1.4 Conclusions of Chapter 1

* Foam film elasticity is not a trivial concept: it is not about surfactant gradients or unsteady diffusion problems but rather finite size effects of the bulk for the sake of populating the interfaces.

* Back in 1967, Prins et al. [5] developed a theoretical model based on Langmuir isotherms taking into account the small thicknesses of the bulk and made predictions for the elasticities in the submicellar regime. They also conducted experiments which validated their model, stretching a vertical film and looking at its thickness profile.

* I extended the model to the micellar regime, which required a numerical resolution not available at the time of Prins. We fitted the model on experimental data made prior to my arrival, deforming a 5 films elementary foam allowing us to draw the stress-strain curves of foam films $\frac{\Delta\sigma}{\sigma_0} = f(\epsilon)$.

* Our thermochemical model captures the qualitative behaviour of these curves but fails to predict them quantitatively with the nominal concentrations of surfactants added to the solutions.

* The not-so-well-known kinetics of hydrolysis of SDS into dodecanol and the sensibility of film elasticity with insoluble impurities make it difficult to actually control and predict the stress-strain curves quantitatively. We fitted our model on an effective dodecanol concentration encompassing both the amount of dodecanol already present with SDS and the insoluble impurities present in the air.

* We managed to evaluate film elasticity $E_f$ for quite a wide range of SDS and dodecanol concentrations. This will be useful later on while looking at the visco-elastic response of an elementary foam in chapter 3, in order to validate the elasticity measurements performed thereupon.
# In-plane motions in a foam film: line tension and damping forces

Foam films display complex 2D motions in their own planes, which can be easily visualised when we look at the reflected dancing colours at their surface. Any child playing with bubbles can notice it! A closer look also reveals that these coloured patterns tend to adopt circular shapes. But what is the driving force at play here? How is it related to the colours? The short answer is: the different colours correspond to different film thicknesses, and the resulting thickness gradient corresponds to a driving force on the interfaces, due to its excess of energy compared to a flat film. Hence the coloured circular arrangement seen at the surface.

Also, the motions implied by these forces must be damped somehow, and we need to clarify the physical ingredients potentially at stake here: it may be due to the intrinsic surface shear viscosity $\eta_s$ and/or bulk viscosities of both the gas and the liquid $\eta_g$ or $\eta_l$. The competition between these ingredients, and more incredibly the existence of a surface viscosity in a certain range of situations, is still open to debate!

In this chapter, I will expose the work of our group around an experimental setup which allowed us to create and measure a line tension $T$ in the plane of a single foam film, using its heterogeneities of thickness. By studying the relaxation of a patch of foam film, and knowing the damping mechanism at play, we were able to quantify and validate our force measurement of $T$. This setup was built a first time during the internship of Marion Berry (spring/summer 2021, supervised by Raphaël Poryles), from which a proof of concept and preliminary results were obtained, and a second time the year after for the internship of Gaëlle Audéoud (spring/summer 2022 supervised by me), from which the definitive experimental data were obtained. I made the data analysis and the theoretical model, under the supervision of Isabelle Cantat, and the results have been submitted to *Physical Review Letters* [99] (currently addressing minor revisions).
# In-plane motions in a foam film: line tension and damping forces

Foam films display complex 2D motions in their own planes, which can be easily visualised when we look at the reflected dancing colours at their surface. Any child playing with bubbles can notice it! A closer look also reveals that these coloured patterns tend to adopt circular shapes. But what is the driving force at play here? How is it related to the colours? The short answer is: the different colours correspond to different film thicknesses, and the resulting thickness gradient corresponds to a driving force on the interfaces, due to its excess of energy compared to a flat film. Hence the coloured circular arrangement seen at the surface.

Also, the motions implied by these forces must be damped somehow, and we need to clarify the physical ingredients potentially at stake here: it may be due to the intrinsic surface shear viscosity $\eta_s$ and/or bulk viscosities of both the gas and the liquid $\eta_g$ or $\eta_l$. The competition between these ingredients, and more incredibly the existence of a surface viscosity in a certain range of situations, is still open to debate!

In this chapter, I will expose the work of our group around an experimental setup which allowed us to create and measure a line tension $T$ in the plane of a single foam film, using its heterogeneities of thickness. By studying the relaxation of a patch of foam film, and knowing the damping mechanism at play, we were able to quantify and validate our force measurement of $T$. This setup was built a first time during the internship of Marion Berry (spring/summer 2021, supervised by Raphaël Poryles), from which a proof of concept and preliminary results were obtained, and a second time the year after for the internship of Gaëlle Audéoud (spring/summer 2022 supervised by me), from which the definitive experimental data were obtained. I made the data analysis and the theoretical model, under the supervision of Isabelle Cantat, and the results have been submitted to *Physical Review Letters* [99] (currently addressing minor revisions).
## 2.1 Capillary force and line tension in foam films

As we discussed in the Introduction, in-plane motions of foam films are only about plug flows for time scales up to tens of seconds (cf. 0.2.1), giving them a sliding puzzle-like dynamics. This has proven to be a crucial piece of information, as it enabled the development of a general theoretical framework for these motions in the form of 2D Navier-Stokes equations. This conception of foam films retraces back to the work of Mysels, Couder et al. [28, 49], part of which being fairly recent.

These equations have since enabled to further advance in the description of these systems: measuring surface viscosities [1, 36], studying gravity drainage and the convection in the plane of foam films [64, 100], or even 2D turbulence phenomena [101]. A proper passage from a 3D to a 2D Stokes equation in the case of a foam film is derived and proposed in Appendix 4, along with its Green function.

In this section, we write our own equation of motion following the same path as these previous studies. The novelty is that we build a stress tensor for the fluid which takes into account thickness gradients in the film. Once we have this constitutive law and the corresponding equation of motion, we will focus on the particular case of a very localised thickness gradient, which will allow us to define a line tension $T$, and the 2D Stokes equation in which it is a source term.

All this preparatory theoretical work will help us describe the experiment of the next section where a line tension is brought to light by a “simple” relaxation phenomenon.

### 2.1.1 Building the capillary stress tensor for an elementary piece of film with a thickness gradient

We consider the piece of film represented in the Figure 2.1. It is limited by the two interfaces and by the vertical planes which intersect the plane $z = 0$ along the elementary lengths $d\xi$ and $ds$. The unit vector $n$ is chosen along the steepest slope of the interface and the unit vector $t$ is perpendicular to $n$ (both in the $(x, y)$ plane):

$$ \mathbf{n} = (h_x, h_y, 0) / \sqrt{h_x^2 + h_y^2}, \quad \mathbf{t} = (h_y, -h_x, 0) / \sqrt{h_x^2 + h_y^2} \qquad (2.1) $$

With the indices accounting for the derivatives along the specified coordinates of the cartesian basis $B_0 = (e_x, e_y)$.

Our goal is now to build the 2D stress tensor for the film $\overline{\sigma}_{cap}$ by considering the forces exerted on the lateral faces. Note that the air pressure is taken as the pressure reference, so that no forces are exerted on the top and bottom interfaces. The norm of the thickness gradient is a small parameter in the problem, as classically used in the lubrication approximation. To build the film stress tensor, we anticipate that the forces exerted on the system are of order 2 in this parameter (as will be shown below), and we thus drop higher order terms. Also, we consider that both interfaces are subject to a surface tension $\gamma = \gamma_0 + \delta\gamma$ deviating from its equilibrium tension $\gamma_0$ by a second order term $\delta\gamma$.

Let us first consider the face 1 defined in Fig.2.1. It is a rectangle of area $dS_1 = 2hds$ of normal $n$, and we denote $dF_1 = dF_{p,1} + dF_{\gamma,1}$ the total force applied on the surface. The local pressure in the film is $P = -\gamma_0\Delta h$, leading to the pressure force $dF_{p,1} = -PdS_1n = 2\gamma_0h\Delta hdn$. Note
## 2.1 Capillary force and line tension in foam films

As we discussed in the Introduction, in-plane motions of foam films are only about plug flows for time scales up to tens of seconds (cf. 0.2.1), giving them a sliding puzzle-like dynamics. This has proven to be a crucial piece of information, as it enabled the development of a general theoretical framework for these motions in the form of 2D Navier-Stokes equations. This conception of foam films retraces back to the work of Mysels, Couder et al. [28, 49], part of which being fairly recent.

These equations have since enabled to further advance in the description of these systems: measuring surface viscosities [1, 36], studying gravity drainage and the convection in the plane of foam films [64, 100], or even 2D turbulence phenomena [101]. A proper passage from a 3D to a 2D Stokes equation in the case of a foam film is derived and proposed in Appendix 4, along with its Green function.

In this section, we write our own equation of motion following the same path as these previous studies. The novelty is that we build a stress tensor for the fluid which takes into account thickness gradients in the film. Once we have this constitutive law and the corresponding equation of motion, we will focus on the particular case of a very localised thickness gradient, which will allow us to define a line tension $T$, and the 2D Stokes equation in which it is a source term.

All this preparatory theoretical work will help us describe the experiment of the next section where a line tension is brought to light by a “simple” relaxation phenomenon.

### 2.1.1 Building the capillary stress tensor for an elementary piece of film with a thickness gradient

We consider the piece of film represented in the Figure 2.1. It is limited by the two interfaces and by the vertical planes which intersect the plane $z = 0$ along the elementary lengths $d\xi$ and $ds$. The unit vector $n$ is chosen along the steepest slope of the interface and the unit vector $t$ is perpendicular to $n$ (both in the $(x, y)$ plane):

$$ \mathbf{n} = (h_x, h_y, 0) / \sqrt{h_x^2 + h_y^2}, \quad \mathbf{t} = (h_y, -h_x, 0) / \sqrt{h_x^2 + h_y^2} \quad (2.1) $$

With the indices accounting for the derivatives along the specified coordinates of the cartesian basis $B_0 = (e_x, e_y)$.

Our goal is now to build the 2D stress tensor for the film $\overline{\sigma}_{cap}$ by considering the forces exerted on the lateral faces. Note that the air pressure is taken as the pressure reference, so that no forces are exerted on the top and bottom interfaces. The norm of the thickness gradient is a small parameter in the problem, as classically used in the lubrication approximation. To build the film stress tensor, we anticipate that the forces exerted on the system are of order 2 in this parameter (as will be shown below), and we thus drop higher order terms. Also, we consider that both interfaces are subject to a surface tension $\gamma = \gamma_0 + \delta\gamma$ deviating from its equilibrium tension $\gamma_0$ by a second order term $\delta\gamma$.

Let us first consider the face 1 defined in Fig.2.1. It is a rectangle of area $dS_1 = 2hds$ of normal $n$, and we denote $dF_1 = dF_{p,1} + dF_{\gamma,1}$ the total force applied on the surface. The local pressure in the film is $P = -\gamma_0\Delta h$, leading to the pressure force $dF_{p,1} = -PdS_1n = 2\gamma_0h\Delta hdn$. Note
<figure><img src="image_46.png" /><figcaption>Figure 2.1: Notations for an elementary piece of foam film confined between two liquid-gas interfaces. The semi-thickness $h$ of the foam film varies along $\xi$.</figcaption></figure>

that here we discarded the second order term $\delta\gamma$ in the Laplace pressure as we anticipate it would yield a contribution of high order.

The forces induced by the surface tension at the top and bottom interfaces $dF_{\gamma,1}$ need more precautions as the force orientations are slightly tilted with respect to $n$ and differ between both interfaces. They are applied along the vectors:

$$ \mathbf{n}^{\text{top/bot}} = \frac{\mathbf{n} \pm ||\nabla h|| \mathbf{e}_z}{\sqrt{1 + ||\nabla h||^2}} \qquad (2.2) $$

Where by definition of our local frame $B_e = (\mathbf{n}, \mathbf{t})$, we have $||\nabla h|| = |\partial_\xi h|$. Thus, we write the forces:

$$ d\mathbf{F}_{\gamma,1} = (\gamma_0 + \delta\gamma)ds (\mathbf{n}^{\text{top}} + \mathbf{n}^{\text{bot}}) = (2\gamma_0(1 - \frac{||\nabla h||^2}{2}) + 2\delta\gamma)ds \mathbf{n} \qquad (2.3) $$

In which the two contributions along $e_z$ compensate for each other, leading to this development at second order. We can now write the total force acting on face 1:

$$ d\mathbf{F}_1 = 2\left(\gamma_0\left(1 - \frac{||\nabla h||^2}{2}\right) + h\Delta h\right) + \delta\gamma \, ds \, \mathbf{n} \qquad (2.4) $$

We now look at face 2 of normal $\mathbf{t}$. In a similar way, we write the pressure force $dF_{p,2} = -PdS_2\mathbf{t} = 2\gamma_0h\Delta h d\xi\mathbf{t}$ which is similar to face 1 at leading order despite the trapezoidal form of $dS_2$.

This time, the direction of the forces due to surface tension is horizontal along $\mathbf{t}$ on both interfaces. The subtlety here lies in the lengths over which are applied the surface tension at the top and bottom interfaces, and which are given by $dl_2 = d\xi(1 + ||\nabla h||^2/2)$ at second order. This leads to the expression:

$$ d\mathbf{F}_{\gamma,2} = 2(\gamma_0 + \delta\gamma)dl_2 \, \mathbf{t} = \left(2\gamma_0\left(1 + \frac{||\nabla h||^2}{2}\right) + 2\delta\gamma\right)d\xi \, \mathbf{t} \qquad (2.5) $$

And we finally have the full expression of the force on face 2:
<figure><img src="image_46.png" /><figcaption>Figure 2.1: Notations for an elementary piece of foam film confined between two liquid-gas interfaces. The semi-thickness h of the foam film varies along ξ.</figcaption></figure>

that here we discarded the second order term δγ in the Laplace pressure as we anticipate it would yield a contribution of high order.

The forces induced by the surface tension at the top and bottom interfaces $dF_{γ,1}$ need more precautions as the force orientations are slightly tilted with respect to $n$ and differ between both interfaces. They are applied along the vectors:

$$ \mathbf{n}^{\text{top/bot}} = \frac{\mathbf{n} \pm ||\nabla h|| \mathbf{e}_z}{\sqrt{1 + ||\nabla h||^2}} \qquad (2.2) $$

Where by definition of our local frame $B_e = (\mathbf{n}, \mathbf{t})$, we have $||\nabla h|| = |\partial_\xi h|$. Thus, we write the forces:

$$ d\mathbf{F}_{\gamma,1} = (\gamma_0 + \delta\gamma)ds (\mathbf{n}^{\text{top}} + \mathbf{n}^{\text{bot}}) = (2\gamma_0(1 - \frac{||\nabla h||^2}{2}) + 2\delta\gamma)ds \mathbf{n} \qquad (2.3) $$

In which the two contributions along $e_z$ compensate for each other, leading to this development at second order. We can now write the total force acting on face 1:

$$ d\mathbf{F}_1 = 2\left(\gamma_0\left(1 - \frac{||\nabla h||^2}{2}\right) + h\Delta h\right) + \delta\gamma \, ds \, \mathbf{n} \qquad (2.4) $$

We now look at face 2 of normal $\mathbf{t}$. In a similar way, we write the pressure force $dF_{p,2} = -PdS_2\mathbf{t} = 2\gamma_0h\Delta h d\xi\mathbf{t}$ which is similar to face 1 at leading order despite the trapezoidal form of $dS_2$.

This time, the direction of the forces due to surface tension is horizontal along $\mathbf{t}$ on both interfaces. The subtlety here lies in the lengths over which are applied the surface tension at the top and bottom interfaces, and which are given by $dl_2 = d\xi(1 + ||\nabla h||^2/2)$ at second order. This leads to the expression:

$$ d\mathbf{F}_{\gamma,2} = 2(\gamma_0 + \delta\gamma)dl_2 \, \mathbf{t} = \left(2\gamma_0\left(1 + \frac{||\nabla h||^2}{2}\right) + 2\delta\gamma\right)d\xi \, \mathbf{t} \qquad (2.5) $$

And we finally have the full expression of the force on face 2:
$$d\bar{F}_2 = 2\left(\gamma_0\left(1 + \frac{\|\nabla h\|^2}{2} + h\Delta h\right) + \delta\gamma\right)d\xi \mathbf{t} \qquad (2.6)$$

The goal now is to finally write $\bar{\bar{\sigma}}_{cap}$ whose general expression is given and developed:

$$\bar{\bar{\sigma}}_{cap} = \left( \begin{matrix} \frac{dF_1}{ds} \cdot \mathbf{n} & \frac{dF_1}{ds} \cdot \mathbf{t} \\ \frac{dF_2}{d\xi} \cdot \mathbf{n} & \frac{dF_2}{d\xi} \cdot \mathbf{t} \end{matrix} \right)_{B_e} = \gamma_0 \left( \begin{matrix} -\|\nabla h\|^2 & 0 \\ 0 & \|\nabla h\|^2 \end{matrix} \right)_{B_e} + 2\left(\gamma_0(1 + h\Delta h) + \delta\gamma\right) \bar{\bar{I}}_2 \quad (2.7)$$

Which can be more easily transcribed in the initial cartesian basis $B_0 = (e_x, e_y)$ in the form $\bar{\bar{\sigma}}_{cap} = \bar{\bar{\sigma}}_{cap}^* + \sigma^f \bar{\bar{I}}_2$, with $\sigma^f = 2(\gamma_0(1 + h\Delta h) + \delta\gamma)$ and:

$$\bar{\bar{\sigma}}_{cap}^* = -\gamma_0 \left( \begin{matrix} \partial_x h^2 - \partial_y h^2 & 2\partial_x h \partial_y h \\ 2\partial_x h \partial_y h & \partial_y h^2 - \partial_x h^2 \end{matrix} \right)_{B_0} \qquad (2.8)$$

This tensor $\bar{\bar{\sigma}}_{cap}$ eq.2.7 accounts for all the capillary forces acting on a piece of foam film with a thickness gradient such as depicted in Fig.2.1. It takes into account both the pressure and surface tension forces projected in the plane of the film.

However, the specific case we want to further describe and study has only very localised regions where $h$ varies. The experimental situation will be described in section 2.2: we create two adjacent regions of constant thicknesses (or varying very slowly spatially) where the variations of $h$ are all contained at the border. This border is so thin compared to the dimensions of the two regions that we will aim at describing this zone as a 1D object subject to a line tension $T$ defined and computed using $\bar{\bar{\sigma}}_{cap}$.

### 2.1.2 Gradients of thickness localised in a very thin region: building a line tension $T$

The goal of this subsection is to consider the limit where the thickness of the foam film varies in a very localised zone only, which we will call thickness jump zone hereafter. We will approximate this thin zone to a closed contour $C$, of any shape, across which we jump from a thickness $h_1$ to $h_2$. By expressing the capillary stress tensor $\bar{\bar{\sigma}}_{cap}$ in this region, we can integrate the forces and exhibit the presence of a line tension $T$ acting along $C$.

We will focus on the force acting on a section line normal to $C$, exerted by one side of the line on the other. The line tension $T$ is defined as the excess of force applied by one side of the line on the other (Fig.2.2, the line is parallel to the thickness gradient) compared to the case where we would have a flat film with interfaces at the equilibrium surface tension $\gamma_0$. It can be expressed as (note that we define $T$ for a single interface whereas $\bar{\bar{\sigma}}_{cap}$ has been already defined for two interfaces):

$$T = \int_{\xi_{min}}^{\xi_{max}} \mathbf{t} \cdot \left( \frac{1}{2} \bar{\bar{\sigma}}_{cap} - \gamma_0 \bar{\bar{I}}_2 \right) \cdot \mathbf{t} \, d\xi \qquad (2.9)$$
$$d\bar{F}_2 = 2\left(\gamma_0\left(1 + \frac{\|\nabla h\|^2}{2} + h\Delta h\right) + \delta\gamma\right)d\xi \mathbf{t} \qquad (2.6)$$

The goal now is to finally write $\bar{\bar{\sigma}}_{cap}$ whose general expression is given and developed:

$$\bar{\bar{\sigma}}_{cap} = \left( \begin{matrix} \frac{dF_1}{ds} \cdot \mathbf{n} & \frac{dF_1}{ds} \cdot \mathbf{t} \\ \frac{dF_2}{d\xi} \cdot \mathbf{n} & \frac{dF_2}{d\xi} \cdot \mathbf{t} \end{matrix} \right)_{B_e} = \gamma_0 \left( \begin{matrix} -\|\nabla h\|^2 & 0 \\ 0 & \|\nabla h\|^2 \end{matrix} \right)_{B_e} + 2\left(\gamma_0(1 + h\Delta h) + \delta\gamma\right) \bar{\bar{I}}_2 \quad (2.7)$$

Which can be more easily transcribed in the initial cartesian basis $B_0 = (e_x, e_y)$ in the form $\bar{\bar{\sigma}}_{cap} = \bar{\bar{\sigma}}_{cap}^* + \sigma^f \bar{\bar{I}}_2$, with $\sigma^f = 2(\gamma_0(1 + h\Delta h) + \delta\gamma)$ and:

$$\bar{\bar{\sigma}}_{cap}^* = -\gamma_0 \left( \begin{matrix} \partial_x h^2 - \partial_y h^2 & 2\partial_x h \partial_y h \\ 2\partial_x h \partial_y h & \partial_y h^2 - \partial_x h^2 \end{matrix} \right)_{B_0} \qquad (2.8)$$

This tensor $\bar{\bar{\sigma}}_{cap}$ eq.2.7 accounts for all the capillary forces acting on a piece of foam film with a thickness gradient such as depicted in Fig.2.1. It takes into account both the pressure and surface tension forces projected in the plane of the film.

However, the specific case we want to further describe and study has only very localised regions where $h$ varies. The experimental situation will be described in section 2.2: we create two adjacent regions of constant thicknesses (or varying very slowly spatially) where the variations of $h$ are all contained at the border. This border is so thin compared to the dimensions of the two regions that we will aim at describing this zone as a 1D object subject to a line tension $T$ defined and computed using $\bar{\bar{\sigma}}_{cap}$.

### 2.1.2 Gradients of thickness localised in a very thin region: building a line tension $T$

The goal of this subsection is to consider the limit where the thickness of the foam film varies in a very localised zone only, which we will call thickness jump zone hereafter. We will approximate this thin zone to a closed contour $C$, of any shape, across which we jump from a thickness $h_1$ to $h_2$. By expressing the capillary stress tensor $\bar{\bar{\sigma}}_{cap}$ in this region, we can integrate the forces and exhibit the presence of a line tension $T$ acting along $C$.

We will focus on the force acting on a section line normal to $C$, exerted by one side of the line on the other. The line tension $T$ is defined as the excess of force applied by one side of the line on the other (Fig.2.2, the line is parallel to the thickness gradient) compared to the case where we would have a flat film with interfaces at the equilibrium surface tension $\gamma_0$. It can be expressed as (note that we define $T$ for a single interface whereas $\bar{\bar{\sigma}}_{cap}$ has been already defined for two interfaces) :

$$T = \int_{\xi_{min}}^{\xi_{max}} \mathbf{t} \cdot \left( \frac{1}{2} \bar{\bar{\sigma}}_{cap} - \gamma_0 \bar{\bar{I}}_2 \right) \cdot \mathbf{t} \, d\xi \qquad (2.9)$$
<figure><img src="image_47.png" /><figcaption>Figure 2.2: Contour C separating two zones of different thicknesses $h_1$ and $h_2$ at equilibrium surface tension $\gamma_0$. Line tension $T$ is defined as the excess force applied by one side (ext) on the other (int) compared to a case where there is no change in $h$ or $\gamma$ along the line.</figcaption></figure>

Where $\xi_{min}$ and $\xi_{max}$ are chosen outside of the transition region, so that the integral does not depend on their precise value. The computation of $T$ needs to be carefully written, as not only the thickness varies along $\xi$, but also the surface tension $\gamma = \gamma_0 + \delta\gamma$.

### Surface tension profile within the thickness jump zone

<figure><img src="image_48.png" /><figcaption>Figure 2.3: Notations for a contour $\mathcal{I}$ in the thickness jump zone symbolised by the colour gradient. By integrating the interface stress along this contour for any $\epsilon$, we get an expression for the local tension $\gamma(\xi)$ in the thickness jump zone, enabling the computation of the line tension $T$.</figcaption></figure>

In order to estimate $\delta\gamma$ in the thickness jump zone, we start by doing a force balance over a contour $\mathcal{I}$ defined in Fig.2.3. It follows the tangential direction of $C$ and encompasses a range of $\xi$ from $\xi_{min} = 0$ the edge of the inner side where $h = h_1$ to $\xi = \epsilon$ arbitrarily taken between 0 and $\xi = \xi_{max}$ the edge of the outer side where $h = h_2$. The choice of $\xi_{max}$ is a practical problem which we address in the experimental section 2.3.4. Another important assumption is that the zone where $\nabla h$ is not negligible is very localised. That means we assume that $\xi_{max}/R \ll 1$, the
<figure><img src="image_47.png" /><figcaption>Figure 2.2: Contour C separating two zones of different thicknesses $h_1$ and $h_2$ at equilibrium surface tension $\gamma_0$. Line tension $T$ is defined as the excess force applied by one side (ext) on the other (int) compared to a case where there is no change in $h$ or $\gamma$ along the line.</figcaption></figure>

Where $\xi_{min}$ and $\xi_{max}$ are chosen outside of the transition region, so that the integral does not depend on their precise value. The computation of $T$ needs to be carefully written, as not only the thickness varies along $\xi$, but also the surface tension $\gamma = \gamma_0 + \delta\gamma$.

### Surface tension profile within the thickness jump zone

<figure><img src="image_48.png" /><figcaption>Figure 2.3: Notations for a contour $\mathcal{I}$ in the thickness jump zone symbolised by the colour gradient. By integrating the interface stress along this contour for any $\varepsilon$, we get an expression for the local tension $\gamma(\xi)$ in the thickness jump zone, enabling the computation of the line tension $T$.</figcaption></figure>

In order to estimate $\delta\gamma$ in the thickness jump zone, we start by doing a force balance over a contour $\mathcal{I}$ defined in Fig.2.3. It follows the tangential direction of $C$ and encompasses a range of $\xi$ from $\xi_{min} = 0$ the edge of the inner side where $h = h_1$ to $\xi = \epsilon$ arbitrarily taken between 0 and $\xi = \xi_{max}$ the edge of the outer side where $h = h_2$. The choice of $\xi_{max}$ is a practical problem which we address in the experimental section 2.3.4. Another important assumption is that the zone where $\nabla h$ is not negligible is very localised. That means we assume that $\xi_{max}/R \ll 1$, the
width $\xi_{max}$ over which $h$ varies is much smaller than the radius of curvature $R$ of contour $C$. We will thus develop our force balance equation in orders of $\xi_{max}/R$.

Let us write this force balance acting on the closed contour $\mathcal{I}$:

$$ \boldsymbol{F}_{\mathcal{I}} = \oint_{\mathcal{I}} \bar{\bar{\sigma}}_{\text{cap}} \cdot \boldsymbol{n}_{\mathcal{I}} \, d\mathcal{I} \qquad (2.10) $$

Where $\boldsymbol{n}_{\mathcal{I}}$ is the local normal vector oriented outward to the contour $\mathcal{I}$ of which we show an example for the segment $[BC]$ in Fig.2.3 (not to be confused with $\boldsymbol{n}$, the local normal to $C$). On the one hand, the integrals over $[AB]$ and $[CD]$ have constant local normal vectors $\boldsymbol{n}_{AB}$ and $\boldsymbol{n}_{CD}$ and a varying stress tensor $\bar{\bar{\sigma}}_{\text{cap}}$ as the thickness $h$ varies. On the other hand, $\bar{\bar{\sigma}}_{\text{cap}}$ is constant over $[BC]$ and $[DA]$, but the local normal vectors $\boldsymbol{n}_{BC/DA}(s_{\mathcal{I}})$ will vary. Note that because we suppose that the local curvature $\kappa = 1/R$ of $C$ varies slowly along the curvilinear abscissa, we approximate the Laplacian in the expression of $\bar{\bar{\sigma}}_{\text{cap}}$ to $\Delta h = \partial_{\xi\xi}h$, and we get, using eq.2.7:

$$ \begin{aligned} \frac{\boldsymbol{F}_{\mathcal{I}}}{\gamma_0} = & \left[ \boldsymbol{n}_{AB} + \boldsymbol{n}_{CD} \right] \int_0^{\epsilon} \left[ (\partial_{\xi}h)^2 + 2h \partial_{\xi\xi}h + 2\left(1 + \frac{\delta\gamma}{\gamma_0}\right) \right] d\xi \\ & + 2\left(1 + \frac{\delta\gamma(0)}{\gamma_0}\right) \int_B^C \boldsymbol{n}_{BC}(s_{\mathcal{I}}) ds_{\mathcal{I}} \\ & + \left[ -\left(\partial_{\xi}h\right)^2 \Big|_{\epsilon} + 2h \partial_{\xi\xi}h \Big|_{\epsilon} + 2\left(1 + \frac{\delta\gamma(\epsilon)}{\gamma_0}\right) \right] \int_D^A \boldsymbol{n}_{DA}(s_{\mathcal{I}}) ds_{\mathcal{I}} \end{aligned} \qquad (2.11) $$

The first line accounts for the integral over $[AB]$ and $[CD]$, the second for $[BC]$ and the third for $[DA]$. Note that we use $\partial_{\xi}h = 0$ and $\partial_{\xi\xi}h = 0$ at $\xi = 0$ as the inner and outer region of $C$ are defined so that $h = \text{cst}$.

We now express the local vectors of $\mathcal{I}$, $\boldsymbol{n}_{AB}$ and $\boldsymbol{n}_{CD}$ in the local basis of $C(t, \boldsymbol{n})$ using the local curvature of $C$ denoted $\kappa = 1/R$, and the length $[BC] = L$:

$$ \begin{aligned} \boldsymbol{n}_{AB} &= t - \frac{L}{2R} \boldsymbol{n} \\ \boldsymbol{n}_{CD} &= -t - \frac{L}{2R} \boldsymbol{n} \end{aligned} $$

And the vector integrals over $[BC]$ and $[DA]$:

$$ \begin{aligned} \int_B^C \boldsymbol{n}_{BC}(s_{\mathcal{I}}) ds_{\mathcal{I}} &= -L \boldsymbol{n} \\ \int_D^A \boldsymbol{n}_{DA}(s_{\mathcal{I}}) ds_{\mathcal{I}} &= +L\left(1 + \frac{\epsilon}{R}\right) \boldsymbol{n} \end{aligned} $$

Furthermore, because the contour $C$ is closed and the capillary forces diverge as $\xi_{max} \to 0$, we neglect inertia and viscous forces and we have the nullity of $\boldsymbol{F}_{\mathcal{I}}$. Projecting eq.2.11 along $\boldsymbol{n}$, dividing by $L$, and setting $\delta\gamma(0) = 0$ (even if it mean redefining the reference of surface tension with an offset to the equilibrium value), we get:
width $\xi_{max}$ over which $h$ varies is much smaller than the radius of curvature $R$ of contour $C$. We will thus develop our force balance equation in orders of $\xi_{max}/R$.

Let us write this force balance acting on the closed contour $\mathcal{I}$:

$$ \mathbf{F}_{\mathcal{I}} = \oint_{\mathcal{I}} \overline{\overline{\sigma}}_{\text{cap}} \cdot \mathbf{n}_{\mathcal{I}} \, d\mathcal{I} \qquad (2.10) $$

Where $\mathbf{n}_{\mathcal{I}}$ is the local normal vector oriented outward to the contour $\mathcal{I}$ of which we show an example for the segment $[BC]$ in Fig.2.3 (not to be confused with $\mathbf{n}$, the local normal to $C$). On the one hand, the integrals over $[AB]$ and $[CD]$ have constant local normal vectors $\mathbf{n}_{AB}$ and $\mathbf{n}_{CD}$ and a varying stress tensor $\overline{\overline{\sigma}}_{\text{cap}}$ as the thickness $h$ varies. On the other hand, $\overline{\overline{\sigma}}_{\text{cap}}$ is constant over $[BC]$ and $[DA]$, but the local normal vectors $\mathbf{n}_{BC/DA}(s_{\mathcal{I}})$ will vary. Note that because we suppose that the local curvature $\kappa = 1/R$ of $C$ varies slowly along the curvilinear abscissa, we approximate the Laplacian in the expression of $\overline{\overline{\sigma}}_{\text{cap}}$ to $\Delta h = \partial_{\xi\xi}h$, and we get, using eq.2.7:

$$ \begin{aligned} \frac{\mathbf{F}_{\mathcal{I}}}{\gamma_0} = & \left[ \mathbf{n}_{AB} + \mathbf{n}_{CD} \right] \int_0^{\epsilon} \left[ (\partial_{\xi}h)^2 + 2h \partial_{\xi\xi}h + 2\left(1 + \frac{\delta\gamma}{\gamma_0}\right) \right] d\xi \\ & + 2\left(1 + \frac{\delta\gamma(0)}{\gamma_0}\right) \int_B^C \mathbf{n}_{BC}(s_{\mathcal{I}}) ds_{\mathcal{I}} \\ & + \left[ -\left(\partial_{\xi}h\right)^2 \Big|_{\epsilon} + 2h \partial_{\xi\xi}h \Big|_{\epsilon} + 2\left(1 + \frac{\delta\gamma(\epsilon)}{\gamma_0}\right) \right] \int_D^A \mathbf{n}_{DA}(s_{\mathcal{I}}) ds_{\mathcal{I}} \end{aligned} \qquad (2.11) $$

The first line accounts for the integral over $[AB]$ and $[CD]$, the second for $[BC]$ and the third for $[DA]$. Note that we use $\partial_{\xi}h = 0$ and $\partial_{\xi\xi}h = 0$ at $\xi = 0$ as the inner and outer region of $C$ are defined so that $h = \text{cst}$.

We now express the local vectors of $\mathcal{I}$, $\mathbf{n}_{AB}$ and $\mathbf{n}_{CD}$ in the local basis of $C(t, \mathbf{n})$ using the local curvature of $C$ denoted $\kappa = 1/R$, and the length $[BC] = L$:

$$ \begin{aligned} \mathbf{n}_{AB} &= t - \frac{L}{2R} \mathbf{n} \\ \mathbf{n}_{CD} &= -t - \frac{L}{2R} \mathbf{n} \end{aligned} $$

And the vector integrals over $[BC]$ and $[DA]$:

$$ \begin{aligned} \int_B^C \mathbf{n}_{BC}(s_{\mathcal{I}}) ds_{\mathcal{I}} &= -L \mathbf{n} \\ \int_D^A \mathbf{n}_{DA}(s_{\mathcal{I}}) ds_{\mathcal{I}} &= +L\left(1 + \frac{\epsilon}{R}\right) \mathbf{n} \end{aligned} $$

Furthermore, because the contour $C$ is closed and the capillary forces diverge as $\xi_{max} \to 0$, we neglect inertia and viscous forces and we have the nullity of $\mathbf{F}_{\mathcal{I}}$. Projecting eq.2.11 along $\mathbf{n}$, dividing by $L$, and setting $\delta\gamma(0) = 0$ (even if it mean redefining the reference of surface tension with an offset to the equilibrium value), we get:
$$0 = -\frac{1}{R} \int_{0}^{\epsilon} \left[ (\partial_{\xi} h)^2 + 2h \partial_{\xi\xi} h + 2 \frac{\delta\gamma}{\gamma_0} \right] d\xi + \left( 1 + \frac{\epsilon}{R} \right) \left[ - (\partial_{\xi} h)^2 \bigg|_{\epsilon} + 2h \partial_{\xi\xi} h \bigg|_{\epsilon} + 2 \frac{\delta\gamma(\epsilon)}{\gamma_0} \right] \quad (2.12)$$

Out of the integral, two terms depending directly on h scale as $O(h^2/\epsilon^2)$ and are balanced by the field of surface tension $\delta\gamma(\epsilon)/\gamma_0$. We also notice the presence of terms of order $O(h^2/\epsilon R)$ which need to be balanced by $\delta\gamma(\epsilon)/\gamma_0$ as well. Writing with the $\nabla = \partial_\xi$ notation again, eq.2.12 leads to an expression of the field of surface tension:

$$\frac{\delta\gamma}{\gamma_0}(\epsilon) = \frac{1}{2} \|\nabla h(\epsilon)\|^2 - h\Delta h(\epsilon) + \Gamma_{sup}(\epsilon) \quad (2.13)$$

Where the first two terms are of lowest order $O(h^2/\epsilon^2)$ and where $\Gamma_{sup}$ is the correction for the higher order $O(h^2/\epsilon R)$. The first two terms of the surface tension field are sufficient to advance and to finally define the line tension $T$, and a discussion about the higher order term $\Gamma_{sup} = O(h^2/\epsilon R)$ awaits the interested reader in subsection 2.1.3.

### Final expression of T

We now have the expression $\delta\gamma(\epsilon)$ for the surface tension anywhere in the thickness jump $\epsilon \in [0, \xi_{max}]$. This is a key information as the general expression of $T$ writes using the expression of $\bar{\sigma}_{cap}$ of eq.2.7:

$$T = \int_{\xi} \mathbf{t} \cdot \left( \frac{1}{2} \bar{\bar{\sigma}}_{cap} - \gamma_0 \bar{\bar{I}}_2 \right) \cdot \mathbf{t} \, d\xi = \gamma_0 \int_{\xi} \left( \frac{1}{2} \|\nabla h\|^2 + \frac{\delta\gamma}{\gamma_0} + h\Delta h \right) d\xi \quad (2.14)$$

Thanks to the expression of $\delta\gamma$ given by eq.2.13, the lowest order is simplified: the gradient term is completed and the laplacian term is cancelled, yielding:

$$T = \gamma_0 \int_{\xi_{min}}^{\xi_{max}} \|\nabla h(\xi)\|^2 d\xi \quad (2.15)$$

This is the expression for the line tension acting along $C$. Interestingly, this quantity is actually twice the excess of energy stored in the excess of area in the thickness jump zone. Another computation based on thermodynamics considerations and a discussion awaits the reader in subsection 2.1.3.

### 2.1.3 Important remarks

Line tension expression through virtual work considerations and factor 2 with the excess of energy per unit length of line

In subsection 2.1.2, we have found an expression of the line tension $T$, which is actually an excess of stress stored into the thickness jump. We found it by considering the mechanical equilibrium of a tilted patch of film at rest in this region. What is implied through the existence of this
$$0 = -\frac{1}{R} \int_{0}^{\epsilon} \left[ (\partial_{\xi} h)^2 + 2h \partial_{\xi\xi} h + 2 \frac{\delta\gamma}{\gamma_0} \right] d\xi \\ + \left( 1 + \frac{\epsilon}{R} \right) \left[ - (\partial_{\xi} h)^2 \bigg|_{\epsilon} + 2h \partial_{\xi\xi} h \bigg|_{\epsilon} + 2 \frac{\delta\gamma(\epsilon)}{\gamma_0} \right] \qquad (2.12)$$

Out of the integral, two terms depending directly on h scale as $O(h^2/\epsilon^2)$ and are balanced by the field of surface tension $\delta\gamma(\epsilon)/\gamma_0$. We also notice the presence of terms of order $O(h^2/\epsilon R)$ which need to be balanced by $\delta\gamma(\epsilon)/\gamma_0$ as well. Writing with the $\nabla = \partial_\xi$ notation again, eq.2.12 leads to an expression of the field of surface tension:

$$\frac{\delta\gamma}{\gamma_0}(\epsilon) = \frac{1}{2} ||\nabla h(\epsilon)||^2 - h\Delta h(\epsilon) + \Gamma_{sup}(\epsilon) \qquad (2.13)$$

Where the first two terms are of lowest order $O(h^2/\epsilon^2)$ and where $\Gamma_{sup}$ is the correction for the higher order $O(h^2/\epsilon R)$. The first two terms of the surface tension field are sufficient to advance and to finally define the line tension $T$, and a discussion about the higher order term $\Gamma_{sup} = O(h^2/\epsilon R)$ awaits the interested reader in subsection 2.1.3.

### Final expression of $T$

We now have the expression $\delta\gamma(\epsilon)$ for the surface tension anywhere in the thickness jump $\epsilon \in [0, \xi_{max}]$. This is a key information as the general expression of $T$ writes using the expression of $\bar{\sigma}_{cap}$ of eq.2.7:

$$T = \int_{\xi} \mathbf{t} \cdot \left( \frac{1}{2} \bar{\bar{\sigma}}_{cap} - \gamma_0 \bar{\bar{I}}_2 \right) \cdot \mathbf{t} \, d\xi = \gamma_0 \int_{\xi} \left( \frac{1}{2} ||\nabla h||^2 + \frac{\delta\gamma}{\gamma_0} + h\Delta h \right) d\xi \qquad (2.14)$$

Thanks to the expression of $\delta\gamma$ given by eq.2.13, the lowest order is simplified: the gradient term is completed and the laplacian term is cancelled, yielding:

$$T = \gamma_0 \int_{\xi_{min}}^{\xi_{max}} ||\nabla h(\xi)||^2 d\xi \qquad (2.15)$$

This is the expression for the line tension acting along $C$. Interestingly, this quantity is actually twice the excess of energy stored in the excess of area in the thickness jump zone. Another computation based on thermodynamics considerations and a discussion awaits the reader in subsection 2.1.3.

### 2.1.3 Important remarks

Line tension expression through virtual work considerations and factor 2 with the excess of energy per unit length of line

In subsection 2.1.2, we have found an expression of the line tension $T$, which is actually an excess of stress stored into the thickness jump. We found it by considering the mechanical equilibrium of a tilted patch of film at rest in this region. What is implied through the existence of this
mechanical equilibrium is that an element of film flattens out through capillary flows at times much longer than the time of our experiment. In this mean time, surface tension enforces surface conservation by being larger in the thickness jump. This is all due to this intrinsic feature of a surfactant-laden interface that is being able to create Marangoni stresses to resist capillary flows.

However, the expression of the excess of stress/line tension $T$ can also be retrieved by considering the virtual work of the flat films neighbouring the thickness jump, which act as reservoirs of surface tension, both sides being set at reference tension $\gamma = \gamma_0$. These works must be computed under the constraints of conserved volume and area of interface.

Let us consider a portion of thickness jump, of initial width $\xi_{max}$ (along $y$), length $L$ (along $x$) (with respect to the notations of subsection 2.1.2), and initial thickness field $h_0$. $A^p = L\xi_{max}$ denotes the projected area of interface. We also consider that the thickness jump is in contact with a reservoir of surface tension set at $\gamma_0$. The portion of thickness jump is deformed, with its width becoming $\xi_{max} + d\xi_{max}$, its length $L + dL$ and its thickness field $h$. These variations verify $d\xi_{max}dL < 0$ as we impose the set of constraints: (i) $V = \int_{A^p} hdxdy = \text{cst}$ conserved volume, and (ii) $A = \int_{A^p}(1 + \frac{1}{2}||\nabla h||^2)dxdy = \text{cst}$ conserved area of interface. We will denote $\delta l = \frac{1}{2} \int_{0}^{\xi_{max}} ||\nabla h_0||^2 dy$ the excess of length stored in the thickness jump along the $y$ axis in the initial state.

<figure><img src="image_49.png" /><figcaption>Figure 2.4: Notations for a segment $L$ of thickness jump.</figcaption></figure>

The work of the reservoir in this transformation must be zero, as there is no dissipative phenomenon at stake here, nor any internal energy variations as area is conserved. This writes:

$$ \delta W^{res} = 0 = \gamma_0 dA^p + T dL \qquad (2.16) $$

Where $T$ is the line tension, defined for one interface, that is an excess of stress stored in the thickness jump. In order to determine $T$, we need to provide an expression for the variation of projected area $A^p$ during the transformation. This variation is the consequence of imposing both volume and interface area conservations, meaning we work at constant distribution of thicknesses (the histograms of $h_0$ and $h$ are the same). Thus, patches of film are redistributed in steeper regions during the transformation (for $dL > 0$, resp. flatter regions for $dL < 0$), and the projected area $A^p$ changes.
mechanical equilibrium is that an element of film flattens out through capillary flows at times much longer than the time of our experiment. In this mean time, surface tension enforces surface conservation by being larger in the thickness jump. This is all due to this intrinsic feature of a surfactant-laden interface that is being able to create Marangoni stresses to resist capillary flows.

However, the expression of the excess of stress/line tension $T$ can also be retrieved by considering the virtual work of the flat films neighbouring the thickness jump, which act as reservoirs of surface tension, both sides being set at reference tension $\gamma = \gamma_0$. These works must be computed under the constraints of conserved volume and area of interface.

Let us consider a portion of thickness jump, of initial width $\xi_{max}$ (along $y$), length $L$ (along $x$) (with respect to the notations of subsection 2.1.2), and initial thickness field $h_0$. $A^p = L\xi_{max}$ denotes the projected area of interface. We also consider that the thickness jump is in contact with a reservoir of surface tension set at $\gamma_0$. The portion of thickness jump is deformed, with its width becoming $\xi_{max} + d\xi_{max}$, its length $L + dL$ and its thickness field $h$. These variations verify $d\xi_{max}dL < 0$ as we impose the set of constraints: (i) $V = \int_{A^p} hdxdy = \text{cst}$ conserved volume, and (ii) $A = \int_{A^p}(1 + \frac{1}{2}||\nabla h||^2)dxdy = \text{cst}$ conserved area of interface. We will denote $\delta l = \frac{1}{2} \int_{0}^{\xi_{max}} ||\nabla h_0||^2 dy$ the excess of length stored in the thickness jump along the $y$ axis in the initial state.

<figure><img src="image_49.png" /><figcaption>Figure 2.4: Notations for a segment $L$ of thickness jump.</figcaption></figure>

The work of the reservoir in this transformation must be zero, as there is no dissipative phenomenon at stake here, nor any internal energy variations as area is conserved. This writes:

$$ \delta W^{res} = 0 = \gamma_0 dA^p + T dL \qquad (2.16) $$

Where $T$ is the line tension, defined for one interface, that is an excess of stress stored in the thickness jump. In order to determine $T$, we need to provide an expression for the variation of projected area $A^p$ during the transformation. This variation is the consequence of imposing both volume and interface area conservations, meaning we work at constant distribution of thicknesses (the histograms of $h_0$ and $h$ are the same). Thus, patches of film are redistributed in steeper regions during the transformation (for $dL > 0$, resp. flatter regions for $dL < 0$), and the projected area $A^p$ changes.
These constraints impose a dilation along the y axis, and the relationship between the gradients of thickness writes:

$$ \nabla h = \nabla h_0 \times \left( \frac{1}{1 + \frac{d\xi_{max}}{\xi_{max}}} \right) \qquad (2.17) $$

Area conservation can be written between both configurations:

$$ L \times (\xi_{max} + \delta\ell) = (L + dL) \times \int_{0}^{\xi_{max} + d\xi_{max}} \left(1 + \frac{1}{2}\|\nabla h\|^2\right) dy \qquad (2.18) $$

Where a change of variable $y' = y/(1 + \frac{d\xi_{max}}{\xi_{max}})$ can be applied in the integral of the left-hand side, as well as eq.2.17 for the gradient, and we can linearise and simplify our expression to get:

$$ 0 = \xi_{max}dL + Ld\xi_{max} - \frac{L}{\xi_{max}}d\xi_{max}\delta\ell + dL\delta\ell \qquad (2.19) $$

Here we recognise $dA^p = \xi_{max}dL + Ld\xi_{max}$ with the first two terms. This shows that $dA^p = O(dL\delta\ell)$, allowing us to write $d\xi_{max}/\xi_{max} = -dL/L$ at leading order. The third term thus becomes: $\frac{L}{\xi_{max}}d\xi_{max}\delta\ell = -dL\delta\ell + O(dL\delta\ell\frac{\delta\ell}{\xi_{max}})$. This is injected in eq.2.19 and yields:

$$ dA^p = -2dL\delta\ell + O(dL\delta\ell\frac{\delta\ell}{\xi_{max}}) \qquad (2.20) $$

Which can be used at leading order in eq.2.16 to finally get:

$$ T = 2\gamma_0\delta\ell = \gamma_0 \int_{0}^{\xi_{max}} \|\nabla h_0\|^2 dy \qquad (2.21) $$

This expression is in agreement with the demonstration of subsection 2.1.2, and corresponds to twice the excess of surface energy per unit length of the line of tension.

### Higher order correction to the surface tension in the thickness jump zone and 2D Laplace law

Let us consider eq.2.13 which gives the profile of surface tension $\delta\gamma$ across the thickness jump zone. We remind:

$$ \frac{\delta\gamma}{\gamma_0}(\epsilon) = \frac{1}{2}\|\nabla h(\epsilon)\|^2 - h\Delta h(\epsilon) + \Gamma_{sup}(\epsilon) $$

for any normal ordinate $\epsilon \in [0, \xi_{max}]$ in the thickness jump zone. This is a development at order $O(\frac{h^2}{\xi_{max}^2})$, but let us now take a closer look at the higher order term $\Gamma_{sup} = O(\frac{h^2}{\xi_{max}R})$ (where we recall that $R \gg \xi_{max}$ is the local radius of curvature of $C$).

By injecting the expression 2.13 of $\delta\gamma$ in eq.2.12, we get:

$$ \Gamma_{sup}(\epsilon) = \frac{1}{R} \int_{0}^{\epsilon} \|\nabla h\|^2 d\xi \qquad (2.22) $$

And if we now look at the difference of surface tension across the whole thickness jump zone, that is $\Delta\gamma_C = \delta\gamma(\xi_{max})$, the first two terms of eq.2.13 on the right hand-side vanish and we have:
These constraints impose a dilation along the y axis, and the relationship between the gradients of thickness writes:

$$ \nabla h = \nabla h_0 \times \left( \frac{1}{1 + \frac{d\xi_{max}}{\xi_{max}}} \right) \qquad (2.17) $$

Area conservation can be written between both configurations:

$$ L \times (\xi_{max} + \delta\ell) = (L + dL) \times \int_{0}^{\xi_{max} + d\xi_{max}} \left(1 + \frac{1}{2} \|\nabla h\|^2\right) dy \qquad (2.18) $$

Where a change of variable $y' = y/(1 + \frac{d\xi_{max}}{\xi_{max}})$ can be applied in the integral of the left-hand side, as well as eq.2.17 for the gradient, and we can linearise and simplify our expression to get:

$$ 0 = \xi_{max}dL + Ld\xi_{max} - \frac{L}{\xi_{max}}d\xi_{max}\delta\ell + dL\delta\ell \qquad (2.19) $$

Here we recognise $dA^p = \xi_{max}dL + Ld\xi_{max}$ with the first two terms. This shows that $dA^p = O(dL\delta\ell)$, allowing us to write $d\xi_{max}/\xi_{max} = -dL/L$ at leading order. The third term thus becomes: $\frac{L}{\xi_{max}}d\xi_{max}\delta\ell = -dL\delta\ell + O(dL\delta\ell\frac{\delta\ell}{\xi_{max}})$. This is injected in eq.2.19 and yields:

$$ dA^p = -2dL\delta\ell + O(dL\delta\ell\frac{\delta\ell}{\xi_{max}}) \qquad (2.20) $$

Which can be used at leading order in eq.2.16 to finally get:

$$ T = 2\gamma_0\delta\ell = \gamma_0 \int_{0}^{\xi_{max}} \|\nabla h_0\|^2 dy \qquad (2.21) $$

This expression is in agreement with the demonstration of subsection 2.1.2, and corresponds to twice the excess of surface energy per unit length of the line of tension.

### Higher order correction to the surface tension in the thickness jump zone and 2D Laplace law

Let us consider eq.2.13 which gives the profile of surface tension $\delta\gamma$ across the thickness jump zone. We remind:

$$ \frac{\delta\gamma}{\gamma_0}(\epsilon) = \frac{1}{2}\|\nabla h(\epsilon)\|^2 - h\Delta h(\epsilon) + \Gamma_{sup}(\epsilon) $$

for any normal ordinate $\epsilon \in [0, \xi_{max}]$ in the thickness jump zone. This is a development at order $O(\frac{h^2}{\xi_{max}^2})$, but let us now take a closer look at the higher order term $\Gamma_{sup} = O(\frac{h^2}{\xi_{max}R})$ (where we recall that $R \gg \xi_{max}$ is the local radius of curvature of $C$).

By injecting the expression 2.13 of $\delta\gamma$ in eq.2.12, we get:

$$ \Gamma_{sup}(\epsilon) = \frac{1}{R} \int_{0}^{\epsilon} \|\nabla h\|^2 d\xi \qquad (2.22) $$

And if we now look at the difference of surface tension across the whole thickness jump zone, that is $\Delta\gamma_C = \delta\gamma(\xi_{max})$, the first two terms of eq.2.13 on the right hand-side vanish and we have:
$$ \Delta\gamma_C = \gamma_0 \times \Gamma_{sup}(\xi_{max}) = \frac{\gamma_0}{R} \int_{\xi_{min}}^{\xi_{max}} ||\nabla h||^2 d\xi = \frac{T}{R} \quad (2.23) $$

Which is a 2D Laplace law linking the jump of surface tension $\gamma$ across a curved line to the line tension $T$.

## 2.2 Creating line tension in a foam film: experimental setup

In the previous section 2.1 we predict the presence of a line tension $T$ in the plane of a foam film with two regions of different thicknesses. We provide an expression for it with eq.2.15, and it appears that the knowledge of the thickness profile in the transition zone between the two regions is the key to compute $T$.

The goal of this section is to create experimentally this situation with two regions, and to measure the thickness profiles in the film to get access to an estimation of the line tension $T$. We achieved this using a simple setup composed of a single horizontal foam film deformed in a controlled way.

### 2.2.1 Principle of the experiment, technical limitations and notations

<figure><img src="image_50.png" /><figcaption>Figure 2.5: Top view of the film, timelapse of the deformation followed by the inner relaxation of the thin coloured patch. Times from A to F: $t = -1, -0.6, 0, 1, 3, 6s$.</figcaption></figure>

The principle of the experiment is detailed in Fig.2.5 of which we describe the different steps:

* **A:** A horizontal foam film is created very elongated in the rectangular aperture formed by a plastic piece with a squared “U” form and a mobile metallic bar. We create it by bathing
$$ \Delta\gamma_C = \gamma_0 \times \Gamma_{sup}(\xi_{max}) = \frac{\gamma_0}{R} \int_{\xi_{min}}^{\xi_{max}} ||\nabla h||^2 d\xi = \frac{T}{R} \qquad (2.23) $$

Which is a 2D Laplace law linking the jump of surface tension $\gamma$ across a curved line to the line tension $T$.

## 2.2 Creating line tension in a foam film: experimental setup

In the previous section 2.1 we predict the presence of a line tension $T$ in the plane of a foam film with two regions of different thicknesses. We provide an expression for it with eq.2.15, and it appears that the knowledge of the thickness profile in the transition zone between the two regions is the key to compute $T$.

The goal of this section is to create experimentally this situation with two regions, and to measure the thickness profiles in the film to get access to an estimation of the line tension $T$. We achieved this using a simple setup composed of a single horizontal foam film deformed in a controlled way.

### 2.2.1 Principle of the experiment, technical limitations and notations

<figure><img src="image_50.png" /><figcaption>Figure 2.5: Top view of the film, timelapse of the deformation followed by the inner relaxation of the thin coloured patch. Times from A to F: $t = -1, -0.6, 0, 1, 3, 6s$.</figcaption></figure>

The principle of the experiment is detailed in Fig.2.5 of which we describe the different steps:

* **A:** A horizontal foam film is created very elongated in the rectangular aperture formed by a plastic piece with a squared “U” form and a mobile metallic bar. We create it by bathing
the frame by hand in a soapy solution. We wait for it to drain until it is thin enough to display colours in reflected light.

* **B:** The preexisting film is stretched very rapidly, extracting some new film from the menisci of the frame (appearing in grey in Fig.2.5). This new grey film is a Frankel film, whose dynamics of extraction is detailed in subsection 0.3.2.

* **C:** At the end of the motor motion, which defines the origin of time $t = 0$, the very elongated arena-shaped thin piece of film is surrounded by the thicker film: a line tension exists at the border which tends to minimise its perimeter. We measure its value over the time of the experiment by looking at the thickness profile along $\mathcal{L}$ with the hyperspectral camera.

* **D-F:** The thin film retracts toward a circular shape to achieve its minimum of perimeter. The timescale of the motion is set by a trade-off between the value of $T$ and the damping forces imposed by the viscous air friction.

All notations used to describe our system are shown in Fig.2.6.

<figure><img src="image_51.png" /><figcaption>Figure 2.6: Experimental setup and notations used in the chapter.</figcaption></figure>

The foam film is formed with an initial width $a = 2.1$ mm. This very narrow configuration only lets a very elongated flat film to exist between the menisci present at each edge of the frame. The film is let to drain for 3 min to the point that it displays coloured patterns (Fig.2.5A), meaning its thickness has decreased down to a few hundreds of nanometres. The stretching imposed by the motor is performed at $V^{mot} = 10$ mm/s for 1 s up till the film has a width $a = 12.1$ mm. The origin of times $t = 0$ is set at the end of the motor motion, and we show a typical film at this time in Fig.2.6A.

The contrast between the coloured thin film and the thick grey film allow us to estimate the position of the contour $C$ along which the line tension is applied. The thickness across $C$ is monitored using a spectral camera along $\mathcal{L}$. The precise localisation of this line from the spatio-spectral image required some work. We found its precise position using a mirror put in the plane of the rectangular frame. With a triangular cache put on the mirror, we were also able to
the frame by hand in a soapy solution. We wait for it to drain until it is thin enough to display colours in reflected light.

* **B:** The preexisting film is stretched very rapidly, extracting some new film from the menisci of the frame (appearing in grey in Fig.2.5). This new grey film is a Frankel film, whose dynamics of extraction is detailed in subsection 0.3.2.

* **C:** At the end of the motor motion, which defines the origin of time $t = 0$, the very elongated arena-shaped thin piece of film is surrounded by the thicker film: a line tension exists at the border which tends to minimise its perimeter. We measure its value over the time of the experiment by looking at the thickness profile along $\mathcal{L}$ with the hyperspectral camera.

* **D-F:** The thin film retracts toward a circular shape to achieve its minimum of perimeter. The timescale of the motion is set by a trade-off between the value of $T$ and the damping forces imposed by the viscous air friction.

All notations used to describe our system are shown in Fig.2.6.

<figure><img src="image_51.png" /><figcaption>Figure 2.6: Experimental setup and notations used in the chapter.</figcaption></figure>

The foam film is formed with an initial width $a = 2.1$ mm. This very narrow configuration only lets a very elongated flat film to exist between the menisci present at each edge of the frame. The film is let to drain for 3 min to the point that it displays coloured patterns (Fig.2.5A), meaning its thickness has decreased down to a few hundreds of nanometres. The stretching imposed by the motor is performed at $V^{mot} = 10$ mm/s for 1 s up till the film has a width $a = 12.1$ mm. The origin of times $t = 0$ is set at the end of the motor motion, and we show a typical film at this time in Fig.2.6A.

The contrast between the coloured thin film and the thick grey film allow us to estimate the position of the contour $C$ along which the line tension is applied. The thickness across $C$ is monitored using a spectral camera along $\mathcal{L}$. The precise localisation of this line from the spatio-spectral image required some work. We found its precise position using a mirror put in the plane of the rectangular frame. With a triangular cache put on the mirror, we were also able to
find the angle $\theta_y = 7^\circ$ between $\mathcal{L}$ and the $y$ axis (which we tried to lower as much as possible). This angle is neglected hereafter.

In the following, we will only consider the measurement range $t \in [0, 1.5]$s where we measure the relaxations of $L$ and $R$ and the line tension $T$. This restriction is due to two main factors: (i) the lamp for the spectral camera radiates quite a lot of heat, we must turn it off before it generates significant thermal Marangoni flows, and (ii) we will later need the hypothesis $L \gg R$, which is less and less true as the relaxation occurs.

Gravity is neglected throughout this whole study, as the film lies horizontally with a very good precision. This is due to the fact that the thin patch acts as a very sensitive level for the foam film, and the slightest tilt angle leads to a movement toward the menisci at the edges during the relaxation, hindering the measurement. We thus put the deformable frame on three adjustable vertical screws in order to get the right corrections to the horizontality of the setup.

Last problem but not least are the parasitic air movements which we limit by covering the setup with an enclosure as soon as we have created the foam film, and before the deformation.

### 2.2.2 Setup

<figure><img src="image_52.png" /><figcaption>Figure 2.7: Experimental setup used to create the foam film with two regions of different thicknesses. **1-A:** black plastic piece with a squared “U” form (corresponding to the black edges in Fig2.6B). **1-B:** metallic mobile edge (corresponding to the green edge in Fig2.6B) attached to a motor with a wedge-shaped plastic piece. **1-C:** plastic bath containing the soapy solution that we raise by hand to form the foam film on the rectangular deformable frame. **2:** Hyperspectral camera mounted on a navitar objective monitoring the thickness along the line $\mathcal{L}$ of Fig.2.5. **3:** Colour camera monitoring the foam film with reflected light. **4:** White powerful light aligned in reflection with the hyperspectral camera to create interferences against the foam film. **5:** White LED panel aligned in reflection with the colour camera in order to image the coloured reflection of the thin parts of the film. **6:** Cardboard enclosure positioned around the setup during the experiments to mitigate parasitic air movements.</figcaption></figure>
find the angle $\theta_y = 7^\circ$ between $\mathcal{L}$ and the $y$ axis (which we tried to lower as much as possible). This angle is neglected hereafter.

In the following, we will only consider the measurement range $t \in [0, 1.5]$s where we measure the relaxations of $L$ and $R$ and the line tension $T$. This restriction is due to two main factors: (i) the lamp for the spectral camera radiates quite a lot of heat, we must turn it off before it generates significant thermal Marangoni flows, and (ii) we will later need the hypothesis $L \gg R$, which is less and less true as the relaxation occurs.

Gravity is neglected throughout this whole study, as the film lies horizontally with a very good precision. This is due to the fact that the thin patch acts as a very sensitive level for the foam film, and the slightest tilt angle leads to a movement toward the menisci at the edges during the relaxation, hindering the measurement. We thus put the deformable frame on three adjustable vertical screws in order to get the right corrections to the horizontality of the setup.

Last problem but not least are the parasitic air movements which we limit by covering the setup with an enclosure as soon as we have created the foam film, and before the deformation.

### 2.2.2 Setup

<figure><img src="image_52.png" /><figcaption>Figure 2.7: Experimental setup used to create the foam film with two regions of different thicknesses. **1-A:** black plastic piece with a squared “U” form (corresponding to the black edges in Fig2.6B). **1-B:** metallic mobile edge (corresponding to the green edge in Fig2.6B) attached to a motor with a wedge-shaped plastic piece. **1-C:** plastic bath containing the soapy solution that we raise by hand to form the foam film on the rectangular deformable frame. **2:** Hyperspectral camera mounted on a navitar objective monitoring the thickness along the line $\mathcal{L}$ of Fig.2.5. **3:** Colour camera monitoring the foam film with reflected light. **4:** White powerful light aligned in reflection with the hyperspectral camera to create interferences against the foam film. **5:** White LED panel aligned in reflection with the colour camera in order to image the coloured reflection of the thin parts of the film. **6:** Cardboard enclosure positioned around the setup during the experiments to mitigate parasitic air movements.</figcaption></figure>
### Chemistry

The soapy solutions used to create our flat foam films are SDS/dodecanol mixtures in a solvent made of 75% water 15% glycerol (NB: contrary to any other solution used in the frame of this Ph.D work, no fluorescein is added here as we do not need any fluorescent light to image the system). The concentration of SDS is always set at $c_{SDS} = 2.4$ CMC = 5.6 g/L while the concentration of dodecanol varies between $c_{DOH} = 0, 15, 35, 50$ mg/L. The protocol to create the soapy solutions is the same as the others.

### Deformable frame and motor

The deformable frame used in this set of experiments is made of one single 3D-printed plastic piece with a squared "U" form (Fig2.7A) to which we add a straight metallic part sliding in its arms. This latter is also attached to a PI micromove C-867 motor (Fig2.7B). The result is a rectangular deformable frame with one side of fixed length $w = 62$ mm and the other side with a length varying between $a = 1$ and $25$ mm.

### Cameras

We use a Resonon Pika L hyperspectral camera mounted on a 12X Navitar objective (Fig2.7(2)) to monitor the thickness profile of the foam film along the line $L$ shown in Fig.2.5C. It is aligned with a white light (Fig2.7(4)) with respect to the angle of reflection at the surface of the foam film, which acts as a Fabry-Pérot interferometre for the sake of this measurement. The angle of incidence $\theta$ with the horizontal plane of the foam film was carefully measured $\theta = 38^{\circ}$.

The other (regular) camera we use is an Imaging Source colour DFK 23UM021 camera mounted on a 24-mm Nikon objective (Fig2.7(3)) monitoring the foam film from above and the dynamics of retraction of the thin patch toward a circular shape. Fig.2.5 is a timelapse of a typical experiment produced using this camera. To get the contrast of colours at the surface of the foam film, an LED panel is attached in reflection of this camera (Fig2.7(5)). In our subsequent usage of this camera, the angle of incidence with the foam film ($\sim 10^{\circ}$) will always be neglected with respect to parallax problems.

## 2.3 Image processing

### 2.3.1 Detection of the thin region contour and kinematics of the relaxation

In order to study the dynamics of relaxation of the thin film, we start by tracking the area of the thin coloured film. We want to binarise colour images to make contrast between two kinds of pixels: coloured ones (thin film) and grey ones (thick film). The former have very high values in their respective colours (either in R, G or B) but very low values in the others, and the latter have intermediate values in all channels.

We create two sets of three masks:
### Chemistry

The soapy solutions used to create our flat foam films are SDS/dodecanol mixtures in a solvent made of 75% water 15% glycerol (NB: contrary to any other solution used in the frame of this Ph.D work, no fluorescein is added here as we do not need any fluorescent light to image the system). The concentration of SDS is always set at $c_{SDS} = 2.4$ CMC = 5.6 g/L while the concentration of dodecanol varies between $c_{DOH} = 0, 15, 35, 50$ mg/L. The protocol to create the soapy solutions is the same as the others.

### Deformable frame and motor

The deformable frame used in this set of experiments is made of one single 3D-printed plastic piece with a squared "U" form (Fig2.7A) to which we add a straight metallic part sliding in its arms. This latter is also attached to a PI micromove C-867 motor (Fig2.7B). The result is a rectangular deformable frame with one side of fixed length *w* = 62 mm and the other side with a length varying between *a* = 1 and 25 mm.

### Cameras

We use a Resonon Pika L hyperspectral camera mounted on a 12X Navitar objective (Fig2.7(2)) to monitor the thickness profile of the foam film along the line *L* shown in Fig.2.5C. It is aligned with a white light (Fig2.7(4)) with respect to the angle of reflection at the surface of the foam film, which acts as a Fabry-Pérot interferometre for the sake of this measurement. The angle of incidence $\theta$ with the horizontal plane of the foam film was carefully measured $\theta = 38^{\circ}$.

The other (regular) camera we use is an Imaging Source colour DFK 23UM021 camera mounted on a 24-mm Nikon objective (Fig2.7(3)) monitoring the foam film from above and the dynamics of retraction of the thin patch toward a circular shape. Fig.2.5 is a timelapse of a typical experiment produced using this camera. To get the contrast of colours at the surface of the foam film, an LED panel is attached in reflection of this camera (Fig2.7(5)). In our subsequent usage of this camera, the angle of incidence with the foam film ($\sim 10^{\circ}$) will always be neglected with respect to parallax problems.

## 2.3 Image processing

### 2.3.1 Detection of the thin region contour and kinematics of the relaxation

In order to study the dynamics of relaxation of the thin film, we start by tracking the area of the thin coloured film. We want to binarise colour images to make contrast between two kinds of pixels: coloured ones (thin film) and grey ones (thick film). The former have very high values in their respective colours (either in R, G or B) but very low values in the others, and the latter have intermediate values in all channels.

We create two sets of three masks:
<figure><img src="image_53.png" /><figcaption>Figure 2.8: Example of contour detection performed on a film with no dodecanol at time $t = 0$. **A:** Image of the film at $t = 0$ recalling the notations used in the text, the detected contour $C$ is highlighted in red, the line $\mathcal{L}$ along which the thickness profiles are measured is in light blue. **B:** Image of the film at $t = 1.5$ s the end of our measurement range. **C:** Perimeter $P$ of $C$ as a function of time. **D:** Area $A$ of the interior of $C$ as a function of time. Mind the ordinate, which is centred around the slightly varying values of $A$. **E:** Semi-length $L$ of the thin film as a function of time, defined as half the distance between the highest point and the lowest point of $C$. **F:** Semi-width $R$ of the arena of thin film as a function of time, estimated with the area and the semi-length $R = A/(4L)$. All the uncertainties are represented with shaded areas and evaluated by taking the standard deviation over 13 experiments.</figcaption></figure>

* Set 1 is three masks for the highest values for each RGB channel, each colour with its own threshold. The pixels below the threshold are put to zero, the pixels above subtracted by the threshold. Each mask is thus white where the thin film is of the given colour, black everywhere else.

* Set 2 is three masks for the lowest values for each RGB channel, each colour with its own threshold. The pixels above the threshold are put to zero, the pixels below are kept as they are. Each mask is thus white in the coloured region except for the thresholded colour which is in black. The threshold is chosen so that the grey regions are above it and appear in black.

By summing the masks of Set 1 and 2 by colour, we have three images where the coloured region appears in white, and the grey region appears in black. We average the three masks to get an image which is binarised using a dedicated MATLAB function `IM2BW`. We fill the gaps in the detected coloured region by dilation-erosion with a small disk of a few pixels in radius.

We then use the contour detection function `CONTOURC` of MATLAB on the binary image we obtained to get $C$. A typical frame of the set of experiments without dodecanol taken at time $t = 0$ is shown with its detected contour in Fig. 2.8A.
<figure><img src="image_53.png" /><figcaption>Figure 2.8: Example of contour detection performed on a film with no dodecanol at time $t = 0$. **A:** Image of the film at $t = 0$ recalling the notations used in the text, the detected contour $C$ is highlighted in red, the line $\mathcal{L}$ along which the thickness profiles are measured is in light blue. **B:** Image of the film at $t = 1.5$ s the end of our measurement range. **C:** Perimeter $P$ of $C$ as a function of time. **D:** Area $A$ of the interior of $C$ as a function of time. Mind the ordinate, which is centred around the slightly varying values of $A$. **E:** Semi-length $L$ of the thin film as a function of time, defined as half the distance between the highest point and the lowest point of $C$. **F:** Semi-width $R$ of the arena of thin film as a function of time, estimated with the area and the semi-length $R = A/(4L)$. All the uncertainties are represented with shaded areas and evaluated by taking the standard deviation over 13 experiments.</figcaption></figure>

* Set 1 is three masks for the highest values for each RGB channel, each colour with its own threshold. The pixels below the threshold are put to zero, the pixels above subtracted by the threshold. Each mask is thus white where the thin film is of the given colour, black everywhere else.

* Set 2 is three masks for the lowest values for each RGB channel, each colour with its own threshold. The pixels above the threshold are put to zero, the pixels below are kept as they are. Each mask is thus white in the coloured region except for the thresholded colour which is in black. The threshold is chosen so that the grey regions are above it and appear in black.

By summing the masks of Set 1 and 2 by colour, we have three images where the coloured region appears in white, and the grey region appears in black. We average the three masks to get an image which is binarised using a dedicated MATLAB function `IM2BW`. We fill the gaps in the detected coloured region by dilation-erosion with a small disk of a few pixels in radius.

We then use the contour detection function `CONTOURC` of MATLAB on the binary image we obtained to get $C$. A typical frame of the set of experiments without dodecanol taken at time $t = 0$ is shown with its detected contour in Fig. 2.8A.
From there we can have access to the geometry of the arena of thin film and, by repeating the process over the whole movie, characterise the whole kinematics of the system. All the quantities we extract are shown in Fig.2.8C-F with their uncertainties being the standard deviation over 13 experiments. The perimeter *P* and the area *A* are directly extracted from the features of the contour *C* and are plotted in Fig.2.8C and D. We see as expected that the perimeter tends to decrease as the thin film relaxes toward a circular shape, while the area remains almost constant (mind the ordinate in graph D) which is a key information allowing us later on to assume the incompressibility of the interfaces.

From *C*, we also deduce the geometrical parameters *L* and *R*. They define an approximate arena-shaped geometry for *C*, with parallel straight lines of lengths 2*L* connected by semi-circles of radii *R*. *L* is estimated by measuring the distance between the highest and lowest point of *C* while *R* is deduced from the measurements of *L* and *A* as the area of the arena shape we suppose has the expression for *L* ≫ *R*: *A* ≈ 4*LR*. We thus see that toward the end of the measurement range we have, we end up with *L*(*t* = 1.5 s) ≈ 12 mm which remains much greater than *R*(*t* = 1.5 s) ≈ 1 mm, and our assumption holds for the whole dynamics we study.

### 2.3.2 Comment on the chemistry

In the previous subsection 2.3.1, we displayed an example of relaxation with a peculiar chemistry that is without dodecanol. We indeed tried to vary this parameter, but ended up with less reliable data and a lack of repeatability arising as soon as we put any concentration of dodecanol, starting from 35 mg/L up to 200 mg/L.

With dodecanol, the film either undergoes the exact same dynamics as the 13 previous experiments shown in Fig.2.8, or at the opposite shows no relaxation at all. An example is shown in Fig.2.9. In the case with no relaxation, which becomes more prevalent as more dodecanol is added, the thin film remains elongated and eventually destabilises, bending along its length. We hypothesise several origins for this wriggling behaviour during the time range of our measurement:

* Solid-like in-plane motions resisting shear due to heterogeneities of dodecanol concentration at the interfaces. Dodecanol is known to undergo a phase transition at a water-air interface and produce rafts of crystalline 2D phase [102] if concentrated enough. The concentration threshold for this transition is not clear however, even more so in our case where the bulk has SDS micelles, and the dodecanol population is distributed between the interface, the bulk and the SDS micellar phase.

* Air movements at longer times, although in this case our repeatable experiments without dodecanol should also be affected as we witness bending during the same time range as our measurements.

This qualitative lack of repeatability makes us exclude dodecanol as a working parameter, and all the results we will show and discuss in the following will come from the same set of 13 experiments without dodecanol shown in Fig.2.8.
From there we can have access to the geometry of the arena of thin film and, by repeating the process over the whole movie, characterise the whole kinematics of the system. All the quantities we extract are shown in Fig.2.8C-F with their uncertainties being the standard deviation over 13 experiments. The perimeter *P* and the area *A* are directly extracted from the features of the contour *C* and are plotted in Fig.2.8C and D. We see as expected that the perimeter tends to decrease as the thin film relaxes toward a circular shape, while the area remains almost constant (mind the ordinate in graph D) which is a key information allowing us later on to assume the incompressibility of the interfaces.

From *C*, we also deduce the geometrical parameters *L* and *R*. They define an approximate arena-shaped geometry for *C*, with parallel straight lines of lengths 2*L* connected by semi-circles of radii *R*. *L* is estimated by measuring the distance between the highest and lowest point of *C* while *R* is deduced from the measurements of *L* and *A* as the area of the arena shape we suppose has the expression for *L* ≫ *R*: *A* ≈ 4*LR*. We thus see that toward the end of the measurement range we have, we end up with *L*(*t* = 1.5 s) ≈ 12 mm which remains much greater than *R*(*t* = 1.5 s) ≈ 1 mm, and our assumption holds for the whole dynamics we study.

### 2.3.2 Comment on the chemistry

In the previous subsection 2.3.1, we displayed an example of relaxation with a peculiar chemistry that is without dodecanol. We indeed tried to vary this parameter, but ended up with less reliable data and a lack of repeatability arising as soon as we put any concentration of dodecanol, starting from 35 mg/L up to 200 mg/L.

With dodecanol, the film either undergoes the exact same dynamics as the 13 previous experiments shown in Fig.2.8, or at the opposite shows no relaxation at all. An example is shown in Fig.2.9. In the case with no relaxation, which becomes more prevalent as more dodecanol is added, the thin film remains elongated and eventually destabilises, bending along its length. We hypothesise several origins for this wriggling behaviour during the time range of our measurement:

* Solid-like in-plane motions resisting shear due to heterogeneities of dodecanol concentration at the interfaces. Dodecanol is known to undergo a phase transition at a water-air interface and produce rafts of crystalline 2D phase [102] if concentrated enough. The concentration threshold for this transition is not clear however, even more so in our case where the bulk has SDS micelles, and the dodecanol population is distributed between the interface, the bulk and the SDS micellar phase.

* Air movements at longer times, although in this case our repeatable experiments without dodecanol should also be affected as we witness bending during the same time range as our measurements.

This qualitative lack of repeatability makes us exclude dodecanol as a working parameter, and all the results we will show and discuss in the following will come from the same set of 13 experiments without dodecanol shown in Fig.2.8.
<figure><img src="image_54.png" /><figcaption>Figure 2.9: Two examples for [DOH] = 200 mg/L. The repeatability is poor: on the left the thin film relaxes with a kinematics quantitatively very similar to what is shown in Fig.2.8, on the right the thin patch does not relax and eventually bends along its length.</figcaption></figure>

### 2.3.3 Extracting the thickness profiles

In order to measure the thickness profiles, we collect the movie of the spectra along the line $L$ of Fig.2.5C, of which an example of frame at $t = 1.5$ s is shown in Fig.2.10A. We start by finding the average spectrum $I_{water}$ of the lamp and exposition we use by averaging our frames over time and position, yielding the green curve shown in Fig.2.10B. Now at a given time and position, we can collect a spectrum $I(\lambda)$ (blue curve in Fig.2.10B) and filter it by defining $\tilde{I}(\lambda) = \frac{I}{I_{water}} - \left(\frac{I}{I_{water}}\right)$ where the top bar is an average over $\lambda$. The result is shown in Fig.2.10C. Given that the foam film behaves for the light as a Fabry-Pérot interferometre of varying thickness $2h(y)$ observed at an angle $\theta = 38^{\circ}$, we can directly have a prediction for the intensity:

$$ \tilde{I}(k, 2h, \theta) = \tilde{I}_0 \cos \left( 4nkh \sqrt{1 - \sin^2 \theta / n^2} \right) \quad (2.24) $$

Where $n$ is the refractive index of water and $k = \frac{2\pi}{\lambda}$ is the wavenumber. Thus, from an intensity profile $\tilde{I}(k, 2h) \propto \cos(\alpha k \times 2h)$ with $\alpha = 2n\sqrt{1 - \sin^2 \theta / n^2} = 2.36$, we can extract the thickness $2h$. The extraction itself is performed by taking the numerical Fourier transform of $\tilde{I}$ from the wavenumber space $k$ into the thickness space $H$:

$$ \mathcal{F}[\tilde{I}](H) = \int_{k_{min}}^{k_{max}} \tilde{I}(k, 2h)e^{ikH}dk \quad (2.25) $$

Where $k_{min}$ and $k_{max}$ are respectively the minimum and maximum wavenumbers given by the maximum and minimum wavelengths detected by the hyperspectral camera. As eq.2.24 suggests, in the thickness space we expect a dirac function centered around $H = \alpha \times 2h$ convoluted by a cardinal sine of width determined by $k_{max} - k_{min}$, which defines our resolution. Taking into account the geometrical factor $\alpha$ of dilation of our thickness space ($\alpha = 2.36$ in our case with $\theta = 38^{\circ}$), the theoretical resolution is $H_{min,th} = \frac{\pi}{\alpha(k_{max}-k_{min})} \approx 150$ nm. However, given the
<figure><img src="image_54.png" /><figcaption>Figure 2.9: Two examples for [DOH] = 200 mg/L. The repeatability is poor: on the left the thin film relaxes with a kinematics quantitatively very similar to what is shown in Fig.2.8, on the right the thin patch does not relax and eventually bends along its length.</figcaption></figure>

### 2.3.3 Extracting the thickness profiles

In order to measure the thickness profiles, we collect the movie of the spectra along the line $\mathcal{L}$ of Fig.2.5C, of which an example of frame at $t = 1.5$ s is shown in Fig.2.10A. We start by finding the average spectrum $I_{water}$ of the lamp and exposition we use by averaging our frames over time and position, yielding the green curve shown in Fig.2.10B. Now at a given time and position, we can collect a spectrum $I(\lambda)$ (blue curve in Fig.2.10B) and filter it by defining $\tilde{I}(\lambda) = \frac{I}{I_{water}} - \left(\frac{I}{I_{water}}\right)$ where the top bar is an average over $\lambda$. The result is shown in Fig.2.10C. Given that the foam film behaves for the light as a Fabry-Pérot interferometre of varying thickness $2h(y)$ observed at an angle $\theta = 38^{\circ}$, we can directly have a prediction for the intensity:

$$ \tilde{I}(k, 2h, \theta) = \tilde{I}_0 \cos \left( 4nkh \sqrt{1 - \sin^2 \theta / n^2} \right) \quad (2.24) $$

Where $n$ is the refractive index of water and $k = \frac{2\pi}{\lambda}$ is the wavenumber. Thus, from an intensity profile $\tilde{I}(k, 2h) \propto \cos(\alpha k \times 2h)$ with $\alpha = 2n\sqrt{1 - \sin^2 \theta / n^2} = 2.36$, we can extract the thickness $2h$. The extraction itself is performed by taking the numerical Fourier transform of $\tilde{I}$ from the wavenumber space $k$ into the thickness space $H$:

$$ \mathcal{F}[\tilde{I}](H) = \int_{k_{min}}^{k_{max}} \tilde{I}(k, 2h)e^{ikH}dk \quad (2.25) $$

Where $k_{min}$ and $k_{max}$ are respectively the minimum and maximum wavenumbers given by the maximum and minimum wavelengths detected by the hyperspectral camera. As eq.2.24 suggests, in the thickness space we expect a dirac function centered around $H = \alpha \times 2h$ convoluted by a cardinal sine of width determined by $k_{max} - k_{min}$, which defines our resolution. Taking into account the geometrical factor $\alpha$ of dilation of our thickness space ($\alpha = 2.36$ in our case with $\theta = 38^{\circ}$), the theoretical resolution is $H_{min,th} = \frac{\pi}{\alpha(k_{max}-k_{min})} \approx 150$ nm. However, given the
<figure><img src="image_55.png" /></figure>

Figure 2.10: Processing of the spectral images obtained for a typical experiment without dodecanol at $t = 1.5$ s toward the end of the measurement range. **A:** Typical image obtained with the spectral camera. The abscissa is the position along the monitored line with its origin set in the middle of the thin film, the ordinate is the wavelength. The central zone displaying a single large interference oscillation and delimited by the vertical red dotted line is the thin film. The red shaded areas are blind zones where it not possible perform the thickness measurement. The blue line is the abscissa $y$ chosen for the example in **B**. **B:** The blue curve is a typical intensity profile taken at a given $y$ shown in **A** by the vertical blue line. The green line is the spectrum of the lamp obtained by averaging the frames in space and time. **C:** Normalised and centred intensity profile $\tilde{I}$ out of which we extract the thickness of the foam film at the given time and position $y$.

exposure and vanishing spectrum of the lamp at the edges of the spectral range of our camera, our experimental resolution is estimated around $H_{min,exp} = 400$ nm.

In the same way, our numerical integration is performed over a wavenumber space with finite steps $\Delta k$. Knowing that the difference of $\lambda$'s from one pixel to the next is always $\Delta\lambda = 1$ nm with a range $\lambda \in [400, 1000]$ nm, the largest step in wavenumber as we integrate is $\Delta k = 2\pi(\frac{1}{\lambda_{min}} - \frac{1}{\lambda_{min}+\Delta\lambda}) = 4.5 \times 10^4$ rad m$^{-1}$. This yields a theoretical maximum detectable value $H_{max,th} = \frac{\pi}{\alpha\Delta k} = 30$ µm which is never attained in practice.

In our measurable range $H_{min,exp} = 400$ nm and $H_{max,th} = 30$ µm, we now work to deduce the profile $2h(y)$ out of the spectra $F[\tilde{I}](H)$, looking at its highest peak. We also define a second criterion which is a threshold to exceed for a peak to prevent noise. We thus consider both the position and height of the peaks of $F[\tilde{I}](H)$ in the thickness space $H$ and have the following cases which can be seen in Fig. 2.10A and Fig. 2.11:

* If the peak is above the threshold and positioned at $H \in [H_{min,exp}, H_{max,th}]$ in the measurable range, we attribute its position $H$ to the thickness $2h = H$. This is the case in
<figure><img src="image_55.png" /></figure>

Figure 2.10: Processing of the spectral images obtained for a typical experiment without dodecanol at $t = 1.5$ s toward the end of the measurement range. **A:** Typical image obtained with the spectral camera. The abscissa is the position along the monitored line with its origin set in the middle of the thin film, the ordinate is the wavelength. The central zone displaying a single large interference oscillation and delimited by the vertical red dotted line is the thin film. The red shaded areas are blind zones where it not possible perform the thickness measurement. The blue line is the abscissa $y$ chosen for the example in **B**. **B:** The blue curve is a typical intensity profile taken at a given $y$ shown in **A** by the vertical blue line. The green line is the spectrum of the lamp obtained by averaging the frames in space and time. **C:** Normalised and centred intensity profile $\tilde{I}$ out of which we extract the thickness of the foam film at the given time and position $y$.

exposure and vanishing spectrum of the lamp at the edges of the spectral range of our camera, our experimental resolution is estimated around $H_{min,exp} = 400$ nm.

In the same way, our numerical integration is performed over a wavenumber space with finite steps $\Delta k$. Knowing that the difference of $\lambda$'s from one pixel to the next is always $\Delta\lambda = 1$ nm with a range $\lambda \in [400, 1000]$ nm, the largest step in wavenumber as we integrate is $\Delta k = 2\pi(\frac{1}{\lambda_{min}} - \frac{1}{\lambda_{min}+\Delta\lambda}) = 4.5 \times 10^4$ rad m$^{-1}$. This yields a theoretical maximum detectable value $H_{max,th} = \frac{\pi}{\alpha\Delta k} = 30$ µm which is never attained in practice.

In our measurable range $H_{min,exp} = 400$ nm and $H_{max,th} = 30$ µm, we now work to deduce the profile $2h(y)$ out of the spectra $F[\tilde{I}](H)$, looking at its highest peak. We also define a second criterion which is a threshold to exceed for a peak to prevent noise. We thus consider both the position and height of the peaks of $F[\tilde{I}](H)$ in the thickness space $H$ and have the following cases which can be seen in Fig. 2.10A and Fig. 2.11:

* If the peak is above the threshold and positioned at $H \in [H_{min,exp}, H_{max,th}]$ in the measurable range, we attribute its position $H$ to the thickness $2h = H$. This is the case in
the outer regions where the intensity profile Fig2.10B is taken, it corresponds to the grey thick film and most of the thickness jump zone.

* If the peak is above the threshold and positioned at $H < H_{min,exp}$, it is below our resolution, the peak is too broad to be precise but intense enough thanks to our thresholding to rule out noise to be its cause. A value of zero is attributed to the thickness $2h = 0$. This is the case in the inner region, between the two vertical red dotted lines, where a single well-defined oscillation is measured along the $\lambda$ axis. It corresponds to the coloured thin flat film, that is located between the red dotted lines in Fig.2.10A, or equivalently in the flat segments of Fig.2.11 set to zero.

* If the peak is below the threshold, or above it but located in $H > H_{max,th}$, it is treated as noise and we attribute the value $2h = NaN$. This is what happens in the blind zones, where the thickness profiles become too steep and hinder the measurement (red shaded areas of Fig.2.10A or dotted lines of Fig.2.11).

This methodology is applied for each spectrum at each $y$ for the given frame, and repeated for each frame of the movie, building a thickness profile at each time of the experiment. Typical thickness profiles obtained throughout a whole experiment are shown in Fig.2.11. The grey shaded area corresponds to the lower resolution $H_{min,exp}$. The dotted parts of the profiles correspond to the blind zones where the thickness profile is too steep, neighbouring the edges of the thin films. They are parabolic interpolations found by imposing the continuity of thickness $2h$ at both the edge of the thin film ($2h = 0$) and the edge of thick film ($2h > 0$), and the continuity of thickness gradient at the edge of the thick film.

<figure><img src="image_56.png" /><figcaption>Figure 2.11: Thickness profiles obtained along $\mathcal{L}$ at different times. The origin of time $t = 0$ is set when the motor stops its motion. The red line corresponds to the spectral image taken as an example in Fig.2.10, and is the last thickness profile of our measurement range $t \in [0, 1.5]$s. At this time $t = 1.5$ s, the foam film is in a state close to what is shown in Fig.2.5D.</figcaption></figure>
the outer regions where the intensity profile Fig2.10B is taken, it corresponds to the grey thick film and most of the thickness jump zone.

* If the peak is above the threshold and positioned at $H < H_{min,exp}$, it is below our resolution, the peak is too broad to be precise but intense enough thanks to our thresholding to rule out noise to be its cause. A value of zero is attributed to the thickness $2h = 0$. This is the case in the inner region, between the two vertical red dotted lines, where a single well-defined oscillation is measured along the $\lambda$ axis. It corresponds to the coloured thin flat film, that is located between the red dotted lines in Fig.2.10A, or equivalently in the flat segments of Fig.2.11 set to zero.

* If the peak is below the threshold, or above it but located in $H > H_{max,th}$, it is treated as noise and we attribute the value $2h = NaN$. This is what happens in the blind zones, where the thickness profiles become too steep and hinder the measurement (red shaded areas of Fig.2.10A or dotted lines of Fig.2.11).

This methodology is applied for each spectrum at each $y$ for the given frame, and repeated for each frame of the movie, building a thickness profile at each time of the experiment. Typical thickness profiles obtained throughout a whole experiment are shown in Fig.2.11. The grey shaded area corresponds to the lower resolution $H_{min,exp}$. The dotted parts of the profiles correspond to the blind zones where the thickness profile is too steep, neighbouring the edges of the thin films. They are parabolic interpolations found by imposing the continuity of thickness $2h$ at both the edge of the thin film ($2h = 0$) and the edge of thick film ($2h > 0$), and the continuity of thickness gradient at the edge of the thick film.

<figure><img src="image_56.png" /><figcaption>Figure 2.11: Thickness profiles obtained along $\mathcal{L}$ at different times. The origin of time $t = 0$ is set when the motor stops its motion. The red line corresponds to the spectral image taken as an example in Fig.2.10, and is the last thickness profile of our measurement range $t \in [0, 1.5]$s. At this time $t = 1.5$ s, the foam film is in a state close to what is shown in Fig.2.5D.</figcaption></figure>
### 2.3.4 Computing the resulting line tension

As we have access to the thickness profiles $2h$ during the relaxation (Fig.2.11), we now aim at measuring the line tension $T$.

Note that we will be able to compute two values of $T$ for each profile, as two thickness jump zones are monitored at the same time by the spectral camera on both sides of the arena. To do so, we provide again the theoretical formula of $T$, that is eq.2.15:

$$T = \gamma_0 \int_{\xi_{min}}^{\xi_{max}} ||\nabla h(\xi)||^2 d\xi$$

The experimental values of the different terms involved in this equation are discussed below.

#### Surface tension $\gamma_0$

The equilibrium surface tension $\gamma_0$ is well known for a given chemistry thanks to a pendant drop campaign already described in the Introduction 0 subsection 0.1.3. For our solutions with SDS alone, it is $\gamma_0 = 35 \text{ mN/m}$.

#### Thickness gradient $||\nabla h||$

First, we identify the $y$ axis of our problem (along which we measure the thickness profiles in Fig.2.11) to the direction of the thickness gradients on both sides of the thin film. We thus define an axis $\xi$ for each semi-profile, one going in the $+y$ direction, the other oriented in the $-y$ direction. Along these axis $\xi$, we compute the gradients of the semi-profiles $||\nabla h||$. They need to be carefully computed, as any numerical derivative, but especially here that our thickness profiles $h$ have been interpolated in the blind zones where the contributions of the gradient is important.

In the blind zones, we interpolate our thickness profiles by a well-defined parabola imposing continuity of $h$ at both ends of the blind zone and the continuity of $||\nabla h||$ at the thick film edge. These parts of the profiles are analytically well defined, and the computation of the gradient is not a problem. In the thicker zones where we are able to measure $h$, computing a gradient requires to smooth the profile. To do so, we chose to perform polynomial fits of order 3 starting at the edges of the blind zones and going up til the end of a maximal length of integration $l_\infty$ discussed right after. As we do not constrain the 3rd order polynomial fit to respect any continuity of thickness or thickness gradient, we will be very careful not to integrate the computed gradient at the junction between the blind zone and the measurement zone.

#### Lower integration boundary

The lower boundary $\xi_{min} = 0$ is always defined as the edges of the thin film, that is the first non-zero values starting from the origin $y = 0$ toward the right or left.

#### Upper integration boundary

The upper boundary $\xi_{max}$ is a parameter to consider with caution in practice as $||\nabla h||$ vanishes smoothly. To build this discussion, we introduce an auxiliary function $T(l_{int})$ defined as:
### 2.3.4 Computing the resulting line tension

As we have access to the thickness profiles $2h$ during the relaxation (Fig.2.11), we now aim at measuring the line tension $T$.

Note that we will be able to compute two values of $T$ for each profile, as two thickness jump zones are monitored at the same time by the spectral camera on both sides of the arena. To do so, we provide again the theoretical formula of $T$, that is eq.2.15:

$$T = \gamma_0 \int_{\xi_{min}}^{\xi_{max}} ||\nabla h(\xi)||^2 d\xi$$

The experimental values of the different terms involved in this equation are discussed below.

#### Surface tension $\gamma_0$

The equilibrium surface tension $\gamma_0$ is well known for a given chemistry thanks to a pendant drop campaign already described in the Introduction 0 subsection 0.1.3. For our solutions with SDS alone, it is $\gamma_0 = 35 \text{ mN/m}$.

#### Thickness gradient $||\nabla h||$

First, we identify the $y$ axis of our problem (along which we measure the thickness profiles in Fig.2.11) to the direction of the thickness gradients on both sides of the thin film. We thus define an axis $\xi$ for each semi-profile, one going in the $+y$ direction, the other oriented in the $-y$ direction. Along these axis $\xi$, we compute the gradients of the semi-profiles $||\nabla h||$. They need to be carefully computed, as any numerical derivative, but especially here that our thickness profiles $h$ have been interpolated in the blind zones where the contributions of the gradient is important.

In the blind zones, we interpolate our thickness profiles by a well-defined parabola imposing continuity of $h$ at both ends of the blind zone and the continuity of $||\nabla h||$ at the thick film edge. These parts of the profiles are analytically well defined, and the computation of the gradient is not a problem. In the thicker zones where we are able to measure $h$, computing a gradient requires to smooth the profile. To do so, we chose to perform polynomial fits of order 3 starting at the edges of the blind zones and going up til the end of a maximal length of integration $\ell_\infty$ discussed right after. As we do not constrain the 3rd order polynomial fit to respect any continuity of thickness or thickness gradient, we will be very careful not to integrate the computed gradient at the junction between the blind zone and the measurement zone.

#### Lower integration boundary

The lower boundary $\xi_{min} = 0$ is always defined as the edges of the thin film, that is the first non-zero values starting from the origin $y = 0$ toward the right or left.

#### Upper integration boundary

The upper boundary $\xi_{max}$ is a parameter to consider with caution in practice as $||\nabla h||$ vanishes smoothly. To build this discussion, we introduce an auxiliary function $T(\ell_{int})$ defined as:
<figure><img src="image_57.png" /><figcaption>Figure 2.12: Example of the dependence $T(\ell_{int})$ with the length of integration $\ell_{int}$ (whose retained value for the sake of computing $T$ in the main graph is $\ell_{\infty} = 3$ mm). $\ell_{80}$ corresponds to the length of integration where $T$ reaches 80% of its maximal value obtained with $\ell_{\infty}$. This profile is obtained with the orange thickness profile of Fig.2.11, at $t = 1$ s.</figcaption></figure>

$$T(\ell_{int}) = \gamma_0 \int_{0}^{\ell_{int}} ||\nabla h(\xi)||^2 d\xi \qquad (2.26)$$

In which the upper bound is a free variable. In Fig.2.12 we show typical measurement $T(\ell_{int})$ made out of the orange curve of Fig.2.11 (averaged on both semi-profiles). We arbitrarily define a length $\ell_{\infty} = 3$ mm beyond which we risk to start integrating the thickness gradient due to the meniscus on the supporting frame. As expected from the smoothly-vanishing thickness gradients, $T(\ell_{int})$ plateaus smoothly as well.

### 2.3.5 Results

#### Measurement of $T$

The retained value for the line tension $T$ at a given time step is $T(\ell_{\infty})$. This is done for each experiment of our set of 13, and the average over them is shown in Fig.2.13 (again the orange dotted line is matched with the previous profiles of $h$ and $T(\ell_{int})$). We thus have an estimate of our line tension $T(t)$ at each time, and we see that we are dealing with a tiny force, of the order of $T \sim 10^{-10}$N.

Also, we see that it tends to increase with time, which is consistent with the fact that the thickness profiles in Fig.2.11 tend to be steeper over time, leading to more important thickness gradients.

The experimental error is the standard deviation over both semi-profiles of the 13 experiments. In Fig.2.12, the error bars account for the difference in measured line tension between the left and right thickness jump zones monitored by the camera, and a difference up to 20% is observed. This remains under our global error due to statistical repetition which is of the order of 50%. The fact that the measurement is performed in the flat regions of the contour, however, raises a good question: do the curved edges undergo a local compression during the relaxation which would change the local value of $T$ where it effectively applies a force? This conundrum is addressed later on in subsection 2.4.6, as well predictions for the evolution of $T$ with time. In the following,
<figure><img src="image_57.png" /><figcaption>Figure 2.12: Example of the dependence $T(\ell_{int})$ with the length of integration $\ell_{int}$ (whose retained value for the sake of computing $T$ in the main graph is $\ell_{\infty} = 3$ mm). $\ell_{80}$ corresponds to the length of integration where $T$ reaches 80% of its maximal value obtained with $\ell_{\infty}$. This profile is obtained with the orange thickness profile of Fig.2.11, at $t = 1$ s.</figcaption></figure>

$$T(\ell_{int}) = \gamma_0 \int_{0}^{\ell_{int}} ||\nabla h(\xi)||^2 d\xi \qquad (2.26)$$

In which the upper bound is a free variable. In Fig.2.12 we show typical measurement $T(\ell_{int})$ made out of the orange curve of Fig.2.11 (averaged on both semi-profiles). We arbitrarily define a length $\ell_{\infty} = 3$ mm beyond which we risk to start integrating the thickness gradient due to the meniscus on the supporting frame. As expected from the smoothly-vanishing thickness gradients, $T(\ell_{int})$ plateaus smoothly as well.

### 2.3.5 Results

#### Measurement of $T$

The retained value for the line tension $T$ at a given time step is $T(\ell_{\infty})$. This is done for each experiment of our set of 13, and the average over them is shown in Fig.2.13 (again the orange dotted line is matched with the previous profiles of $h$ and $T(\ell_{int})$). We thus have an estimate of our line tension $T(t)$ at each time, and we see that we are dealing with a tiny force, of the order of $T \sim 10^{-10}$N.

Also, we see that it tends to increase with time, which is consistent with the fact that the thickness profiles in Fig.2.11 tend to be steeper over time, leading to more important thickness gradients.

The experimental error is the standard deviation over both semi-profiles of the 13 experiments. In Fig.2.12, the error bars account for the difference in measured line tension between the left and right thickness jump zones monitored by the camera, and a difference up to 20% is observed. This remains under our global error due to statistical repetition which is of the order of 50%. The fact that the measurement is performed in the flat regions of the contour, however, raises a good question: do the curved edges undergo a local compression during the relaxation which would change the local value of $T$ where it effectively applies a force? This conundrum is addressed later on in subsection 2.4.6, as well predictions for the evolution of $T$ with time. In the following,
<figure><img src="image_58.png" /></figure>

Figure 2.13: Line tension $T$ computed over time on the set of experiments without decanol. The error is the standard deviation over the computations for 26 semi-profiles of thickness $2h(\xi)$ which consist of 13 experiments, each with 2 monitored sides of the thin film treated independently. The vertical dashed lines indicating particular times are colour-matched with the thickness profiles given as examples in Fig. 2.11.

we will assume that $T(t)$ is a uniform scalar value along the line of tension whose sole dependency is time, given by the experimental data in Fig. 2.13.

### Location of the force $T$

<figure><img src="image_59.png" /></figure>

Figure 2.14: First schematic view of the contour $C^*$ as an arena of semi-width $R^*$, semi-length $L$. The line tension $T$ is applied at any point of $C^*$ (and is supposed uniform along it, the factors 2 accounting for the two interfaces).

As stated in subsection 2.3.1, the thin film contour $C$ is approximated to an arena with parallel straight lines of lengths $2L$ connected by semi-circles of radii $R$. However, the values of $R$ being actually comparable to the typical values $l_{int}$ over which $T$ is integrated, the position where the line tension is applied becomes ill-defined. Thus, we need to distinguish the contour $C$ from the effective contour $C^*$ along which we consider the line tension to be applied.

In Fig. 2.14, we define $C^*$, a similar arena along which we consider that $T$ is effectively applied, with a semi-width $R^*$ defined as:
<figure><img src="image_58.png" /></figure>

Figure 2.13: Line tension $T$ computed over time on the set of experiments without decanol. The error is the standard deviation over the computations for 26 semi-profiles of thickness $2h(\xi)$ which consist of 13 experiments, each with 2 monitored sides of the thin film treated independently. The vertical dashed lines indicating particular times are colour-matched with the thickness profiles given as examples in Fig. 2.11.

we will assume that $T(t)$ is a uniform scalar value along the line of tension whose sole dependency is time, given by the experimental data in Fig. 2.13.

### Location of the force $T$

<figure><img src="image_59.png" /></figure>

Figure 2.14: First schematic view of the contour $C^*$ as an arena of semi-width $R^*$, semi-length $L$. The line tension $T$ is applied at any point of $C^*$ (and is supposed uniform along it, the factors 2 accounting for the two interfaces).

As stated in subsection 2.3.1, the thin film contour $C$ is approximated to an arena with parallel straight lines of lengths $2L$ connected by semi-circles of radii $R$. However, the values of $R$ being actually comparable to the typical values $l_{int}$ over which $T$ is integrated, the position where the line tension is applied becomes ill-defined. Thus, we need to distinguish the contour $C$ from the effective contour $C^*$ along which we consider the line tension to be applied.

In Fig. 2.14, we define $C^*$, a similar arena along which we consider that $T$ is effectively applied, with a semi-width $R^*$ defined as:
$$R^* = R + \ell_{80}/2 \qquad (2.27)$$

Where $\ell_{80}$ is the length for a given profile $T(\ell_{int})$ so that $T(\ell_{80}) = 0.8T(\ell_{\infty})$, which is graphically represented in Fig.2.12. Both $R$ and $\ell_{80}/2$ are of the order of $\sim 0.5 - 1$ mm as shown respectively in Fig.2.8F and Fig.2.12.

## 2.4 Modelling the relaxation of the thin film: we validate our line tension measurement

Now that we have come up with a measurement for the line tension $T$ (Fig.2.13), which has never been measured before to the best of our knowledge, we want to validate the values we found. To do so, a valuable piece of experimental knowledge is the time evolution of the geometry of the thin patch as it relaxes towards a circular shape. All this information is shown in Fig.2.8.

In this section, we detail the equations leading to a prediction of the dynamics of $L(t)$, the semi-length of the thin patch, knowing the forces in the plane of the foam film. The force here is the line tension, and the dynamics is found to be damped by air friction. We have a quantitative agreement which validates our measurement of $T$. We first start by writing a general equation of motion for a foam film with a line tension in its plane.

### 2.4.1 Equation of motion for a foam film with line tension

Let us consider a foam film with the same configuration as in Fig.2.2 Left, except we now keep $C^*$ as a 1D object subject to a line tension $T$ (mind the * we have now due to the correction of the contour detailed above 2.3.5). Besides the capillary forces we described in section 2.1, the foam film is considered to be subject to friction both in its own plane with an intrinsic surface shear viscosity and with air viscosity. An element of film of area $dA$ taken in the inner or the outer region of $C^*$ without crossing it have the force balance:

$$(2\eta_s \Delta v + 2\nabla\gamma + 2f_g + f_\eta) \, dA = 0 \qquad (2.28)$$

Where $v$ is the 2D velocity field in the plane of the foam film, $\eta_s$ is the surface shear viscosity, $f_g$ is the friction force on the air for one interface and $f_\eta$ the friction force due to bulk liquid viscosity integrated over the thickness of the film. The factors 2 account for the two water-air interfaces of the film. This equation is a 2D Stokes equation with exterior forces $2f_g$ and $f_\eta$.

If we now choose our element of film $dA$ so that it crosses $C^*$ at the curvilinear abscissas $s$ and $s + ds$, we must add two terms for the line tension $2T$ (2 interfaces again) acting along $+t(s + ds)$ and $-t(s)$ the tangential directions of $C^*$. This gives the new balance:

$$(2\eta_s \Delta v + 2\nabla\gamma + 2f_g + f_\eta) \, dA = -\frac{\partial(2Tt)}{\partial s} \, ds \qquad (2.29)$$

Which we can rewrite with the normal vector $n$ and the local curvature of $C^*$ denoted $\kappa(s)$. We also consider $T$ to be constant along $C^*$, giving:
$$R^* = R + \ell_{80}/2 \qquad (2.27)$$

Where $\ell_{80}$ is the length for a given profile $T(\ell_{int})$ so that $T(\ell_{80}) = 0.8T(\ell_{\infty})$, which is graphically represented in Fig.2.12. Both $R$ and $\ell_{80}/2$ are of the order of $\sim 0.5 - 1$ mm as shown respectively in Fig.2.8F and Fig.2.12.

## 2.4 Modelling the relaxation of the thin film: we validate our line tension measurement

Now that we have come up with a measurement for the line tension $T$ (Fig.2.13), which has never been measured before to the best of our knowledge, we want to validate the values we found. To do so, a valuable piece of experimental knowledge is the time evolution of the geometry of the thin patch as it relaxes towards a circular shape. All this information is shown in Fig.2.8.

In this section, we detail the equations leading to a prediction of the dynamics of $L(t)$, the semi-length of the thin patch, knowing the forces in the plane of the foam film. The force here is the line tension, and the dynamics is found to be damped by air friction. We have a quantitative agreement which validates our measurement of $T$. We first start by writing a general equation of motion for a foam film with a line tension in its plane.

### 2.4.1 Equation of motion for a foam film with line tension

Let us consider a foam film with the same configuration as in Fig.2.2 Left, except we now keep $C^*$ as a 1D object subject to a line tension $T$ (mind the * we have now due to the correction of the contour detailed above 2.3.5). Besides the capillary forces we described in section 2.1, the foam film is considered to be subject to friction both in its own plane with an intrinsic surface shear viscosity and with air viscosity. An element of film of area $dA$ taken in the inner or the outer region of $C^*$ without crossing it have the force balance:

$$(2\eta_s \Delta v + 2\nabla\gamma + 2f_g + f_{\eta}) \, dA = 0 \qquad (2.28)$$

Where $v$ is the 2D velocity field in the plane of the foam film, $\eta_s$ is the surface shear viscosity, $f_g$ is the friction force on the air for one interface and $f_{\eta}$ the friction force due to bulk liquid viscosity integrated over the thickness of the film. The factors 2 account for the two water-air interfaces of the film. This equation is a 2D Stokes equation with exterior forces $2f_g$ and $f_{\eta}$.

If we now choose our element of film $dA$ so that it crosses $C^*$ at the curvilinear abscissas $s$ and $s + ds$, we must add two terms for the line tension $2T$ (2 interfaces again) acting along $+t(s + ds)$ and $-t(s)$ the tangential directions of $C^*$. This gives the new balance:

$$(2\eta_s \Delta v + 2\nabla\gamma + 2f_g + f_{\eta}) \, dA = -\frac{\partial(2Tt)}{\partial s} \, ds \qquad (2.29)$$

Which we can rewrite with the normal vector $n$ and the local curvature of $C^*$ denoted $\kappa(s)$. We also consider $T$ to be constant along $C^*$, giving:
$$ (2\eta_s \Delta \mathbf{v} + 2\nabla\gamma + 2\mathbf{f}_g + \mathbf{f}_\eta) \, dA = 2T\kappa(s)\mathbf{n}(s) \, ds \qquad (2.30) $$

This equation can be rewritten globally using a well-defined Dirac distribution. Let us define a set of such functions $\delta_{C^*}$ defined over $S$ the whole foam film so that:

$$ \forall \mathbf{r}' \in C^*, \quad \iint_S \delta_{C^*}(\mathbf{r} - \mathbf{r}') d^2\mathbf{r} = 1 \qquad (2.31) $$

And eq.2.28 and 2.30 can be defined as a single equation at any point $\mathbf{r} \in S$:

$$ 2\eta_s \Delta \mathbf{v}(\mathbf{r}) + 2\nabla\gamma(\mathbf{r}) + 2\mathbf{f}_g(\mathbf{r}) + \mathbf{f}_\eta(\mathbf{r}) = \oint_{C^*} 2T\kappa(\mathbf{r}')\delta_{C^*}(\mathbf{r} - \mathbf{r}')\mathbf{n}(\mathbf{r}')d\mathbf{r}' \qquad (2.32) $$

This is the general equation for the in-plane motion of a foam film with a line tension acting along a contour $C^*$ of any shape. This is still a specific case of a more general one, which is a foam film with any thickness field $h$. In our case, the thickness gradients are very localised in a specific zone, but a more general framework awaits the reader in Appendix 5.

Also, it has to be noted that equation 2.32 must be combined with a relationship between the surface tension and the surface deformation in order to be a closed problem. Let us provide one in the form of the incompressibility of the interfaces.

## 2.4.2 Closing the problem: incompressibility of the interfaces

The question of whether the surfactant-laden interfaces we are dealing with are compressible, and to what extent, is one of the main issues we attempt to address in this manuscript for multiple situations. In the previous chapter 1, we considered extensible interfaces and measured their surface elasticity $E_f$.

In the present case of our thin film relaxing toward a circular shape under the effect of line tension, we will consider incompressible interfaces. To support this assumption, a first physical quantity we can look at is the evolution of the area $A$ of the thin region, shown in Fig.2.8D. It varies slightly throughout the relaxation and we neglect these variations. This incompressibility hypothesis is a key assumption for the model we will use later on. It is questionable however, as a global condition over the conservation of the total area of thin film $\frac{dA}{dt} = 0$ do not exclude local compression and extension to occur. A discussion and additional arguments for this hypothesis awaits the interested reader in subsection 2.4.6.

Thus, we write this condition $\nabla \cdot \mathbf{v} = 0$, where we remind that $\nabla$ is a 2D operator in the plane $(Oxy)$ and we have the full set of equations for the relaxation of the arena:

$$ \left\{ \begin{array}{l} 2\eta_s \Delta \mathbf{v}(\mathbf{r}) + 2\nabla\gamma(\mathbf{r}) + 2\mathbf{f}_g(\mathbf{r}) + \mathbf{f}_\eta(\mathbf{r}) = \oint_{C^*} 2T\kappa(\mathbf{r}')\delta_{C^*}(\mathbf{r} - \mathbf{r}')\mathbf{n}(\mathbf{r}')d\mathbf{r}' \\ \nabla \cdot \mathbf{v} = 0 \end{array} \right. \qquad (2.33) $$

Now that the problem is closed, we need to find the solution for our velocity field $\mathbf{v}$. Before looking at the source term on the right-hand side of the first equation, let us have a closer look at the damping forces on the left-hand side. Some simplifications can be made before finding a suitable model for our geometry.
$$ (2\eta_s \Delta \mathbf{v} + 2\nabla\gamma + 2\mathbf{f}_g + \mathbf{f}_\eta) \, dA = 2T\kappa(s)\mathbf{n}(s) \, ds \qquad (2.30) $$

This equation can be rewritten globally using a well-defined Dirac distribution. Let us define a set of such functions $\delta_{C^*}$ defined over $S$ the whole foam film so that:

$$ \forall \mathbf{r}' \in C^*, \quad \iint_S \delta_{C^*}(\mathbf{r} - \mathbf{r}') d^2\mathbf{r} = 1 \qquad (2.31) $$

And eq. 2.28 and 2.30 can be defined as a single equation at any point $\mathbf{r} \in S$:

$$ 2\eta_s \Delta \mathbf{v}(\mathbf{r}) + 2\nabla\gamma(\mathbf{r}) + 2\mathbf{f}_g(\mathbf{r}) + \mathbf{f}_\eta(\mathbf{r}) = \oint_{C^*} 2T\kappa(\mathbf{r}')\delta_{C^*}(\mathbf{r} - \mathbf{r}')\mathbf{n}(\mathbf{r}')d\mathbf{r}' \qquad (2.32) $$

This is the general equation for the in-plane motion of a foam film with a line tension acting along a contour $C^*$ of any shape. This is still a specific case of a more general one, which is a foam film with any thickness field $h$. In our case, the thickness gradients are very localised in a specific zone, but a more general framework awaits the reader in Appendix 5.

Also, it has to be noted that equation 2.32 must be combined with a relationship between the surface tension and the surface deformation in order to be a closed problem. Let us provide one in the form of the incompressibility of the interfaces.

### 2.4.2 Closing the problem: incompressibility of the interfaces

The question of whether the surfactant-laden interfaces we are dealing with are compressible, and to what extent, is one of the main issues we attempt to address in this manuscript for multiple situations. In the previous chapter 1, we considered extensible interfaces and measured their surface elasticity $E_f$.

In the present case of our thin film relaxing toward a circular shape under the effect of line tension, we will consider incompressible interfaces. To support this assumption, a first physical quantity we can look at is the evolution of the area $A$ of the thin region, shown in Fig. 2.8D. It varies slightly throughout the relaxation and we neglect these variations. This incompressibility hypothesis is a key assumption for the model we will use later on. It is questionable however, as a global condition over the conservation of the total area of thin film $\frac{dA}{dt} = 0$ do not exclude local compression and extension to occur. A discussion and additional arguments for this hypothesis awaits the interested reader in subsection 2.4.6.

Thus, we write this condition $\nabla \cdot \mathbf{v} = 0$, where we remind that $\nabla$ is a 2D operator in the plane $(Oxy)$ and we have the full set of equations for the relaxation of the arena:

$$ \left\{ \begin{array}{l} 2\eta_s \Delta \mathbf{v}(\mathbf{r}) + 2\nabla\gamma(\mathbf{r}) + 2\mathbf{f}_g(\mathbf{r}) + \mathbf{f}_\eta(\mathbf{r}) = \oint_{C^*} 2T\kappa(\mathbf{r}')\delta_{C^*}(\mathbf{r} - \mathbf{r}')\mathbf{n}(\mathbf{r}')d\mathbf{r}' \\ \nabla \cdot \mathbf{v} = 0 \end{array} \right. \qquad (2.33) $$

Now that the problem is closed, we need to find the solution for our velocity field $\mathbf{v}$. Before looking at the source term on the right-hand side of the first equation, let us have a closer look at the damping forces on the left-hand side. Some simplifications can be made before finding a suitable model for our geometry.
**Important remark:** In subsection 0.2.5 of the Introduction, I already introduced this limit of incompressible interfaces. There, I also refer to Appendix 2 where I developed a set of equations for compressible interfaces in the limit of high elasticities. The resulting compressible correction to the flow field is computed in Appendix 3, in the specific geometry of the line tension experiment. We found that it is indeed a higher order correction for the sake of studying the relaxation dynamics, and we discard it in the present chapter.

### 2.4.3 Damping forces: the Boussinesq number $Bq$

Before applying the above equation to our arena-shaped geometry, let us focus on the left hand side of eq.2.32. The first, third and fourth terms $2\eta_s\Delta v$, $2f_g$ and $f_\eta$ are dissipative forces respectively accounting for the surface shear viscosity, the friction on the air and the liquid bulk dissipation.

If we want to use it in practice, a natural question about the general framework eq.2.33 is: What is the dominating damping force? This is not an easy question to answer, as surface shear viscosity is highly chemistry-dependent. **In fact, $\eta_s$ is not well quantified in the case of soluble surfactants, which is our case with SDS alone [36].** On the other hand, the bulk viscosities are well known, but the resulting forces (especially in the gas) need to be treated carefully. Let us define them more accurately and start by comparing these two.

#### Bulk liquid friction $f_\eta$

As discussed in subsection 0.2.1 of the Introduction, only plug flows in the $z$ direction are allowed in our foam films, far from the menisci. This means that the only liquid bulk viscous contributions to our equations come from the gradients of velocity in the plane of the foam film. They can be integrated over $z$ to give:

$$f_{\eta} = \int_{-h}^{h} \eta \Delta v dz \qquad (2.34)$$

Where $\Delta$ and $v$ are here and again only defined in the $(Oxy)$ plane of the foam film. The scaling for this force is thus:

$$f_{\eta} \sim \eta h \frac{U}{\ell_{xy}^2} \qquad (2.35)$$

Where $\eta = 10^{-3} \text{Pa} \cdot \text{s}$ is the bulk viscosity of water, $h \sim 10^{-6} \text{m}$ the thickness of the film, $U \sim 10^{-2} \text{mm/s}$ the typical in-plane velocity, and we need here to provide $\ell_{xy}$ the typical length scale over which the flow field varies in the plane of the foam film. **We take the experimental order of magnitude $\ell_{xy} \sim R^* \sim 10^{-3} \text{m}$ (cf. the experimental section 2.2, subsection 2.3.5).**

#### Gas friction $f_g$

The tangential force applied by gas friction on an element of interface can be expressed:

$$f_g = \eta_g \frac{\partial v_g}{\partial z} \Big|_{z=\pm h} \qquad (2.36)$$
**Important remark:** In subsection 0.2.5 of the Introduction, I already introduced this limit of incompressible interfaces. There, I also refer to Appendix 2 where I developed a set of equations for compressible interfaces in the limit of high elasticities. The resulting compressible correction to the flow field is computed in Appendix 3, in the specific geometry of the line tension experiment. We found that it is indeed a higher order correction for the sake of studying the relaxation dynamics, and we discard it in the present chapter.

### 2.4.3 Damping forces: the Boussinesq number $Bq$

Before applying the above equation to our arena-shaped geometry, let us focus on the left hand side of eq.2.32. The first, third and fourth terms $2\eta_s\Delta v$, $2f_g$ and $f_\eta$ are dissipative forces respectively accounting for the surface shear viscosity, the friction on the air and the liquid bulk dissipation.

If we want to use it in practice, a natural question about the general framework eq.2.33 is: What is the dominating damping force? This is not an easy question to answer, as surface shear viscosity is highly chemistry-dependent. **In fact, $\eta_s$ is not well quantified in the case of soluble surfactants, which is our case with SDS alone [36].** On the other hand, the bulk viscosities are well known, but the resulting forces (especially in the gas) need to be treated carefully. Let us define them more accurately and start by comparing these two.

#### Bulk liquid friction $f_\eta$

As discussed in subsection 0.2.1 of the Introduction, only plug flows in the $z$ direction are allowed in our foam films, far from the menisci. This means that the only liquid bulk viscous contributions to our equations come from the gradients of velocity in the plane of the foam film. They can be integrated over $z$ to give:

$$f_{\eta} = \int_{-h}^{h} \eta \Delta v dz \qquad (2.34)$$

Where $\Delta$ and $v$ are here and again only defined in the $(Oxy)$ plane of the foam film. The scaling for this force is thus:

$$f_{\eta} \sim \eta h \frac{U}{\ell_{xy}^2} \qquad (2.35)$$

Where $\eta = 10^{-3} \text{Pa} \cdot \text{s}$ is the bulk viscosity of water, $h \sim 10^{-6} \text{m}$ the thickness of the film, $U \sim 10^{-2} \text{mm/s}$ the typical in-plane velocity, and we need here to provide $\ell_{xy}$ the typical length scale over which the flow field varies in the plane of the foam film. **We take the experimental order of magnitude $\ell_{xy} \sim R^* \sim 10^{-3} \text{m}$ (cf. the experimental section 2.2, subsection 2.3.5).**

#### Gas friction $f_g$

The tangential force applied by gas friction on an element of interface can be expressed:

$$f_g = \eta_g \frac{\partial v_g}{\partial z} \Big|_{z=\pm h} \qquad (2.36)$$
Where $z$ is the normal direction to the median plane (Oxy) of the foam film, $\eta_g$ is the dynamic viscosity of the gas phase, and $v_g$ is the velocity in the exterior gas phase projected in (Oxy). We suppose in our geometry that $v_g(x, y, z) = v_g(x, y, -z)$ for any $(x, y)$. This force scales as:

$$f_g \sim \eta_g \frac{U}{\ell_z} \qquad (2.37)$$

Where $\eta_g = 1.8 \cdot 10^{-5} \text{Pa} \cdot \text{s}$, $U \sim 10^{-2} \text{mm/s}$, and $\ell_z \sim \sqrt{\eta_g \tau}$ is a typical length scale for the vertical gradient of velocity in the gas phase.

Giving an order of magnitude for this latter length raises the question of taking into account gas inertia or not. An estimation of the Reynolds number for the gas in our problem gives:

$$Re = \frac{\rho_g U \ell_{xy}}{\eta_g} \sim 1 \qquad (2.38)$$

With $\rho_g \sim 1 \text{ kg/m}^3$ the density of the air.

In this regime of Reynolds numbers, the distance $\ell_z$ can be chosen as a Blasius length over which a visco-inertial boundary layer develops in the neighbouring gas. Getting an order of magnitude with this scaling means providing a **typical time for our dynamics, which is of the order $\tau \sim 1 \text{ s}$ (Fig.2.8), yielding $\ell_z \sim 10^{-3} \text{m}$**.

In the hypothesis of vanishing Reynolds numbers $Re = 0$, no typical length other than the one imposed by the system can be constructed, and we necessarily have: $\ell_z \sim \ell_{xy} \sim 10^{-3} \text{m}$ which remains of the same order of magnitude! This is important as we will later use in practice a model at $Re = 0$ despite our computation $Re \sim 1$, which is a strong hypothesis in our work.

### Comparison between the bulk frictions: gas friction dominates

With the two scalings provided to $f_g$ and $f_\eta$, we can now compute their relative importance:

$$\frac{f_\eta}{f_g} \sim \frac{\eta h \ell_z}{\eta_g \ell_{xy}^2} \sim 10^{-1} \qquad (2.39)$$

**Meaning from now on we can discard the liquid bulk friction term $f_\eta$ from eq.2.33.**

### The Boussinesq number: comparing gas friction and surface shear viscosity

Thus, the two remaining damping forces are the surface shear viscosity (which depends on the chemistry), and gas friction. In the literature, measurements of $\eta_s$ for soluble surfactants such as SDS, which we are using, are limited by subphase bulk friction [34, 36], and only upper-bounds can be computed. We face the exact same problem in the case of in-plane motions of foam films, except the subphase here is the gas phase rather than the liquid bulk.

Let us proceed with caution, and define a number to compare these damping forces, which is called a Boussinesq number $Bq$. This is going to come in handy later on when looking at the damped dynamics of the in-plane motions. $Bq$ is defined in our case as:

$$Bq = \frac{\eta_s}{\ell_{xy}\eta_g} \qquad (2.40)$$
Where $z$ is the normal direction to the median plane (Oxy) of the foam film, $\eta_g$ is the dynamic viscosity of the gas phase, and $v_g$ is the velocity in the exterior gas phase projected in (Oxy). We suppose in our geometry that $v_g(x, y, z) = v_g(x, y, -z)$ for any $(x, y)$. This force scales as:

$$f_g \sim \eta_g \frac{U}{\ell_z} \qquad (2.37)$$

Where $\eta_g = 1.8 \cdot 10^{-5} \text{Pa} \cdot \text{s}$, $U \sim 10^{-2} \text{mm/s}$, and $\ell_z \sim \sqrt{\eta_g \tau}$ is a typical length scale for the vertical gradient of velocity in the gas phase.

Giving an order of magnitude for this latter length raises the question of taking into account gas inertia or not. An estimation of the Reynolds number for the gas in our problem gives:

$$Re = \frac{\rho_g U \ell_{xy}}{\eta_g} \sim 1 \qquad (2.38)$$

With $\rho_g \sim 1 \text{ kg/m}^3$ the density of the air.

In this regime of Reynolds numbers, the distance $\ell_z$ can be chosen as a Blasius length over which a visco-inertial boundary layer develops in the neighbouring gas. Getting an order of magnitude with this scaling means providing a **typical time for our dynamics, which is of the order $\tau \sim 1 \text{ s (Fig.2.8)}$, yielding $\ell_z \sim 10^{-3} \text{m}$**.

In the hypothesis of vanishing Reynolds numbers $Re = 0$, no typical length other than the one imposed by the system can be constructed, and we necessarily have: $\ell_z \sim \ell_{xy} \sim 10^{-3} \text{m}$ which remains of the same order of magnitude! This is important as we will later use in practice a model at $Re = 0$ despite our computation $Re \sim 1$, which is a strong hypothesis in our work.

### Comparison between the bulk frictions: gas friction dominates

With the two scalings provided to $f_g$ and $f_\eta$, we can now compute their relative importance:

$$\frac{f_\eta}{f_g} \sim \frac{\eta h \ell_z}{\eta_g \ell_{xy}^2} \sim 10^{-1} \qquad (2.39)$$

**Meaning from now on we can discard the liquid bulk friction term $f_\eta$ from eq.2.33.**

### The Boussinesq number: comparing gas friction and surface shear viscosity

Thus, the two remaining damping forces are the surface shear viscosity (which depends on the chemistry), and gas friction. In the literature, measurements of $\eta_s$ for soluble surfactants such as SDS, which we are using, are limited by subphase bulk friction [34, 36], and only upper-bounds can be computed. We face the exact same problem in the case of in-plane motions of foam films, except the subphase here is the gas phase rather than the liquid bulk.

Let us proceed with caution, and define a number to compare these damping forces, which is called a Boussinesq number $Bq$. This is going to come in handy later on when looking at the damped dynamics of the in-plane motions. $Bq$ is defined in our case as:

$$Bq = \frac{\eta_s}{\ell_{xy}\eta_g} \qquad (2.40)$$
Where we again use the experimental coincidence that $l_z \sim l_{xy}$ for the gas phase. This number will be a central parameter for the Saffman & Hughes model we are about to describe and apply to our case in order to predict the relaxation dynamics.

### 2.4.4 Mapping on the Saffman and Hughes model

Effective forces in $C^*$ are applied at the curved edges

<figure><img src="image_60.png" /></figure>

Figure 2.15: Full schematic view of our system. The inner red line is the arena defined by the edge of thin film of semi-width $R$. The outer black dotted line is the arena defined by the edge of thick film of semi-width $R + l_{80}$. The plain black line is the arena of semi-width $R^* = R + l_{80}/2$ retained to be the line of tension along which a tension $2T$ is applied (with a factor 2 for the two interfaces). The domains $\Omega^{\pm}$ are the cylindrical domains of fluid where a force $F_x = 4T$ pulls them toward the centre.

In Fig.2.15, we show the two domains $\Omega^{\pm}$ where an effective force is applied in the fluid, that is where the curvature of $C^*$ is non-zero. These two can be considered as impenetrable cylinders as they both have a homogeneous velocity field of absolute value $|\frac{dL}{dt}|$. Thus, eq.2.33 needs only to be solved outside of these domains $\Omega^{\pm}$, where the line tension forces are zero.

Also, by definition of the tension (cf. Fig.2.2 and eq.2.9), $\Omega^{\pm}$ are both pulled with a force $4T$ (two pulling edges times two interfaces).

The problem of describing the dynamics of relaxation of $L(t)$ can thus be mapped to the problem of predicting the kinematics of two thin cylindrical patches each pulled towards the other with a force $4T$ in a fluid of surface viscosity $\eta_s$ and subject to air friction at their circular bases. The two cylinders are far from one another in the sense that $L \gg R^*$, and we anticipate that they can be treated independently, i.e. they both generate flow fields that are negligible in the vicinity of their distant counterpart. We will validate this hypothesis later on.

The mobility of an individual cylinder is known

Let us start by considering the problem of a single dragged cylinder depicted in Fig.2.16. An infinite liquid sheet of medium plane $(Oxy)$ and of width $2h$ has a viscosity of $\eta$. For scores $|z| > h$ we have a gas phase of viscosity $\eta_g$, and the interfaces $\{z = \pm h\}$ are incompressible and have a surface shear viscosity of $\eta_s$. We consider a cylinder contained in the liquid sheet of radius
Where we again use the experimental coincidence that $l_z \sim l_{xy}$ for the gas phase. This number will be a central parameter for the Saffman & Hughes model we are about to describe and apply to our case in order to predict the relaxation dynamics.

### 2.4.4 Mapping on the Saffman and Hughes model

Effective forces in $C^*$ are applied at the curved edges

<figure><img src="image_60.png" /></figure>

Figure 2.15: Full schematic view of our system. The inner red line is the arena defined by the edge of thin film of semi-width $R$. The outer black dotted line is the arena defined by the edge of thick film of semi-width $R + l_{80}$. The plain black line is the arena of semi-width $R^* = R + l_{80}/2$ retained to be the line of tension along which a tension $2T$ is applied (with a factor 2 for the two interfaces). The domains $\Omega^{\pm}$ are the cylindrical domains of fluid where a force $F_x = 4T$ pulls them toward the centre.

In Fig.2.15, we show the two domains $\Omega^{\pm}$ where an effective force is applied in the fluid, that is where the curvature of $C^*$ is non-zero. These two can be considered as impenetrable cylinders as they both have a homogeneous velocity field of absolute value $|\frac{dL}{dt}|$. Thus, eq.2.33 needs only to be solved outside of these domains $\Omega^{\pm}$, where the line tension forces are zero.

Also, by definition of the tension (cf. Fig.2.2 and eq.2.9), $\Omega^{\pm}$ are both pulled with a force $4T$ (two pulling edges times two interfaces).

The problem of describing the dynamics of relaxation of $L(t)$ can thus be mapped to the problem of predicting the kinematics of two thin cylindrical patches each pulled towards the other with a force $4T$ in a fluid of surface viscosity $\eta_s$ and subject to air friction at their circular bases. The two cylinders are far from one another in the sense that $L \gg R^*$, and we anticipate that they can be treated independently, i.e. they both generate flow fields that are negligible in the vicinity of their distant counterpart. We will validate this hypothesis later on.

The mobility of an individual cylinder is known

Let us start by considering the problem of a single dragged cylinder depicted in Fig.2.16. An infinite liquid sheet of medium plane $(Oxy)$ and of width $2h$ has a viscosity of $\eta$. For scores $|z| > h$ we have a gas phase of viscosity $\eta_g$, and the interfaces $\{z = \pm h\}$ are incompressible and have a surface shear viscosity of $\eta_s$. We consider a cylinder contained in the liquid sheet of radius
$R^*$ and height $2h$ aligned with the $z$ axis, and subject to a force $F_x = 4Te_x$. No inertia is taken into account, ruling out time as a variable. The velocity of the cylinder is thus colinear to $F_x$ and solving the problem is reduced to finding the friction coefficient $\zeta$ (Pa·m·s) defined as:

$$F_x = -\zeta(Bq) U_x \qquad (2.41)$$

$\zeta$ is here a function of the Boussinesq number $Bq$ defined in subsection 2.4.3 comparing surface shear viscosity to gas friction. It comes with the fact that, as demonstrated in 2.4.3, the bulk viscous friction of the cylinder over its lateral area $2\pi R^*h$ on the neighbouring liquid is negligible.

<figure><img src="image_61.png" /><figcaption>Figure 2.16: Notations used to describe the drag force applied to the cylinders $\Omega^{\pm}$ subject to a force $F_x = 4T$.</figcaption></figure>

An exhaustive resolution of this problem has been developed first by Saffman [103] (1976) in the low $Bq$ regime, followed by Hughes et al. [104] (1981) who extended the prediction for any $Bq$. The interested reader will note that what these studies call a surface shear viscosity is defined in our notations by $\eta_s = 2h\eta$ but they end up in the same physical situation as us, where $\eta_s$ exists on its own and $\eta$ is set to zero. They solve for both the rotational and translational problems of a cylinder subject to constant torque and force, but only the latter relates to our situation.

Hughes et al. [104] compute the two limits $Bq = 0$ and $Bq \to +\infty$, and solve numerically the in-between, yielding the black curve in Fig.2.17. The low limit $\zeta_0 = \zeta(Bq = 0)$ is given by:

$$\zeta_0 = 16\eta_g R^* \qquad (2.42)$$

And the asymptotic expression of $\zeta$ becomes, for large $Bq$:

$$\frac{\zeta}{\zeta_0}(Bq \to +\infty) = \frac{\frac{\pi}{4}Bq}{\ln(2Bq) - C_e + \frac{4}{\pi Bq} - \frac{\ln(2Bq)}{2Bq^2}} \qquad (2.43)$$

Where $C_e \approx 0.58$ is the Euler-Mascheroni constant.

These predictions for the friction coefficient $\zeta(Bq)$ will allow us model the relaxation of the thin film, as we measure directly both the force $F_x = 4T$ applied to the edges and their velocity of retraction $U = \frac{dL}{dt}$.
$R^*$ and height $2h$ aligned with the $z$ axis, and subject to a force $F_x = 4Te_x$. No inertia is taken into account, ruling out time as a variable. The velocity of the cylinder is thus colinear to $F_x$ and solving the problem is reduced to finding the friction coefficient $\zeta$ (Pa·m·s) defined as:

$$F_x = -\zeta(Bq) U_x \qquad (2.41)$$

$\zeta$ is here a function of the Boussinesq number $Bq$ defined in subsection 2.4.3 comparing surface shear viscosity to gas friction. It comes with the fact that, as demonstrated in 2.4.3, the bulk viscous friction of the cylinder over its lateral area $2\pi R^*h$ on the neighbouring liquid is negligible.

<figure><img src="image_61.png" /><figcaption>Figure 2.16: Notations used to describe the drag force applied to the cylinders $\Omega^{\pm}$ subject to a force $F_x = 4T$.</figcaption></figure>

An exhaustive resolution of this problem has been developed first by Saffman [103] (1976) in the low $Bq$ regime, followed by Hughes et al. [104] (1981) who extended the prediction for any $Bq$. The interested reader will note that what these studies call a surface shear viscosity is defined in our notations by $\eta_s = 2h\eta$ but they end up in the same physical situation as us, where $\eta_s$ exists on its own and $\eta$ is set to zero. They solve for both the rotational and translational problems of a cylinder subject to constant torque and force, but only the latter relates to our situation.

Hughes et al. [104] compute the two limits $Bq = 0$ and $Bq \to +\infty$, and solve numerically the in-between, yielding the black curve in Fig.2.17. The low limit $\zeta_0 = \zeta(Bq = 0)$ is given by:

$$\zeta_0 = 16\eta_g R^* \qquad (2.42)$$

And the asymptotic expression of $\zeta$ becomes, for large $Bq$:

$$\frac{\zeta}{\zeta_0}(Bq \to +\infty) = \frac{\frac{\pi}{4}Bq}{\ln(2Bq) - C_e + \frac{4}{\pi Bq} - \frac{\ln(2Bq)}{2Bq^2}} \qquad (2.43)$$

Where $C_e \approx 0.58$ is the Euler-Mascheroni constant.

These predictions for the friction coefficient $\zeta(Bq)$ will allow us model the relaxation of the thin film, as we measure directly both the force $F_x = 4T$ applied to the edges and their velocity of retraction $U = \frac{dL}{dt}$.
<figure><img src="image_62.png" /><figcaption>Figure 2.17: Mobility $\zeta$ as a function of the Boussinesq number $Bq$ normalised by the low-$Bq$ limit $\zeta_0$. The black line is the theoretical prediction, the blue horizontal lines are the highest and lowest experimental estimations of $\zeta$ made throughout the whole relaxation.</figcaption></figure>

The retraction of $C^*$ can be treated as two independent moving cylinders

As said above, we have separated the problem of solving eq.2.33 into two different subproblems:

* $\Omega^{\pm}$ are individual impenetrable cylinders each pulled with a force $4T$ towards the other.

* The flow field outside these domains can be solved by superimposing the two solutions of eq.2.33 with source terms reduced to the drag of each cylinder.

The former subproblem needs the latter to yield a flow field which becomes negligible at the distance separating the two cylinders. This allows us to see the dynamics of $\Omega^{\pm}$ as two decoupled problems.

The computation of the flow field in outside of $\Omega^{\pm}$ has been done by Stone & Ajdari [105]. Compared to all the cases detailed in this paper, we have here an infinite subphase ($H/R \to +\infty$ with the notations of the paper), and we anticipate that we are in a subphase friction-dominated regime ($\Lambda \gg 1$ in the paper). Fig.7 of the paper shows that in our case, and with our notations now, the amplitude of the velocity field imposed by one domain $\Omega^{\pm}$ in the vicinity of the other is below $0.1|\frac{dL}{dt}|$ for $2L > 10R^*$, which is always the case in our measurable range of time.

This validates our assumption that we can treat both domains independently and use the Saffman & Hughes model to look at the dynamics of retraction of $L(t)$.

### 2.4.5 Quantitative agreement in the air friction-dominated regime

The previous subsection 2.4.4 provides us with a model for the mobility $\zeta(Bq)$ of the edges of the arena $\Omega^{\pm}$ (cf. Fig.2.15). Experimentally, we measure at each time step both their velocities $U^{exp}(t) = \frac{dL}{dt}$ and the forces applied on them $F_x(t) = 4T(t)$. Thus, for any time $t$, we now want to compare the value $U^{exp}$ to a theoretical value:

$$U^{th}(t) = \frac{F_x(t)}{\zeta(Bq)} \qquad (2.44)$$
<figure><img src="image_62.png" /><figcaption>Figure 2.17: Mobility $\zeta$ as a function of the Boussinesq number $Bq$ normalised by the low-$Bq$ limit $\zeta_0$. The black line is the theoretical prediction, the blue horizontal lines are the highest and lowest experimental estimations of $\zeta$ made throughout the whole relaxation.</figcaption></figure>

The retraction of $C^*$ can be treated as two independent moving cylinders

As said above, we have separated the problem of solving eq.2.33 into two different subproblems:

* $\Omega^{\pm}$ are individual impenetrable cylinders each pulled with a force $4T$ towards the other.

* The flow field outside these domains can be solved by superimposing the two solutions of eq.2.33 with source terms reduced to the drag of each cylinder.

The former subproblem needs the latter to yield a flow field which becomes negligible at the distance separating the two cylinders. This allows us to see the dynamics of $\Omega^{\pm}$ as two decoupled problems.

The computation of the flow field in outside of $\Omega^{\pm}$ has been done by Stone & Ajdari [105]. Compared to all the cases detailed in this paper, we have here an infinite subphase ($H/R \to +\infty$ with the notations of the paper), and we anticipate that we are in a subphase friction-dominated regime ($\Lambda \gg 1$ in the paper). Fig.7 of the paper shows that in our case, and with our notations now, the amplitude of the velocity field imposed by one domain $\Omega^{\pm}$ in the vicinity of the other is below $0.1|\frac{dL}{dt}|$ for $2L > 10R^*$, which is always the case in our measurable range of time.

This validates our assumption that we can treat both domains independently and use the Saffman & Hughes model to look at the dynamics of retraction of $L(t)$.

### 2.4.5 Quantitative agreement in the air friction-dominated regime

The previous subsection 2.4.4 provides us with a model for the mobility $\zeta(Bq)$ of the edges of the arena $\Omega^{\pm}$ (cf. Fig.2.15). Experimentally, we measure at each time step both their velocities $U^{exp}(t) = \frac{dL}{dt}$ and the forces applied on them $F_x(t) = 4T(t)$. Thus, for any time $t$, we now want to compare the value $U^{exp}$ to a theoretical value:

$$U^{th}(t) = \frac{F_x(t)}{\zeta(Bq)} \qquad (2.44)$$
Note that the model does not have time as a variable, as no inertia or time-varying forces are among its physical ingredients. Each time step thus represents an independent measure to compare to the model.

To get $U^{th}$, we now only need to define in what regime of Boussinesq number $Bq$ we are. As discussed in [36], solid experimental clues set an upper bound for $\eta_s$ in the case of soluble surfactants (and micellar SDS in particular) at $\eta_s \lesssim 10^{-9}$ Pa m s, allowing us to estimate in our case:

$$ Bq = \frac{\eta_s}{R^* \eta_g} \lesssim \frac{10^{-9}}{10^{-3} \times 18 \times 10^{-6}} \sim 0.06 \qquad (2.45) $$

Meaning we expect to be in an air friction-dominated regime. With this in mind, the model of Saffman & Hughes et al. [103, 104] of subsection 2.4.4 gives the expression:

$$ U^{th}(t) = \frac{4T(t)}{16\eta_g R^*(t)} \qquad (2.46) $$

A comparison with the experimental values $U^{exp}(t) = \frac{dL}{dt}$ is shown in Fig.2.18. The middle thicker blue line is the ratio $U^{exp}/U^{th}$ computed with $R^* = R + \ell_{80}/2$, the higher blue line with $R^* = R + \ell_{80}$ and the lower one with $R^* = R$. The uncertainties are shown with shaded areas and correspond to the standard deviation over the 13 repetitions of the experiment, the lower bound being taken with the lower line, the upper bound with the higher line.

<figure><img src="image_63.png" /><figcaption>Figure 2.18: Experimental and theoretical comparison for the relaxation speed. The theoretical values $U^{th}$ are computed in the low Boussinesq number limit, which corresponds to an air friction-dominated regime. The three lines correspond each to a different definition of $R^*$, from top to bottom: $R^* = R + \ell_{80}$, $R^* = R + \ell_{80}/2$ and $R^* = R$. The uncertainties are the extremal values obtained considering all three lines with their respective standard deviations.</figcaption></figure>

The quantitative agreement is good, as for $t > 0.5$ s we have $U^{exp}/U^{th} \simeq 1.5 \pm 1.0$. It means we experimentally validate our measure of line tension $T$, assuming we are in the low Boussinesq limit thanks to our chemistry. We can also express this in terms of experimental friction coefficient $\zeta^{exp}$:
Note that the model does not have time as a variable, as no inertia or time-varying forces are among its physical ingredients. Each time step thus represents an independent measure to compare to the model.

To get $U^{th}$, we now only need to define in what regime of Boussinesq number $Bq$ we are. As discussed in [36], solid experimental clues set an upper bound for $\eta_s$ in the case of soluble surfactants (and micellar SDS in particular) at $\eta_s \lesssim 10^{-9}$ Pa m s, allowing us to estimate in our case:

$$Bq = \frac{\eta_s}{R^* \eta_g} \lesssim \frac{10^{-9}}{10^{-3} \times 18 \times 10^{-6}} \sim 0.06 \qquad (2.45)$$

Meaning we expect to be in an air friction-dominated regime. With this in mind, the model of Saffman & Hughes et al. [103, 104] of subsection 2.4.4 gives the expression:

$$U^{th}(t) = \frac{4T(t)}{16\eta_g R^*(t)} \qquad (2.46)$$

A comparison with the experimental values $U^{exp}(t) = \frac{dL}{dt}$ is shown in Fig.2.18. The middle thicker blue line is the ratio $U^{exp}/U^{th}$ computed with $R^* = R + \ell_{80}/2$, the higher blue line with $R^* = R + \ell_{80}$ and the lower one with $R^* = R$. The uncertainties are shown with shaded areas and correspond to the standard deviation over the 13 repetitions of the experiment, the lower bound being taken with the lower line, the upper bound with the higher line.

<figure><img src="image_63.png" /><figcaption>Figure 2.18: Experimental and theoretical comparison for the relaxation speed. The theoretical values $U^{th}$ are computed in the low Boussinesq number limit, which corresponds to an air friction-dominated regime. The three lines correspond each to a different definition of $R^*$, from top to bottom: $R^* = R + \ell_{80}$, $R^* = R + \ell_{80}/2$ and $R^* = R$. The uncertainties are the extremal values obtained considering all three lines with their respective standard deviations.</figcaption></figure>

The quantitative agreement is good, as for $t > 0.5$ s we have $U^{exp}/U^{th} \simeq 1.5 \pm 1.0$. It means we experimentally validate our measure of line tension $T$, assuming we are in the low Boussinesq limit thanks to our chemistry. We can also express this in terms of experimental friction coefficient $\zeta^{exp}$:
$$ \frac{\zeta^{\text{exp}}}{\zeta_0}(t) = \frac{4T(t)}{16\eta_g R^*(t) U^{\text{exp}}(t)} \qquad (2.47) $$

Where time is again acting as a dummy variable listing independent measurements throughout our working range $t \in [0, 1.5]$ s. The maximum and minimum values for this friction coefficient (taking into account all times $t > 0.5$ s and the uncertainties) are shown with blue lines in Fig.2.17, and correspond as expected to the low $Bq$ regime.

### 2.4.6 Important remarks and some calculations

#### Quantitative discrepancy at early times

In Fig.2.18, we see that for $t \in [0, 0.5]$ s, we have a velocity $U^{\text{exp}}$ significantly higher than what we expect with our model in the low Boussinesq model.

We are relaxing “too” fast and the explanation cannot be that we have taken the wrong $Bq$-limit: higher Boussinesq numbers means taking into account the contribution of surface shear viscosity besides air friction, leading to velocities lower than the $U^{\text{th}}$ we computed.

The only qualitatively reasonable explanation we have envisioned is an air displacement along the y-axis coming from both directions towards the centre $y = 0$ induced by the displacement of the motor in the $+x$ direction. This would lead to a decrease in the relative velocities between the moving edges of the thin film and the surrounding air, lowering the vertical velocity gradients in the latter and thus lowering the friction.

As for validating this hypothesis... no complex or outlandish experiments trying to control the air movements around the setup were attempted, and so no undergrad interns conducting the experiments were harmed.

#### Spatial distribution of $T(s)$

The major weakness of the validation of our measurement of the line tension $T$ is that it relies on the bold hypothesis that $T$ is uniform along the contour $C^*$ (of curvilinear abscissa denoted $s$). In subsection 2.4.1 we assume this, supported by the fact that the measurement on both sides of the arena are very close. However the line tension responsible for the retraction of the arena is applied at the edges, where we do not measure it, and we have to rely on the assumption $\frac{\partial T}{\partial s} = 0$.

The risk we incur regarding this hypothesis is a local compression of the thickness jump zone during the relaxation as the edges retract. This would lead to different line tensions at the edges (where it applies the driving force) and in the straight parts of the arena (where we measure it).

In Fig.2.19B, we show a spatiotemporal image of the interference colours along the red line of A and rearranged so that the middle of the arena is at the origin of $x$ (thus correcting for the slight global advection of the arena). The coloured patterns show that the compression $\epsilon_x$ is applied in a uniform way along $x$. This is a solid experimental argument in favour of an equally shared compression of the contour $C^*$ as it relaxes, and thus a uniform line of tension $T$.
$$ \frac{\zeta^{exp}}{\zeta_0}(t) = \frac{4T(t)}{16\eta_g R^*(t)U^{exp}(t)} \qquad (2.47) $$

Where time is again acting as a dummy variable listing independent measurements throughout our working range $t \in [0, 1.5]$ s. The maximum and minimum values for this friction coefficient (taking into account all times $t > 0.5$ s and the uncertainties) are shown with blue lines in Fig.2.17, and correspond as expected to the low $Bq$ regime.

## 2.4.6 Important remarks and some calculations

### Quantitative discrepancy at early times

In Fig.2.18, we see that for $t \in [0, 0.5]$ s, we have a velocity $U^{exp}$ significantly higher than what we expect with our model in the low Boussinesq model.

We are relaxing “too” fast and the explanation cannot be that we have taken the wrong $Bq$-limit: higher Boussinesq numbers means taking into account the contribution of surface shear viscosity besides air friction, leading to velocities lower than the $U^{th}$ we computed.

The only qualitatively reasonable explanation we have envisioned is an air displacement along the $y$-axis coming from both directions towards the centre $y = 0$ induced by the displacement of the motor in the $+x$ direction. This would lead to a decrease in the relative velocities between the moving edges of the thin film and the surrounding air, lowering the vertical velocity gradients in the latter and thus lowering the friction.

As for validating this hypothesis... no complex or outlandish experiments trying to control the air movements around the setup were attempted, and so no undergrad interns conducting the experiments were harmed.

### Spatial distribution of $T(s)$

The major weakness of the validation of our measurement of the line tension $T$ is that it relies on the bold hypothesis that $T$ is uniform along the contour $C^*$ (of curvilinear abscissa denoted $s$). In subsection 2.4.1 we assume this, supported by the fact that the measurement on both sides of the arena are very close. However the line tension responsible for the retraction of the arena is applied at the edges, where we do not measure it, and we have to rely on the assumption $\frac{\partial T}{\partial s} = 0$.

The risk we incur regarding this hypothesis is a local compression of the thickness jump zone during the relaxation as the edges retract. This would lead to different line tensions at the edges (where it applies the driving force) and in the straight parts of the arena (where we measure it).

In Fig.2.19B, we show a spatiotemporal image of the interference colours along the red line of A and rearranged so that the middle of the arena is at the origin of $x$ (thus correcting for the slight global advection of the arena). The coloured patterns show that the compression $\epsilon_x$ is applied in a uniform way along $x$. This is a solid experimental argument in favour of an equally shared compression of the contour $C^*$ as it relaxes, and thus a uniform line of tension $T$.
<figure><img src="image_64.png" /><figcaption>Figure 2.19: Temporal mosaic of the relaxation of the arena. **A:** The red line is the pixel line along which each row of the mosaic is taken at every time step. **B:** Temporal mosaic centred in the frame of the arena (the global advection of the arena in the frame of the camera is corrected).</figcaption></figure>

## Time evolution of $T(t)$

In Fig.2.13, we see that the line tension $T$ increases significantly by a factor 2 during the relaxation. We want to rationalise this trend.

Let us first look at the scalings of $T$ with the geometry of the thickness jump zone. If we recall the expression of the line tension, we see that it scales as:

$$T = \gamma_0 \int_0^{\xi_m} ||\nabla h||^2 d\xi \sim \gamma_0 \frac{\delta h^2}{w} \qquad (2.48)$$

Where $\delta h$ is the typical difference of thickness between the thin and thick films, and $w$ the typical width of the thickness jump zone.

We can have an experimental access to both $\delta h$ and $w$ using the thickness profiles of Fig.2.11. $w$ can be estimated with $w = l_{80}$ defined in Fig.2.13 as the distance of integration from the edge of the thin film where 80% of the value of $T$ is attained. $\delta h$ can be estimated as the thickness of the thick film at the position where $w$ is measured. Both are shown in Fig.2.20A-B, and graph C is the normalised line tension defined as:

$$T^* = \frac{wT}{\gamma_0 \delta h^2} \qquad (2.49)$$

Which is now constant in time and close to 1, meaning our numerical integration and our evaluation of $\delta h$ and $w$ went well.

During the relaxation, we see that both $\delta h$ and $w$ increase by similar factors $\sim 1.5 - 2$, hence the comparable increase of line tension as it scales as $T \sim \delta h^2/w$. What we want now to rationalise is thus how $\delta h$ and $w$ grow as the line of tension relaxes.
<figure><img src="image_64.png" /><figcaption>Figure 2.19: Temporal mosaic of the relaxation of the arena. **A:** The red line is the pixel line along which each row of the mosaic is taken at every time step. **B:** Temporal mosaic centred in the frame of the arena (the global advection of the arena in the frame of the camera is corrected).</figcaption></figure>

### Time evolution of $T(t)$

In Fig.2.13, we see that the line tension $T$ increases significantly by a factor 2 during the relaxation. We want to rationalise this trend.

Let us first look at the scalings of $T$ with the geometry of the thickness jump zone. If we recall the expression of the line tension, we see that it scales as:

$$T = \gamma_0 \int_0^{\xi_m} ||\nabla h||^2 d\xi \sim \gamma_0 \frac{\delta h^2}{w} \qquad (2.48)$$

Where $\delta h$ is the typical difference of thickness between the thin and thick films, and $w$ the typical width of the thickness jump zone.

We can have an experimental access to both $\delta h$ and $w$ using the thickness profiles of Fig.2.11. $w$ can be estimated with $w = l_{80}$ defined in Fig.2.13 as the distance of integration from the edge of the thin film where 80% of the value of $T$ is attained. $\delta h$ can be estimated as the thickness of the thick film at the position where $w$ is measured. Both are shown in Fig.2.20A-B, and graph C is the normalised line tension defined as:

$$T^* = \frac{wT}{\gamma_0 \delta h^2} \qquad (2.49)$$

Which is now constant in time and close to 1, meaning our numerical integration and our evaluation of $\delta h$ and $w$ went well.

During the relaxation, we see that both $\delta h$ and $w$ increase by similar factors ~ 1.5 – 2, hence the comparable increase of line tension as it scales as $T \sim \delta h^2/w$. What we want now to rationalise is thus how $\delta h$ and $w$ grow as the line of tension relaxes.
<figure><img src="image_65.png" /><figcaption>Figure 2.20:</figcaption></figure>

If we get back to our general hypothesis that fluid exchanges between two neighbouring elements of film are infinitely long (due to very slow Poiseuille flows as detailed in 0.2.2) we can think of the thickness jump zone as a closed system, in the sense of volume conservation. Thus, we would assume that $V_{C^*} = \delta h \times w \times P$ is conserved during the relaxation, and, as the perimeter $P$ is a decreasing function of time (cf. Fig.2.8C), we would expect $\delta h$ and/or $w$ to increase over time. However, the volume $V_{C^*}$ contained in the thickness jump zone, estimated with our measurement of $\delta h$ and $w$, is shown in Fig.2.20D and increases as well.

This is puzzling, and hinders going further in the reasoning. A potential explanation is a local compression along the direction tangent to $C^*$ of the thickness jump zone near our line of measurement, which is located in the middle of the arena. Even in this case, what remains unclear is why $\delta h$ grows over time, as it is basically the thickness of the thick film. The areas of both the thin and thick films being conserved (cf. Fig.2.8D), the product $\delta h \times A$ should remain constant as well instead of growing because of $\delta h$. This could mean that not only the thickness jump zone is being locally compressed, but also the thick film near it. To further advance on this matter, we would need the thickness field everywhere in the foam film, or at least at the edges as well.

Therefore, the function $T(t)$ is assumed uniform along $C^*$ and its evolution in time is known but not predicted. These are the main flaws of this novel force measurement.
<figure><img src="image_65.png" /><figcaption>Figure 2.20:</figcaption></figure>

If we get back to our general hypothesis that fluid exchanges between two neighbouring elements of film are infinitely long (due to very slow Poiseuille flows as detailed in 0.2.2) we can think of the thickness jump zone as a closed system, in the sense of volume conservation. Thus, we would assume that $V_{C^*} = \delta h \times w \times P$ is conserved during the relaxation, and, as the perimeter $P$ is a decreasing function of time (cf. Fig.2.8C), we would expect $\delta h$ and/or $w$ to increase over time. However, the volume $V_{C^*}$ contained in the thickness jump zone, estimated with our measurement of $\delta h$ and $w$, is shown in Fig.2.20D and increases as well.

This is puzzling, and hinders going further in the reasoning. A potential explanation is a local compression along the direction tangent to $C^*$ of the thickness jump zone near our line of measurement, which is located in the middle of the arena. Even in this case, what remains unclear is why $\delta h$ grows over time, as it is basically the thickness of the thick film. The areas of both the thin and thick films being conserved (cf. Fig.2.8D), the product $\delta h \times A$ should remain constant as well instead of growing because of $\delta h$. This could mean that not only the thickness jump zone is being locally compressed, but also the thick film near it. To further advance on this matter, we would need the thickness field everywhere in the foam film, or at least at the edges as well.

Therefore, the function $T(t)$ is assumed uniform along $C^*$ and its evolution in time is known but not predicted. These are the main flaws of this novel force measurement.
## 2.5 Conclusions of Chapter 2

* In-plane motions in foam films can be induced by the presence of a line tension $T$. It originates from localised thickness gradients in the film and tend to minimise the perimeter between pieces of film with different thicknesses.

* For instance, this line tension is relevant for the description of marginal regeneration (cf. 0.3.4) where thin patches take circular shapes.

* We managed to create a foam film with a configuration where an elongated arena of thin film is embedded in a thicker film, with a tension line separating them. The thin film relaxes towards a circular shape to minimise the perimeter of the tension line. We are able to measure simultaneously the line tension and the relaxation dynamics of the arena.

* We are speaking of very tiny forces $T \sim 10^{-10}N$, whose precise measurement is a novelty of our work.

* In the frame of our study, using a soluble surfactant (SDS), the in-plane motions are damped by air friction and I analytically predicted the dynamics of relaxation of the arena.

* The measure of our line tension $T$ is in quantitative agreement with the predicted kinematics for the relaxation motion damped by air friction. This validates our new measurement.
2.5 Conclusions of Chapter 2

* In-plane motions in foam films can be induced by the presence of a line tension $T$. It originates from localised thickness gradients in the film and tend to minimise the perimeter between pieces of film with different thicknesses.

* For instance, this line tension is relevant for the description of marginal regeneration (cf. 0.3.4) where thin patches take circular shapes.

* We managed to create a foam film with a configuration where an elongated arena of thin film is embedded in a thicker film, with a tension line separating them. The thin film relaxes towards a circular shape to minimise the perimeter of the tension line. We are able to measure simultaneously the line tension and the relaxation dynamics of the arena.

* We are speaking of very tiny forces $T \sim 10^{-10}N$, whose precise measurement is a novelty of our work.

* In the frame of our study, using a soluble surfactant (SDS), the in-plane motions are damped by air friction and I analytically predicted the dynamics of relaxation of the arena.

* The measure of our line tension $T$ is in quantitative agreement with the predicted kinematics for the relaxation motion damped by air friction. This validates our new measurement.
3

Dissipative phenomena in an elementary foam:
a matter of geometrical frustration

This last chapter is about the core of my experimental work. Here, we will be looking at the rheology of an elementary foam, like my predecessors Adrien Bussonnière and Raphaël Poryles did [4,6], gathering information on the viscoelastic response of this “local scale of liquid foams”. Only now we focus on a system with three films that will allow us to better describe the viscous response of these foams.

In fact, we will be able to quantify the exchanges of surfactants between the films and with the meniscus during the deformation. This is important because a stretched film relaxes by accepting some interface in its plane (cf. Frankel films 0.3.2) whereas a compressed film relaxes by giving up some interface. Thus, quantifying the interface balance (i.e. the surfactant balance) around the meniscus neighbouring these compressed and stretched films is the key experimental piece of information needed to complete the rheological model developed in our group [4] and before [1, 78].

The main result of this study, and thus of this thesis, is that in the range of strain rate 1–10 Hz, and with all our SDS/DOH mixtures, we only witness film-film exchanges of surfactants, and the meniscus does not play any role of reservoir in this balance.
Dissipative phenomena in an elementary foam:
a matter of geometrical frustration

This last chapter is about the core of my experimental work. Here, we will be looking at the rheology of an elementary foam, like my predecessors Adrien Bussonnière and Raphaël Poryles did [4,6], gathering information on the viscoelastic response of this “local scale of liquid foams”. Only now we focus on a system with three films that will allow us to better describe the viscous response of these foams.

In fact, we will be able to quantify the exchanges of surfactants between the films and with the meniscus during the deformation. This is important because a stretched film relaxes by accepting some interface in its plane (cf. Frankel films 0.3.2) whereas a compressed film relaxes by giving up some interface. Thus, quantifying the interface balance (i.e. the surfactant balance) around the meniscus neighbouring these compressed and stretched films is the key experimental piece of information needed to complete the rheological model developed in our group [4] and before [1, 78].

The main result of this study, and thus of this thesis, is that in the range of strain rate 1–10 Hz, and with all our SDS/DOH mixtures, we only witness film-film exchanges of surfactants, and the meniscus does not play any role of reservoir in this balance.
# 3.1 Rheology of an elementary liquid foam: geometric frustration

## 3.1.1 An elementary liquid foam with three films

Our elementary foam is presented schematically in Fig. 3.1, as well as the general principle of its deformation.

<figure><img src="image_66.png" /><figcaption>Figure 3.1: Elementary liquid foam with three films. **A:** Side view with the three films (1 for the top left, 2 for the bottom left, 3 for the right) and the three interfaces shared by the films (top, left and bottom interfaces). **B:** Perspective view of the foam, we impose the velocities $u_i$ for each film using three motors. The distance between the two supporting red frames is fixed at 4.2 cm.</figcaption></figure>

Describing this system is equivalent to looking at the local scale of a liquid foam: one meniscus-neighboured by three films. In the Introduction 0, subsection 0.4.5, we look at the rheology of an elementary foam with 5 films, and see the viscoelastic response of a Maxwell fluid. Going down to three films and imposing deformations so that $u_1 = u_2 = -u_3$, the response of the difference of tension between film 3 and the two others is similar as shown in Fig. 29A.

As stated in subsection 0.4.5, the flat deformed foam films have a non-linear elastic response, and the mechanical dissipation is located elsewhere. We expect this region to be very localised near the meniscus and to exist because of what we call a geometric frustration.

Investigating this is the motivation behind passing from 5 to 3 films. We want to be able to better control the deformation state on both side of the meniscus. This was a problem with the 5-films configuration, as we could not deform the central film significantly.

## 3.1.2 Position of the problem

To understand where lies the problem of the viscous dissipation in our elementary foam, let us look at a peculiar zone, depicted in Fig. 3.2: near the meniscus, where **the symmetry of deformation imposes a stagnation point at the interface** (blue cross). Given the three films configuration, the presence of these stopping points is inevitable, and they result in what we call a **geometric frustration**.
# 3.1 Rheology of an elementary liquid foam: geometric frustration

## 3.1.1 An elementary liquid foam with three films

Our elementary foam is presented schematically in Fig. 3.1, as well as the general principle of its deformation.

<figure><img src="image_66.png" /><figcaption>Figure 3.1: Elementary liquid foam with three films. **A:** Side view with the three films (1 for the top left, 2 for the bottom left, 3 for the right) and the three interfaces shared by the films (top, left and bottom interfaces). **B:** Perspective view of the foam, we impose the velocities $u_i$ for each film using three motors. The distance between the two supporting red frames is fixed at 4.2 cm.</figcaption></figure>

Describing this system is equivalent to looking at the local scale of a liquid foam: one meniscus-neighboured by three films. In the Introduction 0, subsection 0.4.5, we look at the rheology of an elementary foam with 5 films, and see the viscoelastic response of a Maxwell fluid. Going down to three films and imposing deformations so that $u_1 = u_2 = -u_3$, the response of the difference of tension between film 3 and the two others is similar as shown in Fig. 29A.

As stated in subsection 0.4.5, the flat deformed foam films have a non-linear elastic response, and the mechanical dissipation is located elsewhere. We expect this region to be very localised near the meniscus and to exist because of what we call a geometric frustration.

Investigating this is the motivation behind passing from 5 to 3 films. We want to be able to better control the deformation state on both side of the meniscus. This was a problem with the 5-films configuration, as we could not deform the central film significantly.

## 3.1.2 Position of the problem

To understand where lies the problem of the viscous dissipation in our elementary foam, let us look at a peculiar zone, depicted in Fig. 3.2: near the meniscus, where **the symmetry of deformation imposes a stagnation point at the interface** (blue cross). Given the three films configuration, the presence of these stopping points is inevitable, and they result in what we call a **geometric frustration**.
<figure><img src="image_67.png" /></figure>

Figure 3.2: Schematic view of the region close to the meniscus as predicted by Bussonnière et al. [4]. **A:** Plug flows are imposed far from the meniscus, but because of the symmetry, the left interface has a stagnation point (blue cross). **B:** Zoom near the stagnation point, with the flow field $u$ in blue, the surfactant flux $\Gamma u$ in red. The stagnation point implies the existence of a sheared zone, whose length $\ell$ can be estimated [4] $\ell \sim 100~\mu m$. The stagnation point is also a point of vanishing surfactant flux, and it is yet undetermined in [4] whether the influx of surfactants on the left interface comes from the meniscus ($j_m$) or from the other film ($j_F$).

Fig.3.2B represents the flow field (blue) and the surfactant transports (red) in the foam near the stagnation point, localised in the middle of the left interface. Most of our experimental results are obtained in the symmetry of deformation and with the direction of the imposed velocities shown in the figure.

The vanishing velocity at the interface imposes the existence of a sheared region near the meniscus of length $\ell \sim 100~\mu m$ which is described and estimated by Bussonnière & Cantat [4]. The model behind it and its estimation will be adapted to our own geometry in 3.2.4. This is where the actual viscous dissipation occurs in the system.

The prediction of this length $\ell$ relies on the surfactant transport map. Here the stagnation imposes the creation of interface on the left, for which we need to provide for its surfactants. Two sources of surfactants can be envisioned here: either the top interface transfers some of its surfactants with a transverse flux $j_F$, or the meniscus provides for it with a flux $j_m$. This is where lies the main theoretical conundrum: these two fluxes must enforce the surfactant balance in the system, but how shared is burden? Can we neglect one of them?

The relative importance of these two fluxes $j_F$ and $j_m$ determines the profiles of surface concentrations $\Gamma_{t/l}$ of surfactants along both interfaces, setting the Marangoni stress boundary condition for the shear flow. It thus controls the value of the extension $\ell$ of the sheared domain and eventually the amount of viscous dissipation in the system.

### 3.1.3 First hint

In order to get a first qualitative input about these exchanges, we look at the extracted pieces of film in a stretched film and how it may depend on the compression/extension state of its neighbouring films. In Fig.3.3, we impose a fixed deformation $\Delta L_{in}$ for a film and a varying
<figure><img src="image_67.png" /></figure>

Figure 3.2: Schematic view of the region close to the meniscus as predicted by Bussonnière *et al.* [4]. **A:** Plug flows are imposed far from the meniscus, but because of the symmetry, the left interface has a stagnation point (blue cross). **B:** Zoom near the stagnation point, with the flow field *u* in blue, the surfactant flux Γ*u* in red. The stagnation point implies the existence of a sheared zone, whose length *l* can be estimated [4] *l* ~ 100 µm. The stagnation point is also a point of vanishing surfactant flux, and it is yet undetermined in [4] whether the influx of surfactants on the left interface comes from the meniscus (*j*<sub>m</sub>) or from the other film (*j*<sub>F</sub>).

Fig.3.2B represents the flow field (blue) and the surfactant transports (red) in the foam near the stagnation point, localised in the middle of the left interface. Most of our experimental results are obtained in the symmetry of deformation and with the direction of the imposed velocities shown in the figure.

The vanishing velocity at the interface imposes the existence of a sheared region near the meniscus of length *l* ~ 100 µm which is described and estimated by Bussonnière & Cantat [4]. The model behind it and its estimation will be adapted to our own geometry in 3.2.4. This is where the actual viscous dissipation occurs in the system.

The prediction of this length *l* relies on the surfactant transport map. Here the stagnation imposes the creation of interface on the left, for which we need to provide for its surfactants. Two sources of surfactants can be envisioned here: either the top interface transfers some of its surfactants with a transverse flux *j*<sub>F</sub>, or the meniscus provides for it with a flux *j*<sub>m*. This is where lies the main theoretical conundrum: these two fluxes must enforce the surfactant balance in the system, but how shared is burden? Can we neglect one of them?

The relative importance of these two fluxes $j_F$ and $j_m$ determines the profiles of surface concentrations $\Gamma_{t/l}$ of surfactants along both interfaces, setting the Marangoni stress boundary condition for the shear flow. It thus controls the value of the extension $l$ of the sheared domain and eventually the amount of viscous dissipation in the system.

3.1.3 First hint

In order to get a first qualitative input about these exchanges, we look at the extracted pieces
of film in a stretched film and how it may depend on the compression/extension state of its
neighbouring films. In Fig.3.3, we impose a fixed deformation ΔL<sub>in</sub> for a film and a varying
$\Delta L_{ex}$ for the two neighbouring films. The three films have initially the same lengths $L_{init}$. This corresponds to the deformation factors:

$$ \epsilon_{in/ex} = \frac{\Delta L_{in/ex}}{L_{init}} - 1 \qquad (3.1) $$

And we set a constant $\epsilon_{in} = +0.3$ whereas we vary continuously $\epsilon_{ex}$ from $-0.5$ (compression) to $+0.3$ (extension). This latter case thus corresponds to an isotropic stretching of the elementary foam. The deformations are imposed with fixed velocities so that all deformations have ended at 0.1 s thus corresponding to a strain rate of 10 Hz.

<figure><img src="image_68.png" /><figcaption>Figure 3.3: Geometry of the deformations applied to the films. All three films start at the same initial length $L_{init}$. The horizontal film 3 is always stretched in the same way by a distance $\Delta L_{in}$, and we vary the deformation of film 1 and 2 (keeping it with a top-down symmetry) with a distance $\Delta L_{ex}$. The direction of the deformation sets the sign of $\epsilon_{ex}$: negative in compression, positive in extension.</figcaption></figure>

Throughout the deformation, we witness the extraction of pieces of film in the plane of film 3 (undergoing the fixed deformation $\epsilon_{in}$). It is a Frankel film as described in the Introduction in subsection 0.3.2, which has a contrast in fluorescence with the preexisting film 3 due to a difference of thicknesses. This allows us to measure a length $L_{out}$ shown in Fig.3.4A, which is a top view of film 3: the large bright fringe on the right is the motor after the stretch, the thin fringe on the middle-left is the free meniscus linking film 3 to its vertical neighbours, film 1 and 2. Fig.3.4B shows the quantity $L_{out}$ of Frankel film extracted by stretching film 3 as a function of time for all the range of $\epsilon_{ex}$ imposed to its neighbours.

What we immediately see is that the quantity of outgoing film $L_{out}$ heavily depends on the compression state of the neighbouring films. Even more interestingly, $L_{out}$ completely vanishes as the deformation $\epsilon_{ex}$ of the neighbouring films tends towards $\epsilon_{in}$, that is the case of an isotropic stretching of the elementary foam. In this specific configuration, the films are unable to exchange any interface by symmetry (we turn $j_F$ off), and we see that no interface seems to be provided by the meniscus alone, suggesting that the flux $j_m$ is zero.

This is a first qualitative piece of evidence about the relative importance of these surfactant exchanges. To further investigate this, we need to make the balance between the outgoing quantities of interface of the extracted films as well as the quantitites of interface given up by the compressed films.
$\Delta L_{ex}$ for the two neighbouring films. The three films have initially the same lengths $L_{init}$. This corresponds to the deformation factors:

$$ \epsilon_{in/ex} = \frac{\Delta L_{in/ex}}{L_{init}} - 1 \qquad (3.1) $$

And we set a constant $\epsilon_{in} = +0.3$ whereas we vary continuously $\epsilon_{ex}$ from $-0.5$ (compression) to $+0.3$ (extension). This latter case thus corresponds to an isotropic stretching of the elementary foam. The deformations are imposed with fixed velocities so that all deformations have ended at 0.1 s thus corresponding to a strain rate of 10 Hz.

<figure><img src="image_68.png" /><figcaption>Figure 3.3: Geometry of the deformations applied to the films. All three films start at the same initial length $L_{init}$. The horizontal film 3 is always stretched in the same way by a distance $\Delta L_{in}$, and we vary the deformation of film 1 and 2 (keeping it with a top-down symmetry) with a distance $\Delta L_{ex}$. The direction of the deformation sets the sign of $\epsilon_{ex}$: negative in compression, positive in extension.</figcaption></figure>

Throughout the deformation, we witness the extraction of pieces of film in the plane of film 3 (undergoing the fixed deformation $\epsilon_{in}$). It is a Frankel film as described in the Introduction in subsection 0.3.2, which has a contrast in fluorescence with the preexisting film 3 due to a difference of thicknesses. This allows us to measure a length $L_{out}$ shown in Fig.3.4A, which is a top view of film 3: the large bright fringe on the right is the motor after the stretch, the thin fringe on the middle-left is the free meniscus linking film 3 to its vertical neighbours, film 1 and 2. Fig.3.4B shows the quantity $L_{out}$ of Frankel film extracted by stretching film 3 as a function of time for all the range of $\epsilon_{ex}$ imposed to its neighbours.

What we immediately see is that the quantity of outgoing film $L_{out}$ heavily depends on the compression state of the neighbouring films. Even more interestingly, $L_{out}$ completely vanishes as the deformation $\epsilon_{ex}$ of the neighbouring films tends towards $\epsilon_{in}$, that is the case of an isotropic stretching of the elementary foam. In this specific configuration, the films are unable to exchange any interface by symmetry (we turn $j_F$ off), and we see that no interface seems to be provided by the meniscus alone, suggesting that the flux $j_m$ is zero.

This is a first qualitative piece of evidence about the relative importance of these surfactant exchanges. To further investigate this, we need to make the balance between the outgoing quantities of interface of the extracted films as well as the quantitites of interface given up by the compressed films.
<figure><img src="image_69.png" /></figure>

Figure 3.4: Quantifying the length out extracted film in the stretched film 3 depending on the deformation of its neighbours film 1 and 2. **A:** Motor 3 imposes a fixed global deformation $\epsilon_{in} = 0.3$ to film 3, motor 1 and 2 impose the same global deformations $\epsilon_{ex} \in [-0.5, 0.3]$ to film 1 and 2. $L_{out} = f(t)$ is the length of extracted film seen in the plane of film 3 throughout the deformation. **B:** $L_{out} = f(t)$ for the whole range of $\epsilon_{ex}$. The outgoing quantity of film heavily depends on the deformation imposed to the neighbouring films, and vanishes for $\epsilon_{ex} = 0.3 = \epsilon_{in}$ the isotropic deformation. This suggests that the flux $j_m$ (meniscus towards the interfaces) is minimal in the surfactant balance around the meniscus.

First, let us have a closer look at the model for the sheared zone and how $j_F$ and $j_m$ intervene in this theoretical description. It was first developed by Bussonnière & Cantat [4] in the frame of the five films elementary foam. We propose here an adaptation with three films.

## 3.2 Theoretical state of the art: the Bussonnière & Cantat model for the viscoelasticity of elementary liquid foams

### 3.2.1 Theoretical description of the flat film, the sheared zone, the dynamic and static menisci

The model developed in [4] will be useful for interpreting my results, and is briefly recalled here for the sake of consistency. As the geometry is different, I have made the necessary minor adaptations.

The model uses the following assumptions :

* The system is invariant by translation in the direction parallel to the free meniscus.
<figure><img src="image_69.png" /></figure>

Figure 3.4: Quantifying the length out extracted film in the stretched film 3 depending on the deformation of its neighbours film 1 and 2. **A:** Motor 3 imposes a fixed global deformation $\epsilon_{in} = 0.3$ to film 3, motor 1 and 2 impose the same global deformations $\epsilon_{ex} \in [-0.5, 0.3]$ to film 1 and 2. $L_{out} = f(t)$ is the length of extracted film seen in the plane of film 3 throughout the deformation. **B:** $L_{out} = f(t)$ for the whole range of $\epsilon_{ex}$. The outgoing quantity of film heavily depends on the deformation imposed to the neighbouring films, and vanishes for $\epsilon_{ex} = 0.3 = \epsilon_{in}$ the isotropic deformation. This suggests that the flux $j_m$ (meniscus towards the interfaces) is minimal in the surfactant balance around the meniscus.

First, let us have a closer look at the model for the sheared zone and how $j_F$ and $j_m$ intervene in this theoretical description. It was first developed by Bussonnière & Cantat [4] in the frame of the five films elementary foam. We propose here an adaptation with three films.

## 3.2 Theoretical state of the art: the Bussonnière & Cantat model for the viscoelasticity of elementary liquid foams

### 3.2.1 Theoretical description of the flat film, the sheared zone, the dynamic and static menisci

The model developed in [4] will be useful for interpreting my results, and is briefly recalled here for the sake of consistency. As the geometry is different, I have made the necessary minor adaptations.

The model uses the following assumptions :

* The system is invariant by translation in the direction parallel to the free meniscus.
* The shear is mainly localised in a region C (defined in Fig.3.5) where the curvature of the interfaces can be neglected.

* The central piece of the films D undergo plug flows imposed by the motors.

* The dynamic menisci B bridge the sheared zone with the meniscus A. The curvature gradient is localised here.

* The system is supposed in steady state at the time scale of the deformation.

* Most of the surfactants are at the interface, meaning we are in a case where $\Gamma \gg ch$ with $h$ the typical thickness of film and $c$ the bulk concentration of surfactants.

The regions and notations are all shown in Fig.3.5 for our main symmetry of deformation where the stagnation point is on the left interface. Note that all minus/plus signs, in/out exponents and inequalities must be inverted if the directions of the imposed velocities are inverted (if we want $u_{1,2}$ to point inward and $u_3$ to point outward). **The main studied deformation has this symmetry and these directions for the imposed velocities and is called “Push 3” hereafter.**

<figure><img src="image_70.png" /><figcaption>Figure 3.5: Symmetry of the imposed deformation ($u_1 = u_2 = -u_3$) and notations for the different zones considered in the model. $\gamma$'s are the surface tensions of the interfaces and $\sigma$'s are the film tensions defined for zones D where the interfaces are symmetrical. Credits to Bussonnière et al. [4].</figcaption></figure>

It has to be noted that compared to what is shown in [4], the main difference between the three films case and five films case is that the latter suppose that film 3 (the central horizontal film) remains at rest with no deformation and with the equilibrium tension $\gamma_0$. This assumption is supported both by theoretical and experimental arguments detailed in [4]. For the three films
* The shear is mainly localised in a region C (defined in Fig.3.5) where the curvature of the interfaces can be neglected.

* The central piece of the films D undergo plug flows imposed by the motors.

* The dynamic menisci B bridge the sheared zone with the meniscus A. The curvature gradient is localised here.

* The system is supposed in steady state at the time scale of the deformation.

* Most of the surfactants are at the interface, meaning we are in a case where $\Gamma \gg ch$ with $h$ the typical thickness of film and $c$ the bulk concentration of surfactants.

The regions and notations are all shown in Fig.3.5 for our main symmetry of deformation where the stagnation point is on the left interface. Note that all minus/plus signs, in/out exponents and inequalities must be inverted if the directions of the imposed velocities are inverted (if we want $u_{1,2}$ to point inward and $u_3$ to point outward). **The main studied deformation has this symmetry and these directions for the imposed velocities and is called “Push 3” hereafter.**

<figure><img src="image_70.png" /><figcaption>Figure 3.5: Symmetry of the imposed deformation ($u_1 = u_2 = -u_3$) and notations for the different zones considered in the model. $\gamma$'s are the surface tensions of the interfaces and $\sigma$'s are the film tensions defined for zones D where the interfaces are symmetrical. Credits to Bussonnière et al. [4].</figcaption></figure>

It has to be noted that compared to what is shown in [4], the main difference between the three films case and five films case is that the latter suppose that film 3 (the central horizontal film) remains at rest with no deformation and with the equilibrium tension $\gamma_0$. This assumption is supported both by theoretical and experimental arguments detailed in [4]. For the three films
case, film 3 is deformed and the tension conditions for the top/bottom interfaces at infinity in zones C is $\gamma_3 < \gamma_0$. It only requires to redefine the reference tension in the model, which is without important consequences.

### 3.2.2 Set of equations

Let us now write the coupled equations in region C. We need to write the conservation of momentum, volume, surfactant, as well as the Gibbs identity coupling surface tension with surfactant surface excesses, and the chemical equilibrium between bulk and surface concentrations of surfactants. The notations for the local frame as well as the boundary conditions in zone C are all shown in Fig.3.6 which is zone C of film 1, with the top and left interfaces.

<figure><img src="image_71.png" /><figcaption>Figure 3.6: Notations for zone C in film 1 (top and left interfaces). $P_0$ is the junction with zone D, $P_m$ with zone B. The local frame is denoted $(s, \zeta)$ and the direction perpendicular to it the direction $z$ along which our system is supposed invariant by translation. $h(s)$ is the thickness of the film, $v(s, \zeta)$ is the flow field, with $u_{t/l}(s)$ denoting its values in $\zeta = \pm \frac{h}{2}$. $c(s, \zeta)$ is the concentration field of surfactants and $\Gamma_{t/l}$ is the surface excess of surfactants at both interfaces. Credits to Bussonnière et al. [4].</figcaption></figure>

Let us start by the lubrication equation in the bulk, in which the pressure gradient is zero as curvature is negligible in zone C. Also, the boundary condition for the velocity field at the interfaces is given by the Marangoni stress so that:

$$ \partial_{\zeta\zeta} v = 0 \qquad (3.2) $$

$$ \partial_s \gamma_{t/l} = \pm \eta \partial_{\zeta} v \Big|_{\pm \frac{h}{2}} \qquad (3.3) $$

Where the $\pm$ sign in front of the shear accounts for the fact that the surfaces are oriented towards $\pm\zeta$ on the top and left interfaces.

Eq.3.2 comes with the direct consequence that the flow field is a simple shear flow of equation:

$$ v(s, \zeta) = \frac{\Delta u^f}{h} \zeta + u_m \qquad (3.4) $$
case, film 3 is deformed and the tension conditions for the top/bottom interfaces at infinity in zones C is $γ_3 < γ_0$. It only requires to redefine the reference tension in the model, which is without important consequences.

### 3.2.2 Set of equations

Let us now write the coupled equations in region C. We need to write the conservation of momentum, volume, surfactant, as well as the Gibbs identity coupling surface tension with surfactant surface excesses, and the chemical equilibrium between bulk and surface concentrations of surfactants. The notations for the local frame as well as the boundary conditions in zone C are all shown in Fig.3.6 which is zone C of film 1, with the top and left interfaces.

<figure><img src="image_71.png" /><figcaption>Figure 3.6: Notations for zone C in film 1 (top and left interfaces). $P_0$ is the junction with zone D, $P_m$ with zone B. The local frame is denoted $(s, ζ)$ and the direction perpendicular to it the direction $z$ along which our system is supposed invariant by translation. $h(s)$ is the thickness of the film, $v(s, ζ)$ is the flow field, with $u_{t/l}(s)$ denoting its values in $ζ = ±\frac{h}{2}$. $c(s, ζ)$ is the concentration field of surfactants and $Γ_{t/l}$ is the surface excess of surfactants at both interfaces. Credits to Bussonnière et al. [4].</figcaption></figure>

Let us start by the lubrication equation in the bulk, in which the pressure gradient is zero as curvature is negligible in zone C. Also, the boundary condition for the velocity field at the interfaces is given by the Marangoni stress so that:

$$ \partial_{\zeta\zeta} v = 0 \qquad (3.2) $$

$$ \partial_s \gamma_{t/l} = \pm \eta \partial_{\zeta} v \Big|_{\pm \frac{h}{2}} \qquad (3.3) $$

Where the $±$ sign in front of the shear accounts for the fact that the surfaces are oriented towards $±ζ$ on the top and left interfaces.

Eq.3.2 comes with the direct consequence that the flow field is a simple shear flow of equation:

$$ v(s, \zeta) = \frac{\Delta u^f}{h} \zeta + u_m \qquad (3.4) $$
Where $\Delta u^f(s) = u_t - u_l$ is the difference of velocities between the interfaces and $u_m(s) = (u_t + u_l)/2$ is the mean velocity across the film. Mass conservation thus writes:

$$Q = \int_{-\frac{h}{2}}^{\frac{h}{2}} v(s, \zeta) d\zeta = u_m h = \text{cst} \qquad (3.5)$$

Which can be recast using the boundary condition at $s \to -\infty$ (towards domain D) where $v(s, \zeta) = u_\infty$ and $h(s) = h_\infty$:

$$u_\infty h_\infty = \frac{h}{2}(u_t + u_l) \qquad (3.6)$$

Now with eq.3.3, we must balance the stress in the simple shear flow with the Marangoni stress, leading to:

$$\partial_s \gamma_t = \eta \frac{\Delta u^f}{h} \quad \text{and} \quad \partial_s \gamma_l = -\partial_s \gamma_t \qquad (3.7)$$

This is where we are able to see the beauty of shearing a foam film: the opposed gradients of surface tension at the two interfaces means a difference of surfactant populations, and surfactant transport from one interface to the other and at the boundary of C is what will completely determine the shear in the end. And thus the viscous dissipation. We write this further using the Marangoni modulus to link $\gamma_{t/l}$ and $\Gamma_{t/l}$:

$$\gamma_{t/l} = \gamma_0 - E_M \frac{\Gamma_{t/l} - \Gamma_0}{\Gamma_0} \qquad (3.8)$$

Where $\Gamma_0$ is the equilibrium surfactant population at rest and $E_M$ the Marangoni modulus of the interface defined in the Introduction 0.1.4 with eq.11. This leads directly to, using eq.3.7:

$$-\frac{E_M}{\Gamma_0} \partial_s \Gamma_t = \eta \frac{\Delta u^f}{h} \quad \text{and} \quad \partial_s \Gamma_l = -\partial_s \Gamma_t \qquad (3.9)$$

**Meaning that the quantity $\Gamma_t + \Gamma_l = 2\Gamma_\infty$ is conserved along $s$ in region C.**

The next set of equations we can write in order to close the system is the diffusion field of surfactants in the film as well as its conservation at the interface. As discussed in subsection 0.2.2, diffusion along $\zeta$ is considered to be instantaneous while diffusion along $s$ is infinitely long. We thus write:

$$\partial_{\zeta\zeta} c = 0 \qquad (3.10)$$

$$\partial_s(\Gamma_{t/l}u_{t/l}) = \mp D \partial_\zeta c|_{\pm\frac{h}{2}} \qquad (3.11)$$

Eq.3.10 yields a linear profile of $c$ in the bulk for which the boundary conditions are given by providing an additional equation for the chemical equilibrium, that is:

$$c(s, \pm\frac{h}{2}) = c_0 + \frac{\Gamma_{t/l} - \Gamma_0}{h\Gamma} \qquad (3.12)$$
Where $\Delta u^f(s) = u_t - u_l$ is the difference of velocities between the interfaces and $u_m(s) = (u_t + u_l)/2$ is the mean velocity across the film. Mass conservation thus writes:

$$Q = \int_{-\frac{h}{2}}^{\frac{h}{2}} v(s, \zeta) d\zeta = u_m h = \text{cst} \qquad (3.5)$$

Which can be recast using the boundary condition at $s \to -\infty$ (towards domain D) where $v(s, \zeta) = u_\infty$ and $h(s) = h_\infty$:

$$u_\infty h_\infty = \frac{h}{2}(u_t + u_l) \qquad (3.6)$$

Now with eq.3.3, we must balance the stress in the simple shear flow with the Marangoni stress, leading to:

$$\partial_s \gamma_t = \eta \frac{\Delta u^f}{h} \quad \text{and} \quad \partial_s \gamma_l = -\partial_s \gamma_t \qquad (3.7)$$

This is where we are able to see the beauty of shearing a foam film: the opposed gradients of surface tension at the two interfaces means a difference of surfactant populations, and surfactant transport from one interface to the other and at the boundary of C is what will completely determine the shear in the end. And thus the viscous dissipation. We write this further using the Marangoni modulus to link $\gamma_{t/l}$ and $\Gamma_{t/l}$:

$$\gamma_{t/l} = \gamma_0 - E_M \frac{\Gamma_{t/l} - \Gamma_0}{\Gamma_0} \qquad (3.8)$$

Where $\Gamma_0$ is the equilibrium surfactant population at rest and $E_M$ the Marangoni modulus of the interface defined in the Introduction 0.1.4 with eq.11. This leads directly to, using eq.3.7:

$$-\frac{E_M}{\Gamma_0} \partial_s \Gamma_t = \eta \frac{\Delta u^f}{h} \quad \text{and} \quad \partial_s \Gamma_l = -\partial_s \Gamma_t \qquad (3.9)$$

**Meaning that the quantity $\Gamma_t + \Gamma_l = 2\Gamma_\infty$ is conserved along $s$ in region C.**

The next set of equations we can write in order to close the system is the diffusion field of surfactants in the film as well as its conservation at the interface. As discussed in subsection 0.2.2, diffusion along $\zeta$ is considered to be instantaneous while diffusion along $s$ is infinitely long. We thus write:

$$\partial_{\zeta\zeta} c = 0 \qquad (3.10)$$

$$\partial_s (\Gamma_{t/l} u_{t/l}) = \mp D \partial_\zeta c \Big|_{\pm \frac{h}{2}} \qquad (3.11)$$

Eq.3.10 yields a linear profile of $c$ in the bulk for which the boundary conditions are given by providing an additional equation for the chemical equilibrium, that is:

$$c(s, \pm \frac{h}{2}) = c_0 + \frac{\Gamma_{t/l} - \Gamma_0}{h\Gamma} \qquad (3.12)$$