Keywords: Droplet-based microfluidics, droplet, confinement gradient, relaxation, selfsimilarity

13 Eect of temperature on surface tension: the Marangoni eect III Droplet deformation 14 State-of-the-art on the droplet break-up 15 Experimental observations 16 The break-up criteria: experimental check 17 The physical mechanisms at play 17.0.1 Why might the droplet deform when a gradient of surface tension is established at its interface ? . . . . . . . .

• Kari Dalnoki-Veress, Hugues Bodiguel, François Lequeux, Emmanuelle Rio et Denis Bartolo pour la relecture de mon manuscrit et pour leurs commentaires pertinents et constructifs. L'attention qu'ils ont portée à mes travaux est une grande récompense,

• mes directrices de thèse, Marie-Caroline Jullien et Stéphanie Descroix, pour leur soutien, leur bienveillance et leur regard critique,

• Joshua McGraw pour sa grande pédagogie. Sa curiosité et sa rigueur scientique m'ont guidée sur une bonne partie de ma thèse,

• Benjamin Dollet pour les échanges que nous avons eu, toujours très enrichissants,

• Albane, Emilien, Thomas et Aniello pour leurs multiples contributions. C'était très plaisant de travailler avec eux,

• Patrick Tabeling pour son accueil, son ouverture d'esprit et pour toutes les opportunités qu'il m'a oertes,

• Elizabeth Bouchaud pour sa générosité et sa culture. Pouvoir jouer dans son théâtre a été pour nous une immense chance,

• Guillaume, Oliver et Nawel pour leur patience et leur soutien technique.

Enn, j'ai immensément apprécié évoluer aux côtés de Marc, Alex, Lorène, Pierre, Etienne, Hubert, Benjamin, Elian, Charles, Manon, Joshua, Gustavo, Ilham, Maria, Cécile et Marjan. Leur créativité, leur humour et leur bonne humeur contagieuse m'ont marquée pour la vie.

Avec une mention très spéciale pour Marine, évidemment...

Part I Introduction 1 Motivations

This PhD work investigates the eects of connement on the hydrodynamics of micrometric biphasic systems. More precisely, this manuscript studies the response of a micrometric oil droplet in water, conned in a rectangular channel, to a variation of the channel topography. We believe that this study indirectly nds applications in petroleum engineering and in droplet-based microuidics that is thriving in biotechnologies.

Petroleum engineering

Petroleum engineering involves three steps of oil extraction [START_REF] Gurgel | A review on chemical ooding methods applied in enhanced oil recovery[END_REF]. The primary oil recovery step relies on the oil extraction out of the well bore as the result of the natural pressure in the reservoir. During this process, only 10 to 20 % of oil is extracted from the deposit. Then, petroleum engineers proceed to a second oil extraction step that involves the injection of water or steam to drive the oil out of the porous rock. After this second process, the average oil recovery rate is comprised in between 20 to 40 % [START_REF] Nazar | Microemulsions in enhanced oil recovery: A review[END_REF]. Lastly, a third recovery step can be launched that involves the injection of chemical compounds like surfactants and polymers to match the viscosities of the water and oil -to decrease capillary instabilities and to improve the oil mobility - [START_REF] Avendano | Viscoélasticité et récupération assistée du pétrole[END_REF][START_REF] Muggeridge | Recovery rates, enhanced oil recovery and technological limits[END_REF]. This third step is very expensive and can only be developed if the market oil prize is suciently high to make the extraction of the last (∼ 60 %) percent of oil that is trapped protable. Given the signicant loss of hydrocarbons that remain trapped in the deposit, the petroleum industry is eager to improve their understanding of the physical mechanisms that promote oil mobility in the porous media during these recovery phases. The present study may be of interest to understand specic mechanisms of oil mobility in the second step.

During this second phase, it is known that the interface between oil and water can destabilize due to the growing of capillary instabilities like viscous ngering [5,[START_REF] Homsy | Viscous ngering in porous media[END_REF], to fragmentation [START_REF] Nishimura | Breakup of conned drops against a micro-obstacle: an analytical model for the drop size distribution[END_REF][START_REF] Jung | Wettability controls slow immiscible displacement through local interfacial instabilities[END_REF] or to gradients of connement [START_REF] Al-Housseiny | Control of interfacial instabilities using fow geometry[END_REF][START_REF] Keiser | Washing wedges: capillary instability in a gradient of connement[END_REF][START_REF] Dangla | Droplet microuidics driven by gradients of connement[END_REF]. This destabilization results in the formation of a water/oil emulsions that has to ow through the conned porous media.

Droplet-based microuidics

Emulsions owing through pores of the same scale are also present in droplet-based microuidics applications [START_REF] Baroud | Dynamics of microuidic droplets[END_REF][START_REF] Seeman | Droplet based microuidics[END_REF]. The microuidic eld has thrived over the last decades with the promise of miniaturizing chemical and biological assays to reduce their costs and enhance their throughput [START_REF] Brouzes | Droplet microuidic technology for single-cell high-throughput screening[END_REF][START_REF] Guo | Droplet microuidics for high-throughput biological assays[END_REF][START_REF] Hung | Continuous perfusion microuidic cell culture array for high-throughput cell-based assays[END_REF]. Droplet-based microuidics platforms are composed of a network of micrometric channels in which water-in-oil droplets are generated [START_REF] Dangla | Droplet microuidics driven by gradients of connement[END_REF][START_REF] Nakano | Single-molecule pcr using water-in-oil emulsion[END_REF][START_REF] Tawk | Man-made cell-like compartments for molecular evolution[END_REF] and used as biochemical reactors, alternatively to microtitre plates [START_REF] Oldenburg | Assay miniaturization for ultra-high throughput screening of combinatorial and discrete compound libraries: a 9600-well (0,2 microliter) assay system[END_REF]. These droplets constitute cargoes for reagants and can be driven to any location of the platform [START_REF] Griths | Miniaturising the laboratory in emulsion droplets[END_REF][START_REF] Jie | Drop on demand in a microuidic chip[END_REF][START_REF] Song | Reactions in droplets in microuidic channels[END_REF]. To trigger the biochemical assay, the droplets that contain the dierent components of the reaction, are forced to merge [START_REF] Gu | Droplets formation and merging in two-phase ow microuidics[END_REF][START_REF] Sung | Creating, transporting, cutting, and merging liquid droplets by electrowetting-based actuation for digital microuidic circuits[END_REF]. In order to synchronize at best the biochemical reactions that are carried out in microuidic devices, the droplets speed needs to be nely monitored [START_REF] Thorsen | Microuidic large-scale integration[END_REF], which requires to investigate the droplet mobility when the latter is conned in a pore/channel. Interestingly, microuidic systems made of poly(dimethyl)siloxane or glass are often used as models to understand the ow of bi-phasic systems in porous media as these models are transparent. These ideal systems allow to link the emulsion ow pattern to external parameters (pressure drop, dispersed phase velocity, extracted liquid fraction...).

As a whole, understanding the mobility of two phase systems in micrometer size channels nds application in a wide range of domains from microuidics applications up to oil recovery.

To understand the mobility of these emulsions through the pores of the media, many fundamental studies have developed ideal systems based on a complex network of channels [START_REF] Berejnov | Lab-on-chip methodologies for the study of transport in porous media: energy applications[END_REF] or on an isolated droplet that moves in a simply modelled-geometry: in a capillary tube [START_REF] Hodges | The motion of a viscous drop through a cylindrical tube[END_REF] or in a Hele-Shaw cell [START_REF] Huerre | Droplets in microchannels: dynamical properties of the lubrication lm[END_REF][START_REF] Reichert | Topography of the lubrication lm under a pancake droplet travelling in a hele-shaw cell[END_REF][START_REF] Zhu | A pancake droplet translating in a hele-shaw cell: lubrication lm and ow eld[END_REF]. To predict the mobility of the droplet, one has to consider the viscous dissipation that counters the pressuredriven motion. These studies prove that when the droplet is wetting, this viscous dissipation is localized near contact lines [START_REF] Snoeilher | Moving contact lines: scales, regimes, and dynamical transitions[END_REF]. When the droplet is non-wetting, this viscous dissipation is localized near the dynamical meniscus that is dened as the transition region in the between the lubricating at lm and the spherical meniscus of the droplet [START_REF] Bretherton | The motion of long bubbles in tubes[END_REF]. Under some conditions, dissipation may also occur in the at lm [START_REF] Cantat | Liquid meniscus friction on a wet plate: bubbles, lamellae and foams[END_REF]. The description of this viscous dissipation is complex and results from the balance between viscous stresses and surface tension, the viscous stresses being intrinsically dependent on the viscosity ratio in between both phases, the degree of connement [START_REF] Keiser | Washing wedges: capillary instability in a gradient of connement[END_REF] and the boundary conditions at the interface [START_REF] Reichert | Topography of the lubrication lm under a pancake droplet travelling in a hele-shaw cell[END_REF].

All the ideal systems that are mentioned above (Hele-Shaw cells, capillary tube) do not capture the genuine geometry of the pores that droplets can encounter in a rocky soil or in a microuidic device. In this manuscript, we want to take a step further in the understanding of droplet mobility in tortuous porous media by investigating the dynamical response of an isolated conned droplet to a sudden change in the topography of an ideal microuidic channel. We are interested in predicting the variation of the droplet shape and the dynamics of the droplet deformation. Our work is limited to the case of a droplet that is static in the channel. By "static", we mean that the droplet does not travel in the pore but can be deformed by the constraints exerted by the channel walls.

In the studied system, the channel topography is modied on demand with a reversible thermomechanical actuation [START_REF] Selva | Thermocapillary actuation by optimized resistor pattern: bubbles and droplets displacing switching and trapping[END_REF]. The latter relies on the heating of a micropatterned resistance that induces a local dilation of the channel walls. A gradient of connement is then imposed on the droplet such that the latter deforms on the length scale of the gradient and a reservoir of water is trapped in the vicinity of the resistance, see gure 1 for an illustration. This process is called the droplet deformation. When the resistance is switched o, the channel dilation disappears and the droplet relaxes to its initial plug-like shape. The reservoir of water drains towards the droplet extremities. This process is called the droplet relaxation.

General outline

This manuscript investigates the eects of connement on a biphasic system through the relaxation of a droplet that is initially set out-of-equilibrium by a change in the channel topography. It is divided into four chapters:

1. The rst chapter introduces general concepts on the hydrodynamics of biphasic systems and on the eect of connement on these systems. In the next chapters, these concepts will be used to understand and to model our experimental observations.

2. The second chapter describes the experimental set-up and the thermo-mechanical actuation device. The thermal and the mechanical eects that are induced on the droplet as a result of the heating of a micropatterned resistance are experimentally characterized.

3. The third chapter demonstrates the eects of the local modication of the channel geometry induced by the thermo-mechanical device on the droplet. The experimental observations show that a localized tilt of only few degrees in the cavity, giving rise to a deformation of only 15 % of the channel height, can have striking eects on the deformation of the droplet and can even lead to break up. A model based on the surface minimization of the droplet, that takes into account the topography of the channel, manages to capture the deformed shape of the droplet and to predict its break-up. In addition, a power balance between the viscous dissipation and the capillary-driven motion recovers the dynamics of the droplet deformation. We show that our model can be reversibly used to extrapolate the topography of the channel.

4. The last chapter investigates the relaxation of a droplet that has been initially deformed by a local modication of the channel geometry. The droplet relaxation is driven by surface tension and is mediated by viscous dissipation. Interestingly, the dynamics of the droplet relaxation is very dierent from the one of the droplet deformation. A model based on scaling arguments that takes into account the sublinear viscous dissipation in the droplet menisci allows to capture the dynamical features of the droplet relaxation process. This last study evidences the critical contribution of the droplet menisci in such a conned geometry.

3 Overview of the characteristic variables of the studied system

In the majority of the experiments described in this work, a mineral oil droplet in water is conned in the rectangular channel of a microuidic system. By "conned", we mean that the radius of the unconned droplet at equilibrium, i.e. when the droplet takes a spherical shape at constant volume, would be much larger than any of the dimensions of the channel cross section. The channel walls are exposed to oxygen plasma, rendering the channel surface hydrophilic, such that the droplet is non-wetting [START_REF] Dhananjay | Hydrophilization and hydrophobic recovery of pdms by oxygen plasma and chemical treatmentan sem investigation[END_REF]. It means that the droplet interface is always separated from the walls by a lubricating water lm. A meniscus runs along the contour of the droplet, and gutters are formed at the four corners of the channel.

Figure 1 illustrates the droplet geometry and the typical experimental observations that we consider in this manuscript. The interface has a pocket-type prole composed of a reservoir followed by what could be considered as a thin lm in the observation plane, and nally by the menisci at the droplet extremities.

The snapshots shown in gure 1b, illustrates an example of the droplet relaxation.

The orders of magnitude of the dimensions that describe the system are 1. for the typical length, 10 -5 to 10 -4 m 2. for the typical velocity 10 -5 m.s -1

3. for the interfacial tension, 10 -3 N.m -1

4. for the viscosity, 10 -3 Pa.s -1

5. the viscosity ratio between the inner and the outer phase is 25.

Figure 1 denes the equatorial plane of the droplet, at z = 0, as the "in-plane" and shows a cross-sectional scheme of the droplet in the plane (y, z). Geometrical characteristics of the system are illustrated on the gure; h ∞ is the lubricating lms thickness, e/2 is the radius of the droplet spherical menisci and δ is the typical size of the gutters. Lastly, the in-plane distance in between the droplet interface and the side channel wall is described by the function h(x, t) in the reservoir.

Figure 1: a) 3D shape of the droplet set out of equilibrium with typical orders of magnitude of the droplet and channel geometries. b) Snapshot of a droplet relaxation that is typically observed in the system. The observations are made in the equatorial plane (z = 0) for a frame of reference located at the center of the channel cross-section. c) Description of the geometry of the droplet in the rectangular channel in a cross-section plane (y, z) and in an equatorial plane z = 0 dened as the "in-plane". The radius of the droplet meniscus is e/2. The lubricating lm thickness between the droplet and the wall is h ∞ . The interface prole of the droplet in the "in-plane" is described by h(x, t). δ is the typical size of the gutters at the fours corners of the channel. [START_REF] Muggeridge | Recovery rates, enhanced oil recovery and technological limits[END_REF] Introduction to the physics of the studied biphasic system

The studied system, composed of an oil droplet in water, constitutes a biphasic system. In this section, we introduce some general features of the hydrodynamics of biphasic systems that build the background of the further interpretation of our experimental results.

Surface tension

Denition of the interfacial tension

A biphasic system is composed of two uids 1 and 2 that are not miscible. In each of these uids, an internal attractive force maintains the cohesion of molecules. At the interface between 1 and 2, the molecules of uid 1 are partly surrounded by molecules of uid 2 which means that they have lost some of their neighboring molecules (of uid 1). As a result, the forces exerted by molecules 1 at the interface are no longer completely compensated by their neighbors. This absence of compensation is at the origin of the surface tension that makes the creation of an interface energetically unfavorable to the system. Indeed, a biphasic system always tends to minimize its interfacial area given the external constraints that apply on the system (ex: gravity, pressure, geometrical...). Without any of these constraints, the minimum interface that can be adopted by a dispersed phase is spheric [36,[START_REF] Guyon | Hydrodynamique Physique[END_REF].

The work that is required to add an element of interface dA to a biphasic system, ∂W = γdA, denes the value of the surface tension of the system γ. This value depends on the nature of the forces at play in the uids (Van der Waals attractions, hydrogen bonds, etc.) [START_REF] Israechvili | Intermolecular and surface forces[END_REF] Example: When oil is poured in a glass of water, mechanical work is added to the biphasic system and a emulsion composed of oil drops in water is formed (g. 2a). However, this emulsion is not stable and leads to a separation of phases by a succession of droplet coalescence (g. 2b). This nal conguration minimizes the interfacial area between oil and water given the gravity that applies on the system and the value of the surface tension between both oil and water with the glass. Oil lays on top of water since it is the less dense phase.

Near a solid boundary: wetting and non-wetting properties

To determine if a drop of phase 1 surrounded by a uid 2 wets or not on a solid surface, we compare the interfacial energy of the wall surrounded by uid 1 to the interfacial energy of the wall surrounded by uid 2. We dene γ s1 as the surface tension between uid 1 and the wall, γ s2 between the uid 2 and the wall and γ 12 as the surface tension between 1 and 2.

The spreading parameter dened as S = γ s1 -γ s2 -γ 12 compares these quantities. If S > 0, the liquid 2 wets the wall (conguration b, gure 4). On the other hand, if the S is negative, the presence of a lm of phase 2 near the wall is more favorable and the droplet is non-wetting (conguration a, gure 4) [36].

Surface active agents

In order to stabilize an emulsion, surface active agents can be added to the external phase to stabilize droplets and prevent them from merging. This agent is called surfactant and is amphiphilic, see gure 4. The polar head of surfactants is more stable in the aqueous phases while its aliphatic tail (often made of CH2 radicals) has more anity with oil phases. Surfactants adsorb at the interface between the two immiscible liquids. When they are adsorbed at the interface, the creation of new interfaces costs less energy [36]. Thus, surface tension decreases with the concentration of surfactants until it reaches the micellar concentration at which surfactant molecules start to form gather in micelles [START_REF] Bakshi | Micelle formation by sodium dodecyl sulfate in wateradditive systems[END_REF][START_REF] Paula | Thermodynamics of micelle formation as a function of temperature: A high sensitivity titration calorimetry study[END_REF]. Beyond this point, surface tension remains constant as the surfactant concentration increases, see gure 4. At rest, the stability and the thickness of the lm separating two droplets interfaces is set by the disjoining pressure stemming from a combination of several repulsive or attractive potentials [START_REF] Israechvili | Intermolecular and surface forces[END_REF][START_REF] Huerre | Migration de gouttes en microuidique: caractérisation et applications[END_REF]. In the DLVO theory the disjoining pressure is decomposed into structural (Π s ), van der Waals (Π V dW ), and electrostatic interactions (Π el ). When surfactants populate the droplets interfaces, the disjoining pressure increases in the lms that separate them, playing mostly on the electrostatic repulsion [START_REF] Corti | Quasi-elastic light scattering study of intermicellar interactions in aqueous sodium dodecyl sulfate solutions[END_REF]. The resulting exponential decay of the disjoining pressure with the lm thickness leads to the existence of a stable lm between the two interfaces. In section 6.2, we show that surfactants can also signicantly aect the mobility of liquids near interfaces.

In the studied system

The surface active agent that is used is sodium dodecyl sulfate (SDS). SDS is an anionic molecule that is soluble in the external water phase. SDS is added to the system to prevent oil droplets from merging. In the majority of the experiments that are presented in this thesis, the concentration of SDS is set to a constant equal to [SDS] = 2 cmc= 1.610 -2 mol.L -1 at 25 The surface tension in between mineral oil and water mixed with SDS at 2 cmc has been characterized and is equal to

γ = 11 ± 1 mN.m 1 [44].

The Laplace pressure

Laplace has shown that the presence of an interface introduces a pressure drop that is proportional to the surface tension of the biphasic system and to the curvature of the interface. This pressure drop writes

∆P = γ 1 R + 1 R = γC (1)
where R and R are dened as the radii of curvature of the interface in the two orthogonal planes normal to the interface. C is dened as the mean curvature of the interface [START_REF] Guyon | Hydrodynamique Physique[END_REF].

In the absence of external constraints, an mean curvature or a surface tension that is not constant at a uid/uid interface can give rise to a surface stress along the interface. This surface stress induces "capillary-driven" ows that are critical in this study and that are described in the section 6.1.

The eects of connement on the geometry of biphasic systems

When a non-wetting drop of radius R is squeezed in between two plates separated by a distance e, the drop adopts at equilibrium, a pancake-like shape. Park and Homsy have shown that the mean curvature along the droplet interface is constant and veries [START_REF] Homsy | Viscous ngering in porous media[END_REF], [START_REF] Dangla | 2D droplet microuidics driven by connement gradients[END_REF]:

C = 2 e + π 4R (2)
where R is the equatorial in-plane curvature of the squeezed droplet. The calculation of C that recovers this expression is complex and is not explained in this manuscript.

If we consider that the radius of the droplet R is much larger that the cavity height e, the mean curvature simplies as C ∼ 2 e . When the droplet is static in the cavity, the height of the at lm that is separating the non-wetting droplet from the walls can be calculated by balancing the capillary pressure in the droplet P conf ∼ 2γ e with the disjoining pressure in the lm that accounts for the molecular interactions (electrostatic, Van der Waals and steric) between the liquid/liquid interface and the solid boundary.

Figure 6 illustrates the evolution of the disjoining pressure as a function of the distance h between the two interfaces. The equation that balances the capillary pressure in the droplet with the disjoining pressure in the lm Π disj ∼ P conf gives a stable solution for the height of the lubrication lm h ∞ . This height is typically tens of nanometers as observed by Huerre et al. with interferometric measurements [START_REF] Huerre | Droplets in microchannels: dynamical properties of the lubrication lm[END_REF]. ) Typical evolution of the disjoining pressure that combines the electrostatic pressure P el , the steric pressure P S and the Van der Waals pressure P V dW . The solution for the lm thickness results from the electrostatic contribution to the disjoining pressure [START_REF] Huerre | Droplets in microchannels: dynamical properties of the lubrication lm[END_REF]. The gure is adapted from [START_REF] Huerre | Migration de gouttes en microuidique: caractérisation et applications[END_REF] In the studied system

h [nm]

In the rectangular cross-section of the channel, we distinguish the "free" surfaces from the "conned" surfaces. The rst ones are not conned by a solid boundaries while the second ones are. The "conned" surfaces are typically at the level the lubricating lms at the top and bottom interfaces of the droplet as well as the contact lms at the side walls in the equatorial plane (the lm is probably reduced to a line in this plane). The thickness of these lubricating lms has been measured for a mineral oil droplet in water mixed with SDS at 2 cmc and is approximately 20 nm [START_REF] Huerre | Migration de gouttes en microuidique: caractérisation et applications[END_REF].

The "free" surfaces concern all the droplet menisci that are not conned by a wall. They are located in the gutters and at the extremities of the droplet. Near these menisci, the capillary pressure writes

P = γC, (3)
the mean curvature of the droplet menisci being,

• at the extremities of the droplet,

C = 2 w + 2 e ( 4
)
where w is the channel width,

• in the gutters (see gure 1),

C = 2 e (5)
When the droplet is deformed, its menisci at the level of the deformation (in the pocket region) is curved both in the (y, z) and in the (x, y) plane. We recall that, in this region, h(x, t) denes the in-plane distance in between the droplet interface and the channel wall, see gure 1. Thus, the mean curvature of these menisci writes

C = ∂ 2 x h 1 + (∂ x h) 3/2 + 2 e . (6)
The term

∂ 2 x h
√ 1+(∂xh) 3/2 accounts for the curvature of the menisci in the plane (x, y) while the term 2 e accounts for the curvature of the menisci in the plane (y, z).

Hydrodynamics of biphasic systems 6.1 The Navier-Stokes equation

Our work deals with a biphasic system composed of incompressible newtonian liquids: water and mineral oil. Both phases can be described by the Navier-Stokes equation coupled to a boundary condition. This equation derives from the Newton's second law of motion applied to the dynamics of an element of uid. Per unit of volume, the Navier-Stokes equation that describes the motion of uids in each phase can be written as:

ρ ∂v ∂t + ρ(v.∇)v = ρf -∇p + η∇ 2 v (7)
In this equation, ρ is the density of the liquid, v is the velocity eld in the liquid, t is the time, p the pressure in the uid, η the viscosity and f the external forces that apply on the liquid per unit of volume. We assume that uids are incompressible so, by mass conservation, ∇.v = 0. Boundary counditions are required to solve this equation and to get the expression of the velocity eld v(r, t). If we note σ the tensor of stresses at the interface, the boundary conditions write that

• the velocities projected normally to the interface are equal v o .n = v w .n

• the normal stresses at a curved liquid/liquid interface are discontinuous:

([σ] w .n).n- ([σ] o .n).n = γC
• the tangential stresses at the interface are continuous:

([σ] w .n).t = ([σ] w .n).t
where n is the vector normal to the interface and t is the vector tangential to the interface, w is the index for the water phase and o is the index for the oil phase. At the solid boundaries, we dene a no-slip condition: v = 0. The normal stresses at the interface are commonly the pressure and the tangential stresses are typically the viscous stresses.

Dimensionless parameters

From the Navier-Stokes equation, we can dene typical forces that apply on a liquid element:

• the inertial forces deriving from the term ρ(v.∇)v that is integrated over the volume of the uid gives a force that scales as ρU 2 R 2 ,

• similarly, the term η∇ 2 v gives a viscous force that scales as ηU R,

• the term ∇p gives a capillary force that scales as γR, by considering only the Laplace pressure and

• the term ρf times the volume of the liquid gives a volumic force like gravity that scales as gR 3 .

To compare all these forces, dimensionless numbers are commonly used:

• the Weber number: W e = ρU 2 R γ that compares inertia to surface tension eects

• the Reynolds number: R e = ρU R η that compares inertia to viscous stresses

• the Bond number:

B o = ρgR 2 γ
that compares gravity to capillarity

• the Capillary number: C a = ηU γ that compares viscous stresses to capillarity

Example:

As an illustration, if we consider a rain drop that is falling in the air, we can compare the orders of magnitude of these forces to extract the physical mechanisms that prevail in the drop motion. A snapshot of a raining drop during its fall has been captured by Villermaux et al. [START_REF] Villermaux | Single-drop fragmentation determines size distribution of raindrops[END_REF], see gure 7. On theses images, we observe that the 1 mm size drop adopts a large variety of shapes: from a coin to laments to the burst in droplets. The water drop falls under the gravity force ∼ ρgR 3 while it is deformed by the drag force that scales as ρU 2 R 2 . During its fall, extreme deformation rates are achieved since capillary forces γR and viscous forces ηU R are weak compared to the inertial forces applied to millimeter-sized raindrops falling at several meter per second W e = ρU [START_REF] Villermaux | Single-drop fragmentation determines size distribution of raindrops[END_REF].

After the break-up process, droplets are much smaller (R ↓) such that W e ↓ and R e ↓. While droplets are getting smaller and smaller, viscous forces start to compete with inertial forces (R e ∼ 1). When viscous forces overcome inertia, droplets velocity obeys ηRU ∼ ρgR 3 . In this regime, Reyssat et al. [START_REF] Reyssat | Gouttes, lms et jets: quand les écoulements modèlent les interfaces[END_REF] have observed that the diameter of droplets R is around 40 µm and the velocity U decreases down to hundreds of µm.s -1 . As a result, the capillary number that weighs viscous drag versus capillarity Ca ∼ ηU γ ∼ 10 -3 10 -5 10 2 ∼ 10 -6 gets smaller than 1 and droplets are more spheric.

In the studied system

In order to estimate which eects prevail in our micro-system, we estimate the value of all the dimensionless numbers that are mentioned above. Knowing that in our experiments, the typical length is 10 -5 m, the typical velocity is 10 -5 m.s -1 , the surface tension equals to 10 -2 N.m -1 , and the viscosity is at minimum 10 -3 Pa.s -1 , we get:

• W e = ρU 2 R γ ∼ 10 3 10 -10 10 -5 10 -2 ∼ 10 -10 << 1

• R e = ρU R η ∼ 10 3 10 -5 10 -5

10 -3 ∼ 10 -4 << 1 • B o = ρgR 2 γ ∼ 10 3 1010 -10 10 -2 ∼ 10 -4 << 1 • C a = ηU γ ∼ 10 -3 10 -5 10 -2 ∼ 10 -6 << 1.
As a result, we deduce that inertia and gravity are overwhelmed by viscous eects and capillarity. The dimensionless number that is the most relevant for the system is thus the capillary number C a .

Stokes approximation

When viscous and capillary eects prevail, the Navier-Stokes equation simplies as:

0 = -∇p + η∆v. ( 8
)
Figure 8: Scheme of a rectangular channel the dimensions of which are w × e × L

In our experiments, droplets are conned in the rectangular channel of a microuidic system of dimensions w × e × L where the cavity height e is much smaller than the typical lengths L and w over which uids ow (e <<min(L, w)).

Mass is conserved in the system such that the velocity eld v veries ∂vx ∂x + ∂vy ∂y + ∂vz ∂z = 0. Since the length scale in the direction of connement z is small relatively to those in the xy plane, the component v z is negligible: v z ∼ e/min(L, w)v x,y << v x,y . Therefore, ows are considered as nearly parallel in the plane xy.

In the Stokes equation, the viscous stress writes η∆v where ∆v = ∂ After all these simplications (called the lubrication approximation), the Stokes equation writes [START_REF] Guyon | Hydrodynamique Physique[END_REF]

η ∂ 2 v x,y ∂z 2 = ∇p. (9)
As explained in the last section, the pressure gradient ∇p that prevails is the capillary pressure gradient in the x-direction that writes

∂ x p = γ ∂ ∂ x ∂ 2 x h 1 + (∂ x h) 3/2 + 2 e . ( 10
)
We assume that the typical scale of variation of h in the x-direction is L while h varies from 0 to w along the y-axis. Thus, ∂ x h ∼ w/L << 1 and the capillary pressure gradient simplies in the lubrication approximation as

∂ x p ∼ γ∂ 3 x h. ( 11
)
To fully determine the velocity prole from the Stokes equation ( 9), the following boundary conditions

• v = 0 if we consider no-slip at the channel walls

• η o ∂v (o)
∂n = η w ∂v (w) ∂n at the droplet interface are considered, where (o) identies the oil phase and (w) the water phase.

Interfacial rheology

Up to now, we considered ideal interfaces as no stress is applied to them apart from the shear stress in both phases. However, a variation of surface tension can set a stress at the interface. For example, this variation can stem from a thermal gradient since surface tension is known to vary with temperature. If we consider the thermal gradient is set along the x-axis, the stress that is generated at the interface writes dγ dx = dγ dT dT dx . This stress creates ows from low surface tension to high surface tension [START_REF] Selva | Temperature-induced migration of a bubble in a soft microcavity[END_REF] and modies the continuity of the viscous stresses at the interface as follows

η o ∂v (o) ∂n -η w ∂v (w) ∂n = dγ dx (12)
where (o) identies the oil phase and (w) the water phase.

A variation of surface tension can also stem a gradient of concentration of the chemical compounds at the interface. For instance, a gradient of surface concentration of surfactants (Γ) sets a stress dγ dx that is equal to dγ dΓ dΓ dx . This gradient can occur when the interface is convected by pressure-driven ows in both phases. To estimate the value of dΓ dx , one has to compare the adsorption dynamics of surfactants to the interface with the typical convection time of the interface.

The behavior of surfactants at the interface relies on two mechanisms. First, surfactants diuse through the bulk to reach the subsurface, which is the region near the interface. Then, the surfactants have to overcome an energetic barrier to adsorb to the interface [START_REF]Adsorption dynamics of surfactants at the air/water interface: a critical review of mathematical models, data, and mechanisms[END_REF]. If the time of desorption and adsorption of surfactants is smaller than the typical convection time of the interface gure 9a), we can consider that the surfactants are constantly exchanged between the interface and the bulk. In this case, the surface concentration of surfactants at the interface is constant. On the other hand, if the time of desorption of surfactants is much larger than the typical convection time of the interface gure 9b), surfactants are advected by the ow and end up being unevenly distributed at the interface. The resulting gradient of surfactants surface concentration dΓ dx creates a surface stress that counters the convection [START_REF] Pawar | Marangoni eects on drop deformation in an extensional ow: The role of surfactant physical chemistry. i. insoluble surfactants[END_REF] and that rigies the interface. This stress, known as the Gibbs-Marangoni stress, can signicantly reduce the mean velocity of ows in both phases [START_REF] Ybert | Ascending air bubbles in solutions of surfaceactive molecules: Inuence of desorption kinetics[END_REF]. The behavior of surfactants typically depends on their solubility in both phases, from soluble, to poorly soluble up to non soluble for adsorption/desorption times that are much greater than the convection time [START_REF] Cuenot | The eects of slightly soluble surfactants on the ow around a spherical bubble[END_REF]. In the two extreme cases, we say that the interface is "rigid" (insoluble case) or "mobile" (soluble case).

Remark:

The presence of surfactants at the interface can also induce a surface viscosity (both in dilatation and in shear). However, this viscosity is disregarded in our experiments since the surfactant that is used, SDS, is known to add almost no viscosity at interfaces [START_REF] Miralles | Investigating the role of a poorly soluble surfactant in a thermally driven 2d microfoam[END_REF]. Figure 9: a) Scheme of the behavior of soluble surfactants at an advected interface. The quick ability of surfactants to populate and depopulate the interface allows to maintain a homogeneous concentration of agents at the interface. b) Extreme case of insoluble surfactants where the typical time of desorption is much larger than the time of convection of the surfactants along the interface. The unability of surfactants to desorb from the interface during its typical time of convection sets a gradient of surface tension at the interface.

In the studied system

The variation of surface tension can come from the gradient of temperature that is imposed in the channel and from a uneven distribution of surfactants at the interface:

η o ∂v (o) ∂n -η w ∂v (w) ∂n = dγ dT dT dx + dγ dΓ dΓ dx (13)
SDS is an anionic surfactant soluble in water. Its adsorption/desorption typical timescale is around 1 ms which is much smaller than the typical convection time of ows in our experiments which is approximately L/U ∼ 10 -3 10 5 ∼ 100 s. That is why we consider that the interfaces are fully mobile in the studied system (Γ constant along the interface and no surface viscosity).

As a conclusion to this section, the tangential stresses at the interface writes

η o ∂v (o) ∂n = η w ∂v (w) ∂n (14)
when no temperature gradient is imposed at the interface and

η o ∂v (o) ∂n -η w ∂v (w) ∂n = dγ dT dT dx (15)
when a temperature gradient is imposed. N.B. The eect of connement on the continuity of viscous stresses at the interface is discussed in the following section (6.3).

Additional pressure gradient in the dynamical meniscus

In the previous sections, we wrote that the curvature of the droplet menisci in the plane (y, z) was constantly equal to e/2. This assumption implies that the curvature of the menisci is dominated by the Laplace pressure. However, Bretherton has shown that, when a meniscus is advancing or receding on a solid boundary, its curvature could be locally modied by the viscous friction in between the meniscus and the wall. More specically, he has shown that, in the region that separates the lubrication lm (I) from the droplet spherical meniscus (II), the viscous shear had to balance the capillary pressure gradient imposed by this variation of curvature, see gure 10, [START_REF] Bretherton | The motion of long bubbles in tubes[END_REF]. In this region (III), called the dynamical meniscus, the Stokes equation writes:

η o ∂ 2 v ∂ 2 z = ∂p ∂ y (16)
We note H(y) the distance from the wall to the droplet interface such that in the lubrication approximation, the equation ( 16) veries

η o ∂ 2 v ∂ 2 z = γ∂ 3 y H ( 17
)
where γ∂ 3 y H is the capillary pressure gradient in the meniscus. Bretherton nds that, in the case of a conned non-wetting droplet, the viscous friction associated with the droplet motion on the wall is sublinear with the droplet velocity and scales as C 2/3 a .

Figure 10: Scheme of the droplet meniscus conned near a solid boundary. Hodges et al. denes three regions: the spherical meniscus (II), the dynamical meniscus (III) and the at lm (I). The indexes o and i respectively describe the outer phase and the inner phase. The lm thickness is noted h ∞ and l * denes the horizontal extent of the lubrication lm. In their study, it is the solid boundary that is pulled at a velocity U d . r is the half-height of the cavity. The gure is extracted from [START_REF] Hodges | The motion of a viscous drop through a cylindrical tube[END_REF].

In a review [START_REF] Cantat | Liquid meniscus friction on a wet plate: bubbles, lamellae and foams[END_REF], Cantat estimates the numerical expression of the viscous force that prevails in the dynamical meniscus of a bubble, see the Appendix page 114. She shows that the force dependence on the capillary number varies with the boundary condition that is imposed on the liquid/liquid interface at the level of the dynamical meniscus.

In the case of a stress-free condition

The stress-free condition neglects the viscous coupling in between the droplet phase and the outer phase. In this case, the interface is fully mobile and its velocity is set by the viscous dissipation in the outer phase. The boundary condition writes,

η o ∂v ∂z = 0 (18)
Cantat nds that the viscous force per unit of length writes f f sf = 3.84γC

2/3 a if the meniscus is advancing and f r sf = -1.1γC

2/3 a if the meniscus is receding.

Provided that the interface is not rigidied by surface active agents, this condition is by denition true for a bubble/liquid interface since the viscosity of the bubble is negligible.

2. In the case of a rigid interface v = 0 at the interface in the bubble frame. Cantat nds that the viscous forces per unit of length in the meniscus are f f sl = 3.75γC

2/3 a + 0.47γC 1/3 a 2l
e and f r sl = -3.07γC

2/3 a -0.47γC 1/3 a 2l
e , where 2 e is the radius of the meniscus in region II and l is the horizontal extent of the dynamical meniscus.

In order to determine the boundary condition that needs to be considered for the case of a water droplet in oil in the region of the dynamical meniscus, we can rely on the works of Hodges et al. [START_REF] Hodges | The motion of a viscous drop through a cylindrical tube[END_REF]. They have shown that the boundary condition in that region depends mostly on the viscosity ratio between the inner phase and the outer phase λ = ηo ηw and the parameter ∼ h∞ e that scales as C a . By SLIP, Hodges et al. mean that there is a stress-free condition at the interface. By NO-SLIP, they mean that the boundary velocity is set by the ow in the droplet. The gure is extracted from [START_REF] Hodges | The motion of a viscous drop through a cylindrical tube[END_REF].

In the studied system, λ = ηo ηw = 25 and ∼ C 2/3 a ∼ (10 -6 ) 2/3 ∼ 10 -4 . Therefore, we have 1 << λ << -1/2 and the boundary condition is stressfree in the dynamical mensicus (III o ) and in the lm (I o ). In the plane (x, z), this condition writes

η w ∂v (w) ∂z = 0 (19)
In the spherical meniscus region of the droplet (II o ), the boundary condition is dened as 'No-Slip' and veries

η w ∂v (w) ∂z = η o ∂v (o) ∂z (20)
In this region, we can assume that the viscous shear lengthscale is approximately the same for the inner and the outer phase and scales as e, the channel height. However, the viscosity of the inner phase is 25 times more viscous than the outer phase. Therefore, the boundary condition writes

η o ∂v (o) ∂z ≈ 0 (21)
In the experiments, we consider that the ow velocity in the droplet is almost zero such that v = 0 at the droplet interface in region (II).

Summary

We study the dynamical behavior of a mineral oil droplet in water mixed with SDS at 2 cmc. The droplet is conned in the rectangular channel of a microuidic channel and does not wet the walls. Given the geometrical and the physico-chemical characteristics of the experimental system, the droplet behavior is governed by viscous and capillary eects.

In these conditions, the Stokes equation, written in the lubrication approximation, can be invoked to describe the velocity eld in both phases. The motion of the droplet is induced by Laplace pressure gradients (≈ γ∂ 3 x h) in the equatorial plane (x, y) and can in turn induce a capillary pressure gradient (≈ γ∂ 3 z H) in the plane (y, z), along the droplet dynamical menisci (region III), that are moving normally to the ows. These regions concern mainly the pocket of the deformed droplet and the extremities of the droplet. In these dynamical menisci, the viscous force is sublinear with the outer phase velocity and scales as C -stress-free in the dynamical menisci of the droplet and in the lubricating lms

η w ∂v (w) ∂n = 0 (22)
v = 0 at the interfaces of the droplet spherical menisci (mainly in the gutters). In the specic case in which a thermal gradient is imposed along the droplet interface, the boundary condition writes

η o ∂v (o) ∂n -η w ∂v (w) ∂n = dγ dT dT dx (23)
We consider that there is no interfacial rheology induced by the presence of surfactants.

Part II

Materials and Methods -The thermomechanical actuation

Many techniques have already been developed in microuidics to handle droplets based on a large variety of physical mechanisms: electrophoresis [START_REF] Manz | Electroosmotic pumping and electrophoretic separations for miniaturized chemical analysis systems[END_REF], magnetic eld [START_REF] Padovani | Electropermanent magnet actuation for droplet ferromicrouidics[END_REF], acoustic eld [START_REF] Chong | Active droplet generation in microuidics[END_REF], marangoni eects [START_REF] Farahi | Microuidic manipulation via marangoni forces[END_REF][START_REF] Baroud | An optical toolbox for total control of droplet microuidics[END_REF]. In this manuscript, we focus on an additional technique, called the thermomechanical actuation, that can propulse, stop, split, store and sort droplets. All these functionalities have been demonstrated few years ago by Miralles et al. [START_REF] Miralles | Migration of biphasic systems by thermal actuation in microconnement[END_REF]. In this manuscript, we use this technique to impose a local variation of the topography in the microuidic channel and to study, in a more fundamental perspective, the response of a droplet to this topographic variation.

The thermomechanical actuation technique relies on the electrical actuation of a micropatterned resistance that locally heats the microuidic channel by the Joule eect [START_REF] Selva | Temperature-induced migration of a bubble in a soft microcavity[END_REF]. This chapter is dedicated to the description of the microfabrication of the thermomechanical device and to the characterization of the thermal eects that are induced on the droplets: the thermal dilation of the channel walls and the increase of surface tension with temperature.

7 Microfabrication of the thermomechanical device

The PDMS microuidic chip

The microuidic chip is made in a single block of poly(dimehtyl)siloxane (PDMS).

The PDMS block is composed of polymer RTV-615 mixed with a crosslinker at a weight ratio 1/10. The crosslinker/polymer mixture is poured on a silicium wafer (soft-lithography technique [START_REF] Duy | Rapid prototyping of microuidic systems in poly(dimethylsiloxane)[END_REF]) over which the patterns of the microuidic channels are designed in photoresist SU8 as the result of a photolithography process [61] (see gure 14). The mixture is cooked at 70 • C for 2 hours.

Figure 14: Soft-lithography process and photolihtography process.

Substrate with micro-resistances

The substrate is a 700 µm thick borosilicate glass wafer on which are deposited 15nm of chromium and 150 nm of gold. (ACM Grenoble). The two layers of metal are patterned with an etching process using S1818 photoresist [START_REF] Miralles | Migration of biphasic systems by thermal actuation in microconnement[END_REF]. Chromium patterns constitute the heating resistances and gold patterns serve as electrical connectors as their resistance is much smaller. The resistor width is 50 µm and its length is 400 µm. Its typical ohmic value is 300 ohm. The resistances are electrically isolated by spincoating a 30 µm thick layer of PDMS on top of the glass substrate.

1. First, the liquid PDMS at 1:10 is spincoated onto the substrate at 2500 rpm during 40s.

2. Then, the substrate is cooked in the oven at 70 • C during 2h.

Assembly of the device and surface treatment

The microuidic chip and the substrate are exposed to an oxygen plasma for 45 s such that -OH are created at their surface. When the two surfaces are put in contact few seconds after the oxygen plasma, they become bonded with covalent bondings (Si-O-Si). Water is injected in the microuidic channel few minutes after the plasma exposition to ensure that the PDMS channel remains hydrophilic during the experiments [START_REF] Dhananjay | Hydrophilization and hydrophobic recovery of pdms by oxygen plasma and chemical treatmentan sem investigation[END_REF].

Actuation of the thermomechanical device

External wires are connected to the resistances and to an external voltage generator. The device is illustrated on gure 15. Remark:

Fluorescein is added to the water phase at 0.44 g.L -1 . Thus, when the droplet is illuminated by a UV lamp, all the eld of observation appears black, except from the outer phase that appears green (with the eyes) or white (with the camera).

11 Temperature increase in the system When a voltage is applied to the micro-patterned resistance, we measure a local temperature increase in the channel. This temperature increase has been characterized with an infrared camera (FLIR camera) that has a spatial resolution of 5 µm and a time resolution that is smaller than 100 ms, see gure 18.

The infrared camera captures the temperature ony at the surface of an object. In order to measure the temperature increase in the channel induced by the heating resistance, the temperature is captured on top of the sublayer of PDMS that is spincoated on the glass substrate.

To ensure that the temperature that is measured by the infrared camera comes only from the hot substrate and not from reected IR waves of the ambient atmosphere, a painting (a black-body solution) is deposited on the substrate (Velvet black painting) that has an emissivity close to 1 [START_REF] Becker | Surface temperature and emissivity at various scales: Denition, measurement and related problems[END_REF]. Then, the temperature is captured at the surface of this thin layer of painting. In the following experiments, the room temperature is set to 26 • C.

Spatio-temporal prole of the temperature gradient in the system Figure 18a illustrates the temperature prole along the x-axis on one side of the hot resistance (located at x = 0) for electrical powers varying from 0 to 75 mW. Since the length of the resistance in the y-direction is larger than the typical width of the channel, the temperature is considered as homogeneous along the channel width. As suggests gure 18a, the spatial temperature prole is not linear. However, gure 19 shows thar we can extract some bounds for the temperature gradient : 8000 < ∂ x T < 16000 K.m -1 . From now on, we consider that ∂ x T is constant and is of the order of magnitude 10 4 K.m -1 .

-1000

-800 -600 -400 -200 0

Thermal diusion accross the channel height

The time of the thermal diusion accross the channel height scales as τ T = e 2 /D where D is the thermal diusion coecient of mineral oil (D = κ/(ρC) ∼ 0.1333/1900/870 ∼ 10 -7 m 2 .s -1 ), at room temperature [START_REF] Nadolny | Thermal properties of a mixture of mineral oil and synthetic ester in terms of its application in the transformer[END_REF]. We nd that τ T ∼ 10 -10 /10 -7 ∼ 10 -3 s, which is much smaller than the typical time of our experiments (fews tens of seconds). Thus, even though the temperature has only been measured at the surface of the channel bottom wall, the latter can be considered as constant accross the channel height.

N.B. In this calculation, we provide the thermal diusion coecient of mineral oil at room temperature but we expect it to be a bit smaller at higher temperature. Therefore, the calculation t T ∼ 10 -3 s gives an upper bound for the thermal diusion time accross the channel height.

Eect of a ow in the channel on the temperature prole

The temperature prole could be altered by ows in the channel. In the case of a uni-directional stationary ow along the x-axis of mean velocity V , the temperature would obey the advection-diusion law that writes V ∂ x T = D∂ 2 z T , z being the direction along which the dimension of the channel is the smallest.

To know if convection might alter the temperature gradient, we need to compare the time of the heat transfer induced by convection and the time of the heat transfer induced by diusion. Thus, we dene the Peclet Number:

P e = t dif f t adv = V l D .
In our experiments, we recall that V = 10 µm.s -1 l = 1 mm and

D = 10 -7 m 2 .s -1 .
We nd that P e ∼ 10 -1 which means that t dif f << t adv and that the heat transfer is dominated by diusion. Therefore, the temperature gradient is not altered by the ows in the channel.

Variation of the temperature with the electrical power From now on, the variable T describes the maximum temperature in the channel on top of the hot resistance, at x = 0. Figure 20 shows the linear evolution of T as a function of the electrical power in the system. The electrical power P that is applied at the edges of the resistance equals to U 2 R where U is the voltage at the edges of the resistance and R its ohmic value. We nd that within the range 0 < P < 100 mW, the maximum temperature veries T = T amb + β T P where the ambient temperature, 12 Eect of the temperature on the channel topography

T amb , is equal to 26 • C and β T = 0.5 • C.mW -1 .
The thermal expansion of PDMS RTV-615 has been characterized in the literature and its coecient of thermal expansion α T is equal to 3 * 10 -4 K -1 [START_REF] Kim | Thermal expansion and contraction of an elastomer stamp causes position-dependent polymer patterns in capillary force lithography[END_REF][START_REF] Brown | A cantilever-free approach to dot-matrix nanoprinting[END_REF]. In a (x, z) plane, we can estimate the maximum channel deformation, δ(z), that results from the temperature increase in the device. This deformation veries δ(z) = α T hT where h is the initial height of the PDMS layer and T , the maximum temperature in the material.

• For the 30 µm thick PDMS sublayer at the bottom of the channel, we nd that δ(z) = 30 * 10 -6 * 3 * 10 -4 * 80 ∼ 0.1 µm for T = 80 • C.

• For the block of PDMS on top of the channel, we nd that δ(z) = 10 -2 * 3 * 10 -4 * 80 ∼ 100 µm, if we consider that this block is approximately 1 cm thick.

This would mean that the 30 µm high channel could be completely blocked at the level of the hot resistance. We do not observe such a deformation in our experiments, which might be explained by the fact that this calculation is very naive. Indeed, it assumes for example that the temperature is homogeneous across the 1 cm PDMS block which is unlikely true. In order to further characterize the thermal dilation of the channel walls, we carry out a direct experimental calibration.

12.1 In-situ characterization of the channel thermal dilation

Most methods used to measure deformations inside a sealed cavity are optical. Among them, we list:

• Fluorescence measurement using the linear relationship between the uorescent solution layer thickness and intensity [START_REF] Hardy | The deformation of exible pdms microchannels under a pressure driven ow[END_REF].

• Confocal microscopy, very convenient to measure topological shapes but its resolution of about 1 µm makes it unadapted to measure deformations of few µm [START_REF] Serra | A simple and low-cost chip bonding solution for high pressure, high temperature and biological applications[END_REF].

• Refracted images of a pattern produced by a curved interface. Dangla et al. used this optical method to measure the swelling of micro-channels induced by alcane oils [START_REF] Dangla | Microchannel deformations due to solvent-induced pdms swelling[END_REF].

These methods turn out to be complex to implement to measure the thermal dilation of the channel walls in-situ. In the rst method, uorophores bleach with temperature and in the last method, multiple interfaces are curved along the light path and these interfaces might not be curved towards the same direction, which makes the analysis of the refracted patterns quite dicult.

Home-made method to measure the thermal dilation of the channel

To quantify the topological change induced by the thermal dilation of the channel walls, we designed a simple home-made system that we call the calibration step chip. This PDMS chip is composed a 200 µm wide channel that is 4, 5, 6 up to 15 µm high, with incremental steps of 1 µm, see g. 21a. The chip is fabricated with a soft-lithography technique using a brass mold that has been milled with a micro-milling machine (the resolution of which is smaller than 100 nm).

The substrate on top of which is sealed the calibration chip is composed of a glass slide on which micro-resistances are patterned and a 30 µm thick layer of PDMS is spincoated -the substrate is similar to the one that is described in section 1. The channel is full of air (n = 1) to increase the refractive index dierence in between the channel and the PDMS walls (n = 1.42).

When the channel dilates of d z , the roof touches the bottom channel wall such that a contact line in between the two layers appears at step heights that are smaller or equal to d z , see gure 21. By measuring the maximum height of the step at which the contact line is observed for a given temperature in the device, we can quantify the maximum deformation of the channel walls in the z-direction. The variation of this deformation as a function of the maximum temperature in the channel is illustrated in gure 22. The calibration measurements show that the channel deformation is comprised in between 0 µm and 10 µm. Unfortunately, the calibration step chip fails to quantify the horizontal extent of the connement gradient that is established in the channel. An interesting observation might help us to interpret the unexpected large dilation of PDMS When the 30 µm thick sublayer of PDMS is not present in the chip, we do not observe any dilation of the channel, even at the smallest step heights. Oddly enough, we deduce that it is the PDMS sublayer that dilates the most in the channel. Contrary to the main PDMS block that constitutes the chip, the sublayer of PDMS is not degassed during the microfabrication process. Thus, some air bubbles might remain trapped in the thin layer of polymer such that the latter dilates largely when the temperature is increased in the channel. To validate this interpretation, the same calibration measurements should be carried out after having degassed the PDMS sublayer.

Mechanical prolometer measurements

In order to measure the horizontal extent of the channel dilation, we open the thermomechanical device to measure the topography of the bottom wall with a mechanical prolometer (Dektak 6M) when the resistance heats the device. The PDMS sublayer is stiened by two glass slides that are separated by the typical width of the channel 200 µm [START_REF] Miralles | Migration of biphasic systems by thermal actuation in microconnement[END_REF]. The thermal dilation prole of the bottom channel wall is shown in gure 24. We observe that this deformation is comprised in between 500 nm to 2.5 µm for 30 < T < 80 • C and that its horizontal extent is around 60 µm. Figure 25 illustrates the maximum deformation of the sublayer as a function of the maximum temperature in the channel. This deformation is approximately 4 times smaller than the one that has been measured with the calibration step chip. In the following, we only consider the values of d z provided by the step chip that accounts for the dilation of all the channel walls and that has a geometrical conguration more similar to the experimental device. N.B. We measured the deformation prole of the sublayer of PDMS after a succession of cycles of temperature increase. We found the sublayer recovers a nearly at surface when the resistance is switched o. Therefore, we consider that the thermal dilation of the channel is reversible.

13 Eect of temperature on surface tension: the Marangoni eect Surface tension is known to vary with temperature. The sign of dγ dT (∂ T γ) depends on the liquids that are at play. In the case of mineral oil and water mixed with uorescein and sodium dodecyl sulfate, we characterize the surface tension variation with temperature. We observe that surface tension increases with temperature. The gradient ∂ T γ has been measured using the pendant drop method (KRÜSS) coupled with a thermal bath (JILABO), see gure 26. The software provided by Krüss measures the shape of the pendant droplet and extracts a surface tension by balancing the gravitational pressure with the Laplace pressure. For each temperature, a picture of the pendant droplet immersed in the thermal bath is taken at equilibrium and each of the images is associated with a surface tension [START_REF] Berry | Measurement of surface and interfacial tension using pendant drop tensiometry[END_REF]. We nd that dγ dT = 2.3 * 10 -5 N.m -1 .K -1 . Therefore, the stress that is imposed at the interface of the droplet in the vicinity of a hot resistance is equal to

dγ dx = dγ dT dT dx ∼ 2 * 10 -5 * 10 4 ∼ 0.2Pa. (24)

Summary

We describe a technique based on a thermomechanical actuation that allows to change in a reversible way the topography of the channel. This actuation is based on the heating of a micropatterned resistance that establishes a gradient of temperature of around 10 4 K.m 1 on the lengthscale of the droplet. This temperature gradient has two eects on the droplet:

1. it imposes a Marangoni stress at the droplet interface.

The value of the stress is dγ dx = dγ dT dT dx ∼ 2 * 10 -5 * 10 4 ∼ 0.2 N.m -2 (P a ) when the resistance heats at 80 • C (P = 130 mW).

it induces a change of the channel topography because of the thermal expansion

of the latter.

The maximum dilation that is observed in the channel has been characterized in-situ and seems to be bounded in between 5 µm and 10 µm for electrical powers varying from 50 mW to 110 mW.

Part III Droplet deformation

In this chapter, we focus on the droplet deformation induced by a local variation of the channel topography. This topographic variation is induced by the thermal dilation of the channel walls. The temperature increase that creates this dilation stems from the heating of a micro-resistance that is located below the microuidic system, see section Materials and Methods for more details.

We observe that when a droplet is located on top of the heating micro-resistance, it deforms until it can even break. Several thermal eects could account for the droplet deformation. Indeed, beyond the connement gradient that is imposed on the droplet because of the channel dilation, a surface tension gradient sets along the droplet interface that provokes a Marangoni ow in both phases towards the hot regions where the surface tension is higher for the working liquids.

To discriminate the relative contribution of the Marangoni eect and of the connement gradient eect on the experimental droplet deformation, a "mechanical" device has been designed to mimic the topographical variation that is induced by the thermal channel wall dilation without any temperature increase. The connement gradient is indented in the channel during the microfabrication process. By comparing the droplet deformation that is achieved in this "mechanical" device with the one that is observed in the "thermomechanical" device, we are able to show that the thermal Marangoni contribution to the droplet deformation process is negligible. Thus, the droplet deformation is mainly due to the local variation of the channel topography. We show that a variation of only 15 % of the channel height can induce the droplet deformation and even its break-up.

The droplet deformation is driven by a capillary motion and is mediated by viscous dissipation. To model the deformation mechanism, we calculate the minimum surface energy of the droplet given the geometric constraints that are imposed in the channel. Then, we couple this calculation to a power balance between the viscous dissipation power and the increased surface excess energy of the droplet per unit of time that manages to recover the main features of the dynamics of the droplet deformation.

14 State-of-the-art on the droplet break-up Droplet break-up is a basic functionality that is required in emulsion science for droplet production. The break-up mechanism has been extensively studied in the literature. In 1873, Plateau showed that droplets of controlled volume can be produced as the result of the destabilization of a liquid thread; this instability is now known as the Rayleigh-Plateau instability. Since the pioneering study of Taylor on the droplet deformation induced by an extensional ow, break-up has also been investigated in a large variety of ow and geometrical congurations.

When droplets are placed in shear ows [START_REF] Janssen | Boundary-integral method for drop deformation between parallel plates[END_REF] or in extensional ows [START_REF] Stone | Relaxation and breakup of an initially extended drop in an otherwise quiescent uid[END_REF][START_REF] Stone | Dynamics of drop deformation and breakup in viscous uids[END_REF], they deform until they are suciently elongated to allow the growth of capillary instabilities (Rayleigh-Plateau instability). In these ow conditions, it has been demonstrated that droplet breakup occurs when the capillary number exceeds a certain threshold value C * a . This value varies non-monotonically with the viscosity ratio in the biphasic system, λ [START_REF] Vananroye | Eect of connement on droplet breakup in sheared emulsions[END_REF]. The connement also plays a crucial role in these elongationmediated breakups, [START_REF] Ulloa | Eect of connement on the deformation of microuidic drops[END_REF], see Varanoye et al. 2006 for a review. For instance, Guillot et al. have proved that the growth of capillary instabilities could be prevented in conned geometries [START_REF] Guillot | Stability of a jet in conned pressuredriven biphasic ows at low reynolds number in various geometries[END_REF][START_REF] Guillot | Stability of a jet in conned pressure-driven biphasic ows at low reynolds numbers[END_REF].

Since the ow conditions that are required to induce droplet break-up depend on many parameters (shear or extensional ow, viscoity ratio and connement), active techniques have been developed to break droplets on demand using, for instance, an electrical eld [START_REF] Yuehao | Control of the breakup process of viscous droplets by an external electric eld inside a microuidic device[END_REF] or an acoustic eld [START_REF] Chong | Active droplet generation in microuidics[END_REF]. The electrical eld imposes a Lorentz force on the droplet that is polarized as a dipole. It results in the elongation of the droplet until the latter breaks. The acoustic eld deforms the droplet interface under the eect of the radiation pressure.

In parallel, some researchers have designed complex geometric microuidic congurations to break up droplets with a fairly high throughput: T-junction [START_REF] Jullien | Droplet breakup in microuidic t-junctions at small capillary numbers[END_REF][START_REF] Leshansky | Obstructed breakup of slender drops in a microuidic t junction[END_REF][START_REF] Menech | Modeling of droplet breakup in a microuidic t-shaped junction with a phase-eld model[END_REF][START_REF] Leshansky | Breakup of drops in a microuidic t junction[END_REF][START_REF] Link | Geometrically mediated breakup of drops in microuidic devices[END_REF], cross-ow channels [START_REF] Cubaud | Deformation and breakup of high-viscosity droplets with symmetric microuidic cross ows[END_REF], obstacles [START_REF] Nishimura | Breakup of conned drops against a micro-obstacle: an analytical model for the drop size distribution[END_REF], step channels [START_REF] Dangla | The physical mechanisms of step emulsication[END_REF]. In T-junction geometries or in cross-ow channels, that are very often used in microuidic devices, breakup also occurs when the capillary number is higher than a threshold value C * a . In addition, the breakup criteria depends on the geometrical parameters of the system like the initial length of the droplet and the width of the inlet channel [START_REF] Leshansky | Breakup of drops in a microuidic t junction[END_REF].

In the step emulsication, the liquid thread destabilizes when it reaches a step change in the cavity height or when it undergoes a connement gradient [START_REF] Dangla | The physical mechanisms of step emulsication[END_REF][START_REF] Dangla | 2D droplet microuidics driven by connement gradients[END_REF]. In this case, the radius of the droplets that are produced depends on the connement gradient and on the initial height of the cavity. In the step-emulsication process, the thread is injected at a constant ow-rate Q in a stepped cavity. The initial height of the thread is h 0 while h 0 + ∆h is dened as the height of the output cavity. This process results in the production of controlled volume droplets, the radius of which is h 0 when ∆h > h. A variation of this technique consists of injecting the thread into a large cavity where the top wall is tilted with an angle α that is comprised between 0.5 • and 4.5 • . Dangla et al. [START_REF] Dangla | 2D droplet microuidics driven by connement gradients[END_REF] discuss the criteria of break-up of this system and predict that the radius of the droplets that are produced veries

R * = 0.44 1 + 0.2 tan(α) w h 0 h 0 tan(α) . ( 25
)
Our study relies on an additional technique to achieve droplet break-up. This technique -so far called the thermomechanical actuation-has been already developed by [START_REF] Selva | Thermocapillary actuation by optimized resistor pattern: bubbles and droplets displacing switching and trapping[END_REF]44]. In this work, we propose a theoretical framework to predict the droplet deformation and break-up achieved with this thermomechanical device.

Experimental observations

We observe that when the droplet is located on top of a hot resistance, a neck forms at its center. The droplet deforms in the equatorial plane (in-plane) like a peanut (g. [START_REF] Hodges | The motion of a viscous drop through a cylindrical tube[END_REF]. By mass conservation, the outer phase (water) is pulled towards the center of the droplet to occupy the volume that is released by the formation of the neck.

In the meantime, the volume of oil is conserved such that the droplet extremities move slowly away from the resistance.

In order to describe the droplet deformation, several parameters are considered: the droplet length L(t), the neck extension in the y-axis and the neck extension in the x-axis. We dene h(x, t) as the in-plane distance in between the channel wall and the droplet interface. The maximum value of this distance which by denition occurs at x=0, on top of the heating resistance, is noted h 0 (t). Thus, the neck extension in the y-axis writes w -2h 0 (t) where w is the channel width. The neck extension in the x-axis is dened as b(t). By volume conservation, we should be able to relate the neck extension to the droplet elongation. In the following sections, we decide to describe the droplet deformation through the spatio-temporal evolution of the neck instead of the droplet length.

Gutters are located at the four corners of the channel. The typical size of these gutters, schematically indicated in the green-boxed inset in gure 27b, scales linearly with the channel height e. The length of these gutters is noted L g . All these notations are illustrated in Figure 27. When the resistance is switched on, the droplet deforms until it can either reach an equilibrium shape (gure 29) or break into two (gure 28). The fate of the droplet depends on the temperature increase that is induced in the system. Beyond a certain critical value of the temperature increase T * , the droplet deforms until its neck width gets smaller than the channel height (w -2h 0 (t) = e) and the droplet breaks. Snapshots illustrate the droplet break-up process in gure 28. The break-up criteria w -2h 0 (t) = e is discussed further in this chapter.

If the temperature increase in the channel is smaller than the critical temperature, T < T * , the droplet deforms until it reaches an equilibrium shape. Figure 29 illustrates a snapshot of the droplet deformation until the neck width stabilizes to a constant value. The typical timescale of the droplet deformation up to its equilibrium state is few seconds to tens of seconds, for oil-in-water droplets with hundreds of micrometers length, in channels with typically tens of microns in height.

Above T * : Figure 30 shows the time evolution of the neck width dened as w -2h 0 (t) for the breaking (g. 28) and the non-breaking droplet case (g. 29). We recover that in the breaking case (T = 74 • C), the neck width w -2h 0 (t) decreases over time until it reaches the value e = 30 µm and the droplet breaks. In the non-breaking case (T = 54 • C), the neck width decreases until it reaches a plateau. In gure 32, we plot the dynamics of a series of droplets that break. The only geometrical characteristic that dierentiates these droplets is their length. The temperature is the same for all these droplets and is higher than the critical temperature for them to break. On this plot, we put a line for w -2h 0 = e, and we recover once again that the dynamics encounters a transition at this point: the droplets break. Lastly, we observe that the break-up dynamics increases with the droplet length. We now look at the time evolution of the entire interface prole dened as h(x, t), see gure 33a. The latter appears to be self-similar at late times (the last 60 % of the dynamical process) such that we can dene the function p r that veries h(x, t) = h 0 (t)p r (x/b), for -b < x < b. In gure 33b, this function p r is plotted versus the dimensionless variable u dened as u = x/b, b being the extent of the neck in the x-axis. The fact that the interface prole h(x, t) can be expressed as the multiplication of two functions that depend respectively on time and space simplies the model of the droplet deformation process. This separation of variables will be fully elaborated in the next section. 16 The break-up criteria: experimental check

We have observed in the last section that droplets break when the neck width w -2h 0 (t) gets smaller than the channel height e. This criteria is consistent with several studies that deal with droplet break-up in a conned geometry in the literature [START_REF] Leshansky | Breakup of drops in a microuidic t junction[END_REF][START_REF] Dangla | 2D droplet microuidics driven by connement gradients[END_REF].

However, as an experimental check, we show that when w -2h 0 (t) = e, the surface of the deformed droplet gets larger than the surface of two splitted droplets such that it becomes more energetically favorable for the droplet to break. We dene τ break as the experimental time at which the droplet breaks.

Approximately, we calculate the perimeter Φ of the in-plane droplet interface in the neck region at most ∆t = 100 ms before the break-up and 100 ms after the break-up, see g. [START_REF] Selva | Thermocapillary actuation by optimized resistor pattern: bubbles and droplets displacing switching and trapping[END_REF]. We assume that the cavity height prole along this in-plane perimeter is constant during this period of time -that lasts at most 200 ms.

After a simple image processing, we nd that Φ(τ break -∆t) = 512 µm and that Φ(τ break + ∆t) = 460 µm. We can then assume that the surface of the droplet interface in the neck region is approximately 512*e µm 2 just before the break-up and approximately 460*e µm 2 just after the break-up. While this preliminary analysis is highly approximate, we will show more rigorously in the following that droplet breakup is energetically favorable under the conditions of a sloped channel geometry.

Figure 34: a) Zoom on the neck region of a droplet captured just before and just after the breakup. b) The contour of the in-plane droplet interface that is detected over time to measure Φ is highlighted in red.

The following sections provide more robust insights on the physical mechanisms that are at play during the droplet deformation. [START_REF] Nakano | Single-molecule pcr using water-in-oil emulsion[END_REF] The physical mechanisms at play

We identify two physical mechanisms that may account for the droplet deformation: the Marangoni eect and the connement gradient eect (called the mechanical eect). Both eects are discussed in this section. 17.0.1 Why might the droplet deform when a gradient of surface tension is established at its interface ?

The thermal Marangoni eect corresponds to a variation of surface tension with temperature. The surface tension gradient that establishes along the droplet interface sets a stress at the interface such that the continuity of the tangential stresses writes

η o ∂v (o) x ∂n -η w ∂v (w) x ∂n interf ace = dγ dx (26)
where n denes the normal direction at the interface. In the studied system (mineral oil in water and SDS), surface tension increases with temperature. The Marangoni stress that is associated with this increase has been characterized in chapter Materials and Methods.

When the temperature is 70 • C at the level of the resistance, we nd that the order of magnitude of the Marangoni stress is 0.2 N.m -2 .

dγ dx = dγ dT dT dx ∼ 10 -5 * 10 4 ∼ 10 -1 N.m -2 (27)
Figure 35 illustrates the ows that can stem from the thermal Marangoni eect at the droplet interface. These ows are oriented towards the high surface tension regions.

Figure 35: Flow pattern that may arise from the thermal Marangoni eect. The droplet interface is moved by the Marangoni stress and its advection creates a ow near the interface in both phases. This ow is oriented towards the hot regions. By symmetry, there might be ow recirculations in the droplet phase such that the droplet interface can be deformed by the incoming ow of the outer phase towards the neck of the droplet.

We prove in the following that the Marangoni eect is not the prevailing mechanism in the droplet deformation process. However, the Appendix page 111 provides an estimation of the order of magnitude of a break-up time that would only be due to Marangoni ows. 17.0.2 Why might the droplet deform when the channel locally contracts ?

The mechanical eect corresponds to the local variation of topography induced by the thermal dilation of the channel walls. This dilation is illustrated in gure 36.

The gure recaps all the parameters that describe the droplet deformation. More specically, the channel thermal dilation is characterized by two parameters: its horizontal extent, d x and its vertical extent, d z . We recall that the variation of d z with the temperature has been characterized in-situ while we have not characterized the horizontal extent d x of the topographic variation in the channel, see the chapter Materials and Methods. The function e(x) describes the channel height that varies along the droplet length.

Figure 36: Geometrical parameters that are used to describe the droplet deformation.

When the channel height is constant and equal to e, we recall that the radius of the droplet meniscus in the plane (y, z) is constant and equal to e/2, see the chapter Introduction for more details. When the droplet undergoes a local gradient of connement, this radius is decreased as e(x)/2, meaning that the droplet curvature in the (y, z) suddenly increases. Where the connement gradient is localized, the droplet mean curvature writes

C = ∂ 2 x h 1 + (∂ x h) 3/2 + 2 e(x) (28)
where

∂ 2 x h √ 1+(∂xh) 3/2 is its in-plane curvature. At equilibrium, to maintain C constant, ∂ 2
x h √ 1+(∂xh) 3/2 should decrease when 2 e(x) increases. Initially, the in-plane curvature of the droplet is zero at the level of the side channel walls, see gure 36. Thus, when the connement gradient is imposed on the droplet, this curvature is expected to become more and more negative as e(x) decreases. This mechanism could account for the neck formation at the center of the droplet where the connement gradient is established. Indeed, Dangla et al. have shown that a connement gradient of only few degrees could break a thread, as the result of the formation of this negative in-plane curvature in response to the decrease of the channel height [START_REF] Dangla | 2D droplet microuidics driven by connement gradients[END_REF].

In order to determine whether it is the mechanical or the Marangoni eect that prevails in the droplet deformation process, we design an experimental device that allows to isolate the connement gradient eect without any temperature increase.

Comparison with a purely mechanical device

A device that mimics the connement gradient induced by the hot resistance has been designed and fabricated using a micro-milling machine. This device, that is schemed in gure 37, contains a symmetric and linear connement gradient. This connement gradient is dened by two parameters d x and d z that describe respectively its horizontal and vertical extent. Dierent systems were designed in which d x varies from 400 to 800 µm and d z from 3 to 10 µm, e varying from 20 µm to 40 µm and w varying from 200 µm to 400 µm. The values of d x and d z are measured with a mechanical prolometer (Dektak 6M). The ratio dz dx denes the tilt θ of the channel top wall and this tilt ranges from 0.2 • to 1.4 • . In these devices, called "mechanical" devices, the channel height e(x) can be dened as a linear piecewise function:

• for -∞ < x < -d x , e(x) = e • for -d x < x < 0, e(x) = e -d z -θx • for 0 < x < d x , e(x) = e -d z + θx • for d x < x < ∞, e(x) = e
Figure 37: Scheme of a mechanical device in which a connement gradient is indented. The system is called the "mechanical" system and is compared to the "thermomechanical" system. a) 3D view b) Cross view in the plane (x, z).

In the "mechanical" devices, droplets deform when they undergo the gradient of connement, similarly to the "thermomechanical" devices. Figure 38 illustrates a snapshot of a 1600 µm long droplet that deforms in a "mechanical" device in which d z = 5 µm, d x = 400 µm, e = 30 µm and w = 400 µm.

In a "thermomechanical" device, the connement gradient is applied on demand on the droplet once the latter is static in the channel. In the "mechanical" device, the droplet has to be pushed under the connement gradient. The droplet velocity is monitored with a pressure controller that adjusts the pressure at the entrance of the channel. When the droplet is centered under the connement gradient, the latter is stopped -the pressure is decreased to zero at the entrance of the channel. The droplet starts to deform as soon as it undergoes the connement gradient. In the following, we only observe the period of time during which the droplet is stopped in the channel. This period corresponds to the late times of the droplet deformation (typically the last 40-50% of the droplet deformation process).

Figure 38: Snapshots of the droplet deformation in a "mechanical" device in which d z = 5 µm, d x = 400 µm, e = 30 µm and w = 400 µm. The droplet is 1600 µm long. The droplet deformation is considered only after the third frame when the droplet is stopped in the channel. The images are equally spaced in time and the total duration of the snapshot is 104 seconds.

Figure 39 plots the time evolution of the entire interface prole of the droplet. The latter appears to be also self-similar at late times, this represents a further similarity to the "thermomechanical" device. For this device, we can also dene a function p m that veries h(x, t) = h 0 (t)p m (x/b) for -b < x < b. The function p m is plotted in Figure 39b versus u = x/b, b being the extent of the neck in the x-axis that is also constant over time.

Remark:

For this experiment, the neck extension in the x-axis is approximately 350 µm, similarly to the extension of the connement gradient d x that is equal to 400 µm. This observation is important for the model section. The droplet is 1600 µm long. The timelapse that is plotted is around 70s. During this period of time, h(x, t) is self-similar and writes h(x, t) = h 0 (t)p(x). We dene u = x/b where b is the x-axis extent of the neck that is constant over time.

In gure 40, we compare the self-similar function p r and p m respectively for the "thermomechanical" and the "mechanical" devices as a function of u (u = x/b where b is the extent of the droplet deformation in the x-axis). We observe that the self-similar functions in both devices collapse on top of each other with an error of 10%. This observation supports the idea that the droplet is mainly deformed by the connement gradient that is imposed in the channel and that the thermal Marangoni eect can be disregarded. Given these considerations, we write a model in the next section that aims at predicting the droplet equilibrium shape and its break-up given the geometrical constraints of the channel.

Predicting the droplet deformation shape

To predict the droplet deformation shape, we calculate the minimum surface that the droplet can adopt when the connement gradient is imposed.

Surface minimization

The surface energy of the droplet writes E = γS where S is the surface of the droplet. Surface tension leads the droplet to minimize its surface energy. We calculate the excess surface of the droplet that is the dierence between the surface energy of the droplet when the latter is deformed and the surface energy of the droplet when the latter has its initial plug-like shape. This calculation takes into account the variation of the channel height e(x). We write that the excess surface energy of the droplet is E Σ = γ∆S. In the following discussion, we assume that the surface tension is constant along the droplet interface such that minimizing E is equivalent to minimizing ∆S.

Because the droplet is centered at the level of the connement gradient, the axis x = 0 is an axis of symmetry for the droplet. Hence, we only consider the half-surface of the droplet and we distinguish three regions :

• the droplet extremity (1) which only translates in the channel

• the sides of the droplet (2) which are elongated during the evolution and where the gutters are localized

• the connement gradient region (3) where the neck forms.

Figure 41 illustrates the regions 1, 2 and 3. We calculate the excess surface of the droplet interface in each of these regions. In region 1:

By denition, the surface of the droplet interface is not modied in this region.

In region 2

The droplet is elongated by volume conservation. Indeed, the volume that is lost in region 3 for the formation of the neck is moved to region 2. The droplet is then elongated by ∆l.

Since we observe on gure 39 that the extent of the neck is the x-axis, b, is of the same order of magnitude as the connement gradient extent d x , we consider that the cavity height is constant in region 2, equal to e and decreases only in region 3.

The total surface of the droplet including the menisci is given by

∆S 2 = (2(w -e) + πe)∆l (29)
The droplet elongation ∆l is deduced from the volume conservation in the droplet.

The volume that is lost in region 3 is

Ω 1 = (w -e)e + πe 2 4 b -e (w -e)b -2 b 0 h(x, t)e(x)dx - b 0 πe 2 (x) 4 1 + ∂ x h 2 dx (30)
that simplies as

Ω 1 = 2 b 0 h(x, t)e(x)dx - b 0 πe 2 (x) 4 ( 1 + ∂ x h 2 -1)dx. (31)
This volume must be equal to the volume that is increased in region 2

Ω 2 = e (w -e) + πe 4 ∆l (32)
Ω 1 must be equal to Ω 2 by volume conservation such that we get

e (w -e) + πe 4 ∆l = 2 b 0 h(x, t)e(x)dx - b 0 πe 2 (x) 4 ( 1 + ∂ x h 2 -1)dx (33)
By replacing the expression of ∆l obtained with the volume conservation in equation [START_REF] Cantat | Liquid meniscus friction on a wet plate: bubbles, lamellae and foams[END_REF] in the expression of the excess surface in equation ( 29), we get the expression of the surface excess in region 2.

∆S 2 = 2(w -e) + πe (w -e)e + πe 2 4 2 b 0 h(x, t)e(x)dx - b 0 πe 2 (x) 4 ( 1 + ∂ x h 2 -1)dx (34)
In region 3

In this region, the surface of the droplet interface decreases to counter the excess surface induced by the connement gradient. The top channel is considered to be at while the bottom channel wall is locally deformed by the connement gradient tilted with a angle θ << 1.

1. The excess surface of the top interface writes

∆S t 3 = wb -2 b 0 h(x, t) + e(x) 2 dx -(w -e) b (35)
so,

∆S t 3 = -2 b 0 h(x, t) + e(x) 2 dx + eb ( 36
)
∆S t 3 = -2 b 0 h(x, t) + e(x) -e 2 dx (37)
Since ee(x) < d z and d z ∼ 5 µm, e(x)-e 2 << h(x, t) and the expression of ∆S t 3 simplies as

∆S t 3 = -2 b 0 h(x, t)dx (38)
2. The excess surface of the bottom interface writes

∆S b 3 = wb -2 b 0 h(x, t) - e(x) 2 dx -(w -e) b 1 cosθ (39)
simarly to ∆S t 3 , ∆S b 3 simplies as

∆S b 3 = -2 b 0 h(x, t) 1 cosθ dx (40)
As a result, ∆S 3 = ∆S t 3 + ∆S b 3 gives

∆S 3 = -2 b 0 h(x, t) 1 + 1 cosθ dx (41)
3. The excess surface of the droplet menisci writes

∆S m 3 = 2 b 0 πe(x)( 1 + ∂ x h 2 -1)dx (42)

Conclusion

The total excess surface of the droplet writes

∆S = -2 b 0 1 + 1 cos θ h(x, t)dx + 2 b 0 πe(x)( 1 + ∂ x h 2 -1)dx +(2(w -e) + πe)∆l (43)
In this expression, we replace

Minimization of the droplet surface

We minimize the excess surface of the droplet ∆S(h, b) in order to predict the equilibrium shape of the droplet deformation dened by b eq = b(t eq ) and h eq = h 0 (t eq ), t eq being the time at which the droplet reaches an equilibrium shape. We solve the system of equation

∂∆S ∂h (h eq , b eq ) = 0 (46)
and ∂∆S ∂b (h eq , b eq ) = 0 (47) using the matlab function fminsearch.

We nd that b eq and h eq depend on four geometrical parameters of the system: the channel width and height, w and e and the connement gradient extents d x and d z . For example, gure 42 illustrates the theoretical predictions of the droplet deformation for a w = 400 µm wide channel and for a connement gradient that has a d x = 400 µm x-axis extent. To build this phase diagram, we nd the minimum (h eq , b eq ) of ∆S for e varying from 15 µm to 50 µm and for d z varying from 0 µm to 10 µm. If h eq veries w -2h eq < e, we say that the droplet breaks. If h eq < 1 µm, we say that there is no deformation. Otherwise, we draw a transient region in which we predict that the droplet deforms but does not break. or non-breaking (above the orange line). A transient region is plotted above the orange line and corresponds to situations in which droplets deform but reach an equilibrium shape. Then, a darker zone is plotted that corresponds to cases in which droplets do not deform.

We are thus able to draw a phase diagram in gure 43 that predicts the fate of droplets as a function of three dimensionless parameters dz w , e w and dx w . The experimental data obtained with the "mechanical" devices are plotted on this phase diagram, see gure 44. The experimental data seem consistent with theory. To further validate this model, the theoretical predictions of h eq and b eq are compared to Surface Evolver simulations.

Remark:

The experimental data obtained with the "thermomechanical" device can not be illustrated on the diagram since d x has not been properly characterized in this device. Having said this, we give in the next section some details of comparisons of this model with the experimental data to extract values for d x . are added. Dark green dots correspond to breaking devices, light green dots correspond to nonbreaking "mechanical" devices. In the latter devices, the droplets do not even deform.

Validation of the model: Surface evolver simulations

Surface Evolver is a free simulation software that minimizes the surface energy of an object subject to predened geometric and energetic constraints. It uses 2D nite elements to represent 3D bodies and minimizes their energy through a sequence of mesh rening. A detailed description of the Surface Evolver algorithm is given in [START_REF] Brakke | The surface evolver[END_REF]. Our model consists in a xed volume droplet that is bounded in a rectangular microchannel that contains a connement gradient as described in 12. An additional symmetry constraint ensures that the droplet is centered around the deformation.

Surface Evolver calculates the minimum energy geometry of the droplet but gives us no information on the dynamics of the droplet evolution. It allows us to compute equilibrium shapes for a range of geometric parameters and to predict if the droplet will break or not. Figure 45 illustrates an example of a droplet deformation induced by a local gradient of connement that is achieved with Surface Evolver simulations.

Figure 45: a) Example of the surface evolver simulations that are carried out for d z = 5 µm, d x = 400 µm, e = 30 µm and w = 400 µm. These dimensions are similar to the ones of a mechanical device that has been used in our experiments.

Figure 46 shows that the interface prole of the droplet provided by Surface Evolver collapse on top of the one predicted by the model with an error that is smaller than 5 %. The surface minimization of the droplet seems to capture with a good agreement the deformation of the latter. This model neglects the thermal Marangoni eect contribution which thus suggests that its contribution to the equilibrium shapes is minimal. In the next section, we build a model that aims at predicting the dynamics of the droplet deformation.

19 Predicting the droplet dynamics

The droplet deformation that is studied in the last section is not immediate since it is mediated by the viscous dissipation in the biphasic system. In this section, we are interested in predicting the dynamics of the droplet deformation which requires rst to identify the origin of this viscous dissipation. While the analytical description of the viscous distribution in the system is complex, an experimental observation makes us assume that the dynamics of the deformation process is set by the viscous dissipation of water in the gutters.

Localization of the viscous dissipation in the gutters

Figure 48 shows the variation of the break-up time τ break dened by w-2h 0 (τ break ) = e as a function of the droplet length for the channel geometry. We observe that the break-up time varies linearly with the droplet length. We notice that when τ break tends to zero, the droplet length is approximately 300 µm which is approximately equal to b in these experiments, see gure 39a that shows the interface prole of the neck in the same channel geometry: δ = 5µm, b = 400µm, e = 30µm and w = 400µm.

If we now plot τ break as a function of the mean gutter length, we observe that the curve τ break = f (L) is shifted to (0,0), see gure 48b. To calculate the mean gutter length of the droplet, we calculate the equivalent length of the four gutters in parallel through which the neck lls:

1 L g = 1 L lef t g + 1 L right g ( 48
)
where L lef t g and L right g are dened in gure 47. Given these observations, we are motivated to only take into account the viscous dissipation of water that drains through the gutters to model the dynamics of the droplet deformation.

Power balance

To model the droplet deformation dynamics, we write a power balance that compares the viscous dissipation power with the excess surface energy that is increased in the system per unit of time.

The viscous dissipation power

The viscous dissipation power in the system is [START_REF] Guyon | Hydrodynamique Physique[END_REF]

P = η(∇v + T ∇v) 2 dΩ ( 49
)
where η is the viscosity of the uid, v is the velocity eld in the uid, Ω is the volume of the uid in which the viscous power is calculated.

The volume of the gutters scales as e 2 L g , e 2 being the typical section of the gutters. Flows shear on the typical size of the gutters e. Thus, if we only take into account the viscous dissipation in the gutters (see section 21.2), the power scales as

P ∼ η w v 2 e 2 e 2 L g ( 50
)
where v is the mean velocity in the gutters. Hence,

P ∼ η w v 2 L g . (51)
By conservation of mass, the volume of water that is emptying in the neck region per unit of time should be equal to the ow rate in the gutters. Therefore, we must have

∂ t Ω 1 = ve 2 (52)
with Ω 1 given by equation ( 31) that is recalled here,

Ω 1 = 2 b 0 h(x, t)e(x)dx - b 0 πe 2 (x) 4 ( 1 + ∂ x h 2 -1)dx. ( 53
)
Then, the viscous dissipation power is

P ∼ η w L g e 4 [∂ t Ω 1 ] 2 (54)
In order to take into account all the approximations that we make to express this viscous dissipation power, we dene an adjustable parameter K such that

P = K η w L g e 4 [∂ t Ω 1 ] 2 (55)
We replace the expression h(x, t) = h 0 p(x/b) in equation ( 55) such that we get,

P = K η w L g e 4 ∂ t 2 b 0 h 0 p(x/b)e(x)dx - b 0 πe 2 (x) 4 ( 1 + h 2 0 b 2 p 2 (x/b) -1)dx 2 . ( 56
)
In equation ( 56), all terms are constant in time such that the power veries,

P = K η w L g e 4 ∂ 2 t h 0 2 b 0 p(x/b)e(x)dx - b 0 πe 2 (x) 4 h 0 p 2 (x/b) b 2 1 + h 2 0 p 2 (x/b) dx 2 . ( 57
)
For sake of simplicity, we note

I(h 0 ) = 2 b 0 p(x/b)e(x)dx - b 0 πe 2 (x) 4 h 0 p 2 (x/b) b 2 √ 1+h 2 0 p 2 (x/b) dx 2 .
The excess surface energy per unit of time

The excess surface energy of the system per unit of time is obtained by calculating the time derivation of ∆S given by equation ( 43),

∂ t E = γ∂ t (∆S) (58)
In equation ( 58), the interface prole expression h(x, t) is replaced by h 0 (t)p(x/b) and the time derivative of the interface prole is ∂ t h 0 p(x/b).

We have,

∂ t (∆S) = -2 b 0 ∂ t h 0 1 + 1 cos θ p x b x + 2 b 0 πe(x)∂ t h 0 h 0 p 2 (x/b) 2 √ 1 + ∂ x h 2 dx +(2(w -e) + πe)∂ t (∆l) (59)
with

(2(w-e)+πe)∂ t (∆l) = 2(w -e) + πe (w -e)e + πe 2 4 2∂ t h 0 b 0 p x b e(x)dx - b 0 πe 2 (x) 4 ∂ t h 0 h 0 p 2 (x/b) 2 √ 1 + ∂ x h 2 dx . ( 60
)
We factor equation ( 61) by ∂ t h0 such that the equation writes,

∂ t (∆S) = ∂ t h 0 F (h 0 ) (61)
F being a function of h 0 . Indeed, in all the terms of equation ( 61), the only parameter that varies with time is h 0 .

The power balance

The power balance writes

∂ t E = P (62)
Therefore, we must have,

∂ t h 0 F (h 0 ) = Kη w L g e 4 I(h 0 ). ( 63
)
The numerical resolution of equation ( 62) reconstructs the time evolution of h 0 (t) as follows,

h 0 (t + 1) = h 0 (t) + Kη w L g I(h 0 ) e 4 F (h 0 ) (64)
with h 0 (0) = 0.

The adjustable parameter K is set by adjusting the time evolution h 0 (t) predicted by the power balance with the experimental data in the "mechanical" devices in which the geometric parameters (d x , d z , w and e) are well characterized. We nd K = 2.1 * 10 -3 . Figure 49 compares the model prediction to the experimental data. The dierent curves correspond to droplets with varied gutters lengths L g = (17; 37; 60; 68; 100; 150; 155) µm ranked from the droplet that splits the quickest to the one that splits the slowest. The value of K is adjsuted to 2.1 * 10 -3 in the power balance model to t the experimental data (dots) with the theoretical predictions (line).

We set K = 2.1 * 10 -3 for the rest of the discussion and we compare the model to the experimental data obtained with the "thermomechanical" device. The parameter K can be regarded as a permeability coecient by considering the gutters as pores that have a complex geometry. In this device, the value of d z has been characterized as a function of the maximum temperature increase in the channel while the value of d x is unknown. In order to compare the model with the time evolution of the neck width w -2h 0 , we need to adjust the value of d x . For that, we dene the dimensionless function N that calculates the dierence between the model prediction w -2h th (t) and the experimental prole w-2h exp (t) of the neck width, normalized by the typical error of the image processing that we consider to be twice the pixel size, e rr = 10 µm. N writes

N (d x , d z ) = 4 (h exp (t) -h th (t)) 2 e 2 rr . (65)
By minimizing function N (using the matlab function fminsearch), we can obtain the values of (d x , d z ) that best t the experimental data. Figure 50 illustrates the model prediction for two experimental data sets corresponding to the same droplet at two dierent temperatures in the channel. By tting the time evolution of the droplet deformation for a larger range of temperatures, see the Appendix page 114, we can plot the evolution of d z and d x as a function of the maximum temperature in the system. Figure 51a) plots the values of d z that we implemented in the model to recover the dynamics of our experiment data. These values are compared to the ones that we have characterized in gure 22 in the chapter Materials and Methods. The characterization and the tting values of d z seem to be in pretty good agreement with one another, especially at high temperatures (T > 60 • C). Similarly, gure 51b) shows the tting values of d x as a function of the maximum temperature in the system.

By plotting N as a function of d x in the vicinity of the tting value of d x , we can estimate the sensitivity of the model to predict the variation of the channel topography. Figure 52a shows that when N = N min + 1, we have 171 < d x < 179 µm, N min being the minimum value of N . This means the sensitivity of the model on the parameter d x is high (less than ten micrometers). The same considerations are made for d z , see gure 52b. We nd that the sensitivity of the model on the parameter d z is hundreds of nanometers.

Conslusions & Perspectives

• A local variation of the channel height of less than 20 % can lead to the droplet break-up when the latter is conned in a rectangular pore. By volume conservation, the external phase has to drain through the gutters towards the neck region when the droplet deforms.

• Even though, the droplet is 25 times mores viscous that the outer phase, the connement modies the distribution of the viscous stresses such that it is the less viscous phase that seems to set the dynamics of the droplet deformation.

• While a surface minimization seems to capture the droplet deformation features, a power balance based on scaling arguments manages to capture the dynamics of the deformation process. The model recovers that the linear breakup time scales with the length of the gutters L g .

• The geometrical parameters of the connement gradient d z and d x can be set as adjustable parameters in the power balance model such that their values could be extracted by tting the model to experimental droplet deformation data. Consequently, the observation of in-plane droplet deformations could constitute a new technique to indirectly measure in-situ the local variation of a rectangular channel topography.

This work raises the following questions.

On the droplet deformation process

• We see that the droplet deformation is driven by the connement gradient in the channel. However, we can wonder how the thermal Marangoni eect modies the ows in both phases. What is the spatial extent of these ows ?

• Could the droplet deform if there were no gutters -in a cylindrical capillary tube for instance ? How would the deformation dynamics be altered in such a geometry ?

On the thermomechanical actuation

The indirect characterization of the PDMS thermal dilation seems to conrm that the channel contracts of approximately 20%. This large dilation is unexpected. One possible mechanism for this large deformation is the thermal dilation of nanobubbles that would be trapped in the polymer matrix and that would expand with temperature. Of course, other mechanisms are possible and we leave the interpretation of the thermal dilation of the channel as an open question.

Towards the fabrication of droplet shape soft-solid microstructures ?

Figure 53 shows a droplet deformed by a succession of connement gradient indented in a system made out of a micro-milled mold. Figure 54 shows that by monitoring the topography of the channel, we can also deform a droplet made of diethylene glycol diacrylate mixed with 10% of 2-hydroxy-2-methylpropiophenone such that the latter can be solidied under a UV-exposition during the deformation process. Unlike the stop-ow lithography process that manages to build 2D structures with sharp edges -by illuminating a channel full of a photo-polymerisable solution with a UV lamp through a mask- [START_REF] Dendukuri | Stop-ow lithography in a microuidic device[END_REF], we achieve to fabricate 3D soft structures with a deformed droplet-like shape.

Figure 53: Droplet deformed by a succession of connement gradient indented in a system made out of a micro-milled mold.

Figure 54: Solid droplet out of the channel using a scanning electron microscope (x120). The droplet is made of a solution of acrylate based monomers. The internal phase is a solution of diethylene glycol diacrylate with 10% 2-hydroxy-2-methylpropiophenone and the continuous phase is water with 2% SDS. The exposure time to the UV lamp is 500 ms.

Part IV Droplet Relaxation

In this chapter, we report the experimental study concerning the capillary relaxation of a droplet after the channel deformation is released. Initially pinched in its center (gure 55a), the droplet relaxes back to its plug-like shape minimizing its surface energy under the constraints imposed by the channel walls (gure 55b). During this relaxation process, the liquid contained in the central neck drains towards the extremities of the droplet. Surprisingly, the droplet relaxation that is also driven by surface tension and mediated by viscous dissipation, displays a dynamics that is very dierent from the droplet deformation. Figure 55c shows the spatio-temporal evolution for both the droplet deformation (t on < t < t of f ) and relaxation (t > t of f ). t on is the time at which the resistance is switched on and t of f is the time at which the resistance is switched o. While the rate of neck thinning during the deformation process is decreasing, the rate of neck thickening during the relaxation process is constant over time.

To explain the linear dynamics of droplet relaxations, we follow a theoretical approach that is dierent from the one we had for the droplet deformation. This approach integrates the role of the non-linear viscous dissipation in the dynamical meniscus of the neck [START_REF] Bretherton | The motion of long bubbles in tubes[END_REF]. This work allows us to show that classical models incorporating capillary driving with viscous dissipation employing geometrical invariance, whether translation or rotation cannot describe the relaxation in the intrinsically 3-dimensional geometry of the studied system. By "classical models", we refer to the ones that are used to describe for instance, the relaxation of a droplet in an innite bath [START_REF] Stone | Dynamics of drop deformation and breakup in viscous uids[END_REF], [START_REF] Stone | Relaxation and breakup of an initially extended drop in an otherwise quiescent uid[END_REF], of a thin-liquid lm that is invariant by translation [START_REF] Aradian | Marginal pinching in soap lms[END_REF][START_REF] Mcgraw | Self-similarity and energy dissipation in stepped polymer lms[END_REF], of a droplet that has trapped a uid pocket (a dimple) near a solid boundary [START_REF] Bluteau | Water lm squeezed between oil and solid: drainage towards stabilization by disjoining pressure[END_REF][START_REF] Chan | Film drainage and coalescence between deformable drops and bubbles[END_REF], of a droplet in a Hele-Shaw cell [92,[START_REF] Oswald | Droplet relaxation in hele-shaw geometry: Application to the measurement of the nematic-isotropic surface tension[END_REF] By considering the 3D problem, a scaling model incorporating dominant dissipation within the droplet menisci allows a full recovery of the self-similar droplet dynamics. Spatio-temporal evolution of the neck width during the droplet deformation t on < t < t of f and the droplet relaxation t > t of f . t on is the time at which the resistance is switched on and t of f is the time at which the resistance is switched o. In order to prepare the well-controlled, out-of-equilibrium droplet interfaces shown in Figure 57, we use a thermomechanical actuation. All the details of this process are provided in the previous chapter. When the heating resistors are switched o at time t = t off , the PDMS relaxes and the droplet neck then returns to equilibrium. Figure 57a) displays the spatio-temporal evolution of the droplet cross-section at x = 0 for two dierent droplet lengths. Two regimes are observed. First, the neck relaxes abruptly over a typical time of hundreds of milliseconds and second, the relaxation proceeds more slowly and approaches equilibrium.

First temporal regime

The rst temporal regime corresponds to the thermal relaxation of the PDMS dilation. During this regime, the channel thickness e(x) increases such that the gradient of connement disappears and a solid volume is suddenly released. The inset of Figure 57ai shows a zoom of this rst relaxation, as well as a temperature prole taken with a infrared camera (Flir Camera, ATS). The rst regime and the thermal relaxation in the device take place over the same period of approximately 200 ms (at most 500 ms). The newly available volume, schematically indicated in Figure 57aii, is occupied by the phase that ows the most easily. A careful examination of both the volume released by the thermal homogenization and the increase of the inner phase volume in the neck suggests that the inner phase ows more easily than the outer phase. This result is consistent with our observations of the droplet deformation that suggested that it was the external phase that was setting the dynamics of the process, see the previous chapter.

Demonstration: The volume released by the thermal relaxation of the system can be extrapolated from the topography of the channel thermal expansion characterized in chapter Materials& Methods. The bottom channel topography is built from the maximum dilation height that has been experimentally calibrated.

For a temperature of T = 60 C that has been applied to a droplet in a 200 µm wide and 30 µm high channel, the volume (scheme 58) that is suddenly released by the thermal relaxation of the channel bottom layer can be approximated to d z * d x * w ∼ 5 * 100 * 200 ∼ 10 5 µm 3 where d z is the maximal dilation height of the bottom wall, d x is the extent of the dilation in the x-direction that is assumed to be 100 µm and w is the channel width. The increase of the inner phase volume in the neck region, can be estimated by subtracting the top view of a droplet before and after the switch o of the resistance and multiply the surface that derives from this substraction by the channel height.

(see gure 59 which corresponds to a system that also veries T = 60 C, w=200 µm and e=30 µm). The increased volume of the inner phase: V innerphase ∼ 3, 8.10 5 µm 3 in the neck region, is of the same order of magnitude than the volume released by the thermal relaxation,V dilation . This preliminary observation shows that because of the connement, the inner phase (η i = 25 × 10 -3 Pa.s) ows more easily than the external phase (η o = 1 × 10 -3 Pa.s) -that has to ow through the gutters and possibly in the lubricating lms. In the following, we consider that the dissipation occurs mainly in the external phase.

Second temporal regime

The second temporal regime is due to capillary driven ows. In Figures 60b) andc) we show the time evolution of the geometric features describing the neck: rst, the neck extent along y, h 0 (t) and second, the neck extent along x, σ(t) dened in gure 55c. For all the droplets that we have studied, we observe that h 0 (t), varies linearly with time over a signicant period, and extrapolates to zero thickness at a time we call t 0 . We note also that the relaxation time increases with the droplet length for a given channel geometry as demonstrated in Figure 60 -this is a signature of the dissipation of the liquid owing in the gutters, see the previous chapter. Despite the small range of variation that is attainable with these experiments, the neck extent along x, σ(t) varies with a power law that is captured by a scaling (t 0 -t) β , with β = 1/3 providing a good description of the data; this power law is shown by the black line in this latter gure and is discussed in the next section. Having briey described the main features of the prole dynamics, we show in Figure 61a) the time evolution of the full raw proles in the neck region at early times when h 0 (t) > e. Based on the temporal evolutions of the neck features found in gure 57b) and c), we apply the scalings describing h 0 (t) and σ(t) to the entire proles of Figure 61 a). Figure 61 b) thus depicts the rescaled curves of h(x, t)/τ as a function of x/τ 1/3 ≡ u, where we dene τ = t 0 -t . The interface prole h(x, t) writes, in a self-similar way [START_REF] Barenblatt | Scaling[END_REF][START_REF] Mcgraw | Self-similarity and energy dissipation in stepped polymer lms[END_REF][START_REF] Eggers | The role of self-similarity in singularities of partial dierential equations[END_REF]:

h(x, t) = τ α f x τ β (66)
where τ = t 0 -t, u = x (t 0 -t) β . f is dened as the master curve on top of which all the interface proles over time collapse after rescaling (Figure 61 (b)). Experimentally, we nd a good quantitative agreement for α = 1 and β = 1/3.

At late times, when h 0 (t) < e, the interface prole does not follow very well the same self-similar analysis, see the Appendix page 115.

Robustness of the exponent β:

In gure 61, the self-similar analysis shows that β = 1/3, also suggested by the time evolution of σ in gure 60c, is a good candidate to rescale the horizontal extent of the interface prole. Yet, a large range of β seems to give a good agreement with the experimental data as illustrated in gure 62. Since the self-similar analysis does not enable us to discriminate in a robust manner the values of β, we look at another parameter that varies also as a function of β: the volume of water that is trapped in the neck region Ω(t). This volume Ω(t) is approximately

Ω(t) = 2e c -c h(x, t)dx. ( 67
)
where the bound c of the integral is chosen in between the maximum value of the neck extent in the x-axis and the extremity of the droplet. Thus, we ensure that Ω(t) covers the volume of water in the neck at any experimental time, t.

Using the self-similar expression of h(x, t), the volume of the external phase in the neck becomes

Ω(x, t) = 2e c -c τ α f x τ β dx. (68)
Upon a change of variables, u = x/τ β , in the integral, Ω is given by

Ω(x, t) = 2eτ α+β c -c f (u)du. (69)
where c -c f (u)du is constant over time. The boundaries of the integral [-c, c] can be considered as constant upon the change of variables since c has been chosen such c/τ β is larger than the neck x-axis extent at any experimental time and since we consider that h(x, t) = 0 beyond x = ±c.

Figure 63 shows the time evolution of Ω(t) ∼ τ α+β . By tting the curve with a function mτ p , for p = [0; 1/5; 1/4; 1/3; 1/2; 1], we nd that for 2 > p > 3/2 and p = 1, corresponding respectively to 1 > β > 1/2 and β = 0, the time evolution of Ω(t) is no longer in the errorbar of the experiment data.

Model

In this section, we investigate the mechanisms that are at play in the droplet relaxation process. In order to do so, we identify the origin of the viscous dissipation in the system and we balance it with the capillary driving force induced by the initial curvature of the droplet interface.

In the previous section, we have seen that the dynamics of the droplet relaxation is very dierent from the one of the deformation process (discussed in chapter 3): the y-axis extent of the neck varies linearly over time and its x-axis extent seems to have a sublinear evolution with time τ . If we wanted to apply the surface minimization model of chapter 3 to the the droplet relaxation, we would have to integrate a varying boundary b with time in the integrals of the excess surface expression, see equation 43. In addition, the power balance written for the droplet deformation would set only one constraint on α and β while we need a system of two equations to predict both values.

For all these reasons, we rather decide to investigate new theoretical approaches even though both processes deal with a liquid object that relaxes after having been driven out-of-equilibrium.

At rst, we show that classical models as the lubrication equation in a 2D plane of invariance or the Brinkman approximation [START_REF] Bluteau | Water lm squeezed between oil and solid: drainage towards stabilization by disjoining pressure[END_REF][START_REF] Mcgraw | Self-similarity and energy dissipation in stepped polymer lms[END_REF]92] are not appropriate to capture the self-similar exponents of the interface prole. Then, we write a model based on scaling arguments that consider the critical contribution of the droplet menisci and the gutters to the relaxation process. These arguments manage to recover the self-similar exponents of the droplet interface prole α and β.

First attempt: a thin-liquid lm relaxation

At rst sight, we consider the problem with invariance in z as considered, for example in [START_REF] Chan | Hydrodynamics of air entrainment by moving contact lines[END_REF][START_REF] Mcgraw | Self-similarity and energy dissipation in stepped polymer lms[END_REF][START_REF] Bluteau | Water lm squeezed between oil and solid: drainage towards stabilization by disjoining pressure[END_REF]. This assumption discards the presence of the droplet menisci and the connement of the external phase in the z-direction. This consideration is true at most at late times when the interface prole h(x, t) becomes smaller than the cavity height, see gure 60 and 61.

In this approximation, the problem can be reduced to a 2D plane (x, y). The height of the at lm, at equilibrium, is considered to be e, the typical size of the gutters.

Figure 64: 2D approximation of the relaxation problem if we consider that the system is similar to a at thin-liquid lm innite in the z-direction.

We assume that the y-axis deformation extent of the interface h(x, t) is small compared to the scale σ of variations in the x direction, σ being the typical droplet length.

The velocity eld in the external phase writes (v x , v y ) in the (x, y) plane, where (v x , v y ) scale respectively as (V x , V y ). We consider the thin layer of uid as incom-pressible, so: ∂ x v x + ∂ y v y = 0 which thus gives on a scaling level V y ∼ Vxh σ << V x . The velocity can be considered as unidirectional in the x-direction (lubrication approximation, [START_REF] Oron | Long-scale evolution of thin liquid lms[END_REF]).

In the external phase, the velocity v x prole veries the Stokes equation:

0 = -∂ x p + η(∂ xx v x + ∂ yy v x + ∂ zz v x ) (70)
where ∂ x p is the pressure gradient in the x-direction. In the system, the viscous dissipation scales as η Vx σ 2 + Vx h 2 + Vx e 2 . Since we only consider the period of time when, h(x, t) becomes smaller than the channel height e, h < e < σ ensures that the viscous dissipation prevails in the y-direction.

Hence, the Stokes equation can be simplied as:

∂ yy v x = ∂ x P η ( 71
)
where η is the viscosity of the external phase and ∂ x P is the pressure gradient driven by the curvature gradient at the surface of the thin lm and scales as γ∂ 3

x h in the lubrication approximation.

We impose a no-slip boundary condition at the channel wall, v x = 0 on y = 0. At the level of the advancing dynamical meniscus of the droplet, we impose a stress-free boundary condition, η∂ y v x = 0 at y = h(x, t) (see the chapter Introduction), such that equation 71 gives:

v x = ∂ x p 2η (y 2 -2yh) (72)
In these conditions, the ow rate per unit of length, q(x) = h(x) 0 v(y)dy, equals to

γ 3η h 3 (x) ∂ 3 h ∂x 3 .
The mass is conserved in the external phase such that ∂ t h = -∂ x q. If we replace the expression of the ow rate per unit of length in the mass conservation equation, h(x, t) obeys, [START_REF] Mcgraw | Self-similarity and energy dissipation in stepped polymer lms[END_REF][START_REF] Bluteau | Water lm squeezed between oil and solid: drainage towards stabilization by disjoining pressure[END_REF]:

- γ 3η ∂ ∂x h 3 (x) ∂ 3 h ∂x 3 = ∂h ∂t (73)
.

To solve equation 73 that is assumed to rule the interface prole, we use the selfsimilar expression of h(x, t) =τ α f (u) where u = τ -β x.

Resolution of the lubrication equation with a self-similar solution

Using the expression: h(x, t) = τ α f (τ -β x), we obtain the general expression of the temporal and spatial derivatives of h(x, t):

∂h ∂t = ατ α-1 f + τ α -β x τ β+1 ∂f ∂u (74) so ∂h ∂t = ατ α-1 f -βτ α-1 uf (75)
and

∂ n h ∂x n = τ α-nβ f (n) (u). (76)
In equation 73 that is supposed to rule the dynamics of relaxation, we have

∂ ∂x h 3 ∂ 3 h ∂x 3 = 3h 2 ∂ 3 h ∂x 3 + h 3 ∂ 4 h ∂x 4 . (77)
If we combine 73+75+76, the self-similar analysis of equation 77 gives

αf -βuf + γ η f 2 f f τ 4(α-β)-(α-1) + γ 3η τ 4(α-β)-(α-1) f = 0. ( 78
)
For the equation to be right at all times, we have

3α -4β + 1 = 0 (79)
which is not consistent with the experimental data (gure 61) that is much more in agreement with α = 1 and β = 1/3. Indeed, if we take for granted that α = 1 (gure 60b), this equation would give β = 1, which is inconsistent with the experimental data of the time evolution of Ω(t) and of the neck x-axis extension, σ(t), on gures 60c and 63c.

Limitations of this approach

In the previous demonstration, we assumed that there was no shear in the zdirection. In reality, at early stages, h(x, t) is of the same order of magnitude and even larger than the cavity height e, see 61. To take into account the contribution of the height e to the viscous dissipation in our projected 2D problem, we can amend the velocity prole using the Brinkman approximation [START_REF] Boos | Thermocapillary ow in a hele-shaw cell[END_REF][START_REF] Brinkman | A calculation of the viscous force exerted by a owing uid on a dense swarm of particles[END_REF]92].

Brinkman has shown that in a Hele-Shaw conguration, the viscous dissipation is diminished by a term that is proportional to the velocity v x and to k 2 = 12/e 2 , where e is the height of the cavity. The shear term then writes η(∂ 2 y v x -k 2 v x ).

Brinkman approximation

In such approximation, Stokes equation writes

∂ x p = η(∂ yy v x -k 2 v x ) (80)
The general solution is then v x (y) = -1 ηk 2 ∂ x p + Ae ky + Be -ky . The no-slip boundary conditions, v x = 0 on y = 0 and the stress-free boundary condition at the droplet interface, set A =

∂ t h = -γ ηk 2 ∂ x -h + 1 k tanh(hk) ∂ 3 x h (81)
In order to recover the self-similar exponents α and β of h(x, t) and carry out the self-similar analysis of the problem, we discard the term in tanh by making the approximation that kh 1 which is at best true at early times when h e. As a result, h(x, t) veries approximately

∂ t h = γ ηk 2 ∂ x (h∂ 3 x h). (82)
Combining equations 75+76 and 82, we obtain the following condition on the selfsimilar exponents: α -4β + 1 = 0 that is still not consistent with our experimental observations α = 1 and β = 1/3. This equation would give

β = 1/2 if α = 1.
Besides, this model does not achieve to predict the values of both α and β. In the following section, we rene again our model by taking into account the viscous contribution of the droplet menisci to the relaxation dynamics.

Meniscus dissipation model

The fact that neither the lubrication equation 73 nor the Brinkman approximation 82 allows us to recover the dynamical features of droplets relaxation motivates an investigation of the viscous dissipation in the menisci. To take this dissipation into account, we write a scaling model in which we balance the capillary motion of the meniscus in the neck region and the viscous dissipation in the advancing/receding menisci of the droplet. This model, coupled to a volume conservation that takes into account the dissipation in the gutters, nally enables us to predict faithfully the self-similar exponents of the interface prole.

Our approach

In order to predict the self-similar exponents: α and β, a system of two equations is required. The spirit of the model that we write can be likened to the example of a tank of beer that empties through a tube, see the box.

Example of a tank of beer that empties under the eect of gravity and that is connected at its bottom to a small diameter tube.

Fist: a tank of water (without any foam)

It is assumed that the length of the end tiny tube L is long enough or e small enough to consider that the bulk viscous dissipation is larger in the tube than in the tank. The height of water in the tank is ruled by the balance between the gravitational force (F g ∼ ρghπw 2 ) and viscous forces. Among the viscous forces, we list the bulk viscous force in the tube (F v ∼ ηvL), and the viscous dissipation at the level of the moving contact line at the interface water/air/wall. v corresponds to the mean velocity in the tube. If the radius of the tank is wider than the capillary length, the viscous friction at the level of the contact line can be neglected. The viscous dissipation that prevails is localized in the tiny tube.

Second: a tank of beer

To force the comparison, we add a foam on top of the liquid. We imagine that the bubbles of the foam are non-wetting. Therefore, the viscous force that prevails in the foam is due to the friction in the lubricating lms between the wall and the bubbles of the foam [START_REF] Cantat | Liquid meniscus friction on a wet plate: bubbles, lamellae and foams[END_REF][START_REF] Bretherton | The motion of long bubbles in tubes[END_REF]. This force scales as

F c = wγC 2/3 a
where w is the diameter of the tank and C a = η∂th γ .

Figure 65: Scheme of a tank full of beer that is emptying under the eect of gravity and that is connected at its bottom to a small diameter tube. Geometrical parameters of the problem: L is the length of the tube, e is the diameter of the tube, w is the width of the tank, h(t) is the height of beer. The beer is modelled as a liquid on top of which lays a foam.

The force balance in the system writes

F g = F v + F c (83)
We write the conservation of ow rate in the system such that w∂ t h = ev.

The force balance writes: ρghπw 2 = γw η∂th γ 2/3

+ η Lw∂th e . The contribution of the foam to the overall dynamics has been evidenced for a sloshing tank by Gallaire et al. [START_REF] Viola | Foam on troubled water: Capillary induced nite-time arrest of sloshing waves[END_REF]. The authors show that sublinear capillary forces govern the friction at liquid-solid and liquid-liquid interfaces and amend the classical damped harmonic oscillator models for sloshing water even at the macroscale. In the studied system, the tube is the equivalent of the gutters, the gravitational force must be replaced by capillary driving forces and the viscous friction in the moving foam with the tank wall can be compared to the viscous friction in the lubricating lms at the level of the advancing menisci in the neck region.

Local force balance in the advancing menisci

Four menisci are advancing at the neck, both at the top and bottom walls for two sides of the neck. We note that four menisci are also receding at the droplet extremities (top and bottom walls at the two extremities), see gure 66. We assume that the velocities there are proportional to those in the neck. Bretherton showed that, when a meniscus is advancing or receding near a solid boundary, most dissipation is in fact located in the dynamical meniscus [START_REF] Bretherton | The motion of long bubbles in tubes[END_REF][START_REF] Cantat | Liquid meniscus friction on a wet plate: bubbles, lamellae and foams[END_REF],

which connects the at lubricating lm and the static meniscus with characteristic radius of curvature 2/e in the plane (y, z), see gure 67.

Figure 67: Geometry of a moving meniscus when a non-wetting droplet is pushed in a conned cavity. We dene three zones: the static meniscus at the equator, z = 0, the dynamical meniscus close the wall and the constant thickness lubrication lm that separates the liquid boundary from the wall.

The viscous drag force per unit length of the dynamical meniscus (in the x axis) scales with the capillary number, Ca = ηV γ -1 with V a typical interface velocity. Cantat [START_REF] Cantat | Liquid meniscus friction on a wet plate: bubbles, lamellae and foams[END_REF] showed that F diss ≈ γCa 2/3 . The liquid velocity in the meniscus of the neck around the position x = 0 is V ≈ ∂ t h and the subsequent dissipative force in the dynamical meniscus writes F diss ≈ γ (ηγ -1 |∂ t h|)

2/3 . The driving force is generated by the gradient of curvature along the interface, and writes in the limit of small slopes, and per unit length of the meniscus: F cap ≈ γe 2 ∂ 3 x h , by considering that the dominant force is on the uid between the meniscus and the wall, a space of typical lengthscale e.

Remark:

The dissipation, as written in the text, considers the main portion of the dissipation only which is around x = 0 where the uid velocity is normal to the meniscus. There must be a transition between the ows normal to the advancing menisci in the neck and the ows parallel to the droplet interface in the gutters. We do not know what is the ow eld distribution in the neck. However, we could dened an angle φ in the neck such that the meniscus dissipation would be moderated by the projection of the uid velocity normal to the meniscus, see gure 68. In this case, the viscous drag per unit length of the dynamical meniscus would write F diss ≈ γ(cos φ) 2/3 Ca 2/3 , where φ is illustrated in gure 68. In the discussion, we only consider the section of the menisci where ows are normal to the interface and F diss ≈ γCa 2/3 per unit of length of meniscus. By balancing the driving and the dissipation forces, we nd a relation between the temporal evolution of the prole and its current state:

η γ |∂ t h| 2/3 ∼ e 2 ∂ 3 x h . (84)
According to the observed self-similarity, h(x, t) = τ α f x τ β , equation 84 becomes:

η γ 2/3 (-ατ α-1 f + βuτ α-1 f ) 2/3 = e 2 τ α-3β f . (85)
For the equation to be true at all times,

α -9β + 2 = 0 (86)
In order to close the problem, we consider conservation of liquid volume owing from the neck towards the droplet extremities through the gutters.

Volume conservation

The liquid that is expelled from the neck region during the relaxation process has to escape through the gutters by volume conservation. The volume Ω(t) of the external phase in the neck writes

Ω(t) = e c -c 2h(x, t)dx. (87)
The bounds of the integral are -c and c and we take c, a constant that is larger than the maximum half-extent of the neck in the x-axis, see p. . Using the self-similar expression of h(x, t), the volume of the external phase in the neck becomes

Ω(x, t) = e c -c 2τ α f x τ β dx. (88)
Upon a change of variables in the integral, Ω is given by

Ω(x, t) = 2eτ α+β c -c f (u)du. (89)
We consider that the boundaries [-c, c] are chosen such that [-c , c ] still cover the entire volume of the neck. We estimate that the volume lost per unit of time in the neck scales with the mean velocity v of the ow in the gutters times the typical cross-section of the gutters:

dΩ dt = v e 2 (90)
The mean velocity of the ow is estimated using the Stokes equation η∆ 2 v = -∂ x P . The ow is driven through the gutters by the over-pressure induced by the positive curvature at the center of the neck. As a result, the pressure drop is equal to the Laplace pressure gradient induced by the curvature of the neck ∂ x P ∼ γ∂ 2 x h/L g and established along the gutters length L g through which the liquid ows. We consider that the ow in the gutters shears on the typical width of the gutters e. Given all these considerations, the mean velocity scales as:

v = - γ∂ 2 x h ηL g e 2 (91)
The volumetric ow rate in the gutters then scales as:

Consequently, equations ( 90) and ( 91) give:

dΩ dt = - γ∂ 2 x h ηL g e 4 (92)
The second partial derivative of the expression of h(x, t) is, according to the observed self-similarity:

∂ 2 x h = τ α-2β f (u) (93)
The curvature of the neck is considered at x = 0 (at the center of the droplet) where ∂ 2

x h = τ α-2β f (0). Using the expression (93), equation ( 92) then becomes:

dΩ dt = - γτ α-2β f (0) ηL g e 4 . (94)
Another expression of the volume lost in the neck per unit of time can be obtained by simply deriving equation [START_REF] Mcgraw | Self-similarity and energy dissipation in stepped polymer lms[END_REF],

dΩ dt = e(α + β)τ α+β-1 c c f (u)du. (95)
where f (u)du is a constant with time.

We balance equations ( 94) and ( 95) in order to nd a constraint on the temporal self-similar exponents:

- γτ α-2β f (0) ηLg e 4 = e(α + β)τ α+β-1 c c f (u)du. (96)
Figure 69 displays the time evolution of the gutter length L g . The latter expands of 25 µm which corresponds to only 2% of its mean length. In the following, we consider that L g is constant over time. In equation 96 all terms except from τ are independent on time. The exponents of τ must be equal:

α -2β = α + β -1. (97)
Solutions for α and β Equations 86 and 97 lead to the two values of self-similar exponents,

α = 1 (98)
and

β = 1/3 (99)
consistent with the experimental data.

This agreement underlines the dominating eect of the menisci and gutters located at the conning walls on the drainage dynamics.

Characteristic drainage time

Finally, it is possible to dene a characteristic drainage time to couple all of the experimental data h 0 (t). Further mass conservation consideration gives a relation between the velocity of the meniscus at the center of the neck and the mean velocity in the gutters h∂ t h ≈ ev. As a result, the mean velocity of the neck meniscus can be written h e ∆h

τ d = -γ(ηL g ) -1 e 2 ∂ 2
x h, where the time τ d is the typical drainage time during which the neck size reaches the channel width ∆h ≈ w with constant velocity. At rst sight, from Figure 56, the curvature ∂ 2

x h seems to be of the order of w -1 . In order to be more quantitative, gure 70 plots the curvature κ = ∂ 2

x h(0, t) times w versus τ . It shows that κ is of the order of magnitude of w -1 . We therefore dene

τ d = wηL g γe . (100)
The time evolution of the neck height h 0 (τ ) can thus be represented using the dimensionless variables h and τ with τ = τ d τ and h = e h. These dimensionless variables are plotted on gure 71. Remarkably, all the data collapse onto a single linear curve, whatever the droplet length, channel width or cavity thickness. The collapse lies within an errorbar that is calculated in the Appendix 115. As such, the momentum conservation in the advancing neck coupled to a volume conservation in the gutters captures faithfully the experimental observations of the relaxation of the droplet for a wide range of length, channel height and width. To conclude, the 3D geometry of microuidic droplets, shaped by the 2D connement in height and in width gives rise to menisci and gutters of typical size imposed by the channel height. These objects play a major role in relaxation dynamics. Analytical descriptions based on the classical lubrication equations or Brinkman approximation do not achieve to recover the experimental observations. By contrast, we have predicted the relaxation dynamics with a scaling analysis considering capillary driving forces and viscous dissipation mainly in the gutters and menisci. Considering a resulting characteristic relaxation time of droplets depending on the physico-geometrical parameter of the channels, we fully collapse all the dynamics onto a single curve.

Summary

This chapter investigates the droplet relaxation to its plug-like equilibrium shape after the resistance is switched o. We show that the thermal relaxation of the PDMS dilation occurs in a few hundreds of ms, much faster than the droplet relaxation that takes place over tens of seconds. The dynamics of the relaxation process is very dierent from the dynamics that we observe for the droplet deformation. Indeed, the rate of neck thickening is constant over time and the x-axis extent of the neck varies as τ 1/3 where τ is the time distance from the end of the relaxation. The evolution of the droplet interface displays a selfsimlar prole at early times. The droplet relaxation is capillary-driven and is mediated by viscous dissipation.

We write a scaling model that takes into account the Bretherton sublinear friction in the moving menisci of the droplet and the linear friction in the gutters. This model recovers the self-similar exponents of the interface prole.

Our study emphasizes the peculiar role of the droplet menisci in the dynamics of droplet relaxations and raises some questions:

• Why is the nonlinear friction in the moving meniscus evidenced in the droplet relaxation dynamics and not in the droplet deformation one ?

• Why does the x-axis extent of the neck varies with time in the relaxation process and not during the deformation process ?

Part V

Conclusion

In this manuscript, the response of a non-wetting droplet, conned in the rectangular section of a microuidic channel, to a local variation of the channel topography has been investigated. The local connement gradient, that can reversibly be induced in the channel with using a thermomechanical actuation, leads the droplet to deform. If the channel is deformed enough, the droplet can even break. The studied deformations are driven by capillary forces until the surface energy is minimized subject to the constraints imposed by the conning wall. This deformation is not instantaneous since it is mediated by the viscous dissipation that seems to be localized in the gutters at the four corners of the channel. A power balance equating the time derivative of the surface energy of the droplet and the viscous dissipation power manages to capture the time evolution of the droplet deformation. This model can be used to t the experimental observation of the dynamics of the deformation process with the vertical and horizontal dilations of the channel as independent tting parameters, with a micrometric resolution.

When the topography of the channel goes back to at after the thermomechanical actuation is stopped, the droplet relaxes to its plug-like shape. Interestingly, the dynamics of this relaxation process is linear unlike the deformation process, the speed of which decreases over time. By coupling a force balance in the dynamical menisci of the relaxing droplet with a volume conservation of the external phase that has to ow through the gutters, we manage to recover the linear time evolution of the relaxation process. Lastly, a characteristic drainage time that depends on the physico-geometrical parameters of the system can be extracted from this study.

The relaxation dynamics of an initially out-of-equilibrium liquid/liquid interface has been studied for many decades [START_REF] Rallison | The deformation of small viscous drops and bubbles in shear ows[END_REF][START_REF] Chan | Film drainage and coalescence between deformable drops and bubbles[END_REF][START_REF] Josserand | Drop impact on a solid surface[END_REF]. Fundamentally in these studies, simply-modelled geometries lead to quantitative descriptions which can in turn lead to the revelation of molecular scale physics: disjoining pressure and slip length being examples [START_REF] Bluteau | Water lm squeezed between oil and solid: drainage towards stabilization by disjoining pressure[END_REF][START_REF] Mcgraw | Self-similarity and energy dissipation in stepped polymer lms[END_REF][START_REF] Huerre | Migration de gouttes en microuidique: caractérisation et applications[END_REF]. In the geometrical conguration of this work, this small scale physics might not be revealed since ows occurrs mainly in the gutters that are tens of microns large. However, these two studies emphasize the dominating role of capillary forces at the micrometer scale that can even provoke droplet to breakup with a dilation of the channel height of just 15 %. They also emphasize the eects of the connement on the droplet dynamics. For example, the role of the less viscous phase that has been demonstrated in plunging plates [START_REF] Chan | Hydrodynamics of air entrainment by moving contact lines[END_REF] or in bouncing and splashing droplets [START_REF] Richard | Contact time of a bouncing drop[END_REF]104,[START_REF] Jolet De Ruiter | Wettability-independent bouncing on at surfaces mediated by thin air lms[END_REF], wherein the less viscous phase is strongly conned, is one more time illustrated. Similarly, the importance of the dynamical menisci of the droplet [START_REF] Bretherton | The motion of long bubbles in tubes[END_REF], is once again evidenced.

As a perspective, since the boundary conditions at the droplet interface has been proven to modify the viscous friction force in these menisci [START_REF] Cantat | Liquid meniscus friction on a wet plate: bubbles, lamellae and foams[END_REF], we might wonder if we would observe a change in the droplet relaxation dynamics if we rather added insoluble surfactants to the system. We might also wonder what ow patterns are induced by the thermal surface tension gradient at the droplet interface during the deformation process.

If we note V int the velocity of the interface, the tangential stress continuity at the interface writes

η o V int e -η w V int h ∞ = dγ dx (101)
The ratio η o /e ∼ 25 * 10 -3 /(30 * 10 -6 ) ∼ 10 3 while the ratio η w /h ∞ ∼ 10 -3 /(20 * 10 -9 ) ∼ 10 6 . We conclude that the velocity at the interface V int ∼ dγ/dx * h ∞ /η w ∼ 10 -7 m/s and that we can neglect the velocity of the Marangoni ows in the lubricating lms.

• In the gutters

The tangential stress continuity writes

η o V int e -η w V int e = dγ dx (102)
The interface velocity is then In order to estimate a time of break-up, we calculate the ow-rate of water in the gutters induced by the thermal Marangoni eect.

V int = dγ
We dene r as a radial coordinate that veries y 2 +z 2 = r 2 , see gure 73. r identies the distance of a uid element from the droplet interface in the diagonal of the gutter. The velocity eld in the gutter that stems from the Marangoni eect derives from the integration of the boundary condition at the interface written in equation [START_REF] Josserand | Drop impact on a solid surface[END_REF]. We assume that the velocity led in the gutter scales

v(r) ≈ 1 η o -η w dγ dx r + A (103)
A being a constant that must be determined by the boundary conditions. Considering that the typical total section of the gutters is e 2 , the owrate of water that enters in the neck per unit of time scales as Q ∼ e In particular, we note that equation ( 106) largely overestimates the mean velocity since it takes as a typical length scale the channel height, e, which is only true near the center of the gutter. In all other places, the gutter height is smaller. Indeed, in a rectangular channel cross-section, the interface velocity varies from the one given by eq. ( 102) down to the one given at the lubrication lm which is set to zero. A way to take into account the complex contribution of these boundaries to the ow eld is to dene a permeability coecient α. This coecient is commonly dened in the studies that deal with ows in porous media []. The ow-rate is then amended by α and we have Q ∼ αe 2 1 ηo-ηw dγ dx ∼ α10 -3 . The volume of water in the neck is approximately bwe ∼ 10 -12 µm 3 . Therefore, we can dene a time of break-up associated with the thermal Marangoni eect that is around wbe Q ∼ α -1 10 -9 s (107)

In order to recover the time of break-up that is observed in our experimental data, α should be of the order of magnitude 10 -10 . This value is very far from the typical value of the permeability coecient that is calculated in the literature for porous media -in foams this coecient is of the order of magnitude 10 -4 [START_REF] Miralles | Migration of biphasic systems by thermal actuation in microconnement[END_REF]. In addition, the fact that the droplet reaches an equilibrium state while the tangential stress dγ dx is maintained at the interface suggests that the Marangoni ow can hardly be the prevailing mechanism in the system. In the next section, we prove that the local channel dilation is the main motor of the droplet deformation.

Best t of the droplet deformation dynamics data in the "resistance" system

Viscous dissipation in the system

As described by Cantat [START_REF] Cantat | Liquid meniscus friction on a wet plate: bubbles, lamellae and foams[END_REF], the viscous force associated equals to

f diss = -η meniscus ∂v ∂z (y)dy. ( 108
)
The velocity expression and the related viscous force depend on the boundary condition that is set at the droplet interface. In the case of a stress free boundary condition, the viscous force per unit of interface length that is located in the dynamical meniscus and in the lubrication lm scales as:

• for an advancing meniscus, f a diss = 3.84γC

2/3 a

• for a receding meniscus, f r diss = -1.1γC

2/3 a

Droplet relaxation

Late times

Figure 61(a) does not display the interface prole at late times which correspond to the last seconds of the relaxation process for the sake of clarity. However, they are shown on gure 75 that plots that the whole time evolution of h(x, t). We notice that the late times proles are noisier. This noise is amplied as the proles are rescaled for the self-similar analysis. Likely, this noise, comes from the image analysis of the experimental data that relies on the contrast between the inner phase and the outer phase. The quantity of external phase becoming smaller and smaller as the relaxation process ends, we expect the proles to be harder to detect thanks to our image analysis technique.

Calculation of the errorbar on the characteric drainage time

To collapse perfectly the experimental data on a master line in gure 71, we add to adjust the values of w, e, l from the one measured thanks to an image processing by dw, de and dl. On all the experimental images, the pixel size p is smaller than 5 µm. We do not consider the variation of w and e on a same microuidic system induced by the microfabrication process. Then, we consider that de is zero and dw = 2p ∼ 10 µm.

The interface between the droplet and the side channel wall on experimental images is dicult to detect with our image processing since the presence of the menisci blurs the interface. This error is maximum on the datasets corresponding to e = 60 µm where the meniscus radius is e 2 ∼ 30 µm. Therefore, we can estimate that dl ∼ e All these limitations lead to dτ /τ ∼ 14% superior to all our experimental data.

Résumé

Cette thèse porte sur l'étude expérimentale de la déformation et de la relaxation d'une goutte, confinée dans un canal microscopique de section rectangulaire. Cette goutte est initialement déformée en forme de cacahuète grâce à un gradient de confinement induit de façon locale et réversible dans le canal. Lorsque que le gradient de confinement est désactivé, la goutte relaxe vers sa forme d'équilibre. Durant cette phase de relaxation, le liquide contenu dans le réservoir, formé au niveau de la déformation de la goutte, draine vers les extrémités de la goutte. Dans notre étude, la géométrie du système est complexe et ne présente pas d'axe de symétrie ni d'invariance. Pour décrire la dynamique de la déformation et de la relaxation de la goutte, les modèles classiques, qui reposent sur une approximation 2D de la goutte, ne peuvent pas être invoqués. Cependant, des lois d'échelles nous permettent de décrire la dynamique de la goutte. Elles mettent en évidence la contribution significative des ménisques à la dissipation visqueuse dans le système.

Mots-clés

Microfluidique digitale, gouttes, gradient de confinement, relaxation, auto-similarité

Abstract

We report an experimental study concerning the capillary deformation and relaxation of a relatively long droplet confined to a microscopic channel with rectangular crosssection. These droplets, found in numerous microfluidic applications, are centrally pinched into a peanut-like shape thanks to a localized, reversible deformation of the channel. After the channel deformation is released, the droplet relaxes back to its plug-like shape minimizing its surface energy under the constraints imposed by the channel walls. During this relaxation, the liquid contained in the central neck drains towards the extremities of the droplet. Classical models incorporating capillary driving with viscous dissipation employing geometrical invariance, whether translation or rotation cannot describe the deformation and relaxation in such intrinsically 3-dimensional geometries. By considering the 3D problem, scaling models incorporating dominant dissipation within the droplet menisci allows a full recovery of the droplet deformation and relaxation dynamics.

Figure 2 :

 2 Figure 2: a) Emulsion of oil in water created by the mechanical energy added by the pouring process. b) Oil laying on top of water in a glass. Copyright @ https://chemistscorner.com.

Figure 3 :

 3 Figure 3: a) Fluid 1 wets the wall. b) Fluid 2 wets the wall.

Figure 4 :

 4 Figure 4: a) Surfactant conguration b) Typical evolution of the surface tension with the concentration of surfactants.

Figure 5 :

 5 Figure 5: Sodium Dodecylsulfate structure.[43]

Figure 6 :

 6 Figure6: Evolution of the disjoining pressure P = Π disj with the thickness of the lubricating lm: a) Experimental calibration of P as a function of h with interferometric measurements (see inset box) b) Typical evolution of the disjoining pressure that combines the electrostatic pressure P el , the steric pressure P S and the Van der Waals pressure P V dW . The solution for the lm thickness results from the electrostatic contribution to the disjoining pressure[START_REF] Huerre | Droplets in microchannels: dynamical properties of the lubrication lm[END_REF]. The gure is adapted from[START_REF] Huerre | Migration de gouttes en microuidique: caractérisation et applications[END_REF]

2 R γ ∼ 10 3 *

 23 10 -3 * 10 2 >> 1 and Re = ρU R η ∼ 10 3 * 10 -3 * 10 3 >> 1.

Figure 7 :

 7 Figure 7: Snapshot of falling raindrops captured by Villermaux et al. [46].

2 / 3 a

 23 . Figure11recaps all the boundary conditions that can be encountered in the dierent regions of the droplet meniscus, depending on the experimental values of λ and .

Figure 11 :

 11 Figure 11: Table explaining the interface boundary condition in each region of the droplet as a function of λ and the parameter ∼ C 2/3a . By SLIP, Hodges et al. mean that there is a stress-free condition at the interface. By NO-SLIP, they mean that the boundary velocity is set by the ow in the droplet. The gure is extracted from[START_REF] Hodges | The motion of a viscous drop through a cylindrical tube[END_REF].

2 / 3 a

 23 , depending on the boundary conditions at the interface. Based on the work of Hodges et al., we consider that the boundary conditions that are required to solve the Stokes equation are:

Figure 12 Figure 12

 1212 Figure12illustrates the thermomechanical device. The latter is composed of a PDMS microuidic chip[START_REF] Mcdonald | Poly(dimethylsiloxane) as a material for fabricating microuidic devices[END_REF] that is sealed on top of a glass substrate over which are patterned a network of micro-resistances, made of chromium and gold. A crosssection of the chip is schemed in gure 13.

Figure 13 :

 13 Figure 13: Scheme of a cross-section of the device with the dierent materials that are encountered: glass, PDMS, water (the channel), PDMS. The material height scales are not respected on the scheme.

Figure 15 :

 15 Figure 15: Image of the experimental set-up extracted from [106].

Figure 16 :Figure 17 :

 1617 Figure 16: The droplet is experimentally observed in the plane (x, y).

Figure 18b illustrates theFigure 18 :

 18 Figure18billustrates the time evolution of the temperature in the channel when the resistance is successively switched on (at t = 4.6 s) and o (at t = 18.5 s), for a power equal to 50 mW. The typical timescales of the temperature increase and decrease in the channel are smaller than 500 ms. The typical time of our experiments being of few tens of seconds, we can consider that the temperature increase and decrease are instantaneous in the channel. After this transient time of at most 500 ms, the temperature remains constant in the channel as long as the resistance is on. a)

Figure 19 :

 19 Figure 19: Spatial evolution of the temperature prole. Upper (red) and lower bound (orange) of the temperature gradient for T = 80 • C.

Figure 20 :

 20 Figure20: Evolution of T with the electrical power P that is applied at the edge of the micropatterned resistance.

Figure 21 :

 21 Figure 21: a) Scheme of the home-made calibration step device. The device contains steps in its roof to measure the dilation of micro-channels. The channel width is 200µm and its height is comprised in between 1 and 15 µm with steps of 1 µm. b) Typical observation that is made in the calibration chip. A contact line appears when the channel walls dilates such that the top wall touches the bottom wall.

Figure 22 :

 22 Figure 22: Height of the channel deformation versus the maximum temperature in the device, calibrated with the calibration step chip.

Figure 23 :

 23 Figure 23: Scheme of the experimental set-up that is used to characterize a topographical change in the channel with a mechanical prolometer. This image is adapted from Miralles et al. [44]. The sublayer is stiened by two glass slides that are separated by the typical width of the channel 200 µm to reproduce the experimental conguration of a closed channel.

Figure 24 :

 24 Figure24: Spatial prole of the bottom layer of PDMS that is spin-coated on top of the resistance and stiened by two glass slides. The prole is measured with a mechanical prolometer for dierent temperatures in the channel.

Figure 25 :

 25 Figure 25: Maximum deformation the bottom layer of PDMS that is spin-coated on top of the resistance and stiened by two glass slides separated by a distance of 200 µm. This deformation is measured with a mechanical prolometer as a function of the maximum temperature in the channel.

Figure 26

 26 Figure 26: a) Image of a pendant drop b) Prole of the interfacial tension between mineral oil and water mixed for [SDS]=2 CMC.

Figure 27 :

 27 Figure27: a) 3D scheme of the non-wetting mineral oil droplet surrounded by water conned in a rectangular microuidic channel. The neck is dened as the spatial region in which the droplet interface deforms. This region is located in the vicinity of the resistance. b) Geometrical parameters of the system. Gutters are located at the four corners of the channel. The typical size of these gutters, schematically indicated in the green-boxed inset, scales linearly with the channel height e. c) Typical droplet deformation captured with a camera in a top view plane (x, y). The interface prole of the droplet is described by h(x, t), the horizontal extent of the neck region is dened by b.

Figure 28 :Figure 29 :

 2829 Figure 28: Snapshot of the droplet deformation in the vicinity of the heating resistance. 10 seconds after the resistance is switched on, the droplet breaks. In this experiment, the channel width is 200 µm and its height is 30 µm. The temperature at the level of the resistance is approximately 75 • C.

Figure 30 :

 30 Figure 30: Time evolution of the neck width for a non-breaking and for a breaking droplet. In this experiment, the droplet length is 800 µm, the channel height and width are respectively 30 µm and 200 µm. For T = 74 • C, the neck width w -2h 0 (t) decreases over time until it reaches the value e = 30 µm and the droplet breaks. For T = 54 • C, the neck width decreases until it reaches a plateau value that is approximately half of the channel width.

Figure 31 Figure 31 :

 3131 Figure 31 plots the same time evolution of the droplet neck width for a larger range of temperature (47 • C < T < 74 • C). On the gure, we can assume that the critical temperature T * above which the droplet breaks is comprised in between 54 • C and 60 • C.

Figure 32 :

 32 Figure 32: Time evolution of the neck width for a large range of droplet lengths (500 µm < L < 1800 µm). In these experiments, the channel dimensions are e = 30 µm and w = 200 µm. In the experiments, the temperature is T = 60 • C.

Figure 33 :

 33 Figure33: a) Entire interface prole time evolution of the neck during a droplet deformation experiment. Green curves correspond to late times while black curves correspond to early times. The proles h(x, t) are plotted from t = 20 s after the beginning of the droplet deformation up to t = 90 s. The dimensions of the channel are w = 400 µm and e = 30 µm. In this experiment, the droplet length is 3800 µm. b) Same interface proles rescaled by h 0 (t). The x-axis is rescaled by b.

Figure 39 :

 39 Figure 39: Time evolution of the interface prole h(x, t) of the droplet deformation in a "mechanical" device in which d z = 5 µm, d x = 400 µm, e = 30 µm and w = 400 µm.The droplet is 1600 µm long. The timelapse that is plotted is around 70s. During this period of time, h(x, t) is self-similar and writes h(x, t) = h 0 (t)p(x). We dene u = x/b where b is the x-axis extent of the neck that is constant over time.

Figure 40 :

 40 Figure40: Comparison of the self-similar functions p of the droplet deformation in between the "thermomechanical" and the "mechanical" device.

Figure 41 :

 41 Figure 41: Regions (1,2,3) of the droplet surface.

•

  h(x, t) by h 0 p(x/b) and • ∂ x h by h 0 b p (x/b) • e(x) by min(e, ed z dx-x dx ).The function p is extracted from the experimental data. More precisely, p is obtained by tting the experimental proles illustrated in gure 40. We nd for 0 < x < 1,p(x) = -1.6727x 5 + 3.3531x 4 + 0.317x 3 -3.0016x 2 + 0.0033x + 1.0001(44)and p (x) = -5 * 1.6727x 4 + 4 * 3.3531x 3 + 3 * 0.317x 2 -3.0016x + 0.0033.[START_REF] Dangla | 2D droplet microuidics driven by connement gradients[END_REF]

Figure 42 :

 42 Figure 42: Phase diagram of (e, d z ) for d x = 400 µm and w = 400 µm, that predicts the fate of droplets in devices with dimensions (e, w = 400, d z , d x = 400): breaking (below the orange line)or non-breaking (above the orange line). A transient region is plotted above the orange line and corresponds to situations in which droplets deform but reach an equilibrium shape. Then, a darker zone is plotted that corresponds to cases in which droplets do not deform.

Figure 43 :

 43 Figure 43:Phase diagram that predicts the fate of droplets in devices with dimensions (e, w, d z , d x ): breaking (above the line) or non-breaking (below the line). The geometric parameters are adimensionned by the channel width w. The curves correspond to the spectrum of equations w -2h eq = e plotted for dierent ratios dx w .

Figure 44 :

 44 Figure44: Phase diagram 43 on which the experimental data obtained with "mechanical devices" are added. Dark green dots correspond to breaking devices, light green dots correspond to nonbreaking "mechanical" devices. In the latter devices, the droplets do not even deform.

Figure 46 :

 46 Figure 46: Plot of the droplet interface prole in the neck region provided by Surface Evolver (cross markers) and by the semi-theoretical model (black lines using expression 44) for two sets of geometric parameters: (blue) d z = 4µm, d x = 400µm, e = 35µm and w = 400µm and (green) d z = 9µm, d x = 300µm, e = 40µm and w = 200µm

Figure 47 :Figure 48 :

 4748 Figure 47: Denition of the L lef t g and L right g

Figure 49 :

 49 Figure 49: Time evolution of the neck width for a "mechanical" device in which d z = 5 µm, d x = 400 µm, e = 30 µm and w = 400 µm. The dierent curves correspond to droplets with varied gutters lengths L g = (17; 37; 60; 68; 100; 150; 155) µm ranked from the droplet that splits the quickest to the one that splits the slowest. The value of K is adjsuted to 2.1 * 10 -3 in the power balance model to t the experimental data (dots) with the theoretical predictions (line).

Figure 50 :

 50 Figure 50: Time evolution of the neck width for experiments carried out in the "thermomechanical" device for two dierent temperatures: T = 54 • C (non-breaking droplet) and T = 67 • C (breaking droplet). The values of the geometrical parameters (d z , d x ) are adjusted to t the experimental data (dots) with the theoretical predictions (line).

Figure 51 :Figure 52 :

 5152 Figure 51: a) Fitting values of d z (cross markers) that t at best the theoretical model with the experiment prole of the time evolution of the neck width in the resistance device for dierent temperature increase. The predicted values of d z (cross markers) are compared to the characterized values of d z (square) in the chapter Materials and Methods. b) Fitting values of d x that have been chosen to t the experimental data with the model for dierent temperature increase.

Figure 55 :

 55 Figure 55: a) 3D illustration of the peanut-like shape droplet b) Snapshot of a 800 µm long droplet relaxation in a 200µm*30µm rectangular channel. The images are displayed every 5 seconds. c)Spatio-temporal evolution of the neck width during the droplet deformation t on < t < t of f and the droplet relaxation t > t of f . t on is the time at which the resistance is switched on and t of f is the time at which the resistance is switched o.

Figure 56 :

 56 Figure 56: a) 3D schematic of the droplet geometry shaped by the connement in the z-axis and in the y-axis. The undeformed droplet length is L; w and e are respectively the width and the thickness of the channel. The inset schematically shows the form of the gutters and the meniscus of radius e/2. (b) Snapshot of the relaxation of the droplet. The neck full width, half maxima is σ, and h(x, t) is the distance between the channel wall and the deformed interface, of maximal value h 0 (t) = h(0, t).

Figure 57 :

 57 Figure 57: a) Spatiotemporal evolution of the neck for two droplets in a channel of section e × w = 30 × 200 µm 2 with lengths, i ) L d = 510 µm (elapsed time: 15s) and ii ) L d = 1025 µm (15 s). The insets show: i ) experimental temperature versus time for the highlighted 1.4 s interval; ii ) schematics of the thickness prole e(x) around t off , the latter quantity also indicated by the dashed vertical line.

Figure 58 :

 58 Figure 58: Volume of the bottom channel thermal expansion.

Figure 59 :

 59 Figure 59: Calculation of the volume of oil, the inner phase, that occupies the free volume released by the thermal relaxation of the system.

Figure 60 :

 60 Figure 60: b) h 0 as a function of time for three droplet lengths and c) the corresponding time dependence of the x extent of the neck, σ.

Figure 61 :

 61 Figure61: a) Interface prole h(x, t) at the level of the neck over time. Dark blue plots illustrate the neck proles at early stages and light green plots show the later stages; the total elapsed time is 6.5 s and the droplet length was L = 800 µm, with e = 30 µm and w = 200 µm. b) The same proles of the central droplet interface after having rescaled the y-axis by t 0 -t and the x-axis by (t 0 -t) 1/3 .

Figure 62 :

 62 Figure 62: Self-similar prole of the interface after having rescaled the y-axis by τ and the xaxis by τ β with β = 1, 1/2, 1/3, 1/4, 1/5. For more acuity, the interface prole are respectively mulitplied by 5, 4, 3, 2 and 1.

Figure 63 :

 63 Figure63: Time evolution of the volume of water in the neck Ω(t). Experimental data (rectangles) are compared to power evolutions mτ p for for p = [0; 1/5; 1/4; 1/3; 1/2; 1].

1 ηk 2

 12 (1+e 2kh ) ∂ x p and B = e 2kh ηk 2 (1+e 2kh ) ∂ x p Volume conservation writes ∂ t h = v x dy such that we get

Figure 66 :

 66 Figure 66: Direction of the moving droplet menisci in the neck region and at the extremities of the droplet.

Figure 68 :

 68 Figure 68: a) View of the advancing meniscus in the neck. Flows are normal to the interface at the center of the neck. The velocity elds write v y (z) while ows are parallel to the gutters, velocity elds then write v x (z). b) Cross-section of the advancing meniscus in the plane (y, z).

Figure 69 :

 69 Figure 69: Time evolution of the gutter length for a 2100 µm long droplet in a 200 µm wide and a 30 µm high channel

Figure 70 :

 70 Figure 70: Plot of wκ(t) versus τ for dierent datasets: ( ) L g = 446µm e = 30µm w = 200µm, ( )L g = 430µm e = 30µm w = 200µm, ( )L g = 350µm e = 30µm w = 200µm, (Λ)L g = 1000µm e = 29µm w = 400µm, (Λ)L g = 850µm e = 30µm w = 400µm, (Λ)L g = 950µm e = 30µm w = 400µm.

Figure 71 :

 71 Figure 71: a) Plot of h(0, t) versus τ b) Plot of dimensionless variables h(0, t) versus τ for dierent sets of experiments: * L g = 1000µm e = 29µm w = 400µm, * L g = 850µm e = 30µm w = 400µm, L g = 750µm e = 30µm w = 200µm, L g = 549µm e = 30µm w = 200µm, L g = 350µm e = 30µm w = 200µm, •L g = 260µm e = 60µm w = 200µm, •L g = 360µm e = 60µm w = 200µm.

  dx e ηo-ηw ∼ 30 * 10 -6 /(24 * 10 -3 ) ∼ 10 -3 m/s.

Figure 73

 73 Figure73summarizes the distribution of the velocity at the droplet interface in a cross-section view. The gure illustrates that the velocity is nearly zero (see previous discussion) in the lubricating lms and maximum in the gutters.

Figure 73 :

 73 Figure 73: Scheme of the velocity distribution at the interface of the droplet in a cross-section view. The gure illustrates that the velocity is zero in the lubricating lms and maximum in the gutters.

  At the droplet interface v(0) = V int , at the walls, v(e) = 0. Therefore, the velocity eld writes v(r) ≈ 1 η o -η w

∼ 10 - 5 *

 105 10 2 * 0.2 ∼ 10 -4 m 3 .s -1 .

Figure 74 :

 74 Figure 74: Comparison in between the time evolution of the neck width of a 800 µm long droplet deformed by dierent temperature increase in a 200 µm and 30 µm wide and high channel. The values of (d x , d z ) that were implemented in the model to t the experimental data minimize the error function N .

Figure 75 :

 75 Figure75:(a) Interface prole h(x, t) at the level of the neck over 11 seconds, the duration of the relaxation process. (b) Self-similar prole of the interface after having rescaled the y-axis by t 0 -t and the x-axis by (t 0 -t) 1/3 .

Remerciements

Je remercie ...

Summary

In this chapter, we have investigated the droplet deformation induced by a local variation of the channel topography. This variation is induced by the local thermal dilation of the channel walls heated with a micro-patterned resistance that is located below the channel. We show that the droplet deforms to minimize its surface energy when the latter undergoes the local connement gradient until it can even break. The thermal Marangoni eect that is induced in the system does not seem to have a signicant eect on the droplet deformation. The dynamics of the droplet deformation seems to be set by the viscous dissipation of ows that drain through the gutters located at the four corners of the channel. Observing the dynamics of the droplet deformation could constitute a new tool to indirectly measure the local variation of the channel topography, provided that this one can be assimilated to a symmetric and linear connement gradient.

Contents 21 Experimental observations 21.1 Denitions

This paragraph recaps the geometrical parameters of the problem, similar to the ones that are used to describe the droplet deformation. Figure 56 illustrates the droplet geometry. We dene • h(x, t) as the distance between the channel side wall and the droplet interface

• L is the droplet length • L g is the length of the gutters A sequence of droplet proles is shown in Figure 56bi -iv). The neck is dened as the central portion of the peanut-like shape of the droplet. The extent of the neck in the x-axis is described by σ which corresponds to the the full width, half maximum of the interface prole. The extent of the neck in the y-axis is described by h 0 , the maximum value of the interface prole. We observe that the timescale of the droplet relaxation is few seconds.

Appendix Droplet deformation

Early times of the self-similar analysis

Approximative estimation of the time of the droplet break-up in the case of a Marangoni eect

In the lubricating lms (top, bottom, sides of the droplet), the length over which there is a ow shear is h ∞ (see the chapter Introduction). In the gutters, the distance over which ows shear is typically the size of the gutters that scales as e. In the droplet, at the level of the meniscus, ows also typically shear over the height of the cavity e. We can thus estimate the velocity of the Marangoni ow in each regions of the system. For the calculations, we take e = 30 µm and h ∞ = 20 nm. The viscosity of mineral oil is 25 mPa.s -1 and the viscosity of water is 1 mPa.s -1 .

• In the lubricating lm