Foam drainage and Marangoni effect
Although the heart of the study deals with 2D Hele-Shaw cells, the examples in this section are taken in 3D in order to set ideas down and make relevant observations and questions emerge. As mentioned in section 1.3, drainage refers to liquid flowing through the foam’s liquid network, as shown in Fig.1.10 in the case of gravity-driven drainage. In a first part, we will define and model the permeability of a porous medium, and then consider the subtleties inherent to gas/liquid interfaces, such as their fluid or rigid character. Finally we will envision to use surface tension gradients to control foam drainage.
Modelling the permeability of a porous medium
Modeling foam drainage is inspired by the field of flow in solid porous media. The aim of this section is to present some relevant definitions and results derived from this theory. At the scale of the pore
Here we consider a liquid of dynamic viscosity and density l flowing through a pore of radius rp and length l, oriented at an angle to the vertical. In a foam, we can safely assume that the Reynolds number (comparing inertia to viscosity forces) is small compared to one: Re = lul/ << 1, so that the velocity field in the liquid is approximately of the form ~u = u(r)~e, where r is the radial coordinate, see Fig.1.14. The force acting on the liquid due to the pressure p is r2 p(p(0) − p(l)). Along the pore, the gravity force (per unit volume) is equal to l~g · ~e = −lg cos . Since Re << 1, the flow is governed by the equilibrium between the viscous forces and the driving forces (gravity and pressure stemming from gradients of radii of curvature), giving: 1 r d dr r du dr = dp dl + lg cos .
A special porous medium: the foam – Importance of surface mobility
Foams can be considered as porous media, with liquid and bubbles respectively likened to the interstitial phase and grains (in a solid porous medium). Nevertheless, there are substantial differences between a foam and a solid porous medium, namely:
• the geometry of the liquid network in a foam is fixed by Plateau’s laws , which is not the case in a solid porous medium.
• the nature of the gas/liquid interfaces allows the foam to contract and expand contrary to a solid porous medium.
• in a solid porous medium, the liquid obeys a no-slip boundary condition at the walls. Conversely in a foam, the boundary condition at the air-water interface is expressed through a condition on the stress. Hence the rheological behaviour of the interface will be significantly modified depending on the surfactants adsorbed on it, ranging from fluid to rigid interfaces.
In the 1960s, Leonard and Lemlich  considered the flow in an infinitely long Plateau border and investigated the role of the interfacial mobility. The relative contributions of the surface dissipation (i.e. the surface shear stresses) and the bulk dissipation (i.e. the bulk viscous stresses) is described by the dimensionless Boussinesq number: Bq = s r (1.8). where s is the surface shear viscosity  and r is the radius of curvature. Consequently, the value of the Boussinesq number affects the velocity profile in the Plateau border:
• for Bq >> 1, the dissipation is dominated by the effect of the surface shear stresses, and the velocity profile is similar to a Poiseuille flow. In this case, the interface is immobile in the sense that it resists flow by generating high velocity gradients across the Plateau border’s cross section, as presented in Fig.1.16-a.
• for Bq << 1, the dissipation is dominated by the effect of the bulk viscous stresses, and the velocity profile is close to a plug flow. This situation refers to walls presenting a high mobility in response to the bulk flow, going with an increase in the foam permeability, as shown in Fig.1.16-c.
Marangoni effect: a pathway to control drainage ?
The Marangoni effect is a mass transfer along an interface between two fluids due to a surface tension gradient. It was first observed by Lord Kelvin’s brother, James Thomson, in 1855  and called so after the Italian physicist Carlo Giuseppe Matteo Marangoni, who studied it for his doctoral dissertation in 1865.
The tears of wine are one of the most common manifestations of the Marangoni effect (see Fig.1.18). Considering that wine is simply a mixture of alcohol and water, the thin film of wine wetting the edges of a glass will be richer in water than the bulk because alcohol evaporates faster than water in the film. Since alcohol has a lower surface tension than water, a tangential stress will be created at the air/liquid interface in the film, due to the surface tension gradient existing between the top of the film and the bulk. When the typical length on which the liquid film pulled from the bulk reaches the capillary length (typically 2 mm in the present case), an equilibrium between gravity and capillarity establishes, leading to the self-sustained mechanism showed in Fig.1.18 and illustrated by the common ’rolls’ observed in a glass of wine.
Manufacturing a Polydimethylsiloxane (PDMS) system
PDMS devices are cast from the silicon master by pouring a liquid mixture composed of PDMS monomer (RTV, Neyco) and 10 wt% of crosslinker onto the master (Fig.2.5-1). At this stage the system is cured at 70C for 2 hours and then peeled off from the master (Fig.2.5-2). Hence we make holes in the solidified PDMS, corresponding to future inlets and outlets of fluid.
Generating bubbles in microfluidics
Microfluidic devices can allow for high-throughput production of bubbles with very wellcontrolled size and generation frequency: as fast as 100 Hz, with diameters ranging from 10 to 500 μm, and a polydispersity lower than 5 % [18, 25, 26]. The typical geometries involved, such as the T-junction [27, 28, 29, 30], the flow-focusing junction [18, 31, 32] (Fig. 1(b)), the co-flowing junction [33, 34] (Fig.2.6-a), the liquid cross-flow  or the 2.5D geometry  are now well-known implements in the microfluidics toolbox. The formation of bubbles has been extensively investigated, published and reviewed, taking into account various parameters such as the viscosity ratio, flow rates, gas pressure, channel geometry, and channel wettability to cite a few [31, 37, 38, 39].
Table of contents :
I Foam drainage control by Marangoni stresses in a 2D-microchamber
1 Foam drainage
1.1 The Foam: a multiscale physical and physicochemical system
1.1.1 At the scale of a gas/liquid interface
Surface tension and the Young-Laplace law
From interface to film: the role of surfactants and disjoining pressure
1.1.2 At the scale of a film
From film to bubble
1.1.3 At the scale of a bubble
Bubbles, films and Plateau borders
From bubble to foam
1.1.4 At the scale of a foam
Foam under gravity
Quantity of interface
1.2 Geometry of a 2D microfoam
1.3 Foam Ageing
Our positioning/strategy regarding foam ageing
1.4 Foam drainage and Marangoni effect
1.4.1 Modelling the permeability of a porous medium
At the scale of the pore
At the scale of the porous medium: Darcy’s law
A special porous medium: the foam – Importance of surface mobility
1.4.2 Drainage equation
1.4.3 Marangoni effect: a pathway to control drainage ?
Definition & proof of concept: the tears of wine
2 Building the experimental setup
2.1 General framework – experimental setup
2.2 Foam generation
2.2.1 Manufacturing a Silicon Master
2.2.2 Manufacturing a Polydimethylsiloxane (PDMS) system
2.2.3 Generating bubbles in microfluidics
2.3 Materials & Methods
2.3.1 Gas/liquid phases & Stabilization
2.3.2 Dependence in temperature: @ /@T
2.4 Integrating heating resistors to a Hele-Shaw cell & temperature characterization .
2.4.1 Temperature control in microfluidics: context and applications
2.4.2 Manufacturing heating resistors
2.4.3 Integration of heating resistors
2.4.4 Temperature characterisation
Materials & Observation
Reference image –
Power dissipated by the resistors –
Measuring a temperature gradient & dependence with electric power –
Advection Vs Diffusion : thermal Péclet number –
2.5 Thermocapillary tangential stress
3 (Micro)foam drainage control using a thermocapillary stress in a 2D Hele-Shaw cell
3.2 Gravity Vs. Thermocapillarity: on the way to control drainage
3.2.1 Physical chemistry
3.2.2 Time evolution & liquid fraction measurement
Liquid fraction measurement
3.2.3 Gravity drainage
3.2.4 Thermocapillary drainage
3.2.5 Coupling gravity and thermocapillarity to control foam drainage
3.2.6 Mass conservation within the Hele-Shaw cell
3.2.7 The role of the capillary pressure
3.3 Conclusion of Chapter 3
4 Surface Rheology: an insight into solutocapillarity
4.1 General concepts
4.1.1 Interfacial viscoelasticity
4.1.2 Response of an interface to expansion or compression
4.1.3 Response to shear
4.2 Investigating the combined effect of thermocapillarity, solutocapillarity and surface shear viscosity
4.2.1 A brief state of the art
4.2.2 The solutocapillary effect
4.2.3 Physico-chemical characterization
Langmuir-Von Szyszkowski isotherm
Elastic behaviour evidence
4.2.4 Measurements and results
Mass conservation approach
Linearity between the reponse of the system and the temperature gradient .
Evolution of r with the DOH bulk concentration
Evolution of r with geometrical parameters (e and R): possible surfactants’
4.3 Conclusion of Chapter 4
II Development of a thermomechanical actuation system for Labs-onachip
5 State-of-the-art & device microfabrication
5.1 A brief history of droplet handling in microfluidics
5.1.1 State-of-the-art & our positioning
5.1.2 Thermomechanical effect
5.2 Device microfabrication & PDMS dilation characterisation
5.2.1 Device microfabrication
Manufacturing and integrating heating resistors
5.2.2 PDMS dilation characterisation
5.3 Materials & Methods
5.3.1 Bancroft rule & Hydrophilic-lipophilic balance (HLB)
5.3.2 Water/oil phases & stabilisation
Stabilizing oil-in-water droplets
Stabilizing water-in-oil droplets
5.4 COMSOL simulations: temperature control
Geometry, mesh and materials
Physics at play and boundary conditions
6 A versatile technology for droplet-based microfluidics: thermomechanical actuation
6.1 Propelling droplets without an external flow
Proof of concept in 1D-channels
6.2 Directing droplets: stopping, sorting, rearranging
6.2.1 A thermomechanical valve
6.2.3 Storage, release and sequence modification
6.3 Breaking up droplets: thermally-induced hydrodynamic pinching
6.3.1 Droplet breakup
6.3.2 Droplet production
6.4 Conclusion of Chapter 6
7 General Conclusion & Perspectives
7.1 Foam project
7.2 Thermomechanical actuation project
8 Value creation: publications and patents
A Liquid volume fraction measurement
Expressions of rA(z, ) and VA
Expressions of rB(z, ) and VB
Liquid volume fraction
B Why disregarding coarsening ?