This is the simplest model linking c to $\Gamma$, a more refined one was developed in chapter 1 and the physical interpretation of $h_\Gamma$ for SDS/DOH mixtures is discussed in subsection 0.2.3. Solving eq.3.10 with these two boundary conditions allows us to write the full field of bulk concentration:

$$c(s, \zeta) = c_0 + \frac{\Gamma_t + \Gamma_l - 2\Gamma_0}{2h_\Gamma} + \frac{\Gamma_t - \Gamma_l}{hh_\Gamma}\zeta \quad (3.13)$$

Combining this with eq.3.11, we arrive at:

$$\partial_s(\Gamma_t u_t) = -D \frac{\Gamma_t - \Gamma_l}{hh_\Gamma} \quad \text{and} \quad \partial_s(\Gamma_l u_l) = -\partial_s(\Gamma_t u_t) \quad (3.14)$$

**Meaning that the quantity $\Gamma_t u_t + \Gamma_l u_l$ is also conserved along $s$**.

In the end, the conservation laws of momentum and surfactants can be written using the boundary condition at $s \to -\infty$ (zone D) where we have a plug flow between identical interfaces, leading to:

$$\Gamma_t + \Gamma_l = 2\Gamma_\infty \quad (3.15)$$

$$\Gamma_t u_t + \Gamma_l u_l = 2\Gamma_\infty u_\infty \quad (3.16)$$

Which allows us to substitute $\Gamma_l$ and $u_l$ in the other equations, finally yielding a set of four equations bearing on $\Gamma_t$, $u_t$, $\Delta u^f$ and $h$:

$$\partial_s \Gamma_t = - \frac{\eta \Gamma_0 \Delta u^f}{E_M h} \quad (3.17)$$

$$\partial_s(\Gamma_t u_t) = -2D \frac{\Gamma_t - \Gamma_\infty}{hh_\Gamma} \quad (3.18)$$

$$\Delta u^f = (u_t - u_\infty) \frac{2\Gamma_\infty}{2\Gamma_\infty - \Gamma_t} \quad (3.19)$$

$$h = \frac{h_\infty u_\infty (2\Gamma_\infty - \Gamma_t)}{\Gamma_\infty u_\infty + u_t (\Gamma_\infty - \Gamma_t)} \quad (3.20)$$

### 3.2.3 Boundary condition on the meniscus side $s \to s_m$: the surfactant flux $j_m$ as an unknown parameter

In order to close the problem, we need to provide for boundary conditions on the meniscus side of the problem. To do so, we first need to define properly the curvilinear abscissa $s_m$ of point $P_m$ at which these conditions are applied.

This abscissa does not correspond to the point $P_s$ of Fig.3.5, as the vanishing Laplace pressure condition of region C becomes invalid in region B, the dynamic meniscus. $P_m$ defines the frontier between C and B where we suppose that the tension of the top free interface reaches the tension of film 3: $\gamma_t(s_m) = \gamma_3 + \Delta\gamma^{out} + \Delta\gamma^{in}$ besides two additional terms $\Delta\gamma^{out/in}$ corresponding to the tension jump due to the dynamic menisci (cf. Fig.3.5). These two terms have been computed in 0.3.2, and an estimate in [4] allows us to neglect them, leading to the final definition of $P_m$, or abscissa $s_m$ so that: $\gamma_t(s_m) = \gamma_3$. This can be equivalently rewritten as:
This is the simplest model linking c to $\Gamma$, a more refined one was developed in chapter 1 and the physical interpretation of $h_\Gamma$ for SDS/DOH mixtures is discussed in subsection 0.2.3. Solving eq.3.10 with these two boundary conditions allows us to write the full field of bulk concentration:

$$c(s, \zeta) = c_0 + \frac{\Gamma_t + \Gamma_l - 2\Gamma_0}{2h_\Gamma} + \frac{\Gamma_t - \Gamma_l}{hh_\Gamma}\zeta \quad (3.13)$$

Combining this with eq.3.11, we arrive at:

$$\partial_s(\Gamma_t u_t) = -D \frac{\Gamma_t - \Gamma_l}{hh_\Gamma} \quad \text{and} \quad \partial_s(\Gamma_l u_l) = -\partial_s(\Gamma_t u_t) \quad (3.14)$$

**Meaning that the quantity $\Gamma_t u_t + \Gamma_l u_l$ is also conserved along s.**

In the end, the conservation laws of momentum and surfactants can be written using the boundary condition at $s \to -\infty$ (zone D) where we have a plug flow between identical interfaces, leading to:

$$\Gamma_t + \Gamma_l = 2\Gamma_\infty \quad (3.15)$$

$$\Gamma_t u_t + \Gamma_l u_l = 2\Gamma_\infty u_\infty \quad (3.16)$$

Which allows us to substitute $\Gamma_l$ and $u_l$ in the other equations, finally yielding a set of four equations bearing on $\Gamma_t$, $u_t$, $\Delta u^f$ and $h$:

$$\partial_s \Gamma_t = - \frac{\eta \Gamma_0 \Delta u^f}{E_M h} \quad (3.17)$$

$$\partial_s(\Gamma_t u_t) = -2D \frac{\Gamma_t - \Gamma_\infty}{hh_\Gamma} \quad (3.18)$$

$$\Delta u^f = (u_t - u_\infty) \frac{2\Gamma_\infty}{2\Gamma_\infty - \Gamma_t} \quad (3.19)$$

$$h = \frac{h_\infty u_\infty (2\Gamma_\infty - \Gamma_t)}{\Gamma_\infty u_\infty + u_t (\Gamma_\infty - \Gamma_t)} \quad (3.20)$$

### 3.2.3 Boundary condition on the meniscus side $s \to s_m$: the surfactant flux $j_m$ as an unknown parameter

In order to close the problem, we need to provide for boundary conditions on the meniscus side of the problem. To do so, we first need to define properly the curvilinear abscissa $s_m$ of point $P_m$ at which these conditions are applied.

This abscissa does not correspond to the point $P_s$ of Fig.3.5, as the vanishing Laplace pressure condition of region C becomes invalid in region B, the dynamic meniscus. $P_m$ defines the frontier between C and B where we suppose that the tension of the top free interface reaches the tension of film 3: $\gamma_t(s_m) = \gamma_3 + \Delta\gamma^{out} + \Delta\gamma^{in}$ besides two additional terms $\Delta\gamma^{out/in}$ corresponding to the tension jump due to the dynamic menisci (cf. Fig.3.5). These two terms have been computed in 0.3.2, and an estimate in [4] allows us to neglect them, leading to the final definition of $P_m$, or abscissa $s_m$ so that: $\gamma_t(s_m) = \gamma_3$. This can be equivalently rewritten as:
$$ \Gamma_t(s_m) = \Gamma_3 \qquad (3.21) $$

This definition itself is not a sufficient condition to close the problem however, and we also need to provide a condition for the left interface in $s_m$. As beyond $s_m$ the top interface is considered to be at its final value of surface excess of surfactant $\Gamma_3$, the surfactant flux towards the left interface necessarily comes from the meniscus and not the top interface. This takes the form:

$$ \Gamma_l(s_m)u_l(s_m) + j_m = 0 \qquad (3.22) $$

Where $j_m$ is the ad hoc flux from the meniscus bulk to the left interface between $P_m$ and $P_s$ (introduced qualitatively in subsection 3.1.2). A phenomenological description of $j_m$ is developed in [4], by writing it as :

$$ j_m = -U_m(\Gamma_l(s_m) - \Gamma_0) \qquad (3.23) $$

where $U_m = r_m/\tau$ is a typical transport velocity with $r_m$ the meniscus curvature and $\tau$ a typical time of adsorption of surfactants. This time can be estimated with a blasius layer of diffusion of surfactant in the meniscus, knowing the typical interface velocities and diffusion coefficient $D$. All these details are given in [4], but will not be of particular importance for the following of this thesis work.

### 3.2.4 Numerical resolution of the system

The system made of eq.3.17-3.20 along with the boundary conditions eq.3.15, 3.16, 3.21 and 3.22 can eventually be solved numerically. **The control parameter here is the surface excess $\Gamma_\infty$ and the sought quantity is $u_\infty$**.

Fig.3.7 shows the resolution of the system. Blue colours stand for the “Push 3” case we are mostly interested in, where $u_\infty < 0$ (represented in Fig.3.5 and 3.6), orange colours stand for the other case where all velocities in films 1, 2 and 3 are of the opposite sign (same symmetry for the stopping point).

Velocities $U_c = E/\eta$ and $U_d = D/h_\Gamma$ are respectively a capillary and a diffusion velocity defining a normalising velocity $\sqrt{U_cU_d}$. The solid lines stand for the case where $j_m$ is neglected. The dotted lines stand for the case where $j_m$ is non-zero with $U_m$ imposed at: $U_m \sim \sqrt{U_cU_d}$. All curves end in $s = s_m$ where the condition 3.21 is met. **Finding $s_m$ also leads, after redimentionalisation, to an estimate for the length $\ell$ of zone C, and [4] yields $\ell \sim 100$ µm.**

First thing to notice for the vanishing $j_m$ case (solid lines) is that $u_l$ vanishes in $s = s_m$, meaning that we do not have a stopping point in $P_s$ but rather a stagnant cap from $P_m$ to $P_s$. This is expected: without $j_m$, the flux $\Gamma_l u_l(s_m)$ is zero: if not for $\Gamma_l$, then the interface velocity $u_l(s_m)$ must vanish.

Much less intuitive is the fate of $h$ close to the meniscus: it thins out if film 1 is compressed and it thickens if it is stretched! This is an indirect consequence of surfactant fluxes imposing an extension to the mean flux $u_m = (u_t + u_l)/2$ in zone C with a sign opposite to that of zone D.
$$ \Gamma_t(s_m) = \Gamma_3 \qquad (3.21) $$

This definition itself is not a sufficient condition to close the problem however, and we also need to provide a condition for the left interface in $s_m$. As beyond $s_m$ the top interface is considered to be at its final value of surface excess of surfactant $\Gamma_3$, the surfactant flux towards the left interface necessarily comes from the meniscus and not the top interface. This takes the form:

$$ \Gamma_l(s_m)u_l(s_m) + j_m = 0 \qquad (3.22) $$

Where $j_m$ is the ad hoc flux from the meniscus bulk to the left interface between $P_m$ and $P_s$ (introduced qualitatively in subsection 3.1.2). A phenomenological description of $j_m$ is developed in [4], by writing it as :

$$ j_m = -U_m(\Gamma_l(s_m) - \Gamma_0) \qquad (3.23) $$

where $U_m = r_m/\tau$ is a typical transport velocity with $r_m$ the meniscus curvature and $\tau$ a typical time of adsorption of surfactants. This time can be estimated with a blasius layer of diffusion of surfactant in the meniscus, knowing the typical interface velocities and diffusion coefficient $D$. All these details are given in [4], but will not be of particular importance for the following of this thesis work.

### 3.2.4 Numerical resolution of the system

The system made of eq.3.17-3.20 along with the boundary conditions eq.3.15, 3.16, 3.21 and 3.22 can eventually be solved numerically. **The control parameter here is the surface excess $\Gamma_\infty$ and the sought quantity is $u_\infty$**.

Fig.3.7 shows the resolution of the system. Blue colours stand for the “Push 3” case we are mostly interested in, where $u_\infty < 0$ (represented in Fig.3.5 and 3.6), orange colours stand for the other case where all velocities in films 1, 2 and 3 are of the opposite sign (same symmetry for the stopping point).

Velocities $U_c = E/\eta$ and $U_d = D/h_\Gamma$ are respectively a capillary and a diffusion velocity defining a normalising velocity $\sqrt{U_cU_d}$. The solid lines stand for the case where $j_m$ is neglected. The dotted lines stand for the case where $j_m$ is non-zero with $U_m$ imposed at: $U_m \sim \sqrt{U_cU_d}$. All curves end in $s = s_m$ where the condition 3.21 is met. **Finding $s_m$ also leads, after redimentionalisation, to an estimate for the length $\ell$ of zone C, and [4] yields $\ell \sim 100 \mu m$**.

First thing to notice for the vanishing $j_m$ case (solid lines) is that $u_l$ vanishes in $s = s_m$, meaning that we do not have a stopping point in $P_s$ but rather a stagnant cap from $P_m$ to $P_s$. This is expected: without $j_m$, the flux $\Gamma_l u_l(s_m)$ is zero: if not for $\Gamma_l$, then the interface velocity $u_l(s_m)$ must vanish.

Much less intuitive is the fate of $h$ close to the meniscus: it thins out if film 1 is compressed and it thickens if it is stretched! This is an indirect consequence of surfactant fluxes imposing an extension to the mean flux $u_m = (u_t + u_l)/2$ in zone C with a sign opposite to that of zone D.
<figure><img src="image_72.png" /><figcaption>Figure 3.7: Numerical resolutions for: (a) the surface concentrations of surfactants, (b) the interface velocity fields and (c) the thickness profile. Credits to Bussonnière et al. [4].</figcaption></figure>

In the end, the numerical resolution gives a relationship $u_\infty = f(\Gamma_\infty)$ with $\Gamma_\infty$ being the control parameter in the system. This relationship is rather presented in the form of graph $\Delta\sigma = f(U)$ where $\Delta\sigma$ is the difference of tension between film 1 and 3 (directly linked to the difference $\Gamma_\infty - \Gamma_3$ through the Marangoni modulus) and $U = u_t(s_m)$ is found by solving the system and finding $u_\infty$. We show the results for the case of negative $u_\infty$ (stretching film 1) in Fig.3.8. The leftmost blue curve corresponds to vanishing $U_m$ (thus vanishing $j_m$), the rightmost red one to $U_m = 1000\sqrt{U_cU_d}$.

<figure><img src="image_73.png" /><figcaption>Figure 3.8: Numerical resolution of the differences of tension $\Delta\sigma$ as a function of the interface velocity $U = u_t(s_m)$. The colours are for different values of $U_m$ from (blue) $U_m \to 0$, no exchanges with the meniscus, to (red) $U_m = 1000\sqrt{U_cU_d}$ dominating exchanges with the meniscus. Credits to Bussonnière et al. [4].</figcaption></figure>

These stress-strain curves $\Delta\sigma = f(U)$ characterise the rheological response of the elementary liquid foam. Thus, finding the value for the numerical free parameter $U_m$ (and the flux $j_m$ behind it) is the key to predict them.
<figure><img src="image_72.png" /><figcaption>Figure 3.7: Numerical resolutions for: (a) the surface concentrations of surfactants, (b) the interface velocity fields and (c) the thickness profile. Credits to Bussonnière et al. [4].</figcaption></figure>

In the end, the numerical resolution gives a relationship $u_\infty = f(\Gamma_\infty)$ with $\Gamma_\infty$ being the control parameter in the system. This relationship is rather presented in the form of graph $\Delta\sigma = f(U)$ where $\Delta\sigma$ is the difference of tension between film 1 and 3 (directly linked to the difference $\Gamma_\infty - \Gamma_3$ through the Marangoni modulus) and $U = u_t(s_m)$ is found by solving the system and finding $u_\infty$. We show the results for the case of negative $u_\infty$ (stretching film 1) in Fig.3.8. The leftmost blue curve corresponds to vanishing $U_m$ (thus vanishing $j_m$), the rightmost red one to $U_m = 1000\sqrt{U_cU_d}$.

<figure><img src="image_73.png" /><figcaption>Figure 3.8: Numerical resolution of the differences of tension $\Delta\sigma$ as a function of the interface velocity $U = u_t(s_m)$. The colours are for different values of $U_m$ from (blue) $U_m \to 0$, no exchanges with the meniscus, to (red) $U_m = 1000\sqrt{U_cU_d}$ dominating exchanges with the meniscus. Credits to Bussonnière et al. [4].</figcaption></figure>

These stress-strain curves $\Delta\sigma = f(U)$ characterise the rheological response of the elementary liquid foam. Thus, finding the value for the numerical free parameter $U_m$ (and the flux $j_m$ behind it) is the key to predict them.
## 3.3 Experimental challenge: measuring $j_m$

Knowing the value of $j_m$ is thus the experimental challenge we face in the frame of this thesis work. To investigate this, we now need to relate this quantity to experimentally measurable quantities. The surfactant exchanges take the form of creation (resp. destruction) of interfaces in the plane of the stretched film 1 (resp. compressed film 3). Below, we relate the amount of surfactants leaving and entering the films to measurable lengths in these films.

<figure><img src="image_74.png" /><figcaption>Figure 3.9: Schematic view of the time evolution of the elementary liquid foam. The dots are material points attached to the top interface. As film 3 is compressed with a motor velocity $-V_m$, it gives up a certain length $\hat{L}_{in}$ of interface, whereas film 1 (stretched with motor 1 at $+V_m$) accepts a certain length $\hat{L}_{out}$ in its plane.</figcaption></figure>

The two lengths of interface we are interested in in the frame of this balance are $\hat{L}_{out}(t)$ and $\hat{L}_{in}(t)$ (mm) which are represented in Fig. 3.9 during a “Push 3” experiment where the hats indicate the actual lengths we measure. Note that $\hat{L}_{out}$ is directly measurable in the stretched films as it is the length of a well defined piece of film. We define $\hat{L}_{in}$ in the same way, but it has no physical existence as it leaves the compressed film. It is actually the length that would have the piece of film which disappeared if it was still in the film. They are time dependent quantities as the deformation takes place and respectively carry the quantities $N_{out}(t)$ and $N_{in}(t)$ (mol) of surfactants which write:

$$N_{out}(t) = 2\Gamma_{1}(t) \times \hat{L}_{out}(t) \qquad (3.24)$$

$$N_{in}(t) = 2\Gamma_{3}(t) \times \hat{L}_{in}(t) \qquad (3.25)$$

Where $\Gamma_1(t)$ and $\Gamma_3(t)$ are the surface concentrations of surfactants as defined earlier ($\Gamma_1$ corresponds to the $\Gamma_\infty$ of the model). Here, they are time dependent variables, as the central
## 3.3 Experimental challenge: measuring $j_m$

Knowing the value of $j_m$ is thus the experimental challenge we face in the frame of this thesis work. To investigate this, we now need to relate this quantity to experimentally measurable quantities. The surfactant exchanges take the form of creation (resp. destruction) of interfaces in the plane of the stretched film 1 (resp. compressed film 3). Below, we relate the amount of surfactants leaving and entering the films to measurable lengths in these films.

<figure><img src="image_74.png" /><figcaption>Figure 3.9: Schematic view of the time evolution of the elementary liquid foam. The dots are material points attached to the top interface. As film 3 is compressed with a motor velocity $-V_m$, it gives up a certain length $\hat{L}_{in}$ of interface, whereas film 1 (stretched with motor 1 at $+V_m$) accepts a certain length $\hat{L}_{out}$ in its plane.</figcaption></figure>

The two lengths of interface we are interested in in the frame of this balance are $\hat{L}_{out}(t)$ and $\hat{L}_{in}(t)$ (mm) which are represented in Fig. 3.9 during a “Push 3” experiment where the hats indicate the actual lengths we measure. Note that $\hat{L}_{out}$ is directly measurable in the stretched films as it is the length of a well defined piece of film. We define $\hat{L}_{in}$ in the same way, but it has no physical existence as it leaves the compressed film. It is actually the length that would have the piece of film which disappeared if it was still in the film. They are time dependent quantities as the deformation takes place and respectively carry the quantities $N_{out}(t)$ and $N_{in}(t)$ (mol) of surfactants which write:

$$N_{out}(t) = 2\Gamma_{1}(t) \times \hat{L}_{out}(t) \qquad (3.24)$$

$$N_{in}(t) = 2\Gamma_{3}(t) \times \hat{L}_{in}(t) \qquad (3.25)$$

Where $\Gamma_1(t)$ and $\Gamma_3(t)$ are the surface concentrations of surfactants as defined earlier ($\Gamma_1$ corresponds to the $\Gamma_\infty$ of the model). Here, they are time dependent variables, as the central
region (region D in the model) of film 1 and 3 undergo stretching/compression during the deformation. This last point rises a subtlety: $N_{out}(t)$ and $N_{in}(t)$ are not directly proportional to the lengths of interface $\hat{L}_{out}(t)$ and $\hat{L}_{in}(t)$ as time goes by. To solve this problem, we must relate $\Gamma_1(t)$ and $\Gamma_3(t)$ to quantities we can measure experimentally. This is doable by looking back at eq.1.17 from chapter 1 section 1.3, where we define a film elasticity $E_f$, which can be related to the definition of the Marangoni modulus eq.11:

$$ \sigma^f = \sigma_0 + E_f \frac{\epsilon}{1 + \epsilon} = \sigma_0 - 2E_M \frac{\Gamma - \Gamma_0}{\Gamma_0} \quad (3.27) $$

Where we suppose that $E_f = 2E_M$ in presence of poorly soluble surfactants such as dodecanol which dominates the elastic response, as discussed in 1.2.2. This allows us to write directly for both films, 1 and 3:

$$ \Gamma = \frac{\Gamma_0}{1 + \epsilon} \quad (3.28) $$

Thus, the lengths of interface at rest we will have to compare are $L_{out}(t)$ and $L_{in}(t)$ (mm) which are the actual lengths of films corrected by the deformation factor of the film:

$$ L_{out}(t) = \frac{\hat{L}_{out}(t)}{1 + \epsilon_1(t)} = \frac{N_{out}}{2\Gamma_0} \quad (3.29) $$

$$ L_{in}(t) = \frac{\hat{L}_{in}(t)}{1 + \epsilon_3(t)} = \frac{N_{in}}{2\Gamma_0} \quad (3.30) $$

(3.31)

Which are the relevant quantities to measure. **This is experimentally demanding as we need to simultaneously measure the lengths of exchanged films as well as their states of extension/compression.**

On the other hand, we need now to relate the quantities of exchanged surfactants $N_{out}(t)$ and $N_{in}(t)$ to $j_m$. Let us define the time $t = t_1$ when the deformation starts, and where we impose constant and equal velocities $\pm V_m$ at the ends of film 1 and 3. Two precautions: first the motor velocities $\pm V_m$ are not directly the velocities $u_1 = u_\infty$ and $u_3 = u_t(s_m)$ as the zone D undergoes extension (cf. chapter 1 where we study the elastic contribution of zone D). Second, we consider that the velocities $u_i$ are uncontrolled time-dependent variables imposing $u_\infty(t)$ and $u_t(s_m, t)$ in our model, and each time $t$ is instantly at the relevant steady state given the imposed velocities. In this frame, the quantities $N_{out}(t)$ and $N_{in}(t)$ as we defined them can be written as time integrals:

$$ N_{out}(t) = - \int_{t_1}^{t} \left[ \Gamma_t u_t(s_m, t') + \Gamma_l u_l(s_m, t') \right] dt' \quad (3.32) $$

$$ N_{in}(t) = -2 \int_{t_1}^{t} \Gamma_t u_t(s_m, t') dt' \quad (3.33) $$

Where the factor 2 for $N_{in}$ comes from the symmetry of "Push 3" imposing identical top and bottom interfaces and where we identify the second term of the right hand-side of $N_{out}$ using eq.3.22: $\Gamma_l u_l(s_m, t') = -j_m(t')$. This leads to the interface balance:
region (region D in the model) of film 1 and 3 undergo stretching/compression during the deformation. This last point rises a subtlety: $N_{out}(t)$ and $N_{in}(t)$ are not directly proportional to the lengths of interface $\hat{L}_{out}(t)$ and $\hat{L}_{in}(t)$ as time goes by. To solve this problem, we must relate $\Gamma_1(t)$ and $\Gamma_3(t)$ to quantities we can measure experimentally. This is doable by looking back at eq.1.17 from chapter 1 section 1.3, where we define a film elasticity $E_f$, which can be related to the definition of the Marangoni modulus eq.11:

$$ \sigma^f = \sigma_0 + E_f \frac{\epsilon}{1 + \epsilon} = \sigma_0 - 2E_M \frac{\Gamma - \Gamma_0}{\Gamma_0} \quad (3.27) $$

Where we suppose that $E_f = 2E_M$ in presence of poorly soluble surfactants such as dodecanol which dominates the elastic response, as discussed in 1.2.2. This allows us to write directly for both films, 1 and 3:

$$ \Gamma = \frac{\Gamma_0}{1 + \epsilon} \quad (3.28) $$

Thus, the lengths of interface at rest we will have to compare are $L_{out}(t)$ and $L_{in}(t)$ (mm) which are the actual lengths of films corrected by the deformation factor of the film:

$$ L_{out}(t) = \frac{\hat{L}_{out}(t)}{1 + \epsilon_1(t)} = \frac{N_{out}}{2\Gamma_0} \quad (3.29) $$

$$ L_{in}(t) = \frac{\hat{L}_{in}(t)}{1 + \epsilon_3(t)} = \frac{N_{in}}{2\Gamma_0} \quad (3.30) $$

(3.31)

Which are the relevant quantities to measure. **This is experimentally demanding as we need to simultaneously measure the lengths of exchanged films as well as their states of extension/compression.**

On the other hand, we need now to relate the quantities of exchanged surfactants $N_{out}(t)$ and $N_{in}(t)$ to $j_m$. Let us define the time $t = t_1$ when the deformation starts, and where we impose constant and equal velocities $\pm V_m$ at the ends of film 1 and 3. Two precautions: first the motor velocities $\pm V_m$ are not directly the velocities $u_1 = u_\infty$ and $u_3 = u_t(s_m)$ as the zone D undergoes extension (cf. chapter 1 where we study the elastic contribution of zone D). Second, we consider that the velocities $u_i$ are uncontrolled time-dependent variables imposing $u_\infty(t)$ and $u_t(s_m, t)$ in our model, and each time $t$ is instantly at the relevant steady state given the imposed velocities. In this frame, the quantities $N_{out}(t)$ and $N_{in}(t)$ as we defined them can be written as time integrals:

$$ N_{out}(t) = - \int_{t_1}^{t} \left[ \Gamma_t u_t(s_m, t') + \Gamma_l u_l(s_m, t') \right] dt' \quad (3.32) $$

$$ N_{in}(t) = -2 \int_{t_1}^{t} \Gamma_t u_t(s_m, t') dt' \quad (3.33) $$

Where the factor 2 for $N_{in}$ comes from the symmetry of "Push 3" imposing identical top and bottom interfaces and where we identify the second term of the right hand-side of $N_{out}$ using eq.3.22: $\Gamma_l u_l(s_m, t') = -j_m(t')$. This leads to the interface balance:
$$N_{out}(t) = \frac{1}{2}N_{in}(t) + \int_{t_1}^{t} j_{m}(t')dt' \qquad (3.34)$$

Which is directly:

$$L_{out}(t) = \frac{1}{2}L_{in}(t) + \frac{1}{2\Gamma_0}\int_{t_1}^{t} j_{m}(t')dt' \qquad (3.35)$$

This is the key equation to solve our experimental conundrum: quantifying $L_{out}$ and $L_{in}$ allows us to deduce directly $j_m$ and the quantities of surfactant exchanged with the meniscus during the deformation. Besides, two limit cases for these film-meniscus exchanges can be envisioned. Fig.3.10 gives a schematic view of both of them.

<figure><img src="image_75.png" /><figcaption>Figure 3.10: Two extreme cases if: **A** The flux of surfactants from the meniscus to the left interface is zero $j_m = 0$, then we need to create 4 $L_{out}$'s out of 2 $L_{in}$'s, and $L_{out} = \frac{1}{2}L_{in}$; **B** The flux of surfactants from the top/bottom interfaces to the left interface vanishes near the meniscus $j_F(s = s_m) = 0$, then the meniscus provides 2 $L_{out}$'s and we have $L_{out} = L_{in}$.</figcaption></figure>

### Vanishing flux from the meniscus $j_m = 0$

In the case where the meniscus does not provide any surfactant to the frustrated interface, that is $j_m = 0$ (Fig.3.10A), the only ingoing surfactant flux for film 1 comes from the top interface coming from film 3, and we directly have through eq.3.35:

$$L_{out}(t) = \frac{1}{2}L_{in}(t) \qquad (3.36)$$

The simple and intuitive way to interpret this result is to see that if not for the meniscus, we need to create 4 interfaces $L_{out}$ out of 2 interfaces $L_{in}$, hence the factor 1/2.

### Large flux $j_m$

If on the contrary we allow the meniscus to provide as much surfactants as possible (meaning the factor $U_m$ in the model is as large as we want in eq.3.23), the model eq.3.17-3.20 leads to:
$$N_{out}(t) = \frac{1}{2}N_{in}(t) + \int_{t_1}^{t} j_{m}(t')dt' \qquad (3.34)$$

Which is directly:

$$L_{out}(t) = \frac{1}{2}L_{in}(t) + \frac{1}{2\Gamma_0}\int_{t_1}^{t} j_{m}(t')dt' \qquad (3.35)$$

This is the key equation to solve our experimental conundrum: quantifying $L_{out}$ and $L_{in}$ allows us to deduce directly $j_{m}$ and the quantities of surfactant exchanged with the meniscus during the deformation. Besides, two limit cases for these film-meniscus exchanges can be envisioned. Fig.3.10 gives a schematic view of both of them.

<figure><img src="image_75.png" /><figcaption>Figure 3.10: Two extreme cases if: **A** The flux of surfactants from the meniscus to the left interface is zero $j_m = 0$, then we need to create 4 $L_{out}$'s out of 2 $L_{in}$'s, and $L_{out} = \frac{1}{2}L_{in}$; **B** The flux of surfactants from the top/bottom interfaces to the left interface vanishes near the meniscus $j_F(s = s_m) = 0$, then the meniscus provides 2 $L_{out}$'s and we have $L_{out} = L_{in}$.</figcaption></figure>

### Vanishing flux from the meniscus $j_m = 0$

In the case where the meniscus does not provide any surfactant to the frustrated interface, that is $j_m = 0$ (Fig.3.10A), the only ingoing surfactant flux for film 1 comes from the top interface coming from film 3, and we directly have through eq.3.35:

$$L_{out}(t) = \frac{1}{2}L_{in}(t) \qquad (3.36)$$

The simple and intuitive way to interpret this result is to see that if not for the meniscus, we need to create 4 interfaces $L_{out}$ out of 2 interfaces $L_{in}$, hence the factor 1/2.

### Large flux $j_m$

If on the contrary we allow the meniscus to provide as much surfactants as possible (meaning the factor $U_m$ in the model is as large as we want in eq.3.23), the model eq.3.17-3.20 leads to:
$$L_{out}(t) = L_{in}(t) \qquad (3.37)$$

Again, the intuitive interpretation is to see that at best, the meniscus provides for as many surfactants as the ingoing interface $L_{in}$ in order to allow the creation of the interface $L_{out}$ on the frustrated side. Providing more than that (i.e. $L_{out} > L_{in}$) is unphysical as the frustrated interface is slower than the free one $|u_{l}(s_{m})| \le |u_{t}(s_{m})|$ and its surfactant population lower or equal $\Gamma_{l}(s_{m}) \le \Gamma_{t}(s_{m})$.

The next sections 3.4-3.5 are about describing the setup and experimental protocols we used to quantify the exchanges of surfactants, and how we simultaneously monitor the rheological response of the system under deformation. The sections after, 3.6-3.8, detail all the data processing and necessary precautions associated with the quantification of $L_{out}$ and $L_{in}$. **Section 3.9 shows and discusses the results.**

## 3.4 Experimental setup and notations

As the state of the art of elementary liquid foams suggests, the number of coupling between the physical ingredients at play in our system is high and the geometry already complex at this scale. Although being the most elementary foam we can think of, three films connected by a meniscus, this system shows all the couplings we can dream of!

In that respect, collecting a lot of different data in a very well controlled and reliable way is a necessity if we want to better understand these couplings. The price we paid for that is a strong automation of our experimental setup, and a lot of different cameras around it, which needed to be synchronised very accurately (at the µs scale). Also, space management was the key... both in terms of bulkiness of the different elements and in terms of lighting. The experimental set-up allows us to measure the meniscus size and position, the film thicknesses and the velocity fields in the films.

And thus what we reaped has proved to be highly reproducible. This is even surprising, because some phenomena such as marginal regeneration seem to be quite turbulent and non-reproducible. Despite all that, the measurements of deformations, deformation rates, and film tensions in particular, have a good reproducibility for a given chemistry.

### 3.4.1 Chemistries

In the continuity of the experimental campaigns led with the five films rheometer in chapter 1 and [6], we use SDS/dodecanol mixtures to create our three films elementary foam. Thus, we will know what to expect when it will come to measure the elasticities of our films. Note however that contrary to this campaign we removed the glycerol and work with water-based solutions only. Here are the characteristics of the solutions we use followed by the protocol:

* 1 L of demineralised water (with a resistivity as a proxy for its purity of 18 MΩ·cm)

* A concentration of SDS $c_{SDS} = 5.6 g/L = 19.4 mmol/L = 2.4 CMC$
$$L_{out}(t) = L_{in}(t) \qquad (3.37)$$

Again, the intuitive interpretation is to see that at best, the meniscus provides for as many surfactants as the ingoing interface $L_{in}$ in order to allow the creation of the interface $L_{out}$ on the frustrated side. Providing more than that (i.e. $L_{out} > L_{in}$) is unphysical as the frustrated interface is slower than the free one $|u_{l}(s_{m})| \le |u_{t}(s_{m})|$ and its surfactant population lower or equal $\Gamma_{l}(s_{m}) \le \Gamma_{t}(s_{m})$.

The next sections 3.4-3.5 are about describing the setup and experimental protocols we used to quantify the exchanges of surfactants, and how we simultaneously monitor the rheological response of the system under deformation. The sections after, 3.6-3.8, detail all the data processing and necessary precautions associated with the quantification of $L_{out}$ and $L_{in}$. **Section 3.9 shows and discusses the results.**

## 3.4 Experimental setup and notations

As the state of the art of elementary liquid foams suggests, the number of coupling between the physical ingredients at play in our system is high and the geometry already complex at this scale. Although being the most elementary foam we can think of, three films connected by a meniscus, this system shows all the couplings we can dream of!

In that respect, collecting a lot of different data in a very well controlled and reliable way is a necessity if we want to better understand these couplings. The price we paid for that is a strong automation of our experimental setup, and a lot of different cameras around it, which needed to be synchronised very accurately (at the µs scale). Also, space management was the key... both in terms of bulkiness of the different elements and in terms of lighting. The experimental set-up allows us to measure the meniscus size and position, the film thicknesses and the velocity fields in the films.

And thus what we reaped has proved to be highly reproducible. This is even surprising, because some phenomena such as marginal regeneration seem to be quite turbulent and non-reproducible. Despite all that, the measurements of deformations, deformation rates, and film tensions in particular, have a good reproducibility for a given chemistry.

### 3.4.1 Chemistries

In the continuity of the experimental campaigns led with the five films rheometer in chapter 1 and [6], we use SDS/dodecanol mixtures to create our three films elementary foam. Thus, we will know what to expect when it will come to measure the elasticities of our films. Note however that contrary to this campaign we removed the glycerol and work with water-based solutions only. Here are the characteristics of the solutions we use followed by the protocol:

* 1 L of demineralised water (with a resistivity as a proxy for its purity of 18 MΩ·cm)

* A concentration of SDS $c_{SDS} = 5.6 g/L = 19.4 mmol/L = 2.4 CMC$
* A concentration of dodecanol which varies from no dodecanol to $c_{DOH}$ = 50 mg/L = 0.27 mmol/L

* A concentration of fluorescein $c_{fluo}$ = 0.8 g/L = 2.4 mmol/L

We weight and put all the solid parts of the recipe ($m_{SDS}$ = 5.6 g, $m_{DOH}$ = {0, 5, 15, 35, 50}mg and $m_{fluo}$ = 0.8 g) in a bottle thoroughly cleaned with one rinse with acetone followed by several rinse with demineralised water of the same quality as used in the next step. Dodecanol melting at 23 - 24°C, it is cooled down in a fridge in advance if need be to allow a precise weighing (which didn't happen many times thanks to one other parameter being living in Brittany). We then take one litre of demineralised water using a volumetric flask cleaned with the same protocol as the bottle. The litre of water is then carefully added to the bottle containing the solids by ensuring the water is flowing gently along the walls of the bottle to avoid excessive foaming, which could lead to an inhomogeneous mixing by entrapping dodecanol crystals. We add a magnetic rod to the bottle and put it on a heating magnetic stirrer at 50°C for 2 h. The bottle is stored in a dry place at room temperature. Mixtures of SDS/dodecanol being subject to aging in a matter of days (even hours for refined physico-chemistry), the timing which is roughly respected is a solution made between 2PM and 4PM used in the experiment the next day morning at around 10AM.

**NB:** to cope with the aging problem, we tried different chemistries knowing that we need to fill a 1 L tank of solution and allow the creation of centimetric foam films lasting tens of seconds. The use of negatively charged fluorescein (which is conditioned by our cyan laser) prohibits the use of cationic surfactants such as the TABs surfactants, as they interact with fluorescein and hinder the fluorescence. Non-ionic surfactants were ruled out because of the foaming problem, and no non-ageing anionic surfactant was found at a reasonable cost given the rather large quantities we need. Thus, we kept on using SDS.

### 3.4.2 Notations for the 3-films elementary foam

The notations for the axis of the main frame $(x, y, z)$ are shown in Fig.3.11, with $x$ being perpendicular to the free meniscus, parallel to the plane of film 3 and oriented towards motor 3. The system is supposed invariant by translation along $z$, the direction parallel to the free meniscus (this invariance is discussed later in subsection 3.7).

Also, we will need to define local frames $(x_i, y_i)$ for film 1 and 2, with in each case $x_i$ oriented towards motor $i$. All these notations are shown in Fig.3.11A.

Eventually, we need some notations for the positions, angles and lengths in the system, as the motors and the free meniscus all move during the deformations we impose. The origin of the main frame is set at the initial position of the meniscus before we start any deformation. This initial time will be made explicit later in subsection 3.5 while describing the protocol of the deformation. The lengths $L_{m,i}$ between the motors and the origin are the lengths actually imposed in the system by moving the motors (see Fig.3.11B). The meniscus position is denoted $(X_{BP}, Y_{BP})$ and is used to define all the other quantities: the lengths of the films denoted $L_i$, that is the length between the motors and the free meniscus, $\theta$ the bisector angle between film
* A concentration of dodecanol which varies from no dodecanol to $c_{DOH}$ = 50 mg/L = 0.27 mmol/L

* A concentration of fluorescein $c_{fluo}$ = 0.8 g/L = 2.4 mmol/L

We weight and put all the solid parts of the recipe ($m_{SDS}$ = 5.6 g, $m_{DOH}$ = {0, 5, 15, 35, 50}mg and $m_{fluo}$ = 0.8 g) in a bottle thoroughly cleaned with one rinse with acetone followed by several rinse with demineralised water of the same quality as used in the next step. Dodecanol melting at 23 - 24°C, it is cooled down in a fridge in advance if need be to allow a precise weighing (which didn't happen many times thanks to one other parameter being living in Brittany). We then take one litre of demineralised water using a volumetric flask cleaned with the same protocol as the bottle. The litre of water is then carefully added to the bottle containing the solids by ensuring the water is flowing gently along the walls of the bottle to avoid excessive foaming, which could lead to an inhomogeneous mixing by entrapping dodecanol crystals. We add a magnetic rod to the bottle and put it on a heating magnetic stirrer at 50°C for 2 h. The bottle is stored in a dry place at room temperature. Mixtures of SDS/dodecanol being subject to aging in a matter of days (even hours for refined physico-chemistry), the timing which is roughly respected is a solution made between 2PM and 4PM used in the experiment the next day morning at around 10AM.

**NB:** to cope with the aging problem, we tried different chemistries knowing that we need to fill a 1 L tank of solution and allow the creation of centimetric foam films lasting tens of seconds. The use of negatively charged fluorescein (which is conditioned by our cyan laser) prohibits the use of cationic surfactants such as the TABs surfactants, as they interact with fluorescein and hinder the fluorescence. Non-ionic surfactants were ruled out because of the foaming problem, and no non-ageing anionic surfactant was found at a reasonable cost given the rather large quantities we need. Thus, we kept on using SDS.

### 3.4.2 Notations for the 3-films elementary foam

The notations for the axis of the main frame $(x, y, z)$ are shown in Fig.3.11, with $x$ being perpendicular to the free meniscus, parallel to the plane of film 3 and oriented towards motor 3. The system is supposed invariant by translation along $z$, the direction parallel to the free meniscus (this invariance is discussed later in subsection 3.7).

Also, we will need to define local frames $(x_i, y_i)$ for film 1 and 2, with in each case $x_i$ oriented towards motor $i$. All these notations are shown in Fig.3.11A.

Eventually, we need some notations for the positions, angles and lengths in the system, as the motors and the free meniscus all move during the deformations we impose. The origin of the main frame is set at the initial position of the meniscus before we start any deformation. This initial time will be made explicit later in subsection 3.5 while describing the protocol of the deformation. The lengths $L_{m,i}$ between the motors and the origin are the lengths actually imposed in the system by moving the motors (see Fig.3.11B). The meniscus position is denoted $(X_{BP}, Y_{BP})$ and is used to define all the other quantities: the lengths of the films denoted $L_i$, that is the length between the motors and the free meniscus, $\theta$ the bisector angle between film
1 and 2, and $\theta_3$ the angle between film 3 and the bisector line of film 1 and 2. These two last angles will be relevant to project the forces acting on the meniscus and thus to compute the difference of tension in the system.

<figure><img src="image_76.png" /><figcaption>Figure 3.11: **A:** Notations for the local axis for all the films. **B:** Definitions of the different lengths in the system, illustrated for an arbitrary deformation. The motors are the red disks, here placed at arbitrary positions. The three foam films 1, 2 and 3 are the black lines, of respective lengths $L_1$, $L_2$ and $L_3$. The red dotted lines link the three motors to the centre of the deformable frame, their lengths are denoted $L_{m,1}$, $L_{m,2}$ and $L_{m,3}$, and are the lengths imposed experimentally in our setup. The centre of the deformable frame defines the origin of the *x* and *y* axis, and we define the position of the free meniscus with the coordinates $(X_{BP}, Y_{BP})$.</figcaption></figure>

### 3.4.3 Motors, optics and photobleaching setup

#### Motors

The three moving edges of the supporting frame for the foam are attached to three independent and identical motors PI mikromove C-867 PI-line M-663.5U. Macros defining their velocities and position instructions are preloaded on them, and they are all synchronised with a DAQ card National Instruments USB-6343. The velocity instructions can go up to $V = 100 \text{ mm/s}$, but we will keep it below $V = 50 \text{ mm/s}$, as we start having significant discrepancies between the required ramp and the effective movements of the motors at $V = 60 \text{ mm/s}$.

Note that in the following, we extensively use “motor” for “mobile edge of the frame” by metonymy.

#### PF Camera

Our main camera monitoring film 3 by fluorescence, named “PF Camera” hereafter, is a Photon Focus MV1-D1312-160-CL. Its field of view encompasses the whole film 3 with a pixel ratio of 21.8 px/mm, recording 950 × 544 px frames at a rate of 278 Hz. It looks at the system from above, thus, the bath of soapy solution being in its optical axis, we need to cover the latter
1 and 2, and $\theta_3$ the angle between film 3 and the bisector line of film 1 and 2. These two last angles will be relevant to project the forces acting on the meniscus and thus to compute the difference of tension in the system.

<figure><img src="image_76.png" /><figcaption>Figure 3.11: **A:** Notations for the local axis for all the films. **B:** Definitions of the different lengths in the system, illustrated for an arbitrary deformation. The motors are the red disks, here placed at arbitrary positions. The three foam films 1, 2 and 3 are the black lines, of respective lengths $L_1$, $L_2$ and $L_3$. The red dotted lines link the three motors to the centre of the deformable frame, their lengths are denoted $L_{m,1}$, $L_{m,2}$ and $L_{m,3}$, and are the lengths imposed experimentally in our setup. The centre of the deformable frame defines the origin of the *x* and *y* axis, and we define the position of the free meniscus with the coordinates $(X_{BP}, Y_{BP})$.</figcaption></figure>

### 3.4.3 Motors, optics and photobleaching setup

#### Motors

The three moving edges of the supporting frame for the foam are attached to three independent and identical motors PI mikromove C-867 PI-line M-663.5U. Macros defining their velocities and position instructions are preloaded on them, and they are all synchronised with a DAQ card National Instruments USB-6343. The velocity instructions can go up to $V = 100 \text{ mm/s}$, but we will keep it below $V = 50 \text{ mm/s}$, as we start having significant discrepancies between the required ramp and the effective movements of the motors at $V = 60 \text{ mm/s}$.

Note that in the following, we extensively use “motor” for “mobile edge of the frame” by metonymy.

#### PF Camera

Our main camera monitoring film 3 by fluorescence, named “PF Camera” hereafter, is a Photon Focus MV1-D1312-160-CL. Its field of view encompasses the whole film 3 with a pixel ratio of 21.8 px/mm, recording 950 × 544 px frames at a rate of 278 Hz. It looks at the system from above, thus, the bath of soapy solution being in its optical axis, we need to cover the latter
<figure><img src="image_77.png" /></figure>

Figure 3.12: Picture of the whole setup, letters are for cameras, numbers for other experimental tools. **A**: Photon focus fluorescence camera “PF”, looking at film 3 from the top. **B**: Plateau border camera “BP”, looking at the free meniscus from the side. **C**: Top fluorescence camera “TPF”, looking at film 1. **D-D'**: Hyperspectral camera 1 “SP1” looking at the thickness profile along a line contained in the plane of film 1. **D'** is the other position on the experimental setup where “SP1” can be attached to look at film 2. **E**: Hyperspectral camera 2 “SP2” looking at the thickness profile of film 3. **1**: Deformable frame where the elementary foam is created. The foam films fluoresce green-yellow light, and the saturated line connecting the triple points of the three-pointed stars is the free meniscus hanging in the air. **2**: Tank for the soapy solution, lying on a horizontal plate **3** moving up and down using a stepper motor. **4-5**: Blue and white lights used respectively for fluorescence imaging and hyperspectral imaging. A second white lamp aligned with “SP1” is present but hidden by the setup. **6**: Temperature and humidity sensor positioned as close as possible to the elementary foam.

with a printed cache so that we avoid stray fluorescent light. The trigger of the camera and the collection of frames are all done through the DAQ card, with a precision on the synchronisation and the timestamps of the frames on the order of a µs. A typical frame recorded by this camera is shown in Fig.3.13.

## BP Camera (Bord de Plateau)

The camera monitoring the free meniscus by fluorescence from the side, named the “BP Camera” hereafter (for Bord de Plateau), is a Basler 1920-155uc camera mounted on a 12X Navitar objective. Ingoing light is filtered using an orange filter to ensure that we only see the fluoresced light of the elementary foam. The field of view is focused on a portion of a few millimetres of free meniscus at the centre of the foam along the *z* axis, with a pixel ratio in the range 690–1300 px/mm and recording 1920 × 860 px frames at a rate of 150 Hz. The synchronisation and collection of frames is done through the DAQ, with the same precision and in the same way as for the PF camera.
<figure><img src="image_77.png" /></figure>

Figure 3.12: Picture of the whole setup, letters are for cameras, numbers for other experimental tools. **A**: Photon focus fluorescence camera “PF”, looking at film 3 from the top. **B**: Plateau border camera “BP”, looking at the free meniscus from the side. **C**: Top fluorescence camera “TPF”, looking at film 1. **D-D'**: Hyperspectral camera 1 “SP1” looking at the thickness profile along a line contained in the plane of film 1. **D'** is the other position on the experimental setup where “SP1” can be attached to look at film 2. **E**: Hyperspectral camera 2 “SP2” looking at the thickness profile of film 3. **1**: Deformable frame where the elementary foam is created. The foam films fluoresce green-yellow light, and the saturated line connecting the triple points of the three-pointed stars is the free meniscus hanging in the air. **2**: Tank for the soapy solution, lying on a horizontal plate **3** moving up and down using a stepper motor. **4-5**: Blue and white lights used respectively for fluorescence imaging and hyperspectral imaging. A second white lamp aligned with “SP1” is present but hidden by the setup. **6**: Temperature and humidity sensor positioned as close as possible to the elementary foam.

with a printed cache so that we avoid stray fluorescent light. The trigger of the camera and the collection of frames are all done through the DAQ card, with a precision on the synchronisation and the timestamps of the frames on the order of a µs. A typical frame recorded by this camera is shown in Fig.3.13.

## BP Camera (Bord de Plateau)

The camera monitoring the free meniscus by fluorescence from the side, named the "BP Camera" hereafter (for Bord de Plateau), is a Basler 1920-155uc camera mounted on a 12X Navitar objective. Ingoing light is filtered using an orange filter to ensure that we only see the fluoresced light of the elementary foam. The field of view is focused on a portion of a few millimetres of free meniscus at the centre of the foam along the *z* axis, with a pixel ratio in the range 690–1300 px/mm and recording 1920 × 860 px frames at a rate of 150 Hz. The synchronisation and collection of frames is done through the DAQ, with the same precision and in the same way as for the PF camera.
<figure><img src="image_78.png" /><figcaption>Figure 3.13: Typical frame recorded with the PF camera. Film 3 is the rectangle in the middle, bordered on the right by motor 3 (thick saturated vertical fringe) and by the free meniscus on the left (thin saturated vertical fringe).</figcaption></figure>

One goal of this recording is to measure the radius of curvature of the free meniscus by looking at the shape of the intensity profile. In order to get a clear signal, we need to remove the light coming from film 3, whose plane contains the optical axis of the camera, parasiting the fluoresced light from the free meniscus itself. While using this camera, a horizontal cache is placed between film 3 and the blue LED lighting the system, so that only the free meniscus fluoresces toward the BP camera.

An example of a full frame picture is shown in Fig.3.14A, along with the intensity profile *I*(*y*) averaged over *z* in Fig.3.14B.

<figure><img src="image_79.png" /><figcaption>Figure 3.14: **A** Typical frame obtained with the BP camera. The very thin saturated fringe in the middle localise the position of film 3, whose plane contains the optical axis of the camera. **B** Intensity profile *I*(*y*) averaged along *z*. The positions of the shoulders of the signal will allow us to compute the radius of curvature of the meniscus.</figcaption></figure>
<figure><img src="image_78.png" /><figcaption>Figure 3.13: Typical frame recorded with the PF camera. Film 3 is the rectangle in the middle, bordered on the right by motor 3 (thick saturated vertical fringe) and by the free meniscus on the left (thin saturated vertical fringe).</figcaption></figure>

One goal of this recording is to measure the radius of curvature of the free meniscus by looking at the shape of the intensity profile. In order to get a clear signal, we need to remove the light coming from film 3, whose plane contains the optical axis of the camera, parasiting the fluoresced light from the free meniscus itself. While using this camera, a horizontal cache is placed between film 3 and the blue LED lighting the system, so that only the free meniscus fluoresces toward the BP camera.

An example of a full frame picture is shown in Fig.3.14A, along with the intensity profile *I*(*y*) averaged over *z* in Fig.3.14B.

<figure><img src="image_79.png" /><figcaption>Figure 3.14: **A** Typical frame obtained with the BP camera. The very thin saturated fringe in the middle localise the position of film 3, whose plane contains the optical axis of the camera. **B** Intensity profile *I*(*y*) averaged along *z*. The positions of the shoulders of the signal will allow us to compute the radius of curvature of the meniscus.</figcaption></figure>
## SP1 and SP2 Cameras (for SPectral)

We use two hyperspectral cameras to monitor the three films, with "SP1" having two positions, one for film 1, one for film 2, and "SP2" monitoring film 3. These cameras are Resonon Pika L hyperspectral cameras mounted on 23 mm Schneider High Resolution VIS-NIR objectives. Along a physical line of length 1024 px, they monitor the light intensity received for each wavelength, with 600 pixels, each one dedicated to a wavelength $\lambda \in [400, 1000]$nm with a constant step of 1 nm between each. Depending on the zoom applied, the physical line can be seen with a pixel ratio ranging from 20 – 30 px/mm. The scale is found by placing a mirror in the plane of the foam film, on which a narrow strip of tape is affixed, aligned with z and of a well-known width.

## TPF Camera (ToP Fluorescence)

Film 1 is imaged using a Basler 1920-155uc camera mounted on a 25mm Fujifilm objective (Fig.3.12C) with 1920 × 800 px at a rate of 150 Hz. Ingoing light is filtered using a similar orange filter as the one used with the "BP" camera. The "TPF" camera could not be aligned with the normal $y_1$ to film 1 because of a lack of space in the setup due to other cameras. However, it was put as close as possible from it so that the parallax problem remains negligible for the sake of our measurement, which is in the vicinity of the centre of the free meniscus. This is where the pixel ratio is computed (33 or 35 px/mm depending on the set of experiments). The synchronisation and collection of data is ensured using the DAQ just as for the other cameras. This camera does not need any cache to remove stray fluoresced light (however a mat strip of tape was placed on a metallic surface aligned with the optical axis to remove an inopportune reflection). A typical frame recorded by this camera is shown in Fig.3.15.

<figure><img src="image_80.png" /><figcaption>Figure 3.15: Typical frame recorded with the TPF camera. Film 1 is the rectangle in the middle.</figcaption></figure>

## Photobleaching setup

To describe the velocity fields in the planes of the films of the elementary foam, we need to create passive tracers in the film that do not change the state of its interfaces. Thickness heterogeneities which can be imaged by fluorescence could be a lead, but what we need here is a reliable and
## SP1 and SP2 Cameras (for SPectral)

We use two hyperspectral cameras to monitor the three films, with “SP1” having two positions, one for film 1, one for film 2, and “SP2” monitoring film 3. These cameras are Resonon Pika L hyperspectral cameras mounted on 23 mm Schneider High Resolution VIS-NIR objectives. Along a physical line of length 1024 px, they monitor the light intensity received for each wavelength, with 600 pixels, each one dedicated to a wavelength $\lambda \in [400, 1000]$nm with a constant step of 1 nm between each. Depending on the zoom applied, the physical line can be seen with a pixel ratio ranging from 20 – 30 px/mm. The scale is found by placing a mirror in the plane of the foam film, on which a narrow strip of tape is affixed, aligned with z and of a well-known width.

## TPF Camera (ToP Fluorescence)

Film 1 is imaged using a Basler 1920-155uc camera mounted on a 25mm Fujifilm objective (Fig.3.12C) with 1920 × 800 px at a rate of 150 Hz. Ingoing light is filtered using a similar orange filter as the one used with the “BP” camera. The “TPF” camera could not be aligned with the normal $y_1$ to film 1 because of a lack of space in the setup due to other cameras. However, it was put as close as possible from it so that the parallax problem remains negligible for the sake of our measurement, which is in the vicinity of the centre of the free meniscus. This is where the pixel ratio is computed (33 or 35 px/mm depending on the set of experiments). The synchronisation and collection of data is ensured using the DAQ just as for the other cameras. This camera does not need any cache to remove stray fluoresced light (however a mat strip of tape was placed on a metallic surface aligned with the optical axis to remove an inopportune reflection). A typical frame recorded by this camera is shown in Fig.3.15.

<figure><img src="image_80.png" /><figcaption>Figure 3.15: Typical frame recorded with the TPF camera. Film 1 is the rectangle in the middle.</figcaption></figure>

## Photobleaching setup

To describe the velocity fields in the planes of the films of the elementary foam, we need to create passive tracers in the film that do not change the state of its interfaces. Thickness heterogeneities which can be imaged by fluorescence could be a lead, but what we need here is a reliable and
very localised information on the velocity field (at a scale smaller than the typical heterogeneity). Thus, the group developed a photobleaching setup designed to locally deactivate fluorescein using a cyan laser excited at 488 nm to create tiny pieces of non-fluorescing bulk to track (kudos to Jacopo Seiwert, Adrien Bussonnière, former post-docs of our group, and Emmanuel Schaub, the research engineer of the soft matter department at the IPR). This setup was first developed in 2015 and a proof of concept was made with a study of convective transport in vertical foam films by J. Seiwert et al. [106]. The main contribution in the frame of my Ph.D was to develop its automation and to allow drawing arrays of dots with the life-saving help of Emmanuel Schaub.

The optical path of the laser is directed using small mirrors, passing through an acousto-optic modulator AOTfnC-400.650-TN which serves as a very responsive shutter. Toward the end of the path, the laser hits a mirror attached to a galvanometric scanning head dynAXIS to apply tiny changes to its direction and is eventually directed down on film 3 to photobleach it from the top. The shutter and scanning head responsiveness and their synchronisation allow us to shoot in a reliable way down to ~ 10 ms per dot, which happened to be the minimal time it takes to create a dark enough dot in the film.

The laser is used at a power of 120 mW that requires optical protections in the form of glasses and a special cage covering the optical path of the laser. The acousto-optic modulator and the mirror with the galvanometric scanner are synchronised with a dedicated channel of the DAQ card to emit localised flashes at specified positions on film 3 to photobleach them. The result is a 2 × 8 rectangular grid of dark dots of typical diameter ~ 100 µm shot with a precision of the same order of magnitude which we made separate by $\Delta x = \Delta z = 400$ µm in both directions. It is placed near the meniscus and with the two columns aligned with the z-axis. A typical frame recorded by this camera is shown in Fig.3.16.

<figure><img src="image_81.png" /><figcaption>Figure 3.16: Zoomed image of the photobleached dots performed with the PF camera. The saturated vertical fringe on the left is the free meniscus.</figcaption></figure>

### 3.4.4 Interfacing and synchronising the experiment

The cameras, the motors (both for the deformable frames and to lift the soapy solution bath) and all the photobleach-related devices are connected to the DAQ card for the synchronisation and interfaced with LabView.
very localised information on the velocity field (at a scale smaller than the typical heterogeneity). Thus, the group developed a photobleaching setup designed to locally deactivate fluorescein using a cyan laser excited at 488 nm to create tiny pieces of non-fluorescing bulk to track (kudos to Jacopo Seiwert, Adrien Bussonnière, former post-docs of our group, and Emmanuel Schaub, the research engineer of the soft matter department at the IPR). This setup was first developed in 2015 and a proof of concept was made with a study of convective transport in vertical foam films by J. Seiwert et al. [106]. The main contribution in the frame of my Ph.D was to develop its automation and to allow drawing arrays of dots with the life-saving help of Emmanuel Schaub.

The optical path of the laser is directed using small mirrors, passing through an acousto-optic modulator AOTfNc-400.650-TN which serves as a very responsive shutter. Toward the end of the path, the laser hits a mirror attached to a galvanometric scanning head dynAXIS to apply tiny changes to its direction and is eventually directed down on film 3 to photobleach it from the top. The shutter and scanning head responsiveness and their synchronisation allow us to shoot in a reliable way down to ~ 10 ms per dot, which happened to be the minimal time it takes to create a dark enough dot in the film.

The laser is used at a power of 120 mW that requires optical protections in the form of glasses and a special cage covering the optical path of the laser. The acousto-optic modulator and the mirror with the galvanometric scanner are synchronised with a dedicated channel of the DAQ card to emit localised flashes at specified positions on film 3 to photobleach them. The result is a 2 × 8 rectangular grid of dark dots of typical diameter ~ 100 µm shot with a precision of the same order of magnitude which we made separate by $\Delta x = \Delta z = 400$ µm in both directions. It is placed near the meniscus and with the two columns aligned with the z-axis. A typical frame recorded by this camera is shown in Fig.3.16.

<figure><img src="image_81.png" /><figcaption>Figure 3.16: Zoomed image of the photobleached dots performed with the PF camera. The saturated vertical fringe on the left is the free meniscus.</figcaption></figure>

### 3.4.4 Interfacing and synchronising the experiment

The cameras, the motors (both for the deformable frames and to lift the soapy solution bath) and all the photobleach-related devices are connected to the DAQ card for the synchronisation and interfaced with LabView.
The main code is fed with all the parameters: times, motors, cameras, photobleaching setup and save paths. The motors are loaded with individual macros specifying their positions, velocities and when to wait for a trigger to be synchronised together. The DAQ card is linked to all cameras and motors and it triggers the different elements at the times specified by the parameters.

Regarding the cameras, it only launches the beginning of the recording, and the internal clock of the cameras maintains the framerate. Each frame is collected with its timestamp, thus preventing a mismatch between times and frames in case of frame skipping, which can happen if the buffer is overfilled due to random RAM mismanagement events.

The movies, the positions of the motors and their timestamps are collected and stored as binary files.

## 3.5 Detailed protocol for the “Push 3” experiment

<figure><img src="image_82.png" /><figcaption>Figure 3.17: Description of the “Push 3” experiment. **A** is the initial shape of the film assembly, as extracted from the bath. **B** represents the preparation stage, which is required to control the initial thickness of film 3. **C** is the deformation we want to address. All numbers given along the different films are the lengths (in *mm*) between the centre of the frame (triple point of the plain lines for $t < t_0$, of the dotted line for $t > t_0$) and the motors. The red lines indicate which motors are involved at each step, along with their velocities in red as well. The cyan cursor pointing downwards accounts for the photobleaching step occurring at $t = 31$ s. $\theta$ is defined as the semi-angle between the vertical films 1 and 2.</figcaption></figure>

The main results of this chapter are obtained in a specific deformation configuration referred to as “Push 3”, as mentioned earlier and detailed below. In this configuration, films 1 and 2 are stretched, and film 3 is compressed over the same amplitude, at the same velocities, and at the same times. The protocol is illustrated in Fig. 3.17. Other types of deformations have been investigated and will be compared to “Push 3” later on, as we will detail our results.

Before launching the code and the automated part, we start by cleaning the surface of the bath with a custom squeegee (removing bubbles and impurities) and by setting the right light given the choice of camera for the experiment. Note that because of RAM management and lighting, we can only turn on one camera at a time during the experiments.
The main code is fed with all the parameters: times, motors, cameras, photobleaching setup and save paths. The motors are loaded with individual macros specifying their positions, velocities and when to wait for a trigger to be synchronised together. The DAQ card is linked to all cameras and motors and it triggers the different elements at the times specified by the parameters.

Regarding the cameras, it only launches the beginning of the recording, and the internal clock of the cameras maintains the framerate. Each frame is collected with its timestamp, thus preventing a mismatch between times and frames in case of frame skipping, which can happen if the buffer is overfilled due to random RAM mismanagement events.

The movies, the positions of the motors and their timestamps are collected and stored as binary files.

## 3.5 Detailed protocol for the “Push 3” experiment

<figure><img src="image_82.png" /><figcaption>Figure 3.17: Description of the “Push 3” experiment. **A** is the initial shape of the film assembly, as extracted from the bath. **B** represents the preparation stage, which is required to control the initial thickness of film 3. **C** is the deformation we want to address. All numbers given along the different films are the lengths (in *mm*) between the centre of the frame (triple point of the plain lines for $t < t_0$, of the dotted line for $t > t_0$) and the motors. The red lines indicate which motors are involved at each step, along with their velocities in red as well. The cyan cursor pointing downwards accounts for the photobleaching step occurring at $t = 31$ s. $\theta$ is defined as the semi-angle between the vertical films 1 and 2.</figcaption></figure>

The main results of this chapter are obtained in a specific deformation configuration referred to as “Push 3”, as mentioned earlier and detailed below. In this configuration, films 1 and 2 are stretched, and film 3 is compressed over the same amplitude, at the same velocities, and at the same times. The protocol is illustrated in Fig. 3.17. Other types of deformations have been investigated and will be compared to “Push 3” later on, as we will detail our results.

Before launching the code and the automated part, we start by cleaning the surface of the bath with a custom squeegee (removing bubbles and impurities) and by setting the right light given the choice of camera for the experiment. Note that because of RAM management and lighting, we can only turn on one camera at a time during the experiments.
The PF, BP and TPF cameras require blue light to image the system by fluorescence, and we turn on the blue LED right from the start. The SP1 and SP2 cameras require white lights specifically aligned with them for their interferometric measurement of thickness. These white lights produce a lot of heat, thus we only turn them by hand during the main deformation as specified later.

Then we launch the main LabView code with all the parameters we need, which are defined in the following. The motor of the bath rises up til it immerses the deformable frame, for 5s, then goes down at a well-controled and low velocity of 30 mm/s to wet the frame in reproducible way and avoid bubble formation. The trigger sent by the DAQ card to command the motor to go down is the origin of times $t = 0$ for the experiment. Once the bath is down again, we may put on different caches depending on the cameras (as discussed individually in section 3.4.3).

At this point the films are too heterogeneous in thickness to proceed to the main deformation, because of the broken invariance by translation along $z$ and that would lead to a lack of reproducibility of the experiments. The state of the horizontal film 3 is crucial in particular, as the main deformation consists in compressing it after it has been photobleached. Right after the formation of the elementary foam, film 3 can be seen in Fig.3.18A: the thickness distribution in the film ranges from hundreds of nanometers to tens of microns. Patches of same thicknesses tend to regroup for energy reasons (the same as what drives the relaxation of the thin film in chapter 2) and the thicker ones are evacuated towards the menisci at the edges, replaced by thinner patches (by the marginal regeneration instability discussed in 0.3.4).

We first let the film drain until $t = t_0$ (Fig.3.18B and Fig.3.17B), then it is stretched slowly. The result is Fig.3.18C, where the red arrows are the newly created films extracted from the menisci (Frankels films cf. 0.3.2) which have the major advantage to be very homogeneous and reproducible. The photobleaching step and the main deformation can be made from this state at $t = t_1$ as depicted in Fig.3.17C. The deformation performed at $t_1$ is our actual measurement, and varying the velocity of deformation $V_1$ is our second parameter besides chemistry.

All the lengths $L_{m,i}$ imposed between the centre of the frame and the motors $i$ at all times are given in Fig.3.17, as well as the velocities imposed.

**NB:** All the waiting to ensure a controlled state for the horizontal film 3 also has repercussions on the other films 1 and 2:

* As gravity drainage occurs, it tends to make the top film 1 thinner and the bottom film 2 thicker. This has a direct impact on their respective elasticities, as a thinner film is also stiffer as expected with our model of chapter 1 section 1.2.2. This dependency is quite “slow” however, and one order of magnitude in thickness yields a factor two on the elasticity. Boundaries can be set to this problem by monitoring mean thicknesses of both film 1 and 2, which we will do using camera SP1.

* Allowing the menisci to drain for extended periods of time lead to the formation of thin patches near the edges of the film (see 0.3.3). For menisci located at the bottom of their films (the free meniscus for film 1, the meniscus supported by its motor for film 2) the 2D Rayleigh-Taylor instability occurs (cf. section 0.3.5). This leads to the constant presence
The PF, BP and TPF cameras require blue light to image the system by fluorescence, and we turn on the blue LED right from the start. The SP1 and SP2 cameras require white lights specifically aligned with them for their interferometric measurement of thickness. These white lights produce a lot of heat, thus we only turn them by hand during the main deformation as specified later.

Then we launch the main LabView code with all the parameters we need, which are defined in the following. The motor of the bath rises up til it immerses the deformable frame, for 5s, then goes down at a well-controled and low velocity of 30 mm/s to wet the frame in reproducible way and avoid bubble formation. The trigger sent by the DAQ card to command the motor to go down is the origin of times $t = 0$ for the experiment. Once the bath is down again, we may put on different caches depending on the cameras (as discussed individually in section 3.4.3).

At this point the films are too heterogeneous in thickness to proceed to the main deformation, because of the broken invariance by translation along $z$ and that would lead to a lack of reproducibility of the experiments. The state of the horizontal film 3 is crucial in particular, as the main deformation consists in compressing it after it has been photobleached. Right after the formation of the elementary foam, film 3 can be seen in Fig.3.18A: the thickness distribution in the film ranges from hundreds of nanometers to tens of microns. Patches of same thicknesses tend to regroup for energy reasons (the same as what drives the relaxation of the thin film in chapter 2) and the thicker ones are evacuated towards the menisci at the edges, replaced by thinner patches (by the marginal regeneration instability discussed in 0.3.4).

We first let the film drain until $t = t_0$ (Fig.3.18B and Fig.3.17B), then it is stretched slowly. The result is Fig.3.18C, where the red arrows are the newly created films extracted from the menisci (Frankels films cf. 0.3.2) which have the major advantage to be very homogeneous and reproducible. The photobleaching step and the main deformation can be made from this state at $t = t_1$ as depicted in Fig.3.17C. The deformation performed at $t_1$ is our actual measurement, and varying the velocity of deformation $V_1$ is our second parameter besides chemistry.

All the lengths $L_{m,i}$ imposed between the centre of the frame and the motors $i$ at all times are given in Fig.3.17, as well as the velocities imposed.

NB: All the waiting to ensure a controlled state for the horizontal film 3 also has repercussions on the other films 1 and 2:

* As gravity drainage occurs, it tends to make the top film 1 thinner and the bottom film 2 thicker. This has a direct impact on their respective elasticities, as a thinner film is also stiffer as expected with our model of chapter 1 section 1.2.2. This dependency is quite “slow” however, and one order of magnitude in thickness yields a factor two on the elasticity. Boundaries can be set to this problem by monitoring mean thicknesses of both film 1 and 2, which we will do using camera SP1.

* Allowing the menisci to drain for extended periods of time lead to the formation of thin patches near the edges of the film (see 0.3.3). For menisci located at the bottom of their films (the free meniscus for film 1, the meniscus supported by its motor for film 2) the 2D Rayleigh-Taylor instability occurs (cf. section 0.3.5). This leads to the constant presence
<figure><img src="image_83.png" /><figcaption>Figure 3.18: Fluorescence images of the horizontal film 3 (top view using the PF camera) during the preparation phase. **A:** Right after the elementary foam is formed, the film 3 (indicated by the green line in B) is very heterogeneous in thickness (Fig.3.17A). **B:** Right before the preparatory stretch, the film has drained and is more homogeneous in thickness although some patterns are still visible. **C:** Right after the preparatory stretch (Fig.3.17B), some newly created film went out of the menisci, identified with the red arrows. The pre-existing film (which is a bit stretched in the process) is identified by the green line and double arrows. The extracted Frankel films are very homogeneous and invariant along z.</figcaption></figure>

of thin patches advected upwards, which has consequences for image processing in the non-horizontal films.

## 3.6 Data and image processing

In this section, we will detail how we process the different movies to extract all the physical quantities we need to describe our system. All the following numerical methods will be detailed for the “Push 3” experiment, which is the main studied deformation of our elementary liquid foam described in section 3.5. Other deformations such as “Push 1” and “Pull 3” require some adjustments in movie processing, but the numerical methods remain broadly the same.

### 3.6.1 Tracking of the free meniscus in the horizontal plane

**Extracted ingredient: $X_{BP}$ the horizontal position of the meniscus**

When the films are pushed or stretched and the forces they exert on their neighbours change, the free meniscus moves. The meniscus position will be related quantitatively to the tensions in
<figure><img src="image_83.png" /><figcaption>Figure 3.18: Fluorescence images of the horizontal film 3 (top view using the PF camera) during the preparation phase. **A:** Right after the elementary foam is formed, the film 3 (indicated by the green line in B) is very heterogeneous in thickness (Fig.3.17A). **B:** Right before the preparatory stretch, the film has drained and is more homogeneous in thickness although some patterns are still visible. **C:** Right after the preparatory stretch (Fig.3.17B), some newly created film went out of the menisci, identified with the red arrows. The pre-existing film (which is a bit stretched in the process) is identified by the green line and double arrows. The extracted Frankel films are very homogeneous and invariant along z.</figcaption></figure>

of thin patches advected upwards, which has consequences for image processing in the non-horizontal films.

## 3.6 Data and image processing

In this section, we will detail how we process the different movies to extract all the physical quantities we need to describe our system. All the following numerical methods will be detailed for the “Push 3” experiment, which is the main studied deformation of our elementary liquid foam described in section 3.5. Other deformations such as “Push 1” and “Pull 3” require some adjustments in movie processing, but the numerical methods remain broadly the same.

### 3.6.1 Tracking of the free meniscus in the horizontal plane

**Extracted ingredient: $X_{BP}$ the horizontal position of the meniscus**

When the films are pushed or stretched and the forces they exert on their neighbours change, the free meniscus moves. The meniscus position will be related quantitatively to the tensions in
the films in section 3.8. Here we focus on its horizontal motion by computing its position $X_{BP}$ on the x axis.

As discussed in Bussonnière et al. [4] and seen experimentally in Fig.3.19A-B, the free meniscus curves in the z-direction as it moves, taking a roughly parabolic shape $X(z)$, and the position $X_{BP} = \delta_1 + \delta_2$ as we define it in Fig.3.19 is the position of its apex. Although we have $\delta_1 \simeq \delta_2$, taking into account the parabolic shape for the sake of computing the angle $\theta$ between the film leads to a negligible correction which has been computed and represents less than 1% of deviation throughout the measurement. This correction had already been investigated and computed in Bussonnière et al. [4] in a similar configuration, where the interested reader can find its formula with the same notations for $\delta_1$ and $\delta_2$. In this study, we will thus only measure the apex position $X_{BP}$ and not the whole meniscus shape.

<figure><img src="image_84.png" /></figure>

Figure 3.19: Tracking of the free meniscus horizontal displacement, example with [DOH] = 50 mg/L at $V_1$ = 50 mm/s. **A-B:** Top view (PF camera) of the system respectively right before ($t$ = 31.99 s) and just after ($t$ = 32.1 s) the main deformation occurring at $t_1$ = 32 s. The aspect ratio of the picture is deformed to better see the curvature of the free meniscus as it moves. We directly see that the total displacement of the apex of the meniscus $X_{BP} = \delta_1 + \delta_2$ is the sum of $\delta_1$ the displacement of the free meniscus as a whole and $\delta_2$ the displacement of the apex due to the free meniscus curving in the z direction. Both contributions are of comparable amplitudes. **C:** Zoom in the region of the apex of the free meniscus of **A** with an aspect ratio faithful to reality. This zone is where we look for the maximum of correlation of the binary mask **D** along $x$ for each position $z$ to estimate the position of the apex of the parabola.

In order to track the position of the apex of the free meniscus, we use a correlation technique with a correlation mask which is a bright vertical saturated segment with a well chosen width and length on a black background (cf. Fig3.19D). For each position $z$ in the region of the apex of the free meniscus (Fig3.19C), we look for the maximum of correlation along $x$ of our binary mask. The value $X_{BP}$ is obtained with an average over $z$ over the region encompassed by the red rectangle in Fig.3.19A. This process is repeated for each frame individually, with the same region of interest and the same binary mask.
the films in section 3.8. Here we focus on its horizontal motion by computing its position $X_{BP}$ on the x axis.

As discussed in Bussonnière et al. [4] and seen experimentally in Fig.3.19A-B, the free meniscus curves in the z-direction as it moves, taking a roughly parabolic shape $X(z)$, and the position $X_{BP} = \delta_1 + \delta_2$ as we define it in Fig.3.19 is the position of its apex. Although we have $\delta_1 \simeq \delta_2$, taking into account the parabolic shape for the sake of computing the angle $\theta$ between the film leads to a negligible correction which has been computed and represents less than 1% of deviation throughout the measurement. This correction had already been investigated and computed in Bussonnière et al. [4] in a similar configuration, where the interested reader can find its formula with the same notations for $\delta_1$ and $\delta_2$. In this study, we will thus only measure the apex position $X_{BP}$ and not the whole meniscus shape.

<figure><img src="image_84.png" /></figure>

Figure 3.19: Tracking of the free meniscus horizontal displacement, example with [DOH] = 50 mg/L at $V_1$ = 50 mm/s. **A-B:** Top view (PF camera) of the system respectively right before ($t$ = 31.99 s) and just after ($t$ = 32.1 s) the main deformation occurring at $t_1$ = 32 s. The aspect ratio of the picture is deformed to better see the curvature of the free meniscus as it moves. We directly see that the total displacement of the apex of the meniscus $X_{BP} = \delta_1 + \delta_2$ is the sum of $\delta_1$ the displacement of the free meniscus as a whole and $\delta_2$ the displacement of the apex due to the free meniscus curving in the z direction. Both contributions are of comparable amplitudes. **C:** Zoom in the region of the apex of the free meniscus of **A** with an aspect ratio faithful to reality. This zone is where we look for the maximum of correlation of the binary mask **D** along $x$ for each position $z$ to estimate the position of the apex of the parabola.

In order to track the position of the apex of the free meniscus, we use a correlation technique with a correlation mask which is a bright vertical saturated segment with a well chosen width and length on a black background (cf. Fig3.19D). For each position $z$ in the region of the apex of the free meniscus (Fig3.19C), we look for the maximum of correlation along $x$ of our binary mask. The value $X_{BP}$ is obtained with an average over $z$ over the region encompassed by the red rectangle in Fig.3.19A. This process is repeated for each frame individually, with the same region of interest and the same binary mask.
<figure><img src="image_85.png" /><figcaption>Figure 3.20: Typical curve $X_{BP} = f(t)$ obtained for $V_1 = 50$ mm/s and [$DOH$] = 50 mg/L. The error is smaller than the width of the curve. Each step of the deformation is placed at its time of occurrence, see section 3.5. The preparation phase starting at $t_0 = 28$ s ends around $t = 31$ s and we photobleach film 3 between $t = 31.5$ s and 32 s.</figcaption></figure>

Eventually, the origin of $X_{BP}$ is set so that at $t < t_0 = 28$ s we consider it to be at its rest position, the centre of the deformable frame as shown in Fig. 3.20. No drift of the meniscus position has been witnessed before that time, and this initial position appears as the right criterion to define when the films meet with an angle of $120^\circ$. Thus we set $X_{BP}(t = t_0) = 0$.

### 3.6.2 Tracking the vertical position of the free meniscus and its radius by ray tracing

Extracted ingredients:

* $Y_{BP}$ (mm) the vertical position of the meniscus

* $R$ (mm) the curvature radius of the meniscus

To determine $Y_{BP}$ and $R$, we use the BP camera, located on the side of the meniscus and whose a typical frame is displayed in Fig. 3.14A. The intensity profiles are obtained by first averaging each frame along the $z$ direction, as shown in Fig. 3.14B. The maximum of intensity of each frame is tracked and corresponds to the centre of the meniscus, determining its ordinate $Y_{BP}$ (Fig. 3.22A). This tracking is done without any correlation technique, as the resolution is much greater than with the PF camera used for $X_{BP}$, and we only look at the maximum of $I$ for each frame. Also, the displacements along the $y$ direction are much smaller than in the $x$ direction due to the symmetry of the “Push 3” deformation, and the curvature of the meniscus in the $y$ direction does not need to be quantified.

Finding $R$ is more indirect, as we need to look at the shape of the intensity profile and relate
<figure><img src="image_85.png" /><figcaption>Figure 3.20: Typical curve $X_{BP} = f(t)$ obtained for $V_1 = 50$ mm/s and [$DOH$] = 50 mg/L. The error is smaller than the width of the curve. Each step of the deformation is placed at its time of occurrence, see section 3.5. The preparation phase starting at $t_0 = 28$ s ends around $t = 31$ s and we photobleach film 3 between $t = 31.5$ s and 32 s.</figcaption></figure>

Eventually, the origin of $X_{BP}$ is set so that at $t < t_0 = 28$ s we consider it to be at its rest position, the centre of the deformable frame as shown in Fig. 3.20. No drift of the meniscus position has been witnessed before that time, and this initial position appears as the right criterion to define when the films meet with an angle of $120^\circ$. Thus we set $X_{BP}(t = t_0) = 0$.

### 3.6.2 Tracking the vertical position of the free meniscus and its radius by ray tracing

Extracted ingredients:

* $Y_{BP}$ (mm) the vertical position of the meniscus

* $R$ (mm) the curvature radius of the meniscus

To determine $Y_{BP}$ and $R$, we use the BP camera, located on the side of the meniscus and whose a typical frame is displayed in Fig. 3.14A. The intensity profiles are obtained by first averaging each frame along the $z$ direction, as shown in Fig. 3.14B. The maximum of intensity of each frame is tracked and corresponds to the centre of the meniscus, determining its ordinate $Y_{BP}$ (Fig. 3.22A). This tracking is done without any correlation technique, as the resolution is much greater than with the PF camera used for $X_{BP}$, and we only look at the maximum of $I$ for each frame. Also, the displacements along the $y$ direction are much smaller than in the $x$ direction due to the symmetry of the “Push 3” deformation, and the curvature of the meniscus in the $y$ direction does not need to be quantified.

Finding $R$ is more indirect, as we need to look at the shape of the intensity profile and relate
it to the shape of the meniscus. As depicted in Fig. 3.21, a ray-tracing simulation was necessary to link $R$ to the profile $I_{fluo}$. I developed the simulation with the following assumptions:

* The meniscus has its equilibrium section, that is three sectors of three circles of radii $R$ tangentially connecting one another (cf. left of Fig 3.21)

* We only consider the fluoresced light, which is supposed to be emitted in an homogeneous and isotropic way in the volume of the meniscus. Each emitted ray is subject to reflection/refraction laws at the three curved water/air interfaces.

* We only collect the rays going out of the meniscus horizontally, which we retrace back in the volume. Each segment of their trajectories contributes to the intensity proportionally to its length and the number of reflections (where most of the intensity is lost) before going out towards the camera.

The resulting intensity profile $I_{fluo}$ shown in Fig. 3.21A closely resembles the experimental observations depicted in Figure 3.21B, with a sudden drop of intensity beyond an ordinate $y = 0.208R$ from the centre. This decrease is noticeable in the simulation as soon as we consider the first reflection within the meniscus. Including more than the second reflection did not result in a significant modification of the intensity pattern.

<figure><img src="image_86.png" /></figure>

Figure 3.21: Ray tracing simulation for a fluorescent meniscus. **A:** we define an ideal free meniscus composed of three interfaces of curvatures $R$ in contact. The red lines are the outgoing rays seen by the side. On the right, the resulting intensity profile as a function of the ordinate (normalised by $R$), where the shaded coloured areas are the contributions of the different rays: ray without reflection (green), with one reflection (light blue), or two reflections (purple). **B:** Identification of the intensity drop on experimental data.

The global maximum of $I(y)$ sets the origin of $y$ for the sake of computing $R$ (Fig. 3.21B). The intensity drops are then detected on each semi profile at $y_{drop} = 0.208R$ allowing us to measure $R$. This detection is realised by looking at the lowest absolute values of $y$ where the absolute value of the gradient of intensity $\nabla I = \partial_y I$ exceeds a certain threshold, which sets the
it to the shape of the meniscus. As depicted in Fig.3.21, a ray-tracing simulation was necessary
to link *R* to the profile *I*<sub>fluo</sub>. I developed the simulation with the following assumptions:

* The meniscus has its equilibrium section, that is three sectors of three circles of radii *R* tangentially connecting one another (cf. left of Fig3.21)

* We only consider the fluoresced light, which is supposed to be emitted in an homogeneous and isotropic way in the volume of the meniscus. Each emitted ray is subject to reflection/refraction laws at the three curved water/air interfaces.

* We only collect the rays going out of the meniscus horizontally, which we retrace back in the volume. Each segment of their trajectories contributes to the intensity proportionally to its length and the number of reflections (where most of the intensity is lost) before going out towards the camera.

The resulting intensity profile *I*<sub>fluo</sub> shown in Fig.3.21A closely resembles the experimental
observations depicted in Figure 3.21B, with a sudden drop of intensity beyond an ordinate *y* =
0.208*R* from the centre. This decrease is noticeable in the simulation as soon as we consider the
first reflection within the meniscus. Including more than the second reflection did not result in a
significant modification of the intensity pattern.

<figure><img src="image_86.png" /></figure>

Figure 3.21: Ray tracing simulation for a fluorescent meniscus. **A:** we define an ideal free meniscus composed of three interfaces of curvatures *R* in contact. The red lines are the outgoing rays seen by the side. On the right, the resulting intensity profile as a function of the ordinate (normalised by *R*), where the shaded coloured areas are the contributions of the different rays: ray without reflection (green), with one reflection (light blue), or two reflections (purple). **B:** Identification of the intensity drop on experimental data.

The global maximum of *I*(*y*) sets the origin of *y* for the sake of computing *R* (Fig.3.21B).
The intensity drops are then detected on each semi profile at *y*<sub>drop</sub> = 0.208*R* allowing us to
measure *R*. This detection is realised by looking at the lowest absolute values of *y* where the
absolute value of the gradient of intensity ∇*I* = ∂<sub>*y*</sub>*I* exceeds a certain threshold, which sets the
<figure><img src="image_87.png" /></figure>

Figure 3.22: Results of the detection via the side camera “BP” for an experiment at [DOH] = 35 mg/L and $V_{def}$ = 50 mm/s. **A:** Ordinate of the free meniscus as a function of time around the time of deformation. $Y_{BP}$ = 0 is set at $t_0$ = 28 s the time of preparation. **B:** Radius of curvature of the free meniscus measured over time. During the deformation itself (between $t_1$ = 32 s and $t$ = 32.1 s), the free meniscus goes too far off focus, leading to the presence of a blind zone in the detection.

edge of the drop in intensity of the semi-profile. From this detection, we extract for each frame two radii of curvature (for the top and bottom semi-profiles) which are very close in practice, and we define for each frame the radius $R(t)$ as the mean between the two. A typical profile is shown in Fig.3.22B. Note that we have a blind zone during the deformation, as the meniscus goes too far off focus (and probably out of our equilibrium assumption for the geometry), hindering the measurement of $R$.

### 3.6.3 Tracking the edges of Frankel films in the horizontal film 3

Extracted ingredient:

* $L_{Fr,3}$ (mm) the length of Frankel film near the free meniscus in film 3.

* $L_{Fr,3,ext}$ (mm) the length of film between the edge of the free meniscus and the Frankel film bordering motor 3.

After the preparation phase at $t_0$ = 28 s, two Frankel films are created in the plane of film 3, one at the free meniscus, one at the edge of the motor (see Fig.3.23C). Their respective areas decrease during the main deformation as we compress film 3. Our goal is to track the edges of these films during the main deformation. We will define all the positions we extract with respect to the edge of the free meniscus. The boundaries between the thin/thick pieces of film are material points (cf. subsection 0.3.2), they are highly relevant to track the in-plane motions in the $x$ direction.

The exact edge of the free meniscus is hard to find experimentally however, as the meniscus is saturated in fluorescent light, being much thicker than its neighbouring foam films, thus hindering
<figure><img src="image_87.png" /></figure>

Figure 3.22: Results of the detection via the side camera “BP” for an experiment at [DOH] = 35 mg/L and $V_{def}$ = 50 mm/s. **A:** Ordinate of the free meniscus as a function of time around the time of deformation. $Y_{BP}$ = 0 is set at $t_0$ = 28 s the time of preparation. **B:** Radius of curvature of the free meniscus measured over time. During the deformation itself (between $t_1$ = 32 s and $t$ = 32.1 s), the free meniscus goes too far off focus, leading to the presence of a blind zone in the detection.

edge of the drop in intensity of the semi-profile. From this detection, we extract for each frame two radii of curvature (for the top and bottom semi-profiles) which are very close in practice, and we define for each frame the radius $R(t)$ as the mean between the two. A typical profile is shown in Fig.3.22B. Note that we have a blind zone during the deformation, as the meniscus goes too far off focus (and probably out of our equilibrium assumption for the geometry), hindering the measurement of $R$.

### 3.6.3 Tracking the edges of Frankel films in the horizontal film 3

Extracted ingredient:

* $L_{Fr,3}$ (mm) the length of Frankel film near the free meniscus in film 3.

* $L_{Fr,3,ext}$ (mm) the length of film between the edge of the free meniscus and the Frankel film bordering motor 3.

After the preparation phase at $t_0$ = 28 s, two Frankel films are created in the plane of film 3, one at the free meniscus, one at the edge of the motor (see Fig.3.23C). Their respective areas decrease during the main deformation as we compress film 3. Our goal is to track the edges of these films during the main deformation. We will define all the positions we extract with respect to the edge of the free meniscus. The boundaries between the thin/thick pieces of film are material points (cf. subsection 0.3.2), they are highly relevant to track the in-plane motions in the $x$ direction.

The exact edge of the free meniscus is hard to find experimentally however, as the meniscus is saturated in fluorescent light, being much thicker than its neighbouring foam films, thus hindering
<figure><img src="image_88.png" /><figcaption>Figure 3.23: **A:** Geometry of the free meniscus of radius of curvature $R$. The black dot is the centre of the meniscus located in ($X_{BP}$, $Y_{BP}$), the green one the edge of the flat film, the red one the edge of the Frankel film produced during the preparation step. **B:** Intensity profile of **C** averaged in the $z$ direction taken at $t = 31.9$ s. **C:** Film 3 at $t = 31.9$ s (i.e. before the deformation) seen from the fluorescent camera PF from the top. The green line is the edge of the free meniscus. The red rectangle is where the correlation mask is chosen to track the Frankel film $L_{Fr,3}$. Similarly the blue rectangle is the correlation mask used to track the edge of the Frankel film bordering the motor to get $L_{Fr,3,ext}$. **D:** Positions of the edges of the Frankel films detected over the deformation. They are defined with respect to the edge of the free meniscus. The example we give is for a solution of concentration [DOH] = 35 mg/L whose films are deformed at $V_{def} = 20$ mm/s.</figcaption></figure>

a sharp measurement of its edge. To solve this problem, we must rely on the estimation of the curvature radius $R$ (determined in section 3.6.2) to be able to localise the edge. As shown in Fig.3.23A if we consider that the meniscus remains close to its equilibrium geometry, the meniscus edge is located in $X_{BP} + \frac{\sqrt{3}}{4}R$ (green dot in the figure), which we estimate using the radius $R(t_1 = 32 \text{s})$ found right before the deformation. This $\frac{\sqrt{3}}{4}R$ correction represents up to 10% of the values of $L_{Fr,3}$ shown in Fig.3.23D, and thus is better known than let to the uncertainties. Besides, we see with Fig.3.22B that the meniscus radius $R$ has varied little during the deformation (which is a blind zone for the measurement), thus we choose a single value $R$
<figure><img src="image_88.png" /><figcaption>Figure 3.23: **A:** Geometry of the free meniscus of radius of curvature $R$. The black dot is the centre of the meniscus located in ($X_{BP}$, $Y_{BP}$), the green one the edge of the flat film, the red one the edge of the Frankel film produced during the preparation step. **B:** Intensity profile of **C** averaged in the $z$ direction taken at $t = 31.9$ s. **C:** Film 3 at $t = 31.9$ s (i.e. before the deformation) seen from the fluorescent camera PF from the top. The green line is the edge of the free meniscus. The red rectangle is where the correlation mask is chosen to track the Frankel film $L_{Fr,3}$. Similarly the blue rectangle is the correlation mask used to track the edge of the Frankel film bordering the motor to get $L_{Fr,3,ext}$. **D:** Positions of the edges of the Frankel films detected over the deformation. They are defined with respect to the edge of the free meniscus. The example we give is for a solution of concentration [DOH] = 35 mg/L whose films are deformed at $V_{def} = 20$ mm/s.</figcaption></figure>

a sharp measurement of its edge. To solve this problem, we must rely on the estimation of the curvature radius $R$ (determined in section 3.6.2) to be able to localise the edge. As shown in Fig.3.23A if we consider that the meniscus remains close to its equilibrium geometry, the meniscus edge is located in $X_{BP} + \frac{\sqrt{3}}{4}R$ (green dot in the figure), which we estimate using the radius $R(t_1 = 32 \text{s})$ found right before the deformation. This $\frac{\sqrt{3}}{4}R$ correction represents up to 10% of the values of $L_{Fr,3}$ shown in Fig.3.23D, and thus is better known than let to the uncertainties. Besides, we see with Fig.3.22B that the meniscus radius $R$ has varied little during the deformation (which is a blind zone for the measurement), thus we choose a single value $R$
for this correction that is the value at $t_1 = 32$ s.

The tracking of $L_{Fr,3}$ is performed by image correlation on successive frames. We initialise the procedure right before the main deformation at $t = 31.9$ s by taking the intensity profile $I_{avg}$ averaged over the $z$ direction, with $z$ in the range delimited by the coloured rectangles shown in Fig.3.23. Note that the lighting increases from left to right in Fig.3.23B, leading to the intensity gradient of fluoresced light, but both Frankel films at both edges have very similar thicknesses as shown later in section 3.6.5. Using the first and second derivatives $\partial_x I_{avg}$ and $\partial_{xx} I_{avg}$, we look where we have simultaneously a negative enough slope (with a well-chosen threshold) and an inflection point on the second derivative. The lowest $x$ position is retained as the initial position of the Frankel film on the free meniscus side and is shown in Fig.3.23B with the vertical red line. Once this initial position is found, a first correlation mask is defined around it by cropping the image with an area shown in red in Fig.3.23C. Each following frame is then compared with this mask, where the maximum of correlation along the $x$-axis is found and set to be the new position of the edge of Frankel film, allowing us to draw $L_{Fr,3}(t)$. The mask is refreshed every few frames in order to keep an accurate detection.

The detection of $L_{Fr,3,ext}(t)$ by tracking the edge of Frankel film at motor 3 (blue lines and mask in Fig.3.23) is done in a similar fashion. The only difference is the initialisation done by clicking directly on the intensity profile Fig.3.23B while monitoring the frame Fig.3.23C. This is due to a less repeatable initial state: the initial position of this Frankel film can vary by up to a millimetre, and be-neighboured by a central film of different thickness heterogeneities pattern.

### 3.6.4 Tracking the Frankel film in the top left film 1

**Extracted ingredient: $L_{Fr,1}$ (mm) the length of Frankel film going out of the free meniscus at the bottom of film 1 as it is stretched during the main deformation.**

In this subsection, we detail how we detect the outgoing Frankel film in the stretched film 1
using the TPF camera.

In Fig.3.24A-C we detail the relative position of the camera with respect to the films, and
show a typical image obtained as we record. For each frame, we first detect the position of
the free meniscus which we set to be the origin of the local axis $x_1$ as shown in Fig3.24D. This
is done by looking at the position of the middle of the rightmost saturated continuum of the
cropped frame Fig.3.24C. As the image of motor 3 is also saturated, this works only when both
domains are separated (see Fig.3.24C). At longer times, motor 3 becomes too close and only the
right edge of the saturated fringe of the meniscus can be distinguished. Thus we keep memory
of the semi-length of the free meniscus (which varies very little throughout the experiment), and
place the origin of $x_1$ with respect to the rightmost edge of saturation by retrieving the stored
semi-width.

The detection of the edge of the Frankel film located along the free meniscus in film 1 is
based on the same principle as the initialisation of the algorithm of the previous subsection 3.6.3.

The detected edge of Frankel film allows us to define $L_{Fr,1}$ as the length between it and the
edge of free meniscus, whose position is deduced in the same way as for the previous subsection
3.6.3 by taking into account the length $\frac{\sqrt{3}}{4}R$. We plot an example of result in Fig.3.24E where
for this correction that is the value at $t_1 = 32$ s.

The tracking of $L_{Fr,3}$ is performed by image correlation on successive frames. We initialise the procedure right before the main deformation at $t = 31.9$ s by taking the intensity profile $I_{avg}$ averaged over the $z$ direction, with $z$ in the range delimited by the coloured rectangles shown in Fig.3.23. Note that the lighting increases from left to right in Fig.3.23B, leading to the intensity gradient of fluoresced light, but both Frankel films at both edges have very similar thicknesses as shown later in section 3.6.5. Using the first and second derivatives $\partial_x I_{avg}$ and $\partial_{xx} I_{avg}$, we look where we have simultaneously a negative enough slope (with a well-chosen threshold) and an inflection point on the second derivative. The lowest $x$ position is retained as the initial position of the Frankel film on the free meniscus side and is shown in Fig.3.23B with the vertical red line. Once this initial position is found, a first correlation mask is defined around it by cropping the image with an area shown in red in Fig.3.23C. Each following frame is then compared with this mask, where the maximum of correlation along the $x$-axis is found and set to be the new position of the edge of Frankel film, allowing us to draw $L_{Fr,3}(t)$. The mask is refreshed every few frames in order to keep an accurate detection.

The detection of $L_{Fr,3,ext}(t)$ by tracking the edge of Frankel film at motor 3 (blue lines and mask in Fig.3.23) is done in a similar fashion. The only difference is the initialisation done by clicking directly on the intensity profile Fig.3.23B while monitoring the frame Fig.3.23C. This is due to a less repeatable initial state: the initial position of this Frankel film can vary by up to a millimetre, and be-neighboured by a central film of different thickness heterogeneities pattern.

### 3.6.4 Tracking the Frankel film in the top left film 1

**Extracted ingredient: $L_{Fr,1}$ (mm) the length of Frankel film going out of the free meniscus at the bottom of film 1 as it is stretched during the main deformation.**

In this subsection, we detail how we detect the outgoing Frankel film in the stretched film 1
using the TPF camera.

In Fig.3.24A-C we detail the relative position of the camera with respect to the films, and
show a typical image obtained as we record. For each frame, we first detect the position of
the free meniscus which we set to be the origin of the local axis $x_1$ as shown in Fig3.24D. This
is done by looking at the position of the middle of the rightmost saturated continuum of the
cropped frame Fig.3.24C. As the image of motor 3 is also saturated, this works only when both
domains are separated (see Fig.3.24C). At longer times, motor 3 becomes too close and only the
right edge of the saturated fringe of the meniscus can be distinguished. Thus we keep memory
of the semi-length of the free meniscus (which varies very little throughout the experiment), and
place the origin of $x_1$ with respect to the rightmost edge of saturation by retrieving the stored
semi-width.

The detection of the edge of the Frankel film located along the free meniscus in film 1 is
based on the same principle as the initialisation of the algorithm of the previous subsection 3.6.3.

The detected edge of Frankel film allows us to define $L_{Fr,1}$ as the length between it and the
edge of free meniscus, whose position is deduced in the same way as for the previous subsection
3.6.3 by taking into account the length $\frac{\sqrt{3}}{4}R$. We plot an example of result in Fig.3.24E where
<figure><img src="image_89.png" /></figure>

Figure 3.24: Detection process of the outgoing Frankel film in film 1, the example is given for an experiment at [DOH] = 35 mg/L and $V_1 = 50$ mm/s. **A:** Full frame image of film 1 at $t = 32.1$ s at the end of the main deformation. Film 2 is also visible on the left. **B:** Schematic position of the camera with respect to the elementary foam. **C:** Cropped portion of the frame used to detect the outgoing Frankel film. **D:** Intensity profile $I_{avg}$ obtained by averaging **C** over the $z$ axis. The leftmost saturated fringe is motor 3 entering the field of view at the end of the deformation, the rightmost one whose centre is set at $x_1 = 0$ is the free meniscus. The green line is the edge of the free meniscus, the blue one is the detected position of the edge of outgoing Frankel film. **E:** Detected length of outgoing Frankel film $L_{Fr,1}(t)$ as a function of time. It is defined with respect to the edge of the free meniscus. At the earliest times of deformation the detection is hindered by the saturating brightness of the free meniscus.

$t_1 = 32$ s is the time of deformation. It has to be noted however that at earliest times after $t_1 = 32$ s, when the deformation as begun, the edge of Frankel film is not detected as it is too close from the saturated fringe of the free meniscus, hence the presence of a blind zone in Fig.3.24E.

### 3.6.5 Thickness profiles with hyperspectral cameras

**Extracted ingredients:**

* $H_1(x_1, t)$ (µm) the thickness profiles of film 1 before the main deformation (for $t < t_1 = 32$ s).

* $H_2(x_2, t)$ (µm) the thickness profiles of film 2 before the main deformation (for $t < t_1 = 32$ s).
<figure><img src="image_89.png" /></figure>

Figure 3.24: Detection process of the outgoing Frankel film in film 1, the example is given for an experiment at [DOH] = 35 mg/L and $V_1 = 50$ mm/s. **A:** Full frame image of film 1 at $t = 32.1$ s at the end of the main deformation. Film 2 is also visible on the left. **B:** Schematic position of the camera with respect to the elementary foam. **C:** Cropped portion of the frame used to detect the outgoing Frankel film. **D:** Intensity profile $I_{avg}$ obtained by averaging **C** over the $z$ axis. The leftmost saturated fringe is motor 3 entering the field of view at the end of the deformation, the rightmost one whose centre is set at $x_1 = 0$ is the free meniscus. The green line is the edge of the free meniscus, the blue one is the detected position of the edge of outgoing Frankel film. **E:** Detected length of outgoing Frankel film $L_{Fr,1}(t)$ as a function of time. It is defined with respect to the edge of the free meniscus. At the earliest times of deformation the detection is hindered by the saturating brightness of the free meniscus.

$t_1 = 32$ s is the time of deformation. It has to be noted however that at earliest times after $t_1 = 32$ s, when the deformation as begun, the edge of Frankel film is not detected as it is too close from the saturated fringe of the free meniscus, hence the presence of a blind zone in Fig.3.24E.

### 3.6.5 Thickness profiles with hyperspectral cameras

Extracted ingredients:

* $H_1(x_1, t)$ (µm) the thickness profiles of film 1 before the main deformation (for $t < t_1 = 32$ s).

* $H_2(x_2, t)$ (µm) the thickness profiles of film 2 before the main deformation (for $t < t_1 = 32$ s).
* $H_3(x_3, t)$ (μm) the thickness profiles of film 2 before the main deformation (for $t < t_1 = 32$ s).

<figure><img src="image_90.png" /></figure>

Figure 3.25: Example of thickness profiles measurements on [DOH] = 35 mg/L at $t = 31.9$ s just before the deformation. **A:** Spectral image of film 2. $x_2$ is the local abscissa aligned with the axis of deformation of the film defined so that $x_2 = 0$ is the position of the free meniscus (cf. Fig.3.11). The ordinate is the wavelength $\lambda$ for each value of which we measure the light intensity. **B:** In blue, we show a typical spectrum obtained at a given position $x_2 = 3$ mm located on **A** by the blue line. In red, we show the average light spectrum, obtained by averaging the spectra over time and $x_2$. **C:** Filtered spectrum extracted from **B** which varies as $cos(\alpha H/\lambda)$ from which we extract the thickness $H$. **D-E:** Thickness profiles obtained for the 3 different films along their respective local abscissas $x_1$, $x_2$ and $x_3$ (with $x_3 = x$) at $t = 31.9$ s just before the deformation.

We probe the system with our hyperspectral cameras to get the initial distributions of thicknesses of the three different films before deforming them at various velocities. This will allow to monitor the differences of initial states between the films, as gravity drainage occurs before we start deforming the films at $t_1 = 32$ s. However, measuring during the deformation was not possible, and motion blur could be a reason.

In order to measure the profiles, we collect the movies of the spectra for the three films, of which an example of frame right before the deformation at $t = 31.9$ s is shown in Fig.3.25A. The procedure is then similar to what is detailed in chapter 2, subsection 2.3: The raw intensity profiles $I$ (Fig.3.25B) are processed in order to get the normalised and centred profiles $\tilde{I}$ (Fig.3.25C) from which a thickness $H(y, t)$ is extracted.

Fig.3.25D-F show the typical profiles measured for the three films of the elementary foam at $t = 31.9$ s for the chemistry [DOH] = 35 mg/L. These are the profiles right before the main
* $H_3(x_3, t)$ (µm) the thickness profiles of film 2 before the main deformation (for $t < t_1 = 32$ s).

<figure><img src="image_90.png" /><figcaption>Figure 3.25: Example of thickness profiles measurements on [DOH] = 35 mg/L at $t = 31.9$ s just before the deformation. **A:** Spectral image of film 2. $x_2$ is the local abscissa aligned with the axis of deformation of the film defined so that $x_2 = 0$ is the position of the free meniscus (cf. Fig.3.11). The ordinate is the wavelength $\lambda$ for each value of which we measure the light intensity. **B:** In blue, we show a typical spectrum obtained at a given position $x_2 = 3$ mm located on **A** by the blue line. In red, we show the average light spectrum, obtained by averaging the spectra over time and $x_2$. **C:** Filtered spectrum extracted from **B** which varies as $\cos(\alpha H/\lambda)$ from which we extract the thickness $H$. **D-E:** Thickness profiles obtained for the 3 different films along their respective local abscissas $x_1$, $x_2$ and $x_3$ (with $x_3 = x$) at $t = 31.9$ s just before the deformation.</figcaption></figure>

We probe the system with our hyperspectral cameras to get the initial distributions of thicknesses of the three different films before deforming them at various velocities. This will allow to monitor the differences of initial states between the films, as gravity drainage occurs before we start deforming the films at $t_1 = 32$ s. However, measuring during the deformation was not possible, and motion blur could be a reason.

In order to measure the profiles, we collect the movies of the spectra for the three films, of which an example of frame right before the deformation at $t = 31.9$ s is shown in Fig.3.25A. The procedure is then similar to what is detailed in chapter 2, subsection 2.3: The raw intensity profiles $I$ (Fig.3.25B) are processed in order to get the normalised and centred profiles $\tilde{I}$ (Fig.3.25C) from which a thickness $H(y, t)$ is extracted.

Fig.3.25D-F show the typical profiles measured for the three films of the elementary foam at $t = 31.9$ s for the chemistry [DOH] = 35 mg/L. These are the profiles right before the main
deformation at $t_1 = 32$ s, and we see that film 2 (at the bottom) is twice as thick as the other films because of gravity drainage.

### 3.6.6 Tracking of the photobleached dots

Extracted ingredients:

* $L_{c,1}$ (mm) the distance between the edge of the free meniscus and the first column of dots.

* $L_{c,2}$ (mm) the distance between the edge of the free meniscus and the second column of dots.

* $(X_k, Z_k)$ (mm) with $k = [1, 16]$ the positions of the photobleached dots

<figure><img src="image_91.png" /><figcaption>Figure 3.26: Example given on [DOH] = 35 mg/L at $t = 31.99$ s just before the deformation. **A:** Zoomed frame of the PF camera in the region where the photobleached dots are created, near the free meniscus, which appears as the vertical saturated fringe. The yellow disk is the position of a dot, the yellow rectangle is the correlation mask chosen around this position, and the green rectangle is the correlation area where the mask will be moved in the next frame to seek the new position of the dot. **B-C:** Intensity profiles $I$ along the $x$ and $z$ axis taken along the lines of matching colours in **A**. The coloured lines are only here as guides for the eye. The black lines are the affine profiles $I_{grad}$ computed using the average profiles over both directions of the correlation area. This correction is especially necessary in the $x$ direction as the free meniscus fluoresces much more than the foam film. **D-E:** Corrected profiles $I_{corr}$ where we suppress the local gradient along the given axis in order to better localise the local minimum of intensity. The apex of the parabolic fits gives a subpixellar estimation of the position.</figcaption></figure>
deformation at $t_1 = 32$ s, and we see that film 2 (at the bottom) is twice as thick as the other films because of gravity drainage.

### 3.6.6 Tracking of the photobleached dots

Extracted ingredients:

* $L_{c,1}$ (mm) the distance between the edge of the free meniscus and the first column of dots.

* $L_{c,2}$ (mm) the distance between the edge of the free meniscus and the second column of dots.

* $(X_k, Z_k)$ (mm) with $k = [1, 16]$ the positions of the photobleached dots

<figure><img src="image_91.png" /><figcaption>Figure 3.26: Example given on [DOH] = 35 mg/L at $t = 31.99$ s just before the deformation. **A:** Zoomed frame of the PF camera in the region where the photobleached dots are created, near the free meniscus, which appears as the vertical saturated fringe. The yellow disk is the position of a dot, the yellow rectangle is the correlation mask chosen around this position, and the green rectangle is the correlation area where the mask will be moved in the next frame to seek the new position of the dot. **B-C:** Intensity profiles $I$ along the $x$ and $z$ axis taken along the lines of matching colours in **A**. The coloured lines are only here as guides for the eye. The black lines are the affine profiles $I_{grad}$ computed using the average profiles over both directions of the correlation area. This correction is especially necessary in the $x$ direction as the free meniscus fluoresces much more than the foam film. **D-E:** Corrected profiles $I_{corr}$ where we suppress the local gradient along the given axis in order to better localise the local minimum of intensity. The apex of the parabolic fits gives a subpixellar estimation of the position.</figcaption></figure>
After the preparation phase at $t_0 = 28$ s, which ends at around $t = 31$ s, the system is let to relax around one second before the deformation at $t_1 = 32$ s. In the mean time, we use our cyan laser to photobleach two rows of dots in the Frankel film bordering the free meniscus in film 3. Once the last dot is shot, we start tracking them individually via the PF camera. They remain aligned in their respective rows as they are pushed towards the free meniscus when the “Push 3” deformation occurs with a very good repeatability. The initial positions at $t = 31.9$ s are roughly determined by clicking all the dots of each first frame of each experiment. The true initial positions are found by looking for the local minimum of intensity in the close vicinity of the clicked pixels.

Once the initial position of a dot is given, we start by defining a correlation mask around it, shown in the form of a yellow square in Fig.3.26A. We then look in the next frame the position where it correlates the most in a wider correlation area defined by the green rectangle in Fig.3.26A. This gives a first estimate of the next position of the dot, however, we need to look around it for the real local minimum of intensity, as even an offset of one pixel would rapidly lead to a misalignment between the centre of the next correlation mask and the centre of the tracked dot.

In order to do so, we look at the intensity profiles $I_{avg,z}(x)$ and $I_{avg,x}(z)$ averaged and computed over the dimensions of our correlation area and we compute their respective gradients. This leads to the definition of a correcting intensity surface, which is a tilted planar surface we retrieve to the original intensity surface. In Fig.3.26B-C two examples of raw intensity profiles $I(x)$ and $I(z)$ are chosen respectively along the blue and red lines of Fig.3.26A. In black, we display the calculated gradients of the average profiles, and in Fig.3.26D-E, we have the corrected profiles along the same lines. Using these last two, we can find the pixel of the local minimum of intensity, which will be defining the next correlation mask.

Eventually, we estimate the position of the local minimum with a subpixellar resolution by fitting the intensity profiles along $x$ and $z$ with two parabolas around the pixel of minimum of intensity. The coordinated found by doing so are stored as the $X_k$ and $Z_k$ positions of dot $k$ for the given time of the frame.

Using these positions, we are able to evaluate $L_{c,1}$ and $L_{c,2}$ (Fig.3.27) the distance between the edge of the free meniscus and the average position of each column 1 and 2 (containing 8 dots each), with whom we will later try to evaluate the compression of film 3 $\epsilon_3$. The positions of the dots $Z_k$ along $z$ will also allow us to monitor the nullity of the compression $\epsilon_z$ along $z$, which is a key measurement to ensure our invariance by translation along this direction (see section 3.7).

## 3.7 Monitoring the invariance by translation along $z$

All the experiments and result we aim to establish have been thought as invariant by translation along the $z$ axis. The main direct consequence of this sought invariance is to allow us to write all compressions happening in the film as uniaxial compressions, meaning any surface compression can be written discarding the $z$ terms:

$$ \frac{1}{A} \frac{dA}{dt} = \frac{\partial v_x}{\partial x} \qquad (3.38) $$

Where $A$ is an area of a piece of film and $v_x$ the local velocity along the $x$ axis.
After the preparation phase at $t_0 = 28$ s, which ends at around $t = 31$ s, the system is let to relax around one second before the deformation at $t_1 = 32$ s. In the mean time, we use our cyan laser to photobleach two rows of dots in the Frankel film bordering the free meniscus in film 3. Once the last dot is shot, we start tracking them individually via the PF camera. They remain aligned in their respective rows as they are pushed towards the free meniscus when the “Push 3” deformation occurs with a very good repeatability. The initial positions at $t = 31.9$ s are roughly determined by clicking all the dots of each first frame of each experiment. The true initial positions are found by looking for the local minimum of intensity in the close vicinity of the clicked pixels.

Once the initial position of a dot is given, we start by defining a correlation mask around it, shown in the form of a yellow square in Fig.3.26A. We then look in the next frame the position where it correlates the most in a wider correlation area defined by the green rectangle in Fig.3.26A. This gives a first estimate of the next position of the dot, however, we need to look around it for the real local minimum of intensity, as even an offset of one pixel would rapidly lead to a misalignment between the centre of the next correlation mask and the centre of the tracked dot.

In order to do so, we look at the intensity profiles $I_{avg,z}(x)$ and $I_{avg,x}(z)$ averaged and computed over the dimensions of our correlation area and we compute their respective gradients. This leads to the definition of a correcting intensity surface, which is a tilted planar surface we retrieve to the original intensity surface. In Fig.3.26B-C two examples of raw intensity profiles $I(x)$ and $I(z)$ are chosen respectively along the blue and red lines of Fig.3.26A. In black, we display the calculated gradients of the average profiles, and in Fig.3.26D-E, we have the corrected profiles along the same lines. Using these last two, we can find the pixel of the local minimum of intensity, which will be defining the next correlation mask.

Eventually, we estimate the position of the local minimum with a subpixellar resolution by fitting the intensity profiles along $x$ and $z$ with two parabolas around the pixel of minimum of intensity. The coordinated found by doing so are stored as the $X_k$ and $Z_k$ positions of dot $k$ for the given time of the frame.

Using these positions, we are able to evaluate $L_{c,1}$ and $L_{c,2}$ (Fig.3.27) the distance between the edge of the free meniscus and the average position of each column 1 and 2 (containing 8 dots each), with whom we will later try to evaluate the compression of film 3 $\epsilon_3$. The positions of the dots $Z_k$ along $z$ will also allow us to monitor the nullity of the compression $\epsilon_z$ along $z$, which is a key measurement to ensure our invariance by translation along this direction (see section 3.7).

## 3.7 Monitoring the invariance by translation along $z$

All the experiments and result we aim to establish have been thought as invariant by translation along the $z$ axis. The main direct consequence of this sought invariance is to allow us to write all compressions happening in the film as uniaxial compressions, meaning any surface compression can be written discarding the $z$ terms:

$$ \frac{1}{A} \frac{dA}{dt} = \frac{\partial v_x}{\partial x} \qquad (3.38) $$

Where $A$ is an area of a piece of film and $v_x$ the local velocity along the $x$ axis.
<figure><img src="image_92.png" /></figure>

Figure 3.27: Monitoring the positions of the edge of prepared Frankel film and the photo-bleached columns. **A:** Zoom near the meniscus in film 3. The vertical bright fringe on the middle left is the free meniscus, film 3 is on its right. The red line localises the detected edge of prepared Frankel film, whose distance from the edge of free meniscus is $L_{Fr,3}$, the green and yellow lines localise the two columns of photobleached dots, whose distances from the edge of meniscus are $L_{c,1}$ and $L_{c,2}$. **B:** Time evolution of the distances defined in **A**. The main deformation happens at $t_1 = 32$ s, where film 3 is compressed.

<figure><img src="image_93.png" /></figure>

Figure 3.28: Measurement of the compression along the *z* axis on an example of [DOH] = 35 mg/L at $V_1$ = 35 mm/s. **A:** We track the lengths $l_{z,1}$ and $l_{z,2}$ defined as the distances between the upper and lower dots of each column throughout the whole experiment. **B:** We define the deformation along *z*, $\epsilon_z = l_z/l_{z,init} - 1$ and make an average of this quantity over both column and over all the repetitions of the experiments. The error is the standard deviation associated with this average. The vertical black dotted line is the time $t_1 = 32$ s at which the main deformation starts.
<figure><img src="image_92.png" /></figure>

Figure 3.27: Monitoring the positions of the edge of prepared Frankel film and the photo-bleached columns. **A:** Zoom near the meniscus in film 3. The vertical bright fringe on the middle left is the free meniscus, film 3 is on its right. The red line localises the detected edge of prepared Frankel film, whose distance from the edge of free meniscus is $L_{Fr,3}$, the green and yellow lines localise the two columns of photobleached dots, whose distances from the edge of meniscus are $L_{c,1}$ and $L_{c,2}$. **B:** Time evolution of the distances defined in **A**. The main deformation happens at $t_1 = 32$ s, where film 3 is compressed.

<figure><img src="image_93.png" /></figure>

Figure 3.28: Measurement of the compression along the *z* axis on an example of [DOH] = 35 mg/L at $V_1$ = 35 mm/s. **A:** We track the lengths $l_{z,1}$ and $l_{z,2}$ defined as the distances between the upper and lower dots of each column throughout the whole experiment. **B:** We define the deformation along *z*, $\epsilon_z = l_z/l_{z,init} - 1$ and make an average of this quantity over both column and over all the repetitions of the experiments. The error is the standard deviation associated with this average. The vertical black dotted line is the time $t_1 = 32$ s at which the main deformation starts.
The main indicator that it is verified is by monitoring with the photobleached dots that the compression $\epsilon_z$ along $z$ is close to zero. This could be defaulted by two main reasons more or less easy to identify.

## Two possible causes for a broken invariance

* A local one: the meniscus-flat film junction instability, which tends to create semicircular patches of thin film spontaneously going out of the meniscus (see section 0.3.4 relying on the work of Trégouët et al. [66], Gros et al. [107]). This one can be easily seen once it is triggered, as the thin semicircles are regularly dispatched along the $z$ axis and hinder thicker film to approach the meniscus.

* A global one: the relaxation of the thick Frankel film we created next to the free meniscus during the preparation. Due to the presence of a line tension between the thick prepared Frankel film and the thin film preexisting the preparation phase of “Push 3”(see section 3.5 for the preparation step, chapter 2 section 2.2 for the line tension), the Frankel film tends to relax at longer times. This can be directly seen far from the centre, where the lighting is poor, making this cause for a broken invariance less obvious and more dangerous.

In both cases, measuring that the field $\epsilon_z$ is null is the main indicator that these phenomena do not hinder the measurement.

## Control of the invariance

In order to do so, we use our photobleached dots and define the distances $l_z$ between dots aligned along the $z$ axis as illustrated in Fig.3.28A. The deformation along $z$ is thus defined as:

$$ \epsilon_z = \frac{l_z}{l_{z,init}} - 1 \qquad (3.39) $$

Where $l_{z,init}$ is the initial length taken right at the beginning of the dot tracking, that is after the preparation phase. Thus it does not take into account the pre-deformation applied to film 3 during this phase. However, it shows how little it varies anyway during the main deformation. In Fig.3.28B, we see that $\epsilon_z$ remains of the order of a few percents, and does not vary significantly during the main deformation occurring at $t_1 = 32$ s. With the given example, the deformation lasts until $t = 31.14$ s, but tracking the dots becomes impossible before this time, hence the range of time we display in the figure.

These values are to be compared to the deformation seen on the other axis of the foam film, that is along $x$, where we anticipate that the deformations $\epsilon_x \gg \epsilon_z$ will be much greater as shown later in subsection 3.8.2. In the following, we will assume this invariance along $z$ to hold, and we will resume discussing its validity later while computing the $\epsilon_x$'s. We also extend this hypothesis of invariance to the preparation phase, before we are able to create the dots, which is reasonable although not measured directly.
The main indicator that it is verified is by monitoring with the photobleached dots that the compression $\epsilon_z$ along $z$ is close to zero. This could be defaulted by two main reasons more or less easy to identify.

## Two possible causes for a broken invariance

* A local one: the meniscus-flat film junction instability, which tends to create semicircular patches of thin film spontaneously going out of the meniscus (see section 0.3.4 relying on the work of Trégouët et al. [66], Gros et al. [107]). This one can be easily seen once it is triggered, as the thin semicircles are regularly dispatched along the $z$ axis and hinder thicker film to approach the meniscus.

* A global one: the relaxation of the thick Frankel film we created next to the free meniscus during the preparation. Due to the presence of a line tension between the thick prepared Frankel film and the thin film preexisting the preparation phase of “Push 3”(see section 3.5 for the preparation step, chapter 2 section 2.2 for the line tension), the Frankel film tends to relax at longer times. This can be directly seen far from the centre, where the lighting is poor, making this cause for a broken invariance less obvious and more dangerous.

In both cases, measuring that the field $\epsilon_z$ is null is the main indicator that these phenomena do not hinder the measurement.

## Control of the invariance

In order to do so, we use our photobleached dots and define the distances $l_z$ between dots aligned along the $z$ axis as illustrated in Fig.3.28A. The deformation along $z$ is thus defined as:

$$ \epsilon_z = \frac{l_z}{l_{z,init}} - 1 \qquad (3.39) $$

Where $l_{z,init}$ is the initial length taken right at the beginning of the dot tracking, that is after the preparation phase. Thus it does not take into account the pre-deformation applied to film 3 during this phase. However, it shows how little it varies anyway during the main deformation. In Fig.3.28B, we see that $\epsilon_z$ remains of the order of a few percents, and does not vary significantly during the main deformation occurring at $t_1 = 32$ s. With the given example, the deformation lasts until $t = 31.14$ s, but tracking the dots becomes impossible before this time, hence the range of time we display in the figure.

These values are to be compared to the deformation seen on the other axis of the foam film, that is along $x$, where we anticipate that the deformations $\epsilon_x \gg \epsilon_z$ will be much greater as shown later in subsection 3.8.2. In the following, we will assume this invariance along $z$ to hold, and we will resume discussing its validity later while computing the $\epsilon_x$'s. We also extend this hypothesis of invariance to the preparation phase, before we are able to create the dots, which is reasonable although not measured directly.
## 3.8 Computing tensions, deformations and elasticites of the films

This section details how we use the processed data from section 3.6 to extract the relevant physical quantities with respect to the model presented in section 3.2.

The first quantity is the ratio of tensions $\sigma_3/\sigma_1$ between the compressed/stretched films. Theoretically, it gives the boundary conditions $s \to \pm\infty$ for region C in terms of surface tensions $\gamma = \sigma/2$. Experimentally, it is extracted by monitoring the angle $\theta$ between the films 1 and 2.

The second kind of quantities are the deformations of the films $\epsilon_1$ and $\epsilon_3$. Theoretically, they give access to the quantities $L_{out}$ and $L_{in}$ of interface exchanged between the films, enabling the quantification of the flux $j_m$. Experimentally, they are first deduced by tracking the lengths of well-identified pieces of film (seen as closed systems).

The third and last quantity is the film elasticity of our elementary foam $E_f$. It allows us to refine our measurements of the deformations $\epsilon_1$ and $\epsilon_3$, and to have access to the individual absolute tensions $\sigma_1$ and $\sigma_3$, using their ratio found with the angle measurement. Experimentally, it is found with a linearisation of our quantities around a specific time of the experiment, which is more detailed later.

Note that our data processing contains several redundancies, which we consider of high importance to ensure the robustness and consistency of the analysis. The film elasticity is thus measured *in situ*, and compared to the values obtained in chapter 1.

### 3.8.1 Relative tensions between the films

As the films are stretched and compressed, their tensions change, leading to a change in the angle between the films as the meniscus at the triple point has to maintain its force balance. This change in angle results in a displacement of the meniscus, which we measure, as shown in section 3.6. In this subsection, we detail how we start from our displacement measurement ($X_{BP}$, $Y_{BP}$) to be able to compute a ratio of tensions $\sigma_3/\sigma_{1,2}$ between the stretched film 3 and the compressed films 1 and 2.

#### Raw data and minor corrections

Let us start with the horizontal displacement $X_{BP}$ as it carries most of the information on the angle between the film, considering the configuration of the “Push 3” deformation. In Fig.3.29, we show a typical graphs $X_{BP} = f(t)$ for various velocities of deformation $V_1$ at a given chemistry ($[DOH] = 50$ mg/L). The uncertainty on the reproducibility is of the order of 0.5%, which is a key indicator that the experimental reproductibility is excellent and the sub-pixellar correlation techniques detailed in subsection 3.6.1 went well.

A zoom in the very early times of deformations at $t_0$ and $t_1$ in Fig.3.30 reveals two things:

* A very short delay followed by oscillations, mainly due to the inertia of both the motors and the meniscus.
## 3.8 Computing tensions, deformations and elasticites of the films

This section details how we use the processed data from section 3.6 to extract the relevant physical quantities with respect to the model presented in section 3.2.

The first quantity is the ratio of tensions $\sigma_3/\sigma_1$ between the compressed/stretched films. Theoretically, it gives the boundary conditions $s \to \pm\infty$ for region C in terms of surface tensions $\gamma = \sigma/2$. Experimentally, it is extracted by monitoring the angle $\theta$ between the films 1 and 2.

The second kind of quantities are the deformations of the films $\epsilon_1$ and $\epsilon_3$. Theoretically, they give access to the quantities $L_{out}$ and $L_{in}$ of interface exchanged between the films, enabling the quantification of the flux $j_m$. Experimentally, they are first deduced by tracking the lengths of well-identified pieces of film (seen as closed systems).

The third and last quantity is the film elasticity of our elementary foam $E_f$. It allows us to refine our measurements of the deformations $\epsilon_1$ and $\epsilon_3$, and to have access to the individual absolute tensions $\sigma_1$ and $\sigma_3$, using their ratio found with the angle measurement. Experimentally, it is found with a linearisation of our quantities around a specific time of the experiment, which is more detailed later.

Note that our data processing contains several redundancies, which we consider of high importance to ensure the robustness and consistency of the analysis. The film elasticity is thus measured *in situ*, and compared to the values obtained in chapter 1.

### 3.8.1 Relative tensions between the films

As the films are stretched and compressed, their tensions change, leading to a change in the angle between the films as the meniscus at the triple point has to maintain its force balance. This change in angle results in a displacement of the meniscus, which we measure, as shown in section 3.6. In this subsection, we detail how we start from our displacement measurement ($X_{BP}$, $Y_{BP}$) to be able to compute a ratio of tensions $\sigma_3/\sigma_{1,2}$ between the stretched film 3 and the compressed films 1 and 2.

#### Raw data and minor corrections

Let us start with the horizontal displacement $X_{BP}$ as it carries most of the information on the angle between the film, considering the configuration of the “Push 3” deformation. In Fig.3.29, we show a typical graphs $X_{BP} = f(t)$ for various velocities of deformation $V_1$ at a given chemistry ($[DOH] = 50$ mg/L). The uncertainty on the reproducibility is of the order of 0.5%, which is a key indicator that the experimental reproductibility is excellent and the sub-pixellar correlation techniques detailed in subsection 3.6.1 went well.

A zoom in the very early times of deformations at $t_0$ and $t_1$ in Fig.3.30 reveals two things:

* A very short delay followed by oscillations, mainly due to the inertia of both the motors and the meniscus.
* An apparent free slip of the free meniscus of the order of 10 µm in the frame of the cameras as it starts to move.

The inertial oscillations are small in comparison with the main deformation and the meniscus acceleration will be neglected in the following. The apparent slip is negligible too, but its origin is more puzzling. the existence of this slip is clearly due to an initial repositioning of the films with respect to their meniscii at the motor edges. This slip is suspected to occur also at $t_1 = 32$ s in the opposite direction as motor 3 pushes back its film. We choose to correct the horizontal positions and to use $X_{BP,corr}$, where we removed a $+\Delta X_{slip} \sim 30$ µm jump at $t_0 = 28$ s, and a $-\Delta X_{slip}$ jump at $t_1 = 32$ s. This correction is very small in any case. A zoom at $t_0 = 28$ s is shown in Fig.3.30 to illustrate this procedure.

<figure><img src="image_94.png" /><figcaption>Figure 3.29: Horizontal displacement of the free meniscus throughout the whole “Push 3” experiment, for [DOH] = 50 mg/L and a varying velocity of main deformation $V_1$. The figure on the right is just a zoom around $t_1 = 32$ s of the figure on the left. At $t_0 = 28$ s, the horizontal film 3 is prepared, meaning it is being stretched at $V_0 = 2$ mm/s from 4.2 mm to 10 mm, leading to a first displacement of the free meniscus in $+x$. At $t_1 = 32$ s, the main deformation of the “Push 3” experiment is performed at $V_1$ varying between 5 – 50 mm/s.</figcaption></figure>

The other component of the displacement of the meniscus, $Y_{BP}$, which is measured using the BP camera as detailed in subsection 3.6.2. Given the geometry of the “Push 3” deformation, a non-zero vertical displacement represents a breaking of symmetry for the system induced by gravity. In Fig.3.31, we show a typical trajectory for the meniscus throughout the whole experiment, and we make sure that our assumption of a top-down symmetry remains valid as we have $X_{BP} \gg Y_{BP}$. $Y_{BP}$ remains small, and no slipping behaviour as for $X_{BP}$ was observed at our resolution (even though it is one order of magnitude greater than on the $x$ axis, given the configuration of the PF and BP cameras).
* An apparent free slip of the free meniscus of the order of 10 µm in the frame of the cameras as it starts to move.

The inertial oscillations are small in comparison with the main deformation and the meniscus acceleration will be neglected in the following. The apparent slip is negligible too, but its origin is more puzzling. the existence of this slip is clearly due to an initial repositioning of the films with respect to their meniscii at the motor edges. This slip is suspected to occur also at $t_1 = 32$ s in the opposite direction as motor 3 pushes back its film. We choose to correct the horizontal positions and to use $X_{BP,corr}$, where we removed a $+\Delta X_{slip} \sim 30$ µm jump at $t_0 = 28$ s, and a $-\Delta X_{slip}$ jump at $t_1 = 32$ s. This correction is very small in any case. A zoom at $t_0 = 28$ s is shown in Fig.3.30 to illustrate this procedure.

<figure><img src="image_94.png" /><figcaption>Figure 3.29: Horizontal displacement of the free meniscus throughout the whole “Push 3” experiment, for [DOH] = 50 mg/L and a varying velocity of main deformation $V_1$. The figure on the right is just a zoom around $t_1 = 32$ s of the figure on the left. At $t_0 = 28$ s, the horizontal film 3 is prepared, meaning it is being stretched at $V_0 = 2$ mm/s from 4.2 mm to 10 mm, leading to a first displacement of the free meniscus in $+x$. At $t_1 = 32$ s, the main deformation of the “Push 3” experiment is performed at $V_1$ varying between 5 – 50 mm/s.</figcaption></figure>

The other component of the displacement of the meniscus, $Y_{BP}$, which is measured using the BP camera as detailed in subsection 3.6.2. Given the geometry of the “Push 3” deformation, a non-zero vertical displacement represents a breaking of symmetry for the system induced by gravity. In Fig.3.31, we show a typical trajectory for the meniscus throughout the whole experiment, and we make sure that our assumption of a top-down symmetry remains valid as we have $X_{BP} \gg Y_{BP}$. $Y_{BP}$ remains small, and no slipping behaviour as for $X_{BP}$ was observed at our resolution (even though it is one order of magnitude greater than on the $x$ axis, given the configuration of the PF and BP cameras).
<figure><img src="image_95.png" /><figcaption>Figure 3.30: Correction because of the apparent free slip. The dark blue data are the uncorrected positions $X_{BP}$, the light blue data are the corrected $X_{BP,corr}$ taking into account the slips at $t_0 = 28$ s and $t_1 = 32$ s.</figcaption></figure>

<figure><img src="image_96.png" /><figcaption>Figure 3.31: Position of the free meniscus in the *Oxy* plane with scale-matched axis on an example of [DOH] = 50 mg/L at $V_1$ = 35 mm/s. The red dot is the beginning of the preparation step $t_0$ = 28 s which defines equilibrium position of the meniscus. The yellow dot is the end of the preparation phase at $t$ = 30.9 s. The green dot is the deformation time $t_1$ = 32 s at which the main deformation occurs. The black dot is the end of the main deformation, which is $t$ = 32.14 s in this specific case of $V_1$ = 35 mm/s.</figcaption></figure>

## Computing the tension ratio

Now that we have the corrected positions ($X_{BP}(t), Y_{BP}(t)$), we can compute the angle $\theta$ between the vertical films 1 and 2. Knowing at all times the lengths $L_{m,1}(t)$ and $L_{m,2}(t)$ imposed between the edges of the motors and the centre of the deformable frame, we can measure the vectors $L_1(t)$ and $L_2(t)$ linking the centre of the meniscus to the edges of the motors (see Fig. 3.32 for
<figure><img src="image_95.png" /><figcaption>Figure 3.30: Correction because of the apparent free slip. The dark blue data are the uncorrected positions $X_{BP}$, the light blue data are the corrected $X_{BP,corr}$ taking into account the slips at $t_0 = 28$ s and $t_1 = 32$ s.</figcaption></figure>

<figure><img src="image_96.png" /><figcaption>Figure 3.31: Position of the free meniscus in the *Oxy* plane with scale-matched axis on an example of [DOH] = 50 mg/L at $V_1$ = 35 mm/s. The red dot is the beginning of the preparation step $t_0$ = 28 s which defines equilibrium position of the meniscus. The yellow dot is the end of the preparation phase at $t$ = 30.9 s. The green dot is the deformation time $t_1$ = 32 s at which the main deformation occurs. The black dot is the end of the main deformation, which is $t$ = 32.14 s in this specific case of $V_1$ = 35 mm/s.</figcaption></figure>

## Computing the tension ratio

Now that we have the corrected positions ($X_{BP}(t), Y_{BP}(t)$), we can compute the angle $\theta$ between the vertical films 1 and 2. Knowing at all times the lengths $L_{m,1}(t)$ and $L_{m,2}(t)$ imposed between the edges of the motors and the centre of the deformable frame, we can measure the vectors $L_1(t)$ and $L_2(t)$ linking the centre of the meniscus to the edges of the motors (see Fig. 3.32 for
<figure><img src="image_97.png" /><figcaption>Figure 3.32: Definitions of the different lengths in the system, illustrated for an arbitrary deformation. This is a reminder of Fig.3.11B.</figcaption></figure>

angles and lengths notations):

$$L_1 = X_1 e_x + Y_1 e_y = (-L_{m,1} \cos(\theta_0) - X_{BP}) e_x + (L_{m,1} \sin(\theta_0) - Y_{BP}) e_y \quad (3.40)$$

$$L_2 = X_2 e_x + Y_2 e_y = (-L_{m,2} \cos(\theta_0) - X_{BP}) e_x + (-L_{m,2} \sin(\theta_0) - Y_{BP}) e_y \quad (3.41)$$

With $\theta_0 = 60^{\circ}$ being the rest semi-angle between the films. The angle $\theta$ we seek is given through the scalar product of these two vectors, and we have:

$$\theta = \frac{1}{2} \mathrm{Arccos} \left( \frac{L_1 \cdot L_2}{L_1 L_2} \right) \quad (3.42)$$

Typical measurements of $\theta$ are shown in Fig.3.33 Left for various velocities at a given chemistry ($[DOH] = 50$ mg/L).

Eventually, we are able to compute a ratio between the tensions $\sigma_i$ of the films. In the general case, without the assumption of a top/down symmetry (because of the vertical displacement), the angle $\theta$ we defined is the semi-angle between film 1 and 2 and the angle $\theta_3$ is the angle of film 3 with the bisector line of film 1 and 2 as shown in Fig.3.32. A projection along the bisector line of $L_1$ and $L_2$ of the forces acting on the meniscus yields the balance:

$$\sigma_3 \cos \theta_3 = (\sigma_1 + \sigma_2) \cos \theta \quad (3.43)$$

Estimating the small angle $\theta_3$, we reach at max: $\theta_{3,max} = \mathrm{Arcsin}(\frac{Y_{BP}}{L_3}) + \frac{X_1 Y_{BP}}{Y_1^2} \sim 4^{\circ}$ whose cosine remains very close to 1 with a 0.5% error, and so we write:

$$\sigma_3 = 2\bar{\sigma} \cos \theta \quad (3.44)$$

With $\bar{\sigma} = \frac{\sigma_1+\sigma_2}{2}$ the average of the tensions between the films 1 and 2. As we cannot work with both $\sigma_1$ and $\sigma_2$, and because the vertical displacement $Y_{BP}$ remains small compared to $X_{BP}$, **we will assume in the following $\sigma_1 = \bar{\sigma}$**. This choice is consistent with the assumption of top/down symmetry made on some other parameters which are obtained only on film 1 but not on film 2. Under these assumptions, the ratio between the tension of the compressed film 3 and the tension of the stretched films 1 and 2 is simply given by:
<figure><img src="image_97.png" /><figcaption>Figure 3.32: Definitions of the different lengths in the system, illustrated for an arbitrary deformation. This is a reminder of Fig.3.11B.</figcaption></figure>

angles and lengths notations):

$$L_1 = X_1 e_x + Y_1 e_y = (-L_{m,1} \cos(\theta_0) - X_{BP}) e_x + (L_{m,1} \sin(\theta_0) - Y_{BP}) e_y \quad (3.40)$$

$$L_2 = X_2 e_x + Y_2 e_y = (-L_{m,2} \cos(\theta_0) - X_{BP}) e_x + (-L_{m,2} \sin(\theta_0) - Y_{BP}) e_y \quad (3.41)$$

With $\theta_0 = 60^\circ$ being the rest semi-angle between the films. The angle $\theta$ we seek is given through the scalar product of these two vectors, and we have:

$$\theta = \frac{1}{2} \operatorname{Arccos} \left( \frac{L_1 \cdot L_2}{L_1 L_2} \right) \quad (3.42)$$

Typical measurements of $\theta$ are shown in Fig.3.33 Left for various velocities at a given chemistry ($[DOH] = 50$ mg/L).

Eventually, we are able to compute a ratio between the tensions $\sigma_i$ of the films. In the general case, without the assumption of a top/down symmetry (because of the vertical displacement), the angle $\theta$ we defined is the semi-angle between film 1 and 2 and the angle $\theta_3$ is the angle of film 3 with the bisector line of film 1 and 2 as shown in Fig.3.32. A projection along the bisector line of $L_1$ and $L_2$ of the forces acting on the meniscus yields the balance:

$$\sigma_3 \cos \theta_3 = (\sigma_1 + \sigma_2) \cos \theta \quad (3.43)$$

Estimating the small angle $\theta_3$, we reach at max: $\theta_{3,max} = \operatorname{Arcsin}(\frac{Y_{BP}}{L_3}) + \frac{X_1 Y_{BP}}{Y_1^2} \sim 4^\circ$ whose cosine remains very close to 1 with a 0.5% error, and so we write:

$$\sigma_3 = 2\bar{\sigma} \cos \theta \quad (3.44)$$

With $\bar{\sigma} = \frac{\sigma_1 + \sigma_2}{2}$ the average of the tensions between the films 1 and 2. As we cannot work with both $\sigma_1$ and $\sigma_2$, and because the vertical displacement $Y_{BP}$ remains small compared to $X_{BP}$, we will assume in the following $\sigma_1 = \bar{\sigma}$. This choice is consistent with the assumption of top/down symmetry made on some other parameters which are obtained only on film 1 but not on film 2. Under these assumptions, the ratio between the tension of the compressed film 3 and the tension of the stretched films 1 and 2 is simply given by:
<figure><img src="image_98.png" /><figcaption>Figure 3.33: Example for [DOH] = 50 mg/L and a varying velocity of main deformation $V_1$. A: Semiangle $\theta$ between film 1 and 2. At the initial state $t_0 = 28$ s, right before the preparation, the angle meets the Plateau’s law. Its deviation from this law arise because of a difference of the tension between film 1 (or 2), and film 3. B: Tension ratio between film 3 and 1 (or 2 equivalently) deduced from the angle measurement.</figcaption></figure>

$$ \frac{\sigma_3}{\sigma_1} = 2 \cos \theta \qquad (3.45) $$

Using this relation, we compute the ratios in Fig.3.33 Right from the set shown on its Left.

### 3.8.2 Deformation of film 3 $\epsilon_3$ and film 1 $\epsilon_1$

To further describe our system, we now want to look at the states of deformation of our films $\epsilon_i$. In fact, the elastic response of a single foam film due to insoluble surfactants can be captured with the expression used in chapter 1 (eq.1.17):

$$ \sigma = \sigma_0 + E_f \frac{\epsilon}{1 + \epsilon} \qquad (3.46) $$

Thus, measuring the deformations $\epsilon_1$ and $\epsilon_3$ of film 1 and 3 and using the tension ratio $\sigma_3/\sigma_1$, we will be able to determine the elasticity $E_f$ of our films.

#### Determination of $\epsilon_3$

We start by computing the deformation $\epsilon_3$ of the horizontal film 3. Before the preparation step at $t_0 = 28$ s, film 3 is at rest with a repose length $L_{i,3}$. As it is stretched during the preparation phase, two thicker new films are created in the plane of film 3 with a clear contrast with the preexisting film. The result can be seen in Fig.3.23C. During this preparation phase, film 3 has
<figure><img src="image_98.png" /><figcaption>Figure 3.33: Example for [DOH] = 50 mg/L and a varying velocity of main deformation $V_1$. A: Semiangle $\theta$ between film 1 and 2. At the initial state $t_0 = 28$ s, right before the preparation, the angle meets the Plateau's law. Its deviation from this law arise because of a difference of the tension between film 1 (or 2), and film 3. B: Tension ratio between film 3 and 1 (or 2 equivalently) deduced from the angle measurement.</figcaption></figure>

$$ \frac{\sigma_3}{\sigma_1} = 2 \cos \theta \qquad (3.45) $$

Using this relation, we compute the ratios in Fig.3.33 Right from the set shown on its Left.

### 3.8.2 Deformation of film 3 $\epsilon_3$ and film 1 $\epsilon_1$

To further describe our system, we now want to look at the states of deformation of our films $\epsilon_i$. In fact, the elastic response of a single foam film due to insoluble surfactants can be captured with the expression used in chapter 1 (eq.1.17):

$$ \sigma = \sigma_0 + E_f \frac{\epsilon}{1 + \epsilon} \qquad (3.46) $$

Thus, measuring the deformations $\epsilon_1$ and $\epsilon_3$ of film 1 and 3 and using the tension ratio $\sigma_3/\sigma_1$, we will be able to determine the elasticity $E_f$ of our films.

#### Determination of $\epsilon_3$

We start by computing the deformation $\epsilon_3$ of the horizontal film 3. Before the preparation step at $t_0 = 28$ s, film 3 is at rest with a repose length $L_{i,3}$. As it is stretched during the preparation phase, two thicker new films are created in the plane of film 3 with a clear contrast with the preexisting film. The result can be seen in Fig.3.23C. During this preparation phase, film 3 has
been only accepting interface (either from the menisci or the other films) so that all the matter initially present before the preparation remained in the film. The initial film, between the two Frankel films, is thus a closed system that can be used to determine $\epsilon_3$.

The total deformation $\epsilon_3$ of film 3 is a priori defined as the sum of the deformation along both axis $x$ and $z$ of its plane: $\epsilon_3 = \epsilon_{3,x} + \epsilon_{3,z}$. The second term $\epsilon_{3,z}$ can be discarded thanks to the invariance by translation along $z$ which we discussed and verified in section 3.7.

Thus, we only need to compute the first term $\epsilon_{3,x}$. By tracking the edges of Frankel films in film 3, as illustrated in Fig.3.23, we deduce the length $L_{c,3} = L_{Fr,ext,3} - L_{Fr,3}$ of the closed system along $x$. Knowing that it was equal to the length imposed to film 3 before the preparation $L_{i,3} = 4.2$ mm, we have a direct experimental access to $\epsilon_{3,x}$ and $\epsilon_3$:

$$ \epsilon_3(t) = \epsilon_{3,x}(t) = \frac{L_{c,3}(t)}{L_{i,3}} - 1 \qquad (3.47) $$

This quantity is defined as long as we are able to track the edges of both Frankel films. This lasts up till the Frankel film next to the free meniscus completely disappears in it, or until the thin film instability occurs at the edge of the free meniscus as discussed in section 3.7.

A result of the calculation of $\epsilon_3$ is shown in Fig.3.34 (red curve), where we see that film 3 starts from a stretched state due to the preparation phase, and is being compressed as we push on it during the main deformation at $t_1 = 32$ s.

### A unique value for the film elasticity $E_f$?

The full determination of the absolute tensions and deformations relies on the assumption that the film elasticity $E_f$ is the same for the three films.

This hypothesis of unique $E_f$ is not straightforward as the elasticity of a single film depends on its thickness which varies from one film to another due to the gravity drainage for instance. However, by looking at the initial state of our system using the spectral cameras, we see in Fig.3.25D-F that right before the main deformation at $t_1 = 32$ s, the three films have a factor 2 to 4 of difference in thickness. As discussed in Chapter 1 and in Poryles et al. [6], the thinner a foam film, the stiffer it is. This dependency is very slow however, as a factor 4 on the thickness leads to a 15% change in stiffness only.

A direct experimental evidence that the three elasticities are very close in our case is that the angle $\theta_3$ (see Fig.3.32) remains very small compared to $\theta$. An even faster way to control this symmetry between the films is to see in Fig.3.31 that the displacement along $y$ of the meniscus is small compared to the displacement along $x$.

Thus, we will assume that $E_f$ is the same for the three films and that the relationship between the tension and the deformation of each film is the same and given by eq.3.46.

### A first determination $\epsilon_{1,a}$ of $\epsilon_1$

In order to compute $\epsilon_1$, we define a closed system that is the piece of film present in the plane of film 1 when the main deformation starts at $t_1 = 32$ s. At later times, its length along the local abscissa $x_1$ is denoted $L_{1,close}(t)$ and can be written as:
been only accepting interface (either from the menisci or the other films) so that all the matter initially present before the preparation remained in the film. The initial film, between the two Frankel films, is thus a closed system that can be used to determine $\epsilon_3$.

The total deformation $\epsilon_3$ of film 3 is a priori defined as the sum of the deformation along both axis $x$ and $z$ of its plane: $\epsilon_3 = \epsilon_{3,x} + \epsilon_{3,z}$. The second term $\epsilon_{3,z}$ can be discarded thanks to the invariance by translation along $z$ which we discussed and verified in section 3.7.

Thus, we only need to compute the first term $\epsilon_{3,x}$. By tracking the edges of Frankel films in film 3, as illustrated in Fig.3.23, we deduce the length $L_{c,3} = L_{Fr,ext,3} - L_{Fr,3}$ of the closed system along $x$. Knowing that it was equal to the length imposed to film 3 before the preparation $L_{i,3} = 4.2$ mm, we have a direct experimental access to $\epsilon_{3,x}$ and $\epsilon_3$:

$$ \epsilon_3(t) = \epsilon_{3,x}(t) = \frac{L_{c,3}(t)}{L_{i,3}} - 1 \qquad (3.47) $$

This quantity is defined as long as we are able to track the edges of both Frankel films. This lasts up till the Frankel film next to the free meniscus completely disappears in it, or until the thin film instability occurs at the edge of the free meniscus as discussed in section 3.7.

A result of the calculation of $\epsilon_3$ is shown in Fig.3.34 (red curve), where we see that film 3 starts from a stretched state due to the preparation phase, and is being compressed as we push on it during the main deformation at $t_1 = 32$ s.

### A unique value for the film elasticity $E_f$?

The full determination of the absolute tensions and deformations relies on the assumption that the film elasticity $E_f$ is the same for the three films.

This hypothesis of unique $E_f$ is not straightforward as the elasticity of a single film depends on its thickness which varies from one film to another due to the gravity drainage for instance. However, by looking at the initial state of our system using the spectral cameras, we see in Fig.3.25D-F that right before the main deformation at $t_1 = 32$ s, the three films have a factor 2 to 4 of difference in thickness. As discussed in Chapter 1 and in Poryles et al. [6], the thinner a foam film, the stiffer it is. This dependency is very slow however, as a factor 4 on the thickness leads to a 15% change in stiffness only.

A direct experimental evidence that the three elasticities are very close in our case is that the angle $\theta_3$ (see Fig.3.32) remains very small compared to $\theta$. An even faster way to control this symmetry between the films is to see in Fig.3.31 that the displacement along $y$ of the meniscus is small compared to the displacement along $x$.

Thus, we will assume that $E_f$ is the same for the three films and that the relationship between the tension and the deformation of each film is the same and given by eq.3.46.

### A first determination $\epsilon_{1,a}$ of $\epsilon_1$

In order to compute $\epsilon_1$, we define a closed system that is the piece of film present in the plane of film 1 when the main deformation starts at $t_1 = 32$ s. At later times, its length along the local abscissa $x_1$ is denoted $L_{1,close}(t)$ and can be written as:
$$L_{1,close}(t) = L_1(t) - 2L_{Fr,1}(t) \qquad (3.48)$$

Which is the distance between motor 1 and the free meniscus to which we retrieve the length of Frankel film going out of the edges. The factor 2 in front of $L_{Fr,1}$ is there to take into account the Frankel film going out of the supported meniscus of motor 1, which is assumed to be equal to the Frankel film going out of the free meniscus.

**Important remark:** Assuming that the Frankel film at the motor is equal to the Frankel film at the free meniscus is a crude proxy. As we aim at avoiding such hypotheses, $\epsilon_1$ will be re-estimated later on, using only the film elasticity $E_f$ we are about to compute in a time range where $L_{Fr,1}$ is negligible.

Its reference length $L_{1,close,0}$, i.e. the length it would have at the reference tension is unknown, as film 1 is a priori pre-stretched during the preparatory phase. This is not obvious, but stretching film 3 alone at $t_0 = 28$ s (cf. Fig.3.17) means stretching the top and bottom interfaces of the elementary foam, the former being shared with film 1. The deformation $\epsilon_1$ is thus defined as:

$$\epsilon_1(t) = \epsilon_{1,x_1}(t) = \frac{L_{1,close}(t)}{L_{1,close,0}} - 1 \qquad (3.49)$$

The quantity that remains to be found is $L_{1,close,0}$. In order to estimate it, we can see in Fig3.33B that at some crossing time during the deformation $t_c > t_1 = 32$ s, the tension ratio is $\sigma_3/\sigma_1 = 1$, meaning that the tension $\sigma_1$ of film 1 is the same as the tension $\sigma_3$ of film 3, of which we know the deformation $\epsilon_3$ at all times. Knowing with eq.3.46 that we also have $\epsilon_1(t_c) = \epsilon_3(t_c)$ we can write:

$$L_{1,close,0} = \frac{L_1(t_c) - 2L_{Fr,1}(t_c)}{1 + \epsilon_3(t_c)} \approx \frac{L_1(t_c)}{1 + \epsilon_3(t_c)} \qquad (3.50)$$

Where at this particular time $t_c$ which is always at the very beginning of the deformation, the term $L_{Fr,1}(t_c) \ll L_1(t_c)$ can be discarded as the Frankel films are barely out of their menisci. Eventually, we have the expression:

$$\epsilon_{1,a}(t) = (1 + \epsilon_3(t_c)) \frac{L_1(t) - 2L_{Fr,1}(t)}{L_1(t_c)} - 1 \qquad (3.51)$$

This first estimation $\epsilon_{1,a}$ is also shown in Fig.3.34 (light blue). The time at which it crosses the red curve $\epsilon_3$ is $t_c$ the crossing time indicated with a short pointed line. It is important to note that at some time during the deformation, indicated by the tall vertical dashed lines in Fig.3.34, we start detecting $L_{Fr,1}$, leading to a sudden drop in the value of $\epsilon_{1,a}$ as we take $L_{Fr,1} = 0$ before that time.

With these deformations $\epsilon_{1,a}$ and $\epsilon_3$ and with the ratio of the tensions $\sigma_3/\sigma_1$, we now have all we need to compute $E_f$ the elastic modulus of our films.
$$L_{1,close}(t) = L_1(t) - 2L_{Fr,1}(t) \qquad (3.48)$$

Which is the distance between motor 1 and the free meniscus to which we retrieve the length of Frankel film going out of the edges. The factor 2 in front of $L_{Fr,1}$ is there to take into account the Frankel film going out of the supported meniscus of motor 1, which is assumed to be equal to the Frankel film going out of the free meniscus.

**Important remark:** Assuming that the Frankel film at the motor is equal to the Frankel film at the free meniscus is a crude proxy. As we aim at avoiding such hypotheses, $\epsilon_1$ will be re-estimated later on, using only the film elasticity $E_f$ we are about to compute in a time range where $L_{Fr,1}$ is negligible.

Its reference length $L_{1,close,0}$, i.e. the length it would have at the reference tension is unknown, as film 1 is a priori pre-stretched during the preparatory phase. This is not obvious, but stretching film 3 alone at $t_0 = 28$ s (cf. Fig.3.17) means stretching the top and bottom interfaces of the elementary foam, the former being shared with film 1. The deformation $\epsilon_1$ is thus defined as:

$$\epsilon_1(t) = \epsilon_{1,x_1}(t) = \frac{L_{1,close}(t)}{L_{1,close,0}} - 1 \qquad (3.49)$$

The quantity that remains to be found is $L_{1,close,0}$. In order to estimate it, we can see in Fig3.33B that at some crossing time during the deformation $t_c > t_1 = 32$ s, the tension ratio is $\sigma_3/\sigma_1 = 1$, meaning that the tension $\sigma_1$ of film 1 is the same as the tension $\sigma_3$ of film 3, of which we know the deformation $\epsilon_3$ at all times. Knowing with eq.3.46 that we also have $\epsilon_1(t_c) = \epsilon_3(t_c)$ we can write:

$$L_{1,close,0} = \frac{L_1(t_c) - 2L_{Fr,1}(t_c)}{1 + \epsilon_3(t_c)} \approx \frac{L_1(t_c)}{1 + \epsilon_3(t_c)} \qquad (3.50)$$

Where at this particular time $t_c$ which is always at the very beginning of the deformation, the term $L_{Fr,1}(t_c) \ll L_1(t_c)$ can be discarded as the Frankel films are barely out of their menisci. Eventually, we have the expression:

$$\epsilon_{1,a}(t) = (1 + \epsilon_3(t_c)) \frac{L_1(t) - 2L_{Fr,1}(t)}{L_1(t_c)} - 1 \qquad (3.51)$$

This first estimation $\epsilon_{1,a}$ is also shown in Fig.3.34 (light blue). The time at which it crosses the red curve $\epsilon_3$ is $t_c$ the crossing time indicated with a short pointed line. It is important to note that at some time during the deformation, indicated by the tall vertical dashed lines in Fig.3.34, we start detecting $L_{Fr,1}$, leading to a sudden drop in the value of $\epsilon_{1,a}$ as we take $L_{Fr,1} = 0$ before that time.

With these deformations $\epsilon_{1,a}$ and $\epsilon_3$ and with the ratio of the tensions $\sigma_3/\sigma_1$, we now have all we need to compute $E_f$ the elastic modulus of our films.
<figure><img src="image_99.png" /><figcaption>Figure 3.34: Example given on [DOH] = 35 mg/L and $V_1 = 50$ mm/s. Red is the deformation $\epsilon_3$ of film 3, Light Blue the first estimation of the deformation $\epsilon_{1,a}$ of film 1 and Dark Blue the corrected deformation $\epsilon_{1,b}$ of film 1. The leftmost dotted line is $t_c$ the crossing time. The dashed lines encompass the time at which we simultaneously measure the ingoing and outgoing films.</figcaption></figure>

### 3.8.3 Determination of the film elasticity $E_f$

To estimate the film elasticity $E_f$, we linearize all our quantities around $t_c$, defining a new time scale $\hat{t} = t - t_c$:

$$ \left\{ \begin{aligned} \epsilon_{1,a}(\hat{t}) &= \epsilon_0 + \lambda_1 \hat{t} \\ \epsilon_3(\hat{t}) &= \epsilon_0 + \lambda_3 \hat{t} \\ \frac{\sigma_3}{\sigma_1}(\hat{t}) &= 2 \cos \theta = C + \lambda_\theta \hat{t} \end{aligned} \right. \qquad (3.52) $$

In the vicinity of this crossing time $\hat{t} = 0$, which is located in Fig. 3.34 where all curves intersect with an ordinate $\epsilon_0$, we consider that we have $\epsilon_{1,a}$, $\epsilon_3 \ll 1$. Thus, we linearize eq. 3.46 and get:

$$ \frac{\sigma_3}{\sigma_1}(\hat{t}) = \frac{\sigma_0 + E_f \epsilon_0 + E_f \lambda_3 \hat{t}}{\sigma_0 + E_f \epsilon_0 + E_f \lambda_1 \hat{t}} \qquad (3.53) $$

And a development at first order in $\hat{t}$ leads to the expression:

$$ E_f \approx \frac{\sigma_0 \lambda_\theta}{\lambda_3 - \lambda_1} \qquad (3.54) $$

In which all quantities can be estimated. The equilibrium tension $\sigma_0$ depends on the chemistry only and is already characterised in Chapter 1 with a pendant drop method. The slopes $\lambda_1$, $\lambda_3$ and $\lambda_\theta$ are found with a linear regression for $\hat{t}$ comprised between $\hat{t} = 0$ the crossing time and
<figure><img src="image_99.png" /><figcaption>Figure 3.34: Example given on [DOH] = 35 mg/L and $V_1 = 50$ mm/s. Red is the deformation $\epsilon_3$ of film 3, Light Blue the first estimation of the deformation $\epsilon_{1,a}$ of film 1 and Dark Blue the corrected deformation $\epsilon_{1,b}$ of film 1. The leftmost dotted line is $t_c$ the crossing time. The dashed lines encompass the time at which we simultaneously measure the ingoing and outgoing films.</figcaption></figure>

### 3.8.3 Determination of the film elasticity $E_f$

To estimate the film elasticity $E_f$, we linearize all our quantities around $t_c$, defining a new time scale $\hat{t} = t - t_c$:

$$ \left\{ \begin{aligned} \epsilon_{1,a}(\hat{t}) &= \epsilon_0 + \lambda_1 \hat{t} \\ \epsilon_3(\hat{t}) &= \epsilon_0 + \lambda_3 \hat{t} \\ \frac{\sigma_3}{\sigma_1}(\hat{t}) &= 2 \cos \theta = C + \lambda_\theta \hat{t} \end{aligned} \right. \qquad (3.52) $$

In the vicinity of this crossing time $\hat{t} = 0$, which is located in Fig. 3.34 where all curves intersect with an ordinate $\epsilon_0$, we consider that we have $\epsilon_{1,a}$, $\epsilon_3 \ll 1$. Thus, we linearize eq. 3.46 and get:

$$ \frac{\sigma_3}{\sigma_1}(\hat{t}) = \frac{\sigma_0 + E_f \epsilon_0 + E_f \lambda_3 \hat{t}}{\sigma_0 + E_f \epsilon_0 + E_f \lambda_1 \hat{t}} \qquad (3.53) $$

And a development at first order in $\hat{t}$ leads to the expression:

$$ E_f \approx \frac{\sigma_0 \lambda_\theta}{\lambda_3 - \lambda_1} \qquad (3.54) $$

In which all quantities can be estimated. The equilibrium tension $\sigma_0$ depends on the chemistry only and is already characterised in Chapter 1 with a pendant drop method. The slopes $\lambda_1$, $\lambda_3$ and $\lambda_\theta$ are found with a linear regression for $\hat{t}$ comprised between $\hat{t} = 0$ the crossing time and
the detection time of the Frankel film for film 1 (leftmost vertical dotted line in Fig.3.34). We compute the elasticities $E_f$ for all chemistries and velocities of deformation $V_1$ and plot them in Fig.3.35.

<figure><img src="image_100.png" /><figcaption>Figure 3.35: A: Measured film elasticity as a function of the dodecanol concentration of the solution for different velocities of deformation. B: Red is graph A averaged over all velocities, and we compare it to the Black, the elasticities measured with the setup of chapter 1 (cf. [6])</figcaption></figure>

As we can see in Fig.3.35A, our measures seem to depend on $V_1$ with the trend “the faster, the stiffer”. At first glance, we could think that this is coherent with the fact that the faster the interfaces are deformed, the lower is the time given to bulk-interface exchanges, which would tend to compensate for the gain/loss of surface concentration of surfactants, lowering the change in surface tension and thus lowering the apparent elasticity of the interfaces. This reasoning would be valid if the interfaces were in contact with an infinitely thick bulk phase, however as seen in Chapter 1, the elasticity we observe is expected to be caused by the finite amount of dodecanol present in the bulk. Besides, the interface is supposed to be at equilibrium with the bulk at all times given the very short times of diffusion through the foam films for a common surfactant such as SDS or dodecanol $\tau_{eq} \sim L^2/D \sim 1$ ms. This is much smaller than the smaller time of deformation $\tau_{def} \sim 100$ ms, thus ruling out unsteady populations of classical surfactants as an explanation for varying elasticity.

The explanation for this dependency is actually unclear, and could originate from very insoluble and large impurities present in the air which would take an infinite time to diffuse through the bulk.

We see that with the same chemistries and the same range of film thicknesses, we recover values for $E_f$ which are very close to what was measured in previous works (see Poryles et al. [6] and Fig.3.35B). In the following, we take our measurements averaged over the velocities as our reference elasticities $E_f$ depending only on the chemistry.

Also, this elasticity measurement allows us to get the absolute tensions $\sigma$. We first compute $\sigma_3$, as the measurement of $\epsilon_3$ is the most reliable of all deformations. This writes:

$$ \sigma_3 = \sigma_0 + E_f \frac{\epsilon_3}{1 + \epsilon_3} \qquad (3.55) $$

Getting $\sigma_1$ can be performed by then using the ratio $\sigma_3/\sigma_1 = 2 \cos\theta$, which is our reliable force measurement shown in Fig.3.33. This will be useful in the form of re-measuring a
the detection time of the Frankel film for film 1 (leftmost vertical dotted line in Fig.3.34). We compute the elasticities $E_f$ for all chemistries and velocities of deformation $V_1$ and plot them in Fig.3.35.

<figure><img src="image_100.png" /><figcaption>Figure 3.35: A: Measured film elasticity as a function of the dodecanol concentration of the solution for different velocities of deformation. B: Red is graph A averaged over all velocities, and we compare it to the Black, the elasticities measured with the setup of chapter 1 (cf. [6])</figcaption></figure>

As we can see in Fig.3.35A, our measures seem to depend on $V_1$ with the trend “the faster, the stiffer”. At first glance, we could think that this is coherent with the fact that the faster the interfaces are deformed, the lower is the time given to bulk-interface exchanges, which would tend to compensate for the gain/loss of surface concentration of surfactants, lowering the change in surface tension and thus lowering the apparent elasticity of the interfaces. This reasoning would be valid if the interfaces were in contact with an infinitely thick bulk phase, however as seen in Chapter 1, the elasticity we observe is expected to be caused by the finite amount of dodecanol present in the bulk. Besides, the interface is supposed to be at equilibrium with the bulk at all times given the very short times of diffusion through the foam films for a common surfactant such as SDS or dodecanol $\tau_{eq} \sim L^2/D \sim 1$ ms. This is much smaller than the smaller time of deformation $\tau_{def} \sim 100$ ms, thus ruling out unsteady populations of classical surfactants as an explanation for varying elasticity.

The explanation for this dependency is actually unclear, and could originate from very insoluble and large impurities present in the air which would take an infinite time to diffuse through the bulk.

We see that with the same chemistries and the same range of film thicknesses, we recover values for $E_f$ which are very close to what was measured in previous works (see Poryles et al. [6] and Fig.3.35B). In the following, we take our measurements averaged over the velocities as our reference elasticities $E_f$ depending only on the chemistry.

Also, this elasticity measurement allows us to get the absolute tensions $\sigma$. We first compute $\sigma_3$, as the measurement of $\epsilon_3$ is the most reliable of all deformations. This writes:

$$ \sigma_3 = \sigma_0 + E_f \frac{\epsilon_3}{1 + \epsilon_3} \qquad (3.55) $$

Getting $\sigma_1$ can be performed by then using the ratio $\sigma_3/\sigma_1 = 2 \cos\theta$, which is our reliable force measurement shown in Fig.3.33. This will be useful in the form of re-measuring a
corrected $\epsilon_{1,b}$ knowing $\sigma_1$ and $E_f$ precisely.

### 3.8.4 A second estimation of the deformation $\epsilon_{1,b}$

We see that now we have computed $\sigma_3$ we just defined a path to compute the absolute tension $\sigma_1$ without relying on the estimation of a deformation factor $\epsilon_1$ for the vertical films. In fact, the first estimation $\epsilon_{1,a}$ at large times (dashed lines in Fig.3.34) depends on two things:

* Our capability to measure as early as possible the Frankel film $L_{Fr,1}$ going out of the free meniscus as film 1 is stretched.

* A strong assumption that the Frankel film going out on the side of motor 1 is the same in length as the one we measure at the free meniscus.

Thus, we can now try to work the opposite way to avoid making these assumptions: using $\sigma_3$ and the ratio $\sigma_3/\sigma_1 = 2 \cos\theta$ which is known with a high precision, we can deduce the absolute tension $\sigma_1$. With $\sigma_1$, we can now deduce another estimation $\epsilon_{1,b}$ using the equation:

$$ \epsilon_{1,b} = \frac{\sigma_1 - \sigma_0}{E_f - (\sigma_1 - \sigma_0)} \qquad (3.56) $$

Whose computation is shown in Fig.3.34 (dark blue). Note that the earlier points, before the crossing time, are shown with a dotted line as this is where the reliability of our angle measurement is the weakest because of the slip described and discussed at the beginning of section 3.8.1. Between the crossing time and the time of detection of the Frankel films in film 1, both estimations are expected to be reliable, and we indeed see that they remain quite close. These two estimations differ however, and the conclusions we make on the interface transfer properties remains the same using either one estimation of $\epsilon_1$ or the other.

The points in which we are the most interested in are between the two black vertical dotted lines: after the time at which we start measuring some Frankel film going out of the free meniscus in film 1, and before the end of tracking of the edge of Frankel film at the free meniscus in film 3. In this range, $\epsilon_{1,b}$ is the most reliable estimation of $\epsilon_1$.
corrected $\epsilon_{1,b}$ knowing $\sigma_1$ and $E_f$ precisely.

### 3.8.4 A second estimation of the deformation $\epsilon_{1,b}$

We see that now we have computed $\sigma_3$ we just defined a path to compute the absolute tension $\sigma_1$ without relying on the estimation of a deformation factor $\epsilon_1$ for the vertical films. In fact, the first estimation $\epsilon_{1,a}$ at large times (dashed lines in Fig.3.34) depends on two things:

* Our capability to measure as early as possible the Frankel film $L_{Fr,1}$ going out of the free meniscus as film 1 is stretched.

* A strong assumption that the Frankel film going out on the side of motor 1 is the same in length as the one we measure at the free meniscus.

Thus, we can now try to work the opposite way to avoid making these assumptions: using $\sigma_3$ and the ratio $\sigma_3/\sigma_1 = 2 \cos\theta$ which is known with a high precision, we can deduce the absolute tension $\sigma_1$. With $\sigma_1$, we can now deduce another estimation $\epsilon_{1,b}$ using the equation:

$$ \epsilon_{1,b} = \frac{\sigma_1 - \sigma_0}{E_f - (\sigma_1 - \sigma_0)} \qquad (3.56) $$

Whose computation is shown in Fig.3.34 (dark blue). Note that the earlier points, before the crossing time, are shown with a dotted line as this is where the reliability of our angle measurement is the weakest because of the slip described and discussed at the beginning of section 3.8.1. Between the crossing time and the time of detection of the Frankel films in film 1, both estimations are expected to be reliable, and we indeed see that they remain quite close. These two estimations differ however, and the conclusions we make on the interface transfer properties remains the same using either one estimation of $\epsilon_1$ or the other.

The points in which we are the most interested in are between the two black vertical dotted lines: after the time at which we start measuring some Frankel film going out of the free meniscus in film 1, and before the end of tracking of the edge of Frankel film at the free meniscus in film 3. In this range, $\epsilon_{1,b}$ is the most reliable estimation of $\epsilon_1$.
# 3.9 Results: Surfactant balance and the fate of $j_m$

## 3.9.1 Computing $L_{in}$ and $L_{out}$

Now that we have carefully computed the deformations of the films $\epsilon_1$ and $\epsilon_3$, and that we have given ourselves the lengths of film $L_{Fr,1}$ and $L_{Fr,3}$ involved in the interface balance, it is time to compute the proper balance between $L_{out}$ and $L_{in}$ as promised in subsection 3.3.

### Computing the outgoing quantity of interface $L_{out}(t)$

The length of outgoing interface $\hat{L}_{out}(t)$ is directly given by the length of Frankel film detected near the free meniscus in film 1: $\hat{L}_{out} = L_{Fr,1}$ (cf. subsection 3.6.4).

Computing the length of interface at rest $L_{out}$ also requires to know the extension state $\epsilon_1$ of the film:

$$L_{out}(t) = \frac{\hat{L}_{out}}{1 + \epsilon_1} = \frac{L_{Fr,1}(t)}{1 + \epsilon_1(t)} \qquad (3.57)$$

In section 3.8, we estimate $\epsilon_1$ twice: $\epsilon_{1,a}$ is computed with eq.3.51 using the tension ratio between the films and the length of Frankel film $L_{Fr,1}$, whereas $\epsilon_{1,b}$ is computed with eq.3.56 using the absolute tension of film 1 once the film elasticity is computed. Both quantities are represented with an example in Fig.3.34, and we see that these two measurements may differ by up to a factor 2. They make different hypotheses, detailed above, and although $\epsilon_{1,b}$ seems to have the most parsimonious set of assumptions, we keep both of them and will compute $L_{out,a,b}$ twice as well.

### Computing the ingoing quantity of interface $L_{in}(t)$

It is not directly possible to compute $\hat{L}_{in}$, as we track a film which is simultaneously compressed and evacuated from film 3 out of the sight of the camera. To circumvent the problem, we compute the difference of lengths of interface at rest. To do so, we use the prepared Frankel film of length $L_{Fr,3}(t)$ (cf. subsection 3.6.3), and knowing the state of compression $\epsilon_3(t)$ of film 3 (cf. subsection 3.8.2) we write:

$$L_{in}(t) = \frac{\hat{L}_{in}}{1 + \epsilon_3} = \frac{L_{Fr,3}(t_1)}{1 + \epsilon_3(t_1)} - \frac{L_{Fr,3}(t)}{1 + \epsilon_3(t)} \qquad (3.58)$$

The underlying assumption of this method is to consider that the thick prepared Frankel film $L_{Fr,3}$ has the same state of compression as the thin central film $L_{c,3}$ preexisting the preparation from which $\epsilon_3$ is computed through 3.47.

This uniformity of compression state $\epsilon_3$ between the two kinds of film is not obvious but well-founded. As discussed in subsection 0.2.4, tension is uniform in the plane of a foam film, whatever the thickness of the different patches. With eq.3.46, we see that the condition of uniform $\epsilon_3$ is that both kinds of film have the same film elasticity $E_f$. This latter depends on the film thickness!
## 3.9 Results: Surfactant balance and the fate of $j_m$

### 3.9.1 Computing $L_{in}$ and $L_{out}$

Now that we have carefully computed the deformations of the films $\epsilon_1$ and $\epsilon_3$, and that we have given ourselves the lengths of film $L_{Fr,1}$ and $L_{Fr,3}$ involved in the interface balance, it is time to compute the proper balance between $L_{out}$ and $L_{in}$ as promised in subsection 3.3.

#### Computing the outgoing quantity of interface $L_{out}(t)$

The length of outgoing interface $\hat{L}_{out}(t)$ is directly given by the length of Frankel film detected near the free meniscus in film 1: $\hat{L}_{out} = L_{Fr,1}$ (cf. subsection 3.6.4).

Computing the length of interface at rest $L_{out}$ also requires to know the extension state $\epsilon_1$ of the film:

$$L_{out}(t) = \frac{\hat{L}_{out}}{1 + \epsilon_1} = \frac{L_{Fr,1}(t)}{1 + \epsilon_1(t)} \qquad (3.57)$$

In section 3.8, we estimate $\epsilon_1$ twice: $\epsilon_{1,a}$ is computed with eq.3.51 using the tension ratio between the films and the length of Frankel film $L_{Fr,1}$, whereas $\epsilon_{1,b}$ is computed with eq.3.56 using the absolute tension of film 1 once the film elasticity is computed. Both quantities are represented with an example in Fig.3.34, and we see that these two measurements may differ by up to a factor 2. They make different hypotheses, detailed above, and although $\epsilon_{1,b}$ seems to have the most parsimonious set of assumptions, we keep both of them and will compute $L_{out,a,b}$ twice as well.

#### Computing the ingoing quantity of interface $L_{in}(t)$

It is not directly possible to compute $\hat{L}_{in}$, as we track a film which is simultaneously compressed and evacuated from film 3 out of the sight of the camera. To circumvent the problem, we compute the difference of lengths of interface at rest. To do so, we use the prepared Frankel film of length $L_{Fr,3}(t)$ (cf. subsection 3.6.3), and knowing the state of compression $\epsilon_3(t)$ of film 3 (cf. subsection 3.8.2) we write:

$$L_{in}(t) = \frac{\hat{L}_{in}}{1 + \epsilon_3} = \frac{L_{Fr,3}(t_1)}{1 + \epsilon_3(t_1)} - \frac{L_{Fr,3}(t)}{1 + \epsilon_3(t)} \qquad (3.58)$$

The underlying assumption of this method is to consider that the thick prepared Frankel film $L_{Fr,3}$ has the same state of compression as the thin central film $L_{c,3}$ preexisting the preparation from which $\epsilon_3$ is computed through 3.47.

This uniformity of compression state $\epsilon_3$ between the two kinds of film is not obvious but well-founded. As discussed in subsection 0.2.4, tension is uniform in the plane of a foam film, whatever the thickness of the different patches. With eq.3.46, we see that the condition of uniform $\epsilon_3$ is that both kinds of film have the same film elasticity $E_f$. This latter depends on the film thickness!
<figure><img src="image_101.png" /><figcaption>Figure 3.36: Thickness profiles of prepared film 3 right before the main deformation at $t_1 = 32$ s using the hyperspectral cameras SP1 and SP2. The shaded areas are the Frankel films extracted during the preparation phase. $x = 0$ is the free meniscus, $x = 10$ mm is the edge of motor 3.</figcaption></figure>

However, as seen in Fig. 1.3B of subsection 1.2.1, elasticity is a slowly varying function of thickness. In the present case, typical thickness profiles right before $t_1$ the deformation time are shown in Fig. 3.36 for all chemistries. We identify the thick Frankel films $L_{Fr,3}$ and $L_{Fr,3,ext}$ with respectively the left and right coloured shaded areas.

We see that the central film is thinner on average by less than a factor 1/2, which corresponds to a difference of elasticities $E_f$ of less than 10%. This means that at maximum, we make an error of 10% on the estimation of the $\epsilon_3$ for the ingoing Frankel film $L_{Fr,3}$. This remains within our errorbars for the estimation of $L_{in}$, as this relative error is mitigated by the fact that $L_{in}$ depends on $1 + \epsilon_3$, hence our assumption.

## Results

In Fig. 3.37, we show all the computable $L_{in}$ and $L_{out,b}$: we managed to get them for the range $[DOH] = 15, 35, 50$ mg/L in chemistry, and $V_1 = 10, 20, 35$ and $50$ mm/s for the velocity of deformation.
<figure><img src="image_101.png" /><figcaption>Figure 3.36: Thickness profiles of prepared film 3 right before the main deformation at $t_1 = 32$ s using the hyperspectral cameras SP1 and SP2. The shaded areas are the Frankel films extracted during the preparation phase. $x = 0$ is the free meniscus, $x = 10$ mm is the edge of motor 3.</figcaption></figure>

However, as seen in Fig. 1.3B of subsection 1.2.1, elasticity is a slowly varying function of thickness. In the present case, typical thickness profiles right before $t_1$ the deformation time are shown in Fig. 3.36 for all chemistries. We identify the thick Frankel films $L_{Fr,3}$ and $L_{Fr,3,ext}$ with respectively the left and right coloured shaded areas.

We see that the central film is thinner on average by less than a factor 1/2, which corresponds to a difference of elasticities $E_f$ of less than 10%. This means that at maximum, we make an error of 10% on the estimation of the $\epsilon_3$ for the ingoing Frankel film $L_{Fr,3}$. This remains within our errorbars for the estimation of $L_{in}$, as this relative error is mitigated by the fact that $L_{in}$ depends on $1 + \epsilon_3$, hence our assumption.

## Results

In Fig. 3.37, we show all the computable $L_{in}$ and $L_{out,b}$: we managed to get them for the range $[DOH] = 15, 35, 50$ mg/L in chemistry, and $V_1 = 10, 20, 35$ and $50$ mm/s for the velocity of deformation.
<figure><img src="image_102.png" /><figcaption>Figure 3.37: All exchanged lengths $L_{in}(t)$ and $L_{out,b}(t)$ in the ranges of chemistry [DOH] = 15, 35, 50 mg/L and of velocities of deformation $V_1$ = 10, 20, 35 and 50 mm/s.</figcaption></figure>

Now we have ways to compute $L_{in}(t)$ and $L_{out}(t)$, and we show a typical result in Fig.3.38 which contains the averaged curves for $V_1$ = 50 mm/s and [DOH] = 50 mg/L. At this velocity, the deformation takes place between $t_1$ = 32 s and $t$ = 32.1 s.

Our goal is now to compare simultaneously $L_{in}(t)$ and $L_{out}(t)$ over the same time range, which is shown with the shaded orange area. Before this time range, we are limited by the fact that we start detecting the outgoing film $L_{Fr,1}$ during the deformation and $L_{out}$ cannot be defined. The end of this time range is set by the end of the detection of the ingoing Frankel film $L_{Fr,3}$, hindering the measurement of $L_{in}$ as well as the estimation of the corrected deformation $\epsilon_1$, and thus of $L_{out,b}$.

### 3.9.2 Vanishing flux $j_m$

In Fig.3.39 we show the two quantities $L_{out} = f(L_{in})$ plot against each other, for the two computations $L_{out,a}$ (Fig.3.39A) and $L_{out,b}$ (Fig.3.39B). These constitute the main result of this manuscript.

The yellow area corresponds to the expected range for the interface balance. We recall that
<figure><img src="image_102.png" /><figcaption>Figure 3.37: All exchanged lengths $L_{in}(t)$ and $L_{out,b}(t)$ in the ranges of chemistry [DOH] = 15, 35, 50 mg/L and of velocities of deformation $V_1 = 10, 20, 35$ and 50 mm/s.</figcaption></figure>

Now we have ways to compute $L_{in}(t)$ and $L_{out}(t)$, and we show a typical result in Fig.3.38 which contains the averaged curves for $V_1 = 50$ mm/s and [DOH] = 50 mg/L. At this velocity, the deformation takes place between $t_1 = 32$ s and $t = 32.1$ s.

Our goal is now to compare simultaneously $L_{in}(t)$ and $L_{out}(t)$ over the same time range, which is shown with the shaded orange area. Before this time range, we are limited by the fact that we start detecting the outgoing film $L_{Fr,1}$ during the deformation and $L_{out}$ cannot be defined. The end of this time range is set by the end of the detection of the ingoing Frankel film $L_{Fr,3}$, hindering the measurement of $L_{in}$ as well as the estimation of the corrected deformation $\epsilon_1$, and thus of $L_{out,b}$.

### 3.9.2 Vanishing flux $j_m$

In Fig.3.39 we show the two quantities $L_{out} = f(L_{in})$ plot against each other, for the two computations $L_{out,a}$ (Fig.3.39A) and $L_{out,b}$ (Fig.3.39B). These constitute the main result of this manuscript.

The yellow area corresponds to the expected range for the interface balance. We recall that
<figure><img src="image_103.png" /><figcaption>Figure 3.38: Typical graph of interface lengths obtained for [DOH] = 50 mg/L and a velocity of main deformation $V_1$ = 50 mm/s. $L_{in}$ is computed from $\epsilon_3$, $L_{out,a}$ from $\epsilon_{1,a}$ and $L_{out,b}$ from $\epsilon_{1,b}$. The shaded area corresponds to the time range over which both the ingoing and outgoing quantities of surfactants are measurable simultaneously.</figcaption></figure>

<figure><img src="image_104.png" /><figcaption>Figure 3.39: Comparison of the outgoing quantities of surfactants as a function of the ingoing ones $L_{out} = f(L_{in})$ for various chemistries and for the velocities $V_1 = 10, 20, 35$ and $50$ mm/s. **A:** $L_{out,a}$ is computed with $\epsilon_{1,a}$; **B:** $L_{out,b}$ with $\epsilon_{1,b}$. The shaded area corresponds to the range of expected balance, with $y = x$ the limit of no film-film exchanges, and $y = 0.5x$ the limit of no exchanges with the meniscus.</figcaption></figure>

the upper limit $L_{out}(t) = L_{in}(t)$ corresponds to the extreme case where the meniscus provides
<figure><img src="image_103.png" /><figcaption>Figure 3.38: Typical graph of interface lengths obtained for [DOH] = 50 mg/L and a velocity of main deformation $V_1$ = 50 mm/s. $L_{in}$ is computed from $\epsilon_3$, $L_{out,a}$ from $\epsilon_{1,a}$ and $L_{out,b}$ from $\epsilon_{1,b}$. The shaded area corresponds to the time range over which both the ingoing and outgoing quantities of surfactants are measurable simultaneously.</figcaption></figure>

<figure><img src="image_104.png" /><figcaption>Figure 3.39: Comparison of the outgoing quantities of surfactants as a function of the ingoing ones $L_{out} = f(L_{in})$ for various chemistries and for the velocities $V_1 = 10, 20, 35$ and $50$ mm/s. **A:** $L_{out,a}$ is computed with $\epsilon_{1,a}$; **B:** $L_{out,b}$ with $\epsilon_{1,b}$. The shaded area corresponds to the range of expected balance, with $y = x$ the limit of no film-film exchanges, and $y = 0.5x$ the limit of no exchanges with the meniscus.</figcaption></figure>

the upper limit $L_{out}(t) = L_{in}(t)$ corresponds to the extreme case where the meniscus provides
for as many surfactants as the free interface, the lower limit $L_{out}(t) = \frac{1}{2}L_{in}$ to the case where the flux of surfactants from the meniscus is zero. All this is detailed in subsection 3.3.

**In our range of measurement, all velocities and chemistries are on the lower bound, suggesting that the flux of surfactants from the meniscus is null in all cases: $j_m = 0$**.

This is the key information we needed to further advance on the theoretical descriptions of flows in sheared foams. This is also the main experimental result of this thesis work.

### Important remarks

* Our result is verified in quite a narrow range of velocities of deformations, from $V_1 = 10$ to $50$ mm/s, corresponding to strain rates of $\dot{\epsilon} = 2 - 10$ Hz given our geometry. At lower strain rates, we lose the invariance by translation along $z$ before being able to measure $L_{in}$ and $L_{out}$ simultaneously. Higher strain rates cannot achieve reliable measurements, either because the motors are limited at $V_1 = 50$ mm/s or because smaller amplitudes of deformations make it impossible to measure $L_{in}$ and $L_{out}$.

* This result is heavily chemistry dependent to the best of our knowledge, as all hidden time variables allowing/excluding the existence of a $j_m$ are adsorption, desorption or diffusion times of surfactants. Notably, even changing the direction of deformation, and thus the sign of the potential $j_m$, could lead to very different results under the same deformation conditions. Adsoption and desorption dynamics are known to be very different. This is a bit explored in section 3.10, where we look at other deformations, unfortunately without achieving measuring the interface balance as we did here.

* The balance computed with the corrected $\epsilon_{1,b}$ shown in Fig.3.39B is a bit below the lower bound of our model. This is puzzling, as it would suggest a $j_m$ of the opposite sign (from the interface to the meniscus in this case, which is excluded). It might be caused by the fact that we neglect the exchanges in the dynamic menisci, the $\Delta\gamma^{in/out}$ of Fig.3.5 which have been neglected and which correspond both to a $\Delta\Gamma^{in/out}$ not taken into account in the surfactant balance. This assumption is more discussed in [4], and could explain this discrepancy.

## 3.10 Other deformations

### 3.10.1 “Push 1”: Potential influence of gravity

In the “Push 1” experiments the geometry of deformation remains exactly the same as in “Push 3” but the roles of the different films are interchanged so that we look at the same system with the direction of gravity changed. The protocol is shown in Fig.3.40. As discussed in subsections 0.2.4 and 0.3.5, what we expect gravity to change here is:

* Giving different initial thicknesses for the compressed/stretched films. Figs.3.6.5 D-F show how different are the films whose roles have been interchanged between “Push3” and “Push1”.