Keywords: Tensioactifs, Mousses aqueuses, Huile, Etalement de gouttes, Imbibition, Instabilité hydrodynamique, Micro uidique, Surfaces texturées, Milieux poreux Surfactants, Aqueous foams, Oil, Drop spreading, Imbibition, Hydrodynamical instability, Micro uidics, Liquid-infused surfaces, Porous media

L'extraction de matières premières du sol à des ns énergétiques (récupération assistée d'huile) et environnementales (dépollution des sols) fait l'objet de recherches intensives en lien avec des thématiques telles que la séquestration du carbone ou la fracturation hydraulique. L'objectif est de trouver des méthodes moins destructives, moins gourmandes en matériel et en énergie, mais aussi plus e caces et moins coûteuses.

Nous proposons d'étudier une méthode alternative aux moyens conventionnels avec l'utilisation de mousses aqueuses comme agent extracteur d'huile. Les mousses aqueuses sont souvent utilisées en présence d'huile : dans des applications quotidiennes comme la cosmétique et la détergence, mais aussi dans des domaines moins connus comme la décontamination des centrales nucléaires ou l'industrie pétrolière. Ainsi, des tensioactifs et du gaz sont couramment injectés dans le sol a n d'améliorer les procédés de récupération de pétrole.

Nous explicitons deux mécanismes d'extraction que nous quanti ons en termes d'e cacité et de stabilité. Tout d'abord, la mousse peut aspirer de l'huile en son sein, comme le ferait une éponge. Ensuite, lorsque celle-ci est mise en écoulement, elle peut entraîner de l'huile con née dans la rugosité d'une surface par cisaillement. Notre étude s'appuie en particulier sur une analyse théorique et expérimentale, à la fois multi-échelle, statique et dynamique pour laquelle nous avons systématiquement fait varier les paramètres géométriques (con guration de l'huile, taille des bulles et fraction volumique de liquide dans la mousse) et physico-chimiques (tensions interfaciales, rigidité des interfaces entre bulles et viscosité).

First, of course, I would like to thank my parents. You constantly support me to get the best of me. You encourage me for any plan and project I have in and out of my studies. You o er me the best possible conditions to ful ll my dreams and learn so many things in di erent elds: science, sport, music, travels, literature and so on. Now, I have so many options in my life and my job career and I give you credit for that. You know that I like going in every direction, I am very curious about everything and I am picky to make decisions. But the most important thing is that you teach me to ght for everything, to give the maximum I can to have no regrets, to do what I like and this is what I have tried to do. A big thank you, I love you.

Second, I obviously thank my supervisor Prof. Elise Lorenceau for supporting me and helping me during those hard three years. Well, actually ve years, since I met Elise during my Master's internship... You are an incredible scienti c mentor and researcher who gave me meaningful advice on my projects and encouraged me to keep going in the right direction: always stay positive, never give up. You let me carry out my own thinking process and experiments, which I really appreciate. And you let me meet your nice kids, so thanks to Louison, Lucile and Jules for your smiles, your enthusiasm and your questions (I'm sure you got some...), you have a great Mom.

Third, I address my deep acknowledgements to Prof. Howard A. Stone. You hosted me in your Complex Fluid group at Princeton University with open arms. I was so lucky to be able to work with you, I learnt so many things and you made my time in Princeton so enjoyable. I wish every student in uid dynamics and soft matter could learn as many things as I learnt from you and your whole group. I am also very grateful to the following people from the Navier Institute with whom I spent three amazing years at work but also o -work: -Alizée Lehoux: you are a very special friend to me. We had a lot of fun and nice and burning discussions, as well as joking time. We shared unique moments whenever we spent time together: swimming, running, strolling, tasting original food products but not only... I am sure we will nurture this bond. Keep swimming, you are the only speaking dolphin I know! -François Gorlier: my roommate. What can I say? The most enjoyable guy in the lab. You are vi ACKNOWLEDGEMENTS a superhero who speaks to himself but I like that. Do not forget the music quizzes... -Jennifer Fusier: my dear lab delegate. You are the most persevering person. Keep going, you will succeed. Do not forget to work on your arm strength and become a board game master.

-Blandine Feneuil: the professional runner and seamstress. So quiet but full of energy. Keep running, running and running to catch me... -Benjamin Ha ner, Benoit Laborie and Guillaume Chivot: I had a lot of fun while joking with you, my former labmates. You are heavy drinkers but you always had a positive attitude! I also want to give a special credit to Mathilde Maillard, Marine Fourmentin, Lucie Ducloué, Emmanuel Keita, Antoine Morandeau, Thibault Chevalier, Aymeric Mouquet, Xiao Zhang, Dr. Romain Morini and Dr. Keyvan Piroird.

Then, I have a special word for the following members of the Complex Fluids group and students at Princeton University: -Jesse T. Ault: one of the most clever and reactive labmates I have ever met. The imbibition project would not have succeeded without your help. You were the only American left (maybe not now...), so keep carrying the Stars and the Stripes high in the lab! And stop eating food in tin cans! -Dr. Jason S. Wexler: like Jesse, you laid the foundations for the micro uidic project. Bold and curious, you always had inventive ideas with this West Coast relaxed attitude! -Dr. Hyoungsoo Kim and Dr. Sangwoo Shin: the Korean daddies (whisky and micro uidic experts!). You also assisted me for my projects by your advice and your intuition.

-Ching-Yao Lai: you have become one of my nice friends. You are always the rst one to help people be integrated into the group and Princeton's life. Keep doing it, it is unvaluable.

-Jie Feng: the paper machine. I will always remember our wonderful karaoke time in New York and your help on the spreading project! -Kevin Kim: my deskmate. You have also become a close friend. Do not work too hard and do not sleep in the lab! -Suin Shim: you are the best piano player I have ever met and heard. Always quiet but ready for a drink, I love this attitude. -Dr. Janine Nunes: well, you allow this group to run so smoothly. What will happen when you leave... -Mrs. Jill F. Ray: like Janine at another level, you enable the Department of Mechanical and Aerospace Engineering to run perfectly. And you also make visiting scholars and students feel comfortable.

-Ting-Hsuan Chen: my best friend in Princeton. I am so glad to have met you. Not only are you a remarkable young researcher, but you are also one of the most open-minded people I have ever met. Thanks to your dynamism and your support, you make me discover o -campus life xviii LIST OF FIGURES 2.16 Flow morphology of a simple uid ow (air) in a frictional uid or granular mixture in a (log φ -1 , log Q) diagram where φ is the solid fraction and Q the injection ow rate. The gure is extracted from [START_REF] Sandnes | Patterns and ow in frictional uid dynamics[END_REF]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.17

3.12

Wavelength λ max as a function of the slug radius R oil . The coloured dotted lines are the theoretical predictions [START_REF] Tomotika | On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous uid[END_REF]. The black dotted line is the prediction for the inertia-dominated regime [START_REF] Rayleigh | On the instability of a cylinder of viscous liquid under capillary force[END_REF] and the orange dotted line, the prediction for the viscous-dominated regime when the viscosity of the outer uid is neglected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.13

Theoretical ratio wavelength of maximum growth -cylinder radius λ max /R oil as a function of the viscosity ratio η o /η w [START_REF] Tomotika | On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous uid[END_REF]. The solutions for our oil-foaming solution systems are indicated with the same color code as in Fig. 3.12. . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.14

Theoretical radius of the oil droplet R d as a function of (λ max R 2 oil ) 1/3 for para n oil, olive oil and hexadecane. The oil radius R oil is deduced from the volume Ω of the oil slug. The black dashed line represents the theoretical prediction from Eq. 3.7. Inset: Theoretical radius R d as a function of (λ max R oil ) 1/2 by assuming a pancake-like shape. The black dashed line is the theoretical prediction as detailed in the text. We observe that the agreement is not quantitative. . . . . . . . From top to bottom, Ω d = 37.9, 5.8, 4.03, 1.35, 0.35 µL. The data are extracted from [START_REF] Cazabat | Dynamics of wetting on smooth and rough surfaces[END_REF]. . . . . .

4.5

Topology of an oil lens spreading on a water surface in the limit where the gravity e ects are negligible, i.e. the shape of the drops is only dictated by interfacial tension. . . . . . . . . . . . 2g L is for a droplet attached to the needle for 1 h in the aqueous bath before spreading (in case surfactant dynamics matters). The dashed line is a guide line for the power law T * 1/8 . . . . . . .

4.16

Phase diagram (Bo, R) for a given Oh in the surfactant case. The transitions for di erent values of Oh between the short-time and the long-time regimes are plotted in the red dashed line (for Oh >> 1) and in the orange dashed line (for Oh << 1). We have f (Oh) = Oh 2 / -1 + √ 1 + 4Oh 2 .

The curve R = 1/Bo 1/2 de nes the transition between capillary and gravity-dominated regime (capillary length). The curves R = f (Oh) 1/6 Bo 1/6 and R = f (Oh)

5.25

Dimensionless extraction ow rate Q * as a function of the dimensionless foam velocity U * for the imbibition of olive oil with di erent initial liquid fractions φ i in the foam. The orange line is the t with Eq. 5.62 and v o (t) ∼ t 3/2 . The blue and the purple dashed lines are, respectively, the ts with v o (t) ∼ t 1/3 and ∼ t 1/2 deduced from the numerical simulation from §5.2. . . . . . . . 6 Foam -oil interactions in micropatterned structures 6.1 a. Wetting on a smooth surface with the Young contact angle θ Y . b. Wetting on a rough surface with the apparent contact angle θ * . c. Evolution of cos θ * with cos θ Y for a smooth and a rough surface of the same chemical nature [START_REF] Frye | Super water-repellent surfaces resulting from fractal structures[END_REF]. The parameters φ S and r S are respectively the solid fraction and the roughness. The black plain line in the upper right part corresponds to Eq. 6.2. . . [START_REF] Adamson | Physical Chemistry of Surfaces[END_REF] to the Wenzel regime (2). . . . . . . . . . . . . . 6.6 a. Touch down scenario [START_REF] Bartolo | Bouncing or sticky droplets: Impalement transitions on superhydrophobic micropatterned surfaces[END_REF]. b. Sliding scenario [START_REF] Bartolo | Bouncing or sticky droplets: Impalement transitions on superhydrophobic micropatterned surfaces[END_REF]. . . . . . . . . . . . . . . . . . . . . . . xxiv LIST OF FIGURES 6.7 a. Shear-driven drainage of oil in a groove geometry [START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF]. b. Local con guration of the oil-water interface. The parameters L o , L(t), h, w, θ, r min nd u are, respectively, the length of the groove, the retention length, the height of the groove, the spacing in a groove, the receding contact angle, the minimal radius of curvature and the ow velocity. . . . . . . . . . . . . . . . . . . . . . 6.8 Dimensionless retention length L = L/L ∞ as a function of the dimensionless time t = t/t c . The blue, the red and the gray curves correspond to experimental results of drainage of silicon oils by water, respectively for oil viscosity η o = 42.7 mPa.s and injection ow rate Q = 1 mL/min, η o = 201 mPa.s and Q = 2 mL/min and η o = 42.7 mPa.s and Q = 2 mL/min. The dotted lines are the theoretical predictions [START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF]. Both gures are extracted from [START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF]. . . . . . . . . . . . 6.9 Geometry of the micro uidic channel with the liquid-infused micropattern imprinted on the upper side of the channel. a. Analogy between a real microfracture and a liquid-infused substrate. and CAPB -silicon oil 1 (blue dots) combinations. Oil appears red in the confocal microscope and the green line is the laser re ection between the glass and the aqueous phase. Lower right:

Transition from the regime of air cushion

Evolution of the numerical factor c d with respect to θ. . . . . . . . . . . . . . . . . . . . . . 6.17 a. Experimental e ciency α exp as a function of g(t) for CAPB -olive oil, CAPB -silicon oil 1 and CAPB -silicon oil 2 combinations. b. f (α exp ) as a function of g(t) for di erent surfactant-oil combinations. The black dashed line is the theoretical prediction (Eq. (6.19a)) . . . . . . . . . . Yesterday morning, I took a cup of cappucino with my breakfast. Then, I had a refreshing shower and spread a moisturizing cream onto my face. After work, I went back home by the park and saw an artist entertaining the passers-by while dipping a long rope into a bath of liquid. Later, I had a marvelous dinner with a cup of champagne. Why am I telling you what I did yesterday? Because in all these unsigni cant events of anyone's life, I encountered this intriguing physical object lled with gas inside a thin liquid lm that one calls bubble. Philosophers and painters already understood their importance and beauty by portraying bubbles no later than the 16 th century. But bubbles rarely travel alone. Flocked into a group, they become liquid foams. Foams -especially aqueous foams if they are water-based -are everywhere. We need them to shave the face and the skin hair, we nd them in all kinds of cosmetics and creams, we create foams for any cleaning activities from body care to detergency, we eat foams in many desserts from cakes to ice creams. They bring comfort, decorate, enhance taste and enable to sell products for a small cost. Everyday life is not the only place where foams are essential. In civil engineering, the quest for better building insulation focuses on foams as a way to include more air. Fire-ghters sprinkle foams to block the connection between oxygen and re. Mining resorts to foams in separation processes because their interfaces can capture ne particles very easily and their density is lower than any other liquid. Eventually, oil companies and decontamination agencies include foams in their strategy to capture oil and contaminants by decreasing the interfacial tension and bene ting from their high expansion property. Due to their high composition in air, they also generate less waste, which might be interesting for environmental issues. The wide variety of applications is only made possible because foams exhibit original physical features and properties. In the world of hydrodynamics and soft matter, foams are part of the complex uids' branch with polymers, colloidal suspensions, emulsions, gels and yield stress uids (ketchup, toothpaste) as some of their counterparts. The composition, the structure, the motion and the ow (the so-called rheology) of those uids have been and are still the subject of intense research. With the development of new chemical formulations for the surfactants, complex foams, composed of air, aqueous solution and a third phase, which can be a solid, an immiscible liquid or a gel are now commonly encountered in strategic elds, such as civil engineering or plastic manufacturing. Chapter 1 recalls the nature of foams from surfactants to the assembly of bubbles. It insists on important physical and mechanical properties with an emphasis on environmental and energetical applications where foams interact with immiscible phases. Such multiphasic interactions are at stake for the extraction of a liquid phase from a con ned medium, typically in soil remediation and enhanced oil recovery. Yet, in many industrial cases, oil is used as an anti-foaming agent to destroy foams or to avoid their generation. However, in some appropriate conditions, oil droplets can increase the stability of foams and invade them without any damage. The antifoaming feature and the foam collapse strongly depend on the ability of oil to emerge at the air/water interface [START_REF] Denkov | Mechanisms of foam destruction by oil-based antifoams[END_REF][START_REF] Garrett | Preliminary considerations concerning the stability of a liquid heterogeneity in a plane-parallel liquid lm[END_REF]. In Chapter 2, we answer the related questions: what parameters determine the stability of a foam containing emulsi ed oils? How do foams react to the invasion and the ow of oil through the liquid microchannels, called Plateau borders, characteristic of the foam structure? Then, we travel from the global scale of oil-laden foams to the scale of a Plateau border -i.e. the liquid microchannel of the foam in which the ow of oil occurs). The stability of oil-imbibed foams requires to understand the local picture of oil inside the foam components. Hence, we tackle the following questions: what is the geometry of an oil slug propagating through a Plateau border? What happens if an oil-lled Plateau border breaks due to an external perturbation? Chapter 3 describes how the stability of oil-lled Plateau borders can be understood by looking at the local energetical con guration of the three-phase system air-oil-water. We also try to INTRODUCTION understand how this system evolves when an adjacent lm bursts, inducing the rupture of the Plateau border.

Plateau borders drive oil inside the foam but lms can also contain dispersed oil droplets in many situations (antifoaming action, oil-stabilized foams). Oil droplets have the ability to emerge and spread at the air-water interface. The questions here are: what is the local dynamics of oil spreading at the interface from the emergence of the droplet to the formation of an oil lm? How does it compare to the common theories on the wetting of solid and liquid surfaces? Chapter 4 focuses on the interactions between soap lms and oil. We study the dynamics of oil droplets rising to the air-water interface and spreading at this interface.

After looking at the interactions between oil and foams a the local scale, we wonder how oil can be driven inside the foam. In Chapter 5, we consider what happens when a large volume of oil is put into contact with a foam. Indeed, providing that the right chemical composition for the foam and a compatible oil are found, a foam can act as a liquid sponge: oil penetrates the foam through the liquid capillary microtubes (called Plateau borders) of the foam network, located at the bubbles' jonctions. Thus, foams can be seen as physical systems in which imbibition processes can occur. Depending on the con guration of the oil reservoir and the relative motion of the foam compared to the imbibed liquid, di erent dynamics can occur. Hence, we reply to the following issues: how does oil imbibition in foams occur? What impact does the con nement and the characteristic size of the oil reservoir have on the imbibition dynamics? How can oil be transported by a moving foam if one wants to recover it?

In the last part, we move from the millimeter or centimeter scale, at which the imbibition of oilladen foams occurs, down to the micrometer scale corresponding to the size of the pores in rock layers and soils. In real life situations, oil can be trapped in the roughness of con ned media, as well as in the pores. Oil companies usually inject surfactants to untrap the oil phase. Trapping oil in experimental models can be done by using liquid-infused surfaces. These surfaces can be compared to an oil-lled rough medium. As depicted in Fig. 4, they are micropatterned surfaces of controlled size, which are used to create omniphobic objects, to reduce biofouling and to enhance drag reduction. The failure of these surfaces and the loss of the aforementioned properties, owing to the disappearance of the lubricating layer, is analogous to the issue of oil extraction from the roughness of con ned media. We shed a new light on di erent questions: can we evaluate the quantity of oil extracted from a micropatterned and con ned channel when surfactant solutions are injected? What is the extraction e ciency if foams are injected (industries often co-inject a gas phase with surfactants)? Chapter 6 focuses on the oil extraction from micropatterned surfaces, as sketched in Fig. 4. This chapter underlines the importance of textured and liquid-infused surfaces and how they can be used to study recovery processes from oil-lled rough con ned media.

INTRODUCTION

Chapter 1

Aqueous foams and their applications in environment and energy

Aqueous foams are multiscale objects whose structure involves di erent pieces: surfactants, lms and bubbles. Their structure o ers original properties for a multitude of daily and industrial applications, including environment and energy-related activities.

CHAPTER 1: AQUEOUS FOAMS AND THEIR APPLICATIONS IN ENVIRONMENT

AND ENERGY Contents Let us start from the molecular scale. Foams exist because some particular molecules, called surfactants, appear as the key components for the whole structure. Broaching the subject of surfactants rst requires to introduce our reader to the world of interfacial physics and capillarity.

1.1.1 Interfacial energy, tension and force

1.1.1.

Interfacial tension

The existence of foams is related to the interfacial physics and its main characteristic parameter: the interfacial tension [START_REF] De Gennes | Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls and Waves[END_REF][START_REF] Rowlinson | Molecular Theory of Capillarity[END_REF].

A liquid is a condensed state whose molecules are attracted to each other. Molecules do not like lying at the surface of the liquid phase because they lose half of their interactions with their counterparts. Therefore, liquids adapt their shape to let the fewest possible surfaces appear.

We introduce the cohesion energy per molecule -E C . The loss of half of the cohesion energy per molecule at the surface is written as the interfacial energy per molecule e S = E C /2. The interfacial energy can be written as e S = γa 2 , where γ = E C /2a 2 is the interfacial tension and a is the molecular size, with a 2 the area of one exposed molecule at the surface. The total interfacial energy E I of a liquid system is found by adding up the contributions e s of every single molecule at the surface. We can write E I = γΣ where Σ is the total interfacial area.

We can now easily understand that creating interfaces costs the interfacial energy E I . Indeed, an interface tends to reduce and to minimize its area, and to resist any deformation. Generating interfaces require an energy E I per interfacial area Σ, whose value is the interfacial tension γ = dE I /dΣ, expressed in N/m. The higher the attractive interactions, the higher E C , the higher γ. Table 1.1 reports some characteristic air-liquid interfacial tensions γ al for common liquids. For oil, the liquid/vapour transition occurs at ambient temperature, which means that E C is close to the Boltzmann thermal energy k B T where k B = 1.38 * 10 -23 J.K -1 is the Boltzmann constant and T the temperature. For water, hydrogen bonding is dominant, thus the air-water interfacial tension γ aw is higher than the airoil interfacial tension γ ao . Mercury has the highest air-liquid interfacial tension, since metallic bonding is strongly cohesive (E C ≈ 1 eV).

An interfacial tension arises as soon as one phase is in a condensed state. Therefore, an inter- facial tension also exists when two liquids are connected to each other. In that case, the value of the interfacial tension takes into account the attractive interactions of the molecules of the two condensed phases, but also the molecular interaction between the two condensed phases.

A typical value of oil-water interfacial tension is around 20 mN/m. In this thesis, we deal with various interfacial tensions: air/water, air/oil, oil/water, solid/air, solid/liquid and air/liquid. To characterize the related interfacial tensions, we use the notation γ with di erent subscripts indicating the nature of the interface. For instance, γ ao stands for the interfacial tension at the air-oil interface.

Film and capillary force

As sketched in Fig. 1.1, let us imagine that we draw a liquid lm of length dx and width L. Its interfacial energy dE I is dE I = 2γ aw Ldx, since we have two air-water interfaces. This energy corresponds to the work δW done to draw the lm: δW = F dx where F is the applied force. Thus, F = 2γ aw L and γ aw can be interpreted as the lm tension. This capillary sti ness can be very high, which explains why some insects can walk on water or why a needle can oat as well, as seen in Fig. 1.2. Indeed, the mass m gerris of the gerris is small (≈ 2 g) and this insect has tiny hair which allows to pull a water lm at its edge. The balance between the weight m gerris g (g is the constant of gravity acceleration and g = 9.81 m.s -2 ) and the capillary force γ aw L gives a length of hair of 0.2 m, which is a common feature of the gerris.

Interface, drop and bubble

What shape can the interface take: plane, curved? This question is connected to the formation of drops and bubbles.

Drop and bubble formation

If we close the lm containing a given quantity of gas or if we have a given volume of gas in a liquid bath, we create a bubble. If we invert the gas phase and the liquid phase, we obtain a drop. The drop and the bubble adopt a shape that minimize their interfacial energy E I . Fig. 1.3 exhibits successive steps of calculations of the interfacial energy for a given volume of gas which is initially enclosed in a cubical bubble using the software Surface Evolver [START_REF] Brakke | The Surface Evolver[END_REF]. The shape corresponding to the minimum interfacial energy is the sphere when they are isolated and when external forces, such as wind and gravity, do not play any signi cant role.

The Laplace law and pressure

The spherical shape of the drop or the bubble implies a mechanical equilibrium inside the sphere, requiring that the inner pressure be higher than the outer pressure. Indeed, anyone has ever noticed that one needs to blow into a lm to create a bubble, i.e. to induce an overpressure ∆p L . In 1750, Leidenfrost [START_REF] Leidenfrost | Traité sur Certaines Propriétés de l'Eau -De aquae communis nonnullis qualitatibus tractatus[END_REF] showed this overpressure experimentally by pinching a straw into a bubble, which emptied the gas phase. In 1805, Laplace was the rst to estimate ∆p L [START_REF] Laplace | [END_REF]. well-known Laplace law for a spherical drop in the air phase: For bubbles whose lm thickness is negligible, Eq. 1.1 becomes:

∆p L = 2γ al R d . ( 1
∆p L = 4γ al R b , (1.2)
since we have two air-liquid interfaces instead of a single one (R b is the bubble radius). In both cases, the smaller the drop and the bubble, the higher the Laplace pressure. This di erence of pressure explains the coarsening phenomena, i.e. the transfer of matter (air or liquid) from small to big neighbouring drops and bubbles through the continuous phase.

For any type of air-liquid and liquid-liquid interfaces, we can use Eq. 1.1 and 1.2 with the appropriate interfacial tension γ. And for any interfacial shape, we can use the generalized version of the Laplace law:

∆p L = γ 1 R 1 + 1 R 2 , (1.3)
where R 1 and R 2 are the main radii of curvature of the interface. To evaluate the Laplace pressure in the following chapters, we basically estimate or calculate the radii of curvature and use Eq. 1.3.

Measuring the interfacial tension

Several methods exist to measure the interfacial tension γ [START_REF] Adamson | Physical Chemistry of Surfaces[END_REF]. These methods use force calculations and surface minimization described in §1.1.1.2 and §1.1.2.1: -the Wilhelmy method where one measures the γ-dependent capillary force exerted on a blade that is pulled out from a liquid bath [START_REF] Wu | Dynamic surface tension measurement with a dynamic Wilhelmy plate technique[END_REF], -the pendant drop method where the shape of the drop is tted to the theoretical shape dictated by the balance between gravity and capillarity. It is the same procedure for sessile (drop deposited on a surface), rising (for instance when there is a di erence of density between two liquids) and spinning (by elongation of a drop inside another liquid) drops [START_REF] Del Rio | Axisymmetric drop shape analysis: Computational methods for the measurement of interfacial properties from the shape and dimensions of pendant and sessile drops[END_REF], -the capillary rise in a tube where the rising height is related to γ, by using the density, the radius of the tube and the gravity acceleration.

In this work, the technique that we use to measure all the interfacial tensions is the pendant drop method (see Fig. 1.5). The tting of the drop shape is proceeded by two methods. In the rst one, we directly use the result given by the dedicated measurement device Teclis which calculates γ through its own software. In the second one, we use our own code to extract γ from the drop image. AND ENERGY Figure 1.5: Pendant drop method with the tting of the drop shape.

Surfactants: who are you?

Bubbles appear when air is incorporated in the liquid phase by shaking and mixing, but can also nucleate from a solid wall due to impurities. However, those bubbles can coalesce with each other in a fraction of seconds. Indeed, the attractive Van der Waals forces tend to decrease the lm thickness until rupture. Thus, the question is: how do we increase their life expectancy? The answer : add surfactants!

Structure and e ect on interfacial tension

Surfactants are original molecules with a polar head and a long carbon tail. Those two parts have di erent a nities for water. On one hand, the polar head is hydrophilic and likes being surrounded by water molecules. On the other hand, the carbon chain is hydrophobic and hates being in contact with the water phase. Surfactants are often said to be amphiphilic due to this dual character. Most polar heads are anionic (sulfate -OSO - 3 ), cationic (ammonium -N + (CH 3 ) 3 ) or non-ionic (ethylene polyoxyde). The hydrophobic tail is generally a long aliphatic chain (10 carbon atoms). Fig. 1.6 shows the structure of common surfactants including those used in this thesis.

Surfactants adsorb at the air-water interface until saturation with the polar head in the aqueous phase and the hydrophobic tail in the air. The rest of surfactants remains in solution. However, in solution, surfactant molecules can aggregate due to the preference of carbon chains to ock together and to expel the polar heads. These aggregates, called micelles, are in equilibrium with isolated molecules when the bulk concentration of surfactants c is above the critical micellar concentration (cmc) [START_REF] Israelashvili | Intermolecular and surface forces[END_REF]. The cmc is an important parameter to characterize the nature of a surfactant and its e ect on the interfacial tension. Indeed, if one deposits a little droplet of surfactants on a water bath recovered by pepper, this latter will be expelled towards the outer ridge of the bath. In fact, pepper is put into motion by the surfactants that spread at the bath surface. This is only possible because surfactants decrease the interfacial tension. The air-liquid interfacial tension at the droplet location is lower than the air-liquid interfacial tension far from the droplet. Thus, the outer surface of the bath pulls on the inner surface and drags pepper with it. We can introduce the interfacial concentration of adsorbed surfactants Γ. The free enthalpy G S is given by G S = n S µ + γΣ where n S and µ are, respectively, the number of molecules at the surface and the chemical potential of the dissolved surfactant (chemical potentials have to be equal between the surfactant at the interface and in the solution). We di erentiate this relationship for dG S , which gives: with Γ = n S /Σ. When the bulk concentration c is small, we can consider the solution as ideal and write µ = µ 0 ( T ) + k B T ln c, where µ 0 ( T ) is the reference potential. By di erentiating µ, we get the well-known Gibbs' relationship [START_REF] Gibbs | [END_REF]:

dG S = µdn s + n s dµ + γdΣ + Σdγ. ( 1
dγ = -k B T Γ dc c . (1.6)
Thus, for adsorbed surfactants, if c increases, γ decreases. However, for c > cmc, µ remains constant with micelles in equilibrium with isolated molecules. Thus, γ stops decreasing and reaches a plateau value. However, when introducing surfactants into a solution, γ does not reach its equilibrium value instantaneously but slowly decreases with time. This time dependence is due to two factors: the surfactant molecule has to di use to the interface and has to adsorb at the interface. To do so, it has to overcome energy barrier (electrostatic or steric) and, at short times, the process can be kinematically limited. Thus, along with c, time has to be taken into account when measuring γ if the studied dynamics is examined at short times. To illustrate this stabilizing role, Fig. 1.9 shows a lm formed by pulling a frame out of a bath of surfactants [START_REF] De Gennes | Young soap lms[END_REF]. Surfactants immediately adsorb at the air-water interfaces. Since the concentration of surfactants c is higher at the bottom than in the middle of the lm, the interfacial tension γ aw is lower at the bottom. Thus, an interfacial pressure gradient is created upwards. The upper part of the lm "pulls" on the lower part. Therefore, balancing gravity and the interfacial capillary forces on an elementary volume of lm (with two surfactant monolayers on each interface) yields:

ρ w ge = 2 dγ aw dz , (1.7)
where ρ w is the solution density and z is the vertical position. Considering c > cmc and integrating 1.7 between 0 (γ aw = γ cmc which is the value of γ aw at the cmc) and the maximum height z max (γ aw = γ 0 with γ 0 the air-pure water interfacial tension), we have an estimation of z max = 2(γ 0 -γ cmc )/ρ w ge. For e = 1 µm and γ 0 -γ cmc ≈ 50 mN/m, z max ≈ 10 m!, as observed on giant soap lms. The stability of a lm is considerably enhanced by the presence of surfactants due to short-range interactions and interfacial tension gradients.

However, the role of surfactants in stabilizing soap lms cannot be reduced to a simple decrease of interfacial tension: the interfacial tension of tap water is 50 mN/m. According to the theory presented above, we nd z max ≈ 4 m. Yet, it is impossible to withdraw a lm out of tap water. Surfactants modify the interfacial mechanics in a more complex way, as tentatively described below.

E ect of surfactants on the interfacial mechanics

Surfactants increase the stability of interfaces, especially the stability of soap lms. However, all interfaces can still be deformed under normal and shear stresses. We characterize the ability of the interface to deform by viscoelastic parameters. The viscoelastic behaviour of an interface has a strong impact on the mechanical properties of macrostructures, such as foams [START_REF] Cantat | Foams: Structure and Dynamics[END_REF].

Interfacial viscoelasticity

The interfacial area varies when a compression and a dilatation are applied to the interface [START_REF] Cantat | Foams: Structure and Dynamics[END_REF]. The shape of the interface without any area variation can also change: in this case, a shear stress is applied. Fig. 1.10 sketches both situations.

THE PHYSICS OF INTERFACES AND SURFACTANTS

For an isotropic compression-dilatation sequence in small deformations, we can describe the normal stresses per unit length σ = σ xx = σ yy and the shear stress per unit length τ xy by:

σ = γ + E d δΣ Σ + η d 1 Σ dΣ dt , τ xy = 0 , (1.8)
where η d is the interfacial dilatational viscosity, E d the interfacial dilatational elasticity and δΣ the variation of interfacial area. In Eq. 1.8, the second term is the elastic contribution, while the third term represents the viscous contribution to the deformation. An elastic contribution appears because the variation δΣ induces a variation of the interfacial surfactant concentration Γ, which also changes the value of γ and thus the interface "tension". For insoluble surfactants, the variation of Γ is directly related to δΣ and E d is de ned as the well-known Gibbs'-Marangoni elastic modulus E G = dγ/d ln Σ. For soluble surfactants, E d depends on the frequency ω at which the interface is mechanically deformed in the case of periodic oscillations δΣ 0 e iwt . We can de ne a complex interfacial dilatational modulus E * (ω) = E d (ω) + iωη d (ω) and relate it to σ by:

σ = γ + Re(E * (ω) δΣ 0 Σ 0 e iwt ). (1.9)
In the following sections, we refer to |E * (ω)| as the parameter which allows us to characterize the mechanical response of an interface to normal stresses.

For shear deformation, it is also possible to de ne a shear stress σ S with elastic and viscous contributions. Like the dilatational modulus, we can de ne a complex interfacial shear modulus G * (ω) = G shear (ω) + iωη shear , where G shear and η shear are, respectively, the interfacial shear elasticity and the interfacial shear viscosity. More accurately, η shear can depend on the interfacial shear rate when polymers are added to the interface for instance. In that case, η shear also depends on the frequency ω.

The friction between molecules and the surfactant exchange between the interface and the bulk during deformation are in uenced by the excitation of the interface, which explains the variations of the di erent elastic and viscous moduli with ω.

Resistance of a soap lm

Looking at the interfacial viscoelasticity is essential because it plays an important role in the resistance of a soap lm to the deformation, and thus to the stability of bigger structures with multiple interfaces like foams.

We can imagine a uctuation of interfacial area because a lm is stretched, as sketched in Fig. 1.11 [START_REF] Vrij | Rupture of thin liquid lms due to spontaneous uctuations in thickness[END_REF]. The interfacial concentration of surfactants is higher at the edges than in the middle of the AND ENERGY lm. This unfavourable situation creates an interfacial tension gradient which is proportional to the dilatational modulus |E * |. The gradient drives surfactants back to the location where the concentration is the lowest. Moreover, this gradient also induces a liquid back ow to the same location by a Marangoni e ect (i.e. a liquid ow induced by interfacial tension gradients). Eventually, the lm does not break and is able to resist e ciently to the deformation.

The resistance of a lm is mainly characterized by its dilatational and shear modulus |E * | and |G * | de ned previously. By analogy to materials science, they recall the sti ness of a material. "Fracture" in the interfacial layer occurs if |E * | is too high, "plasticity" appears if |E * | is too small. The value of these parameters can in uence the ow dynamics inside the lm for example [START_REF] Cantat | Foams: Structure and Dynamics[END_REF].

Rigid and mobile interfaces

The interfacial rigidity depends on the values of the dilatational modulus |E * | and the shear viscosity η shear . When interfaces are non-rigid, they are called "mobile". In this case, |E * | is basically of the order of a few mN/m and η shear varies from 10 -8 to 10 -7 kg.s -1 [START_REF] Cantat | Foams: Structure and Dynamics[END_REF]. If interfaces are rigid, they are deemed as "immobile". Then, |E * | increases around a few hundreds of mN/m and η shear gets to 10 -5 kg.s -1 . Fig. 1.12 shows measurements of |E * | for a mobile interface (SDS: Sodium dodecylsulfate) and an immobile interface (CAPB -Cocoamidopropylbetain -SLES: Sodium laurylethylsulfate -Mac: Myristic acid) with air and olive oil. To measure |E * |, we apply 10 cycles of periodic oscillations to a pendant drop of volume V d = 1 -2 µL at a frequency f = 0.2 Hz with a variable amplitude from 0.2 µL (for small γ) to 2 µL (for high γ). We can notice a strong di erence from 10 -20 mN/m to 300 mN/m for air-water interfaces in the case of rigid interfaces as γ decreases. The di erence is less signi cant for oil-water interfaces [START_REF] Bon Llon | Viscoelasticity of monolayers at oil-water interfaces[END_REF] but still sti er for CAPB solutions. Demonstrating the immobile character of the CAPB interfaces justi es the model used in Chapter 5. We will see that interfacial mechanics has a strong in uence on foam mechanics at a global scale, as observed in foam rheology or drainage.

Soap lm: a fragile structure

Without any perturbation from other immiscible liquid and solid phases, a naked soap lm is an evolving object that shrinks and that can break. Such phenomena involve di erent buoyancy and capillarity-driven mechanisms [START_REF] Rio | Thermodynamic and mechanical timescales involved in foam lm rupture and liquid foam coalescence[END_REF].

1.1.5.1 How lms shrink... As sketched in Fig. 1.1 and shown in Eq. 1.7, a vertical lm can survive because surfactants bring rigidity to the lm and drive a ow of water upwards. However, water can still drain downwards because of gravity, which could decrease the lm thickness despite the surfactantdriven resistance. The ow rate downwards is relatively small. The ow of liquid is a Poiseuille ow. Balancing viscous forces and pressure gradient gives in scaling law η w v/e 2 ∼ ∆p/L, where η w , v, e, ∆p and L are respectively the water dynamic viscosity, the average ow velocity, the lm thickness, the pressure di erence between the top and the bottom of the lm and the length of the lm. Thus, v ∝ e 2 and the ow rate is Q f ≈ veW (where W is the lm width), which gives Q f ∝ W e 3 , so the drainage process is very slow.

Due to water drainage, the lm thickness can be smaller at one location and larger at another location. In the region of smaller thickness, the volume of water is smaller than the volume in the regions of larger thickness. Thus, because of buoyancy, the region of smaller thickness rises to the top of the lm, as depicted in Fig. 1.13a. Such a movement, analogous to convection of uid, can destabilize the lm faster than classical drainage and entrain its rupture.

These convective uid motions are emphasized by the presence of the edges of a lm. As shown in Fig. 1.13b, the pressure p in the lm around the frame is lower than the pressure close to the center of the lm (atmospheric pressure p atm ). Hence, a ow is induced from the center to the edges. Regions of smaller thickness called black lms appear and rise to the top of the frame due to buoyancy. This creates a turbulent ow from the edges to the middle of the lm. This phenomenon, called marginal regeneration, also exists when the lm is connected to a Plateau border, the curvature of the Plateau border inducing uid motion. Marginal regeneration disturbs the lm stability and can lead to rupture.

These mechanisms can be coupled with thermal uctuations engendering a decrease of the lm thickness, and also a diminution of the disjoining pressure Π. This speeds up the shrinking of the lm and its rupture if the interfacial viscoelasticity de ned in §1.1.4 is not su cient enough. When a lm breaks, the rupture dynamics can be well-described by the Taylor-Culick's law [START_REF] Cantat | Foams: Structure and Dynamics[END_REF][START_REF] Culick | Comments on a ruptured soap lm[END_REF][START_REF] Hutzler | Foam drainage in 2D[END_REF][START_REF] Rio | Thermodynamic and mechanical timescales involved in foam lm rupture and liquid foam coalescence[END_REF][START_REF] Stevenson | Foam Engineering: Fundamentals and Applications[END_REF][START_REF] Taylor | The dynamics of thin sheets of uid. Part III: Disintegration of uid sheets[END_REF]. Fig. 1.14 presents the theoretical analysis of the lm opening. In 1D dimension or in cylindrical coordinates, we can write the mass m rim of the water rim around the edge of the hole as m rim = ρ w er h where ρ w and r h are, respectively, the water density and the hole radius. The velocity v rim of the hole edge is dr h /dt. Then, the capillary force per unit length acting on the rim is 2γ aw . The balance between the quantity of motion d(m rim v rim )/dt, with v rim constant, and the capillary force yields the following equation for the hole velocity:

v rim = 2γ aw ρ w e 1/2
.

(1.10) One can also state that the interfacial energy of the bubble per unit length 2r h γ aw is converted into the kinetic energy (1/2)m rim v 2 rim = (1/2)ρ w er h v 2 rim , which gives v rim = (4γ aw /ρ w e) 1/2 . Eq. 1.10 actually violates energy conservation. However, it does not if one considers that dissipation also arises through the propagation of shock waves in the lm. Recent works [START_REF] Petit | Holes and cracks in rigid foam lms[END_REF] have shown that the rigidity of the air-water interfaces modi es the Culick's law and generates cracks in the breaking of a soap lm.

1.2 Aqueous foams: de nition, structure, characteristics and main properties

In §1.1, we travelled from the nanoscale of surfactants to the millimetric scale of isolated lms and bubbles. We focus now on the scale of several bubbles which pack and interact with each other to generate aqueous foams [START_REF] Cantat | Foams: Structure and Dynamics[END_REF][START_REF] Sadoc | Foams and Emulsions[END_REF][START_REF] Weaire | The Physics of Foams[END_REF].

CHAPTER 1: AQUEOUS FOAMS AND THEIR APPLICATIONS IN ENVIRONMENT

AND ENERGY

Foam structure

An aqueous foam is a dispersion of gas bubbles in a water-based phase (generally an aqueous solution of surfactants). The structure of a foam highly depends on the liquid fraction φ l , which is the ratio between the volume of liquid and the total volume of the foam. For wet and dry foams, we can de ne speci c structural features depicted in Fig. 1.16. When bubbles start being squeezed, thin lms (≈ 100 nm to 1 µm width) between bubbles are therein created and meet each other by groups of three. The liquid microchannel (≈ 10 to 100 µm width and 100 µm to 1 mm length) that appears at their jonction is called Plateau border (named after the physicist Joseph Plateau (1801-1883)). Four Plateau borders meet at the intersections called

Dry foams

When the foam is su ciently dry (φ l < 0.01), we can apply the Plateau laws [START_REF] Almgren | The geometry of soap lms and soap bubbles[END_REF][START_REF] Plateau | Statique Expérimentale et Théorique des Liquides Soumis aux Seules Forces Moléculaires[END_REF] to predict the structure of the foam. We assume that other conditions, such as mechanical equilibrium and incompressibility, are also ful lled. As sketched in Fig. 1.16, the Plateau laws are given as follows:

-Films have a constant mean radius of curvature de ned in Eq. 1.3.

-The angle β between two lms is 120 o . The interfacial forces induced by the three lms and acting on a Plateau border transversely can only be balanced if β = 120 o . -The Plateau borders meet together with an angle of 109.5 o to form a tetraedron.

In this con guration, the Laplace law (Eq. 1.3) determines the curvature of the air-water interfaces and thus the pressure p in the liquid phase, i.e. in the Plateau borders and vertices. Fig. 1.17 shows the curvature radii of a Plateau where r P B designates the typical width of a Plateau border. The negative curvature radius along the Plateau border cross-section is R 1 ∼ r P B while the curvature radius in the Plateau border plane is R 2 ∼ R b where R b is the bubble radius. Since r P B R b , Eq. 1.3 yields:

∆p = P b -p ≈ p atm -p = γ aw r P B , (1.11)
where the gas pressure in the bubbles is P b ≈ p atm , which is the atmospheric pressure if we neglect the gas overpressure in the bubbles. For r P B ≈ 10 µm and γ aw ≈ 30 mN/m, P b -p ≈ 3000 Pa. The liquid phase is thus in strong underpressure compared to the gas phase. We can estimate the value of r P B and thus the capillary pressure, with respect to R b and φ l by considering a Kelvin structure [START_REF] Kelvin | On the division of space with minimum partitional area[END_REF] for the shape of a packed bubble in a dry foam, as sketched in Fig. 1.18. The Kelvin cell is a tetrakaidecahedron with 8 identical hexagonal faces and 6 identical square faces whose side is a Plateau border. Weaire and Phelan have brought corrections to this structure by considering pentagonal faces [START_REF] Weaire | A counter-example to Kelvin's conjecture on minimal surfaces[END_REF]. Meanwhile, for φ l < 0.01, we have, by geometrical calculations [START_REF] Cantat | Foams: Structure and Dynamics[END_REF]:

r P B = 1.74R b φ 1/2 l .
(1.12) Thus, the lower the liquid fraction, the higher the capillary pressure, the lower the pressure in the liquid phase compared to the gas phase in the bubbles. This estimation is especially used in Chapter 5 for the study of the imbibition into aqueous foams.

Main properties of foams

Owing to their peculiar structure presented in §1.2.1.2, foams are endowed with properties of considerable interest, especially in terms of pressure-driven ow inside the foam.

Capillary suction and disjoining pressure

Since the interfaces are parallel to each other in the lms, their curvatures are very high and the subsequent pressure in the lms is p f ≈ P b initially when the foam takes shape. Since p P b , the pressure in the Plateau border is lower than the pressure in the lms, which triggers the motion of the liquid phase from the lms to the Plateau borders. While a lm is emptied from its liquid phase, the pressures get equilibrated and p f decreases to reach p. For the lm to be mechanically stable, a disjoining pressure Π(e) (which depends on the lm thickness e) must be exerted on the interface and its expression is given by:

Π(e) = P b -p ≈ γ aw r P B . (1.13)
From φ l , R b and Eq. 1.12, we deduce Π. Then by reporting this value on the theoretical calibration curves Π(e) for each type of surfactants [START_REF] Bergeron | Disjoining pressures and lm stability of alkyltrimethylammonium bromide foam lms[END_REF][START_REF] Bergeron | Monolayer spreading of polydimethylsiloxane oil on surfactant solutions[END_REF][START_REF] Bergeron | Equilibrium measurements of oscillatory disjoining pressures in aqueous foam lms[END_REF][START_REF] Bergeron | Disjoining pressure and strati cation in asymmetric thinliquid lms[END_REF], we can obtain the lm thickness e and entirely determine the structural parameters of the foam at equilibrium.

Osmosis in foams

Since the pressure in the Plateau borders p < p atm , if the foam is in contact with a reservoir at the atmospheric pressure p atm , liquid rises from the reservoir to the foam through the Plateau border. Physically speaking, a foam wants to reduce its interfacial energy. If a foam is wet, it has fewer interfaces than a dry foam, which explains the imbibition of liquid. This imbibition phenomenon can be quanti ed by the osmotic pressure Π o , which is the applied pressure to prevent liquid from entering the foam [START_REF] Princen | Osmotic pressure of foams and highly concentrated emulsions. 2. Determination from the variation in volume fraction with height in an equilibrated column[END_REF].

For a dry foam, Eq. 1.11 and Eq. 1.12, we deduce that:

Π o = p atm -p ≈ γ aw 1.74R b φ 1/2 l . (1.14)
If the bubbles and the liquid fraction are small, we have more interfaces, thus the osmotic pressure is higher and imbibition more important with the aim to reduce the interfacial energy. As underlined before, the osmotic pressure depends on the variation of interfacial areas with the liq-

CHAPTER 1: AQUEOUS FOAMS AND THEIR APPLICATIONS IN ENVIRONMENT AND ENERGY

uid fraction. Simulations with the software Surface Evolver [START_REF] Brakke | The Surface Evolver and the stability of liquid surfaces[END_REF] allow to estimate this variation and are in good agreement with the empirical law for Π o for φ l < φ * l = 0.26 (thus generalize Eq. 1.14 for wet foams) calculated by Princen et al. [START_REF] Mason | Osmotic pressure and viscoelastic shear moduli of concentrated emulsions[END_REF][START_REF] Princen | Osmotic pressure of foams and highly concentrated emulsions. 2. Determination from the variation in volume fraction with height in an equilibrated column[END_REF]:

Π o = 7.3 γ aw R b φ 1/2 l (φ l -φ * l ) 2 .
(1.15)

For dry foams, φ l φ * l , thus Eq. 1.15 is approximated by

Π o = 7.3γ aw φ * 2 l /R b φ 1/2 l . For φ * l = 0.26, Π o ≈ 0.49γ aw /R b φ 1/2 l
which is close to Eq. 1.14 given that 1/1.74 ≈ 0.57. So the consistency and the continuity between both models for dry and wet foams are ensured. This osmosis and imbibition process in foams is at the origin of the work presented in Chapter 5.

How foams ow...

Along with the interactions between foams and external uids, our work also focuses on the hydrodynamics of owing foams in Chapter 6 especially. Therefore, introducing how a foam deforms and ows (i.e. its rheology) is of considerable interest. The interplay between the structural behaviour and the nature of the ow is particularly relevant.

Foams are complex uids

The classi cation of "matter" is divided into several categories. On one hand, there is the simple uids' world, such as water, oil or honey. These uids are also called newtonian uids. On the other hand, we have the elastic solids (or Hookean solids), such as metals and ceramics. In-between these two categories, one nds the complex uids [START_REF] Coussot | Rheophysics -Matter in All States[END_REF], such as toothpaste, ketchup, mayonnaise, cement, corn starch and liquid foams (as presented in the following). These uids are known as non-newtonian uids. The di erence between the newtonian and the non-newtonian uids lies on the response of the uid to a deformation (which is assumed to be generally small to remain in the eld of the theory of linear mechanics). The response of the uid can be of di erent solid-liquid natures: -elastic: the deformation is reversible. The uid stores energy and can release it when coming back to its initial state.

-plastic: the deformation becomes irreversible above a certain threshold and the uid keeps its nal state without having the possibility to come back to its initial state. -viscous: the uid ows by losing energy (viscous dissipation). Complex uids combine all these features, while simple uids are purely viscous and Hookean solids purely elastic.

AQUEOUS FOAMS: DEFINITION, STRUCTURE, CHARACTERISTICS AND MAIN PROPERTIES 33

The behaviour of a owing complex uid can be described and quanti ed by imposing a shear stress τ xy or a small strain or a strain rate ˙ on an elementary volume of uid. For instance, for foams, this volume consists in several layers of bubbles. By applying a stress ramp at steady-state in a rheometer (respectively a strain ramp) and measuring the strain rate ˙ (respectively the shear stress τ xy ), we can de ne τ xy as:

τ xy = χ( , ˙ , t, λ) , (1.16)
where χ can be a function of ˙ , the strain , the time t and a structural constant λ taking the change of inner structure into account. For a purely viscous newtonian uid, the shear stress only depends on the shear rate through the relationship τ xy = η ˙ where η (Pa.s) is the shear viscosity, which is among the usual values (with the air-liquid interfacial tension for instance and the density) given to characterize a uid. For a purely elastic material, the shear stress only depends on the strain through the relationship τ = G where G is the shear modulus of the material. The most common rheological laws are summarized in Fig. 1.19. For all complex uids, the apparent shear viscosity η app = τ xy / ˙ depends on ˙ , while η app = η and is constant for newtonian uids. The viscoelastic behaviour of a complex uid can also be detected by applying a small oscillatory deformation and measuring the shear elastic modulus G and the loss shear modulus G , and thus the complex shear modulus G * = G + iG . This test is equivalent to the test used to characterize the rheological behaviour of a 2D lm by measuring its interfacial shear modulus |E * | (see Eq. 1.9).

Among the family of complex uids, a 3D foam can be assimilated and well-described as a yield stress uid [START_REF] Coussot | Rheophysics -Matter in All States[END_REF]. A yield stress uid is a uid that does not ow as long as the applied shear stress τ xy is below a threshold value τ Y called yield stress. When τ xy < τ Y , the uid remains in an elastic reversible state (or viscoelastic when viscous e ects are taken into account at small stress amplitudes) and does not ow. However, when τ xy > τ Y , the foam deforms irreversibly and ows. The rheological behaviour can be written as the Herschel-Buckley law: [START_REF] Herzhaft | Measurement and modeling of the ow behavior of aqueous foams using a recirculating pipe rheometer[END_REF]. The liquid fractions are given in the legend. The yield stress tends to disappear for high liquid fractions above 0.5 for which the foam looks more like a suspension of bubbles. The dotted lines are the Herschel-Buckley t with the power law coe cient n varying from 1 for φ l = 0.4 to 0.66 and 0.5 for φ l ≈ 0.2.

˙ = 0 if τ xy < τ Y , τ xy = τ Y + k p ˙ n if τ xy > τ Y , (1.17
For the foam we use in Chapter 6, we determine the yield stress by a creep experiment. In a cone-plate geometry (rheometer Anton-Paar), we apply a constant shear stress for 120 s to a

AQUEOUS FOAMS: DEFINITION, STRUCTURE, CHARACTERISTICS AND MAIN PROPERTIES 35

foam sample (2 mm thick initially) whose bubble radius is 60 µm. We measure the deformation (or strain) as a function of time, from which we deduce the shear rate ˙ at any moment. The shear rate decreases with time, so the apparent viscosity η app = τ xy / ˙ increases. Once the shear stress is removed after 120 s, the deformation and the apparent viscosity (in nite shear rate) drop abruptly to a value close to 0. We run a series of creep experiments for di erent successive shear stresses and record the value of the apparent viscosity at the moment where the shear stress is removed (yield point). Fig. 1.21 plots the apparent viscosity as a function of the applied shear stress. The drop in the value of the apparent viscosity, characteristic of a viscous ow behaviour, corresponds to the equilibrium yield-stress point. We nd a value of τ Y ≈ 20 Pa, which is of the same order of magnitude as the one measured in Fig. 1.20.

Relationship between the structure and the foam rheology

Now, how do τ Y , k p and n vary from one foam to another? [START_REF] Cohen-Addad | Rheology of foams and highly-concentrated emulsions[END_REF][START_REF] Dollet | Rheology of foams[END_REF] It all depends on the initial structure of the foam and on the nature of the surfactants.

When a shear deformation is applied to a foam in a quasistatic con guration, the moment when the foam starts owing (i.e. at the yield stress τ Y ) corresponds to the local situation where four Plateau borders meet with each other at one vertex. According to the Plateau laws ( §1.2.1.2), this is highly unstable. Thus, both adjacent bubbles separate from each other by the creation of a new Plateau border: this rearrangement is called T1 event [START_REF] Reinelt | Simple shearing ow of a dry Kelvin soap foam[END_REF][START_REF] Reinelt | Simple shearing ow of dry soap foams with tetrahedrally close-packed structure[END_REF][START_REF] Weaire | Soap, cells and statistics-random patterns in two dimensions[END_REF] As sketched in Fig. 1.22b, one can show that, when subjected to a deformation , the perimeter length of a bubble in the direction of the shear plane is on the order of R b (1+ 2 ) 1/2 . The associated energy is E S ≈ γ aw R b (1+ 2 ) 1/2 . The related elastic force is F elastic = dE S /du ≈ γ aw /(1+ 2 ) 1/2 , where u = R b is the displacement at the top of a bubble. The shear stress τ is the shear force per horizontal length, i.e τ ≈ F elastic /R b . Hence, τ ≈ γ aw /R b (1 + 2 ) 1/2 . For small deformations 1, we have τ ≈ γ aw /R b at rst order. Since the quasistatic shear modulus G is de ned by τ ≈ G . We have:

G ≈ γ aw R b . (1.18)
At the yield point where four Plateau borders meet with each other (1.22c), the deformation reaches a value of order 1, since a bubble is deformed on a length close to the bubble radius, thus

Y ≈ 1. Thus, we have τ Y ≈ G Y ≈ G and nally: τ Y ≈ γ aw R b . (1.19)
For γ aw = 30 mN/m and R b = 1 mm, τ Y = 30 Pa. This value is small compared to usual plasticity yield values for common solids, such as metals, which varies from 10 9 to 10 11 Pa. Therefore, the word "soft matter" is well-adapted to describe the mechanical behaviour of foams.

This estimation of the yield stress is a good approximate for dry foams in which the liquid phase

APPLICATIONS IN ENVIRONMENT AND ENERGY

has little in uence on the global behaviour of the foam. Yet, when foams are wetter, the liquid fraction φ l has a great importance and τ Y ∼ (γ aw /R b )(φ * l -φ l ) 2 [START_REF] Princen | Rheology of foams and highly concentrated emulsions[END_REF] where φ * l is the ordered (0.26) or disordered (0.36) maximum packing fraction.

The power coe cient n strongly depends on the nature of the air-water interfaces [START_REF] Denkov | Wall slip and viscous dissipation in sheared foams: E ect of surface mobility[END_REF], thus on the nature of the surfactants, as stated in §1.1.4. For mobile interfaces (small dilatational modulus |E * |), such as SDS, the interfaces are compressible and can deform and roll around the bubbles. This means that dissipation mainly occurs in the Plateau borders and n ≈ 0.5. However for rigid interfaces (high dilatational modulus |E * |), such as the CAPB solution mainly used in Chapter 2, 3, 5 and 6, the interfaces are incompressible, move with the bubbles and slide upon each other. This means that dissipation mostly occurs in the lms. In that case, n ≈ 0.2. Typical measurements from [START_REF] Golemanov | Surfactant mixtures for control of bubble surface mobility in foam studies[END_REF] show these two power laws in Fig. 1.23. Eventually, the plastic viscosity k p is highly dependent on the air-water interfacial tension, but also on the bubble radius and the liquid fraction. More generally, the interfacial rigidity has a strong in uence on the foam rheological parameters. The question of viscous dissipation and the local role of the interfaces is tackled into detail in Chapter 6.

Applications in environment and energy

As underlined in §1.1 and §1.2, foams have a complex structure with many physical, chemical and mechanical properties that can be quanti ed experimentally and described theoretically.

CHAPTER 1: AQUEOUS FOAMS AND THEIR APPLICATIONS IN ENVIRONMENT AND ENERGY

These properties allow to use foams in numerous applications with a special emphasis on soil remediation and enhanced oil recovery, which are the potential targets of our study.

Foams in everyday life

Foams are everywhere around us. They naturally appear when air or any gas is mixed with liquid. If surfactants are added, they can hold longer and form a dense structure of bubbles involved with entertaining, practical and industrial applications. Every foam constituent presented in §1.2.1.2 is related to a speci c property. The gas phase represents 99 % of the foam volume. Thus, such a quantity provides a huge insulating reservoir, as air is known to be a very good insulator. Fire ghters use the same idea to sprinkle a carpet of thick foam preventing re from being nurtured by oxygen. They also use the lower density of foams compared to water and hydrocarbons to cover up re.

For detergency issues, surfactants have a strong e ect on the cleaning process but foams do not. They are a marketing symbol which enables to sell more body care products. Foams appear if contaminants and dirt vanish, typically for dishwashing issues. Thus, they are a good indicator of the cleaning e ciency. On the opposite, if foams appear in nature (sea, river), it means that waters are contaminated by waste containing surfactants.

We probably eat foams at least once per day. Creams, beers, champagne, gazei ed drinks are or have foams in their contents. The foamy nature acts as a avor enhancer owing to the number of interfaces. They ensure a stronger and "refreshing" contact between our taste sensors and the product we eat and drink. We can even nd more foam-based products if we consider that cakes and bread were a foamy paste before baking.

The yield stress character of a foam is particularly at stake for shaving activities. Foams o er more comfort and coverage due to their solid-liquid nature. A single motion of razor blade allows to get rid of the shaving foam e ciently.

In industry

Apart from daily activities, foams have considerable interest in industrial applications [START_REF] Stevenson | Foam Engineering: Fundamentals and Applications[END_REF]. Since air is its main constituent, foams are cheap products and companies make a large economic margin by selling air. They also generate less waste. When a facility or a soil is decontaminated with a foam, it is cost-saving and environment-friendly, since less energy is used to separate the noxious agents from the foam. Foams can also be spread very easily and very quickly. This is relevant for re ghting but also insulating any chemical accidents and dampening explosions. The liquid phase can transport and capture di erent matters: solid (in case of a separation process like froth otation), molecular (for medicinal purpose) and liquid (as we see in this study).

The structure of the foam, half-solid, half-liquid, and its ow properties are bene cial to push extractible liquids, such as in enhanced oil recovery. The dual structure enables to use foams as a precursor for the generation of solid foams. Aluminum foams (for car bumpers), polyurethane foams (for seats and mattresses), cement foams (for building insulation) are all intense subjects of fundamental and applied research. In the following sections, we focus on two applications related to the extent of this work: soil remediation and enhanced oil recovery.

Towards a cleaner environment: soil remediation

The pollution of soils is a burning issue of public health and environmental awareness. Recently, an "Atlas de la France toxique" has been published. This book shows how contaminated soils (and waters) are located in France. A map of the polluted and degraded sites worldwide, published by the UNEP (United Nations Environment Program), is shown in Fig. 1.25. Worldwide, the situation is even worse due to population growth, industrial development and intensi ed agriculture. AND ENERGY However, heavy technical solutions exist to tackle this problem. But research on new ones, including foam engineering, is on track.

Common remediation techniques: advantages and drawbacks

We insist on some of the main remediation techniques. Our description is not comprehensive. Thus, more details can be found in [START_REF] Colombano | Quelles Techniques pour quel Traitements -Analyse Coûts-Béné ces[END_REF]. Three di erent categories of processes can be highlighted and avoid the excavation of polluted land, which is cost-e ective, heavy in means and tools, and destructive.

First, we have the physical methods. One can directly extract the pollutants from the soil without losing the treated area. As sketched in Fig. 1.26, many deep wells are dug into the porous soil. From one or several wells, air is injected and the soil is ventilated. This technique is particularly e cient for volatile pollutants. It is also possible to entrain the pollution by air injection into the ground waters and to pump these waters through extraction wells. The non-destruction of the soil is a strong advantage. However, the heterogeneity and the variable permeability of the soil are the main drawback, as well as the entrainment of organic matter during extraction.

Another physical method is to trap the pollution inside the soil. One can con ne the polluted area with a surface coverage or directly solidify the pollution by the injection of concrete. This process is represented in Fig. 1.27. This technique is relevant for a high volume of pollutants. Meanwhile, one does not get rid of the pollution and the con nement requires a long-term control and monitoring. This technique is e cient for soluble and concentrated pollutants, as well as permeable terrains. Yet, it does not hinder the transfer of pollutants from one place to another by adsorption or trapping in the pores. Soil heterogeneity and the high volume of waters to be treated are also real obstacles.

Third, we have biological methods. They resort to plants to remediate soils, also known as phytoremediation. This method is completely environment-friendly and renewable. It is e cient for small areas of pollution close to the surface, as far as the plant roots can penetrate the soil. Thus, Foams could allow to solve some of these issues. Indeed, foams are a cheap product, since air is basically free. It does not require a lot of process and the material is light to transport. It would generate fewer by-products and less waste if the pollutants were to be absorbed and entrained by the foam. Besides, foams could carry chemical agents in the liquid phase and play the role of a transport agent into the porous soil. The transport of such agents is actually much more e cient with foams than with air or any newtonian liquids. Foams are able to span more polluted area and to invade all the defects and pores (providing that bubbles are small enough) thanks to their non-newtonian ow properties and their deformability. Therefore, such a ow allows foams to follow every direction. On the opposite, a pure cleaning solution would follow one direction lead by gravity and the injection pressure, with the capillary pressure preventing liquids from entering the smallest pores. Nuclear industry has already used foam processing to decommission old nuclear plants. The burning question of radioactive waste treatment obviously encourages research teams to nd a method that generates less waste. In terms of e ciency, foams can decontaminate very complex structures, such as a nuclear tank and its related facilities. In this subject, the life expectancy of foams is also at stake to increase their e ciency and to use fewer surfactants (by the addition of plasti ers and stabilizing nanoparticles). More information on this subject of nuclear decommissioning by foams and other methods can be found in [START_REF] Dame | Relations between physicochemical properties and instability of decontamination foams[END_REF][START_REF] Sarrade | Pour des Déchets et des E uents plus Propres[END_REF]. Applying such processes to soil remediation appears very attracting, although the structure of a soil is more disordered and unknown than a tank.

Towards more energy supply: enhanced oil recovery

Soil remediation is a potential application to extract a material for people who would like to recycle or dispose of this material. However, extraction also means extracting resources that can be used in terms of energy. This is the case when one talks about enhanced oil recovery.

Processes of enhanced oil recovery

The extraction of oil (conventional and shale oil) is a research topic of considerable scope for the energy industry. We can describe the recovery of oil in three di erent stages [2].

First, one has the primary recovery. Once the extraction well is drilled into the oil reservoir, the high pressure of gas (which is often dissolved in the oil phase) enables oil to come out to the surface. A system of pumps can also allow to increase the di erence of pressure between the reservoir and the surface. At the end of this stage, one extracts 10 to 20 % of the oil contained initially in the reservoir. While the reservoir is progressively emptied, the underground pressure decreases until being unsu cient to push oil out of the reservoir. At this point, companies have to provide energy to the reservoir to extract the remaining oil. This is done by injecting di erent uids into the soil to increase the pressure in the reservoir and push oil out. This is the secondary recovery. Fluids are water, natural gas and air. This technique is cost-e ective, since energy needs to be provided to inject the aforementioned uids. Meanwhile, the recovery rate is generally improved up to 35 to 45 %.

How about the remaining 55 to 65 %? This is the tertiary recovery or enhanced oil recovery. The idea is to increase the mobility of oil. The low recovery rate is essentially due to the high oil viscosity (1 to 10 8 Pa.s) and the existence of a high capillary pressure induced by the porous scale in the soil (see Eq. 1.1). One method consists in heating up the oil phase to decrease its viscosity and make it ow more easily. Water vapour is injected to increase the temperature. Burning oil (combustion) inside the reservoir is another heating technique.

The second well-known method to increase the mobility of oil is the injection of carbon dioxide. By doing so at high pressure and temperature, carbon dioxide goes to a supercritical state which is used as an organic solvent to dissolve and drag oil out. Then, the injection of water can also complement the process to extract the least viscous oil. With this technique, the recovery rate goes up between 40 and 60 %. We can cite two other techniques to improve the output of oil. Sometimes, micro-organisms are used to decompose oil into subproducts that are easier to extract. More frequently, surfactants are added to decrease the interfacial tension between water and oil and untrap oil from the porous matrix. This method is the one we are interested in. The question is: can we reach a better extraction e ciency above 60 %? This is where foams could help. AND ENERGY

Surfactants, foams and the quest of oil

The injection of gas (especially carbon dioxide) and surfactants are two usual techniques to extract more oil. Why not combining both of them to inject aqueous foams inside the reservoir? This idea has come to the fore and promoted by oil research laboratories, such as the Institut Français du Pétrole Energies Nouvelles. Indeed, this could provide new opportunities: -replace water by foams to push oil out. Like for soil remediation, we use fewer resources and make the most of the ow properties of foams, -increase the mobility of oil by injecting gas, -decrease the oil-water interfacial tension, -ensure a better contact between oil and surfactants by co-transport with a gas phase. Hence, we use foams as a transport vector for chemical reactants.

-use foams as a direct agent to absorb oil. Foams for enhanced oil recovery have been and are still the subject of some research on real soil systems and laboratory porous models. In particular, the injection of foams to push oil out of the porous matrix directly has been quantitatively studied [START_REF] Conn | Visualizing oil displacement with foam in a micro uidic device with permeability contrast[END_REF][START_REF] Farajzadeh | Foam-oil interaction in porous media: Implications for foam assisted enhanced oil recovery[END_REF][START_REF] Jones | Two-dimensional constriction ows of foams[END_REF][START_REF] Quennouz | Micro uidic study of foams ow for enhanced oil recovery (EOR)[END_REF][START_REF] Rossen | Foams in enhanced oil recovery[END_REF][START_REF] Schramm | Foams: Fundamentals and Applications in the Petroleum Industry[END_REF].

Conclusions

In this chapter, we have presented the di erent physical objects that will be part of the following chapters: surfactants, lms, drops, foams and porous media. We have summed up their main properties and drawn the bond between all of them. Surfactants are at the heart of the stability of air-water interfaces. They allow soap lms to hold in time by putting them under "tension". Short-range and electrostatic interactions, as well as interfacial tension gradients enhance the resistance of a lm to deformation. This tension is quanti ed by the interfacial tension γ and the related interfacial energy E I . This interfacial energy is proportional to the interfacial area. As a consequence, bubbles and drops appear when a lm gets enclosed by minimizing the energy E I and adopting a spherical shape.

The di erence of pressure ∆p L through the air-liquid interface (bubbles in air) or the liquidliquid interface (oil drops in water) is determined by the well-known Laplace law, which states that ∆p L is proportional to the interfacial tension and the main radii of curvature of the interface. The existence of an overpressure in a bubble due to this Laplace pressure maintains their shape. However, when bubbles ock together, they deform and form a dense assembly called foam.

Aqueous foams have a complex structure, potentially periodic for ordered bubbles, de ned by 1.4 CONCLUSIONS several typical features: lms, Plateau borders at the bubbles' jonction and vertices at the Plateau borders' jonction. The geometry of foams gives astounding mechanical and physical properties. Among those, we can cite the osmosis e ects that we will study in Chapter 2 and 5, the rheological behaviour similar to a yield stress uid (either solid or liquid), the high interfacial rigidity (for speci c surfactants) enabling foams to interact with other uids, as we will see in Chapter 3, 5 and 6.

Combining these mechanical and physical properties o ers huge opportunities in numerous daily and industrial applications from detergency to food industry. In manufacturing and civil engineering, foams represent the rst stage to add air inclusions. In soil remediation and enhanced oil recovery, they contribute to extract more pollutants and oil. In these latters, the mechanisms at play involve the interactions between foams and immiscible uids in a three-phase system. Dynamical phenomena involving the three-phase system air, water and oil have been studied at the scale of a single Plateau border and an entire aqueous foam. In particular, some studies have shown that oil droplets can enhance foamability (how foams are generated) and foam stability (how foams can hold in time) [START_REF] Koczo | E ect of oil on foam stability: Aqueous foams stabilized by emulsions[END_REF][START_REF] Salonen | Dual gas and oil dispersions in water: production and stability of foamulsion[END_REF], although oil is known to be a good antifoaming agent. Such studies discuss about aqueous foams stabilized by emulsions, which are dispersions of oil in an aqueous phase. These kinds of foams are called "foamulsions" by Salonen et al. [START_REF] Salonen | Dual gas and oil dispersions in water: production and stability of foamulsion[END_REF] and are characterized by the Plateau borders lled with oil droplets.

The situation we study is quite di erent. We want to use a naked foam as a liquid sponge in which oil is initially not present. Oil has to invade the foam while oil droplets play an active role in the formation of the foam for foamulsions. This idea of oil invasion is related to the work of Piroird et al. [START_REF] Piroird | Capillary ow of oil in a single foam microchannel[END_REF] on the propagation of an oil slug in a Plateau border and the simulations of an elongated oil droplet in a Plateau border by Neethling et al. [START_REF] Neethling | Modeling oil droplets in Plateau borders[END_REF]. Such studies have also been carried out for miscible liquids [START_REF] Cohen | Inertial mass transport and capillary hydraulic jump in a liquid foam microchannel[END_REF][START_REF] Cohen | Drop coalescence and liquid ow in a single plateau border[END_REF].

However, oil has the peculiarity that it can interact with the external aqueous phase of the foam and destabilize the system. Thus, studying the hydrodynamics at the air-oil-water interface and the stability of oil-laden foams is the rst step to understand how foams react to mechanical perturbations induced by the contact with an external oil phase and the possible invasion by this immiscible phase.

Chapter 2

Oil-foam interactions and stability

In our work, we try to understand how foams can transport immiscible liquids. Thus, studying how foams interact with these liquids and in particular how mechanically and physically stable this system can be are highly important. We need to consider the interaction between oil and every component of the foam at the local scale. In this chapter, we start by presenting the stability of soap lms and Plateau borders, when put into contact with oil. Then, we discuss experimental results on the collapse of foams induced by oil. Unlike foams containing dispersed oil droplets, what happens if oil is directly put into contact with the foam after generation -by spongeous absorption or forced injection? How do foams react to the invasion and the ow of oil afterwards? Is the oil-laden foam obtained with this experimental procedure stable?

Speci c variables

Stability of a soap lm towards oil

In this section, we rst look at the stability of a soap lm when oil droplets are found inside the lm. We highlight the di erent criteria that come into play to predict the antifoaming behaviour of an oil droplet or the stability of the lm-oil system. We resort to classical thermodynamical arguments presented into great details in [START_REF] Denkov | Mechanisms of foam destruction by oil-based antifoams[END_REF][START_REF] Garrett | Defoaming: Theory and Industrial Applications[END_REF], which do not actually describe the whole picture for foam stability. Meanwhile, di erent breaking mechanisms are presented if the soap lm is destabilized by the presence of oil.

Liquid antifoams

As underlined in §1.1.5, destabilization mechanisms can occur in soap lms due to drainage process and the uctuations of the lm thickness. The life expectancy of a lm depends on the surfactant-induced rigidity of its interfaces, which can dampen any of these mechanisms. However, in most applications described in §1.3.1.2, one wants to break lms and foams quickly (even before their generation). Thus, industries add liquid or solid antifoaming agents (oil-based and particle-based). Studying the di erent mechanisms explaining this antifoaming behaviour also allows us to understand what criteria are required to block this antifoaming action [START_REF] Denkov | Mechanisms of foam destruction by oil-based antifoams[END_REF].

When a miscible liquid is injected into a lm, two situations can occur. If the solution is a surfactant solution, the lm generally remains stable and does not break, since interfaces are constantly saturated with surfactants. However, if a pure solution of water is injected, it is possible to induce a variation of interfacial concentration of surfactants by transfer to the bulk solution. Hence, one decreases the disjoining pressure, which can lead to lm rupture [START_REF] Mensire | Coalescence of dry foam under water injection[END_REF]. Now, if immiscible liquids, typically oil and alkyl components (hexadecane, dodecane), are already dispersed into the foaming solution, they appear as small droplets in the lm when these latters are generated, due to the oil-water interfacial tension [START_REF] Piroird | Capillary ow of oil in a single foam microchannel[END_REF]. This situation is drawn in Fig.

2.1a.

Then, two situations can appear. In one con guration, the oil droplet stays in the bulk phase. In the second con guration, it emerges at the air-water interface, as shown in Fig. 2.1a. This behaviour is controlled by the entry coe cient E. This coe cient compares the gain or the loss of interfacial energy between the rst and the second con guration. From one con guration to the other, an air-water and an oil-water interface are replaced by an air-oil interface of the same area. Subsequently, the entry coe cient is de ned as:

E = γ aw + γ ow -γ ao , (2.1)
where γ aw , γ ow and γ ao are, respectively, the air-water, the oil-water and the air-oil interfacial tensions. An oil droplet emerges at the air-water interface if E > 0 (loss of interfacial energy).

On the opposite, if E < 0 (gain of interfacial energy), the droplet remains in the lm, which does not break when invaded by an immiscible phase. If the droplet emerges at the air-water interface, it has two possibilities afterwards. The oil droplet can spread or not. The spreading behaviour depends on the spreading coe cient S which compares the di erence of interfacial energy between a naked lm and a lm with oil at its air-water interface. From the non-spreading to the spreading con guration, the oil droplet loses an airwater interface but gains an oil-water and an air-oil interface of the same area. This transition is shown in Fig. 2.1b. The spreading coe cient is thus written as:

S = γ aw -γ ow -γ ao . (2.2)
If S > 0, it is energetically favourable for the oil droplet to spread onto the air-water interface. If S < 0, the oil droplet is unlikely to spread.

For antifoaming purpose, E must be positive since the oil droplet has to emerge at the air-water interface. Therefore, if E < 0, oil does not have any antifoaming activity and the lm is stable, this is what we look for in our study. If E > 0, a lm is more likely to break but it is also possible to have no lm rupture, depending on the value of S.

Mechanisms of lm rupture

In this section, we present the di erent mechanisms which destabilize the lm according to the value of the thermodynamical coe cients E, S and B, and cause its rupture [START_REF] Denkov | Mechanisms of foam destruction by oil-based antifoams[END_REF].

Spreading uid entrainment

If E > 0 and S > 0, the oil droplet emerges and spreads at the air-water interface, as sketched in Fig. 2.2. The newly created air-oil interface does not contain any surfactants. Thus, attractive forces and mechanical uctuations can destabilize the lm. Moreover, the spreading of oil triggers the underlying ow of bulk aqueous phase by viscous entrainment. The bulk ow is ejected outwards, which thins the lm and can eventually lead to rupture. The mechanism of spreading uid entrainment has never been proven although this picture is commonly accepted. What has been observed is the existence of interfacial waves induced by the spreading of oil [START_REF] Denkov | Mechanisms of foam destruction by oil-based antifoams[END_REF]. These waves destabilize and break the lm. This is all the more true that the oil phase is able to sweep surfactants from the interface, which decreases the lm elasticity and its resistance to the local deformations of the interface.

Bridging-stretching and bridging-dewetting mechanism

When E > 0 and S < 0, the oil droplet emerges at the air-water interface but does not spread. As drawn in Fig. 2.3, the oil droplet bridges the lm with the angle α ow at the oil-water boundary.

To determine if a bridge is stable, Garrett et al. [START_REF] Garrett | Preliminary considerations concerning the stability of a liquid heterogeneity in a plane-parallel liquid lm[END_REF] consider that the balance of capillary forces at the triple-phase contact line should be satis ed, as well as the pressure equilibrium beyond the oil-water and the air-oil interfaces. Fig. 2.3 represents the geometry of the drop during the initial formation of the bridge. For the system to be in mechanical equilibrium, the capillary pressure jump across the air-oil interface ∆p ao = p o -p atm and the oil-water interface ∆p ow = p o -p w -Π should be equal (p o , p w and Π are the pressures in the oil phase, the aqueous phase and the disjoining pressure). From Garrett's notations, we introduce the curvature radii R ao of the air-oil interface and R ow of the oil-water interface, as well as φ the angle between the normal to the oil-water interface and the axis of revolution. The distance x is the distance to the axis of revolution. By using the Laplace law (see §1), we can estimate the capillary pressure jumps and the condition ∆p ao = ∆p ow yields:

2 γ ow R ow = 2γ ow sin φ x = 2 γ ao R ao . (2.3)
Besides, one has to balance the capillary forces at the three-phase contact line with the Neumann relationship: γ aw + γ ao + γ ow = 0. By projection on the vertical axis, one gets the following relationship:

γ ow sin α ow = γ ao sin α ao , (2.4)
where α ao is the angle made by the contact line with the air-oil interface. Using x = R d , φ = α ow at the three-phase contact line -where R d is the equivalent drop radius -R ao = R d / sin α ao and Eq. 2.4 in Eq. 2.3 gives the condition for mechanical equilibrium:

sin φ x = sin α ow R d = γ ao γ ow sin α ao R d . (2.5)
Eq. 2.5 is the equation of a series of spheres whose radius is R ow ∝ 1/ sin α ow and ∝ 1/ sin α ao .

Let us assume that α ow < 90 o . If α ow decreases, sin α ow also decreases and R ow should increase. At the same time, by volume conservation, α ao and sin α ao have to increase. Thus, R ow has to decrease. We have an incompatibility between both sides of Eq. 2.5. Now if α ow > 90 o and increases, sin α ow decreases and R ow increases. By volume conservation, α ao and sin α ao decrease and R ow increases. This time, Eq. 2.5 can be veri ed. Therefore, an oil bridge exists when the oil-water interface is a spherical segment with α ow > 90 o . A bridge cannot be stable if α ow < 90 o . We relate this condition to the di erent interfacial tensions through the bridging coe cient B:

B = γ 2 aw + γ 2 ow -γ 2 ao . (2.6)
It has been shown that the stability condition on α ow is not ful lled if B > 0 because cos

α ow = (1 + (γ ow /γ aw ) 2 -(γ ao /γ aw ) 2 )/2(γ ow /γ aw ) [72]
. If α ow > 90 o and B < 0, the oil bridge can be stable and the lm does not break.

Fig. 2.4a describes the shape of the oil bridge when α ow < 90 o . We calculate ∆p ao and ∆p ow , which gives:

∆p ao = 2 γ ow R ao , (2.7a
)

∆p ow = γ ow 1 R ao - 1 R ow . (2.7b)
From Eq. 2.7a and Eq. 2.7b, ∆p ow < ∆p ao (the capillary pressure jumps are not equal as expected) and p w = p atm + γ ow (1/R ao + 1/R ow ). Thus, the pressure in the lm around the oil-water interface is higher than the pressure in the lm far from the droplet (≈ p atm ). Then, a ow of water is induced inside the lm from the droplet to the periphery of the lm. The oil bridge is stretched, gets thinner from its edge to its center and eventually breaks into smaller oil droplets. The water lm gets perforated and disappears. This is the bridging-stretching mechanism. This mechanism is summarized in Fig. 2.5b, along with experimental snapshots from [START_REF] Denkov | Mechanisms of foam destruction by oil-based antifoams[END_REF]. It di ers from the mechanism of spreading-uid entrainment because the oil globule is eventually stretched and ruptures instead of the liquid lm.

Another mechanism can also arise. If the oil droplet is not easy to deform (typically for high oil viscosities), the lm dewets around the oil bridge until the triple-phase contact line is reached, which leads to rupture: this is the bridging-dewetting mechanism (see Fig. 2.5a). This mechanism competes with the bridging-stretching mechanism: one can assume that a comparison between the characteristic time of dewetting and the time of oil deformation would give an idea on the mechanism at play. If the dewetting velocity is higher than the deformation velocity, then the bridging-dewetting mechanism is likely to occur. However, the bridging-stretching mechanism is encountered in most situations. This arises above all when lms are covered with a predeposited layer of oil, which enhances oil wetting at the air-water interface by bringing more oil into the bridge.

For α ow > 90 o (see Fig. 2.4b), we have p w = p atm + γ ow (1/R ao -1/R ow ) with R ow < R ao . Thus, p w < p atm , which generates a ow of water from the lm to the droplet and enhances the stability of the bridge.

Stability of oil-lled Plateau borders

Foams can collapse because oil-based antifoams (often coated with solid particles) are able to break lms [START_REF] Lobo | Mechanisms of aqueous foam stability in the presence of emulsied non-aqueous-phase liquids: structure and stability of the pseudoemulsion lm[END_REF]. However, it is not the only location since oil can also be found in the Plateau borders. In fact, in the global picture of the conditions for which foams are unstable, the role of the di erent coe cients E, S and B, presented in §2.1.1, is still unclear.

In-situ interactions between oil and Plateau borders

Oil droplets can also be dispersed into Plateau borders. In this case, di erent mechanisms of rupture come into play. The shape of the oil phase is critical to understand how Plateau borders can resist to oil invasion.

Antifoaming action in Plateau borders

Any oil-induced antifoaming action starts by the emergence of oil drops at the air-water interface. This can happen in the lms, as shown in §2.1.1, but also in the Plateau borders, as suggested by Koczo et al. [START_REF] Koczo | Mechanisms for antifoaming action in aqueous systems by hydrophobic particles and insoluble liquids[END_REF].

When oil stays in the lms and causes foam destruction by breaking them, it occurs very quickly after foam generation: this type of oil is a "fast" antifoam. Fast antifoams break the lms by the bridging-dewetting or the bridging-stretching mechanism described in §2.1.2.2.

It is also possible to see oil drops being expelled from the lms to the Plateau borders, as described in Fig. 2.6. Due to drainage, the lm thickness can uctuate with spots of larger thickness than the rest of the lm, called dimples. These dimples can contain oil droplets. If the interfacial viscoelasticity is high, the dimple is unswollen symetrically from both sides of the lm. If the same viscoelasticity is small, the dimple is quickly expelled asymetrically to the periphery of the lm, due to the capillary suction. In both cases, dimples drive oil out of the lm. Once oil is expelled, the lm reaches its equilibrium thickness (e ≈ 10 nm from an initial thickness of 1 µm). When oil emerges at the wall of the Plateau border, it sweeps surfactants from the wall. Thus, the interfacial concentration of surfactants is lower in the Plateau border than in the connected lms. There is an interfacial tension gradient from the lm to the Plateau border, which generates a uctuation of lm thickness. This uctuation creates capillary waves that can cause the rupture of lms when the interfaces get closer. The timescale of foam destruction by slow antifoams is longer (typically minutes and hours) than foam destruction by fast antifoams (seconds). This di erence is related to the drainage timescale in the lms and the Plateau borders, as con rmed by a microgravity study [START_REF] Yazhgur | How antifoams act: a microgravity study[END_REF].

Criteria for antifoaming action between lms and Plateau borders

According to many studies and the conclusions from §2.1.1 and 2.2.1, oil spreading is not necessary to destabilize a foam through its lms and its Plateau borders. Non-spreading oils and fast antifoams exhibit bridging mechanisms that do not require any spreading behaviour. However, oil spreading is very e cient to reduce the energy required to break the air-water interface and supply oil to the bridges.

The entry coe cient E and the spreading coe cient S do not determine the fast or slow character of oil-based antifoams. Only the bridging coe cient B should be positive to have a destabilizing bridging mechanism. What determines such a character is the disjonction-related entry barrier which controls the entry of oil at the air-water interface. For fast antifoams, the entry barrier is low, below 15 Pa [START_REF] Denkov | Mechanisms of foam destruction by oil-based antifoams[END_REF], since oil should easily enter the interface. Thus, solid-oil compounds are often considered as fast antifoams since solid particles decrease this entry barrier. For slow antifoams, the entry barrier is higher (above 15 Pa) and characteristic of many non particle-coated oil drops.

Pseudoemulsion lm and foam stability

The entry barrier mentioned in §2.2.1.2 is related to the existence of a pseudoemulsion lm between the air phase and the oil phase. This lm has an important role in the stability of foams containing emulsi ed oil droplets.

Pseudoemulsion lm and disjoining pressure

The pseudoemulsion lm is a thin nanometric aqueous lm formed between the air-water interface and an oil droplet in the Plateau border or in the lm, as sketched in Fig. 2.8. Like naked foam lms, a pseudoemulsion lm can drain due to gravity and capillary suction. When drained, a pseudoemulsion lm exhibits micellar structures which determines an oscillating disjoining pressure Π through the thickness of the lm between the air-water interface and the oil-water interface [START_REF] Bergeron | Equilibrium measurements of oscillatory disjoining pressures in aqueous foam lms[END_REF][START_REF] Bergeron | Disjoining pressure and strati cation in asymmetric thinliquid lms[END_REF], as shown in Fig. 2.9. These oscillations originate from the surfactant structuring in both lm and pseudoemulsion lm, including the formation of micelles and bilayers. As the thickness decreases, these structures get closer and the repulsive forces get stronger, as well as the disjoining pressure.

Pseudoemulsion lm and stability mechanisms

The disjoining force induced by the repulsive and attractive amphiphilic interactions, as well as the strati cation of the pseudoemulsion lm with micellar structures, enhances the stability of oil-lled Plateau borders and lms. This happens when the entry coe cient E is negative and characteristic of repulsive interactions in the pseudoemulsion lm .

To enable the destabilization of a Plateau border or a lm, the oil droplet has to break the pseudoemulsion lm when E is positive. However, the disjoining pressure, also characterized by the local entry barrier, can prevent oil from emerging. This is the case when this pressure overcomes the capillary pressure driving the thinning of the pseudoemulsion lm. If the capillary pressure is higher, the pseudoemulsion lm breaks and oil can destabilize the foam structure [START_REF] Bergeron | Disjoining pressure and strati cation in asymmetric thinliquid lms[END_REF].

Other mechanisms are involved when one looks at the foamability of foams with dispersed oil droplets at short times. A higher surfactant concentration increases the foamability. Due to the adsorption of surfactants at the surface of the oil droplet, the fewer surfactants are available, the more di cult the stabilization of the pseudoemulsion lm. Oil droplets accumulate and block the Plateau borders, as shown in Fig. 2.10. This slows down drainage (Plateau borders and lms thin due to gravity and capillary suction) and the thinning of the pseudoemulsion lm, which reduces the probability of rupture [START_REF] Salonen | Dual gas and oil dispersions in water: production and stability of foamulsion[END_REF]. In the general case, foam stability is governed by the stability of the pseudoemulsion lm. For dispersed oil droplets, the drainage dynamics drives its stability in Plateau borders and in lms.

Experimental study of the stability of oil-laden foams

In §2.1.1, we present the stability of oil-laden foams through the stability of oil-lled lms and thermodynamical coe cients. In §2.2.2.2, we consider the stability of emulsion-laden foams through the stability of the pseudoemulsion lm. However, for imbibition into dry foams, the structures that are invaded by oil are the Plateau borders and the oil phase is continuous, so we can not really apply both theories. However, we would like to nd a criterion that allows us to predict whether an aqueous foam would resist to oil invasion.

Testing procedure

In order to do so, we perform testing experiments to check the stability of oil-laden foams and to choose the appropriate oil-foaming solution for the rest of our work.

EXPERIMENTAL STUDY OF THE STABILITY OF OIL-LADEN FOAMS 61 2.3.1.1 Observations of foam stability and collapse

To generate the foam, a slow ow of gaseous nitrogen is blown at imposed and constant pressure (0.7 Bar at the outlet of the gas tank) through a syringe needle (inner diameters vary from 0.6 to 0.8 mm depending on the bubble radii) into a 27 cm-high rectangular column with a 6 cmwide square base lled with the foaming solution. The 5-cm high upper part of this column is removable, thus allowing us to extract foam samples at the top of the column. We use di erent syringe needles to obtain very monodisperse foams with an average bubble radius R b of 0.5 mm, 1.8 mm or 2.5 mm and a polydispersity index of less than 2%. To measure the bubble size and check the monodispersity of the foam, we process an image of the foam taken near the column walls.

As shown in Fig. 2.11a, we deposit an oil droplet on a glass lamella. The oil droplet was previously covered by a surfactant lm to facilitate the entry of oil into the foam. We also carry out the experiments without this covering procedure, which gives the same results presented in the following.

At time t = 0, we put this droplet into contact with a sample of dry aqueous foam and observe the response of the foam. We could have also imagined to deposit the oil droplet on top of a foam for our tests, as presented later in this chapter.

We use di erent oils and foaming solutions whose characteristics are summarized in Table 2.1. The CAPB-SLES-Mac solution, presented in Chapter 1, is a foaming solution of cosurfactants with a fatty acid, which gives a high interfacial viscoelasticity to the air-water interface. The foam generated by this solution can hold for many hours. Without Mac, the CAPB-SLES foam holds for a shorter time, but still higher than a few minutes. Saponin is a natural non-ionic surfactant which has the same properties as the CAPB-SLES-Mac surfactant, except that its airwater interfacial tension is exceptionnally high (≈ 50 mN/m). The lifetime of SDS, TTAB-based foams is too short to be used in our experiments. Three di erent scenarii are observed. First, in the case of sun ower oil/CAPB-SLES-Mac combination, oil is sucked up by the foam with a dynamics which will be described in Chapter 5. Oil is found in the network of the Plateau borders and very few breaking events occur. In that case, we can say that the foam is resistant to oil invasion. Second, in the case of silicon oil/CAPB-SLES-Mac combination, oil is also pumped through the Plateau borders. However, within less than 1 s, we observe many breaking events in the Plateau borders and the imbibed part of the foam has completely collapsed. Last, for sun ower oil/saponin couple, no imbibition occurs, the droplet is only squeezed between the foam and the glass lamella without entering the foam.

Conditions increasing the stability of oil-laden foams

Let us compare our results to the stability theory developed for oil-lled lms. In Fig. 2.12, we report the di erent oil-foaming solution combinations on a (B, E) diagram and a (S, E) diagram. Our experimental data shows a good correlation between the values of the coe cients E, S and B with the stable or unstable character of the oil-laden foams, except for two points. Fig. 2.12 shows that there are three di erent possibilities according to the values of E and S.

First, E < 0 (and S < 0 because S < -2γ ow ) means a stable invasion, this is especially the case for organic oils (olive, sun ower) and miscible liquids (glycerol+foaming solution). Second, E > 0 and S 0 (and B 0 since B = (S + γ ow + γ ao ) 2 + γ 2 ow -γ 2 ao ), as this is the case for the oil-saponin couples, means no invasion. Oil prefers to spread at the air-water interface, rather than invade the foam, this explains why a thin oil layer forms between the bottom of the foam and the glass lamella. Third, the foam collapse occurs for mineral oils which exhibit E > 0 and S > 0 (silicon oils/CAPB-SLES), and also E > 0, S < 0 and B > 0 (silicon oils/CAPB-SLES-Mac). Indeed, for E > 0 and S < 0, as it is the case also for hexadecane, the outcome of the oil-foam interaction is dictated by the value of B. In the unstable case, B > 0 but B < 0 in the stable case, which recalls the criterion for which an oil-laden lm is stable. Stability theories for oil-laden foams and lms seem in good agreement.

The case of the sun ower/CAPB-SLES couple is more intriguing. The entry coe cient E is negative but the oil droplet still destroys the foam. The static coe cients give indications on the stability of oil-laden foams but do not always describe the whole picture, especially when the entry coe cient is close to 0. The uncertainty around the values of interfacial tension ±0.5 mN/m can make E either slightly positive, or slightly negative.

Morphology of gravitational imbibition into foams

Once the right oil-surfactant combination is found to have a stable oil-laden foam, we can visualize the ow of oil through the foam by a simple forced drainage. In this section, we provide preliminary experimental results concerning the front pattern and velocity when oil is injected at the top of a foam sample at a given ow rate.

Flow and collapse patterns

For the experimental set-up, we use a cylindrical column of foam which is extracted by the same method as in §2.3. All the physical and chemical parameters are measured by the same procedure. We inject oil at the top of the sample, as shown in Fig. 2.13, by small capillary tubes. To obtain a uniform front throughout the width of the column, oil is own throughout four injection points. We observe two types of ow pattern, as described in Fig. 2.14. For small ow rates, oil ows throughout the whole network of Plateau borders (see Fig. 2.14a), as observed in classical experiments of forced drainage when a foaming solution is injected at the top of the foam sample [START_REF] Cantat | Foams: Structure and Dynamics[END_REF][START_REF] Weaire | Liquid oil that ows in spaces of aqueous foam without defoaming[END_REF]. For high ow rates, we observe a regime of "fracture". The draining front ows downwards by following a straight path (see Fig. 2

.14b).

To compare these two regimes quantitatively, we plot the position of the draining front h as a function of time t in Fig. 2.15. For the regime of "fracture", the front propagation is linear with time. The draining velocity is constant and corresponds to the injection velocity. For the invasive regime at small ow rates, the front propagation slows down with time.

Analogy to forced drainage and ow in granular media

The existence of di erent ow regimes is still an open question. However, we can try to make two analogies to explain our observations. First, Weaire et al. [START_REF] Weaire | Liquid oil that ows in spaces of aqueous foam without defoaming[END_REF] have shown the existence of a convective forced drainage above a ow velocity threshold. The convective drainage generates a motion of the bubbles upwards for wet foams. They report a threshold of 1 cm/s for a liquid fraction of 0.4-0.5. In our experiments, the ow rate threshold is around 10 mL/h, which gives a velocity threshold of 5 x 10 -4 cm/s for a liquid fraction of 10 -3 . We have a di erence of three orders of magnitude, which is probably due to the range of liquid fractions. The drier the foam, the easier it is to modify the structure of the foam at low ow rates. However, we can make an analogy with air injection into a con ned granular suspension. The injection of air into a water-immersed granular medium at a given ow rate exhibits di erent ow patterns presented in Fig. 2.16 [START_REF] Sandnes | Patterns and ow in frictional uid dynamics[END_REF]. Such regimes were also experimentally observed [START_REF] Chevalier | Morphodynamics during air injection into a con ned granular suspension[END_REF][START_REF] Varas | Air rise through an immersed granular bed -bulk and surface dynamics[END_REF][START_REF] Varas | Morphology of air invasion in an immersed granular layer[END_REF], as shown in Fig. 2.17. For granular suspensions, a phase diagram of the di erent ow morphology can be established by using the solid fraction and the injection ow rate as free parameters. For aqueous foams, equivalent free parameters (by considering the air fraction instead of the solid fraction) are also at stake. If we refer to Fig. 2.16 in the case of high air fractions (dry foams), we should observe capillary ngering at low ow rates and viscous ngering or fracturing at high ow rates, which is quite similar to our experimental observations. We did not systematically vary the liquid fraction of the foam, yet we observe similar features between dry foams (high air fraction) and con ned granular suspensions with high solid fraction. In both cases, there is a transition at a critical ow rate for which the ow pattern evolves from a fracturing pattern to a more connected pattern.

Meanwhile, for granular suspensions, the regime of low ow rates is not exactly equivalent to the one we observe. Indeed, in aqueous foams, the permeability of all the paths is equivalent as long as the capillary forces are dominant. This is a consequence of the ability of the system to deform and to rearrange as needed. In granular suspensions, the system is more con ned and the friction between all the grains induce heterogeneities in the di erent ow paths. Therefore, di erent ow paths are followed by the uid as observed in Fig. 2.17a.

Conclusions

Studying the hydrodynamics of oil-water interfaces is of considerable scope to understand the interactions between foams and immiscible oil-based liquids. Oil is often seen as an antifoaming agent which begets the breaking of soap lms, and the possible collapse of foams. Especially in cleaning activities, if foams appear and hold for a long time, it means that greasy or oily impurities have completely vanished.

To start with, we have considered the stability of oil-lled lms which is related to the values of the di erent interfacial tensions (air-water γ aw , oil-water γ ow and air-oil γ ao ) on which the di erent mechanisms of rupture also depend. One usually de nes three coe cients: the entry coe cient E, the spreading coe cient S and the bridging coe cient B. To break a lm, the oil droplet has to emerge at the air-water interface, which means E > 0. As soon as it does it, destabilization mechanisms arise when S > 0 (spreading-wave generation), or S < 0 and B > 0 (bridging-stretching and dewetting). On the opposite, if E < 0, or E > 0 and B < 0 (which gives S < 0 since S < (γ ao -γ ow )(γ ao + γ ow ) -(γ ao + γ ow )), then the lm does not break. This thermodynamical picture is in good agreement with our experimental observations, except for one system.

We also have a global picture of oil invasion at the scale of several bubbles' layers in the case of forced imbibition. We highlight two di erent ow regimes according to the injection ow rate. These regimes of percolation and "tongues" recall observations made for air injection into a wet granular medium.

To go one step further in the next chapter, we focus on the oil-foam interaction in the speci c geometry of a Plateau border.

Chapter 3 Plateau borders and oil: equilibrium shape and coalescence

As demonstrated in §2.1.1 and §2.2.1, oil drops are able to break lms and Plateau borders. But they do not destabilize them in some physical conditions. Moreover, the study of the oil-induced stability is carried out for systems where antifoams are already present in the aqueous phase during foam generation. In our case, we consider a continuous oil phase forming long cylinders of oil inside the foam. These structures are also fragile and break when a lm bursts. COALESCENCE Contents

Picture of oil invasion in Plateau borders by numerics

As stated in §2.2.1, foam collapse can start from the Plateau borders with slow oil-based antifoams, non-coated with solid particles. The oil droplet has to emerge at the air-water interface. In §2.2.2.2, we consider the breaking of the pseudoemulsion as the triggering event for rupture of the Plateau border. However, we have highlighted mechanisms in which the Plateau border thins with the jamming of oil droplets. In our study, the con guration is quite di erent since the foam is already dry and imbibed with oil afterwards. In this situation, we need to understand how the oil phase is shaped inside the Plateau border. This was done by Neethling et al. by using Surface Evolver simulations [START_REF] Neethling | Modeling oil droplets in Plateau borders[END_REF].

Oil slug in a Plateau border

Neethling et al. [START_REF] Neethling | Modeling oil droplets in Plateau borders[END_REF] performed the numerical simulations of an oil slug elongated inside a Plateau border. Such con guration is similar to the con guration found in [START_REF] Piroird | Capillary ow of oil in a single foam microchannel[END_REF] in which Piroird et al. studied the elongation of an oil slug inside a Plateau border at imposed ow rate of oil or at imposed volume of oil.

Shape of the oil phase

In the case of lm rupture by fast antifoams, we have seen that the entry coe cient E, the bridging coe cient B and the entry barrier are the three major factors that drive the stability of lms macroscopically and locally. In particular, the entry barrier depends on the disjoining forces that appear in the air-water-oil pseudoemulsion lm and how this lm resists to the capillary suction. The pseudoemulsion lm is a thin water lm (nanometric) separating the squeezed oil phase from the air phase. This lm has to break rst to allow the bridge to be formed. If the entry barrier is too high, oil droplets are expelled towards the Plateau borders. They are trapped inside the Plateau border and the node with a pseudoemulsion lm separating the oil phase from the air phase, as simulated in Fig. 3.2a.

Like for lms, the rupture of the pseudoemulsion lm is the rst stage for the failure of Plateau borders, which could lead to foam collapse by the same aforementioned bridging mechanism or the spreading-wave generation ( §2.2.1). A pseudoemulsion lm breaks when the mechanical pressure exerted by the walls of the Plateau borders during drainage gets higher than the disjoining forces.

Neethling et al. [START_REF] Neethling | Modeling oil droplets in Plateau borders[END_REF] de ne di erent dimensionless parameters to characterize the shape of the droplet in the Plateau border and the subsequent stability of the pseudoemulsion lm:

PICTURE OF OIL INVASION IN PLATEAU BORDERS BY NUMERICS

73

-the ratio K between the oil-water and the air-water interfacial tensions: K = γ ow /γ aw , -the ratio r * between the equivalent spherical radius of the oil droplet r o and the curvature radius of the Plateau border r P B (which is also the characteristic size of the Plateau border): r * = r o /r P B , -the ratio p * between the di erent pressures P (pressure in the Plateau border, in the pseudoemulsion lm and the oil phase) and the capillary pressure at the air-water interface: p * = (P -p atm )r P B /γ aw . Fig. 3.1 shows the three di erent con gurations for the oil droplet in the Plateau border according to the size of the droplet.

Interactions between the oil phase and the pseudoemulsion lm

The size of the oil droplet r * and the interfacial tension ratio K have a strong in uence on the pressure p * f exerted in the pseudoemulsion lm and the related pressure di erence with the Plateau border ∆p * f = p * f -p * w , where p * w is the dimensionless pressure in the Plateau border. Fig.

shows the evolution of ∆p *

f with respect to r * . Small droplets and high interfacial tension ratios induce a stronger pressure on the pseudoemulsion lm, due to the curvature of the oil droplet. Since ∆p * f > 1 for most droplet sizes and interfacial tension ratios, the pseudoemulsion lm is curved outwards (it swells the Plateau border). However when the interfacial tension ratio gets lower (≈ 0.1), ∆p * f < 1. We have ∆p * f = 1+(p f -p atm )r P B /γ aw and p f -p atm < 0, thus the pseudoemulsion lm is oriented inwards. As K goes to zero, the pseudoemulsion lm remains curved inwards to follow the curvature of the Plateau border, i.e. ∆p * f = 0 and p f -p atm = -γ aw /r P B . The oil e ect is negligible with no interfacial tension, as though the Plateau border was empty.

As the interfacial tension ratio decreases, the pseudoemulsion lm also gets larger. Indeed, for low K, it is easier to deform the droplet than to swell the Plateau border. The oil phase forms a long slug rather than an elongated droplet of cross-section bigger than the Plateau border's cross-section.

In Chapter 5, the oil-surfactant systems we use exhibit interfacial tension ratios K ≈ 0.1 -0.25. Thus, according to the simulations from Neethling et al. (see Fig. 3.2), the Plateau border keeps its initial shape (very slightly swollen) when invaded by oil and the oil phase is assumed to be entirely squeezed by the Plateau border to form a long slug. This assumption veri ed by numerics plays an important role in the model presented in Chapter 5.

Interfacial energy comparison with Surface Evolver

To help us complement the picture of oil inside a Plateau border in §2.3, we perform numerical simulations corresponding to the cases of stable invasion where E < 0 and S < 0 and no invasion where S 0 and E 0 by considering the geometry of the Plateau border with Surface Evolver. We would like to determine modi ed thermodynamical arguments taking the speci c geometry of the Plateau borders into account.

Numerical con gurations

The idea is to compare how oil distributes in the two con gurations and calculate their interfacial energy: the inner con guration: a Plateau border is lled with an oil slug, the outer con guration: the oil slug lies at the jonction between two adjacent bubbles when oil stays at the surface. The preferred con guration is the one which has the lower interfacial energy. The simulations depend on the interfacial tensions and the ratio between the volume of oil V o and the volume of water V w in the Plateau border, which means the volume of oil compared to the volume of the pseudoemulsion lm for the inner con guration. Fig. 3.4 shows the typical simulations for the inner and the outer con guration. COALESCENCE

Comparison between numerics and experiments

For every con guration, we plot the amount of interfacial energy E I as a function of the volume ratio V o /V w . Fig. 3.4 shows the di erent pro les for several oil-foaming solution systems presented in Table 2.1 and in Fig. 2.11. The ratio V o /V w starts at 1 because the oil slug does not touch and deform the walls of the Plateau border when V o /V w < 1. For organic oils-CAPB+SLES+Mac combinations, E I is lower for the inner con guration than the outer con guration (Fig. 3.5a). Thus, organic oils prefer to stay inside the Plateau border than to spread at the surface. On the contrary, silicon oils prefer to remain at the surface since E I is higher for the inner con guration than the outer con guration (Fig. 3.5b). Oil is absorbed but quickly emerges and breaks the Plateau borders.

If we change the surfactant from CAPB+SLES+Mac to saponin, the di erence of interfacial energy is very important (300 to 500 J) in favour of the outer con guration (Fig. 3.5c). So, oil strongly prefers to stay at the surface, which explains why oil invasion does not occur. Finally, for hexadecane for which the thermodynamical coe cients did capture the experimental observations with B < 0, E I is indeed lower for the inner con guration, so numerics predicts the invasion of the Plateau border, in agreement with the experimental observations. For sun ower oil-CAPB+SLES, E I is lower for the inner con guration, so sun ower oil should stay in the Plateau border (Fig. 3.5d), which is not what we observe experimentally. For this latter case, the numerics is also in discrepancy with the experimental observation. However, the interfacial energies in both con gurations are really close and might overlap. Hence, a small variation of interfacial tension could modify the energy balance.

Numerics allows to take the equilibrium shape of the oil-lled Plateau border into account. A basic calculation of interfacial energy with the static coe cients E, S and B ignores this fact. Such calculations are valid for in nite lms with vanishing curvatures whose shape does not change much with oil penetration. However, numerics is in good agreement with the coe cient-based estimations, providing that E is not too close to zero. Meanwhile, when the entry coe cient E is close to 0, a small variation of interfacial area and interfacial tension has a strong in uence on the total interfacial energy and modi es the stability of the system. The entry barrier and the stability of the pseudoemulsion lm might play in this case.

Here, oil appears as a long continuous slug in the foam. Drainage mechanisms for oil droplets do not come into play. The stability of the pseudoemulsion lm is only governed by the molecular interactions between the oil phase and the air phase. Thermodynamical arguments seem to provide an accurate prediction for the stability of oil-laden foams, except for the smallest values of E where the system is metastable and the local interactions matter.

Breaking dynamics in the oil-lled Plateau borders

In §2.1.1 and §4.3, we show that oil could cause the collapse of foams by interacting with the air-water interfaces (bridging and spreading for instance). Now, let us imagine that the foam is stable in the presence of oil. Films are fragile structures that can break when an external force is applied or when they change con gurations (by T 1 event for instance). What consequences does this forced breaking of a lm have on the adjacent oil-lled Plateau borders?

The Rayleigh-Plateau instability

As seen in Chapter 5, Plateau borders can drive oil inside a foam. When a lm breaks, those oil-lled Plateau borders appear as long liquid cylinders. The question is: how does this cylinder evolve? The answer is similar to the uid dynamics of liquid cylindrical threads, also known as the Rayleigh-Plateau instability.

Breaking observations

Everyone has noticed (at least once) that a water jet breaks into small droplets when one opens a tap [START_REF] Savart | [END_REF]. Any liquid object wants to reduce its interfacial energy. To do so, the interfacial area has to decrease. This is what happens when a liquid cylinder breaks into small droplets. This instability is known as the Plateau-Rayleigh instability. Indeed, one can estimate the condition for which the interfacial area is smaller with drops than with cylinders. By volume conservation, the total volume of drops is equal to the volume of the cylinder [START_REF] Charru | Instabilités Hydrodynamiques[END_REF]:

πr 2 c L c = 4 3 πr 3 d N , (3.1)
where r c , L c , r d and N are, respectively, the cylinder radius, the cylinder length, the droplet radius and the number of droplets.

By using Eq. 3.1, the ratio of interfacial area between the total area S c of the cylinder and the total area S d of the small droplets is written as:

S c S d = 2πr c L c 4πr 2 d N = 2r d 3r c . (3.2)
For r d > 3r c /2, S d is smaller than S c . The bigger the droplets, the higher the loss of interfacial area.

The Rayleigh-Plateau instability is also observed when a ber is wetted and surrounded by a liquid cylinder. The liquid lm spontaneously ondulates and breaks into small droplets. In this problem, gravity is negligible compared to the interfacial tension e ects. Comparing the hydrostatic pressure ρ l gr c and the Laplace pressure γ al /r c gives the Bond number Bo = ρ l gr 2 c /γ al , introduced in §4.3.3. For r c = 100 µm, γ al = 73 mN/m and ρ l = 1000 g/L, Bo ≈ 0.001 and gravity is negligible. Like the jet case, interfacial tension plays a dominant role and destabilizes the cylindrical lm.

In both situations (jet and liquid on ber), a wavy cylinder is unstable. Indeed, we can look at the capillary pressure in the cylinder, presented in Fig. 3.6. The Laplace pressures are decomposed into two terms: one contribution from the curvatures 1/r 1 and 1/r * 1 in the xy plane and one contribution from the curvatures 1/r 2 and 1/r * 2 in the yz plane. From the rst contribution, the pressure in the bumps is higher than in the holes (p 1 > p * 1 ). Indeed, the curvature is positive in the rst case and negative in the second one. However, from the second contribution, r 2 > r * 2 , thus p * 2 > p 2 . A ow of liquid is driven from the hole to the bump, which enhances the instability. For the development of the instability, the unstable contribution has to be stronger than the stable one [START_REF] Plateau | Statique Expérimentale et Théorique des Liquides Soumis aux Seules Forces Moléculaires[END_REF].

More precisely, if one considers a wave perturbation of wave number k = 2π/λ, the unstable capillary pressure is the strongest for k 1/r c (see Appendix B). However, the growth rate σ RP of the instability increases if the wavelength λ decreases (i.e. k increases). Indeed, λ/2 is the distance over which the ow of liquid from the hole to the bump occurs. The shorter this distance, the faster the transfer of uid. Hence, there should be a maximum value of growth rate (and thus an optimal wavelength) for the strongest capillary pressure and the fastest transfer of liquid. This value depends on the dissipation origin: inertial or viscous-dominated regime.

Inertial regime

In this regime, the inertial forces are opposed to the destabilizing capillary forces because large volumes of liquid are di cult to move from the hole to the bump. These forces are dominant over the viscous forces [START_REF] Charru | Instabilités Hydrodynamiques[END_REF][START_REF] Rayleigh | On the instability of jets[END_REF][START_REF] Rayleigh | On the capillary phenomena of jets[END_REF][START_REF] Rayleigh | On the instability of a cylinder of viscous liquid under capillary force[END_REF]. This is especially the case for low viscous uids, such as water.

We introduce the characteristic time t c , the liquid density ρ l , the cylinder radius r c and the airliquid interfacial tension γ al . To determine the characteristic time of formation of the instability, one balances inertia ρ l r 2 c /t 2 c with capillarity γ al /r c . One nds that:

t c ∼ ρ l r 3 c γ al . (3.3)
The time t c is very short, typically 3 ms for a millimetric cylinder radius. This is not surprising since a water jet impacts the bottom of a sink by small successive droplets that were formed very rapidly.

The growth rate of the instability σ RP is proportional to 1/t c . More accurately, it also depends on the product between the wave number and the cylinder radius kr c by the following relationship:

σ 2 RP ≈ k 2 r 2 c 2t 2 c 1 -k 2

Viscous regime

In §3.2.1.2, we assume that inertial e ects dominate viscous e ects. However, what happens if water is replaced by oil whose viscosity can be 100 times higher? In this case, viscous e ects cannot be neglected and get over the inertial e ects [START_REF] Chandrasekhar | Hydrodynamic and Hydromagnetic Stability[END_REF][START_REF] Tomotika | On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous uid[END_REF]. Moreover, Taylor [START_REF] Taylor | The formation of emulsions in de nable elds of ow[END_REF] has noticed that the nature of the surrounding uid, especially its viscosity, in uences the development of the instability. In the viscous regime, the important parameter is the viscosity ratio between the viscosity of the cylindrical uid η c and the viscosity of the surrounding uid η f . If η c /η f → +∞, the inner uid is more viscous than the outer uid. From the exact calculations summarized in Appendix B, the growth rate σ RP of the instability can be expressed as:

σ RP = γ * 2r c η c 1 -k 2 r 2 c k 2 r 2 c Io(krc) 2 I 1 (krc) 2 -(1 + k 2 r 2 c ) ≈ γ * 6r c η c (1 -k 2 r 2 c ) , (3.5)
where γ * is the interfacial tension between the inner and the outer uid, I o and I 1 the modi ed Bessel functions. The characteristic time of instability growth is the viscous -capillary time t c = η c r c /γ * . Fig. 3.8 plots σ RP t c with respect to kr c . The growth rate is maximum when k max r c = 0, thus λ max → +∞. The instability grows the fastest when the wavelength is very large.

If η c /η f → 0, the inner uid is less viscous than the outer uid. The growth rate is given by:

σ RP = γ * 2r c η f 1 -k 2 r 2 c k 2 r 2 c Ko(krc) 2 K 1 (krc) 2 -(1 + k 2 r 2 c ) , (3.6)
where K o and K 1 are other modi ed Bessel functions. Once more, as shown in Fig. 3.8, the growth rate is maximum for k max r c = 0 and λ max → +∞. In the viscous regime, capillarity drives the choice of the wavelength of maximum growth, which is the largest possible.

Breaking of an emulsi ed viscous cylinder

In this section, we ll a Plateau border with oil and study the rupture of this Plateau border when an adjacent lm is broken. By doing so, we mimic the situation where a broken lm (due to antifoaming or mechanical perturbations) destabilizes the network of adjacent Plateau borders, potentially lled with immiscible liquids. When the oil-lled Plateau border is isolated, it can be assimilated to a long cylindrical thread which can break into small droplets. Thus, the similitude COALESCENCE Figure 3.8: Dimensionless growth rate σ RP t c as a function of kr c in the viscous regime. The blue round dots designate the case η c > η f , while the orange triangular dots designate the case η c < η f . In both cases, the maximum growth rate is obtained for k max r c = 0.

to the Rayleigh-Plateau instability is close. We model our experimental results with regard to the analysis on the Rayleigh-Plateau instability in §3.2.1.

Experiments

We perform the experiments with three di erent oils: para n oil (Sigma-Aldrich), olive oil (commercial) and hexadecane (Sigma-Aldrich). The oil characteristics are summarized in Table 3.1. The surfactant is the CAPB-SLES-Mac composition used in §2. Fig. 3.9 depicts the experimental set-up, which is also described in [START_REF] Piroird | Capillary ow of oil in a single foam microchannel[END_REF]. First, we generate a Plateau border by dipping a prism-like frame of length 15 cm and width 2 cm into the foaming solution. We gently pull the frame out of the surfactant bath. A slender Plateau border appears in the middle of the frame at the jonction between three trapezoidal soap lms. The frame is deposited horizontally between two supports in a closed box with saturated humidity: temperature T = 23 -25 o C and humidity rate h w = 90 -95%. This reduces evaporation and increases the lifetime of the Plateau border. We let the lms drain for di erent durations. As the drainage time increases, the curvature radius of the Plateau border decreases: the system "dries". This is very important if one studies the ow dynamics of oil in the Plateau border, which has been done by Piroird et al. [START_REF] Piroird | Capillary ow of oil in a single foam microchannel[END_REF]. In our case, a variation of curvature radius also allows to study the in uence of the shape of the Plateau border in this problem. Oil is preemulsi ed by plunging a capillary tube containing the oil phase into the surfactant solution for 30 s. After the drainage period, we deposit this given volume of oil in the upper lm to avoid any deformation of the Plateau border. The oil droplet slowly slides downwards due to gravity and enters into the Plateau border. At time t = 0, the oil phase forms a slug, aligned with the revolution axis of the Plateau border, and starts propagating symetrically from the injection point due to the di erence of pressure between the center and the tip of the slug. Indeed, the pressure at the center of the slug p + = p atm + (γ ow + γ aw )/r c is higher than the pressure at the tip p -= p atm -γ aw /r P B , which generates a capillary ow of oil outwards [START_REF] Piroird | Capillary ow of oil in a single foam microchannel[END_REF]. We wait for the oil slug to reach di erent lengths before breaking the lower rear lm with a needle. We record the whole sequence from the propagation of oil to the breaking of the oil cylinder at 15 frames per second (camera AVT Marlin).

Fig. 3.10 presents a typical sequence of images after the lm rupture. Once the lm is broken, a single lm is left with the oil slug right in the middle. However, in a fraction of seconds, this slug breaks into small emulsi ed oil droplets embedded within the remaining soap lm. We can also observe some satellite droplets in-between the bigger droplets. In Fig. 3.11, we also illustrate the transverse view of the rupture of the oil-laden Plateau border. COALESCENCE

Instability wavelength

In the image sequence, we only shoot half of the oil slug to get more precision on the measured distances. We note L oil the half-length of the oil slug at t = 0, Ω the half-volume of the oil droplet and R oil , the radius of the oil slug that we assume uniform along the length L oil . We also introduce the wavelength λ max which is the average distance between two consecutive big droplets (i.e. we do not count the satellite droplets which appear due to pinch-o phenomena not discussed here), and R d as the droplet radius. For each experiment, we calculate R oil = Ω/πL oil . We report λ max with respect to R oil in Fig. 3.12.

The experimental wavelength λ max increases when the slug radius R oil increases. We can try to compare the experimental data with the theoretical predictions deduced from the usual Rayleigh-Plateau instability. In the inertia-dominated regime ( §3.2.1.2), λ max is equal to 9R oil . This regime does not collapse with our data. We can also be in the viscous regime. The viscosity ratios Figure 3.12: Wavelength λ max as a function of the slug radius R oil . The coloured dotted lines are the theoretical predictions [START_REF] Tomotika | On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous uid[END_REF]. The black dotted line is the prediction for the inertia-dominated regime [START_REF] Rayleigh | On the instability of a cylinder of viscous liquid under capillary force[END_REF] and the orange dotted line, the prediction for the viscous-dominated regime when the viscosity of the outer uid is neglected.

between the oils and the foaming solution (of viscosity η w = 1.4 mPa.s) are, respectively, 43.6, 18.4 and 2.2 for olive oil, para n oil and hexadecane. If we neglect the viscosity of the outer foaming solution, as shown in §3.2.1.3, λ max should tend to +∞. Yet, we obtain a nite wavelength in our problem. Therefore, we have to consider the viscosity of the outer uid in the wavelength calculations. Tomotika [START_REF] Tomotika | On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous uid[END_REF] solves the whole Rayleigh-instability problem of a viscous cylinder in another viscous uid (i.e. when the viscosity ratio η o /η w is nite). Fig. 3.13 gives the ratio λ max /R oil as a function of η o /η w . From this curve, we deduce the values of λ max for our oilfoaming solution systems. Figure 3.13: Theoretical ratio wavelength of maximum growth -cylinder radius λ max /R oil as a function of the viscosity ratio η o /η w [START_REF] Tomotika | On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous uid[END_REF]. The solutions for our oil-foaming solution systems are indicated with the same color code as in Fig. 3.12.

From the solutions indicated in Fig. 3.13, λ max is linear with R oil with di erent slopes according to the viscosity ratio. Fig. 3.12 shows a reasonable collapse (at least in terms of scaling order) between the experimental data and the predictions (dotted lines). Thus, by knowing the viscosity ratio and the cylinder radius (by the droplet volume and the slug length), we can predict the instability wavelength which is the distance between droplets. oil ) 1/3 for para n oil, olive oil and hexadecane. The oil radius R oil is deduced from the volume Ω of the oil slug. The black dashed line represents the theoretical prediction from Eq. 3.7. Inset: Theoretical radius R d as a function of (λ max R oil ) 1/2 by assuming a pancake-like shape. The black dashed line is the theoretical prediction as detailed in the text. We observe that the agreement is not quantitative. Fig. 3.14 plots R d as a function of (λ max R 2 oil ) 1/3 for our experimental data. It shows a very good collapse with the theoretical prediction from Eq. 3.7. This result also con rms the assumption of spherical droplets. Indeed, we could have thought about a pill-like shape. In this case, Ω = N πR 2 d R oil by assuming a pill thickness equal to the cylinder radius R oil . By the same calculations as the spherical shape, we nd R d = (λ max R oil ) 1/2 . The inset of Fig. 3.14 presents the collapse for the pill case and it does not clearly work. From these results, we can conclude that the lm does not constrain the Rayleigh-Plateau instability and acts as a free deformable surface. This seems in contradiction at rst glance with the assumption of §3.1 where we state that the presence of oil does not strongly modify the shape of the Plateau border. Yet, in this experiment, the two con gurations are not obtained for the same lm thickness. The lms supporting the Plateau border at the beginning of the experiment are very thin, thus they induce an important capillary pressure which is su cient to deform the oil slug. At the end of the experiment, there is a single lm: the aqueous foaming solution initially contained within the Plateau border and the broken lm have been redistributed. Therefore, the remaining soap lm is thick and its disjoining pressure is not su cient to deform the oil slug, which now adopts a cylindrical shape due to the non-zero value of the oil-water interfacial tension. This result is very interesting if one thinks about the Rayleigh-Plateau instability at the scale of an oil-lled Plateau border inside a foam. From [START_REF] Tomotika | On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous uid[END_REF], we nd that λ max is uniquely determined by R oil and η o /η w :

Size of the oil droplets

λ max = R oil Φ η o η w , (3.8)
with Φ the implicit function determined by Tomotika in [START_REF] Tomotika | On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous uid[END_REF]. The radius of the oil slug R oil can be approximated by the curvature radius of the Plateau border r P B = 1.74φ

1/2
l R b for a dry foam in the case of low oil-water interfacial tensions. We have φ l the liquid fraction of water and oil and R b the bubble radius. Therefore, λ max becomes a function of the liquid fraction and the bubble radius

λ max ∼ φ 1/2 l R b Φ(η o /η w ).
Injecting this relationship and R oil ∼ r P B in Eq. 3.7 yield the radius of the oil droplet R f oam d after the rupture of a Plateau border in a foam:

R f oam d ∼ φ 1/2 l R b Φ η o η w 1/3 . (3.9)
Besides, the number of droplets N f oam can be predicted with N f oam = L oil /λ max . The length of the oil slug L oil is equal to the length of a Plateau border L P B = (π/6 √ 2) 1/3 R b for a dry foam. Using 3.8 gives the number of droplets N f oam coming from the rupture of the Plateau border:

N f oam ∼ 1 φ 1/2 l Φ ηo ηw . (3.10)
We can retrieve the volume of oil absorbed in the Plateau border (i.e. the volume of a Plateau border given that we approximate R oil by r P B )

Ω P B = (4/3)πN f oam (R f oam d ) 3 ∼ φ l R 3
b by multiplying Eq. 3.9 and Eq. 3.10. This result is coherent with the estimation deduced from the Kelvin cell with the length of a Plateau border L P B = (π/6 √ 2) 1/3 R b and the volume Ω P B ∼ L P B r 2 P B ∼ φ l R 3 b . However, as we will see in Chapter 4, the radius of the oil slug is not exactly constant and equal to the curvature of the Plateau border. It is actually a function of the local liquid fraction which depends on the position x in the foam, so

R f oam d (x) ∼ φ l (x) 1/2 R b Φ ηo ηw 1/3
. Hence, if we can measure R f oam d experimentally by forcing the rupture of the oil-laden foam and recording the image sequence, we can actually determine the liquid fraction of oil and water at the position x (or the average liquid fraction in a Plateau border whose center of revolution is located at the position x). Knowing the initial liquid fraction of water φ i , we can retrieve the pro le of liquid fraction for the oil phase.

In this section, we try to understand the rupture of an oil-lled Plateau border into small emulsi ed droplets, owing to the rupture of an adjacent lm. Our experimental data and theoretical analysis demonstrate that this problem can be studied under the light of the well-known Rayleigh-Plateau instability. We have to take the viscosity of the pseudoemulsion lm into account to predict the wavelength of maximum growth of the instability. This problem presents all the conditions that allow to consider the Rayleigh-Plateau theory. The system is assimilated to a long cylindrical oil thread in an in nite outer uid. This uid is the foaming solution in the upper lm which encircles the cylinder when the lower lm is broken. The velocity components and the tangential stresses are continuous at the boundary between the oil phase and the pseudoemulsion lm. The di erence in normal stresses is only due to the oil-water interfacial tension γ ow at the interface between the oil and the foaming solution. If we know the viscosity ratio η o /η w , the oil volume Ω and the length of the invaded region L oil , we can thus predict the wavelength and the droplet size.

For a Plateau border in a foam, we can deduce the liquid fraction of oil at any position x (for instance in case of imbibition process) by measuring the droplet radius after rupture and knowing the bubble radius and the viscosity ratio. This method does not require any external apparatus, such as light scattering [START_REF] Durian | Multiple light-scattering probes of foam structure and dynamics[END_REF] or conductimetry [START_REF] Feitosa | Electrical conductivity of dispersions: from dry foams to dilute suspensions[END_REF].

Conclusions

In this chapter, we have considered the local shape of an oil slug inside a Plateau border. In this situation, oil is encapsulated in the network of Plateau borders to form a three-phase system if the air phase in the bubbles is included in the whole picture.

In Chapter 2, we use the entry E, spreading S and bridging B coe cients to characterize the ability of the foam to resist to oil invasion. To con rm our results, we resort to numerics to compare the interfacial energy in two di erent con gurations: one where the oil is inside the Plateau border and the other one where the oil stays on top of a bubble and does not penetrate the foam. The simulations by Surface Evolver show a good agreement between the con guration of lower interfacial energy and the stability of oil-laden Plateau borders, providing that the error margin on the interfacial tension is small (this is critical for E ≈ 0). Calculating the global interfacial energy for speci c geometries (here an oil-lled Plateau border) is actually more relevant than reasoning on a simple at geometry where two interfaces face each other. Our simulations conrm the calculations using the static coe cients. Locally, both Plateau borders and lms exhibit the same con gurations when oil emerges at the air-water interface.

However, what really governs the stability of oil-laden foams is the ability of the oil droplet to break the pseudoemulsion lm, and thus to overcome the disjoining pressure. Energetical arguments are necessary but not always su cient (for instance for particle-laden oils). The role of the pseudoemulsion lm and the entry barrier is still an open question.

Finally, we have wondered what would occur when the oil-lled Plateau border breaks due to lm bursting, bubble coalescence or rearrangements. We nd that the Plateau border is decomposed into small emulsi ed droplets. This dynamics is analogous to the Rayleigh-Plateau instability. A viscous uid cylinder is unstable because this object wants to reduce its interfacial area and can only do so by division into small droplets. We can retrieve the geometrical parameters of the instability, typically the decomposition wavelength λ max , by considering the whole solution to the Rayleigh-Plateau instability, including the in uence of the surrounding uid. Studying this instability also allows us to retrieve the quantity of oil inside the foam by the evaluation of the liquid fraction.

Oil can come out of the Plateau border when this latter breaks but oil can also emerge at the airwater interfaces and spread on this interface, leading to the destabilization of the corresponding lm or Plateau border.

Chapter 4

Soap lms and oil: dynamics of rising oil droplets spreading at the air-water interface

The stability of oil-laden foams strongly depends on the ability of the oil phase to emerge at the air-water interfaces. Dispersed oil droplets that emerge are able to spread at the air-water interface. Such dynamics can destabilize lms, as well as stabilize the formation of oil bridges [START_REF] Denkov | Mechanisms of foam destruction by oil-based antifoams[END_REF]. We recall known theories on the spreading of oil droplets on solid surfaces before making the same analysis for the spreading on liquid surfaces. We study the spreading dynamics of rising oil droplets where oil is already encapsulated in the lm, which is di erent from common situations where the oil layer is deposited onto the lm by the air phase. We use a foam as a liquid sponge which absorbs oil through the Plateau borders. Hence, it seems logical to consider the spreading dynamics of oil droplets when they emerge from the aqueous phase.

Spreading of oil on solid surfaces

We rst recall some known results about the spreading theory on solid surfaces.

Wetting theory on solid surfaces

In this section, we present the basic notions about the wetting and the spreading of liquids on solid surfaces [START_REF] Léger | Liquid spreading[END_REF]. This allows us to introduce the case where the solid surface is replaced by its liquid counterpart.

Surface energy and spreading coe cient

More details for this subsection can be found in [START_REF] De Gennes | Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls and Waves[END_REF]. The wetting behaviour of a liquid lm on a smooth solid surface is characterized by its energy E S based on the interfacial interactions:

E S = (γ sl + γ al + P (e)) Σ , (4.1)
where γ sl , γ al and Σ are, respectively, the solid-liquid, the air-liquid interfacial tensions and the lm area. When the lm thickness e → ∞, we have a thick lm and P (∞) = 0, because we only have solid-liquid and air-liquid interfaces. When e → 0 , we obtain a free surface without lm and only have an energy contribution from the solid-air interface. Thus, P (0) = γ sa -γ sl -γ al = S S with γ sa the solid-air interfacial tension and S S the spreading coe cient for solid surfaces analogous to the spreading coe cient for air-oil-water interfaces S de ned in §2.1.1.

The term P (e) re ects the short-range and long-range interactions between the liquid and the solid phase and between the liquid molecules. More precisely, P (e) is related to the disjoining pressure Π(e) for lms de ned in §1.1.5: Π(e) = -dP/de. For e higher than the molecular size a but below 100 nm, long-range Van der Waals forces are dominant and P (e) can be written as:

P (e) = H k 12πe 2 , ( 4.2)
where H k is an e ective Hamaker constant characteristic of the interactions between two charged interfaces (|H k | ≈ 10 -20 J). In this case, the disjoining pressure is written as:

Π(e) = H k 6πe 3 . (4.3)
The disjoining pressure is interpreted as the pressure to apply to prevent the lm from being thickened by long-range forces.

Once the lm is formed, E S allows to relate the lm thickness e to the wetting properties of the solid surface, as we will see in the following sections. For macroscopic lms, we directly use the spreading coe cient S S to evaluate the spreading ability of a liquid droplet deposited on a solid surface by comparing the interfacial energy between a dry and a wet surface:

S S = γ sa -γ sl -γ al . (4.4)
If S S > 0, a liquid droplet completely spreads on a solid surface: this is the complete wetting. If S S < 0, a liquid droplet does not spread and form an angle θ Y , called Young's equilibrium contact angle with the surface: this is the partial wetting. Glass and metallic oxides exhibit bonds of high energy (ionic, covalent and metallic, i.e. γ sa = 0.5-1 N/m). These solid surfaces are easily wetted by most liquids. On the contrary, polymers, te on and para n exhibit bonds of low energy (Van der Waals and hydrogen bonding, i.e. γ sa = 0.05 N/m). Wetting can be complete or partial.

The determination of the equilibrium shape of the spreading drop ( lm and/or spherical shape) is solved thanks to the evolution of the equilibrium parameters E S and P (e) with the lm thickness e.

Partial wetting

In the partial wetting situation [START_REF] De Gennes | Wetting: Statics and dynamics[END_REF], S S < 0 and the liquid takes a "pancake" shape by making a contact angle with the solid surface at equilibrium, the Young contact angle θ Y , as sketched in Fig. 4.1 [START_REF] Young | An essay on the cohesion of uids[END_REF].

cos θ Y = γ sa -γ sl γ al = 1 + S S γ al . (4.5)
This angle is only de ned for S S < 0. At S S = 0, the transition to the complete wetting occurs

CHAPTER 4: SOAP FILMS AND OIL: DYNAMICS OF RISING OIL DROPLETS SPREADING AT THE AIR-WATER INTERFACE

and θ Y = 0 o . The thickness e eq of a liquid drop at equilibrium is determined by considering the balance between gravity, which tends to make the drop thinner, and the capillary forces, which hinder the thinning process to reduce the interfacial energy. The total mechanical energy of the puddle E puddle is the sum of an interfacial capillary term and a gravitational term:

E puddle = -S S Σ + 1 2 ρ l gΣe 2 , (4.6)
where ρ l is the liquid density and g the gravity acceleration. The volume of the puddle Ω is constant and Ω = Σe by volume conservation, which gives E puddle = -S S Ω/e + (1/2)ρ l gΩe.

The equilibrium thickness is given by dE = 0:

e eq = -2S S ρ l g = 2κ sin θ Y 2 , (4.7)
where κ = γ al /ρ l g is the capillary length. This length re ects which forces are dominant between capillarity and gravity. If the drop size is higher than κ, then gravity forces are dominant and drags the drop thinning. For κ ≈ 2 mm and θ Y ≈ 45 o , one nds e eq ≈ 1.5 mm.

Complete wetting

If S S > 0, it is favourable for the liquid drop to spread as a lm, which means creating two interfaces (air-liquid and solid-liquid) instead of leaving the solid surface dry. This property generally concerns liquids with low air-liquid interfacial tension since it costs energy to create an air-liquid interface. Thus, liquids with surfactants are ideal candidates. Zisman [START_REF] Zisman | Contact Angle, Wettability and Adhesion[END_REF] de ned an empirical criterion that only depends on the nature of the solid material to determine whether a liquid completely wets a solid. This criterion is the critical tension γ c . If γ al < γ c , the liquid spreads on the solid surface. For instance, plastic materials exhibit γ c = 30-40 mN/m. Thus, as shown in Table 1, most oils wet plastic, while water does not. For glass, γ c = 100 mN/m. Thus, common liquids from oil to water spread on glass. However, most surfaces are not completely clean. Any dust and any greasy matter can decrease the value of γ c and favour partial wetting.

The equilibrium thickness e eq of the lm in its nal state is determined by the balance between the long-range forces and the capillary forces [START_REF] De Gennes | Wetting: Statics and dynamics[END_REF]. According to Eq. 4.2, the Van der Waals forces tend to thicken the lm, since the energy potential decreases in 1/e 2 . But the capillary forces cause the thinning of the drop because S S > 0. The variation of energy per unit area of the lm E f ilm is written as E f ilm = -S S Σ + P (e)Σ. Minimizing this energy (dE f ilm /de = 0) gives S S = e eq Π(e eq ) + P (e eq ) = -(d(E S /Σ)/de) eeq e eq + P (e eq ). This is the equation of the tangent crossing the points (0, γ sa ) and (e eq , E S /Σ (e eq )) in the curves (e, E S /Σ) shown in Fig. We deduce e eq by using Eq. 4.2 and Eq. 4.3:

e eq = H k 4πS S . (4.8)
For H k ≈ 10 -20 J and S S ≈ 20 mN/m, e eq ≈ 0.1 nm. This length is close to the molecular size and increases when one gets close to the transition between partial and total wetting for S S → 0.

Pseudo-partial wetting

Partial and total wetting characterize the spreading of a liquid drop on a solid surface. However, these two options are not the only possibilities. It has been demonstrated theoretically [START_REF] Brochard-Wyart | Spreading of nonvolatile liquids in a continuum picture[END_REF] and experimentally [START_REF] Silberzan | Evidence for a new spreading regime between partial and total wetting[END_REF] that a third regime is possible: the pseudo-partial wetting [START_REF] Brochard-Wyart | Spreading of nonvolatile liquids in a continuum picture[END_REF]. Let us have a look at Eq. 4.1. For H k < 0 and S S > 0 or S S < 0, E S must have the pro le depicted in Fig. 4.2a-b. Indeed, P (e) is an increasing negative function and P (e → 0) = γ sa > 0. Thus, the pro le must have a minimum at a value e pp . We can draw a common horizontal tangent between the point (e pp , (E S /Σ)(e pp )) and the point (e → ∞, γ al + γ sl ). Hence, the nal equilibrium state is a liquid droplet sitting on a lm of thickness e pp , as depicted in Fig. 4.2c.

One can calculate the e ective interfacial tension γ pp of the pseudo-partial lm. We have to SPREADING AT THE AIR-WATER INTERFACE determine the variation of energy in the lm when we vary its area at constant volume dΩ = (dΣ/Σ) + (de/e) = 0. By di erentiating Eq. 4.1 with e, we have dE S = (γ al + γ sl + P (e))dΣ -ΣΠ(e)de. We deduce γ pp = dE S /dΣ as: γ pp (e) = γ al + γ sl + P (e) + eΠ(e).

(4.9)

In Fig. 4.2c, the balance between horizontal forces yields:

γ pp (e pp ) = γ sl + γ al cos θ pp , (4.10)
where θ pp is the equilibrium contact angle at the jonction between the pseudo-partial lm and the droplet. Let us also recall that e pp and e eq are respectively the thickness of the precursor lm and the thickness of the total wetting lm. Replacing the left term in Eq. 4.10 by Eq. 4.9, with Π(e pp ) = -(d(E S /Σ)/de) epp = 0 -because (e eq , (E S /Σ) eeq ) is a minimum -gives the e ective interfacial tension γ pp :

γ pp (e pp ) = γ sl + γ al (1 + P (e pp ) γ al ). (4.11)
The pseudo-partial wetting is possible for droplets that are large enough to cover the whole solid surface. If not, a lm of thickness e eq is formed.

In summary, the wetting of liquid droplets on smooth solid surfaces can be decomposed in three di erent regimes:

-If S S > 0 and H k > 0, there is complete wetting with the formation of a thin lm, -If S S > 0 or S S < 0 and H k < 0, there is pseudo-partial wetting with a droplet sitting on a wet solid, -If S S < 0 and H k > 0, there is partial wetting with a droplet sitting on a dry solid.

Dynamics of liquid spreading on air-solid interfaces

The di erent wetting regimes for a liquid droplet on a solid surface are determined by considering an energy balance in a static con guration. In this section, we wonder how a liquid of spreading coe cient S S > 0 spreads on the solid and reaches the shape of a lm of thickness e eq [START_REF] Léger | Liquid spreading[END_REF]. Driven by capillary and gravity forces, the spreading dynamics is divided into di erent regimes according to the value of the spreading radius r.

Precursor lm

One could imagine that the driving force pulling the droplet is the spreading coe cient S S . Yet, this is not the case since Hardy [START_REF] Hardy | III. The spreading of uids on glass[END_REF][START_REF] Lelah | Spreading kinetics of drops on glass[END_REF] has shown that a precursor lm of thickness e eq quickly forms ahead of the droplet tip, as shown in Fig. 4.3. Due to the magnitude of S S , a molecular liquid lm rapidly spreads, as expected from §4.1.1.3. However, at the rear of this lm, the droplet of thickness e still exists by making a dynamic contact angle θ d between the air-liquid and the solid-liquid interfaces. This is what Tanner and Ho mann rst observe [START_REF] Ho Man | A study of the advancing interface: I. Interface shape in liquid-gas systems[END_REF][START_REF] Tanner | The spreading of silicone oil drops on horizontal surfaces[END_REF]. Therefore, the capillary force per unit length exerted on the droplet has two components on the horizontal axis: one from the lm F f ilm = γ al + γ sl and one from the droplet F droplet = -γ sl -γ al cos θ d . Both contributions yield the global driving force F cap :

F cap = F f ilm + F droplet = γ al -γ al cos θ d .
(4.12)

For small angles, F cap ≈ γ al θ 2 d /2. In the following section, we use this approximation by considering the air-liquid interfacial tension only, with θ d ≈ e/r.

Capillarity-dominated regime

In this regime, for thin lms and droplets, we use the lubrication approximation in the Stokes equation. In cylindrical coordinates, the velocity v in the droplet only depends on the vertical coordinate z. The pressure p is only a function of the radial coordinate r. The Stokes equation is given by:

η l ∂ 2 v ∂z 2 = ∂p ∂r , (4.13)
with η l the liquid viscosity and v the horizontal velocity. The boundary conditions are v(0) = 0 and v(e) = U = (1/e) e 0 vdz, with U the mean velocity along the cross-section of the droplet. This problem can be completely solved analytically and a demonstration is given in Appendix A [START_REF] Cazabat | Dynamics of wetting on smooth and rough surfaces[END_REF]. Here, we choose to present the theoretical analysis in terms of scaling laws.

CHAPTER 4: SOAP FILMS AND OIL: DYNAMICS OF RISING OIL DROPLETS SPREADING AT THE AIR-WATER INTERFACE

The tip of the droplet is in under pressure with the center of the droplet. Therefore, liquid is driven from the center to the tip and allows the liquid front to advance. Since e r, we can approximate its curvature by the second derivative of the interface pro le e/r 2 . The capillary pressure gradient is exerted on the spreading radius r. Thus, the pressure gradient is ∂p/∂r ∼ γ al e/r 3 . And the viscous dissipation is η l ∂ 2 v/∂z 2 ∼ η l U/e 2 . By volume conservation, the droplet volume is 2 . By writing U ∼ dr/dt (t is the time) and injecting the di erent contributions into Eq. 4.13, one nds the following spreading law [START_REF] Guyon | Physical Hydrodynamics[END_REF]:

Ω d ∼ er
r(t) ∼ Ω 3/10 d γ al η l t 1/10 . (4.14)
As expected, the spreading rate is higher when the liquids are less viscous (less viscous friction) and the air-liquid interfacial tension is high (higher capillary pressure).

Gravity-dominated regime

The regime in t 1/10 is valid when the capillary forces are dominant over the gravity forces. Indeed, gravity can also drive the spreading process. Its volume contribution is ρ l ge/r since the hydrostatic pressure is applied on the surface of the spherical cap of the droplet. The spreading driving force changes as time goes by. Gravity dominates capillarity if ρ l ge/r > γ al e/r 3 , which means r > κ, the capillary length. In that case, balancing the viscous dissipation with the gravity contribution gives the following spreading law [START_REF] Guyon | Physical Hydrodynamics[END_REF]:

r(t) ∼ Ω 3/8 d ρ l g η l t 1/8 . (4.15)
As time goes by, the spreading rate decreases. The viscous friction gets higher as the thickness of the drop decreases (∝ 1/e 2 ). The driving forces (∝ e/r 3 or e/r) get weaker since r also increases. As a result, a droplet of 1 cm 3 takes more than 1 year to reach a spreading radius r = 10 cm. Fig. 4.4 shows the capillarity-driven and the gravity-driven regimes with the di erent transition times. Both capillarity and gravity-dominated spreading regimes are analytically solved in Appendix A.

Spreading of oil on air-water interfaces from the air phase

Now, what happens if the solid phase is replaced by a liquid phase that is immiscible with the liquid in the droplet? The wetting theory of liquid-liquid systems is similar to the wetting predictions for liquid-solid systems. However, the spreading dynamics is completely di erent. Depending on the viscosity ratio η o /η w where η o and η w are the oil viscosity and the viscosity of the bulk uid, two di erent regimes are observed for liquids in complete wetting.

SPREADING OF OIL ON AIR-WATER INTERFACES FROM THE AIR PHASE 101

Liquid-liquid wetting

An oil droplet breaks a lm by emerging at the air-water interface if the entry coe cient E > 0 and transitions from a spherical shape to a lens shape. From this con guration, the droplet has three di erent options which depend on the value of the spreading coe cient S = γ aw -γ ow -γ ao already de ned in §2.1.1. If S < 0, the oil lens remains as it is without contaminating the rest of the air-water interface. If S > 0, the oil lens has two possibilities: -spread completely to form a thin lm, -keep its shape and spread on a thick lm of oil (but non-macroscopic) or a solubilised lm of oil. These possibilities are summarized in Fig. 4.5 [5,[START_REF] Bergeron | Monolayer spreading of polydimethylsiloxane oil on surfactant solutions[END_REF][START_REF] Kellay | Wetting properties of n-alkanes on AOT monolayers at the brine-air interface[END_REF].

The value of S has to be taken at equilibrium, which means for a time scale longer than the di usion and the adsorption of surfactants at the oil-water interface. -4 mN/m over 24 hours. However, this decrease induces an increase of S. The equilibration of the value of the interfacial tension should actually enhance the spreading process.

Regimes of oil spreading on air-water interfaces

The theoretical and experimental studies of oil spreading at the surface of an aqueous bath with and without surfactants have been done for several systems [START_REF] Bergeron | Monolayer spreading of polydimethylsiloxane oil on surfactant solutions[END_REF][START_REF] Camp | The spreading of oil on water in the surface-tension regime[END_REF][START_REF] Cheng | Solidlike spreading of a liquid/liquid system[END_REF][START_REF] Craster | On the dynamics of liquid lenses[END_REF][START_REF] Fraaije | Dynamics of spreading on a liquid substrate[END_REF][START_REF] Svitova | Spreading of aqueous dimethyldidodecylammonium bromide surfactant droplets over liquid hydrocarbon substrates[END_REF]. We can distinguish di erent regimes according to the viscosity of the oil phase.

Non-viscous to viscous oils

The theory developed here is found in [START_REF] Bergeron | Monolayer spreading of polydimethylsiloxane oil on surfactant solutions[END_REF]. But all the dynamical studies present the same theoretical arguments. For oil spreading to occur, S has to be positive. The driving force is the interfacial tension gradient along the spreading oil lm. This force is resisted by the viscous force engendered by the underlying aqueous phase which is entrained by the lm spreading. Fig. 4.7 depicts an oil lens of negligible thickness which spreads at the air-water interface. Subsequently, we have the following balance of stresses:

SPREADING OF OIL ON AIR-WATER INTERFACES FROM THE AIR PHASE 103

∇γ + τ ow = 0 , (4.16)
where γ and τ ow are, respectively, the interfacial tension and the shear stress at the oil-water interface. In the lm, the variation of interfacial tension is given by S. The interfacial tension gradient varies over the radial spreading distance r. Thus, we have ∂γ/∂r ≈ S/r. We use an exponentially decaying pro le for the velocity in the viscous boundary layer of the aqueous phase, as described in Fig. 4.7. The radial velocity v r is written as v r = v r (r, z = 0, t)e z/δ d = (dr/dt)e z/δ d where z is the vertical coordinate and δ d the penetration length. According to the usual boundary layer theory for a laminar ow [START_REF] Batchelor | An Introduction to Fluid Dynamics[END_REF][START_REF] Guyon | Physical Hydrodynamics[END_REF][START_REF] Landau | Fluid Mechanics[END_REF], δ d = r/ √ Re r where Re r = ρ w v o d/η w is the Reynolds number for the density of the bulk uid ρ w , the viscosity of the bulk uid η w , the characteristic velocity of the upper layer ow v o and d the characteristic length. In our case, for d ≈ r, v o ≈ r/t. Thus, the penetration length is expressed as:

δ d = η w t ρ w . (4.17)
The shear stress at the oil-water interface is τ ow = η w (∂v r /∂z) z=0 . Therefore, Eq. 4.16 with Eq. 4.17 yields:

S r = η w ∂v r ∂r z=0 = η w δ d dr dt = ρ w η w t dr dt . (4.18)
Integrating Eq. 4.18 gives the following spreading law:

r = 4 3 1/2 S ρ 1/2 w η 1/2 w 1/2 t 3/4 . (4.19)
The power law in t 3/4 is completely di erent than the power laws in t 1/10 and t 1/8 found for the equivalent spreading of liquid on solid surfaces. This is due to the "moving" character of the liquid substrate, which causes the creation of a boundary layer ow. The dynamics for liquid-SPREADING AT THE AIR-WATER INTERFACE liquid systems is faster than the dynamics for liquid-solid systems. It depends on the physical nature of the spreading liquid through the spreading coe cient S. The higher S, the stronger the driving force, the faster the spreading goes. It also strongly depends on the properties of the aqueous bath. In particular, if the water viscosity is high, the spreading rate is smaller. Indeed, the water substrate plays the role of a dampening layer. No geometrical parameter is involved, neither is the oil viscosity. The t 3/4 law is valid as long as δ d < r but at short times, one might have δ d > r since δ d evolves as t 1/2 . In that case, the penetration depth is r. One obtains a timelinear regime of spreading. The transition occurs for δ d = r. By equating Eq. 4.19 and Eq. 4.17, the transition time is η 3 w /ρ w S 2 . By considering η w = 10 -3 Pa.s, ρ w = 10 3 g/L and S = 10 -2 N/m, we nd that the transition occurs around 10 -8 s, which means instantaneously as the liquid starts spreading. The only observable regime is the t 3/4 regime. Fig. 4.8 shows the comparison between the theoretical predictions from Eq. 4.19 and some experimental results [START_REF] Bergeron | Monolayer spreading of polydimethylsiloxane oil on surfactant solutions[END_REF]. In the calculations, one assumes that the interfacial tension is constant over the thickness of the oil droplet and that surfactant dynamics is negligible. If not, discrepancies from the expected law could arise [START_REF] Di Pietro | Dynamics of spreading on a liquid substrate[END_REF].

Very viscous oils

These discrepancies can also appear when the oil phase is very viscous [START_REF] Bacri | Experimental study of the spreading of a viscous droplet on a nonviscous liquid[END_REF][START_REF] Brochard-Wyart | Spreading of viscous droplets on a non viscous liquid[END_REF]. In this case, the oil ow is described as a plug ow, as sketched in Fig. 4.9. One also assumes that a precursor lm is formed at the edge of the droplet. In the limit of small angles, the force balance at the leading edge is written as:

SPREADING OF OIL ON AIR-WATER INTERFACES FROM THE AIR PHASE 105

1 2 γ ao θ 2 ao + 1 2 γ ow θ 2 ow = 1 2 γ eq θ 2 eq , (4.20)
where θ ao (respectively θ ow ) are the dynamic contact angles between the air-water (respectively the oil-water) interface and the horizontal axis. The contact angle θ eq is the angle between the air-water and the oil-water interface. Both interfaces can be assimilated as an association of two springs in parallel. Thus, the equivalent interfacial tension γ eq is deduced from 1/γ eq = 1/γ ao + 1/γ ow .

One considers the viscous dissipation D in the droplet with the shear stress τ xy and the shear rate ˙ :

D = Ω d τ xy ˙ dΩ d ≈ Ω d η o ṙ r 2 . ( 4

.21)

The capillarity-induced power P is given by:

P = F cap • ṙ = 2πr 1 2 γ eq θ 2 eq ṙ , (4.22)
where F cap is the driving force deduced from Eq. 4.20. By volume conservation, we can write e ≈ Ω d /r 2 and by the approximation of small angles, θ eq ≈ e/r. Using these estimations and equating Eq. 4.22 to Eq. 4.21 yield the following spreading law:

r = γ eq η o Ω d 1/4 t 1/4 . (4.23)
The power law in t 1/4 obviously gives a slower dynamics than the t 3/4 result for less viscous oils. Like the di erent laws presented before, increasing oil viscosity decreases the spreading rate, while higher interfacial tension has the opposite e ect. The boundary between this regime and the t 3/4 regime is determined by equating the viscous dissipations from both cases. The t 1 for a viscous ow. By setting δ d ≈ r, one nds an upper limit for the regime separation:

η o > η * o = η w r 3 Ω d . (4.24)
For Ω d = 0.1 cm 3 , r = 5 cm and η w = 1 mPa.s, η * o = 1.3 Pa.s, which is three orders of magnitude higher than the water viscosity. At high radius over the capillary length, the t 1/4 regime switches to a gravitational t 1/2 regime where the capillary force is replaced by the gravitational force [START_REF] Bacri | Experimental study of the spreading of a viscous droplet on a nonviscous liquid[END_REF].

Spreading of oil rising from the aqueous phase at airwater interfaces

In many situations, the oil phase emerges at the air-water interface from the aqueous phase: emulsi ed oil-water systems, froth otation, underwater pollution and oil-laden aqueous foams [START_REF] Cantat | Foams: Structure and Dynamics[END_REF][START_REF] Denkov | Mechanisms of foam destruction by oil-based antifoams[END_REF][START_REF] Koczo | E ect of oil on foam stability: Aqueous foams stabilized by emulsions[END_REF][START_REF] Mensire | Capillary imbibition of aqueous foams by miscible and nonmiscible liquids[END_REF][START_REF] Piroird | Capillary ow of oil in a single foam microchannel[END_REF]. The surrounding aqueous phase can a ect the spreading dynamics, in comparison with the air phase which has a negligible role in previous spreading studies. This is all the more true that most relevant and real aqueous systems contain surfactants (or ionic agents acting as surfactants) which can also strongly modify the spreading behaviour.

We study the radial spreading of rising oil droplets at the surface of an aqueous bath containing soluble surfactants or not. We develop a model for short and long-time dynamics and compare with experimental results obtained with mineral and organic oils. We discuss our model with regard to existing spreading theories.

Experimental results

We perform spreading experiments with di erent oils whose characteristics are summarised in Table 4.1. The mineral oils are para n oil (Sigma-Aldrich) and di erent silicone oils (standard oils from Sigma-Aldrich or oils for rheometer calibration from Brook eld). All oils are dyed (1 wt% from Rohm and Haas) to enhance the contrast between the aqueous and the oily phase. The interfacial tensions are measured by the pendant drop method. For the aqueous phase, the surfactant is SDS (sodium dodecyl sulfate from Sigma-Aldrich) with concentrations C SDS varying from 0.5 g/L to 10 g/L (5 times higher than the cmc). The aqueous phase is contained in a transparent polyacrylic box whose dimensions are variable (width: 10.7/9 cm -length: 15.5/9.5 cm -height: 3.7/9.7 cm). As sketched in Fig. 4.10, the oil phase comes through a needle that is directed upwards and immersed inside the box. The tip of the needle can be located at di erent distances from the surface. First, the oil droplet is generated by a small injection rate (0.05 -0.1 mL/min) and can remain attached or detached from the needle when it touches the surface. Before the touching time, we start recording the spreading dynamics at 7.5 frames per second for the standard camera SPREADING AT THE AIR-WATER INTERFACE (AVT Marlin) and at 2000 or 2500 frames per second for the fast camera (Photron v9.7 and SA3).

SPREADING OF OIL RISING FROM THE AQUEOUS PHASE AT AIR-WATER INTERFACES 107

If the entry coe cient E = γ aw -γ ow -γ ao < 0, the droplet remains spherical under the water surface and does not spread. If E > 0 but S = γ aw + γ ow -γ ao < 0, the droplet breaks the surface and keeps a lens shape without spreading. Finally, if E > 0 and S > 0, within less than one second, the oil droplet breaks and penetrates the water surface before spreading radially. In our experiments, we do not encounter the cases E < 0 and E > 0 with S < 0 for which the droplet does not break the surface or spread at all. We focus on the case E > 0 and S > 0. Then, two possible evolutions are observed. Fig. 4.12 shows these two typical image sequences obtained from the spreading observations. With no surfactants in the aqueous bath, the oil droplet spreads completely in a fraction of seconds until forming a thin lm at the bath surface (Fig. 4.12a). With surfactants, the oil droplet initially spreads until forming a lens (Fig. 4.12b), then keeps spreading slowly while holding the lens shape (Fig. 4.12c).

Short-time dynamics

Here, we consider the short-time dynamics of spreading by taking inertial, viscous and capillary e ects into account.

Inertial-viscous-capillary regime

For the early stages of the no-surfactant and surfactant case, we can de ne a short-time spreading regime. When reaching the bath surface, the spherical oil droplet breaks the water surface in a "coalescence"-like scenario [START_REF] Paulsen | Coalescence of bubbles and drops in an outer uid[END_REF], as shown in the inset of Fig. 4.13. The driving force is the di erence of capillary pressure ∆p C at the neck of the droplet. Using a parabolic shape at the neck [START_REF] Eggers | Coalescence of liquid drops[END_REF][START_REF] Paulsen | Coalescence of bubbles and drops in an outer uid[END_REF], we can estimate

∆p C = γ ow (κ 2 -κ 1 ) ≈ γ ow R d /r 2 + γ ow /R d ≈ γ ow R d /r 2
, where R d is the droplet radius and r the spreading radius. This pressure is balanced by the viscous stress with the velocity gradient over r, and the inertial pressure, which gives:

γ ow R d r 2 = η o r dr dt + ρ o dr dt 2 , ( 4

Collapse between experiments and theory at short times

If Oh 1 (non-viscous oil), we have an inertial-capillary regime and R ∼ (T /Oh) 1/2 , similar to [START_REF] Biance | First steps in the spreading of a liquid droplet[END_REF] for the early spreading on a solid wall. On the opposite, if Oh 1 (viscous oil), we have a viscous-capillary regime and R ∼ T 1/2 . We report the experimental data with the dimensionless spreading radius R as a function of

T Oh = -1+ √ 1+4Oh 2 Oh 2
T in Fig. 4.13. We obtain a good collapse between all curves for a short-time regime in the range of T Oh = 1 to 5 (t < 1 s). The transition to the late stages is discussed in the following section.

SPREADING OF OIL RISING FROM THE AQUEOUS PHASE AT AIR-WATER INTERFACES 111

Long-time dynamics

We also study the long-time spreading dynamics and the transition to multiple regimes for both pure aqueous and surfactant solutions in the bath phase.

Transition to long-time dynamics for pure aqueous solutions

For pure aqueous solutions in the bath, the aforementioned short-time di usive regime transitions to the long-time regime in t 3/4 described in §4.2.2.1. This transition is shown in Fig. 4.14. Eq. 4.19 can be written in the dimensionless coordinates R and T :

R = 4 3 1/2 T 3/4 Oh 1/2 ef f , (4.28)
with the e ective Ohnesorge number

Oh ef f = ρ w R d (γ 3/2 ow /S)/(η 3/2 o /η 1/2 w ).
The transition time T ST -LT and radius R SL-LT between the short-time and the long-time regime are de ned by equating Eq. 4.27 and Eq. 4.28, which gives: However, with surfactants, the oil lens keeps spreading at a much slower spreading rate. Fig. 4.12c shows a typical sequence of spreading images for t = 0.66 to 60 s. Since S > 0, it is not surprising to observe a long-time spreading dynamics. Some studies [5,[START_REF] Brochard-Wyart | Spreading of nonvolatile liquids in a continuum picture[END_REF][START_REF] Kellay | Wetting properties of n-alkanes on AOT monolayers at the brine-air interface[END_REF] have shown the existence of a pseudo-partial wetting regime between the total wetting regime (S > 0) and the partial wetting regime (S < 0) for the spreading of liquid on a solid surface. In this regime, the liquid droplet keeps a lens shape and spreads onto a thin precursor lm of liquid. Here, we assume the same situation, as shown in [START_REF] Brochard-Wyart | Spreading of nonvolatile liquids in a continuum picture[END_REF]. At the end of the short-time regime, the oil droplet reaches a lens shape. This lens is in equilibrium with a thin molecular lm of oil and spreads upon it.

T ST -LT = Oh 2 ef f f (Oh) 2 and R ST -LT ≈ Oh ef f f (Oh) 3/2 , ( 4
If r > κ where κ = γ ow /(ρ o g), the capillary pressure gradient γ ow e/r 3 is dominated by the gravity e ects ρ o ge/r, where e is the height of the lens (assuming a quasi-cylindrical shape). The hydrostatic pressure in the oil phase is only considered here. Indeed, the velocity eld is radial in the droplet, thus the pressure is constant in the vertical direction (from the Stokes equation and the lubrication approximation) and equal to the oil hydrostatic pressure at z = -e . Using the volume conservation R 3 d ∼ er 2 and balancing the gravity e ects with the viscous dissipation (η o /e 2 )dr/dt, we nd the spreading radius:

r(t) ∼ ρ o gR 9 d η o 1/8 t 1/8 . (4.30)
We can non-dimensionalize r and t by R = r/R d and T * = t/(η o /ρ o gR d ) and we nd R = T * 1/8 for R > 1/Bo 1/2 where Bo = ρ o gR 2 d /γ ow is the Bond number. However, if r < κ, the capillary pressure gradient dominates the gravity e ects. Thus, in this range of radii, we nd a spreading dynamics similar to the spreading of a microscopic droplet beyond a at solid surface:

r(t) ∼ γ ow R 9 d η o 1/10 t 1/10 , (4.31)
with R = T 1/10 by non-dimensionalizing Eq. 4.31 for R < 1/Bo 1/2 . For our experiments, 1/Bo 1/2 < 1 < R (see Fig. 4.13) at the end of the short-time regime, we should transition to the viscous-gravitational regime. In Fig. 4.15, we plot the experimental data in the dimensionless coordinates (T * ,R).

We nd a good agreement with the model for all curves in Fig. 4.15. The plateau pro le for T * < 1000 corresponds to the transition between the short-time regime and the long-time dynamics. The crossover time between both dynamics is de ned by the characteristic time τ * g and radius r g . In the long-time dimensionless coordinates, τ * g and r g are non-dimensionalized as T * g and R g , in the viscous-gravitational regime. By equating Eq. 4.26 to Eq. 4.30, we nd:

SPREADING OF OIL RISING FROM THE AQUEOUS PHASE AT AIR-WATER INTERFACES 113

T * g ∼ BoOh 2 -1 + √ 1 + 4Oh 2 4/3 , (4.32a) R g ∼ BoOh 2 -1 + √ 1 + 4Oh 2 1/6 . (4.32b)
As reported in Fig. 4.15, a good correlation is found between the expected crossover time and the experimental transition to the long-time regime. If Oh 1 (inertia-dominated short-time regime), T * g = T * i/g ∼ Bo 4/3 Oh 4/3 and R g = R i/g ∼ Bo 1/6 Oh 1/6 . Yet, if Oh 1 (viscousdominated short-time regime), the crossover time and radius are, respectively, T * v/g ∼ Bo 4/3 and R v/g ∼ Bo 1/6 . For a viscous-capillary regime at long times, by equating Eq. 4.26 to Eq. 4.31, we nd the dimensionless characteristic time T c ∼ Oh 2 / -1 + √ 1 + 4Oh 2 5/4 and radius

R c ∼ Oh 2 / -1 + √ 1 + 4Oh 2 1/8
. By reporting the transition radii, we build a phase diagram (Bo, R) which indicates the di erent regimes for a given Oh in Fig. 4 and in the orange dashed line (for Oh << 1). We have f (Oh) = Oh 2 / -1 + √ 1 + 4Oh 2 . The curve R = 1/Bo 1/2 de nes the transition between capillary and gravity-dominated regime (capillary length). The curves R = f (Oh) 1/6 Bo 1/6 and R = f (Oh) 1/8 correspond to the transitions between the short-time and the long-time dynamics.

.16. SPREADING AT THE AIR-WATER INTERFACE

Our theoretical developments di er from previous studies who have shown that the spreading of an oil droplet upon a bath of aqueous liquid with and without surfactants follows a single regime in t 3/4 [START_REF] Bergeron | Monolayer spreading of polydimethylsiloxane oil on surfactant solutions[END_REF][START_REF] Camp | The spreading of oil on water in the surface-tension regime[END_REF]. At short times, the oil droplet coalesces with the bath by the air phase in previous works. Thus, as sketched in Fig. 4.17a, the viscous dissipation only occurs in the bath.

To evaluate this dissipation, an exponential decaying velocity pro le is assumed in the bath, which allows to calculate the shear stress at the oil-water interface and leads to the aforementioned power law while balancing with the capillary pressure gradient. Yet, in our case, the dissipation occurs in the oil phase and the shear stress is zero at the air-oil interface. We estimate this dissipation by considering a velocity gradient in the coalescence neck along the radial direction, which yields the di usive behaviour in t 1/2 . From our study, we demonstrate that the initial location of the droplet with respect to the bath is critical and yields a di erent spreading power law for short-times. We retrieve a long-time regime in t 3/4 when the e ect of the initial capillary pressure of the neck is lost and when the droplet switches to a lens shape.

Not only is the system con guration important, but the addition of surfactants causes a regime separation at long times. The spreading behaviour is similar to the spreading of an oil droplet on a solid surface with the development of a precursor lm. In other studies, the addition of sur- factants does not seem to impact the spreading law in t 3/4 where the dissipation in the aqueous bath is considered. Dynamics of surfactant adsorption leading to potential variations of S between the initial state and the nal equilibrium state might play a role here. However, we carried out experiments after letting the oil droplet equilibrate with the bath from a few minutes to several hours. The oil-water interfacial tension only decreases, thus S increases and the spreading process should be more favourable.

SPREADING OF OIL RISING FROM THE AQUEOUS PHASE AT AIR-WATER INTERFACES 115

A more consistent explanation comes from the existence of an interfacial stress that arises when the oil droplet spreads, as shown in Fig. 4.17b. The droplet can sweep surfactants from the airwater interface. Thus, the interfacial concentration of surfactants at the center is lower than the concentration at the edges. Hence, an interfacial tension gradient is created from the edges to the center and the air-water-oil interface behaves like a spring and slows down the spreading of the droplet. But a precursor lm still propagates at the front of the droplet. From this con guration, we retrieve the long-time dynamics of a droplet sitting on a precursor lm, as described in the SPREADING AT THE AIR-WATER INTERFACE second part of this study. This happens more easily for mobile surfactants, such as SDS, than immobile surfactants with long polymeric chains like AOT and CTAB for instance. Molecular interactions between the surfactants and the oil might also be at stake and allow the system to nd a con guration where a droplet coexists with a precursor lm of molecular thickness.

In conclusion, we can describe the spreading of oil droplets coming from the aqueous phase by two regimes. The early stage is a fast di usive regime in t 1/2 that occurs for bulk solutions with and without surfactants. In the case of pure aqueous phase, we transition to the common longtime regime in t 3/4 . However, in the surfactant case, it is possible to nd a transition to a long-time regime similar to the spreading of a liquid droplet on a solid wall: the air phase is replaced by the water phase and the solid phase by the air phase. It seems that this behaviour strongly depends on the nature of the oil phase and the type of surfactants.

Conclusions

The existence of oil-lled lms is a question of emergence of oil droplets at the air-water interface. However, spreading issues are also important as it enhances the destabilization process. Therefore, we have studied the spreading dynamics of oil droplets at the air-water interface. Most studies have considered the deposition of oil droplets at the water surface by the air phase. In our case, we consider oil droplets that are already in the aqueous phase before rising to the surface, which is closer to the con gurations of oil-lled lms. Our results demonstrate the existence of a di usive short-time dynamics -driven by capillarity and resisted by viscous and inertial forcesthat joins the common spreading law in t 3/4 at long times for spreading on a pure aqueous surface. For surfactant-laden solutions, molecular dynamics at the oil-water interface gives results similar to the spreading of oil on solid surfaces. The motion of surfactants at the interfaces has a strong in uence on the spreading dynamics.

In Chapter 2, 3 and 4, we have talked about oil-foam systems when oil is already present in the microstructures of the foam (stability in lms and Plateau borders) and when oil seeks to come out of the foam (spreading dynamics, rupture instability). Now, the question is: how is oil absorbed into foams?

Chapter 5

Oil imbibition into aqueous foams

Dry aqueous foams can be seen as liquid porous media constituted by a network of liquid microchannels called Plateau borders at the bubbles' junctions, similarly to pores at the junction of granular beads that can be wicked by liquids. In this chapter, we study the dynamics of oil imbibition into aqueous foams. Di erent situations are considered: point-source of oil, oil slick and moving foam. We discuss our results with regard to known theories for liquid ows in foams and solid porous media.

Imbibition through liquid foams

We rst recall some results on the 1D imbibition into solid porous media (undeformable and deformable) and aqueous foams that are later used for comparison with our imbibition dynamics. Then, we present the main model to explain the ow of liquid in foams that we can assimilate as liquid porous media.

1D Imbibition through solid porous media

The 1D imbibition through undeformable and deformable solid porous media is a well-known subject. We present some of the main results.

Jurin's law

Let us consider an undeformable capillary tube of constant radius that we put into contact with an aqueous bath by the bottom of the tube. Liquid rises into the tube if the wet tube has an interfacial energy lower than the dry tube. By using the solid-liquid γ sl and the solid-air γ sa interfacial tensions, one de nes the impregnation parameter I S : Capillary rise lasts until the interfacial energy is balanced by the potential energy. Fig. 5.1 describes the liquid rising into the capillary tube. The energy E of the liquid column can be written as:

I S = γ sa -γ sl . ( 5
E = -2πr t zI S + 1 2 πr 2 t z 2 ρ l g , (5.2)
where r t is the tube radius, z the vertical coordinate and ρ l the liquid density. In Eq. 5.2, the size of the meniscus is neglected, which is valid for r t z. Setting dE/dz = 0 for the energy minimum and injecting I S = γ al cos θ Y yields the maximum height of capillary rise h max :

h max = 2γ al ρ l gr t cos θ Y .
(5.3) Eq. 5.3 is known as the Jurin law [START_REF] Hauksbee | An account of an experiment touching the ascent of water between two glass planes, in an hyperbolic gure[END_REF][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]. This height is maximum for θ Y = 0 o (for a completely wetting droplet). The thinner the tube, the higher h max . Indeed, if one thinks about the pressure balance at the spherical meniscus, the curvature radius r m is r t / cos θ Y and the pressure p m is given by the Laplace relationship:

p m = p atm -2 γ al cos θ Y r t . (5.4)
This pressure is equilibrated with the hydrostatic pressure p o -ρ l gh max . One nds again Eq. 5.3 and can understand that a smaller tube radius induces a higher capillary pressure, and thus a higher maximum height. For γ al = 70 mN/m, r t = 10 µm, ρ l = 1000 g/L and cos θ Y = 0.5, h max ≈ 70 cm. We have r t h max for micrometric tubes and Eq. 5.3 is valid.

Lucas-Washburn's law

Let us now describe the hydrodynamics of capillary rise. One can write Newton's momentum equation applied to the liquid column along the vertical direction e z :

d(m l v m ) dt = F cap -F viscous -P W , (5.5)
where v m is the mean rising velocity and the capillary force F cap is resisted by the viscous force F viscous and the weight of the liquid column P W . The capillary force is applied to the perimeter of the tube cross-section. Its value is proportional to the di erence between the interfacial energy of the dry and the wet solid. Thus, F cap is written as:

F cap = 2πr t I S = 2πr t γ al cos θ Y . (5.6)
The weight P W is equal to ρ l gπr 2 t z. The viscous force F viscous is determined by considering

IMBIBITION THROUGH LIQUID FOAMS 123

a Poiseuille ow in the tube (see Appendix A). Indeed, the liquid rise is slow (the Reynolds number Re = ρ l v m r t /η l 1) and the no-slip condition at the walls imposes a velocity gradient given by Stokes' equation. The vertical velocity v(r) (r is the radial coordinate) across the cross-section of the tube is given by v(r) = 2v m (1 -(r/r t ) 2 ) (see Appendix A). Thus, we have

F viscous = - 2π 0 z 0 η l (
dv/dr) r=rt r t dβdz, where η l is the liquid viscosity and dβ the elementary cross-sectional angle. After integration, one nds that:

F viscous = 8πη l v m z.
(5.7)

Injecting the di erent force contributions in Eq. 5.5 gives:

ρ l r 2 t d(zv m ) dt = 2r t γ al cos θ Y -8η l v m z -ρ l gr 2 t z. (5.8)
We consider di erent cases. First, for t 0, one can neglect the inertial term compared to the viscous term. As stated for the calculations of the viscous force, the ow of liquid is assimilated to a Poiseuille ow due to the low velocity and the no-slip condition at the wall. At the beginning of the capillary rise, the weight term can also be neglected providing that z γ al /ρ l gr t . As a consequence, we only balance capillary and viscous forces in the early stages of the capillary rise with v m = dz/dt. Eq. 5.8 yields after integration:

z = 1 2 γ al r t cos θ Y η l t 1/2 .
(5.9)

Eq. 5.9 is known as the Lucas-Washburn law [START_REF] Lucas | Über das Zeitgesetz des kapillaren Aufstiegs von Flussigkeiten[END_REF][START_REF] Washburn | The dynamics of capillary ow[END_REF]. The imbibition velocity decreases as a function of 1/ √ t, which is typically observed when co ee is imbibed into a piece of sugar for instance. Eq. 5.9 also allows to determine the Young contact angle θ Y for any porous material providing that the characteristic size of the pore r t is known. The t 1/2 regime has been observed in many homogeneous porous media of unique length scale [START_REF] Lago | Capillary rise in porous media[END_REF][START_REF] Marmur | Characterization of porous media by the kinetics of liquid penetration: The vertical capillaries model[END_REF][START_REF] Tampy | Wettability measurements of coal using a modi ed Washburn technique[END_REF]. Now, when t → 0, the imbibition velocity diverges, which is impossible. Thus, we have to consider the inertial term at very short times and balance this term with the capillary force only (since z → 0). Integrating Eq. 5.8 yields [START_REF] Quéré | Inertial capillarity[END_REF]:

z = 2γ al cos θ Y ρ l r t 1/2
t.

(5.10)

The imbibition velocity is constant and the inertial regime joins the Washburn regime for a characteristic time ρ l r 2 t /η l , typically 10 ms to 1 s for standard micrometric and millimetric tubes and water.

Finally, for t → ∞, the capillary height should reach its maximum h max given by Eq. 5.3. Approximating F viscous by 8πη l v m h max around z = h max yields a decreasing exponential relaxation towards h max in Eq. 5.8.

The 1D model to explain imbibition into porous media is actually limited for several reasons. First, modelling a porous medium as an assembly of capillary tubes is limited as, generally speaking, a porous medium cannot be described by using a unique length scale (the tube radius). Indeed, other dynamics have also been reported: the capillary rise of water in a column of glass beads follows a t 1/4 regime for late times due to the rough surfaces of the glass beads or by a dynamical mechanism of wetting and depinning at the contact line between the invading meniscus and the solid beads [START_REF] Delker | Interface pinning and the dynamics of capillary rise in porous media[END_REF][START_REF] Lago | Capillary rise in porous media[END_REF][START_REF] Reyssat | Imbibition in geometries with axial variations[END_REF][START_REF] Shikhmurzaev | Anomalous dynamics of capillary rise in porous media[END_REF][START_REF] Siddique | Capillary rise of a liquid into a deformable porous material[END_REF]. Heterogeneities at a length scale much smaller than the pore size might therefore play a crucial role in the imbibition dynamics. The characteristic size of the porous medium can also vary with the height z, as observed in imbibition into corners and wedges and explained in §5.1.1.3. Last, the characteristic size of the medium can evolve with time as observed when using deformable porous media and seen in §5.1.1.4. Tang et al. [START_REF] Tang | Capillary rise in tubes with sharp grooves[END_REF] and Ponomarenko et al. [START_REF] Ponomarenko | A universal law for capillary rise in corners[END_REF] have derived the imbibition dynamics in such geometries. As the imbibition height h increases, the apparent curvature radius of the meniscus

IMBIBITION THROUGH LIQUID FOAMS 125

r app decreases as one gets closer to the edge of the corner. When solving the dynamics by using a self-smilar solution, the authors show that capillary forces are balanced by gravity, thus r app ∼ γ al /ρ l gh at any position along the liquid column The capillary rise in open corners and wedges is equivalent to the capillary rise in an undeformable tube of radius r app . Replacing r t by r app in Eq. 5.9 gives the following time-dependent imbibition height:

h ∼ γ 2 al ρ l gη l 1/3 t 1/3 . (5.11)
One can also directly use the Stokes equation η l (dh/dt)/r 2 app ∼ (γ al /r app )/h to nd Eq. 5.11. The dynamics in t 1/3 is slower than the di usive Lucas-Washburn dynamics, due to the adaptability of the liquid column to the geometry of the corner or the wedge.

1D imbibition through a deformable solid porous medium

Now, what happens if the pores are deformable? Duprat et al. [START_REF] Duprat | Dynamics of elastocapillary rise[END_REF] and Cambau et al. [START_REF] Cambau | Capillary rise between exible walls[END_REF] have studied this question by using exible walls. They name the imbibition dynamics by the elastocapillary rise for a su ciently high Bond number Bo = ρ l ge W /γ al and high elastocapillary number

E ec = γ al 4 W /B ec e 2
W where W , e W and B ec are, respectively, the height of the walls, the spacing between the walls where they are clamped and the bending sti ness per unit width of the wall. High Bo and E ec mean that capillarity and gravity are equally dominant and that the walls are very exible.

Three di erent regimes are observable and depicted in Fig. 5.3. At early times, the walls are slightly de ected. Gravity is negligible and capillary forces are not important enough to pull and bend the walls. Imbibition is similar to the undeformable case. Thus, one nds the di usive Lucas-Washburn dynamics in h ∝ t 1/2 shown in §5.1.1.2. Then, when the capillary forces dominate the bending resistance of the sheets, the gap between the walls looks like a wedge and the sheets get closer at the bottom of the liquid column. Thus, the imbibition dynamics follows the t 1/3 power law de ned in §5. 1.1.3. The transition between the t 1/2 and the t 1/3 is de ned for a height h equal to the elastocapillary length ec = (B ec e 2 W /γ al ) 1/4 . For later times, the sheets touch at the bottom, which generates a new diverging geometry. Imbibition is slower in t 1/n with n > 3 and gravity-dominated.

Liquid ow in dry aqueous foams

Imbibition in aqueous foams is a complex problem, as they encompass some of these features : evolution of pore size along the vertical axis due to gravity and deformability of channel walls. A dry aqueous foam (typically for a liquid fraction φ l < 0.01) can be assimilated to a porous medium with liquid pore walls. Indeed, a foam is a network of liquid microchannels called Plateau borders in which liquid can ow. In this section, we model the ow of liquid inside an aqueous foam with regard to common theories about ow in solid porous media. We rst recall the driving pressure leading to imbibition, then we focus on modelling the ow of liquid through the foam.

Osmotic pressure and imbibition

In the case of liquid foams, the interfacial energy of a wet foam is lower than the interfacial energy of a dry foam since the bubbles are more spherical than in a dry foam. Thus, when an aqueous foam is put into contact with the same miscible liquid, liquid imbibition is driven by the reduction of interfacial energy that occurs when going from a dry to a wetter foam.

In other terms, the Plateau borders have a curvature creating a capillary underpressure in the liquid phase. For a dry foam, the pressure in the liquid phase is p = P b -γ aw /r P B ≈ p atm -γ aw /r P B by considering the pressure P b in the bubbles equal to the atmospheric pressure p atm (negligible overpressure in the bubbles). Due to the pressure di erence with the outer environment at the atmospheric pressure, an aqueous foam has the ability to absorb liquids, similarly to a sponge.

To prevent liquid from being absorbed, a pressure Π o called osmotic pressure in analogy with aqueous solutions has to be imposed:

Π o = p atm -p.
(5.12)

The osmotic pressure depends on the liquid fraction and this dependence has been quanti ed of the osmotic pressure as a function of φ l has been reported for monodisperse and polydisperse emulsions and monodisperse foams.

Flow of liquid through a porous medium

Due to the osmotic pressure, liquid can invade a dry foam. Due to gravity, the liquid phase wants to pass below the air phase in the drainage process. In both cases, the liquid ows through the Plateau borders. How does it ow? We can consider the ow of liquid in a cylindrical pore, as depicted in Fig. 5.5. The velocity eld u of the liquid in the pore is u = u(r)e β , where r is the radial coordinate in a pore. The unit vector e β is aligned with the revolution axis of the pore and directed towards the top. In this pore, the Reynolds number Re, which characterizes the dominance of inertia compared to viscous forces, is Re = ρ l ur c /η l . We can estimate Re with the density ρ l = 1000 g/L, the cylinder radius r c = 100 µm, η l = 10 -3 Pa.s and u = 10 -4 m/s, which gives Re = 0.01 1. Thus, we can use the Stokes equation to determine the velocity eld:

η l ∆u = ∇p -ρ l g.
(5.16)

Projecting Eq. 5.16 on e β yields:

η l 1 r ∂ ∂r r ∂u ∂r = dp d + ρ l g cos β.
(5.17) Eq. 5.17 is solved into detail in Appendix A without gravity. The calculations give the mean velocity along the cross-section u P B , also known as the Poiseuille law, as:

u P B = - A η l K c dp d + ρ l g cos β e β , (5.18)
where A is the cross-sectional area of the pore and K c the permeability coe cient. Eq. 5.18 can be generalized to a porous medium by averaging the velocity and the pressure over an elementary volume of pores. This is Darcy's law [START_REF] Darcy | Les fontaines publiques de la ville de Dijon: exposition et application des principes à suivre et des formules à employer dans les questions de distribution d'eau[END_REF]. The average liquid velocity in the porous medium u Darcy is written as:

u Darcy = φ l < u P B >= k D η l (-∇p + ρ l g) , (5.19)
where < u P B > is the average velocity over the whole elementary volume and k D the permeability (m 2 ). We can project Eq. 5.18 on the vertical direction e z , which gives:

u P B-z = - A η l K c dp dz + ρ l g cos 2 β. (5.20)
We can express < u P B-z >= π/2 0 u P B-z p(β)dβ where p(β) is the probability to nd a pore between β and β + dβ. This probability is equal to the ratio between the area covered by the pore between β and β + dβ and the area covered by the half-sphere, which gives:

p(β)dβ = 2π 0 2 sin βdβdξ 2π 0 π/2 0 2 sin βdβdξ = 2π 2 sin βdβ 2π 2 = sin βdβ.
(5.21) Thus, the average vertical component of the velocity eld is:

< u P B-z >= - A η l K c dp dz + ρ l g π/2 0 cos 2 β sin βdβ = - A η l K c 3 dp dz + ρ l g . (5.22)
Projecting Darcy's velocity u Darcy on the vertical direction e z from Eq. 5.19 yields:

< u P B-z >= u Darcy-z φ l = - k D η l φ l dp dz + ρ l g .
(5.23)

Identifying Eq. 5.22 and Eq. 5.23 gives:

k D = AK c φ l 3 .
(5.24) l . More general models associate the dissipation in the Plateau borders and the nodes [START_REF] Cohen-Addad | Flow in foams and owing foams[END_REF][START_REF] Koehler | A generalized view of foam drainage: Experiment and theory[END_REF][START_REF] Lorenceau | Permeability of aqueous foams[END_REF]]. In the following experimental studies, we use rigid interfaces, for which Bq 1 and K c = 0.02, and we apply the channel-dominated model.

Drainage equation

For any ow of miscible liquid in a dry foam, there is a driving equation, called drainage equation, which describes the evolution of the liquid fraction φ l with the time t and the spatial coordinates.

For dry foams, the osmotic pressure can be written as:

Π o = p atm -p ≈ γ aw r P B ≈ γ aw δ b R b φ 1/2 l .
(5.26) By combining Eq. 5.19, 5.26 and k = k f oam with the unit vector e z directed upwards, we have:

< u P B >= k f oam η l φ l ∇ γ aw δ b R b φ 1/2 l -ρ l ge z .
(5.27)

Then, we consider mass conservation in the Plateau border for the liquid phase, which is expressed as:

∂φ l ∂t + ∇ • (φ l < u P B >) = 0.
(5.28)

Injecting Eq. 5.27 into Eq. 5.28 yields the comprehensive drainage equation for the liquid fraction

φ l : ∂φ l ∂t -∇ • γ aw k f oam 2δ b R b η l φ 3/2 l ∇φ l + k f oam ρ l g η l e z = 0.
(5.29)

Eq. 5.29 is the driving equation to study the ow of miscible liquids into a dry aqueous foam.

Changing the boundary and the initial conditions changes the type of ow: ow downwards due to gravity for drainage issues, ow upwards due to the capillary underpressure for imbibition. We resort to Eq. 5.29 in the next sections §5.2 and §5.3.

Point-source imbibition from an oil-lled pore

The results presented in this section have been published in [START_REF] Mensire | Point-source imbibition into dry aqueous foams[END_REF]. I am very grateful to Jesse T. Ault, PhD candidate at Princeton University, with whom I worked on this project.

The con guration we highlight in this section is the imbibition from a point-source, which requires to build a 3D model. Not surprisingly, imbibition for undeformable and deformable solid porous media in 2 and 3 dimensions have also been studied [START_REF] Mendez | Imbibition in porous membranes of complex shape: Quasi-stationary ow in thin rectangular segments[END_REF][START_REF] Xiao | Source-like solution for radial imbibition into a homogeneous semi-in nite[END_REF]. Similar developments can be made for imbibition of dimensions in aqueous foams. In our case, we try to mimic the liquid extraction from a micropore. We develop a mathematical model and compare with experimental results for the radial imbibition from a point source (at imposed pressure). Also, we develop a model to explain why the imbibition of oil is possible in some cases despite the creation of new oil-water interfaces.

3D imbibition from a point-source for miscible liquids

We focus on a particular geometry where a point-source lled with liquid -immiscible with the foam or not -is placed below a dry foam. We recall the foam drainage equation [START_REF] Koehler | Dynamics of foam drainage[END_REF][START_REF] Verbist | A soluble model for foam drainage[END_REF][START_REF] Verbist | The foam drainage equation[END_REF], which describes foam imbibition and drainage. We make the following assumptions: -We consider a dry foam with a constant mean bubble radius R b and a low liquid fraction φ w < 5 x 10 -3 , -The size of the point source is much smaller than the typical dimensions of the foam and the foam domain can be considered as an in nite space for imbibition,

POINT-SOURCE IMBIBITION FROM AN OIL-FILLED PORE

133

-We assume isotropy of the foam. We use cylindrical coordinates (radial coordinate r and axial coordinate z).

Di erential equation for the cross-sectional area A of a Plateau border

Like in §5.3, we consider rigid interfaces in the foam and use a channel-dominated model for the foam permeability k f oam [START_REF] Cantat | Foams: Structure and Dynamics[END_REF][START_REF] Cohen-Addad | Flow in foams and owing foams[END_REF][START_REF] Koehler | A generalized view of foam drainage: Experiment and theory[END_REF][START_REF] Neethling | Modeling oil droplets in Plateau borders[END_REF][START_REF] Nguyen | Liquid drainage in single Plateau borders of foam[END_REF]:

k f oam = δ a δ 2 b R 2 b φ 2 w 150 , (5.30)
with φ w the liquid fraction of water , δ a = √ 3 -π/2, a numerical factor deduced from the cross-sectional area of a Plateau border A = δ a r 2 P B , and δ b = 1.74. Combining Eq. 5.29 and Eq. 5.30 yields the time-space dependence of the liquid fraction φ w :

∂φ w ∂t = ∇ • γ aw δ a δ b R b 300η w φ 1/2 w ∇φ w + δ a δ 2 b R 2 b ρ w g 150η w φ 2 w e z . (5.31)
From Eq. 5.31, and

φ w = A/(δ a δ 2 b R 2 b )
, we deduce a non-linear partial di erential equation for the time-space evolution of the cross-sectional area of a Plateau border [START_REF] Koehler | Dynamics of foam drainage[END_REF]:

∂A ∂t = γ aw δ 1/2 a 300η w ∇ • A 1/2 ∇A + ρ w g 150η w ∂A 2 ∂z .
(5.32)

For the initial condition, we assume that the foam is dry, φ w = 0. For the boundary conditions, we assume that the liquid fraction and thus the cross-sectional area of a Plateau border far from the point-source is close to 0. At the point-source, A = δ a R 2 b , which is the cross-sectional area for the maximum packing of undeformed bubbles. Thus, we have the initial and boundary conditions:

A(r > 0, z > 0, 0) = 0 , A(r, z → +∞, t) = 0 , A(r → +∞, z, t) = 0 , A(0, 0, t) = δ a R 2 b , (5.33)
and the no-ux condition at z = 0:

∂A ∂z = -2ρwg δ 1/2 a γaw A 3/2 .
The no-ux condition is deduced from

u Darcy • e z = ∂ ∂z γawδ 1/2 a A 1/2
-ρ w g = 0.

Solution to the partial di erential equation (PDE)

We non-dimensionalize A, r, z and t as α

= A/(δ a R 2 b ), R = r/R b , Z = z/
∂α ∂τ = Bo ∂α 2 ∂Z + 1 2 1 R ∂ ∂R Rα 1/2 ∂α ∂R + ∂ ∂Z α 1/2 ∂α ∂Z , (5.34)
with the initial and boundary conditions:

α(R > 0, Z > 0, 0) = 0 , α(R, Z → +∞, τ ) = 0 , α(R → +∞, Z, τ ) = 0 , α(0, 0, τ ) = 1 , (5.35)
and the no-ux condition in Z = 0: ∂α ∂Z = -2Boα 3/2 . To solve Eq. 5.34 with the conditions 5.35, we use a discretization by nite di erences on a 2D spatial mesh and solve the PDE with Matlab. The numerical solution gives α(R, Z, τ ), as shown in Fig. 5.7a. The front position is determined when α = 0 as α decreases from the point source to the outer boundary of the mesh, which gives the front pro le for di erent times and Bond numbers (Fig. 5.7b-d). The simulations show that the front pro le attens out as Bo and τ increase.

POINT-SOURCE IMBIBITION FROM AN OIL-FILLED PORE 135

3D imbibition from a point-source for immiscible liquids

Now, let us describe the imbibition of immiscible liquids and the di erences with miscible liquids.

Local picture of oil invasion

When a dry aqueous foam is wetted by a miscible liquid, the physical mechanism that enables the imbibition is the reduction of the surface area of the air-water interfaces inside the foam, i.e, the total energy of the system decreases [START_REF] Höhler | Osmotic pressure and structures of monodisperse ordered foam[END_REF]. The liquid ow swells the Plateau borders, creating more surface area. However, if we assume a constant volume of gas in the bubbles, the surface area of the lms needs to decrease.

The reduction of interfacial energy is used by the foam to pump a volume dV of liquid at the osmotic pressure Π o = p atm -p. Thus, we have Π o dV = -γdΣ (with dΣ < 0) where γ is the interfacial tension, dΣ = dS f + dS P B is the variation of the air-water area, and S f and S P B are the lm and Plateau border area, respectively (Fig. 5.8a). For miscible liquids, the decrease of interfacial energy drives the imbibition into a dry aqueous foam. For immiscible liquids, such as organic oils, we sketch the liquid ow (in the dry limit) by an oil slug that penetrates the Plateau border, as shown in Fig. 5.8b; new air-water surfaces in the Plateau border are created and the surface area of the lms decreases. However, new oil-water interfaces are also created, which are energetically costly. Therefore, we develop another model to add the in uence of these oil-water interfaces.

CHAPTER 5: OIL IMBIBITION INTO AQUEOUS FOAMS

E ective interfacial tension

Rewriting the osmotic pressure by using the interfaces dS w and dS o for the variations of the air-water and oil-water interfaces, respectively, yields:

Π o dV = -(γ aw dS w + γ ow dS o )
, with dS w < 0 and dS o > 0.

(5.36)

Eq. 5.36 can also be expressed as:

Π o dV = -γ aw dS w 1 + γ ow γ aw dS o dS w = -γ ef f dS w , (5.37)
with γ ef f an e ective interfacial tension:

γ ef f = γ aw 1 + γ ow γ aw dS o dS w . (5.38)
For miscible liquids, γ ow = 0, so γ ef f = γ aw and the osmotic pressure is related to the curvature of the Plateau borders by Π o ≈ γ aw /r P B . However, if γ ow = 0, then γ ef f accounts for the oilwater interfacial tension and the creation of oil-water interfaces. The osmotic pressure becomes

Π o = γ ef f /r P B ≈ γ ef f /(δ b R b φ 1/2
ow ) where φ ow is the combined liquid fraction for oil and water. Since dS w < 0, then γ ef f < γ aw . With immiscible liquids, the osmotic pressure is reduced compared to miscible liquids, i.e., the driving force for imbibition is weaker. The value of γ ef f can be estimated. In particular, the variation of the air-water interfaces dS w can be decomposed into two contributions, one from the lms dS f , and the other from the Plateau borders dS P B (Fig.

5.8b).

Pitois et al. [START_REF] Pitois | Speci c surface area model for foam permeability[END_REF] and Hilgenfeldt et al. [START_REF] Hilgenfeldt | A generalized view of foam drainage[END_REF] provide estimates of the lm and Plateau border surface areas with regard to the liquid fraction in the foam:

S f ≈ 3.3 R b V f oam (1 -φ ow )(1 -1.52φ 1/2 ow ) 2 ,
(5.39a)

S P B ≈ V f oam 1.5R b π √ 3δ a φ 1/2 ow , (5.39b)
where V f oam is the volume of foam. Di erentiating S f and S P B with respect to φ ow in the dry limit yields:

dS f ≈ - V f oam R b (3.3)(3.04) 2φ 1/2 ow dφ ow , (5.40a
)

dS P B ≈ V f oam R b π 3 √ 3δ a φ 1/2 ow dφ ow .
(5.40b)

In the dry limit, we can assume as a rst approximation that dS o ≈ dS P B and neglect the oilwater interface at the top of the oil slug (Fig. 5.8b). This assumption is in agreement with recent Surface Evolver simulations of the shape of an oil slug in a single Plateau border [START_REF] Neethling | Modeling oil droplets in Plateau borders[END_REF]. In our case, the ratio γ ow /γ aw ≈ 0.2 is low and the ratio between the equivalent spherical radius for an oil slug embedded within the Plateau border and the radius of curvature of the Plateau border is close to 1 due to the very low liquid fraction. Thus, by using (5.40a) and (5.40b), we have:

γ ef f ≈ γ aw + γ ow dS P B dS f + dS P B ≈ γ aw 1 -3 γ ow γ aw .
(5.41)

Driving equation for immiscible liquids

With this e ective interfacial tension, the osmotic pressure is reduced by a factor that depends on the ratio between the oil-water and the air-water interfacial tensions. More precisely, the governing equation for A is the same as developed previously for aqueous liquids in (5.32) with A, the cross-sectional area of the Plateau border for both oil and water phases, γ ef f , instead of γ aw , and η o the oil viscosity:

∂A ∂t = γ ef f δ 1/2 a 300η o ∇ • A 1/2 ∇A + ρ o g 150η o ∂A 2 ∂z .
(5.42)

We only use η o because the viscous dissipation occurs predominantly in the oil phase as shown by Piroird et al. [START_REF] Piroird | Capillary ow of oil in a single foam microchannel[END_REF] in the limit of η o /η w 1. Eq. 5.42 is identical to Eq. 5.32 and can be solved numerically, except with a smaller air-water interfacial tension. The smaller capillary pressure induced by the oil phase is in agreement with our imbibition experiments presented in the following: for the same elapsed time, the vertical front position of the oil is smaller than the aqueous case, as shown in Fig. 5.11.

Experimental imbibition front

Here, we present the imbibition experiments that we compare to the aforementioned theoretical developments. Most of the chemicals used in this section are similar to the chemicals presented in §2.3.1, as well as the generation of the foam.

Experimental set-up

The foaming solution we use is described in [START_REF] Basheva | Role of betaine as foam booster in the presence of silicone oil drops[END_REF][START_REF] Golemanov | Surfactant mixtures for control of bubble surface mobility in foam studies[END_REF][START_REF] Piroird | Capillary ow of oil in a single foam microchannel[END_REF] and consists in a mixture of di erent components (cocoamidopropyl betaine, sodiumlaurylethyl sulfate, myristic acid and glycerol), already presented in §1.1.3 and §2.3. It provides rigid boundary conditions to the air-water interfaces due to the presence of the fatty myristic acid [START_REF] Basheva | Role of betaine as foam booster in the presence of silicone oil drops[END_REF][START_REF] Golemanov | Surfactant mixtures for control of bubble surface mobility in foam studies[END_REF]. The air-water interfacial tension, viscosity and density of this mixture are, respectively, γ aw = 23.7 ± 1 mN/m, η l = 1.4 mPa.s and

ρ l = 1026 kg/m 3 .
As imbibing liquids, we use olive oil and two glycerol solution mixtures, whose wt.% of glycerol are di erent; the properties (oil-water surface tension, dynamic viscosity, density) are summarised in Table 5.1. We also add a small quantity of uorescent dyes (1:50) to the imbibing liquid (Yellow Black from Rohm and Haas, Fluorescein from Sigma-Aldrich and Tracerline) to enhance the contrast between the foam and the imbibing liquid. We checked that the addition of dyes does not change the value of γ ow .

Imbibing We generate foams with a well-controlled average bubble radius R b = 1 -2 mm by injecting nitrogen or compressed air through a needle into the foaming solution by the same procedure presented in §2.3.1. The rigid interfaces owing to the type of surfactants used yield a stable, monodisperse foam (deviations from the mean bubble radius remain below 5 %). After generating the foam in a 20 cm-high rectangular column with a 4.5 cm-wide square base, we let it drain and extract at random times at the top of the column a foam sample for which the volume V f oam and the weight m f oam are known. The sample is turned upside down to invert the drainage process.

γ ow η ρ R b B liquid CAPB+SLES 0 64
The initial liquid fraction of the foam sample is deduced from φ i = m f oam /(ρ w V f oam ), where ρ w is the density of the foaming solution.

For the imbibing reservoir, we use a polycarbonate plate into which a 1 mm-wide hole is drilled.

A small capillary tube of 1 mm diameter is inserted into the hole and slightly displaced upwards into the foam to ensure complete contact between the foam and the reservoir. This capillary tube is connected to a 20 cm-long and a 2 mm-diameter PTFE tube (Fig. 5.9). This set of tubes is lled with the imbibing liquids, which completely wet the tubes. The bigger tube is attached to a motion controller that allows the adjustment of the input liquid level at the same height as the output liquid level while the liquid ows through the foam. This system sets a constant atmospheric pressure p atm at the outlet of the capillary tube. We also use another system without feedback for comparison, by using a 5 cm-wide funnel for which the output level does not vary much when the liquid is imbibed. Both systems give the same results as shown below.

Imbibition picture

At time t = 0, the bottom of the foam touches the point source of the liquid. We record (at 24 frames per second for 3 to 4 minutes) foam imbibition by using uorescent imaging. Fig. 5.10 shows a typical image sequence of the imbibition process.

The same experiment is repeated for the di erent imbibing liquids of Table 5.1. Thus, di erent liquids, di erent viscosities η, bubble radii R b , air-water (γ aw ) and oil-water (γ ow ) interfacial ten- sions, and initial liquid fractions φ i are tested. We determine the imbibition front by applying a threshold at the boundary between the brighter and darker uorescent areas. Using ImageJ software for image processing, we plot the maximum vertical position z f of the front with respect to time.

Time-dependence of the imbibition front

Fig. 5.11 shows two data sets for the evolution of the vertical front position z f (t). The experimental data for two di erent liquids typically have error bars of ±0.5 mm, owing to the determination of the front, especially in the case of miscible liquids, where light di usion by the lms can create a small front width. For both types of liquid, the position of the front moves faster at short times and slows down as time increases. For similar viscosities, oil imbibition is slower than aqueous imbibition.

Comparison with experimental results

We now build a single comparison between the theory for miscible and immiscible liquids and make some remarks on the resolution of the PDE.

Collapse of the theory with the experiments

In Fig. 5.12, we rescale z f by R b and t by the capillary time (150/δ a )η o/w R b /γ ef f , which is the same scaling as in Eq. 5.34 written with γ ef f instead of γ aw , and the viscosity of oil or aqueous solutions η o/w . The log-log plot in the dimensionless variables displays a reasonable collapse between the experimental data and the numerical solution for all miscible and immiscible imbibing liquids for the range of non-zero Bond numbers between 0.5 and 5. The collapse occurs at shorter τ for immiscible liquids, due to the smaller scaling in time induced by γ ef f . Figure 5.12: Vertical front position z f with respect to time in dimensionless coordinates for immiscible oils (open dots) and miscible aqueous liquids (closed dots). The experimental curves are obtained for two R b = 1 and 2 mm, di erent φ i , η o/w and Bo. The numerical solutions of Eq. 5.34 for Bond numbers Bo = 0, 1.6 and 5, deduced from Eq. 5.34, are plotted by the dashed lines. The self-similar power law evolution in τ 1/2 in the no-gravity case (Eq. 5.44) is shown.

In both miscible and immiscible cases, the numerical solutions that include the capillary pressure gradient and gravity are comparable to the experimental data. The numerical solution with Bo = 0 largely deviates from the experimental data: gravitational e ects atten out the imbibition pro les very quickly as shown in Fig. 5.7. However, our numerical solution also deviates from the data at short times. Indeed when the front position is of the order of magnitude of one bubble size (R * < 2), the Darcy model for the average velocity is not adequate, as imbibition occurs in individual Plateau borders. Also, φ ow = 0 around the point source.

The discrepancy between the numerical solution at Bo = 0 and the experimental data demonstrate that gravitational e ects are comparable to the capillary pressure. Yet, our result di ers from the t 1/3 power law developed by Xiao et al. [START_REF] Xiao | Source-like solution for radial imbibition into a homogeneous semi-in nite[END_REF] for radial imbibition in undeformable porous media with a constant permeability k D . We account for the deformability of the Plateau borders (time variations and non-constant k D ) and the slope of the numerical solution depends on Bo in Fig. 5.12, which produces a non-constant power law from one Bond number to another. We assume that viscous dissipation occurs in the Plateau borders and adopt a channel-dominated model for the liquid ow through the foam. Also, we consider gravity e ects, which are neglected in [START_REF] Xiao | Source-like solution for radial imbibition into a homogeneous semi-in nite[END_REF], since Bo 1 due to the micron size of the glass beads in their porous matrix.

Analytical solution with Bo = 0

An analytical scaling for the front position in the no-gravity case (Bo = 0), which is encountered in microgravity conditions, can be found by assuming spherical symmetry with the dimensionless spherical distance R. We introduce the dimensionless self-similar variable ζ = R/τ 1/2 . Substituting ζ into Eq. 5.34, written with γ ef f and η o/w , we nd that α is the solution of the ordinary di erential equation:

ζ 3 dα dζ + d dζ ζ 2 α 1/2 dα dζ = 0. (5.43)
The rst boundary condition is

α(ζ f ) = 0, where ζ f = 150η o/w γ ef f δaR b 1/2 r f t 1/2
is a constant which then yields the spherical front radius r f (t). A local analysis at the front provides a second boundary condition and uniquely determines the solution.

Setting ζ 4

f α * = α yields the same equation as (5.43) except with ζ f = 1. Therefore, we can take ζ f = 1 without loss of generality. The wetting front r f is given by:

r f (t) R b = δ a γ ef f 150η o/w R b t 1/2 .
(5.44)

CAPILLARY RISE INTO AQUEOUS FOAMS FOR MISCIBLE AND IMMISCIBLE LIQUIDS 143

This solution is plotted in Fig. 5.12. The t 1/2 power law result recalls the 1D di usive imbibition in a Hele-Shaw cell observed in [START_REF] Caps | Capillary rise in foams under microgravity[END_REF][START_REF] Saint-Jalmes | Di usive liquid propagation in porous and elastic materials: The case of foams under microgravity conditions[END_REF]. However, the collapse between the PDE solution with Bo = 0 and the self-similar solution in t 1/2 occurs at long times for τ > 10 4 . Indeed, the solution to Eq. 5.43 has a singularity at the point source, where α diverges to +∞ [START_REF] Christov | Resolving a paradox of anomalous scalings in the di usion of granular materials[END_REF]. However, the solution to Eq. 5.34 has a nite boundary condition α = 1 at the point source. Furthermore, the point source has the nite size of the inlet tip (half a bubble radius). Thus, the introduction of a source value and a length scale at the origin causes the lack of agreement between the PDE solution and the t 1/2 power law. Given su cient time (beyond our experimental time) to lose the e ect of the initial and boundary conditions, which is a classical condition in self-similar solutions enclosed in the structure of the solution itself, the collapse of both solutions should occur. The observed good agreement between experiments and the gravity-free analytical solution at short times is thus only a coincidence as this solution is only valid for R * > 2. Moreover, due to the values of the Bond numbers from 0.5 to 5, the di usive solution (and thus the no-gravity approximation) is not valid over our experimental range.

Capillary rise into aqueous foams for miscible and immiscible liquids

This study has been published in [START_REF] Mensire | Capillary imbibition of aqueous foams by miscible and nonmiscible liquids[END_REF] and was presented at the Eufoam 2014 and the CMD 2014 conferences.

In §5.2, we have studied the 3D point-source imbibition into dry aqueous foams from an oil-lled pore for miscible and immiscible liquids. Here, we consider the 1D imbibition of an oil slick into an aqueous foam. This situation can be encountered when one would like to clean a surface where oil has been spilled or a surface polluted by an environmental catastrophe, such as black tides. In some sense, the oil slick can be considered as a multitude of small point-sources which converge towards each other to create a uniform 1D imbibition front. We can also notice that other con gurations have also been studied theoretically and experimentally: pulsed imbibition at constant volumes and 2D foam drainage at constant ow rate [START_REF] Cox | Comment on "Foam imbibition in microgravity. an experimental study[END_REF][START_REF] Hutzler | Foam drainage in 2D[END_REF][START_REF] Koehler | Dynamics of foam drainage[END_REF][START_REF] Koehler | Foam drainage on the microscale -I. Modeling ow through single Plateau borders[END_REF].

In this section, we experimentally study the 1D imbibition dynamics of miscible and immiscible liquids into a dry aqueous foam. The con guration is an aqueous or an oil slick absorbed into the foam. Our experimental set-up is the one used for the testing experiments in §2.3. We raise questions about the dynamics of imbibition in this geometry and compare our observations in the framework previously developed for the 3D imbibition from a point-source.

Oil absorption: characteristics and picture

The foaming solution is the CAPB-SLES-Mac-10% glycerol used in 5.2 and described in [START_REF] Basheva | Role of betaine as foam booster in the presence of silicone oil drops[END_REF][START_REF] Golemanov | Surfactant mixtures for control of bubble surface mobility in foam studies[END_REF][START_REF] Piroird | Capillary ow of oil in a single foam microchannel[END_REF].

The characteristics of the imbibing liquids are summarized in Table 5.1 while the miscible liquids with a viscosity η w ranging from 1.4 to 100 mPa.s are obtained by adding glycerol to the foaming solution. With our foaming solution and organic oils, such as olive oil or sun ower oil, both E and S exhibit negative values, as shown in Tab. 5.1. The viscosity η o and the density ρ o of the two oils are also displayed in Tab. 5.1. As in §5.2, after generation, we let the foam drain. During this drainage period, we extract the driest upper part of the foam at di erent times. Then, we reverse the foam sample to make the liquid fraction homogeneous and stop the drainage process in the sample. The volume V f oam and the mass m f oam of the sample are measured using a precision weighing scale. We deduce φ i the liquid fraction of the sample from φ i = m f oam /V f oam ρ w . Each time corresponds to a speci c liquid fraction and typically 0.7 x 10 -4 < φ i < 1.2 x 10 -3 . With this set-up, we can therefore vary independently φ i and R b . We also obtain the curvature radius of the Plateau border r P B = 1.74φ 1/2 i R b between 15 and 85 µm (see Fig. 5.13), in the limit where

φ i < 10 -2 .
Given that the contrast of optical indexes between the foam and the liquid is small or null, we still add a small quantity of liquid uorescent dye to the imbibing liquid at a concentration of 10 -2 g/g ( uorescein from Sigma-Aldrich for aqueous solutions and Yellow Black from Rohm and Haas for organic oils). Those uorescent markers, which are trapped in the liquid, provide a means of visualizing the swollen part of the foam [START_REF] Piroird | Oil repartition in a foam lm architecture[END_REF]. With this set-up and under illumination with excitation at 488 nm, the liquid is luminous while the aqueous foam -only constituted of air and foaming solution -is not visible. The uorescence intensity is followed with a camera (AVT Marlin) recording at typically 6 frames per second. Fig. 5.13 illustrates such an experiment where a dry dark foam is put into contact with a 20 µL uorescent oil drop sitting on a solid surface. We rst observe a lateral spreading of the oil drop, squeezed between the solid surface and the foam. Yet, after ten seconds, oil wicks into the foam, revealing the foam architecture and forming a complex oil-laden foam structure. After 5 minutes, oil has risen in the foam up to a height h = 10 mm. The foam lm withstands without any problem the oil invasion: we barely observe any lm break-up during the whole process. This suggests that the imbibition only occurs within the Plateau borders and the nodes and does not interfere with the fragile structure of the foam lms as previously observed at the scale of a single Plateau border [START_REF] Piroird | Capillary ow of oil in a single foam microchannel[END_REF]. Yet, the structure of this oil-laden foam is not frozen: elementary topological changes known as T1 are frequently observed due to liquid redistribution [START_REF] Piroird | Oil repartition in a foam lm architecture[END_REF].

Theoretical analysis

We present here the theoretical analysis by comparison with the drainage theory and the determination of a scaling law for the imbibition dynamics.

Time-dependent imbibition front

Our uorescent set-up allows us to measure h, the upper boundary of the imbibition front in the foam as de ned in the fth frame of Fig. 5.13. Fig. 5.14 shows h as a function of time t, when a 20 µL olive oil drop is put into contact with foams at various liquid fractions. As classically observed in porous media, the early times dynamics is fast. Moreover, here, the drier the foam, the faster the early-time dynamics, a feature which underlines the importance of the liquid fraction of the foam prior to imbibition. When plotted in log-log scale, those curves do not exhibit a well-de ned power law as it can be seen on the inset of Fig. 5.14. The data are well tted by two di erent straight lines of di erent slopes corresponding to the t 1/2 early and t 1/4 late-time dynamics. We also emphasize that for the driest foam, the front velocity is small: typically, ḣ ∼ 0.2 mm/s at t = 10 s. Note that the imbibition of glycerol and sun ower oil exhibit the same trends. The plain line corresponds to the t 1/2 dynamics detailed in [START_REF] Caps | Capillary rise in foams under microgravity[END_REF], while the dashed line illustrates the t 1/4 dynamics.

Comparison with the drainage theory

Foam drainage theory provides di erent scaling laws for foam imbibition, which can be confronted to our results. The foam drainage equations allow to study the time-space evolution of the liquid fractions for di erent con gurations: free drainage (when a foam drains due to gravity), pulsed and forced drainage (when liquid is injected into a foam at constant volume and ow rate respectively). In particular, the di usive regime of imbibition observed under microgravity conditions [START_REF] Caps | Capillary rise in foams under microgravity[END_REF][START_REF] Saint-Jalmes | Di usive liquid propagation in porous and elastic materials: The case of foams under microgravity conditions[END_REF] is in very good agreement with our data as it can be seen in the inset of Fig. 5.14 for very short times, thus underlining the negligible role of gravity when the foam is wet at the bottom.

We can write the drainage equation Eq. 5.45 with the permeability

k f oam = δ a δ 2 b R 2 b K c φ 2 l /3
in the channel-dominated model and the e ective interfacial tension γ ef f found in §5.2:

∂φ l ∂t -∇ • γ ef f δ a δ b R b K c φ 1/2 l 6η l ∇φ l + δ a δ 2 b R 2 b K c φ 2 l ρ l g 3η l e z = 0.
(5.45)

CAPILLARY RISE INTO AQUEOUS FOAMS FOR MISCIBLE AND IMMISCIBLE LIQUIDS 147

Projecting Eq. 5.45 on e z yields:

∂φ l ∂t - δ a δ b R b K c 3η l ∂ ∂z γ ef f φ 1/2 l 2 ∂φ l ∂z + δ b R b φ 2 l ρ l g = 0. (5.46)
We can rewrite Eq. 5.46 by replacing φ l by the cross-sectional area A = δ a δ 2 b R 2 b φ l , which gives:

∂A ∂t - γ ef f δ 1/2 a K c 6η l ∂ ∂z A 1/2 ∂A ∂z - ρ l gK c 3η l ∂A 2 ∂z = 0. (5.47)
For the initial condition, we assume that the foam is dry, φ l = 0 = A. For the boundary conditions, we assume that the liquid fraction and thus the cross-sectional area of a Plateau border far from the bottom of the foam is close to 0. At the bottom, A = δ a R 2 b , which is the cross-sectional area for the maximum packing of undeformed bubbles. Thus, we have the following initial and boundary conditions:

A(0, t) = δ a R 2 b , A(z > 0, 0) = 0 , A(z → +∞, t) = 0. (5.48)
We non-dimensionalize A, r, z and t as α = A/(δ a R 2 b ), R = r/R b , Z = z/R b and τ = (δ a K c /3)t/(η l R b /γ ef f ), and introduce the Bond number Bo = ρ l gR 2 b /γ ef f , transforming Eq. 5.47 into:

∂α ∂τ = Bo ∂α 2 ∂Z + 1 2 ∂ ∂Z α 1/2 ∂α ∂Z , (5.49)
with the initial and boundary conditions: At late times, analytical and numerical calculations for the imbibition of in nitely dry foams gives a self-similar solution for Z f [START_REF] Koehler | Dynamics of foam drainage[END_REF]: In dimensionalized parameters, Eq. 5.51 gives the evolution of the imbibition height h with the time t:

α(Z > 0, 0) = 0 , α(Z → +∞, τ ) = 0 , α(0, τ ) = 1. (5.50)
Z f = 12.1 τ Bo 1/3 . ( 5
h = 1.24 γ 2 aw ρ l gη l 1/3 t 1/3 . (5.52)
Unfortunately, Eq. 5.52 neither captures the in uence of φ i nor R b that is observed in the experiments. This can be understood by recalling that Equation 5.52 has been derived i) solving the Partial Di erential Equation (PDE) ruling the evolution of r(z, t) ii) assuming a foam with an initial liquid fraction equal to zero and iii) seeking for self-similar solutions of the PDE of the same form as in the free drainage case. This procedure is particularly useful to describe the asymptotic behavior of solutions in the limit where these no longer depend on the detail of the initial and boundary conditions. However, in our experiment, the foam sample is only 5-cm high, not in nitely dry and the time-scale is limited. Thus, the initial and boundary conditions are bound and cannot be neglected. Plus, the self-similar solution is found with a no-ux condition from free drainage at the top of the foam, which does not provide an appropriate boundary limit in this problem. Oil ows from the bottom of the foam and the ux condition is non-zero at z = 0. We thus never observe any convergence of the dynamics towards the t 1/3 regime. This is shown in Fig. 5.16 which exhibits the imbibition height h deduced from the self-similar solution (Eq. 5.52) and the exact solution calculated by solving Eq. 5.49. We cannot match our experimental data with the solution of the foam drainage equation (either self-similar solution or numerical solution for a given Bond number).

CAPILLARY RISE INTO AQUEOUS FOAMS FOR MISCIBLE AND IMMISCIBLE LIQUIDS 149

Figure 5.16: Dimensionless imbibition height Z f as a function of the dimensionless time τ deduced from numerical resolution of Eq. 5.49 with the comparison to the self-similar solution given by Eq. 5.52 for di erent Bond numbers Bo = 0.4, 1.6 and 4. Experimental data for Bo = 2.1 and two di erent initial liquid fractions φ i are also plotted and do not converge towards a unique solution.

Scaling law

We try another approach to solve the problem by considering the ow of oil locally. For the length scale d = 10 µm, which is the typical width of a Plateau border, the Reynolds number is around Re = ρ l ḣd/η l ∼ 10 -11 << 1. The rigid boundary conditions provided by the foaming solution and the small value of Re suggest that the uid motion in the foam network corresponds to a Stokes ow where the capillary driving force is balanced with the viscous dissipation, as suggested in 5.1.2.2. In the limit where γ ow /γ aw 1, recent experimental works at the scale of a single horizontal Plateau border have revealed the following features [START_REF] Cohen | Inertial mass transport and capillary hydraulic jump in a liquid foam microchannel[END_REF][START_REF] Piroird | Capillary ow of oil in a single foam microchannel[END_REF]. On one hand, the driving capillary pressure that sustains the imbibition in the Plateau border is given by γ aw /r P B , where r P B is the curvature radius of the Plateau border prior to imbibition.

On the other hand, the bulk viscous dissipation only occurs in the oil phase and scales as η o ż/r * 2 (z, t), with the position z along the Plateau border and r * 2 (z, t) is proportional to the swollen Plateau border cross section A = δ a r * 2 with δ a = 0.16. In [START_REF] Piroird | Capillary ow of oil in a single foam microchannel[END_REF], the experiments were either driven at constant ow rate or constant volume of oil, thus r * (z, t) is deduced from mass conservation [START_REF] Warren | Late stages for various wicking and spreading problems[END_REF], while here imbibition occurs from a large reservoir. In our case, the initial squeezing of the oil droplet, as shown in Fig. 5.13, creates an oil slick at the bottom of the foam and generates a reservoir of oil for the foam. The oil spreading is dynamically instantaneous compared to the initial rise of oil. Thus, we can assimilate our imbibition process to a quasi one-dimensional imbibition. This is all the more a good approximation that a change of oil volume from 20 to 80 µL does not involve any change in the imbibition pattern. Moreover, in this experiment, the size of the oil slick -after it has been squeezed by the foam Figure 5.17: Typical pro le of the uorescence intensity I of a single node during olive oil imbibition as a function of the vertical coordinate z for di erent times t. A peak corresponds to an oil-lled node or an oil-lled Plateau border, a hole to air in the bubbles.

-is large compared to the size of the bubbles. This suggests that the imbibition simultaneously occurs from di erent points. The imbibition from each point source is symmetrical radially, yet since the upward velocity is identical in each point source, the front evolves towards a plane. To gain quantitative understanding of the spatial and temporal evolution of the Plateau border radius r * , we take advantage of the uorescence intensity of our images. Fig. 5.17 shows a typical intensity pro le along the vertical axis z.

In Fig. 5.18, we report the spatial evolution of the uorescence intensity of a single node located at x = 1 mm and z = 7.9 mm for di erent times.

The oil front reaches the height z = 7.9 mm at t = 270 s, thus the signal prior to t = 270 s corresponds to noise uctuation. At t = 274 s, a peak in the uorescence intensity is observed. The intensity of this peak grows until reaching a steady shape observed 100 s after the front has gone through the node, as it can be seen in the inset of Figure 5.18 where a saturation of the peak's width and amplitude is observed after t = 400 s. This suggests that for the foam below the oil front, the distribution of oil quickly reaches a steady-state pro le. This steady-state pro le must match the equilibrium pro le given by a balance between capillarity and gravity [START_REF] Cantat | Foams: Structure and Dynamics[END_REF][START_REF] Weaire | The Physics of Foams[END_REF].

At late times and for z < h, r * does not depend anymore on t and the corresponding steadystate curvature radius r * (z) is deduced from a balance between gravity and the vertical capillary pressure gradient ∇p = -∇( γaw+γow r * ), as observed in §5.1.1.3. We consider a small perturbation of order 1 at long times in Eq. 5.27 with the following relationship at order 0: ρg -∇p = 0.

(5.53)

By projecting Eq 5.53 on the vertical direction e z and derivating r * (z) with respect to z, we nd (with ρ the density of the immiscible or miscible liquid):

dr * dz + ρg γ aw + γ ow r * 2 = 0. (5.54)
Introducing the boundary condition r * (0, t) = R b at the bottom of the foam yields:

r * (z, t) = r * (z) ∼ R b 1 + α * z , (5.55)
where α * = ρgR b γaw+γow ∼ ρgR b γaw has the dimension of an inversed length. This classic pro le illustrates the complex multi-scale geometry of foam under gravity: for z 1/α * , the curvature is only set by the bubble size r ∼ R b , while for z

1/α * , r ∼ R b /α * z.
To describe the imbibition dynamics within the aqueous foam, we consider the ow throughout the slender Plateau borders below the front. We study the local ow dynamics by looking at one single average vertical Plateau border, sketched in Fig. 5.19 and use Eq. 5.22 with a permeability determined by the Plateau border cross-section:

v = - K c δ a r * (z) 2 3η l dp dz , (5.56)
where r * (z) is given by Eq. 5.55 and v, η l , K c represent, respectively, the mean velocity of the imbibing liquid, the dynamic viscosity of the imbibing liquid and the permeability coe cient for in nite slender channels, K c = 0.02 [START_REF] Nguyen | Liquid drainage in single Plateau borders of foam[END_REF][START_REF] Piroird | Capillary ow of oil in a single foam microchannel[END_REF], as already stated. We also use the cross-section of the Plateau border A = δ a r * 2 . [START_REF] Cohen-Addad | Flow in foams and owing foams[END_REF][START_REF] Koehler | Foam drainage on the microscale -I. Modeling ow through single Plateau borders[END_REF][START_REF] Nguyen | Liquid drainage in single Plateau borders of foam[END_REF].

Then, we write mass conservation, which yields v(z) = v(h)r * 2 (h)/r * 2 (z) = ḣr * 2 (h)/r * 2 (z). At short times, the gravity e ects are negligible (for h (γ aw + γ ow )/ρg) but at late times, because of the equilibrium between capillarity and gravity, we consider that gravity only blocks the swelling of the Plateau border, as shown in Fig. 5.18, but does not a ect the vertical ow of liquid upwards. Thus, we can use Darcy's equation for the whole time window.

Integrating Eq. 5.56 for 0 < z < h with the boundary conditions p(z = 0) = p o ≈ p atm at the bottom of the foam and p(h) = p o -γ aw /r P B at the imbibition front, and using Eq. 5.55 gives:

ḣ (1 + α * h) 3 - 1 (1 + α * h) 2 = 5α * δ a γ aw K c R 2 b 3η l r P B . (5.57)
Eq. 5.57 is then integrated with the condition h(t = 0) = 0:

(1 + α * h) 4 + 4 1 + α * h -5 = 2δ a 15 √ 3 α * 2 R b γ aw √ φ i η l t. (5.58)
At late times or for α * h >> 1, Eq. 5.58 predicts an evolution of the rising front in t 1/4 , while for α * h << 1, a second-order limited expansion of Eq. 5.58 gives h 2 ∼ R b γ aw t/ √ φ i η l , thus suggesting an early time dynamics of imbibition in t 1/2 . These regimes intersect for h ∼ 1/α * , the typical height for which the curvature radius of the Plateau borders changes from R b to R b /α * h. Moreover, Eq. 5.58 suggests a collapse of the data using the dimensionless variables α * h and t * = α * 2 R b γaw √ φ i η l t, which is in reasonable agreement with our experimental results obtained for three di erent liquids, either organic or aqueous, and foams with various bubble diameters and initial liquid fractions, as can be seen in Fig. 5.20.

CAPILLARY RISE INTO AQUEOUS FOAMS FOR MISCIBLE AND IMMISCIBLE LIQUIDS 153

Furthermore, the data is reasonably tted by

(1 + α * h) 4 + 4 1+α * h -5 = b 2δa 15 √
3 t * , with a coe cient b deduced from the experiments of b ∼ 0.3. This coe cient may be related to the value of the pressure in the contact area between the foam and the oil. Since the foam squeezes the oil droplet initially to create an oil slick, the lower layer of bubbles is deformed. Thus, the pressure at the bottom that is critical in our model might not be the exact atmospheric pressure but a pressure that is lower, due to the concave curvature induced by this deformation. This will imply a shift in the pressure value (3 times as low as the atmospheric pressure according to the value of b), and thus a vertical shift downwards compared to the theoretical predictions.

In this context, our scaling law, where the topology of the foam is kept at its original value in the unwetted area, better captures the in uence of R b , φ i and the long-term dynamics of the process than the numerical and analytical solution of the foam drainage equation. In [START_REF] Reyssat | Imbibition in geometries with axial variations[END_REF], Reyssat et al. have found a similar dynamics for the imbibition of a conical channel. The dimensional axial variations induce a modi cation of the usual Lucas-Washburn di usive dynamics with a t 1/4 dynamics at long times. This con guration is similar to the case of a deformable and swelling Plateau border which self-adjusts during the whole imbibition process.

In this section, we quantify the 1D-imbibition dynamics of a dry aqueous foam by a miscible or immiscible liquid slick. To describe our experimental data, we use the framework of porous media considering steady-state solution and the initial liquid fraction of the foam. We exhibit a new latetime dynamics in t 1/4 in good agreement with the experiments. However, the dependence on the initial liquid fraction remains to be derived from the foam drainage theory. Moreover, we do not really know how oil behaves at the bottom of the foam (initial squeezing of the oil droplet). In the scaling law, we also assume that we can add up the contributions of every Plateau border connected to the oil droplet independantly, but collective phenomena can arise.

We can also notice that the scaling we nd in this section is di erent from the resolution of the 3D problem. Here, we have a strong dependence on the initial liquid fraction, which is not the case for the pore imbibition. The di erence probably comes from the imposed conditions. For the pore con guration, we impose the pressure, while the volume of oil is not completely in nite for the 1D con guration. Despite this relatively slow dynamics at late times, the remarkable ability of dry aqueous foams to displace and drive upward an immiscible liquid highlighted in this section, should be of considerable interest in various industries such as soil remediation, detergency, shampoo industry, enhanced oil recovery and dismantling of nuclear power plants.

Oil recovery by a moving foam

In this section, we study the oil imbibition into the foam from a point-source with the foam moving over the extracting point. In §5.3 and §5.2, we have studied the 1D imbibition of an oil slick and the 3D static imbibition from a point-source. We go one step further by getting close to a real channel connected to a pore from which we need to remove the foam once it is oil-imbibed.

Experimental con guration

We use the experimental setup depicted in Fig. 5.21. First, we generate the foam with the system presented in §5.3 and §5.2. However, the removable part is here a 14.5 cm high and 2.5 cm width polycarbonate cylinder. This cylinder is blocked by a piston at the top. We ll the whole column and then wait for di erent times corresponding to di erent drainage times. Then, we remove the column and ipped the column upside down several times to ensure that the liquid fraction is homogeneous in the removable column. This foam sample is weighed up to determine the liquid fraction by knowing the volume and the mass. The column is reversed at 90 o with its piston onto a syringe pump (Harvard Apparatus). Layers of bubbles are intentionally eliminated at the bottom end of the column (up to 4 cm in the column). Indeed, a 1 mm wide hole is drilled beforehand into the cylinder (at 2 cm from the end of the column) in order to connect the reservoir of imbibing liquid to the moving foam. For the liquid connections, we use the same system as in §5.2: the motion controller allows to set the atmospheric pressure at the liquid input.

Two cameras are used. The rst one is located at the side of the column and enables to take sideview snapshots of the imbibed foam. The second one is pointed to the bottom end of the U-tube: it allows to record the evolution of the liquid level in the tube, which gives access to the absorbed volume.

The foam composition is the same as the one used so far. The tested liquids (olive oil, foaming solution and glycerol, para n oil) are also similar. The bubble radii R b are 1 mm and 2.5 mm and the initial liquid fractions φ i vary between 5 x 10 -4 and 3 x 10 -3 . At time t = 0, the foam is pushed at an imposed ow rate Q. The ow rates vary from 0.5 mL/h to 1000 mL/h (from 10 µm/min to 3 cm/min). The foam advances into the column as a plug-like ow before touching the injection point. The cameras are started right before the touching time and record the imbibition dynamics at 24 frames per second. While imbibition occurs, the level of liquid in the U-tube is kept at its initial level by moving the tube upwards by incremental distance of 0.2 mm. The distance d by which the tube is moved allows to determine the volume V o of liquid absorbed by the foam with V o = πr 2 t d and r t = 1 mm is the tube radius.

Modelization of the imbibed moving foam

From our experiments, we extract the imbibition pro les and try to model and predict the quantity of extracted oil.

Imbibition pro les

We observe the development of an imbibition front, as described in Fig. 5.22 and Fig. 5.23 for both imbibing oil and miscible liquids. Yet, here the front is axisymmetric towards the axis of the channel. c (r c is the column radius) for an experimental time frame between the touching time t = 0 and the nal running time t r = 4 min and 30 sec. What is surprising is that the volume decreases when the ow rate increases. Indeed, we could have imagined that bringing "fresh" foam would have brought more fresh connections between the extracting point and the foam, thus enhancing oil imbibition.

Discretized model

We try to explain the observed trend for the imbibition of immiscible liquids. We assume that the whole imbibition process is the sum of short static imbibitions every time a Plateau border encounters the extracting point.

We introduce the volume v o (t) of oil absorbed by a group of Plateau borders connected to the extracting point for a duration t. Indeed, the foam moves, so we have several Plateau borders that go through the pore and remain connected to the pore for a duration t d = d t /U f where d t = 2r t = 1 mm is the diameter of the capillary tube. Then, we have two possible cases. If the experimental running time t r < t d , it means that one single Plateau border stays in touch with the extracting point within the injection time t r . Thus, we are in the static case and the imbibed volume after a time t r is V o (t r ) = v o (t r ). Now if t r > t d , several Plateau borders can successively be connected to the extracting point while the foam is moving. Hence, we can think about the simplest possible model where the total absorbed volume V o is the sum of di erent volumes v o absorbed every time a Plateau border is connected, which yields:

V o (t) = F tU f d t v o d t U f + v o t -F tU f d t d t U f , (5.59)
where F is the oor function: given a a whole number and b a real number, if a ≤ b < a + 1, then F (b) = a. In the rst term of Eq. 5.59, F (tU f /d t ) is the number of times a Plateau border is connected and v o (d t /U f ) is the absorbed volume during the contact time t d = d t /U f . The second term is the residual volume for the last connection.

Then, we can express the mean ow rate Q o (t r ) over the running time t r by:

Q o (t r ) = V o (t r ) t r = 1 t r F t r U f d t v o d t U f + v o t r -F t r U f d t d t U f . (5.60)
We need to determine an expression for v o (t) at short times, which corresponds to the time scale of one single static imbibition, close to the point-source. In Fig. 5.12, we showed that the experimental data are well-tted at short times by a t 1/2 law even though we did not provide any modelling for this observed feature. We recall that the good agreement between the self-similar no-gravity solution from Eq. 5.43 and our data is a coincidence. The imbibition volume v o (t) is determined by v o (t) = 4π r f 0 φ o r 2 dr with the oil fraction φ o . This integration gives the following dependence for v o (t):

v o (t) ∼ R 3 b γ ef f t η o R b 3/2 , (5.61)
where γ ef f , R b and η o are the only dimensionalizing parameters of our system.

In dimensionless coordinates Q * for the dimensionless ow rate of oil and U * for the dimensionless foam velocity, Eq. 5.60 gives:

                 U * = U f t r d , Q * = Q(t r ) R 3 b ηR b γ ef f 3/2 1 t 1/2 r , Q * ∼ F (U * ) U * 3/2 + 1 - F (U * ) U * 3/2 .
(5.62)

The log-log plot of Fig. 5.25 shows Q * as a function of U * for di erent experimental data with olive oil as the imbibing liquid. We can notice two di erent regimes. At low foam velocities U * , Q * is almost constant. However, as the foam goes faster, we transition to another regime for U * > 1 where Q * decreases when U * increases. The faster we push the foam, the less oil we recover, as underlined before. We retrieve this transition for U * = 1 in our model. The boundary U * = 1 means that one single Plateau border has managed to be in contact with the extracting point, like the static case. In Fig. 5.25, the t is given for Eq. 5.62 and no multiplying coe cient. The collapse is not perfect. However, our model explains our experimental observations in scaling order. We therefore try to use the numerical solution developed for the static imbibition in §5.2. This solution deviates at short times and exhibits a power law behaviour for v o (t) from t 1/2 to t 1/3 (according to the Bond number) from the simulations, as seen in Fig 5 .26. This actually suggests an increase of the imbibed volume with the foam velocity, which is not what is experimentally observed. The short-time dynamics around the point-source has to be taken into account. Obviously, adding up the contributions of successive imbibitions from the same pore is also a rough model that needs to be re ned. But it is clear that injecting a foam at high ow rates is not the ideal situation to extract immiscible liquids from a connected pore. It makes sense because a Plateau border needs to be kept into contact with the extracting point as long as possible for the imbibition process to occur.

For miscible liquids, imbibition is quite fast, so the imbibing liquid invades the rear of the foam quite easily and the Plateau borders are already imbibed when they cross the extracting point, a discretized model is thus not really appropriate. Modelling this problem more accurately is still an open question.

Conclusions

In this chapter, we show that both miscible and immiscible liquids can be absorbed by an aqueous foam due to the capillary underpressure with the outer environment. We gradually travel from the imbibition of oil from a millimetric pore by a static foam to the 1D imbibition of an oil slick and the recovery of oil by a moving foam.

First, the radial imbibition from a point source (at imposed pressure) for di erent types of liquid into a dry aqueous foam has been studied theoretically, numerically and experimentally. Theory, numerics and experiments are in good agreement. The results demonstrate that gravitational e ects are comparable to the capillary pressure. Moreover, we have identi ed two criteria that de ne the imbibition e ciency. The rst one is the ratio between the oil-water and the air-water interfacial tension that predicts the imbibition strength. The resulting e ective interfacial tension γ ef f allows to describe the local in uence of the oil-water interface on the imbibition process. The second one is the Bond number Bo. The lower Bo, the less important gravity and the more liquid is imbibed. Decreasing the bubble radius is the main option for reducing Bo (apart from microgravity).

To go further, we consider a con guration where we have multiple point-sources to form an oil slick and a uniform propagating front of imbibition. To describe our experimental data in this case of a uniaxial vertical imbibition, we use the framework of liquid ow into porous media by considering a steady-state solution and the initial liquid fraction of foam. We exhibit a t 1/4 late-time dynamics in good agreement with the experiments and some observations made for imbibition into deformable solid porous media [START_REF] Reyssat | Imbibition in geometries with axial variations[END_REF][START_REF] Siddique | Capillary rise of a liquid into a deformable porous material[END_REF]. Another study for imbibition into gelli ed foams [START_REF] Deleurence | Time scales for drainage and imbibition in gelli ed foams: application to decontamination processes[END_REF] has reported a di usive behaviour in t 1/2 by considering immobile air-water interfaces. All in all, our scaling remains to be derived from the foam drainage equation (see Eq. 5.29).

Finally, we have tried to build a system where a macroscopic channel is connected to a millimetric pore from which oil needs to be extracted. In the continuity of the static imbibition from a point-source, we have moved the foam on top of the extracting point-source and have looked at the volume of recovered oil. Our main result is that a low injection ow rate of foam is more favourable than high foam velocities because the foam needs to spend as much time as possible in contact with the pore to be e cient. Obviously, in terms of applications, a compromise has to be found between a better volume of absorption per foam unit and a fast injection process.

In conclusion, we have highlighted the remarkable ability of dry aqueous foams to displace and drive upwards immiscible and miscible liquids, as well as di erent dynamics according to the con gurations of the oil reservoir. Actually, oil can also spontaneously penetrate wet foams due to the dynamic decrease of the oil-water interfacial tension [START_REF] Sonoda | Liquid oil that ows in spaces of aqueous foam without defoaming[END_REF]. Macroscopic imbibition into aqueous foams is not the only mechanism enabling the extraction of liquids from a con ned medium. At the microscopic scale, foams can also drag outer liquids by interfacial interactions, as we see in Chapter 6.

Chapter 6

Foam -oil interactions in micropatterned structures

In this chapter, oil is con ned and trapped in the roughness of a surface. We show how oil can be e ciently dragged out of the roughness by aqueous solutions containing surfactants and foams. To analyze our results, we use the framework developed by Wexler et al. [START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF], who studied how oil is dragged out of the roughness of a groove surface by using pure water. We compare the e ciency of the di erent liquids (surfactant solutions and foams versus pure water) and demonstrate that the volume of extracted oil is highly improved by using foams. We also refer to recovery issues from porous media trapping oil, where aqueous solutions and gas are injected to increase the e ciency of the recovery process. To some extent, we wonder how oil can be extracted from the roughness of a con ned porous medium. This part was completed in collaboration with Dr. Jason S. Wexler in the Complex Fluids Group of Prof. Howard A. Stone at Princeton University. In this chapter, oil is con ned in the roughness of a con ned medium. To realize this geometry experimentally, we use a micro uidic device constituted of a wide channel with a textured wall in which oil is con ned due to a favourable wetting. We rst recall some classical features concerning the wetting of rough surfaces and we show that these properties are enhanced when a lubricating lm impregnates the cavities of the texture [START_REF] Quéré | Wetting and roughness[END_REF]. However, these surfaces can also lose their properties when the air-liquid interfaces are subject to external stresses.

Wetting of rough surfaces

Let us rst discuss how a liquid interacts with a rough surface. It has indeed been shown that geometrical defects considerably modify the wetting properties of a surface. In the following, we discuss di erent results from the literature to relate the wetting properties to the roughness of the surface.

Wetting behaviour on rough surfaces: Wenzel model and hydrophilicity

A rough surface is a surface whose thickness varies locally due to the presence of defects. Shibuichi et al. [START_REF] Frye | Super water-repellent surfaces resulting from fractal structures[END_REF] have shown that the apparent contact angle θ * on a rough surface di ers from the value of the Young static contact angle θ Y on a smooth surface of the same chemical nature. Both con gurations are sketched in Fig. 6.1. The same gure presents the variations of cos θ * with cos θ Y . For hydrophilic smooth surfaces (θ Y < 90 o ), the hydrophilicity is enhanced by the roughness. The apparent contact angle rapidly decreases when θ Y decreases and gets close to a value of 0 o . For hydrophobic smooth surfaces (θ Y > 90 o ), the hydrophobicity is improved by the roughness. The apparent contact angle strongly increases when θ Y increases.

How can we relate the apparent contact angle on a rough surface to the Young contact angle? In 1936, Wenzel was the rst one to relate both angles [START_REF] Wenzel | Resistance of solid surfaces to wetting by water[END_REF]. He calculates θ * by considering the horizontal displacement dx of the air-liquid contact line, as depicted in Fig. 6.1b. The total variation of interfacial energy dE is written as:

dE = r S (γ sl -γ sa ) dx + γ al dx cos θ * , (6.1)
where r S is the surface roughness, γ sl , γ sa and γ al the solid-liquid, the solid-air and the air-liquid interfacial tensions respectively. At equilibrium, dE = 0 and Eq. 6.1 yields the Wenzel state with: The Wenzel model is indeed only valid for the linear part in the plot of Fig. 6.1 at small cos θ Y . However, for cos θ Y and cos θ * close to 1 (complete wetting), a liquid lm can escape from the droplet and invade the pores of the textured surface. This lm creates a lubricating layer which gives a "smooth" character to the textured surface. As a consequence, the liquid droplet sits on a composite solid-liquid surface drawn in Fig. 6.2. Let us calculate the apparent contact angle θ * in this con guration. One considers a textured surface whose size pore and spacing between pores are uniform. We de ne φ S as the solid fraction of projected area, which corresponds to the ratio between the areas at the top of every plot and the whole projected area A P of the surface (see Fig. 6.3). When the air-liquid contact line moves by a distance dx over the solid-liquid area pattern, the virtual movement of the droplet covers the air-liquid interface of area per unit length (1 -φ S )dx by the liquid phase and replaces the air-solid interface by a solid-liquid interface of area φ S dx. An air-liquid interface of area cos θ * dx is also created. Thus, the variation of interfacial energy dE is written as:

dE = dx (φ S (γ sl -γ sa ) -(1 -φ S )γ al + γ al cos θ * ) . (6.3)
By injecting the Young-Dupré equation cos θ Y = (γ sa -γ sl )/γ al and setting dE = 0 at equilibrium in Eq. 6.3, one nds [START_REF] Bico | Rough wetting[END_REF]:

cos θ * = 1 -φ S + φ S cos θ Y . (6.4)
It is impossible to induce a complete wetting transition (cos θ * = 1) since θ Y = 0 o in a situation of partial wetting, which is supported by the experimental results.

Film impregnation

For Eq. 6.4 to be valid, a liquid lm has to impregnate the roughness of the textured surface. For a sponge to impregnate liquid, the solid-liquid interfacial energy γ sl has to be lower than the solid-air interfacial energy γ sa , which means that θ Y < 90 o by the Young-Dupré relationship. The case of textured surfaces is more complicated since air-liquid interfaces are created and increase the total interfacial energy of the system. Bico et al. [START_REF] Bico | Rough wetting[END_REF] have calculated the condition for which a lm impregnates a regular textured surface of uniform pore size, as sketched in Fig. 6.3.

One imposes a small horizontal displacement dx of the lm over a repetitive solid-liquid composite interface. The wetted area per unit length is (r S -φ S )dx because the dry area at the top of the plots is φ S dx. A free air-liquid area (1 -φ S )dx is also created. The variation dE is given by: dE = (r S -φ S )(γ sl -γ sa )dx + (1 -φ S )γ al dx.

(6.5)

A lm impregnates the textured surface if dE < 0 by loss of interfacial energy. By considering the Young-Dupré equation once more and Eq. 6.5, the condition dE < 0 means [START_REF] Bico | Rough wetting[END_REF]:

cos θ Y > 1 -φ S r S -φ S = cos θ c , (6.6)
where θ c represents the transition angle to the Wenzel state. Indeed, we have the following wetting behaviour on a hydrophilic textured surface: -for 0 < θ Y < θ c , which means cos θ Y > cos θ c , a liquid lm impregnates the surface and the droplet sits on a solid-liquid composite surface. This corresponds to the upper right part of Fig. 6.1c and the curve is described by Eq. 6.4; -for θ Y > θ c , which means cos θ Y < cos θ c , the wetting behaviour accurately follows the Wenzel model. The droplet follows the defects of the surface but the solid surface remains dry at the front of the droplet.

Hydrophobic textured surface

For a hydrophobic smooth surface, θ Y > 90 o with γ sa < γ sl . The interfacial energy of the dry surface is lower than the interfacial energy of the wet surface. Providing that the Young-Dupré relationship is veri ed for every air-liquid-solid contact line, a textured surface can also be hydrophobic. In this situation, a liquid droplet sits on a solid-air composite surface by leaving air cushion in-between the pores. This con guration is sketched in Fig. 6.4. One can determine the apparent contact angle θ * by considering once more the variation of interfacial energy dE when the air-liquid contact line advances by a small displacement dx. An air-solid interface of area φ S dx is replaced by a solid-liquid interface of the same area. An airliquid interface of area (1-φ S )dx is also created, as well as an air-liquid interface of area cos θ * dx. The sum of these contributions gives:

dE = (γ sl -γ sa )φ S dx + γ al (1 -φ S )dx + γ al cos θ * dx. (6.7)
Setting dE = 0 at equilibrium and using the Young-Dupré relationship yields:

cos θ * = -1 + φ S (cos θ Y + 1). (6.8)
If one considers that the top of the plots is not at, Eq. 6.8 becomes cos θ * = -1+φ S (r φ cos θ Y +1), where r φ is the texture roughness given by the ratio between the wetted area and the projected area at the top of a plot. This comprehensive equation is known as the Cassie-Baxter equation [START_REF] Marmur | Wetting on hydrophobic rough surfaces: To be heterogeneous or not to be?[END_REF].

For any value of θ Y > 90 o , θ * > 90 o and the surface is hydrophobic. Eq. 6.8 shows that cos θ * jumps to a value below φ S -1 which can be close to -1, enhancing hydrophobicity.

Generation of liquid-infused surfaces

In §6.1.1.2, we have seen that a liquid is able to impregnate the roughness of the surface. By using an oil-based liquid of low air-liquid interfacial tension, we can indeed impregnate the texture to create a liquid-infused surface.

These liquid-infused surfaces, also known as SLIPS or LIPS, exhibit more interesting properties than common superhydrophobic surfaces, such as the non-pinning and the roll-o of liquid droplets (see Appendix D). Eq. 6.6 gives the condition for which lm impregnation occurs. A lubricating lm appears if the

172CHAPTER 6: FOAM -OIL INTERACTIONS IN MICROPATTERNED STRUCTURES

Young contact angle θ Y < θ c where θ c is determined by cos θ c = (1 -φ S )/(r S -φ S ). Playing on the roughness r S and the solid fraction φ S allows to nd the right oil-texture combination. We always have 1/r S > (1 -φ S )/(r S -φ S ), thus cos θ Y > 1/r S is a su cient condition to have lm impregnation [START_REF] Lafuma | Slippery pre-su used surfaces[END_REF]. So the higher the roughness, the less constraining the condition for lm impregnation, the easier the generation of liquid-infused surfaces for common types of oil (silicon oils, alkanes and siloxanes).

In our experiments, the equilibrium contact angles at the air-solid interface are pretty low below 20 o and the roughness is 2 (10 µm x 10 µm x 10 µm posts). The previous condition is always ful lled and we can generate the desired liquid-infused surface.

Failure of textured surfaces

Owing to their repellent properties, textured surfaces are used in many applications from self-cleaning to drag reduction and biofouling (see Appendix D). However, because of failure mechanisms, they can lose their properties. In the case of superhydrophobic surfaces, we want to keep air pockets in the Cassie-Baxter regime, while for liquid-infused surfaces, we would like to keep the oil lm impregnated in the texture. If one of these elements vanishes, textured surfaces lose their interest. The failure of both air and oil layers is mainly due to two types of stresses: normal and tangential.

Transition from the Wenzel to Cassie-Baxter regime

For Eq. 6.8 to be valid, air must be trapped in the pores of the textured surface. To determine the condition for which the trapping of air occurs, one writes the variation of interfacial energy dE for a unit area A P between the con guration with air and the con guration without air under the liquid droplet, as described in Fig. 6.5. From one con guration to the other, an air-liquid interface of area (1 -φ S )A P disappears and solid-liquid interfaces of area (r S -φ S )A P replace solid-air interfaces of the same area. Thus, dE is given by: dE = (γ sl -γ sa )(r S -φ S )A P -γ al (1 -φ S )A P .

(6.9)

The regime of air cushion is favourable if dE > 0, which means that:

cos θ Y < cos θ c = φ S -1 r S -φ S . (6.10)
Therefore, the wetting behaviour on hydrophobic textured surface is described by two regimes:

-when θ c < θ Y < 180 o , the regime of air cushion is stable and a liquid droplet sits on a solid-air composite surface. This corresponds to the lower left part of Fig. 6.1; -when 90 o < θ Y < θ c , the regime of air cushion is metastable and can transition to the Wenzel state although the Young angle is above 90 o .

Indeed, in this latter case, the air-liquid interface can fail and fully wets the surface, as Eq. 6.9 suggests it. Calculations of interfacial energy actually provide a rough argument. Failure is rather induced by distortions of the air-liquid interface that can lead to its sagging. During a sagging event, the air-liquid interface can ultimately touch the bottom of the pores and transitions to a fully-wetted state. Sagging can be induced by an applied external pressure, the impact of a droplet or the mere Laplace pressure in the liquid droplet [START_REF] Tuteja | Robust omniphobic surfaces[END_REF].

Impalement of superhydrophobic surfaces

The failure mechanisms for the sagging of superhydrophobic surfaces have been studied by Bartolo et al. [START_REF] Bartolo | Bouncing or sticky droplets: Impalement transitions on superhydrophobic micropatterned surfaces[END_REF]. They suggest that a drop pressure (due to the ow of liquid or a drop impact [START_REF] Reyssat | Bouncing transitions on microtextured materials[END_REF]) can push the air-liquid interface downwards. The drop pressure P d has to overcome the capillary forces at the top of an array of posts for instance. The pressure force F P is given by

F P = P d A P (1 -φ S )
where A P is still the projected area of the post sample and φ S the solid fraction. The capillary force is F C = N P 2πγ ow r p | cos Θ| where N P = A P φ S /πr 2 p for cylindrical posts of radius r p is the number of posts and Θ the angle made by the air-liquid interface with the vertical axis in Fig. 6.6. The pressure P d becomes:

P d = 2φ S 1 -φ S | cos Θ| γ ow r p . (6.11)
Then, two mechanisms are possible and presented in Fig. 6.6. The rst one is the "touch down" scenario. The curvature of the air-liquid interface increases with the pressure. The distance between the lowest point of the interface and the bottom of the cavity decreases up to zero. At this point, air pockets cannot be trapped any more and the superhydrophobic surface loses its repellent property. In Eq. 6.11, one can determine the critical contact angle Θ (which depends on the height h) for which the touchdown occurs, and thus deduce the pressure threshold for touchdown P t imp .

Figure 6.6: a. Touch down scenario [START_REF] Bartolo | Bouncing or sticky droplets: Impalement transitions on superhydrophobic micropatterned surfaces[END_REF]. b. Sliding scenario [START_REF] Bartolo | Bouncing or sticky droplets: Impalement transitions on superhydrophobic micropatterned surfaces[END_REF].

The second mechanism is the "sliding" scenario for higher posts. Due to the increasing pressure, the interface curves more and more downwards and the contact angle Θ increases until reaching the advancing contact angle Θ a . When Θ = Θ a , the air-liquid interface spontaneously slides downwards to the bottom of the cavity. Once more, by setting Θ a as the threshold value of contact angle for impalement, one can determine a pressure threshold for sliding P s imp . As a consequence, a criterion for the impalement of superhydrophobic surfaces is that the external pressure P d > min(P t imp , P s imp ).

Shear-driven drainage of liquid-infused surfaces: scaling arguments

The failure of superhydrophobic surfaces originates from potential normal stresses and contact angle hysteresis applied to the air-water interface in-between the roughness of the surface. As a consequence, air pockets are not sustainable and the surface gets fully wet. For liquid-infused surfaces, applying a normal stress to the lubricating layer deforms the air-oil interface downwards but would certainly not lead to the complete disappearance of the oil lm.

However, liquid-infused surfaces are sensitive to another type of failure mechanisms that involves the tangential stresses exerted by the ow of an outer uid in which they are immersed. This is what Wexler et al. [START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF] call the shear-driven drainage of liquid-infused surfaces. All the calculations stated in the following can be found in [START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF] and we summarize here the main ideas.

For experiments with a groove geometry (see Fig. 6.7), Wexler et al. explained the drainage dynamics of the oil lm for pure aqueous solutions as the outer phase [START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF]. The ratio between the channel width W and height H is high, 40:1 or 20:1, and the channel is much deeper than the pattern (H h). With the low viscosity ratio λ = η w /η o 1 (η w and η o are respectively the water and the oil density) and the no-slip boundary condition, we can consider a parabolic velocity pro le in the outer ow. This pro le corresponds to a planar Poiseuille-like ow [START_REF] Shah | Laminar Flow Forced Convection in Ducts[END_REF] (see Fig. 6.7a) for which the shear stress at the oil-water boundary is τ xy ≈ 6η w Q/(W H 2 ), where Q is the injection ow rate. As sketched in Fig. 6.7, the external ow shears the oil-water interface and drags oil downstream out of the pattern. At short times, the tangential stresses in the water and the oil phase are equal, thus τ xy ∼ -ηo h dL dt , where h is the height of the posts and L the retention length.

Locally, the applied shear deforms the oil-water interface, as sketched in Fig. 6.7 and Fig. 6.16. When the dynamic contact angle at this pinned interface reaches the receding contact angle θ, the interface slides to the bottom of the posts [START_REF] Bartolo | Bouncing or sticky droplets: Impalement transitions on superhydrophobic micropatterned surfaces[END_REF]. Figure 6.7: a. Shear-driven drainage of oil in a groove geometry [START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF]. b. Local con guration of the oil-water interface. The parameters L o , L(t), h, w, θ, r min nd u are, respectively, the length of the groove, the retention length, the height of the groove, the spacing in a groove, the receding contact angle, the minimal radius of curvature and the ow velocity.

The deformation of the liquid-liquid interface induces a di erence of curvature between the upstream and the downstream ends for longer times. The capillary pressures are ∆p C = p w -p o = γ ow /r min upstream (p w and p o are, respectively, the pressures in the outer uid and the oil phase) and ∆p C = 0 downstream. The pressure gradient in the aqueous phase can be written as dp w /dx = -12Qη w /W H 3 . Thus, the pressure di erence ∆p w between the inlet and the out- let of the channel of length L channel is ∆p w ≈ -12Qη w L channel /W H 3 ≈ 4 x 10 2 Pa for Q = 2 mL/min. This value is small compared to the capillary pressure and the pressure di erence in the oil phase is directly ∆p o = γ ow /r min . Thus, the pressure p o in the trapped uid is lower upstream than downstream and the pressure gradient drives an oil counter ow within the liquid trapped between the posts. Oil is being extracted so long as the shear-driven ux is higher than the capillary pressure-driven ux. The maximum retention length L ∞ is reached when the shear-driven ux is exactly balanced by the pressure-driven ux. In terms of stresses, τ xy /h ∼ (γ ow /r min )/L ∞ and L ∞ ∼ hγ ow /(r min τ xy ) with r min the minimum radius of curvature of the oil-water interface.

Shear-driven drainage of liquid-infused surfaces: analytical resolution

The comprehensive resolution of the shear-driven drainage, and particularly the time dependence, is done by solving the Stokes' equation. Indeed, the Reynolds number Re is low with

Re = ρ o Qh/(W Hη o ) ≈ 0.01
1 and the Bond number Bo = w 2 g(ρ o -ρ w )/γ ow = 10 -6 (ρ o is the oil density and ρ w the water density) is also low, so gravity is negligible. The ow is unidimensional along the x direction, so the streamwise velocity u is solution of:

η o ∆u = - dp o dx , u(- h 2 , z) = 0 , u(y, - w 2 ) = 0 , u(y, - w 2 ) = 0 , η o ∂u ∂y y=h/2 = τ xy , (6.12)
where p o is the pressure in the oil phase. We apply the no-slip condition at the side and bottom walls and the prescribed shear stress τ xy at the oil-water interface. The velocity u is a superposition of two components, a shear-driven contribution u s and a pressure-driven contribution u p . Thus, we can solve the following system of equations for each component by separation of variables:

η o ∆u s = 0 , η o ∂u s ∂y y=h/2 = τ xy , η o ∆u p = - dp o dx , η o ∂u p ∂y y=h/2 = 0. (6.13)
We nd the following solutions for the shear-driven oil ux q s and the pressure-driven oil ux

q p : q s = c s τ xy wh 2 η o with c s = 1 2 - 4h w ∞ n=0 (-1) n λ 4 n tanh( λ n w 2h ) , q p = -c p wh 3 η o dp o dx with c p = 1 3 - 4h w ∞ n=0 (-1) n λ 5 n tanh( λ n w 2h ) , (6.14)
where λ n = (n + 1/2)π are the eigenvalues deduced from the development in Fourier series. We also have dp o /dx ≈ ∆p/L = γ ow /r min L, thus q p = -c p wh 3 γ ow /r min L.

By volume conservation, the sum of both uxes q s and q p must be equal to the drained ux of oil q d . The drained ux q d is equal to dV /dt with V the volume of oil in the grooves and the time t. This volume is determined by integrating the cross-sectional area along the x direction and considering a linear variation of the curvature radius from -1/r min upstream to 0 downstream. We nd that:

q d = -c d wh dL dt , with c d = 1 - r min h 1 - 1 4 - w 2 16r 2 min + r 2 min wh csc -1 2r min w . (6.15)
Equating Eq. 6.15 with the sum of q s and q p from Eq. 6.14 leads to the following dependence on the time t for the retention length L(t):

q d = q s + q p , c d wh dL dt -c p wh 3 γ ow η o r min 1 L +c s τ xy wh 2 η o = 0. (6.16)
When dL/dt = 0, q s = q p and we nd the maximum retention length L ∞ = cphγow csr min τxy , which we have already demonstrated by scaling arguments. The retention length L is solution of:

L(t) = L o - L ∞ t c t + L ∞ t c ln Lo L∞ -1 L L∞ -1 , (6.17)
with the characteristic time of drainage t c = c d c p η o γ ow /(c 2 s r min τ 2 xy ). When t → 0, L → L o , L is linear with time, as predicted by scaling law. Fig. 6.8 shows the dimensionless collapse between Eq. 6.17 and experimental measurements.

By looking at L ∞ , the retention of oil inside the textured surface can be controlled by playing on the oil-water interfacial tension, the contact angle (since cos θ = w/2r min when the interface does not touch the bottom of the cavities) and the aspect ratio w/h. It is typically easier to manipulate [START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF]. Both gures are extracted from [START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF].

Surfactant-driven oil extraction from micropatterned channels

The results concerning the surfactant and the foam-driven (next section) oil extraction from micropatterned channels have been submitted to the Langmuir journal [START_REF] Mensire | Surfactant and aqueous foam-driven oil extraction from micropatterned surfaces[END_REF]. They were presented at the Droplets 2015 and Eufoam 2016 conferences. This project was carried out with the help of Dr. Jason S. Wexler and Augustin Guibaud.

The properties of liquid-infused surfaces, highlighted in §D.2.1, depend on the robustness of the lubricating layer, similarly to the existence of air trapping for superhydrophobic textured surfaces. As shown in §6.1.2.3, the lubricating layer can be subject to an external shear ow when the surface is immersed in an another uid environment (typically in drag reduction and biofouling). Subsequently, the impregnated lm disappears and the surface loses its interest. In this section, we study the shear-driven extraction of oil from liquid-infused surfaces when surfactants -usually used in many extraction processes to decrease the oil-water interfacial tensionare added to the aqueous solutions. We use the framework of Wexler et al. [START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF] described in §6.1.2.3 and §6.1.2.4. To some extent, we also wonder how it is possible to extract oil from the roughness of a surface, which has potential applications in oil recovery issues. We are also aware that the nature of the interactions is only interfacial between the texture and the main shear ow. However, in the general context of oil recovery, the similitude to oil extraction is to be underlined.

Materials

We choose an array of microfabricated and equally-spaced posts as the texture to study the drainage of liquid-infused surfaces when the external shear ow contains surfactants. Compared to a groove geometry, cavities are connected to each other in the post geometry, which allows to look at collective phenomena within the posts. The patterned geometry is imprinted on the upper side of a micro uidic channel, as shown in Fig. 6.9. The system mimics an oil-lled porous microfracture where the assembly of posts represents a set of oil-lled and connected pores, which is a feature of randomly rough systems with open pores. Figure 6.9: Geometry of the micro uidic channel with the liquid-infused micropattern imprinted on the upper side of the channel. a. Analogy between a real microfracture and a liquid-infused substrate. b. Top-view of the channel, with the oil uorescing green-yellow. c. Side-view of the channel consisting of one wall with an array of posts.

Characteristics of the textured surface and the tested uids

The materials and the microfabrication are the same as those used by Wexler et al. [START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF] and are sketched in Fig. 6.9b-c. The pattern consists of cubic posts (w = h = 10 µm on a side), which are equally spaced at a distance of 10 µm. The pattern is generated by depositing an epoxy adhesive on a PDMS (poly(dimethyl)siloxane) mold. We rst etch a silicon wafer to draw the model for the texture. The etching is done inside the wafer by removing silicon. On this wafer, we deposit some PDMS that covers the cavities created by the etching. After baking, we obtain a mold of post pattern on which we deposit the epoxy.

The pattern and the channel, lying on black borosilicate glass, are molded in an epoxy adhesive (Norland Optical Adhesive 81) by the "sticker technique" [START_REF] Bartolo | Micro uidic stickers[END_REF]. The details of this technique is 180CHAPTER 6: FOAM -OIL INTERACTIONS IN MICROPATTERNED STRUCTURES described in Appendix C. The channels are 7 mm wide, 178 or 89 µm high, and 45 mm long. The channel has two ports: one inlet port at the upstream end and one outlet port at the downstream end. The 36 mm long and 1 mm wide pattern is located near the centerline of the channel and aligned with the inlet and outlet ports at a distance of 5 mm from both ports. The pattern consists of cubic posts (10 µm on a side), which are equally spaced at a distance of 10 µm, and ends with a 1 mm by 1 mm open well for ow entry (Fig. 6.9b).

The characteristics of the tested uids are summarized in Table 6.1, with the dynamic viscosities measured in a rheometer (Anton-Paar) with a cone-plate geometry. The outer uid is a surfactant solution. We use two types of surfactants: either SDS (sodium dodecyl sulfate) with di erent concentrations from 0.1 g/L to 10 g/L (cmc ≈ 2 g/L), or a combination of co-surfactants, CAPB (cocoamidopropylbetaine), SLES (sodium laurylethyl sulfate), Mac (myristic acid) and 10 wt% glycerol, already presented in Chapter 2 and 5 [START_REF] Basheva | Role of betaine as foam booster in the presence of silicone oil drops[END_REF][START_REF] Golemanov | Surfactant mixtures for control of bubble surface mobility in foam studies[END_REF], with di erent surfactant concentrations from 0.005 wt% to 2 wt% (critical micellar concentration cmc ≈ 0.1 wt%). The lubricating oils, which initially ll the patterned substrate, are olive oil (commercial and Sigma-Aldrich) and two silicon oils (Gelest PDM -7040 and 7050) containing a small amount of uorescent dyes (from Tracerline) (1:100). In the following, all the interfacial tensions are measured by the pendant drop technique with a precision of ± 0.5 mN/m. Fig. 6.10 presents the variations of the equilibrium oil-water interfacial tension γ ow with the surfactant concentration c. As expected, γ ow decreases when c increases with the cmc ≈ 2 g/L in both SDS-olive oil and CAPB-SLES-Mac-olive oil cases. As explained below, we clear the channel with the surfactant solution to leave oil trapped in the texture of the surface at a very slow velocity (0.002 mL/min). Thus, we give time for the system to equilibrate in terms of surfactant dynamics, which allows us to use the values presented in Fig. 6.10. Table 6.2 summarizes the di erent values for the interfacial tensions and the entry E and S spreading coe cients de ned in §2.1.1.

Fluid injection

At the start of an experiment, the channel is completely lled with oil from the downstream port. Then, the entire channel is cleared by the ow of the outer aqueous phase at a low injection rate (0.002 mL/min), which leaves oil trapped between the posts, while the oil in excess is evacuated to a reservoir from the outlet port. This procedure prevents oil extraction from being disturbed by an excess of oil. Once the main channel is cleared of oil, the outer uid is injected at a constant ow rate between 0.05 mL/min and 8 mL/min, corresponding to average ow velocities.

Experimental observations and results

The experimental observations are made under UV or blue light (with a yellow lter from Ti en in front of the camera objective) to ensure a strong contrast between the black glass background and the uorescent oil trapped in the pores. Snapshots of the oil extraction process are taken every minute.

Extraction patterns

A typical time series of the oil extraction pattern is shown in Fig. 6.11. Depending on the ow rate Q and the oil-water interfacial tension γ ow , the process shows two di erent patterns. In one case, for su ciently low Q and high γ ow , the draining front advances uniformly, as observed previously by Wexler et al. with pure aqueous solutions [START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF]. We de ne a retention length L(t) (t is the time) and a steady-state retention length L ∞ (Fig. 6.9a). In a second case, for su ciently high Q and low γ ow , the draining front still propagates along the ow direction but is more irregular, leaving oil patches behind it. Since it is di cult to de ne a retention length in this second situation, we introduce the extraction e ciency α to quantify oil extraction from the porous substrate. Speci cally, we de ne α = V d /V 0 which is the ratio between the volume of extracted oil V d and the initial volume of oil V 0 between the posts. The parameter α varies from 0, when the pores are completely lled, to 1, when they are completely empty.

E ciency curves

To measure α(t), we record the uorescence intensity I in the pattern by using the sum of grey levels after image processing with ImageJ software. Neglecting photobleaching that occurs for a timescale longer than our experimental time, we assume that the uorescence level is approximately proportional to the volume of the remaining oil V 0 -V d , since the depth of the posts is small. Given that light conditions can change from one experiment to another, we only have access to α ≈ 1 -I/I 0 , where I 0 is the initial uorescence intensity between the posts for each experiment. In Fig. 6.12, we report α as a function of time t for the di erent liquids.

When varying one parameter (Q, γ ow , η o or the channel height H) during ow of a surfactant solution, α follows one typical evolution, as shown in Fig. 6.12. As time increases, α increases quickly before slowing down and eventually reaching a maximum value α ∞ at long times. For a given time and considering the di erent possible material properties, we nd that oil recovery is more e cient if interfacial tension is low and the aqueous phase ow rate is high. E ciency is also enhanced if the oil viscosity and channel height are low.

Next, we report α as a function of the injected volume Qt (Fig. 6.13). The data is organized by the capillary number Ca = η w Q/(W Hγ ow ), where W is the channel width. The six di erent data sets in Fig. 6.13 correspond to two capillary numbers Ca ≈ 0.005 and Ca ≈ 0.015. The experimental data, for a given capillary number, collapses on one single curve, suggesting that α depends on Qt and Ca.

E ect of surfactants on the extraction dynamics

We explain the experimental observations and results from §6.2.2 by developing and using the model introduced by Wexler et al. [START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF] in §6.1.2.3. We extend their work by considering aqueous liquids containing surfactants rather than pure aqueous solutions. This step builds upon earlier work by Jacobi et al. [START_REF] Jacobi | Over ow cascades in liquid-infused substrates[END_REF], who highlighted that traces of surfactants could induce unusual drainage behavior. The comprehensive resolution of the shear-driven drainage, and particularly the time dependence, refers to §6.1.2.3 and is used in the following.

SURFACTANT-DRIVEN OIL EXTRACTION FROM MICROPATTERNED CHANNELS 185 6.2.3.1 Variations of the contact angle

According to Wexler et al. [START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF], we can write the extraction e ciency α(t) = 1 -L(t)/L o ∼ τ xy ht/η o for short times, where L o is the length of the pattern (see Fig. 6.9) and L(t) the residual length of oil in the pattern. We assume that the oil layer percolates through the connected posts. Thus, it is always possible to de ne a pressure path within the porous system. Oil is being extracted because the shear-driven ux is higher than the capillary pressure-driven ux. The maximum e ciency α ∞ = 1 -L ∞ /L o is reached when the shear-driven ux is exactly balanced by the pressure-driven ux. Rewriting the complete steady-state solution of Wexler et al. [START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF], we nd that:

α ∞ = 1 - c p hγ ow c s r min τ xy L o , (6.18)
where c p and c s are the numerical factors linked to the geometry of the channel, de ned in Eq. 6.14.

The details of the comprehensive original time-dependent drainage theory explained in §6.1.2.4 allow to determine the e ciency α = 1 -L/L ∞ as a function of time t. Rewriting the timedependent theory with α(t) and α ∞ as the main variables, we nd:

α + (1 -α ∞ ) log α ∞ α ∞ -α = 6c s hη w W H 2 c d L 0 η o Qt , (6.19a
)

α ∞ = 1 - c p Hh cos θ 3c s L o wCa , (6.19b)
where c d is the numerical factor depending on the receding contact angle θ at the oil-water interface (see Eq. 6.15). We also introduce the capillary number Ca and cos θ = w/2r min . For cos θ < 0, θ > 90 o and the pressure-driven ux does not create a counter ow but enhances the ow downwards, which leads to a complete oil extraction. For our experiments, w = h. Thus, c d can be expressed as:

c d = 1 - 1 2 cos θ 1 - 1 2 √ 1 -cos 2 θ + 1 4 cos 2 θ csc -1 1 cos θ . (6.20)
Subsequently, we deduce θ from the confocal microscope measurements of the interface de ection at di erent points along the porous substrate, when the pinned oil-water contact line starts receding. A local close-up on the oil-water interface is shown in Fig. 6.14.

We relate the curvature radius r min to the de ection δ by the following geometrical relationship: Solving Eq. 6.21 yields r 2 min = (1/2δ) 2 ((w/2) 2 -δ 2 ) 2 + (w/2) 2 . We nally nd that:

r min = δ 2 1 + w 2δ 2 . (6.22)
Thus, cos θ = w/2r min enables to determine θ from Eq. 6.22. We report the values of θ with respect to Ca in Fig. 6.16 and we notice a systematic variation of θ between 0 o and 90 o . Such a variation has been observed when the contact line is perpendicular to the ow direction and moves streamwise with the ow: this is the well-known Tanner-Ho mann law [START_REF] De Gennes | Wetting: Statics and dynamics[END_REF][START_REF] De Gennes | Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls and Waves[END_REF][START_REF] Ho Man | A study of the advancing interface: I. Interface shape in liquid-gas systems[END_REF][START_REF] Tanner | The spreading of silicone oil drops on horizontal surfaces[END_REF] which relates the variation of the dynamic contact angle θ d at an air-liquid interface moving across a solid substrate with respect to the capillary number with the nding that θ d ∝ Ca 1/3 .

For liquid-liquid interfaces, Fermigier et al. [START_REF] Fermigier | An experimental investigation of the dynamic contact angle in liquid-liquid systems[END_REF] have observed the same kind of variation and use the Cox law [START_REF] Cox | The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous ow[END_REF][START_REF] Huh | Hydrodynamic model of steady movement of a solid/liquid/ uid contact line[END_REF] to model the movement of a glycerin-silicone contact line in a capillary tube, which depends on Ca and the viscosity ratio λ = η w /η o , as sketched in Fig. 6.15a. In our case, it is di cult to apply the aforementioned model since the contact line recedes transversely to the main ow and θ is de ned in the post and ow cross-section, as sketched in Fig. 6.15b.

The velocity pro le in the vicinity of the contact line is unknown and might not be parallel to the motion of the contact line. However, from our contact angle observations and measurements, a dependence of θ on Ca can be similarly reported.

Comparison between the predictions and the experimental results

The coe cient c d does not change very much, from 0.893 to 0.993, when θ varies from 0 o to 90 o . Considering the small variations of c d , Eqs. (6.19a -6.19b) are in good agreement with our experimental observations in terms of the dependence in Fig. 6.12: α depends on Qt, η o , the geometry of the channel and Ca. With the measured values of θ and the theoretically deduced numerical factors c s = 0.0764, c p = 0.0573, which enable the calculation of α ∞ in Eq. (6.19b) and c d from Eq. (6.20), we rescale the data sets. In Fig. 6.17a, we represent the experimental e ciency α exp as a function of g(t) = 6cshηw W H 2 c d L 0 ηo Qt, which is the right-hand side of Eq. (6.19a), with CAPB-SLES-Mac as the outer phase and three di erent oils as the inner phases. All curves collapse onto a master curve for α exp < 0.4 and g(t) < 0.5, which suggests that the theory reasonably predicts the linear time dependence of extraction dynamics at short times. To go further, Fig. 6.17b shows the collapse between f (α exp ) = α exp + (1 -α ∞ ) log α∞ α∞-αexp and g(t) for the entire time frame. The predictions of Wexler et al. [START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF] account for the observed dynamics if we consider a variation of the receding contact angle due to the addition of surfactants, which was not observed with pure solutions.

Extraction instability

As seen in Fig. 6.11b, the drained front loses its uniformity as Q increases and γ ow decreases, leaving patches of oil between the posts. Fig. 6.19a shows the instability for both connected posts and independent grooves, indicating that this instability occurs for multiple types of surface texture.

Instability of viscous strati cation

The appearance of these stable and unstable regimes, reported in Fig. 6.19b, could be caused by the ow of two superposed layers of uids of di erent viscosities (the aqueous solution and the underlying oil layer trapped in-between the posts). Indeed, it has been long known that such a two-uid shear ow can be unstable [START_REF] Charru | Phase diagram" of interfacial instabilities in a two-layer couette ow and mechanism of the long-wave instability[END_REF][START_REF] Hooper | Shear-ow instability at the interface between two viscous uids[END_REF][START_REF] Yih | Instability due to viscous strati cation[END_REF]. In particular, when gravity and interfacial tension are neglected, Yih [START_REF] Yih | Instability due to viscous strati cation[END_REF] and Hooper et al. [START_REF] Hooper | Shear-ow instability at the interface between two viscous uids[END_REF] have shown that the inertial terms of the Navier-Stokes equations are responsible for the development of the instability, even in the limit of low Reynolds numbers. Instabilities appear because of the jump in the slope of the velocity pro le at the interface between both uids. The discontinuity engenders the development of velocity disturbances, i.e. inertia-related vorticities that disturb the interface from the base ow. Long-wave and short-wave instabilities appear if η o > η w and h < H -h.

Surfactant-induced instability

However, in our case, the capillary number is low, so we expect that interfacial tension plays a signi cant role. The destabilizing role of surfactants on the interface recalls the work of Frenkel et al. [START_REF] Frenkel | Stokes-ow instability due to interfacial surfactant[END_REF] and Blyth et al. [START_REF] Blyth | E ect of surfactants on the stability of two-layer channel ow[END_REF] who showed that the presence of insoluble surfactants at the interface can trigger the growth of interfacial waves in a two-layer channel ow, even in the absence of inertia, provided that shear ow occurs. From their calculations, a surfactant-free system is stable, which corresponds to the system used by Wexler et al. [START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF] where the extracting front remains uniform along the section of the post pattern. In the two-layer geometry, the growth rate of the instability increases with the Marangoni number M ∝ C * /γ * ow where C * and γ * ow are, respectively, the initial interfacial concentration of surfactants and the initial oil-water interfacial tension. We also observe that very low capillary numbers can be stable as the range of unstable wavenumbers is reduced [START_REF] Blyth | E ect of surfactants on the stability of two-layer channel ow[END_REF]. But the instability is ampli ed for very low interfacial tension, like in our experimental observations. Thus, this would be the rst experimental observation of the destabilizing role of surfactants in a two-layer channel ow. Yet, other types of two-uid instabilities can also be considered, such as the ngering instabilities [START_REF] Troian | Fingering instabilities of driven spreading lms[END_REF] appearing in the transverse direction to the ow for the displacement of a thin lm by body or shear forces.

Foam-driven oil extraction from micropatterned channels

In §6.2.3, we study oil extraction when surfactants are added to the external phase. In this section, we investigate the case where gas is co-injected with surfactants, thus creating an aqueous foam, which also acts to extract the infused liquid during ow.

FOAM-DRIVEN OIL EXTRACTION FROM MICROPATTERNED CHANNELS 191

Controlled foam generation

We rst generate the aqueous foam through a ow-focusing device by building a micro uidic system.

Flow-focusing

The ow-focusing technique allows to control the bubble size by co-injecting nitrogen gas at a given pressure p and the liquid phase (CAPB -SLES -Mac described for surfactant solutions) at a given ow rate q through a constriction [START_REF] Garstecki | Formation of monodisperse bubbles in a micro uidic ow-focusing device[END_REF][START_REF] Garstecki | Mechanism for ow-rate controlled breakup in con ned geometries: a route to monodisperse emulsions[END_REF].

The ow-focusing device is fabricated by soft photolithography. Photolithography is the basic technique to build micro uidic devices [START_REF] Tabeling | Introduction to Micro uidics[END_REF]. Fig. 6.20 depicts the whole process. First, we need a black lm through which the pattern of the device is imprinted: the lm is transparent where the channels of the micro uidic device are. Then, the lm is deposited on a photoresisting material lying on a silicon wafer. UV light is ashed through the lm and the composite surface. The photoresisting material is reticulated where the lm is transparent and the channels are drawn. The dissolution of the non-reticulated part leaves blocks of photoresist of precised thickness (depending on the quantity of initial deposited photoresist). The silicon wafer with these polymer blocks on top of it is our mold. Afterwards, PDMS is poured onto the mold and baked. By removing the mold, we obtain a negative of the mold etched in the PDMS with channels of controlled thickness. We close the channels by sticking a glass lamella.

.2 Foam calibration

To slow down bubble coarsening, gas is blown through a solution of tetradeca uorohexane (C 6 F 14 ) [START_REF] Saint-Jalmes | Physical chemistry in foam drainage and coarsening[END_REF] before entering the device because uorinated gas is insoluble in water. We calibrate the mean bubble radius R b as a function of the liquid ow rate q and the gas pressure p. The characteristics of the ow-focusing device are summarised in Fig. 6.21. Garstecki et al. [START_REF] Garstecki | Formation of monodisperse bubbles in a micro uidic ow-focusing device[END_REF][START_REF] Garstecki | Mechanism for ow-rate controlled breakup in con ned geometries: a route to monodisperse emulsions[END_REF] have followed the rst steps of bubble formation. First, the gaseous thread enters the constriction and the liquid phase is squeezed within a thin lm between the gas phase and the wall. For a Poiseuille-like viscous ow, the liquid ow rate q is proportional to ∆p e 4 where ∆p is the pressure di erence between the upstream and the downstream end of the liquid ow, and e is the characteristic width of the ow, as shown by scaling arguments in §1.1.5. In the constriction, e decreases. Thus, to maintain a constant q, ∆p has to increase. The increase of pressure gradient induces the squeezing of the liquid-gas interface, leading to the break-up of the thread and the formation of bubbles. The characteristic time of break-up is T ≈ 1/q which corresponds to the time the liquid phase needs to ll the constriction. The bubble volume is V b ≈ q g T where q g is the gas ow rate. This ow rate q g is given by the Poiseuille equation and q g ∝ pH 4 /(η w l c ) where H and l c are the channel height and the constriction length. Using this expression and the volume estimation, one nds that V b scales as: V b ∝ p η w q . (6.23) Fig. 6.22 reports the correlation between V b and p/q. With this plot, we can build a calibration curve that determines the bubble volume and radius for each couple (p, q). We typically generate bubbles with R b ≈ 30 µm by setting p = 7 Psi = 48.3 kPa and q = 10 µL/s.

FOAM-DRIVEN OIL EXTRACTION FROM MICROPATTERNED CHANNELS 193

Other methods, such as T-junction [START_REF] Garstecki | Formation of droplets and bubbles in a micro uidic T-junction-scaling and mechanism of break-up[END_REF] and step emulsi cation [START_REF] Li | Step-emulsi cation in a micro uidic device[END_REF], could have been used. Nevertheless, ow-focusing is an easy method to implement that gives a polydispersity index (standard deviation of the bubble volume) below 2 %. This allows a good reproducibility in our experiments. It is also important to notice that the change of bubble size also induces a change of liquid fraction (identical to T-junction systems but not to step emulsi cation).

Experimental results and discussion

Once the bubbles are generated, the downstream reservoir of the ow-focusing device is connected to a 12 or 24 mL syringe with a 0.3 mm diameter PTFE tube. Bubbles accumulate in the syringe, which is capped at its end. Then, the piston is added at the top of the lled syringe. By weighing the syringe and measuring the foam volume, we deduce the foam liquid fraction which varies from 0.15 to 0.25.

For the injection stage, we use the pre-generated foam cartridge on the experimental set-up presented in the rst section. We reproduce the same protocol with a range of foam ow rates Q 194CHAPTER 6: FOAM -OIL INTERACTIONS IN MICROPATTERNED STRUCTURES between 15 µL/min (0.9 mL/h) and 1000 µL/min (60 mL/h). We also use the same oils in the porous substrates, as well as the same channel geometry.

Typical extraction pattern with foams

We report a typical time series of the extraction process by an aqueous foam in Fig. 6.23. We observe a non-uniform extraction front for all of our experiments. In Fig. 6.24, we represent α as a function of Qt by varying Q, R b and H. Fluorescent dyed oil appears green-yellow. The ow is from left to right. b. Foam ow in the channel with D, the lm thickness between the substrate and the bubbles.

In particular, in Fig. 6.24a-b, we report a surprising feature: an increase of ow rate engenders a decrease of α for a given injected volume with all other parameters constant. In other words, the slower the injection, the more e cient the oil extraction, which is a feature characteristic of extraction by foams. Moreover, Fig. 6.24c shows that the bubble radius has no in uence on the extraction pro le. If we compare α in the foam-driven case to the surfactant-driven case (Fig. 6.24d), α is higher in the foam case for a given injected volume. This result is all the more surprising as the injection ow rates Q used for the aqueous foams are one to two orders of magnitude lower than those used for surfactant solutions.

Shear modeling at the oil-water interface

To understand these results, we note that with aqueous foams, the applied shear stress is higher than with surfactant solutions. Indeed, as sketched in Fig. 6.23b, the existence of a thin lm between the foam bubbles and the underlying oil layer creates a higher shear stress, since the velocity gradient does not apply on the channel height H but rather occurs over the lm thickness D. Owing to their chemical composition, the bubbles have rigid air-water interfaces [START_REF] Golemanov | Surfactant mixtures for control of bubble surface mobility in foam studies[END_REF] that act as a rigid wall for the oil layer. Thus, we model the shear stress exerted by the foam on the oil phase as τ f oam xy = η w Q/(W HD). Replacing τ xy in Eqs. (6.19a -6.19b) with τ f oam xy , we obtain a new expression for α(t):

α + (1 -α ∞ ) log α ∞ α ∞ -α = c s h W HDL o η w η o 1 c d Qt , (6.24a

Dissipation length and lm thickness

The tting parameter D is found to vary between 1 µm and 7 µm, as shown in the inset of Fig. 6.26a. The values of D are in reasonable agreement with common orders of magnitude found for a bubble moving accross a solid wall [START_REF] Cantat | Foams: Structure and Dynamics[END_REF][START_REF] Denkov | Wall slip and viscous dissipation in sheared foams: E ect of surface mobility[END_REF][START_REF] Denkov | Foam-wall friction: e ect of air volume fraction for tangentially immobile bubble surface[END_REF][START_REF] Denkov | The role of surfactant type and bubble surface mobility in foam rheology[END_REF][START_REF] Merrer | Linear and non-linear wall friction of wet foams[END_REF]. Also, we notice that D and τ f oam xy do not depend strongly on R b , since α(Qt) maintains the same pro le when we increase R b by 10 times (Fig. 6.24c). Thus, we choose to introduce the dimensionless lm thickness D * = D/H , respectively, by H and by γ aw /H. In Fig. 6.26, we plot D * and τ f oam * xy as a function of the outer capillary number Ca * = η w Q/(W Hγ aw ) that di ers from Ca by replacing γ ow by the air-water interfacial tension γ aw . We compare the results to the estimates of lm thickness provided by mathematical studies of the liquid meniscus at a solid wall. We nd a good collapse with the exact solution for wet foams, made with the combination of surfactants and fatty acids used in our experiments, as developed by Denkov et al. [START_REF] Denkov | Wall slip and viscous dissipation in sheared foams: E ect of surface mobility[END_REF][START_REF] Denkov | Foam-wall friction: e ect of air volume fraction for tangentially immobile bubble surface[END_REF][START_REF] Denkov | The role of surfactant type and bubble surface mobility in foam rheology[END_REF]. We use two channel heights H = 89 and 178 µm and di erent Ca * for two orders of magnitude from 10 -5 to 10 -3 . In their study, they nd: where ξ and χ are functions of the liquid fraction .

D * = Ca * 1/2 , ( 6
Common surfactant-based foams with poor rigidity and shorter lifetime, such as SDS or TTAB, exhibit the classical Bretherton theory [START_REF] Bretherton | The motion of long bubbles in tubes[END_REF] for D ∝ Ca * 2/3 on a bubble moving in a channel of comparable size. This scaling can be derived by balancing the pressure gradient in the meniscus at the bubble front with the viscous dissipation at the wall. In the exact calculations, friction is neglected in the central area of the lm. However, the calculations by Denkov et al. are based on the assumption of rigid tangentially immobile interfaces, which means that the velocities of the upper lm surface and the wall are di erent and creates friction. Thus, it gives a di erence of power laws (Eqs. (6.25a -6.25b)). While they consider the bubble radius R b and the capillary pressure in the bubbles γ aw /R b as the rescaling parameters, we use H as the characteristic length. This is quite surprising but one possible reason is that the con nement of the bubbles in the channel sets a di erent and unique characteristic length, which is the channel height H.

Since D is smaller when Ca * decreases, this explains why smaller ow rates are more favourable to drain oil out for a given injected volume (see Eq. (6.24a)). A similar trend is observed when oil is removed from a macroscopic channel by imbibition, as described in Chapter 5, due to a longer contact time between the foam and the lubricating liquid when injection is slow. [START_REF] Denkov | Wall slip and viscous dissipation in sheared foams: E ect of surface mobility[END_REF] with H as a rescaling length for a liquid fraction = 0.2.

Conclusions

Liquid-infused surfaces are rough and patterned surfaces, in which a lubricating liquid is infused. Like superhydrophobic textured surfaces, they have been studied for their amazing properties: omniphobicity, biofouling or drag reduction. These properties vanish if the trapping of air (for superhydrophobic surfaces) or oil (for liquid-infused surfaces) into the cavities is lost. The failure of such surfaces is thus at stake. For superhydrophobic surfaces, an external applied pressure, normal to the air-liquid interface can lead to the sagging of this interface. For liquidinfused surfaces, a tangential stress induced by the shear ow of pure aqueous uids can drag oil out of the cavities [START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF]. This failure mechanism also occurs when the nature of the outer uid is changed. We have shown that surfactants in the outer phase adsorb at the oil-water interface. Thus, they change the dynamics of the contact line between the posts that make up the porous substrate. They actually enhance extraction compared to pure water. Then, the adsorption of surfactants can trigger long-wave and short-wave instabilities at the oil-water interface, leading to the non-uniform propagation of the extraction front. Furthermore, using aqueous foams in the outer phase enhances the failure dynamics drastically. The existence of a thin water lm between the foam and the oil phase increases the shear exerted by the outer phase. For comparable volumes of sheared oil, the volume of products used from pure solutions to foams is reduced by a factor 10 to 20. It also decreases by up to two orders of magnitude the typical injection ow rates required to drag oil completely out of the channel. The decreasing value of the lm thickness when the outer capillary number Ca * = η w Q/(W Hγ aw ) is reduced also means that lower ow rates increase the e ciency of oil extraction for a given injected volume of foam.

The failure of liquid-infused surfaces induced by surfactants and aqueous foams can be related to recovery issues if one imagines that oil is trapped within the roughness at the surface of a rock layer. Rocks are often fractured and have a porosity (larger than the typical size of the roughness) with a high speci c surface area from which oil can be di cult to extract because of the high capillary pressure. In this case, surfactants are often injected into the porous soil to reduce the interfacial tension with the oil phase, which causes a decrease of the capillary pressure and enhances oil recovery. Moreover, surfactants are often co-injected with gas, such as carbon dioxide, which is a transport agent. The higher mobility of the resulting foam enables the injected materials to span more areas in the porous soil than a single aqueous phase [START_REF] Kornev | Foam in porous media: Thermodynamic and hydrodynamic peculiarities[END_REF]. In both cases, industrials apply a normal stress on the pores and the oil phase by increasing the extracting pressure (which can also lead to fracking issues) [START_REF] Lenormand | Mechanisms of the displacement of one uid by another in a network of capillary ducts[END_REF].

From our work, we show that it is also possible to use the tangential shear at the wall to extract oil con ned in the roughness. We model the microporous and rough rock layer by a micro uidic liquid-infused surface. However, the chemical and physical nature of the external ow modies the dynamics of the receding contact line and/or the nature of the applied shear stress. In both cases, we account for these e ects by introducing a modi ed dependence on the capillary numbers Ca = η w Q/(W Hγ ow ) and Ca * = η w Q/(W Hγ aw ). For recovery e ciency, it is more bene cial to increase the interfacial capillary number Ca when surfactant solutions are used, especially by the reduction of the oil-water interfacial tension, as expected. In the foam case, it is more bene cial to decrease Ca * for a given injected volume of foam. In other words, extraction e ciency is maximized if the injection ow rate is reduced. In both cases, we suggest a model that can predict the quantity of oil and pollutant that can be extracted.

To compare with, we can evaluate the range of capillary numbers Ca and Ca * for real soil systems. By the Darcy's law, Ca = η l u Darcy /γ ow = k D /(r p L) where u Darcy is the mean ow velocity, [START_REF] Carman | Fluid ow through granular beds[END_REF][START_REF] Kozeny | Über kapillare Leitung des Wassers im Boden[END_REF]. For an assembly of spherical grains:

k D = φ 3 d 2
g /(180(1 -φ) 2 ) where φ and d g are the porosity and the grain diameter. We can take φ = 0.4 and d g ≈ 100 µm, which gives k D ≈ 4 x 10 -11 m 2 . Given that r p ≈ 10 µm and L ≈ 1 km [START_REF] Dake | Fundamentals of Reservoir Engineering[END_REF], we nd that Ca ≈ 4 x 10 -9 and Ca * = (γ ow /γ aw )Ca ≈ 4 x 10 -10 . This is de nitely well below our experimental Ca and Ca * (10 -5 to 10 -3 ). Yet, in both cases, the physics is mainly ruled by capillarity. We thus believe that the understanding gained in this chapter should still be valid for ranges of ow rates and velocities more compatible with soil extraction.

Conclusions and future work

We present here the main conclusions of this thesis and suggest ideas for future work and progress. and the experimental tests from Chapter 2. When an adjacent lm breaks, the oil-lled Plateau border breaks into small emulsi ed droplets. The breaking dynamics can be understood as the Rayleigh-Plateau instability of a viscous uid in another viscous uid. We can predict the size of the droplets from the volume of oil, the length of the oil slug and the wavelength of maximum instability growth.

In Chapter 4, we study locally how oil interacts with the air-water interface by spreading experiments. We consider a con guration where the oil phase rises to the air-water interface by the aqueous phase t reproduce the entry conditions of oil droplets at the air-water interface of soap lms. At short times, we determine a comprehensive time-dependent dynamics for the spreading radius for viscous and non-viscous oil droplets. For pure aqueous phases and surfactant-laden phases, this dynamics follows a t 1/2 behaviour. However, we observe a di erence at long times for both types of aqueous phases. For pure water, the short-time regime joins a long-time regime in t 3/4 already found for spreading at liquid-liquid interfaces, based on the existence of a viscous boundary layer between oil and water. However, for surfactant solutions, we observe a solid-like spreading dynamics driven by gravity in t 1/8 , which still raises questions. Potential molecular interactions and adsorption dynamics might be at stake with surfactants.

In Chapter 2, 3 and 4, we study the consequences of oil-foam interactions at di erent scales. In Chapter 5, we determine how oil is absorbed into foams, and thus how oil-foam interactions can occur. Due to the capillary underpressure (or osmotic pressure) in the foam, oil can be imbibed into aqueous foams, providing that the conditions de ned in Chapter 2 are ful lled. We consider three di erent con gurations. First, we study the 3D imbibition from a point-source at imposed atmospheric pressure to reproduce oil extraction from a pressure-controlled pore. In this case, we numerically solve the drainage equation in cylindrical coordinates. We include the e ect of gravity and the creation of oil-water interfaces (through the Bond number B) in our resolution by introducing an e ective interfacial tension. The higher the Bond number, the slower the imbibition dynamics. the 1D imbibition of an oil slick. The rising oil front follows a t 1/4 dynamics at very long times. The initial liquid fraction of the foam plays an important role as it determines the driving pressure di erence for the oil slug to invade the Plateau borders. Then, we consider the 1D imbibition of an oil slick. The rising oil front follows a t 1/4 dynamics at very long times. The initial liquid fraction of the foam plays an important role as it determines the driving pressure di erence for the oil slug to invade the Plateau borders. Finally, we move the foam on top of the extracting point to simulate a recovery process. Our preliminary results show that a slow injection rate is more favourable to extract oil for a given volume of oil. This can be related to a longer contact time between the Plateau border and the oil reservoir, although this remains to be proven.

In Chapter 6, we trap oil in the roughness of a microfracture. To model this microporous system, we use liquid-infused surfaces. The lubricating layer of oil can be sheared out of the texture by injecting a pure aqueous phase through a micro uidic channel, on one side of which the textured pattern is imprinted. In our case, we study how surfactant solutions and foams can also drag oil out of the texture by comparison with the theory developed by Wexler and al. [START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF] for the shear-induced failure of liquid-infused surfaces. For a given volume of injected uid, surfactants enhance oil extraction and induce a variation of the dynamic contact angle with the capillary number Ca. Oil extraction is also improved with foams. These latters require a lower injection ow rate and a lower injected volume to extract as much oil as with pure aqueous solutions. This is due to an increase of the shear stress at the oil-water interface. Indeed, the viscous dissipation occurs upon the length scale of a lm thickness, which is much smaller than the length scale of the channel height, thus the shear stress is higher for foams than for pure water. The lm thickness we deduce is in good correlation with theoretical estimations for a liquid lm with rigid interfaces lying at the wall.

In this work, we combine di erent aspects and approaches of the hydrodynamics of foams in contact with oil. We model the multiphasic system constituted by the air, the water and the oil phase locally and we try to understand the interplay between the di erent phases and its consequences on the dynamical phenomena at stake in oil-laden foams. We hope that this study could be an interesting starting point for global studies at the scale of bigger foam volumes and macroscopic soil samples. In particular, soil remediation, enhanced oil recovery and any application involving the extraction of a non-aqueous phase might be concerned.

Perspectives and outcomes

The results presented in this thesis ask for more developments and suggest new leads to a better understanding of the interactions between aqueous foams and immiscible liquids.

First, nding a comprehensive argument to predict whether a foam will be destabilized by the oil phase needs to be completed. This requires to understand the behaviour of the air-water-oil pseudoemulsion lm, which strongly depends on the competition between the disjoining pressure and the pressure distributions in the owing oil phase.

Second, the e ect of surfactants on the spreading of oil on a liquid surface is still an open question. Surfactant dynamics certainly play a role and a microscopic and molecular analysis at the contact line could complete the work done so far.

Third, imbibition studies were conducted for a de ned foam sample and an open reservoir. What will happen in the case of a real soil system, whose pores trap oil? The same question arises for the extraction from micropatterned surfaces. We model an ideal system with a controlled roughness and produce a theoretical explanation based on well-de ned parameters. How does it change when the surface con guration is more random and involve di erent features for which it is di cult to de ne a unique length scale?

In this situation, we would need to carry out recovery experiments and injection tests in porous models (glass beads) and macroscopic soil samples that are oil-laden beforehand. In-situ observations can be made by MRI (Magnetic Resonance Imaging) for example. MRI is a non-destructive technique that traces the spin resonance of the hydrogen protons in a molecule. An electromagnetic wave impulsion is applied and changes the magnetization of the system. At the end of the impulsion, the magnetization relaxes to its initial position. Depending on the chemical environment, the relaxation time (T 1 and T 2 relaxation) changes. Preliminary results have shown that it is possible to distinguish the oil phase from the aqueous phase by this technique. By recording the relaxation time and measuring the size of the relaxation peak for each phase, one can map the quantity of oil and water in the system, and thus deduces the quantity of oil left and extracted. A typical T 1 signal given by nuclear magnetic resonance for an oil-laden foam is given in Fig. 6.27. Yet, due to the lack of time, we could not perform a systematic study using MRI. Figure 6.27: Typical NMR spectrum for an oil-laden foam with the intensity I as a function of the relaxation time T 1 . The relaxation times are T 1 ≈ 0.5 ms for water in foams and T 1 ≈ 100 ms for oil. We use gadolinium as a contrasting agent (and EDTA as a chelating agent). Gadolinium is able to shorten the T 1 -typically for a pure water solution from 2 s to 10 ms according to the concentration of gadolinium.

In our study, we use basic types of oil (organic and mineral). We can wonder whether other types of oils and immiscible liquids give the same results. For instance, tar oils are closer to industrial applications. Oil can also carry small particles that might have some e ect on the foam stability.

Combining the e ect of particles and oils is an interesting question. Studies have been carried out for particle-laden foams [START_REF] Ha Ner | Flow and jamming of granular suspensions in foams[END_REF][START_REF] Ha Ner | The drainage of foamy granular suspensions[END_REF] but the subject of "foamulsions" with particles at the air-water interfaces is still open.

Finally, in this work, we look for a stable oil-laden foam for which oil does not exhibit an antifoaming activity. An interesting question is to study the antifoaming mechanisms and in particular the breaking avalanches when an oil droplet is deposited at the top or inside an aqueous foam. This second-order equation has an exponential solution of the form exp(σt) with σ given by:

∂Q(x, t

σ 2 ≈ γ 2ρ l r 3 c k 2 r 2 c 1 -k 2 r 2 c . (B.7)
We nd the expression presented in 3.2.1.2, for which the maximum growth rate corresponds to k max = 0.697/r c and λ max = 2π/k max ≈ 9r c .

B.2 Viscous-dominated regime

Here, we study the Rayleigh-Plateau instability of a viscous uid in another viscous uid. We need to solve the complete Navier-Stokes equation in both inner cylindrical uid and outer uid. We give some hints for the resolution. The comprehensive problem has been solved by Tomotika et al. and more details can be found in [START_REF] Tomotika | On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous uid[END_REF].

We consider the ow dynamics in one of the uids in cylindrical coordinates and the motion is symmetrical around the axis x. Thus, we introduce the velocity eld U 1 = u 1 e r + w 1 e x . (u We introduce the Stokes' current function Ψ which satis es the following relationships and Eq. B.9: where a sinusoidal solution in e i(σt+kx) is considered with the wave number k 1 satisfying k 2 1 = k 2 + iη 1 σ ρ 1 . The functions I 1 and K 1 are Bessel functions. Then, we need boundary conditions to nd the values of the constants A 1 , B 1 , A 2 and B 2 in both inner and outer uids. The following conditions are considered: -Continuity of the velocity components at the interface between both uids in r = r c : in the textured surface and avoids the transition to the Wenzel state, as underlined in §6.1.1.3. In order to do so, the main idea is to sharpen the defects and increase the number of discontinuities.

D.1.3 Tuning the oleophobicity

The superhydrophobicity of textured surfaces (θ * close to 180 o ) is induced for water droplets exhibiting a non-wetting behaviour on a smooth surface with θ Y > 90 o . However, other liquids with low air-liquid interfacial tensions, such as oils, exhibit an oleophilic behaviour (θ Y < 90 o ) and wet most surfaces. According to §6.1.1.2, the textured surface should also present an oleophilic (oil-attractive) character and a fully-wetted surface. However, one can wonder whether it is possible to tune the textured surface and make it oleophobic (oil-repellent) by trapping air pockets in a Cassie-Baxter regime. We can write the Young relationships γ s2 -γ s1 = γ a1 cos θ 1 -γ a2 cos θ 2 where γ a1 and γ a2 are the air-lm and air-droplet interfacial tensions, with θ 1 and θ 2 the Young's contact angles for both lm and droplet. Setting dE > 0 for oating droplets yields the following condition from Eq. D.1:

γ a1 cos θ 1 -γ a2 cos θ 2 > γ 12 r or γ a1 cos θ 1 -γ a2 cos θ 2 > (1 -φ S )γ 12 r -φ S . (D.2)
This condition for oating droplets and thus slippery surfaces can be ful lled if the lubricant exhibits a complete wetting (θ 1 ≈ 0) with the solid surface being hydrophobic (θ 2 > 90 o ) towards the immiscible droplet. Decreasing the interfacial tension between the lubricant and the droplet, as well as increasing the roughness of the texture are also necessary conditions.

The no-pinning character of liquid-infused surfaces is considerably useful in many applications [START_REF] Anand | Enhanced condensation on lubricant-impregnated nanotextured surfaces[END_REF][START_REF] Busse | Change in drag, apparent slip and optimum air layer thickness for laminar ow over an idealised superhydrophobic surface[END_REF][START_REF] Epstein | Liquid-infused structured surfaces with exceptional anti-biofouling performance[END_REF][START_REF] Kim | Liquid-infused nanostructured surfaces with extreme anti-ice and anti-frost performance[END_REF][START_REF] Smith | Droplet mobility on lubricant-impregnated surfaces[END_REF][START_REF] Xiao | Enhanced condensation on lubricantimpregnated nanotextured surfaces[END_REF]. The slippery nature of the surface allows to remove stains and to reduce drag. The small contact angle hysteresis hinders the well-known co ee ring e ect for the evaporation of particle-laden liquid droplets [START_REF] Deegan | Capillary ow as the cause of ring stains from dried liquid drops[END_REF]. When liquid evaporates, particles accumulate where the contact line is pinned. However, on liquid-infused surfaces, particles accumulate uniformly at the center of the evaporating droplet because the receding air-liquid contact line cannot pin the deposition surface. This is particularly useful if one wishes to concentrate particles in small aggregates (for chemical synthesis for instance).

Unlike omniphobic textured surfaces, liquid-infused surfaces cannot fail by sagging phenomena. They are robust to any pressure-driven distortions. Yet, liquid-infused surfaces can lose their properties when they interact dynamically with an external ow dragging the lubricating lm out of the texture. In the following sections, we focus on the shear-driven failure of these surfaces with the aim to understand how impregnated oil can be extracted from a micropatterned porous medium.

  Experimental observations of air injection into a con ned granular suspension. a. Percolation regime at low ow rate. b. "Fracture" regime at high ow rate. The pictures are extracted from [186]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Plateau borders and oil: equilibrium shape and coalescence 3.1 Con gurations of the oil droplet in the Plateau border for di erent droplet sizes r * , where r * c = 2/ √ 3-1 is the critical dimensionless radius for which the oil droplet rst touches the walls of the Plateau border. In the far right image, the dotted lines indicate that oil can also swell the Plateau border and invert the curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Oil droplets in Plateau borders [132]. a. Side-view. b. Cross-section. From top to bottom: K = 1, 0.5, 0.2 and 0.1; the oil-water interfacial tension γ ow decreases. From left to right: r * = 0.2, 0.4, 0.7 and 1; the volume of oil increases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Dimensionless pressure di erence between the pseudoemulsion lm and the Plateau border ∆p * f with respect to the dimensionless size of the oil droplet r * [132]. . . . . . . . . . . . . . . . . 3.4 Simulations by Surface Evolver. a. Inner con guration. b. Outer con guration. . . . . . . . . . 3.5 Energy calculations using simulations by Surface Evolver: total interfacial energy E I as a function of the volume ratio V o /V w . a. Sun ower oil (Sun) and olive oil (Olive)-CAPB+SLES+Mac. b. Silicon oil with η o = 100 mPa.s (Si 100) and silicon oil with η o = 10 mPa.s (Si 10)-CAPB+SLES+Mac. c. Silicon oil with η o = 10 mPa.s (Si 10) and sun ower oil (Sun)-CAPB+SLES+Mac. d. Silicon oil with η o = 10 mPa.s (Si 10) and hexadecane (Hex)-CAPB+SLES. . . . . . . . . . . . . . . . . . 3.6 Mechanism of the Rayleigh-Plateau instability. From top to bottom: cylindrical thread, wave perturbation, breaking into small droplets. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Growth rate σ RP t c as a function of kr c in the inertial regime. The maximum growth rate is obtained for k max r c ≈ 0.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Dimensionless growth rate σ RP t c as a function of kr c in the viscous regime. The blue round dots designate the case η c > η f , while the orange triangular dots designate the case η c < η f . In both cases, the maximum growth rate is obtained for k max r c = 0. . . . . . . . . . . . . . . . . . . 3.9 Experimental set-up presenting the slender Plateau border lled with oil at the jonction between three soap lms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Snapshots of the breaking dynamics. Oil: olive oil; drop volume: Ω = 1.5 µL. . . . . . . . . . . LIST OF FIGURES xix 3.11 Side-view of the breaking process. a. Initial oil-lled Plateau border [137]. b. Breaking of the lower rear lm. The Plateau border is ejected to the opposite direction. c. Oil-lled lm before the Rayleigh-Plateau instability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

  Soap lms and oil: dynamics of rising oil droplets spreading at the air-water interface 4.1 Liquid droplet on a solid surface in partial wetting. . . . . . . . . . . . . . . . . . . . . . . 4.2 Energy of the lm E S as a function of the thickness e for pseudo-partial wetting. . . . . . . . . 4.3 Spreading droplet and its precursor lm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Spreading regimes for a liquid droplet on a solid hydrophilic surface for PDMS with η l = 1 Pa.s.

4. 6

 6 Oil-water interfacial tension γ ow as a function of time t for the silicon oil (η o = 100 mPa.s) -SDS system. The legend indicates the bulk concentration of surfactants. . . . . . . . . . . . . . . . 4.7 Oil lens spreading at the air-water interface. . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Spreading radius r as a function of time t for oil spreading on water-surfactants solutions [13]. Oil: PDMS (Polydimethylsiloxane η o = 100 mPa.s). Aqueous bath: C 16 TAB, Water, AOT (Dioctyl sulfosuccinate sodium) and C 10 E 5 . The black lines are the t from Eq. 4.19. . . . . . . . . . . . 4.9 Viscous oil lens spreading at the surface of a water bath. . . . . . . . . . . . . . . . . . . . . 4.10 Experimental set-up. The oil droplet rises from the immersed needle and spreads at the bath surface. xx LIST OF FIGURES 4.11 Con gurations of an oil droplet at the air-water interface according to the values of the entry E and spreading S coe cient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Snapshots of the spreading dynamics. a. No-surfactant case (aqueous solution = water and oil = silicon oil, η o = 5 mPa.s); R d = 3.72 mm and Oh = 73.3. b. Surfactant case in the short-time regime from t = 0 s to 0.4 s (aqueous solution = SDS solution, C SDS = 2 g/L and oil = silicon oil, η o = 20 mPa.s); R d = 2.84 mm and Oh = 7.7. c. Surfactant case in the long-time regime from t = 0.66 s to 60 s (aqueous solution = SDS solution -C SDS = 2 g/L and oil = silicon oil, η o = 20 mPa.s); R d = 3.76 mm and Oh = 8.9. The thick black line is the injection needle. . . . . . . . . 4.13 Dimensionless spreading radius R as a function of the dimensionless time T Oh . The open dots correspond to a bath solution without surfactants and the close dots with surfactants. The type of oilη o (mPa.s) -C SDS (per liter), Bo and Oh are indicated in this order in the legend. A corresponds to an attached droplet. The dashed line is Eq. 4.27. The black arrow in the snapshot indicates the direction of the oil movement. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.14 Dimensionless spreading radius R as a function of the dimensionless time T Oh for the spreading of para n oil on water. Dimensionless numbers: Oh = 13, Bo = 2.7 and Oh ef f = 3.6. The transition time and radius are T Oh-ST -LT ≈ Oh 2 ef f /f (Oh) 3 ≈ 3.8 and R ST -LT ≈ Oh ef f /f (Oh) 3/2 ≈ 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.15 Dimensionless spreading radius R as a function of the dimensionless time T * . The type of oilη o (mPa.s -C SDS (per liter), Bo, Oh and R g are indicated in this order in the legend. L in Si100

1 / 8 7 x 10 - 4 . 7 x 10 - 3

 1871047103 correspond to the transitions between the short-time and the long-time dynamics. . . . . . . . . . . . . . . . . . . . 4.17 a. Schematic representation of the shear stress contributions at the air-oil interface τ air/oil and at the oil-water interface τ oil/water when the oil droplet initially spreads. b. Side-view of a typical short-time spreading dynamics (para n oil with surfactants) highlighting an oscillatory and a spring-like behaviour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Oil imbibition into aqueous foams 5.1 Liquid rising in a capillary tube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LIST OF FIGURES xxi 5.2 a. Imbibition in an open corner. b. Imbibition in an open wedge. . . . . . . . . . . . . . . . . 5.3 a. Di usive Lucas-Washburn's regime. b. Wedge regime. c. Regime in a diverging channel. . . . . 5.4 a. Osmotic cell: the pressure p in the foam is controlled by the distance h f oam between the membrane and the location of the meniscus in the U-tube. b. Dimensionless osmotic pressure measured from an osmotic cell as a function of the liquid fraction φ l for a monodisperse, polydisperse emulsion (oil droplets in water) and an ordered monodisperse foam (φ * l = 0.26) [88]. . . . . . . 5.5 Porous model: network of cylindrical pores randomly oriented by the angle β. . . . . . . . . . 5.6 Coe cient K c as a function of the Boussinesq number Bq for di erent velocity pro les [42]. . . 5.7 a. Typical numerical solution of Eq. 5.34 for the dimensionless Plateau border area α for Bo = 0 and τ = 1000. b-d. Numerical solutions of the imbibition front for di erent times τ = 10, 50, 100 and 500 with Bond numbers Bo = 0, 1.6 and 5. . . . . . . . . . . . . . . . . . . . . . . . . 5.8 a. Cross-section of a Plateau border for a dry aqueous foam. b. Cross-section of a Plateau border lled with oil for a dry aqueous foam. S P B and S f , respectively, denote the areas of the side surfaces of the Plateau borders and the surface of the lms that are connected to the Plateau borders. S o represents the area of the oil-water interfaces. . . . . . . . . . . . . . . . . . . . 5.9 Experimental set-up. A 30 cm-long PTFE tube (2 mm-diameter) is lled with the imbibing liquid (mixtures of foaming solution-glycerol or oil) and terminated by a 1 cm-long capillary tube (1 mm diameter) that enables the connection with the foam. A motion controller allows maintainence of the pressure of the liquid phase input at atmospheric pressure p atm . . . . . . . . . . . . . . . 5.10 Snapshots of the imbibition front for a 30-second timescale. The imbibing liquid (glycerol-foaming solution) is uorescent and the front is de ned as the boundary between the brighter and darker uorescent areas. The liquid-lled tube is displaced upwards by a millimeter in the rst snapshot to ensure contact between the source and the foam. . . . . . . . . . . . . . . . . . . . . . . 5.11 Evolution of the vertical front position z f with respect to time t for two data sets. The round dots correspond to a glycerol-foaming solution and the square dots to olive oil. The bubble radius R b , the initial liquid fraction φ i , and the viscosity η o/w of the imbibing oil or aqueous solution are indicated, respectively, in the legend. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Vertical front position z f with respect to time in dimensionless coordinates for immiscible oils (open dots) and miscible aqueous liquids (closed dots). The experimental curves are obtained for two R b = 1 and 2 mm, di erent φ i , η o/w and Bo. The numerical solutions of Eq. 5.34 for Bond numbers Bo = 0, 1.6 and 5, deduced from Eq. 5.34, are plotted by the dashed lines. The self-similar power law evolution in τ 1/2 in the no-gravity case (Eq. 5.44) is shown. . . . . . . . . . . . . . xxii LIST OF FIGURES 5.13 a. Experimental set-up. b. Geometrical elements of foams: vertices, Plateau borders and foam lms. c. Slender Plateau border with a radius of curvature r P B . d. Capillary rise of oil in a dry aqueous foam. The olive oil drop appears white thanks to a uorescent dye and invades the network of Plateau borders. The time interval between each frame is 67 s. The scale bar is 3 mm. . 5.14 Height of the rising front for olive oil as a function of time for di erent initial liquid fractions and one mean bubble radius R b = 1.8 mm. Inset: log-log scale of the data corresponding to φ i = 0.The plain line corresponds to the t 1/2 dynamics detailed in [29], while the dashed line illustrates the t 1/4 dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.15 Drainage solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.16 Dimensionless imbibition height Z f as a function of the dimensionless time τ deduced from numerical resolution of Eq. 5.49 with the comparison to the self-similar solution given by Eq. 5.52 for di erent Bond numbers Bo = 0.4, 1.6 and 4. Experimental data for Bo = 2.1 and two di erent initial liquid fractions φ i are also plotted and do not converge towards a unique solution. . . . . 5.17 Typical pro le of the uorescence intensity I of a single node during olive oil imbibition as a function of the vertical coordinate z for di erent times t. A peak corresponds to an oil-lled node or an oil-lled Plateau border, a hole to air in the bubbles. . . . . . . . . . . . . . . . . . . . 5.18 Typical evolution of the uorescence intensity I of a single node during sun ower oil imbibition as a function of the node's vertical coordinate z. Initially, the node is located at x = 1 mm and z = 7.9 mm . The intensity of uorescence is in arbitrary units and the di erent curves correspond to uorescence intensity measurements taken at di erent times. The black arrow indicates the evolution of time. Inset: Width of the peak of intensity σ I (de ned at half of the maximum intensity) as a function of time t. The red arrow highlights the time at which the oil front has reached the node. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.19 Oil imbibing a Plateau border. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.20 Normalized imbibition front α * h as a function of normalized time t * in log-log plot. In the legend, AS1, AS10 and AS100 respectively stand for Aqueous Solution with η l =1.4,10 and 100 mPa.s, while OO and SO stands for Olive Oil and Sun ower Oil. The black line corresponds to (1 + α * h) 4 + 4 1+α * h -5 = b 2δa 15 √ 3 t * with b ∼ 0.3, while the two dashed lines represent the power laws: t 1/2 at early times and t 1/4 at late times. . . . . . . . . . . . . . . . . . . . . . . . . 5.21 Experimental set-up for the oil recovery by a moving foam. The foam is rst generated through a column. The upper part of the column is removable and ipped at 90 o onto a syringe pump with a piston at one end. Then, the oil reservoir is connected to the reversed column at the atmospheric pressure p atm before starting the injection of foam on top of the extraction point. . . . . . . . . LIST OF FIGURES xxiii 5.22 Imbibition pro les for immiscible liquids (olive oil) for successive times t. a. Initial liquid fraction φ i = 3.4 x 10 -3 , injection ow rate Q = 100 mL/h and bubble radius R b = 1 mm. b. φ i = 2.and Q = 200 mL/h. c. φ i = 2.4 x 10 -3 and Q = 400 mL/h. The red arrow indicates the extraction point and the green arrow the movement of the foam. . . . . . . . . . . . . . . . . 5.23 Imbibition pro les for miscible liquids (CAPB+SLES+Mac-glycerol) for successive times t. a. Initial liquid fraction φ i = 2.6 x 10 -3 , injection ow rate Q = 200 mL/h and bubble radius R b = 1 mm. b. φ i = 1.9 x 10 -3 and Q = 200 mL/h. c. φ i = 1.7 x 10 -3 and Q = 400 mL/h. The red arrow indicates the extraction point and the green arrow the movement of the foam. . . . . . . 5.24 Mean extraction ow rate Q o , for an injection time of 4 min and 30 s, as a function of the foam velocity U f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5. 26

 26 Log-log plot of the dimensionless imbibed volumeV * = R f 0 Z f 0 α(R, Z, τ )RdRdZ (where R f = r f /R b and Z f = z f /R b are the dimensionless coordinates of the front) for the static imbibition of §5.2 as a function of the dimensionless time τ for two di erent Bond numbers Bo = 1.6 and 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6. 2

 2 Liquid droplet sitting on a solid-liquid composite surface. . . . . . . . . . . . . . . . . . . . 6.3 Impregnation of a lm in a textured surface. The solid fraction φ S and the projected area A P are indicated in a sub gure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Liquid droplet sitting on a solid-air composite surface. . . . . . . . . . . . . . . . . . . . . .

  b. Top-view of the channel, with the oil uorescing green-yellow. c. Side-view of the channel consisting of one wall with an array of posts. . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Equilibrium oil-water interfacial tension γ ow (mN/m) as a function of the surfactant concentration c (g/L) for SDS-Olive oil (O) and CAPB-SLES-Mac-Olive oil (O) combinations. The blue and the red dashed lines are guide lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Top views of typical oil extractions (olive oil) by a surfactant solution (CAPB -SLES -Mac -10 wt% glycerol). Fluorescent dyed oil appears green-yellow. The small green-yellow dots correspond to some oil droplets stuck on the epoxy surface when the channel was initially cleared of oil in the rst place. The ow is from left to right. a. Extraction with a uniform front; Q = 0.05 mL/min and γ ow = 4.2 mN/m, time interval: 5 h. b. Extraction with an unstable pattern; Q = 5 mL/min and γ ow = 4.2 mN/m, time interval: 5 min. . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 Extraction e ciency α as a function of time t for di erent draining liquid -oil combinations with the following variable parameters: a. γ ow (mN/m). b. Q (mL/min) (Inset: α as a function of Qt).

  c. η o (mPa.s). d. H (µm). S and O are, respectively, the surfactant type, the oil type. . . . . . . . 6.13 Extraction e ciency α as a function of the injected volume Qt for SDS -olive oil combination, with the capillary number Ca ≈ 0.015 and 0.005 de ned in Eq. 6.19b. . . . . . . . . . . . . . 6.14 Confocal snapshot of the oil-water interface in the texture cavity. The geometrical parameters are indicated on the picture: δ is the de ection. . . . . . . . . . . . . . . . . . . . . . . . . 6.15 a. Movement of the A-B interface parallel to the ow of mean velocity U with the viscosity ratio η B /η A > 1. b. Movement of the same A-B interface for our post pattern with the interface receding perpendicular to the main ow of mean velocity U . . . . . . . . . . . . . . . . . . LIST OF FIGURES xxv 6.16 Measurements of the receding contact angle θ as a function of Ca for CAPB -olive oil (red dots)

6. 18 1 cd

 181 Unstable two-layer Couette ow with layers of di erent viscosities η w and η o and di erent thickness h and H -h. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.19 a. Typical unstable extraction pattern for posts and grooves with the same height and width. The surfactant solution -oil combination is SDS (2 g/L) -olive oil; Q = 2 mL/min and t = 60 min. Contrast is enhanced in the snapshots. b. Phase diagram where each point (γ ow , Q) represents an experiment with the CAPB -olive oil, the CAPB -silicon oil 1 and the SDS -olive oil combinations. Unstable extraction is represented by red dots and stable extraction by green dots. The dashed black line de nes the stable regime for pure aqueous solutions -olive oil (the silicon oil 1 case is further to the right of the diagram). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.20 Description of the micro uidic fabrication by photolithography. Colors are here to distinguish each component. They do not correspond to a physical reality. . . . . . . . . . . . . . . . . . 6.21 Flow-focusing device with the main geometrical features: w g = 200 µm, w l = 300 µm, w c = 60 µm, l c = 100 µm and w o = 500 µm. The height H of the channel is 27 µm or 72 µm. . . . . . . 6.22 Bubble volume V b (nL) as a function of the ratio pressure-ow rate p/q (Psi/m 3 /s). Below: bubbles (radius R b ≈ 30 µm) accumulating in the outlet reservoir. . . . . . . . . . . . . . . . . . . . 6.23 a. Top-view of olive oil extraction by an aqueous foam with R b = 100 µm and Q = 50 µL/min. Fluorescent dyed oil appears green-yellow. The ow is from left to right. b. Foam ow in the channel with D, the lm thickness between the substrate and the bubbles. . . . . . . . . . . . 6.24 E ciency α as a function of the injected volume Qt. a. Olive oil extraction by a foam with R b = 30 µm. b. Silicon oil 1 extraction by a foam with R b = 30 µm. c. Extraction for silicon oil 2 and R b = 30, 100 and 200 µm, and H = 179 µm. d. Comparison between surfactant solutions and foams with a varying channel height (89 or 178 µm). In the legend, O, Q (µL/min), R b (µm) and H (µm) are, respectively, the oil type, the injection ow rate, the mean bubble radius and the channel height. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi LIST OF FIGURES 6.25 a. Measurements of the receding contact angle θ as a function of Ca. b. Experimental e ciency α exp as a function of G(t) = csh W HDLo ηw ηo Qt for some oils, Q (µL/min) and R b (µm). c. f (α exp ) as a function of G(t) for the same parameters. f (α exp ) = G(t) is represented by the black dashed line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.26 a. Dimensionless lm thickness D * = D/H as a function of the outer capillary number Ca * for all data sets. Inset: Film thickness D as a function of Ca * for the same data set. b. Dimensionless applied shear stress τ f oam * xy = τ f oam xy H/γ aw as a function of Ca * for all data sets. The green dashed line represents the values predicted by Denkov et al. [57] with H as a rescaling length for a liquid fraction = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.27 Typical NMR spectrum for an oil-laden foam with the intensity I as a function of the relaxation time T 1 . The relaxation times are T 1 ≈ 0.5 ms for water in foams and T 1 ≈ 100 ms for oil. We use gadolinium as a contrasting agent (and EDTA as a chelating agent). Gadolinium is able to shorten the T 1 -typically for a pure water solution from 2 s to 10 ms according to the concentration of gadolinium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Examples of ow dynamics by resolution of Stokes' equation A.1 Spreading a liquid droplet on a solid surface. . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Poiseuille ows in circular and rectangular pipes. . . . . . . . . . . . . . . . . . . . . . . . B Rayleigh-Plateau instability of a viscous uid in another viscous uid B.1 Geometry of the axisymmetric unstable jet. . . . . . . . . . . . . . . . . . . . . . . . . . . C Micro uidic fabrication: the "sticker technique" C.1 a. Fabrication of the micro uidic channel. b. Fabrication of the textured pattern. . . . . . . . . C.2 a. PDMS mold for the main channel. b. Negative of the textured pattern on a silicon wafer. c. Micro uidic channel ready for experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . D Properties of textured surfaces D.1 a. Gull feather repelling water (Source: matthewwills.com). b. Water droplets rolling down a lotus leaf (Source: Hydrobead). c. Water (in blueγ aw = 72.1 mN/m) and rapeseed oil (in redγ aw = 35.5 mN/m) on a shelduck feather dip-coated in a solution of 50/50 uorodecyl POSS/Tecno on [173]. d. Water (blue) and octane (red) droplets on a textured surface (Source: Scienti c American). LIST OF FIGURES xxvii D.2 Anchored air-liquid contact line around a defect on a solid surface. . . . . . . . . . . . . . . . D.3 Di erent texture geometries enhancing hydrophobicity and oleophobicity (images from Scanning Electron Microscopy). a. Electrospun bers of PMMA (poly(methyl)methacrylate)-uorodecyl POSS [185]. b. Barbules of a duck feather dip-coated in uorodecyl POSS/Tecno on [37]. Electrospun beads on strings of uorodecyl POSS [185]. d. Mesh geometry dip-coated in uorodecyl POSS/Tecno on [36]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.4 a. Texture without any reentrant curvature: Ψ > 90 o and θ Y < Ψ. b. Texture with a reentrant curvature: Ψ < 90 o and θ Y > Ψ [185]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.5 Liquid drop sliding down an inclined surface. . . . . . . . . . . . . . . . . . . . . . . . . . D.6 Contact angle hysteresis ∆θ ( o ) as a function of the air-liquid interfacial tension γ al (mN/m) [199]. The red squares, the orange rounds and the green triangles correspond to three di erent liquidinfused surfaces ( uorinated lubricant and te on/epoxy material). The blue diamonds refer to measurements on omniphobic surfaces in [185]. . . . . . . . . . . . . . . . . . . . . . . . . D.7 a. Immiscible droplet oating on the lubricant layer covering the whole texture (complete wetting). b. Immiscible droplet oating on the liquid-infused surface. c. Sinking of the droplet towards the bottom of the cavities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviii LIST OF FIGURES List of Tables 1 Aqueous foams and their applications in environment and energy 1.1 Typical air-liquid interfacial tensions γ al for some common liquids. . . . . . . . . . . . . . . 2 Oil-foam interactions and stability 2.1 Oil and foaming solutions used for the characterization of the stability of oil-laden foams. The air-water γ aw , air-oil γ ao and γ ow interfacial tensions (in mN/m) are given, as well as the entry E, spreading S and bridging B coe cient (in mN/m) de ned in §2.1.1. We also indicate the oil viscosity η o (mPa.s) and the oil density ρ o . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Plateau borders and oil: equilibrium shape and coalescence 3.1 Characteristics of the di erent oils used to ll the Plateau border. The interfacial tensions γ ao (air-oil), γ ow (air-water)(mN/m), the oil viscosity η o (mPa.s), the oil density ρ o and the coe cients E, S and B (mN/m) are given. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Soap lms and oil: dynamics of rising oil droplets spreading at the air-water interface 4.1 Interfacial tensions γ aw , γ ao and γ ow (mN/m) with calculated entry E and spreading coe cients S o of the oils and aqueous solutions. The dynamic viscosity η o (mPa.s) and the density ρ o (for each oil), and the surfactant concentration C SDS (for each aqueous solution) are given. . . . . . xxx LIST OF TABLES 5.1 Characteristics of the imbibing liquids and foaming solution. γ ow (mN/m) is the oil-water interfacial tension, η (mPa.s) the dynamic viscosity, ρ (g/cm 3 ) the density, R b (mm) the average bubble radius of the foam and B the Bond number de ned in (5.34). . . . . . . . . . . . . . . . . . .

Figure 2 :

 2 Figure 2: a. Child with a soap bubble by Rembrandt (1606-1669). b. Soap bubbles by Edouard Manet (1832-1883).

Figure 3 :

 3 Figure 3: For the better: a. Cappucino. b. Beer foam. For the worse: c. foam-induced river pollution.

Figure 4 :

 4 Figure 4: Liquid-infused surface in a micrometer channel. Surfactants and foams are injected to drag oil out of the patterned surface.

Figure 1 . 1 :

 11 Figure 1.1: Formation of a liquid lm inside a frame.

Figure 1 .

 1 Figure 1.2: a. A gerris "walking on water". b. A needle oating at the surface of a water bath.

From Fig. 1 . 4 ,Figure 1 . 3 :

 1413 Figure 1.3: Surface minimization of the interfacial area of a given volume of gas comprised inside a cubical bubble.It evolves towards a sphere, which is the shape corresponding to the minimization of the interfacial energy, as shown by the di erent values of the dimensionless interfacial area Σ d . Numerical simulations by the software Surface Evolver[START_REF] Brakke | The Surface Evolver[END_REF][START_REF] Brakke | The Surface Evolver and the stability of liquid surfaces[END_REF]. The dimensionless interfacial area Σ d is indicated in every image and decreases from the cube of unit length 1 to the sphere.

. 1 )Figure 1 . 4 :

 114 Figure 1.4: Variations of the drop radius R d for the calculations of the Laplace pressure.

Figure 1 . 6 :

 16 Figure 1.6: Chemical composition and structure of common surfactants. The red area is the polar hydrophilic head and the grey area is the hydrophobic tail.

. 4 )CHAPTER 1 :

 41 Di erentiating G S with the rst thermodynamical identity also gives dG S = ΩdP -Sd T + (∂G S /∂n S )dn S + (∂G S /∂Σ)dΣ = ΩdP -Sd T + µdn S + γdΣ. The di erent physical parameters are the pressure P , the temperature T , the volume Ω, the entropy S, the number of molecules n S , the chemical potential µ, the interfacial area Σ and the interfacial tension γ. Equating this expression with Eq. 1.4 yields ΩdP -Sd T -n s dµ-Σdγ = 0. Constant pressure and temperature give: AQUEOUS FOAMS AND THEIR APPLICATIONS IN ENVIRONMENT AND ENERGY dγ = -Γdµ , (1.5)

Fig. 1 .

 1 7 summarizes this dependence of γ on c. In this gure, we present the measurements of the air-water interfacial tension γ aw for some of the surfactants used in our work (SDS for Sodium Dodecyl Sulfate and CAPB-SLES-Mac for CocoAmidoPropylBetaine-Sodium Laureth Sulfate-Myristic Acid).

Figure 1 . 7 :

 17 Figure 1.7: Typical evolution of the air-water interfacial tension γ aw with the bulk surfactant concentration c for SDS and CAPB-SLES-Mac solutions.

Fig. 1 .

 1 Fig. 1.8 depicts a soap lm with the adsorption of surfactants at the air-water interfaces. When the polar head of the surfactant is charged, this geometry induces two charged monolayers that face each other with the polar heads oriented towards the aqueous phase. The Van der Waals attractions are compensated by the electrostatic repulsions induced by the charged heads. The balance between these forces de nes an equilibrium thickness e. The repulsion between both interfaces is materialized by the disjoining pressure Π [4, 12].

Figure 1 . 8 :

 18 Figure 1.8: Soap lm with surfactants at the air-water interfaces.

CHAPTER 1 :Figure 1 . 9 :

 119 Figure 1.9: Liquid lm pulled from a surfactant solution.

Figure 1 .

 1 Figure 1.10: a. Interface under compression -dilatation. b. Interface under shear.

Figure 1 . 11 :

 111 Figure 1.11: Deformation of a soap lm under dilatation: "healing" process by air-water interfacial tension gradient from γ - aw to γ + aw .

Figure 1 .

 1 Figure 1.12: a. Measurements of the air-water dilatational modulus |E * | for two types of surfactants: CAPB-SLES-Mac and SDS. b. Measurements of the oil-water dilatational modulus |E * | for two types of surfactants: CAPB-SLES-Mac and SDS. c. Typical variations of air-water interfacial tension γ aw d. Typical variations of interfacial area with respect to time t. Curves are given for CAPB-SLES-Mac with the following parameters: γ aw = 23.7 mN/m (c = 10 g/L) at equilibrium, initial drop volume V d = 2 µL, oscillation amplitude δV d = 0.2 µL and frequency f = 0.2 Hz.

Figure 1 .

 1 Figure 1.13: a. Shrinking of a lm by the motion of regions of smaller thickness. b. Marginal regeneration in a lm. The black lms (regions of smaller thickness) generate a recirculation of water inside the lm. Below: Top view of a lm close to the frame.

1. 2 27 1. 1 . 5 . 2

 227152 AQUEOUS FOAMS: DEFINITION, STRUCTURE, CHARACTERISTICS AND MAIN PROPERTIES How lms break...

Figure 1 . 14 :

 114 Figure 1.14: Opening dynamics of a lm.

1. 2 . 1 . 1

 211 In uence of the liquid fraction φ l on the structure and Plateau laws According to the liquid fraction, a foam adopts di erent structures: -If φ l > φ * l = 0.26 or 0.36, the bubbles are spherical with no contact. The foam appears as a suspension of bubbles. -If φ * l > φ l > φ * * l , the bubbles touch each other and get packed together by deforming their interfaces. The foam is wet. -If φ l < φ * * l , the bubbles are closely packed together and their geometry is polyhedral. The liquid phase is negligible and the foam is dry. The transition value φ * l depends on the bubble assembly. This value is 0.26 for an ordered closed packing of bubbles and 0.36 for a random closed packing. The value of φ * * l = 0.05 for dry foams is an arbitrary value, it generally varies from 0.01 to 0.05. Fig. 1.15 presents these three con gurations.

Figure 1 . 15 :

 115 Figure 1.15: Foam topology: dry foam (a), wet foam (b) and suspension of bubbles (c).

1. 2

 2 AQUEOUS FOAMS: DEFINITION, STRUCTURE, CHARACTERISTICS AND MAIN PROPERTIES 29vertices or nodes.

Figure 1 . 16 :

 116 Figure 1.16: Plateau borders (red), lms (green), vertices (blue) and the Plateau laws.

CHAPTER 1 :Figure 1 . 17 :

 1117 Figure 1.17: Plateau border and its curvature radii R 1 and R 2 for a dry foam.

Figure 1 . 18 :

 118 Figure 1.18: Kelvin cell and its typical geometrical features.

Figure 1 . 19 :

 119 Figure 1.19: Rheological laws of complex and simple uids.

  ) with k p the constant plastic viscosity and n a power coe cient. Foams are shear-thinning yieldstress uids since n < 1. The typical variations of the shear stress with respect to deformation and shear rate (rheogram -ow curve) are given in Fig. 1.20.

Figure 1 .

 1 Figure1.20: a. Variation of the shear stress τ xy with respect to the deformation for a foam of witconate-PEO in a plate-plate geometry[START_REF] Khan | Foam rheology: III. measurement of shear ow properties[END_REF]: φ l = 0.03, ˙ = 0.042 s -1 . b. Flow curve τ ( ˙ ) for a foam of commercial surfactant in a Couette geometry[START_REF] Herzhaft | Measurement and modeling of the ow behavior of aqueous foams using a recirculating pipe rheometer[END_REF]. The liquid fractions are given in the legend. The yield stress tends to disappear for high liquid fractions above 0.5 for which the foam looks more like a suspension of bubbles. The dotted lines are the Herschel-Buckley t with the power law coe cient n varying from 1 for φ l = 0.4 to 0.66 and 0.5 for φ l ≈ 0.2.

Figure 1 . 21 :

 121 Figure 1.21: Apparent viscosity η app as a function of the applied shear stress τ xy in a creep experiment for a CAPB-SLES-Mac foam with a 60 µm bubble radius (see Chapter 6). The equilibrium yield stress τ Y is indicated by the red arrow.

  Figure 1.22: T1 event in 2D with the strain and Y the yield strain. a. Initial state. b. Shear state where u is the displacement at the top of a bubble. c. Unstable state. d. Final state.

Figure 1 . 23 :

 123 Figure 1.23: Dimensionless ow curves for di erent wet foams (φ l = 0.1) generated from surfactant solutions of SLES-CAPB (Mac is myristic acid and Lac is lauristic acid) with mobile (without fatty acids) and rigid interfaces (with fatty acids Mac and Lac) [81].

1. 3 . 1 . 1

 311 Around us...

  Some of these aforementioned examples are illustrated in Fig.1.24.

Figure 1 . 24 :

 124 Figure 1.24: Examples of industrial applications.

Figure 1 . 25 :

 125 Figure 1.25: Map of the degraded sites worldwide (Source: UNEP).

Figure 1 . 26 :

 126 Figure 1.26: Physical method for soil remediation: soil ventilation.

Figure 1 . 27 :

 127 Figure 1.27: Physical methods for soil remediation: a. Pollution con nement. b. Pollution solidi cation.

Figure 1 . 28 :

 128 Figure 1.28: Remediation by chemical reactions.

Figure 2 .

 2 Figure 2.1: a. Emergence of oil at the air-water interface. b. Spreading of oil at the air-water interface.

Figure 2 . 2 :

 22 Figure 2.2: Spreading uid entrainment mechanism. The red arrows indicate the oil spreading while the green arrows designate the outward ow of water inside the lm.

Figure 2 . 3 :

 23 Figure 2.3: Geometry of the oil drop during bridge formation.

Figure 2 .

 2 Figure 2.4: a. Unstable bridge: α ow < 90 o . b. Stable bridge: α ow > 90 o .

Figure 2 .

 2 Figure 2.5: a. Bridging-dewetting mechanism. b. Bridging-stretching mechanism (black and white snapshots showing experimental observations [56]). c. Stable bridging.

Figure 2 . 6 :

 26 Figure 2.6: Stages of lm thinning with oil droplets expelled to the Plateau border.

Figure 2 . 7 :

 27 Figure 2.7: Spreading-wave generation.

Figure 2 . 8 :

 28 Figure 2.8: Pseudoemulsion lm at the frontier between the oil phase and the air phase

Figure 2 . 9 :

 29 Figure 2.9: Disjoining pressure Π (Pa) as a function of lm thickness e (nm) for SDS (concentration = 0.1 M) and dodecane [15].

Figure 2 .

 2 Figure 2.10: a. "Foamulsion" with rapeseed oil droplets -oil fraction: 70 % [161]. b. "Foamulsion" with octane droplets [103].

Figure 2 .

 2 Figure 2.11: a. Experimental set-up for the stability experiments. b. Typical response of the foam to oil invasion for three di erent oil-foaming solution combinations: sun ower oil -CAPB+SLES+Mac, silicon oil -CAPB+SLES+Mac and sun ower oil -saponin (from top to bottom). Oil appears white due to a uorescent agent (Yellow Black from Rohm and Haas).

Figure 2 .

 2 Figure 2.12: a. (S, E) diagram for the tested oils. b. (B, E) diagram for the tested oils. Two points correspond to miscible liquids: CAPB+SLES+Mac+Glycerol with η w = 10 mPa.s and η w = 100 mPa.s. The symbol * designates the surfactant CAPB-SLES-Mac.

Figure 2 .

 2 Figure 2.13: Experimental set-up for the forced drainage experiment.

Figure 2 .

 2 Figure 2.14: a. Invasive regime for Q = 5 mL/h (olive oil). b. "Fracture" regime for Q = 20 mL/h (sun ower oil).

Figure 2 . 15 :

 215 Figure 2.15: Draining front h as a function of time t for the forced drainage of olive oil (OO) and sun ower oil (SO). The injection ow rate Q (mL/h) and the initial liquid fraction φ i are indicated in the legend.

Figure 2 .

 2 Figure 2.16: Flow morphology of a simple uid ow (air) in a frictional uid or granular mixture in a (log φ -1 , log Q) diagram where φ is the solid fraction and Q the injection ow rate. The gure is extracted from [162].

Figure 2 . 17 :

 217 Figure 2.17: Experimental observations of air injection into a con ned granular suspension. a. Percolation regime at low ow rate. b. "Fracture" regime at high ow rate. The pictures are extracted from [186].

Figure 3 . 1 :

 31 Figure 3.1: Con gurations of the oil droplet in the Plateau border for di erent droplet sizes r * , where r * c = 2/ √ 3-1 is the critical dimensionless radius for which the oil droplet rst touches the walls of the Plateau border. In the far right image, the dotted lines indicate that oil can also swell the Plateau border and invert the curvature.

Figure 3 . 2 :

 32 Figure 3.2: Oil droplets in Plateau borders [132]. a. Side-view. b. Cross-section. From top to bottom: K = 1, 0.5, 0.2 and 0.1; the oil-water interfacial tension γ ow decreases. From left to right: r * = 0.2, 0.4, 0.7 and 1; the volume of oil increases.

Figure 3 . 3 :

 33 Figure 3.3: Dimensionless pressure di erence between the pseudoemulsion lm and the Plateau border ∆p * f with respect to the dimensionless size of the oil droplet r * [132].

Figure 3 . 4 :

 34 Figure 3.4: Simulations by Surface Evolver. a. Inner con guration. b. Outer con guration.

Figure 3 . 5 :

 35 Figure 3.5: Energy calculations using simulations by Surface Evolver: total interfacial energy E I as a function of the volume ratio V o /V w . a. Sun ower oil (Sun) and olive oil (Olive)-CAPB+SLES+Mac. b. Silicon oil with η o = 100 mPa.s (Si 100) and silicon oil with η o = 10 mPa.s (Si 10)-CAPB+SLES+Mac. c. Silicon oil with η o = 10 mPa.s (Si 10) and sun ower oil (Sun)-CAPB+SLES+Mac. d. Silicon oil with η o = 10 mPa.s (Si 10) and hexadecane (Hex)-CAPB+SLES.

Figure 3 . 6 :

 36 Figure 3.6: Mechanism of the Rayleigh-Plateau instability. From top to bottom: cylindrical thread, wave perturbation, breaking into small droplets.

  demonstrated in Appendix B. Fig.3.7 shows the evolution of σ RP t c as a function of kr c . There is a maximum growth rate for k max r c ≈ 0.7. This means that the wavelength of maximum growth is λ max = (2π/0.7)r c ≈ 9r c . Donnelly and Glaberson[START_REF] Donnelly | Experiments on the capillary instability of a liquid jet[END_REF] have experimentally measured the growth rate and con rmed the result of Eq. 3.4.

Figure 3 . 7 :

 37 Figure 3.7: Growth rate σ RP t c as a function of kr c in the inertial regime. The maximum growth rate is obtained for k max r c ≈ 0.7.

165 Table 3 . 1 :

 16531 3.1.2 with the air-water interfacial tension γ aw = 23.7 mN/m. Liquids γ ao γ ow η o ρ o Characteristics of the di erent oils used to ll the Plateau border. The interfacial tensions γ ao (air-oil), γ ow (air-water)(mN/m), the oil viscosity η o (mPa.s), the oil density ρ o and the coe cients E, S and B (mN/m) are given.

Figure 3 . 9 :

 39 Figure 3.9: Experimental set-up presenting the slender Plateau border lled with oil at the jonction between three soap lms.

Figure 3 . 10 :

 310 Figure 3.10: Snapshots of the breaking dynamics. Oil: olive oil; drop volume: Ω = 1.5 µL.

Figure 3 . 11 :

 311 Figure 3.11: Side-view of the breaking process. a. Initial oil-lled Plateau border [137]. b. Breaking of the lower rear lm. The Plateau border is ejected to the opposite direction. c. Oil-lled lm before the Rayleigh-Plateau instability.

Figure 3 . 14 :

 314 Figure 3.14: Theoretical radius of the oil droplet R d as a function of (λ max R 2

CHAPTER

Figure 4 . 1 :

 41 Figure 4.1: Liquid droplet on a solid surface in partial wetting.

Figure 4 . 2 :

 42 Figure 4.2: Energy of the lm E S as a function of the thickness e for pseudo-partial wetting.

Figure 4 . 3 :

 43 Figure 4.3: Spreading droplet and its precursor lm.

Figure 4 . 4 :

 44 Figure 4.4: Spreading regimes for a liquid droplet on a solid hydrophilic surface for PDMS with η l = 1 Pa.s. From top to bottom, Ω d = 37.9, 5.8, 4.03, 1.35, 0.35 µL. The data are extracted from [31].

Fig. 4 .Figure 4 . 5 :

 445 Figure 4.5: Topology of an oil lens spreading on a water surface in the limit where the gravity e ects are negligible, i.e. the shape of the drops is only dictated by interfacial tension.

Figure 4 . 6 :

 46 Figure 4.6: Oil-water interfacial tension γ ow as a function of time t for the silicon oil (η o = 100 mPa.s) -SDS system. The legend indicates the bulk concentration of surfactants.

Figure 4 . 7 :

 47 Figure 4.7: Oil lens spreading at the air-water interface.

Figure 4 . 8 :

 48 Figure 4.8: Spreading radius r as a function of time t for oil spreading on water-surfactants solutions [13]. Oil: PDMS (Polydimethylsiloxane η o = 100 mPa.s). Aqueous bath: C 16 TAB, Water, AOT (Dioctyl sulfosuccinate sodium) and C 10 E 5 . The black lines are the t from Eq. 4.19.

Figure 4 . 9 :

 49 Figure 4.9: Viscous oil lens spreading at the surface of a water bath.

  ao and γ ow (mN/m) with calculated entry E and spreading coe cients S o of the oils and aqueous solutions. The dynamic viscosity η o (mPa.s) and the density ρ o (for each oil), and the surfactant concentration C SDS (for each aqueous solution) are given.

Figure 4 . 10 :

 410 Figure 4.10: Experimental set-up. The oil droplet rises from the immersed needle and spreads at the bath surface.

Fig. 4 .

 4 11 summarizes these three possible con gurations.

Figure 4 . 11 :

 411 Figure 4.11: Con gurations of an oil droplet at the air-water interface according to the values of the entry E and spreading S coe cient.

4. 3 109 Figure 4 . 12 :

 3109412 Figure 4.12: Snapshots of the spreading dynamics. a. No-surfactant case (aqueous solution = water and oil = silicon oil, η o = 5 mPa.s); R d = 3.72 mm and Oh = 73.3. b. Surfactant case in the short-time regime from t = 0 s to 0.4 s (aqueous solution = SDS solution, C SDS = 2 g/L and oil = silicon oil, η o = 20 mPa.s); R d = 2.84 mm and Oh = 7.7. c. Surfactant case in the long-time regime from t = 0.66 s to 60 s (aqueous solution = SDS solution -C SDS = 2 g/L and oil = silicon oil, η o = 20 mPa.s); R d = 3.76 mm and Oh = 8.9. The thick black line is the injection needle.

Figure 4 . 13 :

 413 Figure 4.13: Dimensionless spreading radius R as a function of the dimensionless time T Oh . The open dots correspond to a bath solution without surfactants and the close dots with surfactants. The type of oilη o (mPa.s) -C SDS (per liter), Bo and Oh are indicated in this order in the legend. A corresponds to an attached droplet. The dashed line is Eq. 4.27. The black arrow in the snapshot indicates the direction of the oil movement.

. 29 )

 29 with f (Oh) = Oh 2 / -1 + √ 1 + 4Oh 2 .

Figure 4 . 14 :

 414 Figure 4.14: Dimensionless spreading radius R as a function of the dimensionless time T Oh for the spreading of para n oil on water. Dimensionless numbers: Oh = 13, Bo = 2.7 and Oh ef f = 3.6. The transition time and radius are T Oh-ST -LT ≈ Oh 2 ef f /f (Oh) 3 ≈ 3.8 and R ST -LT ≈ Oh ef f /f (Oh) 3/2 ≈ 2.

Figure 4 . 15 :

 415 Figure 4.15: Dimensionless spreading radius R as a function of the dimensionless time T * . The type of oilη o (mPa.s -C SDS (per liter), Bo, Oh and R g are indicated in this order in the legend. L in Si100 2g L is for a droplet attached to the needle for 1 h in the aqueous bath before spreading (in case surfactant dynamics matters). The dashed line is a guide line for the power law T * 1/8 .

Figure 4 . 16 :

 416 Figure 4.16: Phase diagram (Bo, R) for a given Oh in the surfactant case. The transitions for di erent values of Oh between the short-time and the long-time regimes are plotted in the red dashed line (for Oh >> 1)and in the orange dashed line (for Oh << 1). We have f (Oh) = Oh 2 / -1 + √ 1 + 4Oh 2 . The curve R = 1/Bo 1/2 de nes the transition between capillary and gravity-dominated regime (capillary length). The curves R = f (Oh) 1/6 Bo 1/6 and R = f (Oh) 1/8 correspond to the transitions between the short-time and the long-time dynamics.

Figure 4 .

 4 Figure 4.17: a. Schematic representation of the shear stress contributions at the air-oil interface τ air/oil and at the oil-water interface τ oil/water when the oil droplet initially spreads. b. Side-view of a typical shorttime spreading dynamics (para n oil with surfactants) highlighting an oscillatory and a spring-like behaviour.

. 1 )

 1 If I S > 0, capillary rise occurs in the tube. If I S < 0, liquid goes down the tube. The criterion I S > 0 is equivalent to the contact angle θ Y < 90 o since I S = γ al cos θ Y by the Young-Dupré relationship (with γ al the air-liquid interfacial tension).

Figure 5 . 1 :

 51 Figure 5.1: Liquid rising in a capillary tube.

5. 1

 1 .1.3 1D imbibition through an undeformable solid porous medium of variable pore size This con guration is typically found for imbibition into open corners and imbibition into open wedges between two thin plates of height-dependent spacing, as sketched in Fig. 5.2.

Figure 5 .

 5 Figure 5.2: a. Imbibition in an open corner. b. Imbibition in an open wedge.

Figure 5 .

 5 Figure 5.3: a. Di usive Lucas-Washburn's regime. b. Wedge regime. c. Regime in a diverging channel.

Figure 5 . 5 :

 55 Figure 5.5: Porous model: network of cylindrical pores randomly oriented by the angle β.

Figure 5 . 6 :

 56 Figure 5.6: Coe cient K c as a function of the Boussinesq number Bq for di erent velocity pro les [42].

134 CHAPTER 5 :

 1345 R b and τ = (δ a /150)t/(η w R b /γ aw ), and introduce the Bond number Bo = ρ w gR 2 b /γ aw , transforming Eq. OIL IMBIBITION INTO AQUEOUS FOAMS 5.32 into:

Figure 5 .

 5 Figure 5.7: a. Typical numerical solution of Eq. 5.34 for the dimensionless Plateau border area α for Bo = 0 and τ = 1000. b-d. Numerical solutions of the imbibition front for di erent times τ = 10, 50, 100 and 500 with Bond numbers Bo = 0, 1.6 and 5.

Figure 5 .

 5 Figure 5.8: a. Cross-section of a Plateau border for a dry aqueous foam. b. Cross-section of a Plateau border lled with oil for a dry aqueous foam. S P B and S f , respectively, denote the areas of the side surfaces of the Plateau borders and the surface of the lms that are connected to the Plateau borders. S o represents the area of the oil-water interfaces.

Figure 5 . 9 :

 59 Figure 5.9: Experimental set-up. A 30 cm-long PTFE tube (2 mm-diameter) is lled with the imbibing liquid (mixtures of foaming solution-glycerol or oil) and terminated by a 1 cm-long capillary tube (1 mm diameter) that enables the connection with the foam. A motion controller allows maintainence of the pressure of the liquid phase input at atmospheric pressure p atm .

Figure 5 . 10 :

 510 Figure 5.10: Snapshots of the imbibition front for a 30-second timescale. The imbibing liquid (glycerol-foaming solution) is uorescent and the front is de ned as the boundary between the brighter and darker uorescent areas. The liquid-lled tube is displaced upwards by a millimeter in the rst snapshot to ensure contact between the source and the foam.

Figure 5 . 11 :

 511 Figure 5.11: Evolution of the vertical front position z f with respect to time t for two data sets. The round dots correspond to a glycerol-foaming solution and the square dots to olive oil. The bubble radius R b , the initial liquid fraction φ i , and the viscosity η o/w of the imbibing oil or aqueous solution are indicated, respectively, in the legend.

5. 3 145 Figure 5 .

 31455 Figure 5.13: a. Experimental set-up. b. Geometrical elements of foams: vertices, Plateau borders and foam lms. c. Slender Plateau border with a radius of curvature r P B . d. Capillary rise of oil in a dry aqueous foam. The olive oil drop appears white thanks to a uorescent dye and invades the network of Plateau borders. The time interval between each frame is 67 s. The scale bar is 3 mm.

Figure 5 . 14 :

 514 Figure 5.14: Height of the rising front for olive oil as a function of time for di erent initial liquid fractions and one mean bubble radius R b = 1.8 mm. Inset: log-log scale of the data corresponding to φ i = 0.7 x 10 -4 .The plain line corresponds to the t 1/2 dynamics detailed in[START_REF] Caps | Capillary rise in foams under microgravity[END_REF], while the dashed line illustrates the t 1/4 dynamics.

Fig. 5 .

 5 Fig. 5.15 plots α as a function of Z for Bo = 0.4 and di erent successive times. By nding the value of Z for which α = 0, we deduce the imbibition front Z f = h/R b as a function of time t.

. 51 )Figure 5 . 15 :

 51515 Figure 5.15: Drainage solution.

5. 3 151 Figure 5 . 18 :

 3151518 Figure 5.18: Typical evolution of the uorescence intensity I of a single node during sun ower oil imbibition as a function of the node's vertical coordinate z. Initially, the node is located at x = 1 mm and z = 7.9 mm . The intensity of uorescence is in arbitrary units and the di erent curves correspond to uorescence intensity measurements taken at di erent times. The black arrow indicates the evolution of time. Inset: Width of the peak of intensity σ I (de ned at half of the maximum intensity) as a function of time t. The red arrow highlights the time at which the oil front has reached the node.

Figure 5 . 19 :

 519 Figure 5.19: Oil imbibing a Plateau border.

Figure 5 . 20 : 15 √ 3 t

 520153 Figure 5.20: Normalized imbibition front α * h as a function of normalized time t * in log-log plot. In the legend, AS1, AS10 and AS100 respectively stand for Aqueous Solution with η l =1.4,10 and 100 mPa.s, while OO and SO stands for Olive Oil and Sun ower Oil. The black line corresponds to (1+α * h) 4 + 4 1+α * h -5 = b 2δa 15 √ 3 t * with b ∼ 0.3, while the two dashed lines represent the power laws: t 1/2 at early times and t 1/4 at late times.

Figure 5 . 21 :

 521 Figure 5.21: Experimental set-up for the oil recovery by a moving foam. The foam is rst generated through a column. The upper part of the column is removable and ipped at 90 o onto a syringe pump with a piston at one end. Then, the oil reservoir is connected to the reversed column at the atmospheric pressure p atm before starting the injection of foam on top of the extraction point.

Figure 5 . 22 :

 522 Figure 5.22: Imbibition pro les for immiscible liquids (olive oil) for successive times t. a. Initial liquid fraction φ i = 3.4 x 10 -3 , injection ow rate Q = 100 mL/h and bubble radius R b = 1 mm. b. φ i = 2.7 x 10 -3 and Q = 200 mL/h. c. φ i = 2.4 x 10 -3 and Q = 400 mL/h. The red arrow indicates the extraction point and the green arrow the movement of the foam.

Figure 5 . 23 :

 523 Figure 5.23: Imbibition pro les for miscible liquids (CAPB+SLES+Mac-glycerol) for successive times t. a. Initial liquid fraction φ i = 2.6 x 10 -3 , injection ow rate Q = 200 mL/h and bubble radius R b = 1 mm. b. φ i = 1.9 x 10 -3 and Q = 200 mL/h. c. φ i = 1.7 x 10 -3 and Q = 400 mL/h. The red arrow indicates the extraction point and the green arrow the movement of the foam.

Fig. 5 .

 5 Fig. 5.24 also shows the mean ow rate of absorbed oil Q o = V o /t r as a function of the foam velocity U f = Q/πr 2c (r c is the column radius) for an experimental time frame between the touching time t = 0 and the nal running time t r = 4 min and 30 sec. What is surprising is that the volume decreases when the ow rate increases. Indeed, we could have imagined that bringing "fresh" foam would have brought more fresh connections between the extracting point and the foam, thus enhancing oil imbibition.

Figure 5 . 24 :

 524 Figure 5.24: Mean extraction ow rate Q o , for an injection time of 4 min and 30 s, as a function of the foam velocity U f .

Figure 5 . 25 :

 525 Figure 5.25: Dimensionless extraction ow rate Q * as a function of the dimensionless foam velocity U * for the imbibition of olive oil with di erent initial liquid fractions φ i in the foam. The orange line is the t with Eq. 5.62 and v o (t) ∼ t 3/2 . The blue and the purple dashed lines are, respectively, the ts with v o (t) ∼ t 1/3 and ∼ t 1/2 deduced from the numerical simulation from §5.2.

Figure 5 .

 5 Figure 5.26: Log-log plot of the dimensionless imbibed volume V * = R f 0

164CHAPTER 6 :

 6 FOAM -OIL INTERACTIONS IN MICROPATTERNED STRUCTURES Contents 6.1 Textured and liquid-infused surfaces . . . . . . . . . . . . . . . . . . . . . 167 6.1.1 Wetting of rough surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.1.

168CHAPTER 6 :

 6 FOAM -OIL INTERACTIONS IN MICROPATTERNED STRUCTURES cos θ * = r S cos θ Y . (6.2) For a rough surface (r > 1), hydrophobicity and hydrophilicity are enhanced since cos θ * > cos θ Y and θ * < θ Y for θ Y < 90 o , and cos θ * < cos θ Y and θ * > θ Y for θ Y > 90 o . If r = 1 (smooth surface), one retrieves the Young contact angle. Eq. 6.2 shows that it is possible to reach a complete wetting or a non-wetting state for a speci c value of the roughness r * S . Indeed, this value is given by r * S = 1/| cos θ Y |. One nds r * S = 2 for a value of θ Y = 60 o . This result is in contradiction with the experimental observations from Fig. 6.1 since the apparent contact angles never reach 180 o and 0 0 .

Figure 6 .

 6 Figure 6.1: a. Wetting on a smooth surface with the Young contact angle θ Y . b. Wetting on a rough surface with the apparent contact angle θ * . c. Evolution of cos θ * with cos θ Y for a smooth and a rough surface of the same chemical nature [71]. The parameters φ S and r S are respectively the solid fraction and the roughness. The black plain line in the upper right part corresponds to Eq. 6.2.

Figure 6 . 2 :

 62 Figure 6.2: Liquid droplet sitting on a solid-liquid composite surface.

170CHAPTER 6 :Figure 6 . 3 :

 663 Figure 6.3: Impregnation of a lm in a textured surface. The solid fraction φ S and the projected area A P are indicated in a sub gure.

Figure 6 . 4 :

 64 Figure 6.4: Liquid droplet sitting on a solid-air composite surface.

Figure 6 . 5 :

 65 Figure 6.5: Transition from the regime of air cushion (1) to the Wenzel regime (2).

174CHAPTER 6 :

 6 FOAM -OIL INTERACTIONS IN MICROPATTERNED STRUCTURES

176CHAPTER 6 :

 6 FOAM -OIL INTERACTIONS IN MICROPATTERNED STRUCTURES

178CHAPTER 6 :

 6 FOAM -OIL INTERACTIONS IN MICROPATTERNED STRUCTURESthe aspect ratio to design the most resistant liquid-infused surface. Ideally, the cavities should be high and large to get the smallest aspect ratio as possible.

Figure 6 . 8 :

 68 Figure 6.8: Dimensionless retention length L = L/L ∞ as a function of the dimensionless time t = t/t c . The blue, the red and the gray curves correspond to experimental results of drainage of silicon oils by water, respectively for oil viscosity η o = 42.7 mPa.s and injection ow rate Q = 1 mL/min, η o = 201 mPa.s and Q = 2 mL/min and η o = 42.7 mPa.s and Q = 2 mL/min. The dotted lines are the theoretical predictions[START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF]. Both gures are extracted from[START_REF] Wexler | Shear-driven failure of liquid-infused surfaces[END_REF].

6. 2

 2 SURFACTANT-DRIVEN OIL EXTRACTION FROM MICROPATTERNED CHANNELS 179

Aqueous solution -Oil η w η o

 o CAPB+SLES+MAc+10% glycerol -Olive oil (O) 1.4 68.5 CAPB+SLES+MAc+10% glycerol -Silicon oil 1 (PDM-7050 -S1) 1.4 201 CAPB+SLES+MAc+10% glycerol -Silicon oil 2 (PDM-7040 -S2) 1.4 42.7 SDS -Olive oil 1 68.5

6. 2 181 Figure 6 . 10 :

 2181610 Figure 6.10: Equilibrium oil-water interfacial tension γ ow (mN/m) as a function of the surfactant concentration c (g/L) for SDS-Olive oil (O) and CAPB-SLES-Mac-Olive oil (O) combinations. The blue and the red dashed lines are guide lines.

182CHAPTER 6 :

 6 FOAM -OIL INTERACTIONS IN MICROPATTERNED STRUCTURES

6. 2 183 Figure 6 . 11 :

 2183611 Figure 6.11: Top views of typical oil extractions (olive oil) by a surfactant solution (CAPB -SLES -Mac -10 wt% glycerol). Fluorescent dyed oil appears green-yellow. The small green-yellow dots correspond to some oil droplets stuck on the epoxy surface when the channel was initially cleared of oil in the rst place. The ow is from left to right. a. Extraction with a uniform front; Q = 0.05 mL/min and γ ow = 4.2 mN/m, time interval: 5 h. b. Extraction with an unstable pattern; Q = 5 mL/min and γ ow = 4.2 mN/m, time interval: 5 min.

184CHAPTER 6 :Figure 6 . 12 :

 6612 Figure 6.12: Extraction e ciency α as a function of time t for di erent draining liquid -oil combinations with the following variable parameters: a. γ ow (mN/m). b. Q (mL/min) (Inset: α as a function of Qt). c. η o (mPa.s). d. H (µm). S and O are, respectively, the surfactant type, the oil type.

Figure 6 . 13 :

 613 Figure 6.13: Extraction e ciency α as a function of the injected volume Qt for SDS -olive oil combination, with the capillary number Ca ≈ 0.015 and 0.005 de ned in Eq. 6.19b.

186CHAPTER 6 :

 6 FOAM -OIL INTERACTIONS IN MICROPATTERNED STRUCTURESr min = δ + r 2 min -

Figure 6 . 14 :

 614 Figure 6.14: Confocal snapshot of the oil-water interface in the texture cavity. The geometrical parameters are indicated on the picture: δ is the de ection.

6. 2 187 Figure 6 .

 21876 Figure 6.15: a. Movement of the A-B interface parallel to the ow of mean velocity U with the viscosity ratio η B /η A > 1. b. Movement of the same A-B interface for our post pattern with the interface receding perpendicular to the main ow of mean velocity U .

Figure 6 . 16 :

 616 Figure 6.16: Measurements of the receding contact angle θ as a function of Ca for CAPB -olive oil (red dots) and CAPB -silicon oil 1 (blue dots) combinations. Oil appears red in the confocal microscope and the green line is the laser re ection between the glass and the aqueous phase. Lower right: Evolution of the numerical factor c d with respect to θ.

188CHAPTER 6 :

 6 FOAM -OIL INTERACTIONS IN MICROPATTERNED STRUCTURES

Figure 6 . 17

 617 Figure 6.17: a. Experimental e ciency α exp as a function of g(t) for CAPB -olive oil, CAPB -silicon oil 1 and CAPB

  Figure 6.17: a. Experimental e ciency α exp as a function of g(t) for CAPB -olive oil, CAPB -silicon oil 1 and CAPB -silicon oil 2 combinations. b. f (α exp ) as a function of g(t) for di erent surfactant-oil combinations. The black dashed line is the theoretical prediction (Eq. (6.19a))

Figure 6 . 18 :

 618 Figure 6.18: Unstable two-layer Couette ow with layers of di erent viscosities η w and η o and di erent thickness h and H -h.

190CHAPTER 6 :

 6 FOAM -OIL INTERACTIONS IN MICROPATTERNED STRUCTURES

Figure 6 .

 6 Figure 6.19: a. Typical unstable extraction pattern for posts and grooves with the same height and width. The surfactant solution -oil combination is SDS (2 g/L) -olive oil; Q = 2 mL/min and t = 60 min. Contrast is enhanced in the snapshots. b. Phase diagram where each point (γ ow , Q) represents an experiment with the CAPB -olive oil, the CAPB -silicon oil 1 and the SDS -olive oil combinations. Unstable extraction is represented by red dots and stable extraction by green dots. The dashed black line de nes the stable regime for pure aqueous solutions -olive oil (the silicon oil 1 case is further to the right of the diagram).

Figure 6 . 20 :

 620 Figure 6.20: Description of the micro uidic fabrication by photolithography. Colors are here to distinguish each component. They do not correspond to a physical reality.

192CHAPTER 6 :

 6 FOAM -OIL INTERACTIONS IN MICROPATTERNED STRUCTURES 6.3.1

Figure 6 .

 6 Figure 6.21: Flow-focusing device with the main geometrical features: w g = 200 µm, w l = 300 µm, w c = 60 µm, l c = 100 µm and w o = 500 µm. The height H of the channel is 27 µm or 72 µm.

Figure 6 . 22 :

 622 Figure 6.22: Bubble volume V b (nL) as a function of the ratio pressure-ow rate p/q (Psi/m 3 /s). Below: bubbles (radius R b ≈ 30 µm) accumulating in the outlet reservoir.

Figure 6 .

 6 Figure 6.23: a. Top-view of olive oil extraction by an aqueous foam with R b = 100 µm and Q = 50 µL/min.

6. 3 Figure 6 .

 36 Figure 6.24: E ciency α as a function of the injected volume Qt. a. Olive oil extraction by a foam with R b = 30 µm. b. Silicon oil 1 extraction by a foam with R b = 30 µm. c. Extraction for silicon oil 2 and R b = 30, 100 and 200 µm, and H = 179 µm. d. Comparison between surfactant solutions and foams with a varying channel height (89 or 178 µm). In the legend, O, Q (µL/min), R b (µm) and H (µm) are, respectively, the oil type, the injection ow rate, the mean bubble radius and the channel height.

196CHAPTER 6 :HDLo ηw ηo 1 c

 61 FOAM -OIL INTERACTIONS IN MICROPATTERNED STRUCTURESIn order to use Eqs. (6.24a -6.24b), we measure θ and use D as an adjustable parameter. Again, like for surfactant solutions, θ depends on Ca. More precisely, as shown in Fig.6.25a, we can collapse all the angle measurements from surfactants and foams on the same plot by introducing a new capillary number Ca = Ca(H/ ) where is the characteristic length for viscous dissipation. If = H as it is the case for surfactant solutions, then Ca = Ca. However, for foams, = D. Fig.6.25b-c compares α exp and f (α exp ) to G(t) = csh W d Qt. We are able to obtain good agreement between Eqs. (6.24a -6.24b) and our data sets.

Figure 6 .

 6 Figure 6.25: a. Measurements of the receding contact angle θ as a function of Ca. b. Experimental e ciency α exp as a function of G(t) = csh W HDLo ηw ηo 1 c d Qt for some oils, Q (µL/min) and R b (µm). c. f (α exp ) as a function of G(t) for the same parameters. f (α exp ) = G(t) is represented by the black dashed line.

6. 3

 3 FOAM-DRIVEN OIL EXTRACTION FROM MICROPATTERNED CHANNELS 197 and the dimensionless applied shear stress τ f oam * xy = τ f oam xy H/γ aw by non-dimensionalizing D and τ f oam

  xy

  .25a)τ f oam * xy = 4.63ξ( )Ca * 1/2 + 7.35χ( )Ca * 2/3 ,(6.25b)

198CHAPTER 6 :

 6 Figure 6.26: a. Dimensionless lm thickness D * = D/H as a function of the outer capillary number Ca * for all data sets. Inset: Film thickness D as a function of Ca * for the same data set. b. Dimensionless applied shear stress τ f oam * xy

200CHAPTER 6 :

 6 FOAM -OIL INTERACTIONS IN MICROPATTERNED STRUCTURES r p the typical pore radius, L the depth of the extraction well and k D the permeability. We can approximate k D by using the Carman-Kozeny model

≈ γ 2ρ l r 3 c k 2 r 2 c 1 -k 2 r 2 c

 312 c + (t) cos(kx)) 2 ≈ 2πr c cos(kx) d (t) dt . (B.5)Derivating Eq. B.4 and Eq. B.5 with respect to x and t, and equating them, gives the following time-dependence for (t):d 2 dt 2

  . B.8a and Eq. B.8b with Ψ and neglect the squares and products of the velocity components for small perturbations, thus Ψ is solution of:where D is the di erential operator D = ∂ 2 ∂r 2 -1 r ∂ ∂r + ∂ 2 ∂x 2, ρ 1 and η 1 are the density and the viscosity of one uid.One can decompose Ψ into two subfunctions Ψ * and Ψ * * for which the general solution is:Ψ = Ψ * + Ψ * * = (A 1 rI 1 (kr) + B 1 rK 1 (kr) + A 2 rI 1 (k 1 r) + B 2 rK 1 (k 1 r)) e i(σt+kx) , (B.12)

--

  Figure C.1: a. Fabrication of the micro uidic channel. b. Fabrication of the textured pattern.

Figure C. 2 :

 2 Figure C.2: a. PDMS mold for the main channel. b. Negative of the textured pattern on a silicon wafer. c. Micro uidic channel ready for experiments.

Fig. D. 2

 2 Fig. D.2 shows the transition of an air-liquid contact line moving in the vicinity of a solid defect.Before reaching the defect, the contact line makes the Young contact angle θ Y with the solid surface. While reaching the defect, the contact angle can take any value between θ Y and π -Ψ + θ Y , where Ψ is the opening angle of the defect. This phenomenon is called canthotaxia: the contact line is anchored at the defect and sweeps every value in the interval mentioned above.

Figure D. 2 :

 2 Figure D.2: Anchored air-liquid contact line around a defect on a solid surface.

Fig. D. 7

 7 Fig. D.7 shows two possible initial con gurations: with and without the lubricating lm covering the top of the texture. This latter is possible if the lubricant completely wets the solid surface. We introduce the solid-droplet γ s2 , the solid-lm γ s1 and the lm-droplet γ 12 interfacial tensions. From Fig. D.7a to the nal state (Fig. D.7c), one nds that the loss of interfacial energy is rγ s1 +γ 12 (respectively (r -φ S )γ s1 + (1 -φ S )γ 12 when we start from Fig. D.7b). The gain of interfacial energy is rγ s2 (respectively (r -φ s )γ s2 ). The variation of interfacial energy ∆E from the oating to the pinning con guration is written as:

  Characteristics of the aqueous solution -oil combinations. Here, η w (mPa.s) and η o (mPa.s) are, respectively, the shear viscosity of water and oil. . . . . . . . . . . . . . . . . . . . . . . . 6.2 Interfacial tensions γ aw , γ ao and γ ow (mN/m) with calculated entry E and spreading coe cients S of the oils and surfactant solutions. The dynamic viscosity η o (mPa.s), the density ρ

	2			MAIN PARAMETERS
	Measured parameters
	Variables Parameters	Variables Parameters
	γ al	Air-liquid interfacial tension	γ sl	Solid-liquid interfacial tension
	6 Foam -oil interactions in micropatterned structures γ sa Solid-air interfacial tension γ aw Air-water interfacial tension Main parameters γ ow Oil-water interfacial tension γ ao Air-oil interfacial tension γ Interfacial tension ρ Density ρ l Liquid density ρ o Oil density ρ w Water density η l Liquid shear viscosity η Shear viscosity θ Y Young contact angle η o Oil shear viscosity η w Water shear viscosity p atm Atmospheric pressure P b Bubble pressure φ l Liquid fraction φ o Oil fraction φ w Water fraction φ i Initial liquid fraction R b Bubble radius R d Drop radius r t Tube radius r c Cylinder radius g Gravity acceleration t Time x Horizontal coordinate y -z Vertical coordinate τ xy Shear stress Strain ˙ Strain rate Σ Interfacial area V Volume of liquid 6.1 D Properties of textured surfaces Main variables
	Variables Parameters	Variables Parameters
	E	Entry coe cient	S	Spreading coe cient
	B	Bridging coe cient	Π	Disjoining pressure
	Π o	Osmotic pressure	r P B	Curvature radius of the Plateau border
	p	Mechanical pressure	p o	Pressure in the oil phase
	p w	Pressure in the water phase

o (for each oil), and the surfactant concentration C (for each surfactant solution) are given. . . . . . . . . . A Examples of ow dynamics by resolution of Stokes' equation B Rayleigh-Plateau instability of a viscous uid in another viscous uid C Micro uidic fabrication: the "sticker technique" P B INTRODUCTION "The world's a bubble: and the life of Man less than a span. " The Life of Man, Sir Francis Bacon (1561-1626).

  Film and capillary force . . . . . . . . . . . . . . . . . . . . . . 14 1.1.2 Interface, drop and bubble . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.1.2.1 Drop and bubble formation . . . . . . . . . . . . . . . . . . . . 15 Applications in environment and energy . . . . . . . . . . . . . . . . . . . 1.3.1 Foams in everyday life . . . . . . . . . . . . . . . . . . . . . . . . . . . . What foams can improve... . . . . . . . . . . . . . . . . . . . . 1.3.3 Towards more energy supply: enhanced oil recovery . . . . . . . . . . .

	. . 31 Capillary suction and disjoining pressure . . . . . . . . . . . . 31 1.3.1.2 1.2.2.1 1.2.2.2 In industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Towards a cleaner environment: soil remediation . . . . . . . . . . . . . AND ENERGY 1.3.2.1 Common remediation techniques: advantages and drawbacks 1.3.2.2 CHAPTER 1: AQUEOUS FOAMS AND THEIR APPLICATIONS IN ENVIRONMENT AND ENERGY E C Cohesion energy per molecule e S Interfacial energy per molecule a Molecular size T Temperature L Film length m gerris Mass of the gerris ∆p L Laplace pressure Σ d Dimensionless interfacial area F Tension force δW Work R 1 First radius of curvature R 2 Second radius of curvature c Bulk concentration of surfactants Γ Interfacial concentration of surfac-tants G S Interfacial free enthalpy n S Number of surfactant molecules at the surface P Thermodynamical pressure Ω Volume S Entropy k B Boltzmann constant e Film thickness z max Maximal height γ cmc Interfacial tension at the cmc σ Normal stress σ xx Horizontal normal stress σ yy Vertical normal stress τ xy Shear stress Σ 0 Initial interfacial tension E d Interfacial dilatational elasticity η d Interfacial dilatational viscosity E G Gibbs-Marangoni elasticity G shear Interfacial shear elasticity η shear Interfacial shear viscosity γ 0 Air-pure water interfacial tension w Oscillatory pulsation f Frequency Q f Flow rate in the lm v Flow velocity in the lm r h Hole radius m rim Mass of the rim v rim Velocity of the rim Y Yield strain λ Structural constant η app Apparent viscosity τ Y Yield stress k p Plastic viscosity G Shear elastic modulus G Loss shear modulus G Quasistatic shear modulus Strain τ Shear stress τ Y Yield stress 1.1 THE PHYSICS OF INTERFACES AND SURFACTANTS Osmosis in foams . CHAPTER 1: AQUEOUS FOAMS AND THEIR APPLICATIONS IN ENVIRONMENT Around us... . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variables Parameters Variables Parameters 1.3 1.3.1.1 Speci c variables 1.1 The physics of interfaces and surfactants	11 13

1.1 The physics of interfaces and surfactants . . . . . . . . . . . . . . . . . . 13 1.1.1 Interfacial energy, tension and force . . . . . . . . . . . . . . . . . . . . 13 1.1.1.1 Interfacial tension . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.1.1.2 1.1.2.2 The Laplace law and pressure . . . . . . . . . . . . . . . . . . 15 1.1.2.3 Measuring the interfacial tension . . . . . . . . . . . . . . . . 17 1.1.3 Surfactants: who are you? . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.1.3.1 Structure and e ect on interfacial tension . . . . . . . . . . . . 18 1.1.3.2 Stabilizing role . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.1.4 E ect of surfactants on the interfacial mechanics . . . . . . . . . . . . . 22 1.1.4.1 Interfacial viscoelasticity . . . . . . . . . . . . . . . . . . . . . 22 1.1.4.2 Resistance of a soap lm . . . . . . . . . . . . . . . . . . . . . 23 1.1.4.3 Rigid and mobile interfaces . . . . . . . . . . . . . . . . . . . . 24 1.1.5 Soap lm: a fragile structure . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.1.5.1 How lms shrink... . . . . . . . . . . . . . . . . . . . . . . . . 25 1.1.5.2 How lms break... . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.2 Aqueous foams: de nition, structure, characteristics and main properties 27 1.2.1 Foam structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.2.1.1 In uence of the liquid fraction φ l on the structure and Plateau laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.2.1.2 Dry foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.2.2 Main properties of foams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.2.3 How foams ow... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.2.3.1 Foams are complex uids . . . . . . . . . . . . . . . . . . . . . 32 1.2.3.2 Relationship between the structure and the foam rheology . . 35 1.3.3.1 Processes of enhanced oil recovery . . . . . . . . . . . . . . . 1.3.3.2 Surfactants, foams and the quest of oil . . . . . . . . . . . . . 1.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . In the following, foams mean aqueous foams (water-based) if not speci ed.

Table 1 .

 1 1: Typical air-liquid interfacial tensions γ al for some common liquids.

	CHAPTER 1: AQUEOUS FOAMS AND THEIR APPLICATIONS IN ENVIRONMENT
						AND ENERGY
	Liquids	Helium ( T = 4 K) Silicone oil Ethanol Glycerol Water Mercury
	γ al (mN/m)	0.1	20	23	63	72 -73	485

Table 2 .

 2

1: Oil and foaming solutions used for the characterization of the stability of oil-laden foams. The air-water γ aw , air-oil γ ao and γ ow interfacial tensions (in mN/m) are given, as well as the entry E, spreading S and bridging B coe cient (in mN/m) de ned in §2.1.1. We also indicate the oil viscosity η o (mPa.s) and the oil density ρ o .

  3.1 Picture of oil invasion in Plateau borders by numerics . . . . . . . . . . . 72 3.1.1 Oil slug in a Plateau border . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.1.1.1 Shape of the oil phase . . . . . . . . . . . . . . . . . . . . . . . 72 3.1.1.2 Interactions between the oil phase and the pseudoemulsion lm 73 3.1.2 Interfacial energy comparison with Surface Evolver . . . . . . . . . . . . 75 3.1.2.1 Numerical con gurations . . . . . . . . . . . . . . . . . . . . . 75

	CHAPTER 3: PLATEAU BORDERS AND OIL: EQUILIBRIUM SHAPE AND
	COALESCENCE			71
	Speci c variables
	Variables Parameters	Variables Parameters
	K	Interfacial tension ratio	r o	Radius of oil droplet
	∆p * f	Dimensionless pressure di erence	r *	Dimensionless droplet radius
	r * c	Critical dimensionless radius	p * f	Dimensionless pressure in the pseudoemulsion lm
	p * w	Dimensionless pressure in the Plateau border	p f	Pressure in the pseudoemulsion lm
	r c	Cylinder radius	L c	Cylinder length
	S c	Cylinder area	S d	Droplet area
	r 1	In-plane curvature radius in a bump	r * 1	In-plane curvature radius in a hole
	r 2	Transverse curvature radius in a bump	r * 2	Transverse curvature radius in a hole
	p 1	In-plane pressure in a bump	p * 1	In-plane pressure in a hole
	p 2	Transverse pressure in a bump	p * 2	Transverse pressure in a hole
	t c	Characteristic time	k	Wave number
	σ RP	Growth rate of the Plateau-Rayleigh instability	η c	Cylinder viscosity
	γ *	Interfacial tension between the in-ner and the outer uid	η f	Viscosity of the surrounding uid
	k max	Wave number of maximum insta-bility	λ max	Wavelength of maximum instabil-ity
	L oil	Oil length	R oil	Radius of the oil slug
	N	Number of droplets for the break-ing of a water jet	N f oam	Number of droplets from the rup-ture of a Plateau border
	R f oam d	Droplet radius from the rupture of a Plateau border	Ω P B	Volume of a Plateau border
	h w	Humidity rate	P	Pressure	88
	Bo	Bond number

3.1.2.2 Comparison between numerics and experiments . . . . . . . . 76 3.2 Breaking dynamics in the oil-lled Plateau borders . . . . . . . . . . . . 77 3.2.1 The Rayleigh-Plateau instability . . . . . . . . . . . . . . . . . . . . . . . 78 3.2.1.1 Breaking observations . . . . . . . . . . . . . . . . . . . . . . 78 3.2.1.2 Inertial regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.2.1.3 Viscous regime . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.2.2 Breaking of an emulsi ed viscous cylinder . . . . . . . . . . . . . . . . . 81 3.2.2.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.2.2.2 Instability wavelength . . . . . . . . . . . . . . . . . . . . . . 84 3.2.2.3 Size of the oil droplets . . . . . . . . . . . . . . . . . . . . . . 86 3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  4: SOAP FILMS AND OIL: DYNAMICS OF RISING OIL DROPLETS SPREADING AT THE AIR-WATER INTERFACE 93

	CHAPTER 4: SOAP FILMS AND OIL: DYNAMICS OF RISING OIL DROPLETS
	94		SPREADING AT THE AIR-WATER INTERFACE
	Speci c variables
	Variables	Parameters		Variables Parameters
	S S	Spreading coe cient for liquid-solid spreading	U	Mean velocity
	a	Molecular size		H k	Hamaker constant
	θ Y	Young contact angle	dE	Variation of interfacial energy
	E S	Energy of a lm		E f ilm	Variation of the lm energy
	Ω	Film volume		e eq	Equilibrium thickness
	κ	Capillary length		γ c	Critical interfacial tension
	E f ilm	Film energy		e pp	Pseudo-partial thickness
	r	Spreading radius		F f ilm	Capillary force in the lm
	F droplet	Capillary force in the droplet	θ d	Dynamic contact angle
	F cap	Total capillary force	v	Horizontal velocity
	τ ow	Shear stress at the oil-water inter-face	v r	Radial velocity
	δ d	Penetration depth		v o	Velocity of the upper layer ow
	d	Characteristic length	θ aw	Air-water dynamic contact angle
	θ ow	Oil-water dynamic contact angle	θ eq	Equivalent contact angle
	γ eq	Equivalent interfacial tension	D	Viscous dissipation
	Re r	Reynolds number in the bulk ow	P	Capillarity-induced power
	C SDS	SDS concentration		Oh	Ohnesorge number
	∆p C	Capillary pressure di erence	R	Dimensionless spreading radius
	T	Dimensionless time (capillarity)	T Oh	E ective dimensionless time
	Bo	Bond number		Oh ef f	E ective Ohnesorge number
	T ST -LT	Transition time between short-time and long-time regime	R ST -LT	Transition radius between short-time and long-time regime
	T Oh-ST -LT E ective transition time	T *	Dimensionless time (gravity)
	τ g	Characteristic gravitational transition time inertial-viscous-	r g	Characteristic gravitational transition radius inertial-viscous-
	T * g	Dimensionless gravitational transition time inertial-viscous-	R g	Dimensionless gravitational radius inertial-viscous-
	T c	Dimensionless capillary time	inertial-viscous-	R c	Dimensionless capillary radius	inertial-viscous-
	Ω d	Droplet volume

  1/4 regime dominates if Ω d η o ( ṙ/r) 2 > δ d r 2 η w ( ṙ/δ d ) 2 e z/δ d . For this latter term, the dissipation length is the width of the boundary layer δ d and the volume is δ d r 2 . We have r δ d = r/ √ Re r with Re r

Table 4 .

 4

	Interfacial	Solution	Water	SDS -SDS -SDS -SDS -
	tension			0.5 g/L 1 g/L	2 g/L 10 g/L
	γ aw	Air	73	58.8	52	39	36.9
	Interfacial	Oil	Par 25.7	Si 5	Si 20 Si 100 Si 350
	tensions	Solution	0.838	0.95	0.97	0.965	0.97
	γ ao	Air	19.5	20.9	19.1 19.6 19.1
		Water	37.7	36.7	40.2 36.3 40.2
		SDS -0.5 g/L	21.6		19.8 21.6
	γ ow	SDS -1 g/L	16.7		14.6 18.8
		SDS -2 g/L	8.9		8.7	10.1	8.7
		SDS -10 g/L	7.3		6.2	8.4
		Water	91.2	88.8	94.1 89.7 94.1
			15.8	15.5	13.7 17.1 13.7
	E -S	SDS -0.5 g/L	60.9		59.5 60.8
			17.7		20	17.6
		SDS -1 g/L	49.1		47.5 51.1
			15.8		18.3 13.6
		SDS -2 g/L	28.4		28.6 29.5 28.6
			10.6		11.2	9.3	11.2
		SDS -10 g/L	24.6		24	25.7
			10.1		11.6	8.8

1: Interfacial tensions γ aw , γ

  We non-dimensionalize r and t as R = r/R d and T = t/(η o R d /γ ow ). We also introduce the Ohnesorge number Oh = √ ρ o γ ow R d /η o . Eq. (4.26) becomes:

	CHAPTER 4: SOAP FILMS AND OIL: DYNAMICS OF RISING OIL DROPLETS
	110	SPREADING AT THE AIR-WATER INTERFACE
	r(t) =	-η o + η 2 o + 4ρ o γ ow R d ρ o	1/2	t 1/2 .	(4.26)
	R =	-1 +	√ 1 + 4Oh 2 Oh 2	1/2	T 1/2 .	(4.27)

.

[START_REF] Busse | Change in drag, apparent slip and optimum air layer thickness for laminar ow over an idealised superhydrophobic surface[END_REF]

where η o and ρ o are, respectively, the oil dynamic viscosity and the oil density. By solving (4.25), we nd that:

Table 5 . 1 :

 51 Characteristics of the imbibing liquids and foaming solution. γ ow (mN/m) is the oil-water interfacial tension, η (mPa.s) the dynamic viscosity, ρ (g/cm 3 ) the density, R b (mm) the average bubble radius of the foam and B the Bond number de ned in(5.34).

				1.21	1 0.5
	MAc+80% gly
	CAPB+SLES	0	15	1.17	2 1.8
	MAc+65% gly
	CAPB+SLES	0	1.4 1.016 2 1.7
	MAc+10% gly
	Olive oil 1	6	61	0.88	1 1.5
	Olive oil 2	6.2 68.5 0.88	2	5
	Sun ower oil 4.5	55	0.91	2	3

  1.1 Wetting behaviour on rough surfaces: Wenzel model and hydrophilicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.1.1.2 Film impregnation . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.1.1.3 Hydrophobic textured surface . . . . . . . . . . . . . . . . . . 170 6.1.1.4 Generation of liquid-infused surfaces . . . . . . . . . . . . . . 171 6.1.2 Failure of textured surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 172 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 6.1.2.4 Shear-driven drainage of liquid-infused surfaces: analytical resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Surfactant-induced instability . . . . . . . . . . . . . . . . . . 189 6.3 Foam-driven oil extraction from micropatterned channels . . . . . . . . 190 6.3.1 Controlled foam generation . . . . . . . . . . . . . . . . . . . . . . . . . 191 CHAPTER 6: FOAM -OIL INTERACTIONS IN MICROPATTERNED STRUCTURES165 6.3.1.1 Flow-focusing . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.3.1.2 Foam calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 192 6.3.2 Experimental results and discussion . . . . . . . . . . . . . . . . . . . . 193 Dissipation length and lm thickness . . . . . . . . . . . . . . 196 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

	166CHAPTER 6: FOAM -OIL INTERACTIONS IN MICROPATTERNED STRUCTURES 6.1 TEXTURED AND LIQUID-INFUSED SURFACES	167
	Speci c variables 6.1 Textured and liquid-infused surfaces
	Variables Parameters		Variables Parameters
	θ * r S	6.3.2.1 6.3.2.2 Apparent contact angle Typical extraction pattern with foams . . . . . . . . . . . . . . 194 Variation of interfacial energy per dE unit length Shear modeling at the oil-water interface . . . . . . . . . . . . 195 Surface roughness φ S Solid fraction of projected area
	A P P d	Projected area 6.3.2.3 Drop pressure	r φ F P	Texture roughness Pressure force
	F C	Capillary force	N P	Number of posts
	r p	Post radius		Θ	Angle made by the air-liquid inter-face with the post
	Θ	Advancing contact angle	P t imp	Impalement pressure for touch-down
	P s imp	Impalement pressure for sliding	h	Post height
	W λ	6.1.2.1 Channel width Transition from the Wenzel to Cassie-Baxter regime . . . . . . 172 H Channel height Viscosity ratio Q Injection ow rate
	w	6.1.2.2 Spacing between posts Impalement of superhydrophobic surfaces . . . . . . . . . . . 173 θ Receding contact angle
	L r min	6.1.2.3 Retention length Shear-driven drainage of liquid-infused surfaces: scaling ar-L o Initial length of oil Minimum radius of curvature L ∞ Steady-state retention length
	Re	Reynolds number	Bo	Bond number
	u	Streamwise velocity	u s	Shear-driven velocity
	u p	Pressure-driven velocity	q s	Shear-driven ux
	q p	Pressure-driven ux	q d	Drained ux
	t c	Characteristic time of extraction	L	Dimensionless retention length
	6.2 Surfactant-driven oil extraction from micropatterned channels . . . . . 178 t Dimensionless time V d Volume of extracted oil
	V 0 Ca α ∞	Initial volume of oil 6.2.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 α Extraction e ciency 6.2.1.1 Capillary number (with oil-water interfacial tension) I Fluorescence intensity Characteristics of the textured surface and the tested uids . . 179 6.2.1.2 Fluid injection . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Steady-state extraction e ciency De ection of the oil-water inter-δ face
	M	6.2.2 Experimental observations and results . . . . . . . . . . . . . . . . . . . 182 Marangoni number Initial interfacial concentration of C * surfactants
	γ * ow ∆p	6.2.2.1 Initial oil-water interfacial tension Extraction patterns . . . . . . . . . . . . . . . . . . . . . . . . 182 q Flow rate through the constriction 6.2.2.2 E ciency curves . . . . . . . . . . . . . . . . . . . . . . . . . 182 Pressure di erence through the constriction Characteristic time of bubble T break-up
	q g w c	Gas ow rate 6.2.3 E ect of surfactants on the extraction dynamics . . . . . . . . . . . . . . 183 l c Constriction length Constriction width w o Reservoir width
	w l V b	6.2.3.1 Width of the liquid channel Variations of the contact angle . . . . . . . . . . . . . . . . . . 185 w g Width of the gas channel Bubble volume D Film thickness at the wall
	6.2.3.2 Apparent capillary number Comparison between the predictions and the experimental re-α exp Experimental extraction e ciency Shear stress exerted by the foam D * Dimensionless lm thickness sults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Dimensionless shear stress Outer capillary number (with air-Ca * water interfacial tension) 6.2.4 Extraction instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 τ f oam Ca xy τ f oam * xy r p Pore radius L Depth of the extraction well
	φ	6.2.4.1 Porosity	Instability of viscous strati cation . . . . . . . . . . . . . . . . 189 d g Grain diameter
		6.2.4.2

guments

Table 6 .

 6

1: Characteristics of the aqueous solution -oil combinations. Here, η w (mPa.s) and η o (mPa.s) are, respectively, the shear viscosity of water and oil.

Table 6 .

 6 2: Interfacial tensions γ aw , γ ao and γ ow (mN/m) with calculated entry E and spreading coe cients S of the oils and surfactant solutions. The dynamic viscosity η o (mPa.s), the density ρ o (for each oil), and the surfactant concentration C (for each surfactant solution) are given.

	Interfacial	Solution	Water	CAPB-SLES-Mac SDS -
	tension			10% gly	2 g/L
	γ aw	Air	73	23.7	39
	Interfacial	Oil	Olive 68.5	Si 42.7	Si 201
	tensions	Solution	0.92	1.06	1.09
	γ ao	Air	32.2	22.6	22.6
		Water	24.8	29	28.2
	γ ow	CAPB-SLES-Mac-10% gly	5.2	4.2	4.2
		SDS -2 g/L	4.7
		Water	65.6	79.4	78.6
			16	21.4	22.2
	E -S	CAPB-SLES-Mac-10% gly	-3.3	5.3	5.3
			-13.7	-3.1	-3.1
		SDS -2 g/L	11.5
			2.1

  1 , w 1 ) are solutions of the Navier-Stokes equations:

									∂u 1 ∂r	+	u 1 r	+	∂w 1 ∂x	= 0.	(B.9)
	∂u 1 ∂t	+ u 1	∂u 1 ∂r	+ w 1	∂u 1 ∂x	= -	1 ρ 1	∂p ∂r	+	η 1 ρ 1	∂ 2 u 1 ∂r 2 +	1 r	∂u 1 ∂r	-	u 1 r 2 +	∂ 2 u 1 ∂x 2	,	(B.8a)
		∂w 1 ∂t	+ u 1	∂w 1 ∂r	+ w 1	∂w 1 ∂x	= -	1 ρ 1	∂p ∂x	+	η 1 ρ 1	∂ 2 w 1 ∂r 2 +	1 r	∂w 1 ∂r	+	∂ 2 w 1 ∂x 2 .	(B.8b)
	Due to uid incompressiblity, mass conservation is given by:

2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

ACKNOWLEDGEMENTS v

Acknowledgements

This thesis represents the end of my student path but the beginning of a new chapter in my life. This work would not have been possible without the help and the support of many people to whom I am very grateful.

ACKNOWLEDGEMENTS vii

and many cultural stu s in Princeton, New Jersey and many other places!

by using a dedicated set-up with a membrane impermeable to the bubbles and permeable to the liquid only [START_REF] Höhler | Osmotic pressure and structures of monodisperse ordered foam[END_REF], as sketched in Fig. 5.4. Figure 5.4: a. Osmotic cell: the pressure p in the foam is controlled by the distance h f oam between the membrane and the location of the meniscus in the U-tube. b. Dimensionless osmotic pressure measured from an osmotic cell as a function of the liquid fraction φ l for a monodisperse, polydisperse emulsion (oil droplets in water) and an ordered monodisperse foam (φ * l = 0.26) [START_REF] Höhler | Osmotic pressure and structures of monodisperse ordered foam[END_REF].

For dry foams, we have Π o = γ aw /r P B = γ aw /(1.74R b φ 1/2 l ) [START_REF] Cantat | Foams: Structure and Dynamics[END_REF]. More generally, the reduction of interfacial energy is used by the foam to pump a volume dV of liquid at the osmotic pressure Π o . Thus, we have:

where γ aw is the air-water interfacial tension and dΣ the variation of the air-water interfacial area. However, dV is a function of the variation of liquid fraction dφ l :

with V g the volume of gas in the foam. Thus, Eq. 5.13 becomes:

(5.15)

The osmotic pressure depends on the liquid fraction. The lower the liquid fraction, the higher the variation of interfacial area with the liquid fraction, the higher the osmotic pressure, the faster the imbibition process, as seen in Fig. 5. 4 where data corresponding to the measurement

CHAPTER 5: OIL IMBIBITION INTO AQUEOUS FOAMS

The permeability increases when the liquid fraction and the pore radius r c increase since A ∝ r 2 c . This is expected as bigger pores make ow easier. This relationship will be used in §5.2.

Permeability models

In the case of a dry foam for which the Plateau borders carry most of the liquid volume and have a length P B r P B (their characteristic width), we can make an analogy with the solid porous media to evaluate the permeability of the foam: this is the channel-dominated model [START_REF] Verbist | The foam drainage equation[END_REF]. The cross-section of a Plateau border is A = δ a r 2 P B with δ a = √ 3 -π/2 and r

with δ b = 1.74. Eq. 5.24 gives the permeability for a dry aqueous foam:

(5.25)

In Eq. 5.25, the coe cient K c is unknown. For an undeformable cylindrical pore, K c = 1/8π (see Appendix A). However, we cannot use the same value for three major reasons. First, the structure of the foam is governed by the Plateau laws presented in §1.2.1.2. Second, the pores of the foam are liquid and deformable. Third, air-water interfaces are mobile. The bulk ow of liquid within the pores can entrain the air-water interface by tangential shear (so the boundary condition is a condition of interfacial stress and not a no-slip condition), which makes it di cult to evaluate the coe cient K c , and thus the permeability k f oam .

Thus, to quantify the importance of the bulk viscous ow compared to the interfacial viscous ow, one de nes the Boussinesq number Bq = η s /η l r P B where η s is the interfacial shear viscosity and r P B is the characteristic length of viscous dissipation of the bulk ow [START_REF] Cohen-Addad | Flow in foams and owing foams[END_REF].

If Bq 1, the interface is less "viscous" and o ers less resistance to uid entrainment. The interface is highly mobile and the velocity pro le is a plug ow.

On the opposite, if Bq 1, the interface o ers enough resistance to the bulk ow to induce a strong velocity gradient, similar to a Poiseuille ow. Fig. 5.6 presents some typical velocity pro les in a Plateau border and the evolution of K c with Bq. For very rigid interfaces, Bq → ∞ and K c = 0.02 [START_REF] Nguyen | Liquid drainage in single Plateau borders of foam[END_REF].

Other permeability models exist, especially when the viscous dissipation in the vertices dominates the viscous dissipation in the Plateau borders (for big bubbles and mobile interfaces): this is the node-dominated model [START_REF] Koehler | Liquid ow through aqueous foams: the nodedominated foam drainage equation[END_REF]. In this case, one has to consider the volume fraction of liquid in the vertices or nodes V n /(V n + V P B ) = r P B /(1 + /r P B ) with V n ∝ r 3 P B , the volume in the nodes, and V P B ∝ r 2 P B , the volume in the Plateau borders. For a dry foam, r P B and the

Concluding remarks

In this thesis, we study how aqueous foams interact with immiscible and oily liquids at di erent scales: -microscopic scale: size of a surface roughness, -millimetric scale: soap lms and Plateau, borders -centimetric scale: macroscopic foam samples.

We understand our experimental results by a static and dynamical theoretical analysis. In every part of this thesis, we play on the chemistry and the physical parameters of our experimental systems, including the nature of the foam phase and the oil phase.

In Chapter 1, we show how the physical and the mechanical properties of aqueous foams o er wide opportunities in many applications where one would like to nd an alternative method to extract a liquid phase from a con ned medium (soil remediation, enhanced oil recovery). Foams are light, have a very little water content and o er a high speci c surface area, which generates less waste, less energy supply and requires easier processes.

In Chapter 2, we study the di erent criteria on which the stability and the collapse of foams in contact with oil depends. Common examples of oil interaction with foams refer to the incorporation of oil droplets (emulsi ed droplets) during foam generation to generate "foamulsions". Here, we consider the case where oil is directly absorbed by the foam due to the capillary underpressure. As a consequence, oil appears as long slugs invading the Plateau borders of a dry foam. Oil-laden foams do not collapse if the entry coe cient E, the spreading coe cient S and the bridging coe cient B are all negative, similarly to common theories on antifoaming action induced by oil droplets in soap lms. However, for values of the coe cients close to 0 (and considering the error margins on the interfacial tension measurements), the correlation between the coe cient values and the foam stability is less clear. The destabilization of a Plateau border starts by the breaking of the pseudoemulsion lm which is the thin aqueous lm between the air phase and the oil phase. This can only happen if the entry of the oil phase at the air-water interface overcomes an entry barrier related to the pressure required to thwart the disjoining pressure in the pseudoemulsion lm. The relationship between the entry barrier and the static coe cients is still an open question.

In Chapter 3, we try to understand how oil locally behaves when propagating inside a Plateau border. Numerical simulations and results on the capillary ow of oil inside a Plateau border show that the oil slug swells the Plateau border and that the viscous dissipation is determined by the oil phase. We compare the interfacial energies of con gurations where oil remains in the Plateau border and spreads out of the Plateau border. Our results con rm the stability analysis

Examples of ow dynamics by resolution of Stokes' equation

We give two examples of exact calculations describing the ow dynamics based on Stokes' equation. First, we derive the dynamics for the capillarity and gravity-driven spreading of a liquid drop on a solid surface. Second, we analytically solve the laminar pressure-driven ows in circular and rectangular in nite pipes. EQUATION

A.1 Spreading dynamics of a liquid droplet on a solid surface

We solve the spreading dynamics of a liquid droplet on a solid surface in the capillarity and gravity-dominated regimes.

A.1.1 Lubrication approximation

We consider the viscous spreading of a thin lm on a plane surface (Reynolds number Re = ρ l ur/η l 1), sketched in Fig.

A.1 with the spreading radius r very small compared to its width hu is the velocity eld, ρ l the liquid density and η l its viscosity. From Eq. A.1, we have v ∼ (e/r)u. Thus, since e r, v is negligible compared to u. The Stokes' equation with gravity negligible compared to capillarity is given by (with the pressure p):

Projecting Eq. A.2 on e x and e y , we nd:

The pressure p only depends on the coordinate x and using the boundary condition u(x, 0, t) = 0

A.1 SPREADING DYNAMICS OF A LIQUID DROPLET ON A SOLID SURFACE

209

(t is the time) at the wall (no-slip condition) and the free surface condition ∂u ∂y h(x,t) = 0 yields:

A.1.2 Thin lm equation

Then, we can write the advancing velocity U = (1/h) h 0 udy, which gives:

Mass conservation gives the following relationship:

The pressure gradient close to the edge of the spreading front is given by the Laplace pressure

, where γ al is the air-liquid interfacial tension. Injecting this pressure in Eq. A.5 and Eq. A.6 give the thin lm equation:

The asymptotic solution of Eq. A.7 is [START_REF] Cazabat | Dynamics of wetting on smooth and rough surfaces[END_REF]:

where k = 10 4 is an integration constant in the vicinity of the contact line. Considering a small dynamic contact angle θ ≈ h/x, we deduce the well-known Tanner-Ho mann's law:

This law describes the variation of the dynamic contact angle with the capillary number Ca and thus the velocity U of the contact line.

A.1.3 Spreading laws

We introduce the spreading radius r. We nd the spreading velocity U = dr/dt. The volume of the spreading drop is Ω d = (π/4)θr 3 . From Eq. A.9, we deduce that: EQUATION

(A.10) By integrating A.10, we nally nd that:

We use Eq. A.11 in Chapter 4 to describe our long-time spreading regime for liquid-liquid interfaces.

When gravity dominates capillarity, Eq. A.7 is replaced by:

Mass conservation gives the drop volume

12 can be non-dimensionalized by the variables

with the mass conservation equation written as 1 = 2π

H(X, T )XdX. One can look for a self-similar solution of the form H(X, T ) = T α H * (X/T β ) = T α H * (µ) with µ the dimensionless self-similar variable. Injecting the self-similar solution in Eq. A.13 and mass conservation, one nds that α = -1/4 and β = 1/8 for non-dimensionalization. One deduces that the spreading radius r is written as:

where µ f is a constant determined by studying the asymptotic behaviour at the spreading front of Eq. A.13. Eq. A.7 could also have been solved by self-similarity. Eq. A.14 is also used in Chapter 4 but deduced by scaling arguments.

A.2 Poiseuille ow

We assume a steady laminar pressure-driven ow in the following two examples.

A.2.1 In a cylindrical tube

We consider that the velocity eld u is uniaxial and has the form u = u(r)e x . This problem is invariant by rotation (coordinate θ) and translation (coordinate x). Eq. A.2 becomes in cylindrical coordinates (r, θ, x):

We have the following boundary conditions: u(0) is bounded and u(r c ) = 0 (no-slip condition) where r c is the cylinder radius. Integrating A.15 yields a solution u = (1/4η l )r 2 dp/dx + A 1 ln r + A 2 . From the boundary conditions, we have A 1 = 0 and A 2 = -(1/4η l )r 2 c dp/dx. But dp/dx = -∆p/L where L is the length of the cylinder, so the axial velocity u is given by:

We can deduce the ow rate Q = 2π 0 rc 0 u(r)rdrdθ from Eq. A.16, which yields the well-known Hagen-Poiseuille ow rate in a cylindrical tube:

Eq. A.17 can be rewritten in a more general form Q = AK c ∆p/η l L where A is the cross-sectional area and K c the permeability constant. For the laminar ow in a cylindrical pipe, K c = 1/8π (for a Plateau border, K c = 1/50). Eq. A.17 is used in Chapter 1 and 6 (but demonstrated by scaling arguments).

A.2.2 Between two plates

Here, we consider the steady laminar pressure-driven ow between two parallel plates separated by a distance H. Once more, the velocity eld is uniaxial and u = u(y)e x . We have the EQUATION following projected Stokes' equation:

with the boundary conditions u(0) = 0 and u(H) = 0 (no-slip condition). The solution u has the form u = (1/2η l )(dp/dx)r 2 + A 1 r + A 2 . One nds that A 2 = 0 and A 1 = -(1/2η l )(dp/dx)H. Thus, u is given by:

where we have replaced dp/dx by -∆p/L.

We can determine the ow rate

-W/2 u(y)dydz by integrating the velocity eld beyond the cross-section of a wide rectangular channel of width W . One nds that:

Moreover, we can deduce the shear stress at the wall τ xy = η l (∂u/∂y) y=0 :

Finally, by using Eq A.20 to nd an expression for ∆p/L as a function of Q, one nds that:

Eq. A.22 is the exerted shear stress in the micro uidic channel used in Chapter 6.

Appendix B Rayleigh-Plateau instability of a viscous uid in another viscous uid

We explicit the comprehensive solution of the Rayleigh-Plateau instability in the case of the inertia-dominated regime and give some indications to solve the whole problem in the case of a viscous-dominated regime. ANOTHER VISCOUS FLUID

B.1 Inertia-dominated regime

First, we explain why the maximum instability wavelength λ max is 9r c (r c is the cylinder radius) in the case of an inertia-dominated regime.

We consider an axisymmetric jet of sinusoidal shape de ned by its local radius r 2 (x, t) depicted in The capillary pressure p -p atm is given by the in-plane r 1 and the transverse r 2 curvatures of the air-water interface (γ aw is the air-water interfacial tension), which gives:

by derivating the rst term and developing at rst order the second term with r c .

Then, we use the Navier-Stokes equation by neglecting the viscous e ects and considering a uniform pressure gradient and axial velocity v x along the transverse direction e x , which yields the momentum equation (ρ l is the liquid density):

Integrating Eq. B.3 along the cross-section of the jet and using Eq. B.2 to express the pressure gradient gives the ow rate Q(x, t):

Solving this whole set of equations allows to nd a relationship between the growth rate σ and the wave number k, which especially depends on the viscosity ratio η 2 /η 1 and to deduce the di erent relationships presented in §3.2.1.

Appendix C Micro uidic fabrication: the "sticker technique"

We describe how the micro uidic channels in Chapter 6 are built, by using the "sticker technique" [START_REF] Bartolo | Micro uidic stickers[END_REF].

C.1 Fabrication of the channel

The micro uidic channel is synthesized by using polymeric stickers.

C.1.1 Fabrication of the sticker

A sticker is a layer of polymer material of controlled size. The polymer we use is the thiolenebased resin NOA 81 (Norland Optical Adhesive). NOA is more resistant to swelling than PDMS (polydimethylsiloxane used for most micro uidic chips) and has an elastic modulus higher than PDMS, which allows to reduce the deformation of the polymer when an external force is applied.

We rst generate a mold of our channel by curing PDMS over a silicon wafer on which the channel pattern is drawn. We deposit a drop of NOA on the resulting PDMS stamp and we press the liquid polymer with a black borosilicate glass lamella. PDMS is permeable to air, which allows to obtain a thin polymer lm with no trapped air bubbles. Then, we cure the polymer by UV light through the transparent PDMS stamp before removing the PDMS stamp. We obtain the black glass with the epoxy sticked to it and representing the main channel.

We also punch holes beforehand through the black glass and the PDMS stamp for the inlet and the outlet ports. Afterwards, we treat the channel under plasma to oxydize the surface of the channel and make it hydrophilic.

C.1.2 Fabrication of the pattern

The mold for the pattern is created by soft photolithography. We deposit small droplets of epoxy on the mold. We cover the polymer with a thin glass lamella and let trapped air leave the epoxy through the PDMS stamp during 2 hours at least. Then, we cure the epoxy to print the pattern in the epoxy material. We stick the pattern to the top of the channel by curing once more under UV.

C.2 Sketch of the microfabrication

Appendix D Properties of textured surfaces

We come back on the main properties of superhydrophobic and liquid-infused surfaces and how to tune their properties. By doing so, we underline their great interest and why the understanding of failure mechanisms in Chapter 6 is of considerable scope.

D.1 Properties of superhydrophobic surfaces D.1.1 Self-cleaning

The roughness of a surface enhances its hydrophobicity. This property is well-described in the regime of air cushion (also known as the Cassie-Baxter regime) where air pockets are trapped under the deposited liquid droplet. The hydrophobicity is characterized by the apparent contact angle θ * whose value is higher than the Young contact angle θ Y , as shown in Fig. 6.1.

Thus, on a textured surface, it is possible to reach a superhydrophobic state for which θ * is close to 180 o . Such a property is very common in nature. We can cite two examples pictured in Fig. D.1. The rst one is the duck feather. The feather can be assimilated as a ber textured surface that allows ducks to remain "dry" in spite of the wet environment. The second one is the lotus leaf which is covered by a multitude of bumps of micrometric size. These microstructures give a self-cleaning character to the leaf. Subsequently, water droplets deposited on a lotus leaf roll over the leaf.

D.1.2 Controlling the hydrophobicity

To control the hydrophobicity, one can think about increasing the Young contact angle, which means playing on the chemical properties of the liquid droplet. Another interesting solution consists in increasing the roughness of the surface. The ideal con guration is to trap air pockets This situation corresponds to a re-entrant texture curvature for which Ψ < 90 o . The total force exerted on the air-liquid interface is directed upwards and tends to lift the interface to the top of the plots. This allows to trap air pockets under the liquid droplet with θ * > 90 o even though θ Y < 90 o . Thus, for wetting oils, textured surfaces must have a re-entrant curvature Ψ < 90 o to be oleophobic. For instance, beads on strings and meshes create such con gurations. The Cassie-Baxter regime is metastable and the air-oil interfaces can also be subject to local sagging. The wetting of liquid droplets can be considerably modi ed on liquid-infused surfaces compared to smooth and common textured surfaces. In particular, liquid-infused surfaces have a strong in uence on the contact angle hysteresis.

On an ideal smooth surface, a liquid droplet makes the Young equilibrium contact angle θ Y with the solid surface. However, when chemical (di erence of chemical treatment) or physical (roughness) defects appear, the static contact angle can take di erent values. The droplet is pinned on the defect. If one swells the droplet, the contact angle increases without any movement of the contact line. Above a certain threshold value θ a , the contact line advances: θ a is called the advancing contact angle. On the opposite, if one unswells the droplet, the contact angle decreases without any movement of the contact line. Below a certain threshold value θ r , the contact line recedes: θ r is the receding contact angle. The di erence between the advancing and the receding contact angle is the contact angle hysteresis: ∆θ = θ a -θ r [START_REF] Johnson | Contact angle hysteresis. III. Study of an idealized heterogeneous surface[END_REF].

We are able to observe the e ect of ∆θ by tilting the solid surface. Due to the capillary forces (for millimetric droplets) induced by this hysteresis, the droplet does not fall with gravity. However, as depicted in Fig. D.5, when the contact angle at the head of the droplet reaches θ a and the contact angle at the tail of the droplet reaches θ r , the droplet slides down the inclined surface. Therefore, the contact angle hysteresis allows to characterize the ability of a droplet to slide or to remain stuck to a surface. The lower ∆θ, the more easily it slides. The higher ∆θ, the more easily it remains stuck. [START_REF] Wong | Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity[END_REF].

The red squares, the orange rounds and the green triangles correspond to three di erent liquid-infused surfaces ( uorinated lubricant and te on/epoxy material). The blue diamonds refer to measurements on omniphobic surfaces in [START_REF] Tuteja | Robust omniphobic surfaces[END_REF].

Due to the small contact angle hysteresis, it is very easy for an immiscible liquid droplet to roll o a liquid-infused surface without leaving a lot of residual liquid behind it. For smooth and omniphobic surfaces, the droplet remains pinned on the defects or the texture and leaves a stain when sliding down a tilted surface [START_REF] Wong | Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity[END_REF].

D.2.2 Conditions for slippery surfaces

The hysteresis ∆θ is small providing that the droplet oats on the liquid-infused surface. If it sinks, pinning can arise and the slippery character is lost. One can determine the condition for which pinning does not happen. Wong et al. [START_REF] Wong | Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity[END_REF] and Lafuma et al. [START_REF] Lafuma | Slippery pre-su used surfaces[END_REF] have suggested to compare the interfacial energy of a droplet sitting on the liquid-infused surface and a droplet sinking to the bottom of a texture cavity, as sketched in