transition from dripping to jetting[END_REF], such as the one visualized in figure 4.3.a, many unsolved questions are still waiting for an answer? Because of this incredible richness, droplet formation has gathered for a few decades the interest of many researchers, both from an academic perspective and an industrial one. A lot of industrial applications involve liquid pinching, where practical issues are ad-dressed [START_REF] Osman A Basaran | Small-scale free surface flows with breakup: Drop formation and emerging applications[END_REF]: how long will the breakup process last? How many droplets will be formed? What will be their sizes? In 1833, Savart was already able to observe the destabilization of a liquid jet leading to the formation of droplets of different sizes that oscillate. This destabilization depends on many parameters, such as the frequency of excitation, as represented in figure 4.3.a.

Irrigation processes, as illustrated in figure 4.3.b, benefit from such a precise description of the droplet size distribution generated by a specific irrigation sprinkler [START_REF] Kincaid | Drop size distributions for irrigation sprinklers[END_REF]. The area covered by the sprinkler is also a key aspect. The small drops may be carried away by the wind and disperse pesticides, while the bigger ones may impact the plants too fast and damage them. As a consequence, a sprinkler head has to be chosen according to climate constraints or topography in order to maximize the irrigation efficiency.

The development of inkjet printing devices, nowadays broadly commercialized, is also the result of the progress made in the understanding of droplet production. This technology has continued to improve and is by no means only limited to inkjet printing on paper: transistor circuits [START_REF] Sirringhaus | High-resolution inkjet printing of all-polymer transistor circuits[END_REF] or even mammalian cells [START_REF] Xu | Inkjet printing of viable mammalian cells[END_REF] can be produced with similar techniques, as illustrated by figure 4.4.

In that example, cortical neurons were deposited in a ring pattern by thermal inkjet printing [START_REF] Boland | Application of inkjet printing to tissue engineering[END_REF]. After a few days, the cells started to differentiate and to grow processes, proving their viability. The cells remain able to fire action potentials after printing. Cells and proteins can even be deposited on 3-D hydrogel structures, opening the path towards the manufacture of spatially controlled living structure in well-defined patterns.

Fundamental physical questions

But besides theses numerous practical repercussions of liquid pinch-off, various fundamental questions are raised, whether it is theoretically, numerically, or experimentally, well summarized in various reviews [5,[START_REF] Eggers | Nonlinear dynamics and breakup of free-surface flows[END_REF] that take stock of the current state of the art. A liquid pinch-off can be mathematically described as a finite-time singularity: the radius of the liquid thread formed between the detaching droplet and the rest of the fluid reaches zero in a finite amount of time. Being highly non-linear, such dynamics are for now only described by asymptotic solutions. Numerically speaking, the very nature of a singularity is enough to understand the intrinsic challenge created: because several orders of magnitude separate the size of the drop from the radius of the liquid neck generated, simulations has to be refined a lot both in space and in time. As a consequence, liquid breakup, and more generally interfacial flows [START_REF] Scardovelli | Direct numerical simulation of free-surface and interfacial flow[END_REF], require to carefully choose the physical modeling as well as the numerical methods.

The same argument can be made to illustrate the difficulty of experimental visualizations of liquid breakup, a process that can be at the same time very fast and very spatially localized. The development of high-speed imaging in the past few decades has given a new breath to the study of liquid pinch-off, enabling people to conduct more and more precise experiments.

Despite all the practical limitations described above, numerical and experimental results are decisive because no exact solution is known. Moreover, as stated before, and from a broader perspective, liquid pinch-off belongs to the large family of finite-time singularities, present in numerous physical systems. Understanding not only the asymptotic regimes but also the transitions between them could shed light on other systems, such as the coalescence of two droplets for instance, or even at a larger scale the structure of galaxy clusters.

We are first going to explore the main theoretical results on the formation of a droplet, and compare them to experimental measurements of the detachment of an oil droplet from a nozzle, first of all in the linear approximation, and then by describing the non-linear regimes.

Some results of this chapter can be found in [START_REF] Lagarde | Oscillating path between self-similarities in liquid pinch-off[END_REF].

Formation of a viscous droplet

Before going back to the sinking of a granular raft, we are going to explore the literature on droplet pinch-off. We will later use these results to analyze the thinning of the oil thread formed during the sinking of the raft.

A first historical step: the linear stability analysis

Dispersion relation for an infinite viscous cylinder

The very first study on liquid jets seems to date back to Leonardo da Vinci in the beginning of the sixteenth century [START_REF] Da | 1508 the notebooks of leonardo da vinci ed and transl[END_REF], but similarly to his successors, he missed the decisive role of surface tension, described only in the early nineteenth century by Young [START_REF] Young | Iii. an essay on the cohesion of fluids[END_REF] and Laplace [148]. They both understood its intrinsic dual nature: surface tension can act not only has a cohesive force but also as a driving destabilizing force. Basically, the minimization of the system surface area can lead either to the spherical shape of a droplet or to the destabilization of a cylinder of fluid. This instability, described experimentally by Plateau [2], inspired Rayleigh to develop a linear stability analysis of this problem, and allowed him to understand precisely the first moments of destabilization of a liquid cylinder [149,[START_REF] Rayleigh | on the instability of a cylinder of viscous liquid under capillary force[END_REF], giving rise to what we call today the Plateau-Rayleigh instability. We will present here the basic ingredients. As illustrated in figure 4.5, Rayleigh considered an infinite cylinder of fluid, with small perturbations of its surface that can be expressed as the sum of elementary perturbations of the form:

R(x, t) = R 0 (1 + exp(ikx + ωt)) (4.1)
with k = λ 2π the wavenumber, R 0 the initial radius of the cylinder, the amplitude of the perturbation, λ its wavelength, and ω the growth rate.

In the axisymmetric configuration of figure 4.5, and for a small perturbation, the pressure jump at the interface can be expressed by:

∆P = γ 1 R - d 2 R dx 2 (4.2)
with γ the surface tension of the liquid. Depending on the wavelength of the perturbation, it will either grow until pinch-off of the cylinder, or disappear. For a viscous liquid (which will constitute our main system of interest in this chapter), if we can neglect both inertia and the surrounding fluid, we get the following dispersion relation [START_REF] Tomotika | On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid[END_REF]:

ω = γ 2R 0 µ k 2 R 2 0 -1 k 2 R 2 0 + 1 -k 2 R 2 0 I 2 0 (kR 0 )/I 2 1 (kR 0 ) (4.3)
where I 0 and I 1 are the modified Bessel functions of order 0 and 1, and µ the viscosity of the fluid. In the current set of approximations, the most unstable mode that maximizes the growth rate ω is reached for k = 0, leading to the following relation:

ω = γ 6µR 0 (4.4)
This linear stability analysis has been generalized in 1935 by Tomotika by taking into account the viscosity of the surrounding fluid [START_REF] Tomotika | On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid[END_REF], and by Chandrasekhar in 1970 by including inertia in the previous analysis [START_REF] Chandrasekhar | Hydrodynamic and hydromagnetic stability[END_REF]. In these situations, the wavelength of the most unstable mode will be a function of the viscosity ratio.

Formation of a droplet: the experiment

To check the relevance of equation (4.4), we need to be able to measure the evolution of the radius of a similar viscous filament. Nevertheless, we are interested here in the detachment of a droplet from a needle, a geometry completely different from the infinite liquid jet described above. Will the previous reasoning still hold in our situation? The previous analysis is linear, and as a consequence may not describe the dynamics all the way until breakup of the liquid filament. For all these reasons, we need a visualization of an actual experiment, as presented in figure 4.6 with two images of the same oil droplet at two different moments of its formation.

We place at the bottom of our water tank a needle with a diameter 2R 0 in the range 1 mm < 2R 0 < 5 mm. Using a syringe pump (Harvard Apparatus PHD 2000), we extrude silicone oil at a very low flow rate Q = 0.1 ml/min, in order to form drops quasi-statically. Just like for a granular raft, we record the detachment with a high-speed camera.

Exponential decay of the minimum radius

In a typical experiment, just as the one displayed in figure 4.7.a, we record the detachment of the droplet, and then measure the evolution of r min , z(r min ) (see figure 4.7.b and 4.7.c), but also of the whole profile of the viscous neck. The two curves should be read from right to left, with a monotonous decrease of r min as τ gets closer to zero. τ being the remaining time before breakup, τ = 0 corresponds to the instant of pinch-off, when r min a) b) Following [START_REF] Clanet | Transition from dripping to jetting[END_REF], we fit the first regime of detachment with an exponential function of the form:

r min = R 0 (1 -e ω(-τ -t 0 ) ) (4.5)
with R 0 the radius the needle, ω the growth rate of the instability, and t 0 a time shift accounting for the fact that the linear stability analysis leading to an exponential regime is not valid all the way until breakup.

As we can see in figure 4.7.b, the exponential decay perfectly fits the right side of the curve. Moreover, when varying R 0 , we recover the dependence of ω with respect to R 0 , as predicted by equation (4.4), indicating that the detachment of a droplet from a needle does indeed start with a Plateau-Rayleigh instability. A more thorough study would be needed in order to check the dependency of ω with respect to the viscosity, but this is not our aim here.

It is worth noticing that for τ < 10 ms, the linear approach fails as expected, and the minimum radius is no longer described by an exponential decay. This is not surprising, since deformations can no longer be considered as small. Moreover, the linear analysis cannot explain several features observed in experiments, such as the presence and the number of satellite droplets. This short time non-linear dynamics will be at the center of the next section.

Non linear dynamics near pinch-off

Definition of the equations

Near pinch-off, several orders of magnitude can separate the radius of the liquid thread from the dimensions of the boundary conditions (for instance the diameter of the nozzle in our situation). The dynamics become self-similar [START_REF] Howell Peregrine | The bifurcation of liquid bridges[END_REF], and depending on the balance of forces, various scalings will become valid and succeed one another, giving rise to a vast variety of shapes, as illustrated in figure 4.8. These shapes are self-similar, and do not depend on boundary and initial conditions.

The dynamics of a droplet detaching from a needle are a priori controlled by several dimensionless numbers. Nevertheless, because the drops are formed quasi-statically at a very low flow rate, the Weber number W e = ρ o U 2 R/γ verifies W e 1, with U the velocity of the fluid at the tip of the needle. As a consequence, the flow rate does not impact the dynamics. Similarly, since gravity does not play a role in these self-similar dynamics, it will be neglected. Therefore, the dynamics will only be described through the Ohnesorge number based on the oil properties:

Oh = µ o √ ρ o γR 0 (4.6)
with R 0 the initial radius of the nozzle, µ o the oil dynamic viscosity, ρ o its density, and γ the oil-water interfacial tension. In this section, we limited ourselves to Ohnesorge numbers between 0.3 and 0.5, corresponding to R 0 varying between 0.8 and 2.5 mm. This number compares the effect of inertia and viscosity with surface tension. [START_REF] Cohen | Testing for scaling behavior dependence on geometrical and fluid parameters in the two fluid drop snap-off problem[END_REF], with pictures coming from [START_REF] Cohen | Testing for scaling behavior dependence on geometrical and fluid parameters in the two fluid drop snap-off problem[END_REF], [START_REF] Shi | A cascade of structure in a drop falling from a faucet[END_REF] and [START_REF] Cohen | Two fluid drop snap-off problem: Experiments and theory[END_REF].

As explained before, because several orders of magnitude separate the diameter of the needle from the radius of the liquid neck, R 0 is not the relevant length scale. In fact, as we deal with a viscous liquid, the relevant viscous length scale and time scale will be exclusively based on the oil properties, and defined by:

l µ = µ 2 o γρ o t µ = µ 3 o γ 2 ρ o (4.7)
The computation of the neck radius r(z, t) is based on the Navier-Stokes equations, written in an axisymmetric configuration for an infinite cylinder of fluid, under the assumption of lubrification. The flow being mainly directed along the z-axis, we can expand in Taylor series in r the pressure and velocity fields, giving rise to the following set of equations [START_REF] Eggers | Drop formation in a one-dimensional approximation of the navier-stokes equation[END_REF] (without any assumptions on the respective predominance of one force over the others):

     ρ o (∂ t v + vv ) = -γ 1 R 1 + 1 R 2 + 3µ o (r 2 v ) r 2 -ρ o g ∂ t r + vr = -v r/2 (4.8a) (4.8b)
with R 1 and R 2 the two radii of curvature, and g the acceleration of gravity.

Then, assuming self-similar profiles in the shrinking region, r and v can be described with two self similar functions φ and ψ and a single variable ξ = z /τ β [START_REF] Eggers | Universal pinching of 3d axisymmetric free-surface flow[END_REF]:

r(z, t) = l µ τ α 1 φ(ξ) v(z, t) = (l µ /t µ )τ α 2 ψ(ξ) (4.9a) (4.9b)
with τ the dimensionless time before breakup and z the dimensionless vertical position. Inserting 4.9a and 4.9b in 4.8a and 4.8b leads to a system of differential equations on φ and ψ and to the determination of the self-similar exponents α 1 , α 2 and β.

Self-similar regimes for pinch-off

For viscous liquids, when the linear analysis of the Plateau-Rayleigh instability is no longer valid, thinning becomes governed by a competition between viscosity and surface tension [START_REF] Demetrios | On the breakup of viscous liquid threads[END_REF], where the minimum radius r min and its vertical position z(r min ) evolve according to equation (4.10):

r min l µ = 0.0709 (τ -t 0 ) t µ z(r min ) l µ ∼ τ t µ β (4.10)
with β = 0.175. Just as for the Plateau-Rayleigh instability, t 0 accounts for the fact that this approach is not valid all the way to the pinch-off. Indeed, at some point, the velocity of the fluid increases until inertia can no longer be neglected, and a new scaling arises. The previous regime, either called viscous regime or Papageorgiou regime, is also characterized by the symmetry of the profiles around the minimum radius. For low viscosity fluids, the Papageorgiou regime is not observed. Instead, the balance between capillarity and inertia leads to a regime where r min ∼ τ 2/3 ( γ ρ ) 1/3 . Because we limit ourselves to viscous liquids, this regime will not be explored in the rest of the chapter.

As explained above, at some point the axial velocity of the fluid increases until inertia becomes of the same order as viscosity and capillarity. This leads to equation (4.11), characteristic of the inertial-viscous regime, or Eggers regime [START_REF] Eggers | Universal pinching of 3d axisymmetric free-surface flow[END_REF]:

r min l µ = 0.0304 τ t µ z(r min ) l µ ∼ τ t µ 0.5 (4.11)
In contrast to the viscous regime, we lose here the symmetry of the profile around r min . Moreover, the Eggers regime is observed only for r < l µ , which makes it experimentally inaccessible for water, for which l µ is of the order of ten nanometers. But for a silicone oil similar to the one we used, numerous experimental studies have confirmed that whatever boundary and initial conditions restricting the system, at a given time a viscous system will transit from the viscous regime to the inertial-viscous one. Various experimental proofs have confirmed these scalings [START_REF] Rothert | Formation of a drop: viscosity dependence of three flow regimes[END_REF] and described the shape of the viscous thread for both regimes [START_REF] Rothert | Transition from symmetric to asymmetric scaling function before drop pinch-off[END_REF]: symmetric for the viscous regime, asymmetric for the inertial-viscous one.

For now, we have completely neglected the dynamics of the external fluid, despite the obvious viscous stress it exerts on the liquid thread. When taking into account its influence, as soon as the viscosity of the ambient fluid is not exactly zero, another final regime replaces the Eggers regime. Numerical simulations have first shown that the viscosity ratio of the two fluids had an effect on several aspects of the dynamics, including for instance the duration of breakup [START_REF] Zhang | Drop formation in viscous flows at a vertical capillary tube[END_REF]. Lister and Stone [START_REF] John | Capillary breakup of a viscous thread surrounded by another viscous fluid[END_REF] have shown that if there is viscosity in the external fluid, inertia will at some point become negligible again, and a new balance of forces between capillarity, internal extensional stresses and external shear stress will lead to the final Stokes regime, where both r min and z(r min ) evolve linearly with τ :

r min l µ ∼ τ t µ m 1/2 z(r min ) l µ ∼ τ t µ (4.12)
with m the ratio between the viscosity of the external and the internal fluid. This scaling has been confirmed experimentally and numerically, and several studies have explored the dependence of the self-similar function describing the shape of the liquid thread with the viscosity ratio m [START_REF] Cohen | Two fluid drop snap-off problem: Experiments and theory[END_REF][START_REF] Sierou | Self-similar solutions for viscous capillary pinchoff[END_REF]. The self similar profile is asymmetric and conical [START_REF] Zhang | Similarity solutions for capillary pinch-off in fluids of differing viscosity[END_REF].

All these different scalings are summarized in figure 4.9, inspired from [START_REF] John | Capillary breakup of a viscous thread surrounded by another viscous fluid[END_REF].

Potential regime Viscous regime

Inertial-viscous regime

Stokes regime

Figure 4.9: Self-similar scalings in drop pinch-off. Summary of the different paths a system can follow between the different self-similar dynamics. m stands for the viscosity ratio between the external and internal fluid.

The figure is inspired by [START_REF] John | Capillary breakup of a viscous thread surrounded by another viscous fluid[END_REF].

Self-similar regimes in the experiment

We can now plot on figure 4.7 the theoretical slopes of the two self-similar regimes, and check if the scalings are valid. The results are displayed in figure 4.10. The thinning of the viscous thread, after the initial Plateau-Rayleigh instability, is governed by the succession of two self-similar asymptotic regimes: the viscous regime and the inertial-viscous one. Experimentally, we cannot access the final Stokes regime described above. An important feature of what we have described until now is the asymptotic nature of all these different scalings. For now, nothing has been explained on the dynamical path followed by the system to converge towards these solutions. Moreover, theoretical studies have shown that many other solutions exist, in addition to the specific self-similar solutions of equations (4.10) and (4.11). Brenner et al. [START_REF] Michael P Brenner | Pinching threads, singularities and the number 0.0304[END_REF] have shown that the two similarity solutions described by Eggers and Papageorgiou are both part of a countably infinite family of solutions, and have conjectured that these two solutions are selected just because they are less unstable to perturbations. Interested in the viscous regime, Eggers showed the solution presented in equation (4.10) appears to be the only stable one, whereas all the others are linearly unstable and thus cannot be observed [START_REF] Eggers | Stability of a viscous pinching thread[END_REF].

Inquiries on the transition between regimes have also shown that the final inertialviscous regime is observed at a radius r min far below theoretical predictions [START_REF] Chen | Computational and experimental analysis of pinch-off and scaling[END_REF]. More recently, Castrejon-Pita et al. have demonstrated both numerically and experimentally that a filament thins through several intermediate transient regimes that delay the apparition of the inertial-viscous regime [START_REF] Rafael Castrejón-Pita | Plethora of transitions during breakup of liquid filaments[END_REF]. Li and Sprittles conducted computations on the pinch-off of liquid bridges and identified an oscillatory convergence in log-scale towards this self-similar regime [START_REF] Li | Capillary breakup of a liquid bridge: identifying regimes and transitions[END_REF].

Finally, we can mention the very specific case of a very low viscosity drop (air for example) breaking inside a highly viscous ambient fluid, as reported in [START_REF] Doshi | Persistence of memory in drop breakup: The breakdown of universality[END_REF]. In such a situation, and in contrast to everything we said previously, the universal behavior of pinchoff is not recovered, and the final shape still bears the stigmas of the boundary and initial conditions.

Thanks to this section on simple oil droplets extruded by a nozzle, we know what are the thinning regimes commonly expected. Hence, we can now go back to our problem, the sinking of a granular raft, and see if a similar pinching can be found.

Granular rafts pinch-off

Minimum radius: first measurements

Influence of the size of the beads on the sinking

The experimental procedure is very similar to the one employed in figure 4.6, with a high-speed camera filming the detachment. Nonetheless, unlike the detachment of an oil droplet from a needle, here the liquid thread is far more stretched. Therefore, it becomes difficult to record in a single experiment the long-time dynamics of the whole thread and the short-time dynamics of the neck near pinch-off, for which it is compulsory to use a different camera lens. To overcome this limitation, we conduct several times the same experiment, with different magnification factors and different sampling frequencies. Because the experiments are really reproducible, we can then gather the different data and plot a single curve r min with both the long-time and the short-time dynamics, up to an accuracy of 3 µm per pixel.

Experimental visualizations are displayed in figure 4.11, for three different sizes of beads that all lead to the destabilization of the granular raft and its sinking.

As illustrated by figure 4.11, even when the raft sinks, several distinct behaviors can be observed. For light enough particles, the raft sinks by forming a viscous jet that destabilizes through a Plateau-Rayleigh instability (see figure 4.11.c). The wavelength of the instability is even visible on the last photo.

For slightly heavier particles (figure 4.11.b), an oil droplet is encapsulated inside a shell of particles, but beads are also distributed all along the viscous thread connecting the droplet to the oil layer. In such a situation, the dynamics of pinching will strongly depend on the positions of the individual beads with respect to the minimum radius. This situation will be tackled in section 4.6 of the current chapter.

From now on, the beads properties (size and density) will be carefully chosen to avoid the two previous cases. We will always consider situations for which all the particles of the granular raft are encapsulating the oil droplet, and the liquid thread formed during the sinking process is free of particles (figure 4.11.a, where R part = 0.45 mm). This situation is the closest one to the formation of a droplet from a nozzle, just as in the previous section (see figure 4.7), and enables us to investigate the impact of a gravitational acceleration on the detachment of an oil droplet in a liquid bath.

Minimum radius for a granular raft

Similarly to the formation of an oil droplet from a nozzle, we measure the evolution of the neck during the sinking of the raft, for different particle radii R part and densities ρ part , and compare it with the experimental and theoretical results of the previous section.

In figure 4.12.a, we compare the evolution of the minimum radius r min of a liquid neck in the classic case of an oil droplet detaching from a needle (blue circles) and a granular raft detaching from an oil-water interface (black triangles, R part = 0.45 mm), as a function of time before breakup τ . The curves should be read from right to left, with a monotonous decrease of r min as τ gets closer to zero. For the oil droplet extruded from a needle, we recover precisely the different regimes described previously: first an exponential decay of r min reminiscent of the Plateau-Rayleigh linear instability, followed by two successive regimes where r min decreases linearly, adequately fitted by the theoretical predictions of equations (4.10) (slope u v = 0.0709 γ/µ) and (4.11) (slope u iv = 0.0304 γ/µ). Concerning the granular raft, we also observe in the final stages of the pinching the same two linear decays, with slopes comparable to u iv and u v (inset of figure 4.12). But interestingly, for τ > 5 ms, the dynamics differ drastically. Instead of the classic exponential decay, we observe a succession of linear evolutions, with slopes similar to u iv (5 ms< τ <20 ms) and u v (τ >30 ms). In this system, because the liquid filament does not have a minimum radius for τ >0.05 s, we cannot explore larger times.

The evolution of the vertical position of r min is represented in figure 4.12.b. Here again, the difference between the two experiments is more than clear: for a granular raft, z(r min ) is not even monotonous. Indeed, r min first slowly moves downwards, until τ ≈ 25 ms, at which point it begins to move upwards. Furthermore, the maximum value of z(r min ) is almost four times higher for a granular raft than for an oil droplet extruded from a nozzle: the pinching dynamics is therefore far less localized for the granular raft, because the viscous thread is stretched by the acceleration due to the dense particles.

Influence of the shape of the raft on the reproducibility of the experiment

To understand this intriguing dynamics, we change the geometry of the granular raft, to see its impact on the thinning dynamics. Moreover, it constitutes also a way for us to check the reproducibility of our experiments: because the arrangement of the particles inside a raft will never be completely identical between two successive experiments, if it impacts greatly the destabilization and the pinching dynamics, then it will be hard to conclude anything. Concerning the reproducibility, we can already say that for two similar axisymmetric raft, the r min curves are very well superimposed. But as shown by figure 4.13, this reproducibility is far from being only limited to quasi-identical rafts: figure 4.13.b highlights that two drastically different geometric configurations lead to almost the same pinch-off, with the same succession of linear evolutions for r min (here, the short-time regime is not really visible because the resolution of the data is not high enough). As demonstrated by figure 4.13.a, we also recover the succession of four linear evolutions if we stack the particles on top of each other instead of forming a monolayer. The duration of each regime can change a little, which is why the two curves are not superimposed, but the slopes are almost the same. Therefore, the geometrical configuration of the granular raft does not seem to impact greatly its sinking dynamics.

Oscillation between self-similar regimes

A new representation of the thinning dynamics: ṙmin Before we start to change the parameters of our experiment (size and density of the beads), let's get back to the representation and characterization of the various regimes during the detachment of a granular raft. For a viscous detachment, we have already seen that r min evolves linearly with τ , with different slopes depending on the balance of forces.

As a consequence, if we want to identify the different regimes correctly, instead of r min , we can look at ṙmin , the slope of r min [START_REF] Li | Capillary breakup of a liquid bridge: identifying regimes and transitions[END_REF]. Using this representation, each linear evolution of r min now appears as a constant (u v for the viscous regime, u iv for the inertial-viscous one), allowing us to identify them more easily. Moreover, we plot the curves in log-scale ṙmin as a function of r min instead of τ , since the measurement of r min is simpler than the exact determination of the time of breakup, making comparisons between experiments easier. Figure 4.14 displays such measurements. Just as before, it has to be read from right to left.

For the drop detaching from a needle (blue circles), we recover the three expected regimes: an exponential increase of ṙmin , followed by a first plateau at ṙmin ≈ u v , and a final regime at ṙmin ≈ u iv . The black curve is however drastically different: ṙmin presents a quasi-sinusoidal oscillation in log-scale, and only the last stages of the dynamics appear comparable between both experiments. As intuited with figure 4.12, we observe an oscillation between the two self-similar regimes. This result also emphasizes the existence of a given period of oscillation in the log(r min ) variable.

Influence of the size and density of the particles

To check the robustness of these preliminary observations, we change the size and density of the beads, and plot ṙmin for a broad range of parameters. The results are displayed in figure 4.15. Between the different curves, we change both the density and the radius of the particles. These two parameters will be gathered in the dimensionless number D 1 introduced in equation (2) in the introduction. Its definition is reminded here:

D 1 = ρ part ρ w -ρ o 2R part c (4.13)
The heavier the particle is (larger or denser), the larger D 1 will be. It is clearly visible that whatever diameter and density of the beads, the pinching dynamics are identical. Indeed, for all granular rafts, ṙmin oscillates between the slopes of the two self-similar regimes, with a period of oscillation in log-scale that does not seem to depend on the particles properties. If we want to reinterpret figure 4.15 in term of r min (and not its derivative), it means that for all the granular raft tested, during their sinking, the minimum radius of the stretched liquid thread passes through four successive linear evolutions, twice with a slope u v (red dotted line, Papageorgiou regime), and twice with a slope u iv (green dotted line, Eggers regime). For all D 1 , as expected, the final regime is the inertial-viscous one, just as for an oil drop detaching from a needle.

The phase shift in log-scale between the different curves here in figure 4.15 is related to the sizes of the rafts before destabilization. Indeed, for a larger granular raft, the initial radius of the liquid thread is higher, so r min starts at a higher value. Because the size of a granular raft at destabilization decreases with D 1 (see [START_REF] Abkarian | Gravityinduced encapsulation of liquids by destabilization of granular rafts[END_REF]), we get that for low D 1 , r min is initially higher, leading to an apparent ranking of the curves according to their D 1 : from left to right, D 1 decreases. As an illustration, one can look at the dark blue triangles (D 1 = 5.73), for which r min starts at 3 mm, whereas for the orange curve (D 1 = 12.34), the initial value of r min equals 1 mm. Of course, this explanation only clarifies the initial shift, and not the conservation of this shift throughout the entire dynamics.

Differences between a rising oil droplet and a sinking granular raft For now, several differences between a granular raft and a drop extruded from a nozzle can already be highlighted to account for the different pinching behaviors, as illustrated in figure 4. [START_REF] Peter A Kralchevsky | Particles with an undulated contact line at a fluid interface: interaction between capillary quadrupoles and rheology of particulate monolayers[END_REF]. First, the granular raft before sinking has a larger diameter than the needle (up to ten times larger): for most of our experiments, R 1 > R 0 with the notations of figure 4.16. Nonetheless, this initial diameter is not sufficient to explain the unexpected pinch-off dynamics. Indeed, for the steel beads, due to the very high density of the particles, the raft sinks when its radius reaches approximately 2 mm, an initial value similar to some of our experiments with needles. Schematic representation of the differences between a rising oil droplet extruded from a needle, and a granular raft sinking from an oil layer. The vertical speed is higher for the raft (V 1 > V 0 ), and the lateral extension of the liquid thread is usually higher for the raft (R 1 > R 0 )

The direction of motion of the oil droplet also differs between the two experiments (downwards for the granular raft, upward for the needle), as well as the geometry of the droplet at one end of the viscous neck, encapsulated inside a cocoon of particles or not. The contact angle with the particles could also affect the whole interface by setting different boundary conditions. Finally, the vertical velocity of the droplet, as measured in figure 4.17, is ten times higher for a granular raft, since the weight of the particle shell drags the raft down (

V 1 V 0 ).
But despite all these different external conditions, one fact remains surprising: the self-similar regimes describing a viscous pinching are only based on the oil properties, and are supposed to be completely independent on initial and boundary conditions. As a consequence, we need to carefully study the impact of each parameter on the dynamics. However, whether it is for a granular raft or an oil droplet forming from a needle, we can neither act on the direction of motion nor on the vertical velocity of the detaching droplet, and the range of initial sizes accessible remains limited. oil droplet rising in water (2R 0 = 2.9 mm). Black triangles: granular raft sinking from an oil-water interface (ρpart = 2,500 kg.m -3 , 2Rpart = 0.9 mm).

Pulling the interface

Stretched meniscus thinning dynamics

The experiment

To overcome the limitation we reached in the previous section, we design a new experiment reported in figure 4.18: a circular aluminum cylinder, pinned to an oil-water interface, is pulled downward (similarly to [START_REF] Marmottant | Fragmentation of stretched liquid ligaments[END_REF], where an air-water interface is pulled upwards). Here we now control both the radius of the cylinder R cyl and its vertical velocity v cyl . We use the same oil-water interface as for the granular rafts. We pull the interface downwards with aluminum cylinders within a range of diameters 2R cyl =1-30 mm, with a Thorlabs Linear Translation Stage mounted with a stepped motor, that can reach a velocity v cyl =30 mm/s along a travel range of 150 mm. To move the cylinder quasistatically (v cyl =0), the cylinder vertical motion is controlled in 10 µm increments close to pinch-off. A stretched meniscus is formed between this moving cylinder and the layer of oil, thus Even qualitatively, the difference between the thread profiles in the two sequences of figure 4.19 is quite obvious. For the small cylinder at v cyl = 1 mm.s -1 (figure 4.19.b), the neck keeps a symmetry around its minimum radius throughout the whole dynamics, and the thread breaks on the upper side only 0.2 ms before the second pinch-off near the cylinder. On the contrary, for a larger cylinder with a higher vertical velocity (figure 4.19.a), the thread loses its symmetry more than 50 ms before pinching. Moreover, because of this asymmetry, the second pinch-off happens 40 ms after the first one. These two very different thread aspects are reminiscent of the various shapes of detaching droplets already seen in figure 4.8.

Oil Water

Direction of motion

With this new configuration, we first want to check if the direction of motion has an influence on the pinching dynamics. To that end, we choose a set of parameters really similar to the needle experiment of figure 4.14, where we extruded oil from a nozzle (2R 0 = 1.65 mm) at a very low flow rate. As a consequence, we use a cylinder with a 2 mm diameter that we pull at 1 mm.s -1 , to mimic the very slow motion of the rising oil droplet. Just as before, we measure the minimum radius of the thread, and add on the previous figure 4.14 its time derivative as a function of r min (see figure 4.20). Then, we compare these new results to both the droplet pinch-off and the granular raft sinking.

The stretched oil meniscus pinches following almost exactly the same dynamics as the oil droplet rising up from a nozzle. Here again, we recover the expected exponential regime at high r min , attributable to the Plateau-Rayleigh instability, followed by the two linear evolutions of Papageorgiou and Eggers. Since the two curves are similar, we can exclude the direction of motion as a possible explanation for the different evolution of r min when a raft detaches from an oil-water interface: as expected, gravity does not play a role in the self-similar regimes observed at short time scales. Before exploring the size-velocity space parameters, we need to check if the initial position of the cylinder has an influence on the dynamics. Indeed, unlike an oil droplet attached to a needle or a granular raft confined at an interface, a continuous range of initial positions is accessible for the cylinder, from the flat horizontal interface to a stretched but still stable meniscus (see inset of figure 4.21).

Yet, as highlighted by figure 4.21, this initial position only slightly affects the long-time dynamics of the thinning, and has no effect on the self-similar regimes. For the rest of the experiments, we will place the cylinder at an intermediate position, (middle drawing of figure 4.21), but there is no real need to begin each experiment at the exact same vertical position.

Oscillation between self-similar regimes

Influence of the radius of the cylinder and the pulling velocity

We now vary v cyl and R cyl when pulling the oil-water interface, within the range 0 mm.s -1 < v cyl < 30 mm.s -1 and 1 mm< 2R cyl < 30 mm. Some results are presented in figure 4 We observe a log-oscillation of ṙmin when R cyl or v cyl increase. For the quasistatic pulling (orange circles in figure 4.22), the oscillation appears and grows when 2R cyl > 8 mm. When 2R cyl = 5 mm, ṙmin oscillates only for the highest pulling velocity (v cyl = 30 mm/s). Finally, for 2R cyl > 8 mm, oscillations are always present. For the smallest cylinder (2R cyl = 2 mm), we do not observe oscillations using our experimental setup. However, since an oscillation can be measured for very small granular rafts (R raf t ≈ 2 mm, for a sedimentation velocity of about 300 mm/s), we think ṙmin could present a log-oscillation even when 2R cyl = 2 mm if the interface could have been pulled down sufficiently rapidly. This result is reminiscent of the numerical observations of Li and Sprittles [START_REF] Li | Capillary breakup of a liquid bridge: identifying regimes and transitions[END_REF], where they find an oscillatory convergence in log-scale towards a self-similar regime.

Figure 4.22 leads us to the following conclusion: the diameter of the cylinders and their vertical velocity are the two fundamental parameters acting on the onset of the oscillation of ṙmin during the thinning of an oil thread surrounded by water. Besides, they both act exactly in the same way: for low values, we recover the classic and expected succession of self-similar regimes, whereas for higher values of either the diameter or the vertical velocity, the thinning undergoes a more complex behavior. The ṙmin curves are shifted to the right as v cyl and R cyl increase. The self-similar regimes, supposedly independent of boundary and initial conditions, are therefore astonishingly perturbed by external conditions far from the position of the minimum radius. Their slopes are preserved, but the path followed by the system to reach the final Eggers regime seems trickier.

Reproducing the sinking dynamics of a granular raft with a cylinder

By rapidly pulling down an oil-water interface, we can reproduce the dynamics of detachment of a granular raft from an interface, as underlined by figure 4.23. We achieve to recover a very similar behavior for r min (figure 4 This result shows that the particles only act as a strong vertical force pulling the oil filament downward. We always find an oscillation of ṙmin during the sinking of a granular raft only because it gathers the two conditions leading to this behavior: a large diameter, and a high vertical velocity.

Furthermore, because we can modify independently two parameters for a cylinder pulling the interface (size and velocity), we can also create similar thinning evolutions between two cylinders of different sizes, as illustrated by figure 4.23. Here, we experimentally show that pulling an oil-water interface with a 13 mm cylinder at 10 mm.s -1 leads to the same thinning as pulling with a 20 mm cylinder at 1 mm.s -1 .

Period of the log-oscillation

The existence of a log-oscillation for various parameters is a striking feature that we wish to understand. Even without undertaking a theoretical study of the thinning of a stretched oil filament surrounded by water, the presence of an oscillation in log-scale for such a selfsimilar process can be explained. Indeed, it corresponds to the natural time dependency arising when a perturbation analysis is performed around the self-similar dynamics.

We consider the equations leading to the inertial-viscous regime, and follow the procedure developed by Eggers [START_REF] Eggers | Universal pinching of 3d axisymmetric free-surface flow[END_REF]: with R 1 and R 2 the two radii of curvature, r the radius of the fluid filament, ρ its density, µ its viscosity, γ its surface tension, and v the velocity of the fluid. In the pinching region, r and v can be described with two self similar functions φ and ψ and a single variable ξ = z/τ 1/2 , with τ the time remaining before pinch-off and z the vertical position. Such an approach leads to the asymptotic inertial-viscous regime, but if we want to capture the transition towards this regime, we need to keep a time dependency in φ and ψ, as in equation (4.15a) and (4.15b):

     ρ(∂ t v + vv ) = -γ 1 R 1 + 1 R 2 + 3µ (r 2 v ) r 2 -ρg ∂ t r + vr = -v r/2
r(z, t) = l µ τ φ(z /τ 1/2 , τ ) v(z, t) = (l µ /t µ )τ -1/2 ψ(z /τ 1/2 , τ ) (4.15a) (4.15b)
with τ the dimensionless time before breakup and z the dimensionless vertical position.

Inserting 4.15a and 4.15b in 4.14a and 4.14b leads to a system of equations almost similar to the one obtained by Eggers, with only an additional term:

         - ∂ψ ∂ log τ + ψ 2 + ξ ψ 2 + ψψ = φ φ 2 + 3ψ + 6 ψ φ φ - ∂φ ∂ log τ + φ ψ + ξ 2 = φ 1 - ψ 2 (4.16a) (4.16b)
As we can see, the time τ only intervenes in the equations through log(τ ) so that the dynamics around the self-similar solutions involve the log of the time only. Now that the existence of a log-oscillation has been made clearer, we can measure the log-pulsation ω, and study its dependence with the different parameters (using the following fit ṙmin = A + B sin(ω log t + C) exp(D log t)), as displayed in figure 4 Invariance of the log-pulsation with size and vertical velocity. Pulsation in log-scale of the sinusoidal oscillation of ṙmin as a function of the radius of the cylinders, for different pulling velocities (red circles: 0 mm/s, black crosses: 1 mm/s, dark blue squares: 10 mm/s, green triangles: 30 mm/s), and various granular rafts (light blue stars). For the granular rafts, R stands for the maximum radius of the raft before destabilization. ω is measured only when a complete period of oscillation is present. The black dotted line displays the mean value of ω, and proves the log-pulsation is globally constant for all the experiments.

For the range of parameters we have explored, ω is constant and does not depend on the various parameters of the different experiments we have investigated. However, the exact meaning of this log-pulsation still remains to be explored.

Self-similar shapes

Profiles of the meniscus when ṙmin does not oscillate Up to now, we have only considered the slope of r min , and we have found out that under some conditions, this minimum radius passes through a succession of four linear evolutions that correspond to the predictions of Eggers and Papageorgiou. Nevertheless, the self-similarity does not only imply a given evolution of r min , but rather of the whole profile. As a consequence, if we want to investigate the self-similarity during this oscillation of ṙmin , we need to measure the whole profile of the liquid thread, and see if within each linear evolution, several profiles taken at different times can be rescaled on the same curve.

To that end, we define rescaled variables according to equations (4.10) and (4.11),

         ξ 1 = (z -z(r min )) l 0.5 µ r 0.5 min H(ξ 1 ) = r r min ξ 2 = (z -z(r min )) l 0.825 µ r 0.175 min H(ξ 2 ) = r r min (4.17a) (4.17b)
Since r min decreases linearly with time (in each linear evolution), we use r min instead of τ to rescale both axes. As a consequence, the minimum radius of the thread is located at ξ = 0 and H = 1. ξ 1 and ξ 2 are made dimensionless using the characteristic length scale l µ (see equation (4.7)). The profiles are relatively flat around r min , which can sometimes make the determination of z(r min ) tricky. To overcome this difficulty, we shift the profiles in the ξ direction to superimpose the rescaled profiles [START_REF] Cohen | Testing for scaling behavior dependence on geometrical and fluid parameters in the two fluid drop snap-off problem[END_REF]. When ṙmin = u v (resp. ṙmin = u iv ), the profiles can be rescaled using equation (4.17b) (resp. equation (4.17a)). We begin by examining the dynamics of pinch-off of the 1 mm cylinder, for which no oscillation is observed. Figure 4.25 shows the rescaled profiles for this cylinder, during each of the linear regions ( ṙmin constant).

In a and b, R cyl = 1 mm and no log-oscillation is observed: the profiles are perfectly superimposed in each of the linear region. We also recover the symmetry of the profile during the viscous regime (figure 4.25.b). As expected, we confirm the self-similarity of the pinching dynamics, with an oil thread that transitorily thins following the viscous regime, before reaching a final inertial-viscous regime which lasts until pinch-off.

Profiles of the meniscus when ṙmin oscillates

Keeping in mind these results for a simple droplet, we use the same procedure to analyze the successive profiles of a liquid thread stretched by a bigger cylinder (2R cyl = 13 mm), at v cyl = 1 mm.s -1 , for which ṙmin presents an oscillation. The results are displayed in figure 4. [START_REF] Loudet | Capillary interactions between anisotropic colloidal particles[END_REF].

Near pinch-off (figure 4.26.a), the rescaling is excellent, the three profiles are well superimposed, and the slope is the one predicted by Eggers. This result is somehow reassuring, since the dynamics of pinching must end with the inertial-viscous regime. Similarly, after the exponential regime, the system is well described by the viscous regime (figure 4.26.d), with the expected value of ṙmin (upper dotted line), the clear symmetry of the profiles around r min , and a self-similar shape (the rescaled profiles are well superimposed).

Between these two regimes, ṙmin oscillates and reaches successively u iv and u v . Nevertheless, in figure 4.26.c for instance, even though the slope of r min equals u iv , the corresponding rescaling fails to superimpose the profiles. The same statement is even more obvious for figure 4.26.b, where we expect the profiles to be rescaled by equation (4.17b) (viscous regime), based on the measured slope of r min . Here again, the three rescaled profiles are clearly not superimposed. Moreover, the shapes measured are not even symmetrical, even though this condition is essential for the calculation leading to equation (4.10).

In conclusion, the system oscillates between the self-similar viscous solution and the self-similar inertial-viscous solution, but the transient regimes reached during this process do not correspond to self-similar behaviors, despite the slope of r min indicating so. The simple knowledge of ṙmin during a linear regime is not sufficient to conclude that selfsimilarity is indeed observed. Our result demonstrates the need to perform a complete analysis of the whole profile in order to confirm without a doubt the presence of a selfsimilar regime, contrarily to many existing statements in literature which limit themselves to the measurement of r min .

However, it is surprising that the slopes of the minimum radius in the oscillation are identical to those of the linear viscous and inertial-viscous regimes. We do not know yet if this result is only due to the fact that ṙmin oscillates between u iv and u v , and as a consequence is necessarily close to these values, or if there is a deeper reason. Now that we have identified the two self-similar regimes, we can measure the duration of the transition between them. As highlighted by figure 4.27, this duration increases dramatically both with the cylinder radius R cyl and the speed v cyl as it can reach up to 50 ms, hiding somehow the self-similar features of the pinch-off for most of the process.

We have shown that boundary conditions can lead to a transient loss of self-similarity, during which an oscillation of the slope of r min appears. This reminds us of the behavior described by Lindner et al. [START_REF] Lindner | Single particles accelerate final stages of capillary break-up[END_REF], where they study the influence of isolated particles on the detachment dynamic of a droplet. They conclude that an individual bead can increase the slope of r min , because it deforms the interface to a point where a self similar solution cannot exist anymore. The thinning accelerates until the profile becomes again compatible with a self-similar solution. Here, the situation is completely different, because the neck is free of particles. Nevertheless, we can imagine that the large size of the cylinder and its high vertical velocity prevent the neck from thinning with the inertial-viscous solution, triggering an oscillation until the profile becomes compatible with the Eggers self-similar solution.

Numerical simulations of droplet Pinch-off

Up to now, we have presented many experimental results on pinching dynamics in various experimental configurations. In this section, we wish to complete our results with a numerical study of the pinch-off of an oil droplet surrounded by water. The goal is to see if we can numerically recover one of our main result, that is the presence of an oscillation of ṙmin under specific boundary conditions. We claim neither to realize a vast and complete numerical study of the phenomenon where we would change all the parameters, nor to identify clearly the conditions required to see the oscillations. We will also limit ourselves to the variation of r min , without looking at the profiles of the liquid thread, to see if the self-similarity is conserved throughout the whole thinning. Our only goal is to check some of our previous conclusions.

Numerical methods

The algorithm

We use the free software Basilisk [173] that solves partial differential equations on a Cartesian grid. A VOF method is used to track the fluid-fluid interface. For more details on the code, see [START_REF] Popinet | Gerris: a tree-based adaptive solver for the incompressible euler equations in complex geometries[END_REF][START_REF] Popinet | An accurate adaptive solver for surface-tension-driven interfacial flows[END_REF], where Gerris, the ancestor of Basilisk, is extensively presented.

We simulate the formation of an oil droplet surrounded by water. To that end, we inject oil at a low flow rate at one side of the box, and let the droplet slowly grow until pinchoff. We solve the two-phase Navier-Stokes equations with gravity and surface tension, in an axisymmetric geometry, with an adaptive mesh. The mesh is refined whenever the curvature of the interface or the velocity of the fluid become too high.

We need seven parameters as inputs for our simulations: µ o , µ w , ρ o , ρ w , γ, R 0 , g. As a consequence, four non-dimensional numbers will control the dynamics: the Ohnesorge number, the Bond number, and the viscosity and density ratios:

       Oh = µ o √ ρ o γR 0 δ µ = µ o µ w B = (ρ w -ρ o )gR 2 0 γ δ ρ = ρ o ρ w (4.18a) (4.18b)
with µ o (resp. µ w ) the viscosity of the oil (resp. of the water phase), ρ o (resp. ρ w ) the oil density (resp. the water density), γ the oil-water interfacial tension, R 0 the radius of the needle, and g the acceleration of gravity. These four dimensionless numbers will be calculated according to the experiment we want to reproduce. Then in the simulation, since we have seven parameters for four dimensionless numbers, we can choose arbitrarily the value of three parameters, and then define the other four using the first three ones and the four dimensionless numbers, as follows:

   γ = 1 R 0 = 1 ρ o = 1 =⇒        ρ w = ρ o /δ ρ = 1/δ ρ µ o = Oh √ ρ o γR 0 = Oh µ w = µ o /δ µ = Oh/δ µ g = γB/((ρ w -ρ o )R 2 ) = B/(ρ w -ρ o ) (4.19)

Typical time scale and length scale

This could appear as trivial, but it is going to be fundamental if we want to compare our numerical results with the experiments. Indeed, since we are not putting the real physical parameters inside the code, the results we will get are going to be dimensionless. Therefore, we need to find the characteristic length and time scales. For the length, the result is obvious: since we impose R 0 = 1, we will only need to multiply all lengths by the radius of the needle R 0 . The rescaling in time requires more work. The equations Basilisk solve in each fluid phase are the incompressible Navier-Stokes equations [173]:

   ∂ t u + ∇.(u ⊗ u) = - ∇p ρ + µ ρ ∇.(∇u + (∇u) T ) + a ∇.u = 0 (4.20a) (4.20b)
with u the dimensionless velocity field, p the pressure field, a any external acceleration (for instance gravity), and µ and ρ the dimensionless viscosity and density that we impose: µ = Oh and ρ = 1 for the oil phase (see system (4.19)). All the bars on every quantity mean we are in the unit system of the simulation.

To understand the time rescaling, we need to write the non-dimensionalization in the classic unit system of the experiments. We introduce a typical velocity v c , a typical time t c , a typical length R 0 , a typical pressure p c , and a typical external acceleration a c :

ρ o v c t c ∂ t u + v 2 c R 0 ∇.(u ⊗ u) = p c R 0 ∇p + µv c R 2 0 ∇.(∇u + (∇u) T ) + ρ o a c a (4.21)
This can be written in the same form as equation (4.20a) as follows:

∂ t u + v c t c R 0 ∇.(u ⊗ u) = p c t c ρR 0 v c ∇p + µt c ρR 2 0 ∇.(∇u + (∇u) T ) + a c t c v c a (4.22)
By identifying the factor in front of the viscosity term ∇.(∇u + (∇u) T ) between the two equations (4.20a) and (4.22), we can deduce the typical time t c of the simulation:

µ o t c ρR 2 0 = µ ρ = Oh 1 t c = Oh ρR 2 0 µ o (4.23a) (4.23b) Using Oh = µo √ ρoγR 0
, we end up with the following expression for t c :

t c = ρR 3 0 γ (4.24)
R 0 and t c are given by the experiments. By multiplying in the simulations all lengths by R 0 and all times by t c , we end up with quantities that can be compared to the experiments. Qualitatively, we can already see that for two very different sizes, we recover very different dynamics: for a small radius (figure 4.28.a), the thinning is symmetric, and the two breakups (near the droplet and near the nozzle) happen almost exactly at the same time, whereas for a larger nozzle radius (figure 4.28.b), the symmetry is lost, and the oil neck pinches at its upper side a long time before the second pinch-off. These observations are very similar to the experimental differences we have noticed between a large and a small cylinder stretching an interface (see figure 4.19).

A few results

Visualization of a typical numerical result

Oscillation of ṙmin

After these preliminary qualitative considerations, let's try to characterize precisely the dynamics of thinning. First, we want to check if for the smaller needle, we can recover the result of figure 4.10, where r min decreases through an exponential decay followed by the viscous and finally the inertial-viscous regime, without any oscillation. To that end, we measure the minimum radius of the oil thread at each step of the simulation, for a needle radius similar to one of our experiment, and plot ṙmin as a function of r min , as it is done in figure 4.29. The agreement between the numerics and the experiment is more than clear, since the two curves are almost superimposed. Moreover, we recover with this numerical simulation the succession of the three classic regimes: an exponential decay, typical of the Plateau-Rayleigh instability, followed by a first linear evolution of r min , representative of the viscous regime, and finally a last plateau of ṙmin for the inertial-viscous regime. This result gives us confidence in our simulations, and allows us to proceed with the measurement of r min for larger R 0 . Such a numerical simulation is displayed in figure 4. [START_REF] Amar | Fabrication, assembly, and application of patchy particles[END_REF], where we study the detachment of an oil droplet from a needle of higher R 0 (4 mm). We find an oscillation of ṙmin throughout the thinning of the neck, which is compatible with our experimental results. Indeed, we observe this very same oscillation for a cylinder of the same size pulling an oil-water interface downwards. Here, however, we do not capture the final Eggers regime (the curve does not reach the green dotted line), probably because the level of refinement of the simulation was not high enough. The first plateau of ṙmin is furthermore located almost at the middle of the two selfsimilar regimes, instead of being close to the viscous regime. We know neither the reason nor the implications of such a behavior. But despite this numerical result, we do get what we were looking for: for low needle radius, r min goes straight from the viscous regime to the inertial-viscous one, whereas for higher needle radii, ṙmin presents an oscillation between the two regimes. We also have to note that by increasing the radius of the needle, we increase at the same time its vertical velocity, which is higher for bigger droplets. Changing R 0 thus acts on the two parameters piloting the onset of the oscillation at the same time: the size of the system, and its vertical velocity. As a consequence, an oil droplet detaching from a large needle is doubly more susceptible to present an oscillation in its thinning dynamics than an oil drop forming from a smaller needle.

Our results present a good agreement with the numerical results of Li et al. [START_REF] Li | Capillary breakup of a liquid bridge: identifying regimes and transitions[END_REF], as demonstrated in the inset of figure 4.30 where ṙmin is measured for two different Ohnesorge numbers. In their paper, ṙmin is calculated by deriving the minimum radius with respect to the time t, while in our work, we derive with respect to the time before pinch-off τ , which explains the opposite sign between their work and ours. They report the existence of the same oscillation as what we found experimentally and numerically, and identify a transition between two types of thinning regimes at a critical Ohnesorge number. The succession of unexpected regimes has also been reported in the work of Castrejón-Pita et al. [START_REF] Rafael Castrejón-Pita | Plethora of transitions during breakup of liquid filaments[END_REF].

When particles disturb the pinch-off

Suspensions: a short overview

Up to now, we have focused only on the sinking of granular rafts constituted of large particles, for which the particles remain around the encapsulated droplet. In that situation, the granular nature of the raft does not directly play a role: the beads merely act as a vertical force pulling the oil filament downwards. Nevertheless, as briefly seen in the photos of figure 4.11, when the size of the particles decreases, exotic behaviors can arise: the particles start to be distributed all along the oil neck, disturbing the detachment dynamics. For small enough particles, we can even witness the formation of an oil jet instead of a single droplet. The latter case will not be studied here.

The situation we described present some similarities with the pinch-off of granular suspensions, where particles are present everywhere in the bulk. The results already obtained by the suspension community can therefore guide us through the analysis of our own experiments.

The formation of granular suspension droplets began to be investigated in 2004 by Furbank and Morris [START_REF] Roy | An experimental study of particle effects on drop formation[END_REF][START_REF] Roy | Pendant drop thread dynamics of particle-laden liquids[END_REF], where the idea of an effective viscosity is proposed to describe the dynamics far from breakup. Nevertheless, near pinch-off, an acceleration of the dynamics is observed, and the thinning becomes faster than for the same fluid without particles [START_REF] Bonnoit | Accelerated drop detachment in granular suspensions[END_REF]. The volume fraction of particles inside the fluid thread fluctuates locally [START_REF] Bertrand | Dynamics of drop formation in granular suspensions: the role of volume fraction[END_REF][START_REF] Mcilroy | Modelling capillary break-up of particulate suspensions[END_REF], leading to a succession of various thinning regimes [START_REF] Mathues | Capillary breakup of suspensions near pinch-off[END_REF][START_REF] Pan | Drop formation in shear-thickening granular suspensions[END_REF][START_REF] Zhao | Inhomogeneity in breakup of suspensions[END_REF]. For low volume fractions, as the one presented in figure 4.31, an acceleration of the thinning is found compared to the pure liquid pinching [START_REF] Lindner | Single particles accelerate final stages of capillary break-up[END_REF]. The pinching point appears to be confined between the two closest particles, leading to a thinning faster than the expected viscous regime (blue curves in figure 4 Because individual particles increase locally the curvature of the liquid neck, selfsimilarity can even be transitorily lost [START_REF] Merlijn S Van Deen | Particles accelerate the detachment of viscous liquids[END_REF][START_REF] Lindner | Single particles accelerate final stages of capillary break-up[END_REF], a behavior very similar to what we found for stretched meniscus pulled at a high velocity. One of the major difficulties for these experiments is the intrinsic dependency of the pinching dynamics with the position of each individual bead, leading to non-reproducible results [START_REF] Yong Moon | Heterogeneity in the final stage of filament breakup of silicone oil/pmma suspensions[END_REF].

Our situation is of course different, because for a granular raft, particles are confined at the interface. Moreover, they are far heavier than the liquid phases in our situation, whereas for a suspension, particles are often matched in density with the inner fluid. Yet, the results derived for suspensions can bring us some useful insight on the role of individual particles on the pinching dynamics.

We will first focus on granular rafts made of particles large enough so that only a few beads can be found along the neck. In that situation, most of the results we derived will remain valid. We will then tackle the more intricate situation where a lot of beads are present along the neck, even at pinch-off. Only preliminary results will be presented.

Isolated particles: a moderate effect

Evolution of the minimum radius When particles are along the filament, they can perturb the pinching dynamics. But this perturbation will depend on the experiment, meaning that for a given type of particles, their distribution along the oil filament will change from one experiment to the other. As a consequence, similarly to the formation of suspension droplets, we do not expect the results to be completely reproducible.

Up to now, in order to access both long-time and short-time dynamics, we conducted the same experiment several times with different magnification factors and different sampling frequencies. Thanks to that, we could then gather the different data onto a same curve with both the long-time and short-time behavior.

Here, since we cannot assume any reproducibility, we need to record at the same time both time scales and length scales. To overcome this difficulty, we use two high-speed cameras, filming the detachment from two different sides of the water tank. The first camera records the whole dynamics, with a low sampling frequency and a large field of view, while the other one records at a higher sampling frequency the pinching dynamics in a narrow window centered at the point where the pinch-off will occur. An example of the results obtained with such a procedure is presented in figure 4.32.b and 4.32.c.

In this section, we use ceramic beads with a density ρ part = 3,800 kg.m -3 and a radius R part = 0.25 mm. As illustrated in figure 4.32, for this kind of particles, only a few beads are present along the filament as the granular raft goes down. Visually, we expect the pinching dynamics to be perturbed for large τ (right side of the series of photos), since the minimum radius is located right in the middle of the portion of the filament where particles are present.

This first observation is confirmed by the measurement of the minimum radius and its vertical position. z(r min ) even appears to be discontinuous, because of the numerous particles present: the different jumps observed are the signatures of the position of the beads during the thinning process. But as the sinking progresses, we seem to recover an evolution quite similar to the one displayed by a granular raft with large particles, as extensively described in the previous sections, with a succession of linear regimes with slopes comparable to the theoretical self-similar regimes of Eggers and Papageorgiou.

Reproducibility of the oscillation

We can analyze these results using the same framework as before, by looking at the slope of the minimum radius ṙmin , as a function of r min . The results are displayed in figure 4.33 for three different experiments with the same particles.

The three curves are rather well superimposed, except for their initial portion (right side of the curves). This could be expected, since as we stated before, this part of the dynamics corresponds to a situation where a lot of beads are present at the position of the minimal radius. But for the rest of the pinching, the three granular rafts display the same behavior. In that situation, the particles have only an effect on the initial regime.

The next step is therefore to decrease again the radius of the beads, and see if an enhanced effect can be observed. Contrarily to the series of photos, the curves have to be read from right to left.

Multiple particles: an intricate behavior

Minimum radius

Only preliminary results will be presented here. We use again ceramic beads, with the same density (ρ part = 3,800 kg.m -3 ), but a slightly lower radius: R part = 0.175 mm instead of 0.25 mm. Quite surprisingly, this small decrease is enough to completely modify the behavior under study. A typical experiment is presented in figure 4.34. Qualitatively, it is already clear that the filament is now almost entirely covered by particles throughout the major part of the sinking, contrarily to the various situations studied before. This leads to a much longer oil filament, that can extend on more than 70 mm, while for larger ceramic particles, as for instance in figure 4.11.a where R part = 0.45 mm, the length of the filament does not exceed 15 mm.

The second main difference concerns the vertical localization of the minimum radius, as emphasized by figure 4.34.a and 4.34.c. r min moves downwards during most of the dynamics, staying close to the encapsulated droplet, until a time when the shrinking slowdowns, as showed by the small plateau in figure 4.34.b. After that, a second minimum of the radius appears near the oil-water interface. From there, the sinking proceeds until the filament detaches from its upper part near the oil-water interface. Later, a second pinch-off occurs near the droplet.

Measuring r min for the whole liquid thread, as we did in figure 4.34.b, has therefore only a limited interest, since the two parts of the curve, delimited by the red arrows, correspond to two different pinching events. For the rest of this section, we will only focus on the upper r min , close to the oil-water interface, because of a practical issue that we encountered when trying to visualize the lower one. As one may see in the last photos of figure 4.34.a, the lower minimum radius not only moves downwards, but also out of the plane of observation. Of course, this motion completely changes from one experiment to the other, leading to a very intricate visualization that we could not perform with our setup.

The upper r min appears quite belatedly, but we can still study its evolution as a function of time. The usual self-similar regimes are plotted in the inset, but except for the end of the pinching, we do not seem to recover them very clearly.

Position of the particles with respect to r min

To explore the existence of these regimes, we plot the slope of the minimum radius ṙmin , as a function of r min , for three different experiments (see figure 4 Clearly, the dynamics are not reproducible. This result is not completely surprising, since the distribution of particles along the oil filament changes from one experiment to the next. It is also completely different from the pinch-off of a granular raft with larger particles, as studied before. We no longer recover an oscillation of ṙmin between the two self-similar regimes.

For some experiments, as for instance the one presented in figure 4. [START_REF] Jambon-Puillet | Folds in floating membranes: from elastic sheets to granular rafts[END_REF].c, we do not even observe a viscous linear regime with a slope u v (red dotted line). Only figure 4.35.a seems to present some kind of oscillation of ṙmin , but the right side of the curve here again does not reach the theoretical slope predicted by the viscous regime. The only reassuring feature is the thinning dynamics just before pinch-off: for all experiments, the last stages are always characterized by a decrease of the minimum radius r min at the speed of the inertial-viscous regime (green dotted line, ṙmin = u iv ). This was expected, since near pinch-off, the length scale of the liquid neck is so small that the particles cannot have an effect anymore.

The thinning dynamics seem to be correlated to the position of the particles with respect to the minimum radius. At least qualitatively, we can see on the three series of photos that the distance between r min and the closest particle to r min in the filament, encircled in red, increases from figure 4.35.a to 4.35.b and to 4.35.c. Simultaneously, the maximum of ṙmin = u iv decreases. For the moment, this is only an assumption based on merely a correlation, but the following argumentation can be conducted: at a position where a particle is present, the radius of the filament cannot decrease beyond R part . This means that the particle encircled in red in figure 4.35 acts as some kind of boundary condition: at its position, the radius of the filament is at least of the order of R part . The higher this particle will be, the more confined the pinching filament will be. We believe this confinement of length l conf , defined in figure 4.35.a and 4.35.b as the vertical distance between the point of pinch-off and the closest particle of the oil filament, taken at the moment of pinch-off, could play a role in the velocity at which the filament thins.

To check this assumption, we plot the maximum of ṙmin as a function of l conf . The results are displayed in figure 4.36. Some kind of correlation seems to appear, but more experiments are needed to go beyond these observations. We stop at these preliminary results for the pinch-off dynamics of a liquid thread covered in particles during the sinking of a granular raft.

Conclusion

The dynamics of sinking of a granular raft have been extensively studied, in the case where particles remain far from the minimum radius and encapsulate a unique droplet. In such a situation, an oil filament is stretched between the oil-water interface and the sinking armored droplet, and thins until pinch-off. The dynamics of pinching of a viscous thread is usually controlled by the succession of two self-similar regimes, independent of initial and boundary conditions. But surprisingly for a granular raft, the evolution of the neck minimum radius r min undergoes a succession of four linear evolutions whose slopes equal alternatively those of the viscous and inertial-viscous regime.

By designing a model experiment where the very same oil-water interface is pulled by a cylinder whose size and vertical velocity can be chosen independently, we have reproduced the same thinning dynamics, thus showing that the particles only act as a vertical force directed downwards and pulling the oil filament. Here again, an oscillation in log-scale of the slope of r min has been found, during the transition from the viscous regime to the inertial-viscous one, delaying the onset of the final Eggers regime. This constitutes to our knowledge the first experimental proof of a log-oscillation of ṙmin in a viscous pinching process. One of the very interesting result lies in the nature of the parameters controlling the onset of this transient oscillation, namely the radius of the sinking object and its vertical velocity. Indeed, one would expect that external conditions play no role in this oscillatory behavior, but we find out that such phenomenon appears when one end of the viscous filament is connected to either a large enough structure, or a moving one (at a sufficient velocity). The self-similar regimes remain the same, but the transition between them is astonishingly controlled by boundary conditions.

An important consequence of this oscillation concerns the shape of the liquid filaments during the transition, which does not follow any of the self-similar regimes along this transient dynamics, despite the values of ṙmin alone suggesting otherwise. This result has huge implications, because it questions numerous studies that deduced from the only knowledge of ṙmin the nature of the self-similarity. Here, we show that ṙmin can totally equal the theoretical slope of either the viscous or the inertial-viscous regime, despite the neck profile not being self-similar. Nevertheless, for now we do not know if these values of ṙmin during the transition are only a coincidence due to the fact that ṙmin oscillates between u iv and u v , and thus remains close to these theoretical slopes. The link between these slopes and the lack of self-similar profiles in this region needs therefore to be clarified. A theoretical study of these intriguing results is needed. A deeper understanding could bring some insight to many self-similar mechanisms and finite-time singularities.

Because of both the size of a granular raft and its vertical velocity, the oil filament formed during its sinking always undergoes the transient oscillation between the two selfsimilar regimes. When particles are distributed all along the surface of the oil filament, a situation similar to a granular suspension is observed, where the thinning is modified depending on the position of the individual beads with respect to the minimum radius. The existence of a log-oscillation appears to be perturbed, but more experiments are needed to fully characterize the thinning dynamics in that kind of situation.

Conclusion

Summary of the results

Capillary interactions are central in many systems at the millimeter scale. Objects placed at an interface interact, and this interaction may lead to the aggregation of the different particles. Yet, no complete description exists as soon as several non-identical non-spherical objects are present. In this thesis, we focus on the life of axisymmetric monolayers of heavy particles at an oil-water interface, called granular rafts, from their formation up to their destabilization.

In the first chapter, we investigate the respective motion of two granular rafts at an interface at rest. To do so, we record their speed as they are attracted by one another, and by changing the number of particles in each raft from one experiment to another, we are able to act on both the capillary and the drag forces simultaneously. For such a low Reynolds number system, the speed measurement directly gives us information on the balance of forces.

We generalize a two-particle model in order to express the attractive force between two granular rafts, as well as their individual drag. These two forces appear to strongly depend on the number of particles, a behavior that could have been expected. Indeed, even visually, the deformation imposed on the interface by the raft clearly increases with the size of the raft. The drag appears to be proportional to the square root of the number of particles in the raft considered: at first order, we recover the drag expected for a sphere with the same radius as the raft. The capillary force, in contrast, increases linearly with the number of particles in both interacting rafts. This behavior is explained by going back to the physical origin of the capillary forces: the interfacial deformation created by the granular rafts. By side views of the granular rafts, we measure this vertical deflection, and find a linear dependence with the number of particles. This experimental result is moreover confirmed by numerical simulations where a granular raft is modeled as an axisymmetric heavy membrane. The motion of a granular raft along a curved interface is also investigated, but even though the interaction between a single particle and a static object appears to be satisfactorily explained, the model developed fails to account for the velocity profiles exhibited by a large granular raft.

In chapter two, we no longer limit ourselves to the study of only two rafts. We want to understand the aggregation dynamics of numerous particles initially randomly scattered at an interface. For such a situation where many objects interact, the dynamics of each particle cannot be solved analytically, and only a statistical approach is able to bring some insight into the clustering process. By conducting experiments and numerical simulations, we show that the time scale of the aggregation strongly depends on the initial concentration of particles at the interface: when the initial surface density is increased, the clustering is going faster.

We characterize the distribution of sizes as a function of time, from the initial configuration when all the particles are isolated, up to the end of the aggregation when a single raft has collected all the beads. This distribution evolves with time, and at a given time, it is a function of the number of particles. Two regimes of evolution are highlighted, with a transition that has been well characterized thanks to a thorough analysis of the distributions of sizes. Inspired by the work done on various clustering dynamics in the past decades, we find a self-similar solution for the distributions of sizes that seems to account relatively well for the aggregation. Yet, no scaling law is found to describe the evolution of average quantities, as it would have been expected for a self-similar aggregation.

In a third chapter, we explore the structural changes experienced by a granular raft already formed during its motion at the interface. More precisely, we characterize the conditions under which erosion can be observed during the raft own motion along a curved interface. The cohesion measured exceeds by more than one order of magnitude the expected cohesion. To understand this high capillary attraction, we perform a second experiment where a pair of beads is moved horizontally at an interface. The critical velocity at which the two beads initially in contact separate directly gives us information on their cohesion. Here again, their interaction is far stronger than expected.

A theoretical description of the exact position of the contact line around the particles is performed, along with experimental measurements of the vertical position of this contact line around isolated particles. The tilting of the contact line, combined with small variations in the pinning of the contact line, seems to explain the high cohesive force.

In the last chapter, we study the sinking of the granular raft after its radius exceeds a critical size. During its destabilization, the monolayer encapsulates an oil droplet that sinks until it reaches the bottom of the water tank, where it will remain stable. During the sinking event, an oil thread connecting the armored droplet to the upper oil layer is formed. We focus on the dynamics of thinning of this viscous filament vertically pulled by the heavy particle-covered droplet. The minimum radius of this viscous neck appears to follow four linear decreases, a behavior that is not found during the formation of a droplet. The slopes of these regimes are coherent with the viscous and inertial-viscous self-similar regimes expected, but their chronological succession is surprising.

To further explore this observation, we mimic the sinking of a granular raft by pulling the oil-water interface with a cylinder. For low vertical velocities or small cylinders, the thinning of the oil meniscus does not differ from the classic result of a droplet. But as soon as we increase the radius of the cylinder or its pulling velocity, the succession of the four linear evolutions is recovered. Boundary conditions appear to have an impact on the self-similar regimes, even though they should only depend on the fluid properties. To go further in the analysis, we measure the whole shape of the liquid thread during its thinning, and find out that even if the slope of the minimum radius suggests a self-similar regime, the whole profiles are not self-similar at all. The boundary conditions do not act on the self-similarity itself, but rather on the transition between the two expected regimes.

Finally, we tackle the situation of an oil filament covered with particles. In that specific case, the thinning is perturbed by the position of the beads along the thread. Depending on the distance between the minimum radius and the different particles, the detachment can either slowdown or accelerate.

Future work

There are still a lot of unanswered questions in aggregation dynamics at an interface, erosion of granular aggregates, and pinching dynamics. We hope our work will bring some useful insights to these vast subjects and help future studies to better understand these complex physical systems. They are still a few points of our work that resist our inquiries and desperately need further investigations.

Concerning the motion of a granular raft along a curved interface, our model fails to account for the interaction between a cylinder deflecting the interface downwards and a large granular raft, even though it was able to describe the interaction between two rafts. Experiments focused on the motion at a curved interface need to be performed, with various interfacial deformations. For instance, we could deflect the interface with a long rectangle instead of a cylinder, in order to replace the axisymmetric deformation by a two-dimensional one. Furthermore, the capillary forces for objects floating at a curved interface need a theoretical description. Finally, the drag force around a granular raft would benefit from a few more explanations, because the scaling obtained is far from being obvious, despite its simplicity. Numerical simulations of the flow around a fixed granular raft could be undertaken, as well as experiments in which the raft is replaced by a 3D printed shell with a similar shape, in order to understand how the velocity field of the surrounding fluid behaves around such a curved 2D object placed at an interface.

The many-body aggregation was extensively studied, but we lack a complete understanding of the evolution of the cluster-mass distributions. Here again, a model should be developed to account for the aggregation dynamics in the specific case of heavy particles at an interface. The relevance of the Smoluchowski equations for our system needs to be questioned, as well as the existence of a self-similar clustering. Experiments in a much larger domain with many more particles could also be performed, in order to maximize the possibility of reaching a steady-state regime. A similar remark can be done for the numerical simulations: a more efficient code allowing us to perform simulations with many more particles needs to be developed. Once the clustering of many identical particles better understood, one could also explore the aggregation of a large variety of particles, with different sizes and densities. This situation would be closer to what can be found if sand is sprinkled at the surface of water.

The erosion of the granular raft would also require some further investigations. Indeed, we have explained the cohesion between two beads by looking at the position of the contact line, but we still lack a convincing explanation to account for the differences between the experimental depth angle and its theoretical prediction. A detailed visualization of the contact line around a sphere should therefore be undertaken. Moreover, we need to go back to the situation of a whole granular raft, in order to see how this enhanced cohesion is affected by the overall curvature of the interface around a raft. In the present work, the cohesion of a raft was deduced from the measurement of the velocity needed to detach a particle. Another experiment could be developed, in which the raft is kept still, and a particle of its edge is extracted. By measuring the force needed to detach the particle, and reproducing the measurement all around the raft, we could directly deduce an averaged cohesion force of the raft, explore its dependency with the type of particles or the size of the raft. This would also allow us to measure a local cohesion that probably depends on the numbers of neighboring particles and their respective positions, and to identify the weakest points of the structure.

The pinch-off dynamics also bear their batch of unanswered questions. We described an oscillation in log-scale of the slope of the minimum radius, but no theoretical derivation of the period of this oscillation has been conducted. The influence of the boundary conditions on the transition between the two self-similar regimes requires some clarification, as well as the slope of the minimum radius during these transient regimes. More experiments should be conducted with different viscosities, in order to see if the behavior is found with other liquids. A complete numerical study on pinching dynamics should also be undertaken, where all the parameters are varied. This could allow us to better identify the conditions leading to an oscillatory transition. Finally, the thinning of particle-covered filaments has only been briefly investigated, and would benefit from a large study, in comparison with the literature on suspension droplets. A lot of experiments need to be performed in order to explore the link between the positions of the beads and the thinning dynamics of the thread. As briefly presented in this work, for smaller particles, the raft destabilizes into a particle-covered jet, instead of a single droplet pulling a liquid thread. The wavelength of destabilization of this jet could be studied, in comparison to the expected wavelength for pure liquid cylinders.

In the long run, many other studies could be undertaken. In our work, we only focused on the aggregation of passive heavy spherical particles at an interface, and even though it is already quite challenging, it remains restrictive compared to the countless variety of self-assembling phenomena that one can think of. First of all, we could investigate the clustering of heavy anisotropic particles. Similar studies have been conducted for micrometer particles, but a good understanding of the interactions between numerous non-spherical large objects interacting through gravity is still lacking. This could allow us to control more finely the aggregation process and turn it into a directed assembly. Similarly, the clustering of active heavy particles could be a matter of interest: instead of a final state with only one raft, we would end up with a stationary state resulting from the equilibrium between the attractive forces and the escapes of individual particles from the clusters.

We could also try to characterize the behavior of an armored droplet after detachment from the oil layer. The sinking velocity of such an object is far from being obvious. More interestingly perhaps, we could study how this particle-covered droplet behaves in a flow, in order to see how the coverage gives cohesion to the oil droplet. What is the critical stress at which the droplet breaks, compared to a pure oil droplet? How far can we stretch an armored droplet? How does its cohesion depend on the size of the individual particles, or on the coverage percentage?

We could try to optimize the encapsulation process, and determine what are the optimal sizes and densities of the particles to sink the biggest volume of oil, or to sink it as quickly as possible. This is typically a question that could ask an engineer, and its answer is fundamental to determine if this technique is sufficiently efficient to be used in order to encapsulate oil, or to sink a large layer of oil floating on top on water. In previous work, this was indeed suggested as a solution to contain oil spills in the ocean, by sinking the oil spill in multiple droplets before it can reach the coasts.

After the layer of oil is resting at the bottom of the ocean, we can expect marine microorganisms to consume the oil droplets, but here again, many questions can be raised: what would be the speed of this process? Is there an optimal coverage of the droplets in order to facilitate their degradation?

Finally, the aggregation of particles at an interface perturbed by waves could also be studied (such as waves at the surface of the ocean). The influence of the amplitude of the waves or their frequency on the capillary forces between several objects can be explored, as well as the destabilization threshold of a granular raft resting at a perturbed interface or its cohesion before sinking? If the perturbation is too large, we may expect erosion to happen so that the different rafts are never able to reach their critical size of sinking.

CONTENTS 2.5. 3

 3 Going back to the modified forces . . . . . . . . . . . . . . . . . . . . 2.5.4 Transition between the two regimes . . . . . . . . . . . . . . . . . . .

Figure 1 :

 1 Figure 1: Capillarity in life: various examples. a) Destabilization of a liquid jet in multiple droplets because of a Plateau-Rayleigh instability. The photo is taken from [5]. b) Water strider Gerris. Scale bar: 1 mm. The photo is taken from [3]. c) Tears of wine. The photo is taken from the french Wikipedia article "Larmes de vin", and is in the public domain.

Figure 2 :

 2 Figure 2: Normal capillary forces between particles. The liquid is represented in blue, while the color white stands for the surrounding fluid. A capillary bridge connects the two interacting particles, leading to an attractive normal force. These normal forces can lead to the erection of a castle. Image by Stephane Abando from Pixabay.

Figure 5 :

 5 Figure 5: Examples of aggregated systems. a) Ant raft resisting submersion by a twig. Photo taken from [19].b) Aggregated bubbles floating at the surface of water in a petri dish. Photo taken from[START_REF] Vella | The "cheerios effect[END_REF]. c) Cristalline aggregate generated by the assembly of objects of various shapes. Photo taken from[START_REF] Bowden | Self-assembly of mesoscale objects into ordered two-dimensional arrays[END_REF].

Figure 7 :

 7 Figure 7: Effect of size on the shape of a resting droplet. Water droplets on a superhydrophobic substrate.From left to right, the size of the droplets increases. Image taken from[START_REF] Séon | Les Lois d'échelle: La physique du petit et du grand[END_REF], gracefully given by T. Séon.

Figure 8 :

 8 Figure 8: Birth, life and death of a granular raft. Schematic representation of the different steps leading to the sinking of a granular raft. Particles are added progressively at an oil-water interface from every direction around the raft to form an axisymmetric monolayer, until a critical size is reached and the raft begins to sink. A viscous thread is formed and shrinks until pinch-off, giving birth to an oil droplet encapsulated inside a shell of particles. This droplet finally reaches the bottom of the water tank, where it remains stable. Inset: Experimental visualizations of the different steps represented in the main figure. a) Side view of a granular raft made of 60 ceramic particles (density ρpart = 4,800 kg.m -3 , radius Rpart = 0.35 mm). Scale bar: 3 mm. b) Sinking of a granular raft made of ceramic beads (density ρpart = 3,800 kg.m -3 , radius Rpart = 0.45 mm). The photo is taken 40 ms before pinching of the oil filament. Scale bar: 3 mm. c) Encapsulated oil droplet resulting from the destabilization of a granular raft made of glass beads (density ρpart = 2,500 kg.m -3 , radius Rpart = 0.45 mm). Scale bar: 5 mm. d) Encapsulated droplets at the bottom of the water tank, made of ceramic beads (density ρpart = 3,800 kg.m -3 , radius Rpart = 0.45 mm). Scale bar: 2 cm.

Contents 1 . 1

 11 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Granular rafts dynamics . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 The experiment: first results . . . . . . . . . . . . . . . . . . . . 1.2.2 Dynamics of two particles . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Kinematics of two granular rafts . . . . . . . . . . . . . . . . . . 1.3 Granular raft morphology . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Modified aspect ratio of a raft . . . . . . . . . . . . . . . . . . . . 1.3.2 Morphology of a raft: experiments . . . . . . . . . . . . . . . . . 1.3.3 Morphology of a raft: numerics . . . . . . . . . . . . . . . . . . . 1.3.4 Morphology of a raft: theory . . . . . . . . . . . . . . . . . . . . 1.3.5 The drag force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.6 Limitations of the model . . . . . . . . . . . . . . . . . . . . . . . 1.4 Motion of a raft along a curved interface . . . . . . . . . . . . . 1.4.1 Deflecting the interface . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1: Aggregation of two granular rafts. a) Time lapse of the motion of two granular rafts at an oil-water interface, filmed from the side. b) Top view of the same aggregation process. Each raft is made of ceramic spherical beads (density ρpart = 3,800 kg.m -3 , radius Rpart = 0.45 mm), in an axisymmetric configuration, with 50 particles for the raft on the left (raft A), and 30 particles for the one on the right (raft B). The final image corresponds to the exact moment of contact between the two rafts. Time between two images: 0.13 s. Scale bars: 5 mm.

Figure 1 . 2 :

 12 Figure 1.2: Schematic side view of a granular raft. Deformation of the interface around a single raft B, described by two quantities: the radius of the raft R raf t B , and the depth of the interface at the edge of the raft h raf t B . The radial coordinates are defined from the center of the raft, while the origin of the vertical axis h corresponds to the height of the undisturbed flat interface.

Figure 1 . 3 :

 13 Figure1.3: Kinematics of two interacting granular rafts, compared with two interacting particles. a) Distance between the centers of the two rafts L (made dimensionless using the radius of a bead Rpart), as defined in figure1.1, as a function of time before contact. Each raft is made of ceramic spherical beads (density ρpart = 3,800 kg.m -3 , radius Rpart = 0.45 mm). In blue, the two rafts are made of one particle each, while in red and black, one is made of 30 particles, the other of 50 (see figure1.1 for the experimental visualization). b) Velocity of the two rafts in the same two experiments, as a function of the dimensionless distance between the centers of the two rafts. Red circles: velocity of the 30-particle raft; black squares: velocity of the 50-particle raft, blue squares : velocity of a single particle attracted by another isolated particle of same size. For both figures, the error bars are of the order of the thickness of the curves, and as a consequence are not represented.

Figure 1 . 4 :

 14 Figure 1.4: Theoretical model for the calculation of the capillary force between two particles a) Geometrical parameters of a sphere of radius Rpart floating at a fluid-fluid interface. ψs is the slope of the interface at the contact line, θ the three-phase contact angle, ϕs the position angle of the contact line with respect to its south pole, and ξs the vertical position of the contact line with respect to the undisturbed interface at infinity. b) Schematic representation of the generalized Archimedes' principle, taking into account the geometry of the interface around the sphere.

Figure 1 . 6 :

 16 Figure1.6: Kinematics of two interacting particles. Experimental aggregation of two particles (plastic spherical beads, density ρpart = 1,420 kg.m -3 , radius Rpart = 2.5 mm) in red circles, compared to the theoretical prediction given by equation(1.16) in black dotted line, with only one fitting parameter k accounting for the fact that the particles are only partially immersed in oil. a) Distance between the centers of the two particles L (made dimensionless using the radius of a bead Rpart), as a function of time before contact. b) Velocity of the two same particles, as a function of the dimensionless distance between the centers of the two particles. The error bars are of the order of the thickness of the curves, and as a consequence are not represented.

Figure 1 . 7 :

 17 Figure1.7: Forces acting on a given axisymmetric granular raft. Schematic representation of the interaction between two granular rafts A and B (n A and n B stand for the number of particles in each raft), at an oil-water interface. L is the distance between the centers of the two rafts, while l is the distance between the two nearest particles of each raft. The raft A (respectively B) is subjected to a capillary attraction F cap B→A and a viscous drag F drag A (respectively F cap A→B and F drag B ).

Figure 1 . 8 :

 18 Figure 1.8: Agreement between experimental results and theoretical model. Speed of the raft A for different numbers of particles in A and B, as a function of the distance between the centers of the two rafts (made dimensionless using the radius of a bead Rpart), fitted by the theoretical prediction of equation (1.19) (black dotted curves). For all these experiments, the particles are ceramic spherical beads (density ρpart = 3,800 kg.m -3 , radius Rpart = 0.45 mm). a) Speed of a single particle A for different sizes of the other raft, n B = 1, 5, 10, 30, 60. b) Speed of a raft of increasing size (n A = 1, 5, 10, 20, 40), attracted by a raft of fixed size (n B = 60). The error bars are of the order of the thickness of the curves.

Figure 1 . 9 :

 19 Figure 1.9: Forces acting on a granular raft. Fitting coefficient of the speed of the granular raft A, as a function of n B √ n A , for different densities and sizes of particles (green right-pointing triangles: plastic beads, ρpart = 1,420 kg.m -3 , Rpart = 2.5 mm; blue left-pointing triangles: plastic beads, ρpart = 1,420 kg.m -3 , Rpart = 2 mm; black squares: ceramic beads, ρpart = 4,800 kg.m -3 , Rpart = 0.45 mm; pink stars: ceramic beads, ρpart = 4,800 kg.m -3 , Rpart = 0.35 mm; blue circles: ceramic beads, ρpart = 3,800 kg.m -3 , Rpart = 0.45 mm; red upward-pointing triangles: ceramic beads, ρpart = 3,800 kg.m -3 , Rpart = 0.35 mm). The solid line has a slope of 1.

Figure 1 . 10 :

 110 Figure 1.10: Morphology and weight of a granular raft: experiments. Experimental relation between the deformation of the interface and the weight of the granular raft, for a broad variety of densities, radii and numbers of particles. a) Side view of a granular raft for an increasing number of ceramic particles (density ρpart = 4,800 kg.m -3 , radius Rpart = 0.35 mm). The blue arrows indicate the measurement of R raf t and h raf t . Scale bars: 3 mm in each photo. b) Modified aspect ratio h raf t / c K 0 (R raf t / c ) as a function of the effective weight of the raft Fvert for experiments (green right-pointing triangles: plastic beads, ρpart = 1,420 kg.m -3 , Rpart = 2.5 mm; blue left-pointing triangles: plastic beads, ρpart = 1,420 kg.m -3 , Rpart = 2 mm; black squares: ceramic beads, ρpart = 4,800 kg.m -3 , Rpart = 0.45 mm; pink stars: ceramic beads, ρpart = 4,800 kg.m -3 , Rpart = 0.35 mm; blue circles: ceramic beads, ρpart = 3,800 kg.m -3 , Rpart = 0.45 mm; red upward-pointing triangles: ceramic beads, ρpart = 3,800 kg.m -3 , Rpart = 0.35 mm). The solid line has a slope of 1. The vertical error bars are of the order of the size of the individual points, and therefore are not displayed for clarity reasons.

Figure 1 . 11 :

 111 Figure 1.11: Modelization of a granular raft as an axisymmetric heavy membrane. a) The granular raft is considered as a membrane of thickness b, in an axisymmetric coordinate system. t is the tangential vector, n the normal one, and s the curvilinear abscissa. b) Elementary volume considered for the local balance of the tension forces. c) Schematic representation of a section of a raft at a given azimuthal angle, for the calculation of the hydrostatic pressure.

Figure 1 . 12 :

 112 Figure 1.12: Morphology and weight of a granular raft: experiments and simulations. Relation between the deformation of the interface and the weight of the granular raft, for a broad variety of densities, radii and numbers of particles. a) Side view of a granular raft for an increasing number of ceramic particles (density ρpart = 4,800 kg.m -3 , radius Rpart = 0.35 mm). In the top three photos, the red line represents the shape of the interface calculated by the numerical simulation for a membrane of a radius similar to the background photo (dotted line: oil-water interface, solid line: granular raft). Scale bars: 3 mm in each photo. b) Modified aspect ratio h raf t / c K 0 (R raf t / c ) as a function of the effective weight of the raft Fvert, for experiments (black diamonds), and numerical simulations (colored circles). The solid line has a slope of 1. The vertical error bars for the experiments are of the order of the size of the individual points, and therefore are not displayed for clarity reasons. The colors for the numerical simulations correspond to different sets of parameters.

  .8.b for n A = 20 and 40).

Figure 1 . 13 :

 113 Figure 1.13: Radius of a granular raft. Radius of a granular raft, divided by the radius of an individual particle, as a function of the number of particles inside the raft, for different types of beads (green rightpointing triangles: plastic beads, ρpart = 1,420 kg.m -3 , Rpart = 2.5 mm; blue left-pointing triangles: plastic beads, ρpart = 1,420 kg.m -3 , Rpart = 2 mm; black squares: ceramic beads, ρpart = 4,800 kg.m -3 , Rpart = 0.45 mm; pink stars: ceramic beads, ρpart = 4,800 kg.m -3 , Rpart = 0.35 mm; blue circles: ceramic beads, ρpart = 3,800 kg.m -3 , Rpart = 0.45 mm; red upward-pointing triangles: ceramic beads, ρpart = 3,800 kg.m -3 , Rpart = 0.35 mm). The solid line has a slope of 1.

Figure 1 . 14 :

 114 Figure 1.14: Example of the failure of the linear superposition assumption. Comparison between a side view of two granular rafts during their motion towards one another (background photo) and the expected interfacial shape obtained if the assumption of small deformations was valid (blue dotted curve). Each raft is made of 50 ceramic spherical beads (density ρpart = 3,800 kg.m -3 , radius Rpart = 0.45 mm), in an axisymmetric configuration. The photo is taken 0.1 s before contact. Scale bar: 5 mm.

Figure 1 . 15 :

 115 Figure 1.15: Motion of an isolated particle along a curved interface. A ceramic particle (density ρpart = 3,800 kg.m -3 , radius Rpart = 0.45 mm) is attracted by a cylinder that deforms the interface (radius R cyl = 7.5 mm, depth H cyl = 7.5 mm). The experiment is visualized from above (a) or from the side (b).For both images, we superimpose the position of the particle along its motion, with a time step of 0.1 s between two successive positions. The scale bar is given by the size of the cylinder. In b), we plot on top of the experiment visualization the computed interface, either calculated from equation (1.42) (blue dotted curve) or from a linearized version of the same equation (red dotted curve). c) Velocity of the particle, as a function of its distance to the center of the cylinder. In red, we plot the horizontal component of the velocity, directly measured on a top view, while in black and green, we represent the total velocity of the particle, either measured on the side view (green) or computed thanks to the equation of the interface (1.42) and the horizontal position of the particle (black).

Figure 1 . 16 :

 116 Figure1.16: Motion of a single particle along a curved interface, for different deformations of the interface. Velocity of a ceramic particle (density ρpart = 3,800 kg.m -3 , radius Rpart = 0.45 mm) attracted by a cylinder of various radii and depths, as a function of the horizontal distance between the particle and the center of the cylinder. In a) and b), the radius of the cylinder is kept constant (2R cyl = 3 mm in a, 2R cyl = 5 mm in b) while the depth of the cylinder increases from one curve to another (dark blue: H cyl = 0.4 mm, blue: H cyl = 1.2 mm, light blue: H cyl = 2 mm, green: H cyl = 2.8 mm, orange: H cyl = 3.6 mm, brown: H cyl = 5.2 mm). In c) and d), the depth of the cylinder is fixed (H cyl = 1.2 mm in c, H cyl = 3.6 mm in d) while the radius of the cylinder increases (black: 2R cyl = 3 mm, red: 2R cyl = 5 mm, blue: 2R cyl = 8 mm). The error bars are of the order of the size of the individual points, and are not displayed for clarity reasons.

Figure 1 . 17 :

 117 Figure 1.17: Forces acting on a granular raft. Fitting coefficient of the speed of a single ceramic particle (density ρpart = 3,800 kg.m -3 , radius Rpart = 0.45 mm) attracted by a cylinder deflecting the interface, as defined in equation (1.43), for different radii of cylinder. a) Fitting coefficient as a function of the depth of the cylinder H cyl . b) Fitting coefficient, as a function of the modified aspect ratio

Figure 1 . 18 :

 118 Figure 1.18: Motion of a granular raft along a curved interface. a) Time lapse of the motion of a granular raft at a curved oil-water interface, filmed from the side. b) Top view of the same motion. The raft is made of 10 ceramic spherical beads (density ρpart = 3,800 kg.m -3 , radius Rpart = 0.45 mm), and is attracted by a cylinder with a radius 2R cyl = 5 mm, placed at a depth H cyl = 2.1 mm. The final image corresponds to the exact moment of contact between the raft and the cylinder. Time between two images: 0.4 s. Scale bar: 5 mm.

Figure 1 . 19 :

 119 Figure 1.19: Motion of a granular raft along a curved interface, for different number of particles inside the raft. Velocity of a granular raft made of ceramic particles (density ρpart = 3,800 kg.m -3 , radius Rpart = 0.45 mm) attracted by a cylinder with a radius 2R cyl = 5 mm, placed at a depth H cyl = 3.6 mm, as a function of the horizontal distance between the center of the raft and the center of the cylinder L/R raf t . a) Velocity of the raft for various numbers of particles (red: n = 1, black: n = 5, dark blue: n = 30, light blue: n = 50.) b) and c) Velocity of a single particle (red curve in b) and of a granular raft made of 50 particles (blue curve in c), compared to the theoretical prediction of equation (1.43) (dotted line).

Figure 1 . 20 :

 120 Figure 1.20: Motion of a granular raft subjected to different attractive systems. Velocity of a granular raft made of 30 ceramic particles (density ρpart = 3,800 kg.m -3, radius Rpart = 0.45 mm), either attracted by a cylinder with a radius R cyl = 2.5 mm at a depth H cyl = 1.9. mm (red circles), or attracted by another granular raft (blue squares) with parameters similar to the cylinder (R raf t = 2.7 mm, h raf t = 2.1. mm).

Contents 2 . 1

 21 Collective motion and clustering dynamics . . . . . . . . . . . . 47 2.2 Experimental aggregation . . . . . . . . . . . . . . . . . . . . . . 50 2.2.1 Setting up the experiment . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Preliminary measurements . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Distribution and probabilities . . . . . . . . . . . . . . . . . . . .2.3 Numerical aggregation . . . . . . . . . . . . . . . . . . . . . . . . 59 2.3.1 Setting up the numerics . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Preliminary results: influence of the surface density . . . . . . . 2.3.3 Characterizing the numerical aggregation . . . . . . . . . . . . . 2.3.4 Changing the forces . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Smoluchowski coagulation equation . . . . . . . . . . . . . . . . 70 2.4.1 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Comparison with other aggregating systems . . . . . . . . . . . . 2.4.4 Moment of order 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 A thorough analysis of the distributions . . . . . . . . . . . . . . 80 2.5.1 Time dependency of the distributions . . . . . . . . . . . . . . . 2.5.2 Maximum number of rafts and surface density . . . . . . . . . . 2.5.3 Going back to the modified forces . . . . . . . . . . . . . . . . . 2.5.4 Transition between the two regimes . . . . . . . . . . . . . . . .

Figure 2 . 1 :

 21 Figure 2.1: Clustering dynamics: a few examples. a) Artistic view from the Asteroid Belt, made for the Nasa mission Dawn. Image credit: NASA/UCLA/William K. Hartmann. b) High velocity liquid sheet destabilizing into multiple droplets. Photo taken from [71].

Figure 2

 2 Figure 2.2: Examples of collective motion in various systems. a) Active nematic phase of vibrated copperwire segments. Photo taken from [81]. b) Steady-state of the clustering process inside a solution of bacteria M. xanthus. Photo taken from [83].

Figure 2 . 3 :

 23 Figure 2.3: Structures of clusters of colloidal particles. Clusters of gold particles of radius 7.5 nm, for two different regimes of aggregation, visualized by transmission electron microscopy. a) Diffusion-limited aggregation. b) Reaction-limited aggregation. The two photos are taken from[START_REF] My Lin | Universality in colloid aggregation[END_REF], where more details on the experiment are available.

Figure 2 . 4 :

 24 figure 2.4.a. Using this procedure, particles are confined inside a circular domain of radius R domain . We try to make sure that the initial distribution of particles at the interface is rather homogeneous. Four experimental visualizations are presented in figure 2.4, for the same set of parameters.

Figure 2

 2 Figure 2.5: Aggregation process: the visualization. Time-lapse of the aggregation of 260 particles distributed inside a domain of radius R domain = 5.15 cm. The first image corresponds to the initial configuration. Time between two images: 40 min. Scale bar: 5 cm.

Figure 2 . 6 :

 26 Figure 2.6: Validation of the measurement of the number of particles. Measurement of the total number of particles during the aggregation of 33 mg of particles inside a circular domain of radius R domain = 3.25 cm, for five different experiments with the same input parameters.

Figure 2 . 7 :

 27 Figure 2.7: Reproducibility of the experiments. a) Size of the biggest cluster as a function of time. b) Number of clusters as a function of time. In both graphs, the quantities are divided by the total number of particles. The five colored curves represent five successive experiments using the same set of parameters ntot ≈ 968, R domain = 3.25. The black dotted line is the mean value of the five colored curves.

Figure 2 . 8 :

 28 Figure 2.8: Influence of the number of particles. Influence of ntot on the aggregation dynamics, for a constant domain of radius R domain = 3.25 cm. Two quantities are plotted: the size of the biggest cluster in a, and the number of clusters in b.

Figure 2 . 9 :

 29 Figure 2.9: Influence of the size of the domain. Influence of R domain on the aggregation dynamics, for a constant number of particles ntot ≈ 270. Two quantities are plotted: the size of the biggest cluster in a, and the number of clusters in b.

Figure 2 . 10 :

 210 Figure 2.10: Dynamics of clustering and surface density. Evolution of the size of the biggest cluster in a and the number of clusters in b, as a function of time, for different surface densities of particles in the initial configuration. The color bar displays the surface density φ surf of each experiment. To distinguish the different curves even for two almost identical φ surf , the scale of the color bar is not linear. Inset: empirical rescaling of nmax and n clusters . We multiply the abscissa t by the surface density φ surf . Yellow: R domain = 3.25 cm, ntot = 968; orange: R domain = 2 cm, ntot = 286; green: R domain = 5.15 cm, ntot = 915; turquoise: R domain = 2 cm, ntot = 123; light blue: R domain = 3.25 cm, ntot = 248; blue: R domain = 3.25 cm, ntot = 117; dark blue: R domain = 5.15 cm, ntot = 278.

Figure 2 . 11 :

 211 Figure 2.11: Distribution of sizes during aggregation. Distribution of sizes for a given set of parameters (R domain = 3.25 cm, ntot = 968). The abscissa is the number of particles inside a cluster, while the ordinate is the number of clusters of the corresponding size, normalized by the total number of particles. The four different colors stand for four different times after the beginning of the experiment, with 200 s of difference between each curve: dark blue: t = 20 s, light blue: t = 220 s, green: t = 420 s, orange: t = 620 s. The horizontal dotted line corresponds to the lowest possible value of N (n), inversely proportional to the number of experiments performed.

Figure 2 . 12 :

 212 Figure 2.12: Probability density functions. Two different kinds of probabilities for a given set of parameters (R domain = 3.25 cm, ntot = 968). The four different colors stand for four different times after the beginning of the experiment, with 200 s of difference between each curve: dark blue: t = 20 s, light blue: t = 220 s, green: t = 420 s, orange: t = 620 s. a) Probability Ppart(n) that a given particle belongs to a raft of size n. The horizontal dotted line corresponds to the lowest possible value of Ppart(n), inversely proportional to the number of experiments performed. b) Probability P raf t (n) that a given raft has a size n.

Figure 2 . 13 :

 213 Figure 2.13: Time step in the simulation. Comparison between the results of several simulations, for exactly the same initial distribution of particles, but different time steps (red curve: ∆t = 20 s, black curve: ∆t = 30 s, blue curve: ∆t = 50 s, green curve: adaptive time step). a) Size of the biggest cluster as a function of time. b) Number of clusters as a function of time.

Figure 2 . 14 :

 214 Figure 2.14: Aggregation process: the visualization. Time-lapse of the numerically simulated aggregation of 200 particles distributed inside a domain of radius R domain = 5.8 cm. The first image corresponds to the initial configuration. Time between two images: 50 min. Scale bar: 5 cm.

Figure 2 . 15 :

 215 Figure 2.15: Experiments and simulations. Comparison between experimental (red circles) and numerical (black squares) results, for similar initial surface density (φ surf ≈ 7.5 cm -2 in both cases). a) Size of the biggest cluster as a function of time. t 60 is the time at which nmax = 0.6ntot. b) Number of clusters as a function of time. c, d and e) Comparison of the distributions at three different times, located on the top two figures by dotted lines.

  Figure 2.16: Dynamics of clustering and surface density in simulations. Evolution of the size of the biggest cluster in a and the number of clusters in b, as a function of time, for different surface densities of particles in the initial configuration. The color bar displays the surface density φ surf of each simulation. To distinguish the different curves even for two almost identical φ surf , the scale of the color bar is not linear. Yellow: R domain = 2.3 cm, ntot = 400; orange: R domain = 2.9 cm, ntot = 300; green: R domain = 3.5 cm, ntot = 400; light blue: R domain = 2.9 cm, ntot = 200; blue: R domain = 5.8 cm, ntot = 400; dark blue: R domain = 5.8 cm, ntot = 200.

Figure 2 . 17 :

 217 Figure2.17: PDF of the aggregation for a numerical simulation. Evolution of the distributions of sizes and probabilities of existence, at seven successive times after the beginning of the clustering, up to t 60 = 190 min. The data come from the averaging of 980 simulations with 200 particles inside a domain of radius R domain = 5.8 cm. b) Probability Ppart(n) that a given particle belongs to a raft of size n. c) Probability P raf t (n) that a given raft has a size n.

Figure 2 . 18 :

 218 Figure 2.18: Statistical quantities for the characterization of the aggregation process. The data come from the averaging of 980 simulations with 200 particles inside a domain or radius R domain = 5.8 cm. a) Probability P (X ≥ n) for a particle to be in a raft of size more than or equal to n. b) Mean size of the clusters < n > (red circles) and standard deviation σ of the distribution (black circles), as a function of time.

Figure 2 . 19 :

 219 Figure 2.19: Comparison between two types of simulations. a) Time-lapse of the numerically simulated aggregation of 200 particles distributed inside a domain of radius R domain = 5.8 cm, for forces independent of the cluster size. The first image corresponds to the initial configuration. Time between two images: 330 min. Scale bar: 5 cm. band c) Evolution of the size of the biggest cluster and the number of clusters, as a function of time, for two types of simulations: in red, the simulations are performed with a non-physical interaction between the rafts (see equation (2.12)), while in black, we use the real interaction of equation (2.9). For both curves, R domain = 5.8 cm and ntot = 200.

Figure 2 . 20 :

 220 Figure 2.20: Dynamics of clustering and surface density in simulations with a modified interaction. Evolution of the size of the biggest cluster in a and the number of clusters in b, as a function of time, for different surface densities of particles in the initial configuration. The simulations are performed with a non-physical interaction between the rafts (see equation (2.12)). The color bar displays the surface density φ surf of each simulation. To distinguish the different curves even for two almost identical φ surf , the scale of the color bar is not linear. Yellow: R domain = 3.5 cm, ntot = 1000; orange: R domain = 2.3 cm, ntot = 400; green: R domain = 3.5 cm, ntot = 400; light blue: R domain = 5.8 cm, ntot = 1000; blue: R domain = 3.5 cm, ntot = 200; dark blue: R domain = 5.8 cm, ntot = 200.

Figure 2 . 21 :

 221 Figure2.21: PDF of the aggregation for a numerical simulation with a modified interaction. Evolution of the distributions of sizes and probabilities of existence, at eight successive times after the beginning of the clustering, for a non-physical interaction between the rafts (see equation (2.12)). The data come from the averaging of 832 simulations with 200 particles inside a domain or radius R domain = 5.8 cm. a) Distribution of sizes N (n): number of rafts of size n. b) Probability Ppart(n) that a given particle belongs to a raft of size n. c) Probability P raf t (n) that a given raft has a size n.

Figure 2 . 22 :

 222 Figure 2.22: Rescaling of the distributions of sizes. Distribution of sizes rescaled according to equation (2.[START_REF] Nathan J Mlot | Fire ants self-assemble into waterproof rafts to survive floods[END_REF], for an experiment in a (ntot = 915, R domain = 5.15 cm), and a simulation with physical capillary forces in b (ntot = 300, R domain = 2.9 cm). The input parameters between the experiments are different, but the surface density is similar. The color bars represent the time. The points encircled in the dotted circle correspond to statistically non-relevant data, as explained section 2.2.3. In c, we superimpose the two curves of a and b, in red for the experiment and black for the simulation. In the three figures, the solid lines have a slope -2, and are only a guideline to the eye.

Figure 2 . 23 :

 223 Figure 2.23: Rescaling of the distributions and surface density. Distribution of sizes rescaled according to equation (2.19), for all the experiments in a and all the simulations in b. The color bar displays the surface density φ surf for each set of data. To distinguish the different curves even for two almost identical φ surf , the scale of the color bar is not linear.

Figure 2 . 24 :

 224 Figure2.24: Rescaling of the distributions, for modified forces where the capillary force does not depend on the number of particles. Distribution of sizes rescaled according to equation (2.17), for a simulation with a modified interaction between the rafts (see equation (2.12)). The data come from the averaging of 830 simulations with 200 particles inside a domain of radius R domain = 5.8 cm. The color bar represents the time.

Figure 2 . 25 :

 225 Figure 2.25: Rescaling of the distributions for DLA and RLA. Cluster-mass distribution during the aggregation of nanometer-sized gold particles, for diffusion-limited aggregation in a, and reaction-limited aggregation in b, at different times after the beginning of the aggregation (a: squares t = 1 min, plusses t = 10 min, asterisks t = 30 min; b: squares t = 2 h, plusses t = 6 h, asterisks t = 8 h).The main figures represent the raw distributions, while in the inset, the data have been rescaled according to equation (2.17). The figures are taken from[START_REF] Da Weitz | Dynamic scaling of cluster-mass distributions in kinetic colloid aggregation[END_REF], where more details and results can be found.

Figure 2 . 26 :

 226 Figure 2.26: Function P 2 . Evolution of the function P 2 , defined as the ratio between two successive moments of the ditribution (see equation (2.28)), for all the experiments in a and all the simulations in b. The color bar displays the surface density φ surf for each set of data. To distinguish the different curves even for two almost identical φ surf , the scale of the color bar is not linear.

2.5. 1

 1 Time dependency of the distributions Distributions as a function of time Up to now, we have looked at the distributions of sizes N for specific times, as a function of n. Yet, since it is a function of two variables, nothing prevents us from studying N as a function of time, for given values of n. Such a representation is displayed in figure 2.27.

Figure 2

 2 Figure 2.27: PDF of the aggregation for a numerical simulation. Evolution of the distributions of sizes as a function of time, for different sizes of cluster n identified by different colors. The data come from the averaging of 980 simulations with 200 particles inside a domain of radius R domain = 5.8 cm. The color bar has a logarithmic scale. For clarity reasons, not all the different sizes are plotted.

Figure 2 . 28 :

 228 Figure 2.28: Position of the maxima of the distributions. a) Maximal value of N (n, t) as a function of time, for different sizes of cluster n identified by different colors. The color bar has a linear scale. b) Times tmax at which N (n, t) is maximal, as a function of n. The color bar represents the corresponding values Nmax, in a logarithmic scale. c) Maximal value of N (n, t) as a function of n. The color bar represents the corresponding values tmax, in a logarithmic scale. The solid line has a slope -2, and is only a guideline to the eye. The data come from the averaging of 980 simulations with 200 particles inside a domain or radius R domain = 5.8 cm.

Figure 2 . 29 :

 229 Figure 2.29: Position of the maxima of the distributions, in experiments and simulations. Maximal value of N (n, t) as a function of n for several experiments (colored circles) and one simulation (black squares: ntot = 200, R domain = 5.8 cm, φ surf = 1.9 cm -2). The color bar displays the surface density φ surf of each experiment. To distinguish the different curves even for two almost identical φ surf , the scale of the color bar is not linear.

Figure 2 . 30 :

 230 Figure 2.30: Position of the maxima of the distributions. Maxima of the distributions for different simulations with different surface densities. Yellow: R domain = 2.9 cm, ntot = 300; green: R domain = 2.9 cm, ntot = 200; blue: R domain = 5.8 cm, ntot = 400; dark blue: R domain = 5.8 cm, ntot = 200. a) Maximal value of N (n, t) as a function of time, for different sizes of cluster n. b) Times tmax at which N (n, t) is maximal, as a function of n. c) Maximal value of N (n, t) as a function of n. The solid line has a slope -2, and is only a guideline to the eye.

  Figure2.31: Comparison between two types of simulations. Maxima of the distributions as a function of n, for two types of simulations: in red, the simulations are performed with a non-physical interaction between the rafts (see equation (2.12)), while in black, we use the real interaction of equation (2.9). For both curves, R domain = 5.8 cm, ntot = 200. The solid line has a slope -2, and is only a guideline to the eye.

Figure 2 . 32 :

 232 Figure 2.32: Rescaling and maxima of the distributions Distribution of sizes rescaled according to equation (2.19) for a simulation with physical capillary forces (ntot = 200, R domain = 5.8 cm). In black dots, we plot the rescaled positions of the maxima of the distributions with respect to time, for each value of n. The color bar represents the time.

Figure 2 . 33 :

 233 Figure 2.33: Moment of order 2 and maxima of the distributions. Comparison between the time evolution of the function P 2 in black, as defined in equation (2.28), and the position of the maxima n(tmax) in red, for a simulation with real capillary forces (ntot = 200, R domain = 5.8 cm).

Figure 2 . 34 :

 234 Figure 2.34: Trajectories of particles during aggregation. Superimposition of the successive positions of three large ceramic beads (density ρpart = 3,800 kg.m -3 , radius Rpart = 0.45 mm), for two different initial configurations. Time between two successive positions: 3 s. Scale bar: 5 mm.

Figure 2 . 35 :

 235 Figure 2.35: Trajectories of particles during aggregation. Superimposition of the successive positions of eighteen large ceramic beads (density ρpart = 3,800 kg.m -3 , radius Rpart = 0.45 mm) initially placed along the sides of an equilateral triangle, with an initial spacing of 1 cm between the particles. Scale bar: 3 cm.

Contents 3 . 1

 31 Structural changes of a monolayer . . . . . . . . . . . . . . . . . 89 3.2 Erosion of a granular raft . . . . . . . . . . . . . . . . . . . . . . 92 3.2.1 A typical experiment . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.2.2 First results: existence of erosion . . . . . . . . . . . . . . . . . . 93 3.2.3 Experimental cohesive forces of a raft . . . . . . . . . . . . . . . 95 3.3 Cohesion between two aggregated particles . . . . . . . . . . . . 100 3.3.1 In search of an experimental setup . . . . . . . . . . . . . . . . . 101 3.3.2 Experimental result for the cohesion between two beads . . . . . 103 3.3.3 Calculation of the capillary forces . . . . . . . . . . . . . . . . . . 105 3.3.4 Vertical position of the contact line . . . . . . . . . . . . . . . . . 108 3.3.5 Cohesion between two beads: the explanation . . . . . . . . . . . 110

Figure 3 . 1 :

 31 Figure 3.1: Rupture of a 3D flock. a and b) Breakup of a cluster in simple shear. The cluster is initially spherical. After a certain time, particles detach until they fill the entire streamline region. Image taken from [111]. c, d and e) Rupture of a flock by pulling back the pipette on the right. Scale bars: 10 µm. Image taken from [112].

Figure 3 . 2 :

 32 Figure 3.2: Pickering emulsions. a) Water droplet coated with 3.2 µm diameter latex particles, in a cyclohexane solution. Scale bar: 50 µm. b) Zoom in on the surface of the droplet, to show the geometrical arrangement of the particles. Scale bar: 15 µm. The two images are taken from [119].

Figure 3 . 3 :

 33 Figure 3.3: Examples of the structural changes a monolayer of particle at an interface can undergo under flow. a) Reorganization of the internal structure of the raft. b) Propagation of a crack inside the monolayer. c) Erosion: the raft looses a few particles. d) Fragmentation: the raft breaks into several pieces of comparable sizes.

Figure 3 . 4 :

 34 Figure 3.4: Erosion of a raft: experimental setup and visualization of a typical experiment. a) Schematic representation of the experimental configuration.A cylinder of radius R cyl deflects downwards an oil-water interface at a depth H cyl , the interface being pinned at the perimeter of the cylinder. A granular raft of radius R raf t is formed far away from the cylinder, and is attracted by the cylinder through the deflection of the interface.

Figure 3 . 5 :

 35 Figure 3.5: Visualization of a typical experiment. Time-lapse of a granular raft experiencing erosion during its motion along a deformed oil-water interface. The raft is made of approximately 375 ceramic beads (ρpart = 3, 800 kg.m -3 , Rpart = 0.125 mm), and is moving towards a cylinder of radius R cyl = 15 mm at a depth H cyl = 10 mm. The movie is taken from above. Time between two images: 0.15 s. Scale bar: 10 mm. Inset: Speed of the granular raft, as a function of its distance from the center of the cylinder L, non-dimensionalized by the radius of a particle. Red curve: horizontal component of the speed. Black curve: Total speed of the raft. The arrow indicates the moment of the first detachment of a particle from the raft.

Figure 3 . 6 :

 36 Figure 3.6: Non-reproducibility of the erosion experiment. Top view of two successive experiments with the same parameters: same position of the cylinder (R cyl = 15 mm, H cyl = 8 mm), same ceramic beads (ρpart = 3, 800 kg.m -3 , Rpart = 0.125 mm), same number of particles in each raft. The two photos are taken 0.3 s before contact with the cylinder. In a, not a single particle detaches, while in b, erosion can be observed. Scale bar: 10 mm.

Figure 3 . 7 :

 37 Figure 3.7: Phase diagram for the existence of erosion.Presence or absence of erosion for the experiment described in figure3.4, depending on both the depth of the cylinder (R cyl is fixed at 15 mm) and the number of particles in each raft. Each point is the result of two or three experiments (if no motion of the particles is observed during the first two experiments, we do not repeat the experiment a third time). Red circles: erosion is never observed, green triangles: erosion is always observed, blue squares: erosion is sometimes observed (see figure3.6 for an explanation on the non-reproducibility of the results). Between a and b, the properties of the ceramic particles change: a: { ρpart = 3, 800 kg.m -3 , Rpart = 0.125 mm }, b: { ρpart = 6, 000 kg.m -3 , Rpart = 0.1 mm }.

Figure 3 . 8 :

 38 Figure 3.8: Critical velocity of erosion. Velocity of the raft at the moment of detachment of the first particle, as a function of the radius of the raft. The colors stand for the depth of the cylinder in the corresponding experiment (purple: H cyl = 6 mm, red: H cyl = 7 mm, green: H cyl = 8 mm, light blue: H cyl = 9 mm, dark blue: H cyl = 10 mm, black: H cyl = 11 mm). Between a and b, the properties of the ceramic particles change: a: { ρpart = 3, 800 kg.m -3 , Rpart = 0.125 mm }, b: { ρpart = 6, 000 kg.m -3 , Rpart = 0.1 mm }.

Figure 3 . 10 :

 310 Figure 3.10: Inertial force. Comparison between the Stokes drag, calculated according to equation (3.2), and the inertial forces linked to the transient nature of the motion of the granular raft. The experimental data considered here are the one of figure 3.5: the raft is made of approximately 375 ceramic beads (ρpart = 3, 800 kg.m -3 , Rpart = 0.125 mm), and is moving towards a cylinder of radius R cyl = 15 mm at a depth H cyl = 10 mm.

Figure 3 . 11 :

 311 Figure 3.11: Length of a granular raft. Arclength of a granular raft during its motion towards a cylinder of radius R cyl = 15 mm, as a function of its distance from the center of the cylinder L, non-dimensionalized by the radius of a particle, for two different experiments. In both cases, the granular raft is approximately constituted by 1550 ceramic beads (ρpart = 3, 800 kg.m -3 , Rpart = 0.125 mm). The depth of the cylinder changes between the two curves. Red circles: H cyl = 6 mm. Blue circles: H cyl = 10 mm.

Figure 3 . 12 :

 312 Figure 3.12: Cohesion force of a raft. Minimal cohesion force of a raft (circles), compared to the theoretical capillary force a raft should exert on a particle at its edge, as defined in equation (3.1) and (3.5) (stars), as a function of the radius of the raft, for two different types of ceramic beads (black: ρpart = 3, 800 kg.m -3 , Rpart = 0.125 mm, red: ρpart = 6, 000 kg.m -3 , Rpart = 0.1 mm). The circles are obtained by taking for each raft radius the minimal value of V crit in figure 3.8.

Figure 3 . 13 :

 313 Figure 3.13: Vertical position of a pair of particles. Method used to position the wire-connected bead at the correct depth. In a first experiment, a photo of a pair of freely floating particles is taken from the side (a), the depth of the particles h 2part is measured, and then after having removed the two particles, a bead glued to a wire is placed at the same depth h 2part . Finally, a new bead is attracted by the latest, creating a two-bead aggregate at the depth h 2part . The red arrow indicates the presence of the wire. The particles are ceramic beads (ρpart = 3, 800 kg.m -3 , Rpart = 0.45 mm). Scale bar: 1 mm.

Figure 3 . 14 :

 314 Figure 3.14: Cohesion of a pair of particles: first experiment.A metallic wire is glued on a bead which is then placed at the interface, and attracts another particle. The force applied on the second particle is produced by the local curvature of an interface deflected by a cylinder of radius R cyl placed at a depth H cyl .

Figure 3 . 15 :

 315 Figure 3.15: Cohesion of a pair of particles: second experiment. Here again, a metallic wire is glued on a bead which is then placed at the interface, and attracts another particle. The wire is then moved horizontally at an imposed velocity V , creating a drag on the second particle. In b, c and d, three examples of the bonding between the bead and the wire are shown, for three different sizes of particles (c: Rpart = 0.25 mm, d: Rpart = 0.175 mm, e: Rpart = 0.125 mm). Scale bar in the three images: 0.2 mm.

Figure 3 . 16 :

 316 Figure 3.16: Separation of a pair of particles during a translation at constant speed. Time-lapse of the translation of a couple of plastic particles (ρpart = 1, 420 kg.m -3 , Rpart = 1.5 mm) at an oil-water interface, at the critical speed of detachment V crit = 6 mm/s. The red arrow indicates the separation between the two particles. The movie is taken from the side. Time between two images: 1.6 s. Scale bar: 5 mm.

Figure 3 . 17 :

 317 Figure 3.17: Cohesion between two spheres at an interface. Experimental cohesion force of a pair of particles calculated thanks to equation (3.6) (colored stars), compared with the theoretical capillary force between two beads (dotted lines), as defined in equation (3.7), as a function of the radius of the beads, for different types of particles (black: ceramic beads ρpart = 3, 800 kg.m -3 , red: high density ceramic beads ρpart = 6, 000 kg.m -3 , blue: plastic beads ρpart = 1, 420 kg.m -3 , green: glass beads ρpart = 2, 500 kg.m -3 ). The black diamond at Rpart = 0.45 mm corresponds to an experiment done with a larger wire (diameter of 1.1 mm). The error bars account for the variability of the experiment (for a given set of parameters), and for the influence of the depth of the particles.

Figure 3 . 18 :

 318 Figure 3.18: Model of cohesion between two particles. Schematic representation of a pair of particles in contact, and of an isolated bead, along with the different notations.

dϕp dα cos ϕ p cos α -sin ϕ p sin α dϕp dα cos ϕ p sin α + sin ϕ p cos α dϕp dα sin ϕ p

  α -sin ϕ p cos ϕ p cos α dϕp dα cos α -sin ϕ p cos ϕ p sin α -sin 2 ϕ p λ defined in equation (3.10c). This finally gives us the expression of the horizontal component of F st : -→ F st . -→ e x = γR part 2π 0 (λ sin θ sin ϕ p cos α -cos θ( dϕ p dα sin α + sin ϕ p cos ϕ p cos α))dα (3.13)

2π 0 λ

 0 sin θ sin ϕ p cos αcos θ dϕ p dα sin α + sin ϕ p cos ϕ p cos α dα (3.19a) (3.19b)

Figure 3 . 19 :

 319 Figure 3.19: Depth angle of particles at an oil-water interface. a, b, c, d) Side views of particles at an oilwater interface, for different types of particles (a: plastic bead ρpart = 1, 420 kg.m -3 and Rpart = 2 mm, b: ceramic bead ρpart = 3, 800 kg.m -3 and Rpart = 0.45 mm, c: glass bead ρpart = 2, 500 kg.m -3 and Rpart = 0.25 mm, d: high-density ceramic bead ρpart = 6, 000 kg.m -3 and Rpart = 0.5 mm). The scale bars change from one photo to the other: a: 2 mm, b: 0.5 mm, c: 0.25 mm, d: 0.5 mm. e ) Theoretical depth angle ϕ s theo as a function of the experimental one ϕs exp, for different types of particles (black: ceramic beads ρpart = 3, 800 kg.m -3 , red: high density ceramic beads ρpart = 6, 000 kg.m -3 , blue: plastic beads ρpart = 1, 420 kg.m -3 ). The dotted line of slope one is only a guideline for the eye.

Figure 3 . 20 :

 320 Figure 3.20: Cohesion between two beads: the complete picture. Experimental cohesion force of a pair of particles calculated thanks to equation (3.6) (colored stars), compared with three theoretical predictions: the classical capillary force between two beads Fcap (plain line), as defined in equation (3.7), the full solving of the force F cohez from equation(3.20), taking into account the tilting of the contact line, with the theoretical depth angle ϕ s theo given by equation (3.32) (black squares), and finally in dotted lines the same horizontal force F cohez but with a depth angle of ϕ s theo + 3 • . The data is plotted as a function of the dimensionless number B 5/2 Σ 2 , as defined in equation(3.33), for different types of particles (black: ceramic beads ρpart = 3, 800 kg.m -3 , red: high density ceramic beads ρpart = 6, 000 kg.m -3 , blue: plastic beads ρpart = 1, 420 kg.m -3 , green: glass beads ρpart = 2, 500 kg.m -3 ).

Figure 3 . 21 :

 321 Figure 3.21: Cohesion force and slope of the interface. Theoretical horizontal force F cohez between two high density ceramic beads (ρpart = 6, 000 kg.m -3 ), as a function of the slope of the interface ψs, for three different radii (magenta: Rpart = 0.5 mm, light blue Rpart = 0.25 mm, green Rpart = 0.1 mm). The dotted lines indicate the theoretical slopes ψ s theo for each radius (the corresponding theoretical force can be read in ordinate).

Figure 4 . 1 :

 41 Figure 4.1: Granular raft before destabilization. Side view of granular raft made of 60 ZrO particles (density ρpart = 3,800 kg.m -3 , Rpart = 0.45 mm). Scale bar: 5 mm. This photo corresponds to the maximum interfacial deformation before destabilization of the raft.

Figure 4 .

 4 Figure 4.2: Sinking of a granular raft. Series of snapshots of a sinking granular raft, made of ZrO particles (density ρpart = 3,800 kg.m -3 , Rpart = 0.45 mm). Time between two images: 8 ms. Scale bar: 2 mm. The last image corresponds to the moment of detachment of the thread from the oil layer.

Figure 4 . 3 :

 43 Figure 4.3: Formation of droplets by a jet. a) Breakup of a liquid jet for two different frequencies of excitation, taken from [137]. b) Irrigation by a sprinkling system. Photo in the public domain.

Figure 4 . 4 :

 44 Figure 4.4: Inkjet printing of neurons. Primary cortical neurons printed in a ring pattern, at different days after the printing. Initially, the cells are globally circular (a), but after a few days (three days in b, seven in c), the neurons differentiate and develop processes. Magnifications: 40× in a, 100× in b and c. Images taken from [142].

Figure 4 .

 4 Figure 4.5: Schematic representation of the Plateau-Rayleigh instability. A sinusoidal perturbation is applied on an infinite cylinder of fluid.

Figure 4 . 6 :

 46 Figure 4.6: Oil droplet detaching from a nozzle. Snapshot of an oil droplet during its detachment from a needle (diameter 2R 0 =1.65 mm), 70 ms before pinching (a) and at the exact moment of pinch-off (b). The origin of the vertical axis, for the measurement of z(r min ), is taken at the position of breakup. Oil kinematic viscosity: νo = 1.10 -4 mm.s -2 . Time between the two images: 70 ms. Scale bar: 1 mm.

Figure 4 . 7 :

 47 Figure 4.7: Pinch-off of an oil droplet: visualization and measurements. a) Series of snapshot of an oil droplet detaching from a needle (diameter 2R 0 =1.65 mm), surrounded by water. Time between two images: 5 ms. Scale bar: 1 mm. b) Variation of the minimum neck radius r min with time until breakup τ for the pinch-off of an oil droplet from a nozzle (2R 0 = 1.65 mm). The dotted line represents an exponential fit (see equation (4.5)) of the first portion of the dynamics. Inset: Exponential decay ω as a function of the radius of the needle. The solid line has a slope -1. c) Vertical position of the minimum neck radius z(r min ) with time until breakup for the same experiment.

Figure 4 . 8 :

 48 Figure 4.8: Shapes diversity of a detaching droplet. Plurality of self-similar shapes, depending on the different regimes obtained for various densities and viscosities of both the internal and the external fluid. The figure is taken from[START_REF] Cohen | Testing for scaling behavior dependence on geometrical and fluid parameters in the two fluid drop snap-off problem[END_REF], with pictures coming from[START_REF] Cohen | Testing for scaling behavior dependence on geometrical and fluid parameters in the two fluid drop snap-off problem[END_REF],[START_REF] Shi | A cascade of structure in a drop falling from a faucet[END_REF] and[START_REF] Cohen | Two fluid drop snap-off problem: Experiments and theory[END_REF].

Figure 4 . 10 :

 410 Figure 4.10: The three regimes leading to a viscous pinch-off. Variation of the minimum neck radius r min with time until breakup τ for the pinch-off of an oil droplet from a nozzle (2R 0 = 1.65 mm). The light blue dotted line represents an exponential fit (see equation (4.5)) of the first portion of the dynamics, the red and green dotted lines represent the self-similar linear evolution describing respectively the viscous (equation (4.10)) and inertial-viscous regimes (equation (4.11)).

Figure 4 . 11 :

 411 Figure 4.11: Detachment of a granular raft: various behaviors. Series of snapshots of a sinking granular raft, made of ZrO particles (density ρpart = 3,800 kg.m -3 ) from a layer of silicone oil (νo = 1.10 -4 mm.s -2 ), for different sizes of beads. a) Diameter of the beads 2Rpart = 0.9 mm. Time between two images: 8 ms. Scale bar: 2 mm. b) 2Rpart = 0.5 mm. Time between two images: 30 ms. Scale bar: 10 mm. c) 2Rpart = 0.35 mm. Time between two images: 50 ms. Scale bar: 10 mm.

Figure 4 . 12 :

 412 Figure 4.12: Experimental evidence of a variation in the thinning dynamics for different external constraints during the breakup of an oil filament in water. a) Variation of the minimum neck radius with time until breakup for the detachment of a granular raft from an oil-water interface (black triangles: ρpart=3,800 kg.m -3 , 2Rpart = 0.9 mm) and for the pinch-off of an oil droplet from a nozzle (blue circles: 2R 0 = 1.65 mm). The dotted lines represent the self-similar linear evolution describing respectively the viscous (red dotted line, equation (4.10)) and inertial-viscous regimes (green dotted line, equation (4.11)). Inset: zoom on the evolution of r min in the final moments before pinch-off (τ < 4.10 -3 s), as a function of time before breakup. b) Vertical position of the minimum neck radius z(r min ) with time until breakup, for the two same experiments.

Figure 4 . 13 :

 413 Figure 4.13: Influence of the shape of the granular raft on its detachment. Variation of the minimum neck radius with time until breakup for the detachment of a granular raft, for different experimental geometries. a) Comparison between a classic granular raft (black circles) and a stack of particles, made by adding the beads directly on top of each other (blue triangles) (glass beads, ρpart=2,500 kg.m -3 , 2Rpart = 0.9 mm). Inset: schematic representation of the two configurations. b) Comparison between an axisymmetric granular raft (black circles) and a non-axisymmetric monolayer of particles with a preferential direction (blue triangles) (ZrO beads, ρpart=3,800 kg.m -3 , 2Rpart = 0.9 mm). Inset: photo of the two configurations just before destabilization.

Figure 4 . 14 :

 414 Figure 4.14: Presence of an oscillation between linear regimes when a raft pulls down an oil-water interface with its weight Time derivative of the minimum radius ṙmin as a function of r min , for two typical experimental setups. Blue circles: oil droplet rising in water and finally detaching from a needle (2R 0 = 1.65 mm). Black triangles: a granular raft detaching from an oil-water interface (ρpart = 3,800 kg.m -3 , 2Rpart = 0.7 mm). The dotted lines represent the theoretical linear self-similar regimes describing the viscous regime (red dotted line, equation (4.10), ṙmin = uv) and the inertial-viscous regime (green dotted line, equation(4.11), ṙmin = u iv ).

Figure 4 . 15 :

 415 Figure 4.15: Oscillation of the radial velocity ṙmin for the pinch-off of various types of granular rafts. Time derivative of the minimum radius ṙmin as a function of r min , for seven different types of granular rafts. Dark blue triangles: glass beads, ρpart = 2,500 kg.m -3 , 2Rpart = 0.9 mm; Blue crosses: ZrO beads, ρpart = 6,000 kg.m -3 , 2Rpart = 0.4 mm; Blue squares: ZrO beads, ρpart = 3,800 kg.m -3 , 2Rpart = 0.7 mm; Light blue stars: ZrO beads, ρpart = 6,100 kg.m -3 , 2Rpart = 0.55 mm; Green triangles: ZrO beads, ρpart = 4,800 kg.m -3 , 2Rpart = 0.7 mm; Green circles: ZrO beads, ρpart = 3,800 kg.m -3 , 2Rpart = 0.9 mm; Orange triangles: steel beads, ρpart = 7,850 kg.m -3 , 2Rpart = 0.6 mm. The color bar displays the corresponding dimensionless number D 1 for each granular raft. To distinguish the different curves even for two almost identical D 1 , the scale of the color bar is not linear. The dotted lines represent the theoretical linear self-similar regimes describing the viscous regime (red dotted line, equation (4.10), ṙmin = uv) and the inertial-viscous regime (green dotted line, equation (4.11), ṙmin = u iv ). For clarity, the error bars are not represented here, but they are very similar to the one of figure 4.14.

Figure 4 . 16 :

 416 Figure 4.16: Rising droplet VS sinking granular raft. Schematic representation of the differences between a rising oil droplet extruded from a needle, and a granular raft sinking from an oil layer. The vertical speed is higher for the raft (V 1 > V 0 ), and the lateral extension of the liquid thread is usually higher for the raft (R 1 > R 0 )

Figure 4 . 17 :

 417 Figure 4.17: Sedimentation velocity of a detaching droplet, for two different experimental configurations. Vertical velocity of an oil droplet during its formation, as a function of time until breakup. Blue circles: oil droplet rising in water (2R 0 = 2.9 mm). Black triangles: granular raft sinking from an oil-water interface (ρpart = 2,500 kg.m -3 , 2Rpart = 0.9 mm).

Figure 4 . 18 :

 418 Figure 4.18: Pulling an oil-water interface: the experimental setup. Schematic representation of the experiment: an oil-water interface is pulled downwards with a cylinder of radius R cyl at a given velocity v cyl , until the oil meniscus formed during the process breaks.

  reproducing external conditions similar to a sinking granular raft. Similarly to an armored droplet, the oil thread thins until it finally breaks up (figure 4.19.a and 4.19.b).

Figure 4 . 19 :

 419 Figure 4.19: Thinning of a stretched meniscus. a) and b) Time-lapse showing the breakup of an oil filament formed by pulling an oil-water interface with a cylinder of radius R cyl at a vertical velocity of v cyl . The cylinder is located at the bottom of each photo, and moves downwards. For each image sequence, the last photo corresponds to the moment of pinch-off. a) R cyl = 15 mm, v cyl = 30 mm.s -1 . Time between two consecutive images: 25 ms. Scale bar: 10 mm. b) R cyl = 1 mm, v cyl = 1 mm.s -1 . Time between two consecutive images: 2 ms. Scale bar: 1 mm.

Figure 4 . 20 :Figure 4 . 21 :

 420421 Figure 4.20: Influence of the direction of motion on the presence of an oscillation Time derivative of the minimum radius ṙmin as a function of r min , for three typical experimental setups. Dark blue circles: oil droplet rising in water and finally detaching from a needle (2R 0 = 1.65 mm). Black triangles: a granular raft detaching from an oil-water interface (ρpart = 3,800 kg.m -3 , 2Rpart = 0.7 mm). Light blue stars: interface slowly pulled by a cylinder (2R cyl = 2 mm, v cyl = 1 mm.s -1 ). The dotted lines represent the theoretical linear self-similar regimes describing the viscous regime (red dotted line, equation (4.10), ṙmin = uv) and the inertial-viscous regime (green dotted line, equation (4.11), ṙmin = u iv ). The error bars are not represented for clarity reasons, but they are similar to the ones of figure 4.14.

Figure 4 . 22 :

 422 Figure 4.22: Role of external conditions on the evolution of the thinning of a liquid thread pulled from an oil-water interface. Time derivative ṙmin as a function of r min , the minimum radius, for different cylinder diameters ( a: 2R cyl = 2 mm, b: 2R cyl = 5 mm, c: 2R cyl = 8 mm, d: 2R cyl = 13 mm), and different pulling velocities (orange circles: v cyl = 0 mm/s, red crosses: v cyl = 1 mm/s, dark red triangles: v cyl = 30 mm/s). The dotted lines represent the theoretical linear self-similar regimes describing the viscous regime (red dotted line, equation (4.10), ṙmin = uv) and the inertial-viscous regime (green dotted line, equation (4.11), ṙmin =u iv ). The error bars are not represented for clarity reasons, but they are similar to the ones of figure 4.14.

  .23.b), ṙmin (figure 4.23.a), and even for z(r min ) (figure 4.23.c), between a granular raft made of steel beads and two sets of parameters for a cylinder experiment.

Figure 4 . 23 :

 423 Figure 4.23: Reproducing the pinch-off of a granular raft with cylinders. Experimental evidence of the similarity between a sinking granular raft and a cylinder moving downwards. Black triangles: steel beads, ρpart = 7,850 kg.m -3 , 2Rpart = 0.6 mm. Dark blue circles: cylinder, 2R cyl = 13 mm, v cyl = 10 mm.s -1 . Light blue stars: cylinder, 2R cyl = 20 mm, v cyl = 1 mm.s -1 . The dotted lines represent the theoretical linear self-similar regimes describing the viscous regime (red dotted line, equation (4.10), ṙmin = uv) and the inertial-viscous regime (green dotted line, equation (4.11), ṙmin = u iv ). a) ṙmin as a function of r min , the minimum radius, for the three experiments. The error bars are not represented for clarity reasons, but they are similar to the ones of figure 4.14. b) Variation of the minimum neck radius with time until breakup. Inset: zoom on the evolution of r min in the final moments before pinch-off (τ < 3.10 -3 s). c) Vertical position of the minimum neck radius with time until breakup.

Figure 4 .

 4 Figure 4.24: Invariance of the log-pulsation with size and vertical velocity. Pulsation in log-scale of the sinusoidal oscillation of ṙmin as a function of the radius of the cylinders, for different pulling velocities (red circles: 0 mm/s, black crosses: 1 mm/s, dark blue squares: 10 mm/s, green triangles: 30 mm/s), and various granular rafts (light blue stars). For the granular rafts, R stands for the maximum radius of the raft before destabilization. ω is measured only when a complete period of oscillation is present. The black dotted line displays the mean value of ω, and proves the log-pulsation is globally constant for all the experiments.

Figure 4 . 25 :

 425 Figure 4.25: Self-similar profiles during the classic thinning of a viscous thread. Rescaled profiles of an oil meniscus formed by a cylinder (R cyl = 1 mm) pulling down quasistatically (v cyl = 0 mm.s -1 ) an oil-water interface. The insets show ṙmin as a function of r min in each case, with arrows indicating the profiles chosen in the main figure. Here again, the dotted lines represent the theoretical linear self-similar regimes describing the viscous regime (red dotted line, equation (4.10), ṙmin = uv) and the inertial-viscous regime (green dotted line, equation (4.11), ṙmin = u iv ). The rescaled profiles represent H as a function of ξ 1 if the expected regime is inertialviscous (a), or ξ 2 if it is expected to be only viscous (b). The profiles identified by black arrows in each inset are represented from left to right by green triangles, light blue crosses, and dark blue circles. The photos above each figure show actual experimental visualizations of the oil filament during both regimes. Scale bars: 1 mm.

Figure 4 . 26 :Figure 4 . 27 :

 426427 Figure 4.26: Self-similar profiles during the thinning of a viscous thread. Rescaled profiles of an oil meniscus formed by a cylinder (2R cyl = 13 mm) pulling down at v cyl = 1 mm.s -1 an oil-water interface. The insets show ṙmin as a function of r min in each case, with arrows indicating the profiles chosen in the main figure. Here again, the dotted lines represent the theoretical linear self-similar regimes describing the viscous regime (red dotted line, equation (4.10), ṙmin = uv) and the inertial-viscous regime (green dotted line, equation (4.11), ṙmin = u iv ). The rescaled profiles represent H as a function of ξ 1 if the expected regime is inertial-viscous (a and c), or ξ 2 if it is expected to be only viscous (b and d). The profiles identified by black arrows in each inset are represented from left to right by green triangles, light blue crosses, and dark blue circles. The photos on the side of each figure shows actual experimental visualizations of the oil filament during the different regimes. Scale bars: 0.2 mm for a and b, 2 mm for c and d.

Figure 4 .

 4 Figure4.28 displays visualizations of the typical results we get with our simulations, for two given sizes of needle (which is basically the only parameter we change in this numerical study). This experimental setup is different from the one used in the previous section, where we pulled an interface with a cylinder. Here, we come back to the more classic situation of the pinching of a droplet detaching from a needle, and we want to see if by changing only the diameter of the nozzle, we can manage to trigger oscillations between self-similar regimes.

Figure 4 . 28 :

 428 Figure 4.28: Numerical simulation of an oil droplet detaching from a nozzle. Image sequence of the numerical simulation of the motion of an oil droplet (νo = 1.10 -4 mm.s -2 ) extruded from needles of two different sizes at a low flow rate, surrounded by water. The oil is injected at the bottom of each image, and gravity is directed downwards. The first image corresponds to the initial geometry of the simulation. The following four images display the evolution of the viscous neck near pinch-off. a) R 0 = 0.72 mm. Time between two images: 1.5 ms. Scale bar: 1 mm. b) R 0 = 7 mm. Time between two images: 18 ms. Scale bar: 10 mm.

Figure 4 . 29 :

 429 Figure 4.29: Agreement between experiments and numerical simulations. Time derivative of the minimum radius ṙmin as a function of r min , for an experiment and the corresponding numerical simulation. Blue circles: oil droplet rising in water and finally detaching from a needle (2R 0 = 1.65 mm). Red squares: numerical simulation of the formation of a droplet from a needle (2R 0 = 1.44 mm). The dotted lines represent the theoretical linear selfsimilar regimes describing the viscous regime (red dotted line, equation (4.10), ṙmin = uv) and the inertial-viscous regime (green dotted line, equation (4.11), ṙmin = u iv ). The error bars are not represented for clarity reasons, but they are similar to the ones of figure 4.14.

Figure 4 . 30 :

 430 Figure 4.30: Oscillation of ṙmin in numerical simulations. Time derivative of the minimum radius ṙmin as a function of r min , for the numerical simulation of the formation of a droplet from a large needle (R 0 = 4 mm). The dotted lines represent the theoretical linear self-similar regimes describing the viscous regime (red dotted line, equation (4.10), ṙmin = uv) and the inertial-viscous regime (green dotted line, equation (4.11), ṙmin = u iv ). Inset: Oscillation of the minimum radius for two Ohnesorge numbers, for numerical simulations. The figure is taken from the work of Li et al. [169].

  .31.b), up to a point where the final inertial-viscous regime appears. a) b)

Figure 4 . 31 :

 431 Figure 4.31: Thinning of a granular suspension. Pinching of a viscous suspension, made of polystyrene beads with a diameter of 80 µm dispersed in silicon oil at a volume fraction of 3 %. a) From right to left, time before pinch-off τ = 17.53, 5.19, 1.29 ms. Scale bar: 0.2 mm. b) Minimum neck radius as a function of time (black triangles), compared with the thinning of a pure silicon oil solution. The red and black dotted lines respectively represent the viscous and inertial-viscous regimes, while the blue line is a linear fit with a higher slope. Photos and figure taken from [172].

Figure 4 . 32 :

 432 Figure 4.32: Detachment of a granular raft. Pinching dynamics of an oil filament during the sinking of a granular raft made of ceramic particles (density ρpart = 3,800 kg.m -3 , radius Rpart = 0.25 mm). a) Snapshot of the sinking of a granular raft. The successive images have to be read from left to right. Time between two images: 25 ms. Scale bar: 5 mm. The red arrows designate the position of the minimum radius in each photo. The corresponding measurements for each photo are identified by blue arrows in the two figures under. b) Minimum neck radius as a function of time until breakup. The dotted lines represent the self-similar linear evolution describing respectively the viscous (red dotted line, equation (4.10)) and inertial-viscous regimes (green dotted line, equation (4.11)). Inset: zoom on the evolution of r min in the final moments before pinch-off (τ < 7.10 -3 s), as a function of time before breakup. c) Vertical position of the minimum neck radius z(r min ) with time until breakup, for the same experiment. Contrarily to the series of photos, the curves have to be read from right to left.

Figure 4 . 33 :

 433 Figure 4.33: Reproducibility of the pinching dynamics. Time derivative of the minimum radius ṙmin as a function of r min , for the three same experiments using ceramic particles (density ρpart = 3,800 kg.m -3 , radius Rpart = 0.25 mm). The dotted lines represent the theoretical self-similar regimes describing the viscous regime (red dotted line, equation (4.10), ṙmin = uv) and the inertial-viscous regime (green dotted line, equation (4.11), ṙmin = u iv ). For clarity reasons, the error bars are not represented here. The vertical dotted line indicates the radius of an individual particle.

  .35).

Figure 4 . 34 :

 434 Figure 4.34: Detachment of a granular raft made of small particles. Pinching dynamics of an oil filament during the sinking of a granular raft made of ceramic particles (density ρpart = 3,800 kg.m -3 , radius Rpart = 0.175 mm). a) Snapshot of the sinking of a granular raft. The successive images have to be read from left to right. Time between two images: 37.5 ms. Scale bar: 10 mm. The red arrows designate the position of the minimum radius in each photo. The corresponding measurements for each photo are identified by black arrows in the two figures under. The photo framed in red corresponds to the exact instant at which the upper r min equals the lower one, and the corresponding points are indicated by red arrows in b and c. b) Minimum neck radius as a function of time until breakup. Inset: zoom on the evolution of r min in the final moments before pinch-off (τ < 20.10 -3 s), as a function of time before breakup. The dotted lines represent the viscous regime (red dotted line, equation (4.10)) and inertial-viscous regime (green dotted line, equation (4.11)). c) Vertical position of the minimum neck radius z(r min ), for the same experiment. Contrarily to the series of photos, the curves have to be read from right to left.

Figure 4 . 35 :

 435 Figure 4.35: Variability of the pinching between experiments. a, b and c) Time derivative of the minimum radius ṙmin as a function of r min , for the three same experiments. The dotted lines represent the theoretical linear self-similar regimes describing the viscous regime (red dotted line, equation (4.10), ṙmin = uv) and the inertialviscous regime (green dotted line, equation (4.11), ṙmin = u iv ). The three series of photos on top correspond to specific moments of the sinking, located on the curves thanks to the black solid lines. Time between two images: 10 ms. Scale bar: 1 mm (the three series of photos have the same scale). The particle of the filament closest to r min is encircled in red.

Figure 4 . 36 :

 436 Figure 4.36: Relation between thinning and confinement of the thread. Maximum shrinking velocity max( ṙmin ), as a function of the confinement length, as defined in figure 4.35.a and 4.35.b.

3

  Erosion and cohesion of a granular raft 3.1 Structural changes of a monolayer . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Erosion of a granular raft . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 A typical experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 First results: existence of erosion . . . . . . . . . . . . . . . . . . . . 3.2.3 Experimental cohesive forces of a raft . . . . . . . . . . . . . . . . . 3.3 Cohesion between two aggregated particles . . . . . . . . . . . . . . . . . . . 3.3.1 In search of an experimental setup . . . . . . . . . . . . . . . . . . . 3.3.2 Experimental result for the cohesion between two beads . . . . . . . 3.3.3 Calculation of the capillary forces . . . . . . . . . . . . . . . . . . . . 3.3.4 Vertical position of the contact line . . . . . . . . . . . . . . . . . . . 3.3.5 Cohesion between two beads: the explanation . . . . . . . . . . . . .

Table 1 . 1 :

 11 Characteristics of the beads used in experiments

	Type of particle	ρ part (kg.m -3 )	R part (mm)	n sink
	Plastic	1,420	2	16
	Plastic	1,420	2.5	7
	Ceramic	3,800	0.35	160
	Ceramic	3,800	0.45	60
	Ceramic	4,800	0.35	60
	Ceramic	4,800	0.45	25

Table 2 .

 2

	R domain (cm)	Mass (mg)	< n tot >	φ surf (cm -2 )
	2	5	123	9.8
	2	10	286	22.8
	3.25	5	117	3.5
	3.25	10	248	7.5
	3.25	33	968	29.2
	5.15	10	278	3.3
	5.15	33	915	11.0

1: Space of parameters explored in the experiments

Table 2 .

 2 2: Space of parameters explored in the simulations. Above the line, only 40 simulations are performed for each parameter. Below the line, between 700 and 1000 simulations are performed.

	R domain (cm)	n tot	φ surf (cm -2 )	Number of simulations
	2.3	400	24.1	40
	3.5	400	10.4	40
	11.6	200	0.47	40
	2.9	200	7.6	890
	2.9	300	11.4	680
	5.8	200	1.9	980
	5.8	400	3.8	970

Remerciements