and is not a mere consequence of the balance between viscous and capillary forces. The resulting tension in this case writes [48]:

$$ \Delta\gamma^{in} = \gamma_0 (3Ca)^{2/3} (2.55\alpha^{1/3} - 2.68) \quad (28) $$

Where $\alpha = (h\infty/r_m) (3Ca)^{-2/3}$.

This is an important result. Tension has a unique value in the film (as seen in subsection 0.2.4), and a unique value in the meniscus. The gap between these two values can be filled either by a $\Delta\gamma^{in}$ or a $\Delta\gamma^{out}$ with respectively a negative or positive $U$ and for different thicknesses $h_\infty$ (the nullity of $U$ is no practical case when a film is being deformed). This leads to an instability which spontaneously breaks the symmetry along the dimension parallel to the meniscus: this is called marginal regeneration, and we will describe it later in subsection 0.3.4.

**Important remark:** The problem of taking into account the visco-elasticity of the interfaces (meaning we have $U(x)$ no longer uniform) is quite a recent theoretical advance, and can be found in Seiwert et al. [51, 52].

### 0.3.3 Capillary suction near the meniscus: Aradian's marginal pinching

Let us look now at the instationary problem of a flat film at rest ($U = 0$) put into contact with the meniscus. We will call this problem the Aradian problem [58]. It has the same configuration as previously, but now we keep the time varying term of eq.24 and discard the convection term:

$$ \left\{ \begin{aligned} \partial_t h - \frac{1}{3} \frac{\gamma_0}{\eta} \partial_x (h^3 \partial_{xxx} h) &= 0 \\ \partial_{xx} h (x \to -\infty) &= 1/r_m \\ v_x (x \to +\infty, y) &= 0 \\ h (x \to +\infty) &= h_\infty \end{aligned} \right. \quad (29) $$

It describes what happens to a film at rest when put into contact with a meniscus. As the film is flat (at air pressure) and the meniscus is curved with a negative pressure, the meniscus sucks back fluid from the film, which is what has already been called **capillary drainage** in this manuscript. However, the film does not drain uniformly along $x$, and we rather see the formation of a depleted zone near the meniscus whose geometry includes different lengths, as shown in Fig.16.

The main problem leading to such a multiplicity of lengths is that although the draining Poiseuille flow develops over a large length $W$, the majority of the gradient of curvature $\partial_{xxx}h$ linking the meniscus to the flat film, and which is the main driving force, develops over a very localised zone $w$. All that means the problem needs a careful handling, and a mathematical treatment which involves an asymptotic matching between the two scales. If we now want to find the main results using scaling laws, we also need to do it carefully, by considering the following:

* $h_\infty$ and $W$ are the typical scales for large $x$. $h_\infty$ is a given parameter from the initial conditions, $W$ is a variable of which we seek the scaling. They are linked by the conservation of flow rate (line 1 of eq.29), leading to: $W \sim h_\infty (\frac{\gamma}{\eta h_\infty})^{1/4} t^{1/4}$.
<figure><img src="image_21.png" /><figcaption>Figure 16: Notations for the Aradian problem. A flat film of thickness $h_\infty$ is connected in $x \to -\infty$ to a meniscus of constant curvature $\partial_{xx}h = 1/r_m$. The flat film is immobile in $x \to +\infty$ and we look now at the unsteady problem of the time evolution of the film at the junction. Because of capillary suction acting alone, the initial flat film locally thins out with a non-trivial multi-scale dynamics.</figcaption></figure>

* $h^p$ and $w$ are the typical scales for small $x$ around the position of minimal $h \sim h^p$. Both of them are sought variables, linked by the curvature condition of the meniscus (line 2 of eq.29), leading to: $h^p/w^2 \sim 1/r_m$.

* The matched asymptotic expansion linking the two scales, developed in [58], imposes to match the slopes so that: $h^p/w \sim h_\infty/W$.

We thus have three equations for three variables of which we write the scalings:

$$ \begin{cases} W \sim h_{\infty} \left( \frac{\gamma}{\eta h_{\infty}} \right)^{1/4} t^{1/4} \\ w \sim r_m \left( \frac{\gamma}{\eta h_{\infty}} \right)^{-1/4} t^{-1/4} \\ h^p \sim r_m \left( \frac{\gamma}{\eta h_{\infty}} \right)^{-1/2} t^{-1/2} \end{cases} \quad (30) $$

These scalings reveal a complex dynamics (whose complete calculation is full of subtleties) which is at the origin of the dimple observed in thin film balance or AFM experiments [59–62].

This process is directly linked to the ability of the interface to create tension gradients. If we remove this from the interface, theoretical arguments show that no pinching occurs [63].

### 0.3.4 Marginal regeneration

Marginal regeneration is a phenomenon occurring at the junction between a foam film and its bordering menisci where we observe simultaneously thin patches of film going out of the meniscus and thicker ones going in the meniscus through channels separating the thin patches. It has been first observed and named by Mysels et. al [47] in 1959. It appears spontaneously in vertical foam films where it was first observed, in which gravity set patches of film in motion, producing
<figure><img src="image_21.png" /><figcaption>Figure 16: Notations for the Aradian problem. A flat film of thickness $h_\infty$ is connected in $x \to -\infty$ to a meniscus of constant curvature $\partial_{xx}h = 1/r_m$. The flat film is immobile in $x \to +\infty$ and we look now at the unsteady problem of the time evolution of the film at the junction. Because of capillary suction acting alone, the initial flat film locally thins out with a non-trivial multi-scale dynamics.</figcaption></figure>

* $h^p$ and $w$ are the typical scales for small $x$ around the position of minimal $h \sim h^p$. Both of them are sought variables, linked by the curvature condition of the meniscus (line 2 of eq. 29), leading to: $h^p/w^2 \sim 1/r_m$.

* The matched asymptotic expansion linking the two scales, developed in [58], imposes to match the slopes so that: $h^p/w \sim h_\infty/W$.

We thus have three equations for three variables of which we write the scalings:

$$ \left\{ \begin{array}{l} W \sim h_{\infty} \left( \frac{\gamma}{\eta h_{\infty}} \right)^{1/4} t^{1/4} \\ w \sim r_{m} \left( \frac{\gamma}{\eta h_{\infty}} \right)^{-1/4} t^{-1/4} \\ h^{p} \sim r_{m} \left( \frac{\gamma}{\eta h_{\infty}} \right)^{-1/2} t^{-1/2} \end{array} \right. \qquad (30) $$

These scalings reveal a complex dynamics (whose complete calculation is full of subtleties) which is at the origin of the dimple observed in thin film balance or AFM experiments [59–62].

This process is directly linked to the ability of the interface to create tension gradients. If we remove this from the interface, theoretical arguments show that no pinching occurs [63].

### 0.3.4 Marginal regeneration

Marginal regeneration is a phenomenon occurring at the junction between a foam film and its bordering menisci where we observe simultaneously thin patches of film going out of the meniscus and thicker ones going in the meniscus through channels separating the thin patches. It has been first observed and named by Mysels et. al [47] in 1959. It appears spontaneously in vertical foam films where it was first observed, in which gravity set patches of film in motion, producing
mushroom-shaped contrasts of colours in foam films and bubbles with the right lighting (the coupling between gravity and the thickness of foam films and the existence of a Raleigh-Taylor-like instability is discussed in subsection 0.3.5). Since then, it has been mostly studied in vertical situations [46, 64, 65].

However, it is actually best observed with horizontal foam films undergoing a uni-axial compression, as illustrated in Fig. 17. This is an experiment conducted by Trégouët et al. [66]. Pictures **A-D** are a timelaspe of a deformed foam film with marginal regeneration seen with fluorescence imaging (the brighter the thicker).

<figure><img src="image_22.png" /></figure>

Figure 17: Foam film undergoing an initial uniaxial compression (scale of the film: 6.5 cm large in the horizontal dimension) with a growing marginal regeneration instability (times from A to D: $t = 0, 2, 4, 7s$). The film is observed under fluorescence imaging (the brighter the thicker): **A-B:** The film is being compressed (white arrows) and small semicircular patches of thin film starts going out. **C-D:** Ripening of the patches, as they grow by more thin film extraction and coalescence of adjacent patches. **E:** Zoom at a local scale of D, white upward arrows bordering the meniscus (white horizontal fringe at the bottom) are the direction of thin film extraction. Blue downward arrows are for the ingoing thick patches channelled between the thin patches.

**A** and **B** show the film being compressed, thus forcing either the contraction of the foam film, or the evacuation of patches of film toward the edges at the menisci. Because film elasticity $E_f$ is high (cf. subsection 0.2.4), the drop in tension is significant enough to ensure that the foam film does not withstand the whole compression with a global thickening of the pre-existing film, but rather by evacuating some film toward the menisci.

In pictures **B-D**, we see the formation of circular thin patches at the edges perpendicular to the axis of deformation. As time goes by, they grow and even coalesce.

Picture **E** is a zoom at one of these edges: it shows how some thick film goes in the meniscus (which is the white fringe at the bottom) following the blue arrows, and how some thin film goes out of it following the white arrows. The thick ingoing film is the film pre-existing the
mushroom-shaped contrasts of colours in foam films and bubbles with the right lighting (the coupling between gravity and the thickness of foam films and the existence of a Raleigh-Taylor-like instability is discussed in subsection 0.3.5). Since then, it has been mostly studied in vertical situations [46, 64, 65].

However, it is actually best observed with horizontal foam films undergoing a uni-axial compression, as illustrated in Fig. 17. This is an experiment conducted by Trégouët et al. [66]. Pictures **A-D** are a timelaspe of a deformed foam film with marginal regeneration seen with fluorescence imaging (the brighter the thicker).

<figure><img src="image_22.png" /></figure>

Figure 17: Foam film undergoing an initial uniaxial compression (scale of the film: 6.5 cm large in the horizontal dimension) with a growing marginal regeneration instability (times from A to D: *t* = 0, 2, 4, 7s). The film is observed under fluorescence imaging (the brighter the thicker): **A-B:** The film is being compressed (white arrows) and small semicircular patches of thin film starts going out. **C-D:** Ripening of the patches, as they grow by more thin film extraction and coalescence of adjacent patches. **E:** Zoom at a local scale of D, white upward arrows bordering the meniscus (white horizontal fringe at the bottom) are the direction of thin film extraction. Blue downward arrows are for the ingoing thick patches channelled between the thin patches.

**A** and **B** show the film being compressed, thus forcing either the contraction of the foam film, or the evacuation of patches of film toward the edges at the menisci. Because film elasticity *E*<sub>f</sub> is high (cf. subsection 0.2.4), the drop in tension is significant enough to ensure that the foam film does not withstand the whole compression with a global thickening of the pre-existing film, but rather by evacuating some film toward the menisci.

In pictures **B-D**, we see the formation of circular thin patches at the edges perpendicular to the axis of deformation. As time goes by, they grow and even coalesce.

Picture **E** is a zoom at one of these edges: it shows how some thick film goes in the meniscus (which is the white fringe at the bottom) following the blue arrows, and how some thin film goes out of it following the white arrows. The thick ingoing film is the film pre-existing the
deformation, prepared like so before the experiment, imposing a $h_{thick}$ to the system. The thin film goes out of the meniscus spontaneously with their own thickness and velocity.

It is puzzling to see this spontaneous symmetry breaking along the pushed menisci, all the more when it involves the extraction of patches of film in the plane of an already-compressed foam film. This weird behaviour can be rationalised by considering the Frankel's problems identifiable in this situation. In fact, in subsection 0.3.2, we describe the 2D problem of an advected flat foam film connected to a meniscus depending on a virtually imposed velocity $U$, which can be either positive (outgoing film) or negative (ingoing film). Two points must be raised with respect to the experiment of Fig. 17: the Frankel's model is 2D, supposing an invariance by translation along the meniscus, while in practice we cannot impose $U$ but rather a difference of tension $\Delta\gamma$ with the menisci by compressing the film.

Thus, the advection velocity toward the menisci is actually an experimentally free parameter that has to adapt to the imposed tension $\Delta\gamma$ (which must be uniform in the film cf. 0.2.4), and it can do it in multiple ways, possibly choosing different ones for different positions along the meniscus! As long as the area conservation is preserved.

Depending on the sign of $U$, the tension admits two different expressions: eq. 27 $\Delta\gamma^{out}$ for positive $U$ (extraction of film), eq. 28 $\Delta\gamma^{in}$ for negative $U$ (evacuation of film). The two tensions can have the same values for different velocities $U$ and thicknesses $h_\infty$ and it is possible to find a working point where $\Delta\gamma^{out}(U^{out}) = \Delta\gamma^{in}(U^{in}, h_\infty^{in}) = \Delta\gamma$ (Note that here $h_\infty^{out}$ is completely determined by $U^{out}$, and $h_\infty^{in} = h_{thick}$ depends on the initial film undergoing compression).

However, the existence of this working point is obviously not the whole story, as the thin patches develop with regular patterns: semicircular growing shapes that coalesce as they get closer to one another. The initialisation of this dynamics is still a work in progress, a solid lead being to start from an Aradian's marginal pinching profile (see subsection 0.3.3) invariant by translation along the meniscus, which is unstable and destabilises with the forcing imposed by the compression. The ripening dynamics of these patches on the other hand is already well understood, as the width of the thin patches, the width of the thick film channels and the ingoing/outgoing velocities must obey surface conservation and the uniformity of film tension at all times. This last point is developed in [66].

Finally, and to motivate why I elaborate so much about marginal regeneration, I have to say that later on this instability will be quite a recurrent limiting factor for my measurements. In fact, it eventually always appears whenever a film is being compressed. The whole chapter 3, that is the core of the experimental work of my thesis, is about how deformed foam films can relax by exchanging pieces of interface through their meniscus. In this regard, marginal regeneration will be close to my worst enemy as it breaks the main translational symmetry our models rely on. Thus, the general goal from now on is to keep the experimental conditions so that we work before the initialisation of this instability (by trial and error mostly).

**Important Remark:** The semicircular shape of the thin patches is the signature that a line tension exists at the border with the thicker film. Chapter 2 is dedicated to this line tension, of which a theoretical description and a measurement are detailed.
deformation, prepared like so before the experiment, imposing a $h_{thick}$ to the system. The thin film goes out of the meniscus spontaneously with their own thickness and velocity.

It is puzzling to see this spontaneous symmetry breaking along the pushed menisci, all the more when it involves the extraction of patches of film in the plane of an already-compressed foam film. This weird behaviour can be rationalised by considering the Frankel's problems identifiable in this situation. In fact, in subsection 0.3.2, we describe the 2D problem of an advected flat foam film connected to a meniscus depending on a virtually imposed velocity $U$, which can be either positive (outgoing film) or negative (ingoing film). Two points must be raised with respect to the experiment of Fig. 17: the Frankel's model is 2D, supposing an invariance by translation along the meniscus, while in practice we cannot impose $U$ but rather a difference of tension $\Delta\gamma$ with the menisci by compressing the film.

Thus, the advection velocity toward the menisci is actually an experimentally free parameter that has to adapt to the imposed tension $\Delta\gamma$ (which must be uniform in the film cf. 0.2.4), and it can do it in multiple ways, possibly choosing different ones for different positions along the meniscus! As long as the area conservation is preserved.

Depending on the sign of $U$, the tension admits two different expressions: eq. 27 $\Delta\gamma^{out}$ for positive $U$ (extraction of film), eq. 28 $\Delta\gamma^{in}$ for negative $U$ (evacuation of film). The two tensions can have the same values for different velocities $U$ and thicknesses $h_\infty$ and it is possible to find a working point where $\Delta\gamma^{out}(U^{out}) = \Delta\gamma^{in}(U^{in}, h_\infty^{in}) = \Delta\gamma$ (Note that here $h_\infty^{out}$ is completely determined by $U^{out}$, and $h_\infty^{in} = h_{thick}$ depends on the initial film undergoing compression).

However, the existence of this working point is obviously not the whole story, as the thin patches develop with regular patterns: semicircular growing shapes that coalesce as they get closer to one another. The initialisation of this dynamics is still a work in progress, a solid lead being to start from an Aradian's marginal pinching profile (see subsection 0.3.3) invariant by translation along the meniscus, which is unstable and destabilises with the forcing imposed by the compression. The ripening dynamics of these patches on the other hand is already well understood, as the width of the thin patches, the width of the thick film channels and the ingoing/outgoing velocities must obey surface conservation and the uniformity of film tension at all times. This last point is developed in [66].

Finally, and to motivate why I elaborate so much about marginal regeneration, I have to say that later on this instability will be quite a recurrent limiting factor for my measurements. In fact, it eventually always appears whenever a film is being compressed. The whole chapter 3, that is the core of the experimental work of my thesis, is about how deformed foam films can relax by exchanging pieces of interface through their meniscus. In this regard, marginal regeneration will be close to my worst enemy as it breaks the main translational symmetry our models rely on. Thus, the general goal from now on is to keep the experimental conditions so that we work before the initialisation of this instability (by trial and error mostly).

**Important Remark:** The semicircular shape of the thin patches is the signature that a line tension exists at the border with the thicker film. Chapter 2 is dedicated to this line tension, of which a theoretical description and a measurement are detailed.
### 0.3.5 Rayleigh-Taylor-like instability with thickness heterogeneities and gravity

When marginal regeneration was first described [47], the thin patches were observed not as growing semicircular patterns near the border (where they are produced as discussed in subsection 0.3.4), but rather as rising mushroom-shaped pieces of film. This is not a feature of marginal regeneration itself, although it is often what comes to mind when we use this term, but of the in-plane rearrangements of the patches of film depending on their thickness in order to minimise the gravity potential energy of the foam film.

The mushroom shapes are in fact a feature of a Rayleigh-Taylor-like instability happening at the border between a thick piece of film that is on top of a thinner one. This instability is well known in the context of thin patches moving up in a film, and contributing to drainage [46, 67] but is easier to quantify with a more controlled initial thickness distribution. Such a controlled experiment from Shabalina et al. [68] is represented in Fig. 18 A-D, showing such patterns. A foam film is stretched from top to bottom (the white arrows indicate the position of the moving bottom of the film). What we see in the coloured film is that the thin patches produced by marginal regeneration before the stretching rise to the top (e.g. with C, the yellow mushrooms are thinner than the purple film in which they rise, the latter being thinner than the grey Frankel films extracted at the top and bottom edges). In the same way, the thick grey film at the top in D (highlighted by the red rectangle) starts destabilising with an identifiable wavelength.

<figure><img src="image_23.png" /><figcaption>Figure 18: Stretching of a vertical film (width of (a)-(d): 10.3 mm). A 2D Rayleigh-Taylor instability occurs where thicker films are on top of thinner ones. The white arrows show the bottom of the film being stretched from (a) to (d) (resp. (a) 0 ms, (b) 67 ms, (c) 83 ms and (d) 167 ms</figcaption></figure>
### 0.3.5 Rayleigh-Taylor-like instability with thickness heterogeneities and gravity

When marginal regeneration was first described [47], the thin patches were observed not as growing semicircular patterns near the border (where they are produced as discussed in subsection 0.3.4), but rather as rising mushroom-shaped pieces of film. This is not a feature of marginal regeneration itself, although it is often what comes to mind when we use this term, but of the in-plane rearrangements of the patches of film depending on their thickness in order to minimise the gravity potential energy of the foam film.

The mushroom shapes are in fact a feature of a Rayleigh-Taylor-like instability happening at the border between a thick piece of film that is on top of a thinner one. This instability is well known in the context of thin patches moving up in a film, and contributing to drainage [46, 67] but is easier to quantify with a more controlled initial thickness distribution. Such a controlled experiment from Shabalina et al. [68] is represented in Fig. 18 A-D, showing such patterns. A foam film is stretched from top to bottom (the white arrows indicate the position of the moving bottom of the film). What we see in the coloured film is that the thin patches produced by marginal regeneration before the stretching rise to the top (e.g. with C, the yellow mushrooms are thinner than the purple film in which they rise, the latter being thinner than the grey Frankel films extracted at the top and bottom edges). In the same way, the thick grey film at the top in D (highlighted by the red rectangle) starts destabilising with an identifiable wavelength.

<figure><img src="image_23.png" /><figcaption>Figure 18: Stretching of a vertical film (width of (a)-(d): 10.3 mm). A 2D Rayleigh-Taylor instability occurs where thicker films are on top of thinner ones. The white arrows show the bottom of the film being stretched from (a) to (d) (resp. (a) 0 ms, (b) 67 ms, (c) 83 ms and (d) 167 ms</figcaption></figure>
This instability can really be thought as a 2D analogue of the 3D Rayleigh-Taylor instability, swapping fluid density $\rho$ for thickness of film $h$, pressure $p$ for film tension $\sigma$, and bulk viscosity $\eta$ for $\eta_s$ a surface shear viscosity. One more exotic ingredient to add which has already been introduced in subsection 0.3.4 is the line tension $T$ existing at the border between the two films because of the thickness jump (a full theoretical description followed by an experimental measurement is developed in chapter 2). Here, it holds the same role as surface tension in the 3D instability, as it tends to hinder the existence of the small spatial wavelengths for the border.

A linear stability analysis can give the spatial wavelength $\lambda_c$ maximising the growth rate $n$ of the instability. The latter is dominated by inertia at low $\lambda$ and by viscous forces at high ones:

$$ \begin{cases} n_{\mu s} \sim \lambda \\ n_{in} \sim \lambda^{-1/2} \end{cases} \qquad (31) $$

These two limits imply the existence of an optimal growth rate $n_c$ at a given critical $\lambda_c$ that can be computed with the full calculation of the linear stability analysis and which is in quantitative agreement with the wavelength measured in Fig. 18(e).

This instability occurs quite rapidly at our typical time scale $\tau_{def} \sim 1$s (cf. time scales Fig. 18), meaning that stretching a vertical film can be a problem depending on where we are looking at. As a Frankel film is most of the time thicker than the stretched film pulling it out of the meniscus, **it will not be possible to measure properly the outgoing quantities of films at a meniscus located on top of a vertical stretched film**. This will be another limitation for our experiments later on.

### 0.3.6 Evaporation

Speaking of important flows for the sake of describing our experiments, we need not to forget the influence of evaporation, which tends to uniformly thin out our foam films. As unconditioned air is never at water saturation, it remains a mechanism of great importance. The experimental work of the present thesis is taking place in this context.

In the literature, the dynamics of evaporation-induced ruptures are well rationalised for a single film [67, 69], yielding the order of magnitude for the thinning of $\sim 20$ nm/s in normal conditions of temperature and average humidity (rH $\sim 30-70\%$). However, the exact mechanism triggering the rupture of the thinned film is still only conjectured to be a fast local loss of mass in the film due to evaporation becoming important compared to film drainage. This is supposed to take place in a short amount of time which depends on the relative humidity.

With regard to our foam films, evaporation is important in the sense that their lifespan was seen to be highly dependent on the relative humidity, which was kept at a minimum threshold of rH = 55% using an air humidifier. Living in Brittany also helped out.

## 0.4 Elementary liquid foams

Now that we have discussed what happens with a foam film and what occurs when we connect it to a meniscus, we will take a bottom-up approach towards liquid foams by looking at one scale
This instability can really be thought as a 2D analogue of the 3D Rayleigh-Taylor instability, swapping fluid density $\rho$ for thickness of film $h$, pressure $p$ for film tension $\sigma$, and bulk viscosity $\eta$ for $\eta_s$ a surface shear viscosity. One more exotic ingredient to add which has already been introduced in subsection 0.3.4 is the line tension $T$ existing at the border between the two films because of the thickness jump (a full theoretical description followed by an experimental measurement is developed in chapter 2). Here, it holds the same role as surface tension in the 3D instability, as it tends to hinder the existence of the small spatial wavelengths for the border.

A linear stability analysis can give the spatial wavelength $\lambda_c$ maximising the growth rate $n$ of the instability. The latter is dominated by inertia at low $\lambda$ and by viscous forces at high ones:

$$ \begin{cases} n_{\mu s} \sim \lambda \\ n_{in} \sim \lambda^{-1/2} \end{cases} \qquad (31) $$

These two limits imply the existence of an optimal growth rate $n_c$ at a given critical $\lambda_c$ that can be computed with the full calculation of the linear stability analysis and which is in quantitative agreement with the wavelength measured in Fig. 18(e).

This instability occurs quite rapidly at our typical time scale $\tau_{def} \sim 1$s (cf. time scales Fig. 18), meaning that stretching a vertical film can be a problem depending on where we are looking at. As a Frankel film is most of the time thicker than the stretched film pulling it out of the meniscus, **it will not be possible to measure properly the outgoing quantities of films at a meniscus located on top of a vertical stretched film**. This will be another limitation for our experiments later on.

### 0.3.6 Evaporation

Speaking of important flows for the sake of describing our experiments, we need not to forget the influence of evaporation, which tends to uniformly thin out our foam films. As unconditioned air is never at water saturation, it remains a mechanism of great importance. The experimental work of the present thesis is taking place in this context.

In the literature, the dynamics of evaporation-induced ruptures are well rationalised for a single film [67, 69], yielding the order of magnitude for the thinning of $\sim 20$ nm/s in normal conditions of temperature and average humidity ($rH \sim 30-70\%$). However, the exact mechanism triggering the rupture of the thinned film is still only conjectured to be a fast local loss of mass in the film due to evaporation becoming important compared to film drainage. This is supposed to take place in a short amount of time which depends on the relative humidity.

With regard to our foam films, evaporation is important in the sense that their lifespan was seen to be highly dependent on the relative humidity, which was kept at a minimum threshold of $rH = 55\%$ using an air humidifier. Living in Brittany also helped out.

## 0.4 Elementary liquid foams

Now that we have discussed what happens with a foam film and what occurs when we connect it to a meniscus, we will take a bottom-up approach towards liquid foams by looking at one scale
above: elementary liquid foams. These are arrangements of several foam films connected by menisci, defining a mesoscopic scale between the individual film and the liquid foam. **Chapters 1 and 3 concern this scale.** Thus, our goal is now to look at the hydrodynamics of these systems by probing their mechanical response around their equilibrium state.

All the following subsections constitute a state of the art for elementary liquid foams. **Notably, I detail them in a chronological way, so that it is possible to see how the hypotheses made on the surfactant exchanges in the system have gotten more and more refined over the last 20 years. The following papers represent important milestones for the problem we address in chapter 3.**

### 0.4.1 Plateau's laws: the local equilibrium state for the shape of liquid foams

Dry liquid foams are by no means at thermodynamic equilibrium. In fact, if we wait long enough, evaporation, gravity or capillary drainage will force rearrangements to occur. However, for intermediate times, the local structures they adopt obey several rules dictated by a local mechanical equilibrium. These are the Plateau's laws, stated by Joseph Plateau in 1873 [70] and demonstrated by Almgren & Taylor in 1976 [71]:

* Foam films have a uniform mean curvature, meaning the gas volumes they separate have each a uniform pressure given by Laplace's law (eq.5).

* Menisci are where three foam films intersect, and they do so with an angle of 120° which minimises the resulting interface.

* Vertices are where three bubbles meet, or equivalently where four menisci intersect. They do so with a tetrahedral symmetry, with angles of 109.47°.

These laws hold as long as we keep the foam as ideal as possible, that is very dry, at a global mechanical equilibrium, with a uniform surface tension. In the following most of the dynamics we will be looking at are variations around this mechanical equilibrium state with respect to normal motions of the film. These variations are small, and usually are deviations of the order of 1° around the 120° of the second law.

Lastly, mechanically stressing an elementary foam out of the Plateau's law induces some dissipation, as it relaxes back to another state also prescribed by these laws [72, 73]. Rationalising and predicting this is what requires the biggest efforts from our community, and what represents a pathway towards better rationalising the effective viscosities of liquid foams.

### 0.4.2 T1 events and the Durand-Stone experiment

If a liquid foam respecting Plateau's law undergoes shear or compression at a global scale, bubbles are rearranging at the local scale. This is the scale of elementary liquid foams. Notably, a singular event at this scale is what is called a *T1 event* [74–77] of which an illustration is shown in Fig. 19. It consists of a local change of rules for the neighbourhood of bubbles, which are prescribed by the Plateau's laws again. For a 2D monodisperse hexagonal foam, it is a displacement of one
above: elementary liquid foams. These are arrangements of several foam films connected by menisci, defining a mesoscopic scale between the individual film and the liquid foam. **Chapters 1 and 3 concern this scale.** Thus, our goal is now to look at the hydrodynamics of these systems by probing their mechanical response around their equilibrium state.

All the following subsections constitute a state of the art for elementary liquid foams. **Notably, I detail them in a chronological way, so that it is possible to see how the hypotheses made on the surfactant exchanges in the system have gotten more and more refined over the last 20 years. The following papers represent important milestones for the problem we address in chapter 3.**

### 0.4.1 Plateau's laws: the local equilibrium state for the shape of liquid foams

Dry liquid foams are by no means at thermodynamic equilibrium. In fact, if we wait long enough, evaporation, gravity or capillary drainage will force rearrangements to occur. However, for intermediate times, the local structures they adopt obey several rules dictated by a local mechanical equilibrium. These are the Plateau's laws, stated by Joseph Plateau in 1873 [70] and demonstrated by Almgren & Taylor in 1976 [71]:

* Foam films have a uniform mean curvature, meaning the gas volumes they separate have each a uniform pressure given by Laplace's law (eq.5).

* Menisci are where three foam films intersect, and they do so with an angle of 120° which minimises the resulting interface.

* Vertices are where three bubbles meet, or equivalently where four menisci intersect. They do so with a tetrahedral symmetry, with angles of 109.47°.

These laws hold as long as we keep the foam as ideal as possible, that is very dry, at a global mechanical equilibrium, with a uniform surface tension. In the following most of the dynamics we will be looking at are variations around this mechanical equilibrium state with respect to normal motions of the film. These variations are small, and usually are deviations of the order of 1° around the 120° of the second law.

Lastly, mechanically stressing an elementary foam out of the Plateau's law induces some dissipation, as it relaxes back to another state also prescribed by these laws [72, 73]. Rationalising and predicting this is what requires the biggest efforts from our community, and what represents a pathway towards better rationalising the effective viscosities of liquid foams.

### 0.4.2 T1 events and the Durand-Stone experiment

If a liquid foam respecting Plateau's law undergoes shear or compression at a global scale, bubbles are rearranging at the local scale. This is the scale of elementary liquid foams. Notably, a singular event at this scale is what is called a *T1 event* [74–77] of which an illustration is shown in Fig. 19. It consists of a local change of rules for the neighbourhood of bubbles, which are prescribed by the Plateau's laws again. For a 2D monodisperse hexagonal foam, it is a displacement of one
line of bubbles with respect to another one with an amplitude of one bubble. This results in the full compression of the central film of an elementary assembly of five films (represented in red in Fig.19), the temporary merging of the two menisci at its edges, and the stretching of a new film separating the two new bubble neighbours (in yellow). This is the most elementary event of rearrangement involving the merging and splitting of entire menisci (NB: T2 events also exist, consisting of a bubble disappearing and creating a vertex).

<figure><img src="image_24.png" /><figcaption>Figure 19: T1 rearrangement in a 2D foam. Credits to Titta et al. [3]</figcaption></figure>

Thus, both the starting and ending points of this T1 process represent a local minimum of energy for the system, respecting Plateau’s laws. The goal now is to probe how to pass from one state to another, and if we can get some rheological information about the foam films by doing so. The idea is that the dynamics takes place in two steps: first we load elastically the system by compressing the central film until it disappears, putting the system in the unstable configuration of four films connected by one meniscus. At this point, we stop adding work to the system, which relaxes on its own towards a similar configuration as its initial state, at a lower energy. Quantifying the time of this relaxation was the missing key to get a proper loss modulus for the system.

This has been done by Durand et al. [1] in 2006 by looking with a high speed camera at a 2D liquid foam inducing rearrangements by pumping air against it. The notations are shown in Fig.20.

I will detail their approach a bit, as it is full of problems and ideas we will later face again for our own experiments and models.

They look at the dynamics of relaxation of $x_B(t)$, writing the force balance at both menisci:

$$2\gamma_{\text{sides}}(t) \cos \alpha(t) - \gamma_{\text{central}}(t) - (\mu_s + \kappa_s) \frac{\partial U}{\partial x}\Big|_{x=x_B} = 0 \quad (32)$$

Where $\gamma_{\text{sides}}$ is the surface tension of the four films at the edges, $\gamma_{\text{central}}$ the surface tension of the central film and $U$ is the interface velocity in the central film. The third term combines both the shear and dilatational viscous contributions of the interfaces, which take this form in this 1D model. To further advance, the authors have to make a few important assumptions:
line of bubbles with respect to another one with an amplitude of one bubble. This results in the full compression of the central film of an elementary assembly of five films (represented in red in Fig.19), the temporary merging of the two menisci at its edges, and the stretching of a new film separating the two new bubble neighbours (in yellow). This is the most elementary event of rearrangement involving the merging and splitting of entire menisci (NB: T2 events also exist, consisting of a bubble disappearing and creating a vertex).

<figure><img src="image_24.png" /><figcaption>Figure 19: T1 rearrangement in a 2D foam. Credits to Titta et al. [3]</figcaption></figure>

Thus, both the starting and ending points of this T1 process represent a local minimum of energy for the system, respecting Plateau’s laws. The goal now is to probe how to pass from one state to another, and if we can get some rheological information about the foam films by doing so. The idea is that the dynamics takes place in two steps: first we load elastically the system by compressing the central film until it disappears, putting the system in the unstable configuration of four films connected by one meniscus. At this point, we stop adding work to the system, which relaxes on its own towards a similar configuration as its initial state, at a lower energy. Quantifying the time of this relaxation was the missing key to get a proper loss modulus for the system.

This has been done by Durand et al. [1] in 2006 by looking with a high speed camera at a 2D liquid foam inducing rearrangements by pumping air against it. The notations are shown in Fig.20.

I will detail their approach a bit, as it is full of problems and ideas we will later face again for our own experiments and models.

They look at the dynamics of relaxation of $x_B(t)$, writing the force balance at both menisci:

$$2\gamma_{\text{sides}}(t) \cos \alpha(t) - \gamma_{\text{central}}(t) - (\mu_s + \kappa_s) \frac{\partial U}{\partial x}\Big|_{x=x_B} = 0 \quad (32)$$

Where $\gamma_{\text{sides}}$ is the surface tension of the four films at the edges, $\gamma_{\text{central}}$ the surface tension of the central film and $U$ is the interface velocity in the central film. The third term combines both the shear and dilatational viscous contributions of the interfaces, which take this form in this 1D model. To further advance, the authors have to make a few important assumptions:
<figure><img src="image_25.png" /><figcaption>Figure 20: Notations used by the authors to describe the T1 experiment. Credits to Durand et al. [1]</figcaption></figure>

* Bulk viscosities (air and liquid frictions) were discarded right off.

* Surface tension responds to deformation with its Marangoni modulus $E_M$ (eqs. 11). It means we suppose here that we are in the insoluble limit.

* The four vertical films on the sides will be supposed at rest at all times, meaning the we impose $\gamma_{sides}(t) = \gamma_{eq}$. That means that we virtually fix the surface tension somewhere in the system so that the side films get rid of as many interface as they need as $L(t)$ decreases to enforce this condition. It thus also sets the interface velocity $U$ at $x = x_B$. This is a bold guess, but a necessary one here, which seems reasonable as the central film is the one undergoing the biggest deformation in the process. However, it is artificial, and the next models will not make this assumption (including the one used in chapter 3 detailed in 0.4.5), and the experiments will show that it could not be made.

Solving the problem like so, they end up with two quantities both deductible from $x_B(t)$ and $\dot{x}_B(t)$ linked by an affine relationship whose coefficients give $E_M$ and $\mu_s + \kappa_s$. They thus make an estimate of both the surface elasticity and viscosities for SDS solutions and a mixture of polymer and proteins (bovine serum albumine and propylene glycol alginate):

$$ \begin{cases} E_{M,SDS} = 32 \pm 8 \, \text{mN/m} ; (\mu_s + \kappa_s)|_{SDS} = 1.3 \pm 0.7 \, \text{mPa} \cdot \text{m} \cdot \text{s} \\ E_{M,mix} = 65 \pm 12 \, \text{mN/m} ; (\mu_s + \kappa_s)|_{mix} = 31 \pm 12 \, \text{mPa} \cdot \text{m} \cdot \text{s} \end{cases} \quad (33) $$

These first measurements of the Marangoni modulus and the surface viscosities is a valuable lesson. They all are very high values compared to the existing literature at the time, and to more recent measurements [6, 36, 78].
<figure><img src="image_25.png" /><figcaption>Figure 20: Notations used by the authors to describe the T1 experiment. Credits to Durand et al. [1]</figcaption></figure>

* Bulk viscosities (air and liquid frictions) were discarded right off.

* Surface tension responds to deformation with its Marangoni modulus $E_M$ (eqs. 11). It means we suppose here that we are in the insoluble limit.

* The four vertical films on the sides will be supposed at rest at all times, meaning the we impose $\gamma_{sides}(t) = \gamma_{eq}$. That means that we virtually fix the surface tension somewhere in the system so that the side films get rid of as many interface as they need as $L(t)$ decreases to enforce this condition. It thus also sets the interface velocity $U$ at $x = x_B$. This is a bold guess, but a necessary one here, which seems reasonable as the central film is the one undergoing the biggest deformation in the process. However, it is artificial, and the next models will not make this assumption (including the one used in chapter 3 detailed in 0.4.5), and the experiments will show that it could not be made.

Solving the problem like so, they end up with two quantities both deductible from $x_B(t)$ and $\dot{x}_B(t)$ linked by an affine relationship whose coefficients give $E_M$ and $\mu_s + \kappa_s$. They thus make an estimate of both the surface elasticity and viscosities for SDS solutions and a mixture of polymer and proteins (bovine serum albumine and propylene glycol alginate):

$$ \left\{ \begin{array}{l} E_{M,SDS} = 32 \pm 8 \, \text{mN/m} ; (\mu_s + \kappa_s)|_{SDS} = 1.3 \pm 0.7 \, \text{mPa} \cdot \text{m} \cdot \text{s} \\ E_{M,mix} = 65 \pm 12 \, \text{mN/m} ; (\mu_s + \kappa_s)|_{mix} = 31 \pm 12 \, \text{mPa} \cdot \text{m} \cdot \text{s} \end{array} \right. \qquad (33) $$

These first measurements of the Marangoni modulus and the surface viscosities is a valuable lesson. They all are very high values compared to the existing literature at the time, and to more recent measurements [6, 36, 78].
Notably, more recent and refined models do not make the assumption $\gamma_{sides}(t) = \gamma_{eq}$ for the vertical films. This sets artificially the quantity of interface exchanged by them towards the central film. **This raises a problem however: the two interfaces of each vertical film do not undergo the same dynamics at all. In fact, as seen in Fig. 20, the top and bottom interfaces are free to slide from the vertical films to the central film, but the right and left interfaces have a stopping point at the menisci by symmetry.** Although it is taken into account that each vertical film feeds the central film with only half the quantity of interface it gives up, the lost half is implicitly completely ditched into the bulk of the meniscus if we want to ensure that the tensions in the vertical films remain the same. This is a hidden hypothesis whose validation/invalidation is at the core of the main chapter of this thesis, chapter 3.

### 0.4.3 Axisymmetric elementary liquid foam under small deformation: the Besson-Debrégeas experiment

The experiment by Besson *et al.* [78] described in this subsection examines a configuration similar to the previously observed T1, but in an axisymmetric geometry. Unlike in Durand *et al.* [1], where the setup solely allows us to explore the dynamics between two static states, we have here the ability to control the deformation of the central film, compression or stretching, by observing an oscillatory regime.

<figure><img src="image_26.png" /><figcaption>Figure 21: Besson-Debrégeas experimental setup: (a) Two identical bubbles being slowly brought into contact. (b) The contact is made and the radius of the flat film rises. (c-d) Notations used to describe the geometry of the problem. Credits to Besson *et al.* [78]</figcaption></figure>
Notably, more recent and refined models do not make the assumption $\gamma_{sides}(t) = \gamma_{eq}$ for the vertical films. This sets artificially the quantity of interface exchanged by them towards the central film. **This raises a problem however: the two interfaces of each vertical film do not undergo the same dynamics at all. In fact, as seen in Fig. 20, the top and bottom interfaces are free to slide from the vertical films to the central film, but the right and left interfaces have a stopping point at the menisci by symmetry.** Although it is taken into account that each vertical film feeds the central film with only half the quantity of interface it gives up, the lost half is implicitly completely ditched into the bulk of the meniscus if we want to ensure that the tensions in the vertical films remain the same. This is a hidden hypothesis whose validation/invalidation is at the core of the main chapter of this thesis, chapter 3.

### 0.4.3 Axisymmetric elementary liquid foam under small deformation: the Besson-Debrégeas experiment

The experiment by Besson *et al.* [78] described in this subsection examines a configuration similar to the previously observed T1, but in an axisymmetric geometry. Unlike in Durand *et al.* [1], where the setup solely allows us to explore the dynamics between two static states, we have here the ability to control the deformation of the central film, compression or stretching, by observing an oscillatory regime.

<figure><img src="image_26.png" /><figcaption>Figure 21: Besson-Debrégeas experimental setup: (a) Two identical bubbles being slowly brought into contact. (b) The contact is made and the radius of the flat film rises. (c-d) Notations used to describe the geometry of the problem. Credits to Besson *et al.* [78]</figcaption></figure>
In this configuration and in the theoretical model developed with it, different time regimes exist where the surface tensions of the vertical films are now not artificially kept equal to the equilibrium value but are rather time varying variables, which are decoupled between the outer and inner interfaces. This is a very important point as it allows to take into account regimes where we expect interfaces of the same foam film to have a relative motion, meaning there is a bulk shear to take into account for the effective loss modulus of the elementary foam! **This will be a key ingredient for this thesis work, as this paper from Besson et al. [78] is the first to consider a refined model for the surfactant exchanges between the three different interfaces neighbouring a meniscus, unveiling the existence of dissipative phenomena in elementary liquid foams that are not due to the surface rheology of single interfaces alone.** Also, this paper brings novelty in terms of experimental access to these differences of tensions, by measuring the angle deviation from the Plateau's law.

To achieve this, the authors create two bubbles with the same diameters of $2R = 7$ mm, facing each other, and mobile along a common vertical axis. This configuration is depicted in Fig.21. The pressure in the bubbles is determined by the Laplace's law, and they gradually bring them into contact. As a result, they create a contact zone, a third flat film connected to the two bubbles by a circular meniscus (see Fig.21B). Since the pressures of the bubbles remain the same and identical, Laplace's law dictates that the surfaces of revolution for each bubble have principal curvatures that satisfy $κ₁ + κ₂ = 2/R$, and the central film between them must be flat.

<figure><img src="image_27.png" /><figcaption>Figure 22: (a) Contact radius $r_c(t)$ as a function of time for a typical experiment. (b) Contact angle $θ(t)$ as a function of time. Credits to Besson et al. [78] (Pardon their french for the axis names)</figcaption></figure>

Once the contact is made, the system undergoes rapid changes, akin to a T1 relaxation with a contact radius $r_c$ for the central film growing over time, and the outer contact angle $θ$ opening up. Fig.22 illustrates a typical experiment. After approximately 10 s, they reach mechanical equilibrium: $θ$ plateaus, and $r_c$ reaches its maximum before decreasing at longer times because of bubble ageing through gas escaping due to the inner overpressure. This typical time scale
In this configuration and in the theoretical model developed with it, different time regimes exist where the surface tensions of the vertical films are now not artificially kept equal to the equilibrium value but are rather time varying variables, which are decoupled between the outer and inner interfaces. This is a very important point as it allows to take into account regimes where we expect interfaces of the same foam film to have a relative motion, meaning there is a bulk shear to take into account for the effective loss modulus of the elementary foam! **This will be a key ingredient for this thesis work, as this paper from Besson et al. [78] is the first to consider a refined model for the surfactant exchanges between the three different interfaces neighbouring a meniscus, unveiling the existence of dissipative phenomena in elementary liquid foams that are not due to the surface rheology of single interfaces alone.** Also, this paper brings novelty in terms of experimental access to these differences of tensions, by measuring the angle deviation from the Plateau's law.

To achieve this, the authors create two bubbles with the same diameters of $2R = 7$ mm, facing each other, and mobile along a common vertical axis. This configuration is depicted in Fig.21. The pressure in the bubbles is determined by the Laplace's law, and they gradually bring them into contact. As a result, they create a contact zone, a third flat film connected to the two bubbles by a circular meniscus (see Fig.21B). Since the pressures of the bubbles remain the same and identical, Laplace's law dictates that the surfaces of revolution for each bubble have principal curvatures that satisfy $κ₁ + κ₂ = 2/R$, and the central film between them must be flat.

<figure><img src="image_27.png" /><figcaption>Figure 22: (a) Contact radius $r_c(t)$ as a function of time for a typical experiment. (b) Contact angle $θ(t)$ as a function of time. Credits to Besson *et al.* [78] (Pardon their french for the axis names)</figcaption></figure>

Once the contact is made, the system undergoes rapid changes, akin to a T1 relaxation with a contact radius $r_c$ for the central film growing over time, and the outer contact angle $θ$ opening up. Fig.22 illustrates a typical experiment. After approximately 10 s, they reach mechanical equilibrium: $θ$ plateaus, and $r_c$ reaches its maximum before decreasing at longer times because of bubble ageing through gas escaping due to the inner overpressure. This typical time scale
is governed by surface rheology, surfactant exchanges through the meniscus, and possibly other damping forces (inertia, air friction).

<figure><img src="image_28.png" /><figcaption>Figure 23: Static and dynamic measurements: (a) Experimental static angle $\theta_{\infty}$ plotted against the static angle $\theta_{stat}$ predicted by the decorated meniscus model of Fortes et al. [79]. Credits to Besson et al. [78]</figcaption></figure>

The transient regimes towards mechanical equilibrium shown in Fig.22 are not the main objects of study of this paper however, as they are mainly used to define the static state around which the system will be dynamically probed. To achieve this, the bubbles are brought closer together and moved apart, varying the pulsation $\omega \in [10^{-1}, 10^2]$ rad/s while keeping the amplitude constant. This makes $r_c$ and $\theta$ vary as:

$$ \begin{cases} r_c(t) = r_{c,0} + \Delta r_c(\omega) \cos(\omega t) \\ \theta(t) = \theta_0 + \Delta \theta(\omega) \cos(\omega t \phi(\omega)) \end{cases} \qquad (34) $$

Where the equilibrium angle $\theta_0(t)$ depends on time in all generality as $r_c(t)$ varies. A typical measurement is shown in Fig.23. The phase shift between the two signals allows us to define the elastic and loss moduli $A^* = A'(\omega) + iA''(\omega)$:

$$ \begin{cases} A'(\omega) = -\frac{\Delta\theta}{\Delta r_c} r_{c,0} \cos(\phi(\omega)) \\ A''(\omega) = -\frac{\Delta\theta}{\Delta r_c} r_{c,0} \sin(\phi(\omega)) \end{cases} \qquad (35) $$

Which are plotted in Fig.24(b). It is very important to note that these are **not** the moduli associated with the elasticity and viscosities of a single film! In order to estimate the latter, the authors also perform experiments on a single bubble which undergoes sinusoidal volume variation using half of their setup. In a similar way as previously, the phase shift between the recorded bubble volume and the measured inner pressure allows the authors to define an interface dilatational modulus $E^*(\omega) = E'(\omega) + iE''(\omega)$. They link this modulus to the evolution of the surface tension of a single interface undergoing a deformation $\Delta S$ of its initial surface $S_0$ as:
is governed by surface rheology, surfactant exchanges through the meniscus, and possibly other damping forces (inertia, air friction).

<figure><img src="image_28.png" /><figcaption>Figure 23: Static and dynamic measurements: (a) Experimental static angle $\theta_{\infty}$ plotted against the static angle $\theta_{stat}$ predicted by the decorated meniscus model of Fortes et al. [79]. Credits to Besson et al. [78]</figcaption></figure>

The transient regimes towards mechanical equilibrium shown in Fig.22 are not the main objects of study of this paper however, as they are mainly used to define the static state around which the system will be dynamically probed. To achieve this, the bubbles are brought closer together and moved apart, varying the pulsation $\omega \in [10^{-1}, 10^2]$ rad/s while keeping the amplitude constant. This makes $r_c$ and $\theta$ vary as:

$$ \begin{cases} r_c(t) = r_{c,0} + \Delta r_c(\omega) \cos(\omega t) \\ \theta(t) = \theta_0 + \Delta \theta(\omega) \cos(\omega t \phi(\omega)) \end{cases} \qquad (34) $$

Where the equilibrium angle $\theta_0(t)$ depends on time in all generality as $r_c(t)$ varies. A typical measurement is shown in Fig.23. The phase shift between the two signals allows us to define the elastic and loss moduli $A^* = A'(\omega) + iA''(\omega)$:

$$ \begin{cases} A'(\omega) = -\frac{\Delta\theta}{\Delta r_c} r_{c,0} \cos(\phi(\omega)) \\ A''(\omega) = -\frac{\Delta\theta}{\Delta r_c} r_{c,0} \sin(\phi(\omega)) \end{cases} \qquad (35) $$

Which are plotted in Fig.24(b). It is very important to note that these are **not** the moduli associated with the elasticity and viscosities of a single film! In order to estimate the latter, the authors also perform experiments on a single bubble which undergoes sinusoidal volume variation using half of their setup. In a similar way as previously, the phase shift between the recorded bubble volume and the measured inner pressure allows the authors to define an interface dilatational modulus $E^*(\omega) = E'(\omega) + iE''(\omega)$. They link this modulus to the evolution of the surface tension of a single interface undergoing a deformation $\Delta S$ of its initial surface $S_0$ as:
$$ \gamma(t) = \gamma_0 + E'(\omega) \frac{\Delta S}{S_0} \cos(\omega t) + E''(\omega) \frac{\Delta S}{S_0} \sin(\omega t) \quad (36) $$

In this configuration $E''$ is the signature of the intrinsic and extrinsic viscosities of a single interface deformed with a spherical dilation. $E'$ and $E''$ are plotted in Fig.24(a). The goal is now to link the modulus of the elementary foam $A^*$ to the dilatational modulus of a single interface $E^*$. Both of them differ because of in-plane exchanges of interface at the meniscus, from one film to another. This difference is what makes the rheology of elementary liquid foams a bridge between the surface rheology of single foam films and the rheology of liquid foams, the former being not sufficient to describe the latter.

<figure><img src="image_29.png" /><figcaption>Figure 24: (a) Dilatational moduli of a single interface as function the pulsation. Closed squares for the elastic modulus $E'$, open squares for the viscous modulus $E''$. The line is a model by Lucassen [80]. (b) Elastic and loss moduli of the elementary liquid foam. Closed circles for $A'$, open circles for $A''$. Credits to Besson et al. [78]</figcaption></figure>

The authors thus develop a model based on the conservation of surfactants of the three different interfaces present in the system denoted 1, 2 and 3 and represented in Fig.25. The key ingredient here is the flux $J_s$, which takes into account the surfactant exchanges between the films, which is coupled to the transverse exchanges $J_v$ between the two interfaces of a same film. They can be expressed as:

$$ \left\{ \begin{aligned} J_s &= D_s \frac{\Gamma_2 - \Gamma_3}{L} 2\pi r_c \\ J_v &= S \frac{d\Gamma}{dt} \left(1 - \frac{E_0}{E^*}\right) \end{aligned} \right. \quad (37) $$

Where $L$ is an unpredicted typical length over which the gradient of surface excess between $\Gamma_2$ and $\Gamma_3$ is developed, $D_s$ a surface diffusion coefficient and $E_0$ the Marangoni modulus of the interfaces (i.e. the surface elasticity of the interface in the insoluble limit, see eq.11).

**NB:** Durand et al. [1] also have this $J_s$ with their model where it is a slave parameter $\Gamma_{eq} \times U$ ensuring $\Gamma_2(t) = \Gamma_{eq}$. With Besson et al. [78] both $J_s$ and $\Gamma_2$ are allowed to vary, and the fate of the surfactants of $S_1$ is also taken into account through $J_{v1}$.
$$ \gamma(t) = \gamma_0 + E'(\omega) \frac{\Delta S}{S_0} \cos(\omega t) + E''(\omega) \frac{\Delta S}{S_0} \sin(\omega t) \quad (36) $$

In this configuration $E''$ is the signature of the intrinsic and extrinsic viscosities of a single interface deformed with a spherical dilation. $E'$ and $E''$ are plotted in Fig.24(a). The goal is now to link the modulus of the elementary foam $A^*$ to the dilatational modulus of a single interface $E^*$. Both of them differ because of in-plane exchanges of interface at the meniscus, from one film to another. This difference is what makes the rheology of elementary liquid foams a bridge between the surface rheology of single foam films and the rheology of liquid foams, the former being not sufficient to describe the latter.

<figure><img src="image_29.png" /><figcaption>Figure 24: (a) Dilatational moduli of a single interface as function the pulsation. Closed squares for the elastic modulus $E'$, open squares for the viscous modulus $E''$. The line is a model by Lucassen [80]. (b) Elastic and loss moduli of the elementary liquid foam. Closed circles for $A'$, open circles for $A''$. Credits to Besson et al. [78]</figcaption></figure>

The authors thus develop a model based on the conservation of surfactants of the three different interfaces present in the system denoted 1, 2 and 3 and represented in Fig.25. The key ingredient here is the flux $J_s$, which takes into account the surfactant exchanges between the films, which is coupled to the transverse exchanges $J_v$ between the two interfaces of a same film. They can be expressed as:

$$ \left\{ \begin{aligned} J_s &= D_s \frac{\Gamma_2 - \Gamma_3}{L} 2\pi r_c \\ J_v &= S \frac{d\Gamma}{dt} \left(1 - \frac{E_0}{E^*}\right) \end{aligned} \right. \quad (37) $$

Where $L$ is an unpredicted typical length over which the gradient of surface excess between $\Gamma_2$ and $\Gamma_3$ is developed, $D_s$ a surface diffusion coefficient and $E_0$ the Marangoni modulus of the interfaces (i.e. the surface elasticity of the interface in the insoluble limit, see eq.11).

**NB:** Durand et al. [1] also have this $J_s$ with their model where it is a slave parameter $\Gamma_{eq} \times U$ ensuring $\Gamma_2(t) = \Gamma_{eq}$. With Besson et al. [78] both $J_s$ and $\Gamma_2$ are allowed to vary, and the fate of the surfactants of $S_1$ is also taken into account through $J_{v1}$.
<figure><img src="image_30.png" /><figcaption>Figure 25: Notations used by the authors for the three interfaces in the problem. Credits to Besson et al. [78]</figcaption></figure>

The next step is thus to write the surfactant conservation for the three interfaces and to relate it to the expression of the surface tension, which is linked to the change of surfaces through eq.36. This yields the three variations $\Delta\gamma_i(\omega)$, giving access to the variations of angle with:

$$ \Delta\theta = \frac{1}{\sqrt{3}\gamma_0}(\Delta\gamma_1 + \Delta\gamma_2 - 2\Delta\gamma_3) \qquad (38) $$

This allows to finally write the expression of the modulus $A^*$ as defined by eq.35:

$$ A^* = \frac{2E^*}{\sqrt{3\pi\gamma_0}} \left( \frac{iE_0\omega\tau}{E^* + iE_0\omega\tau} \frac{4R^2}{2R^2 - r_{c,0}^2} + \frac{E^*}{E^* + iE_0\omega\tau} \frac{r_{c,0}^2}{2R^2 - r_{c,0}^2} \right) \qquad (39) $$

Where $\tau = (2\pi \frac{r_{c,0}}{L} D_s \frac{S_{2,0}+S_{3,0}}{S_{2,0}S_{3,0}})^{-1}$ is a typical time of equilibrium through surface diffusion between interfaces 2 and 3 (mind the missing factor $1/\pi$ in the eq.25 of the original paper [78]).

This typical time sets two different regimes for $A^*$: $\omega\tau \ll 1$ where interfaces 2 and 3 are at equilibrium and their total surface $S_2 + S_3$ is conserved so that only $S_1$ undergoes the compression/stretching cycles; $\omega\tau \gg 1$ the three interfaces withstand the compression/stretching cycles independently from one another. These two limits are plotted in Fig.24(b) as the bottom dashed line ($\omega\tau \ll 1$) and the top dash-dotted line ($\omega\tau \gg 1$). Two fits with the full equation 39 are also plotted, fitting with $\tau = 3 \times 10^{-4}$ s. There is a quantitative match at low frequencies, whereas it starts deviating for frequencies higher than $\omega > 1$ rad/s.

This paper concentrates most of the ideas at the foundation of the main contribution of this thesis, that is presented in chapter 3. As developed at the end of the last subsection 0.4.2, the main problem we face when we build a rheological model for an elementary foam, that is here three interfaces around a meniscus, is solving properly the surfactant exchanges occurring between the interfaces, the film bulks, and the meniscus bulk. Both papers from Durand et al. [1] and Besson et al. [78] make two completely different assumptions regarding the role of the meniscus: the former authors implicitly use it to get rid of an excess of surfactants to enforce their equilibrium condition for the side films, the latter do not write any surfactant transfers towards/from it, artificially blocking bulk surfactant diffusion from this region.
<figure><img src="image_30.png" /><figcaption>Figure 25: Notations used by the authors for the three interfaces in the problem. Credits to Besson et al. [78]</figcaption></figure>

The next step is thus to write the surfactant conservation for the three interfaces and to relate it to the expression of the surface tension, which is linked to the change of surfaces through eq.36. This yields the three variations $\Delta\gamma_i(\omega)$, giving access to the variations of angle with:

$$ \Delta\theta = \frac{1}{\sqrt{3}\gamma_0}(\Delta\gamma_1 + \Delta\gamma_2 - 2\Delta\gamma_3) \qquad (38) $$

This allows to finally write the expression of the modulus $A^*$ as defined by eq.35:

$$ A^* = \frac{2E^*}{\sqrt{3\pi\gamma_0}} \left( \frac{iE_0\omega\tau}{E^* + iE_0\omega\tau} \frac{4R^2}{2R^2 - r_{c,0}^2} + \frac{E^*}{E^* + iE_0\omega\tau} \frac{r_{c,0}^2}{2R^2 - r_{c,0}^2} \right) \qquad (39) $$

Where $\tau = (2\pi \frac{r_{c,0}}{L} D_s \frac{S_{2,0}+S_{3,0}}{S_{2,0}S_{3,0}})^{-1}$ is a typical time of equilibrium through surface diffusion between interfaces 2 and 3 (mind the missing factor $1/\pi$ in the eq.25 of the original paper [78]).

This typical time sets two different regimes for $A^*$: $\omega\tau \ll 1$ where interfaces 2 and 3 are at equilibrium and their total surface $S_2 + S_3$ is conserved so that only $S_1$ undergoes the compression/stretching cycles; $\omega\tau \gg 1$ the three interfaces withstand the compression/stretching cycles independently from one another. These two limits are plotted in Fig.24(b) as the bottom dashed line ($\omega\tau \ll 1$) and the top dash-dotted line ($\omega\tau \gg 1$). Two fits with the full equation 39 are also plotted, fitting with $\tau = 3 \times 10^{-4}$ s. There is a quantitative match at low frequencies, whereas it starts deviating for frequencies higher than $\omega > 1$ rad/s.

This paper concentrates most of the ideas at the foundation of the main contribution of this thesis, that is presented in chapter 3. As developed at the end of the last subsection 0.4.2, the main problem we face when we build a rheological model for an elementary foam, that is here three interfaces around a meniscus, is solving properly the surfactant exchanges occurring between the interfaces, the film bulks, and the meniscus bulk. Both papers from Durand et al. [1] and Besson et al. [78] make two completely different assumptions regarding the role of the meniscus: the former authors implicitly use it to get rid of an excess of surfactants to enforce their equilibrium condition for the side films, the latter do not write any surfactant transfers towards/from it, artificially blocking bulk surfactant diffusion from this region.
To focus on Besson's paper again, where the description of the surfactant field is more exhaustive, a few points can be raised:

* The model is geometry dependent, and it does not describe the local flow fields in the films and the meniscus, meaning it is hard to predict the robustness of it if R is taken much bigger or much shorter.

* The amplitude of the mechanical solicitations is small, with deformation of the order of $\epsilon \sim 5 \times 10^{-2}$, allowing the authors to neglect the potential exchanges of interface with the two bubble supports. For bigger amplitudes of deformation, which locally exist in a macroscopic foam even for small global deformations (during a T1 event for instance), this assumption cannot be made.

* The ad hoc time $\tau$ hides the variables L, which is unpredicted, making the problem free of the local detail (i.e. the flow field and the surfactant field in the vicinity of the meniscus). However, if L starts being of the order of R for instance, it adds some geometry-dependence to the problem. Besides, L might very well be a function of the pulsation $L(\omega)$ in all generality, making it a possible candidate to explain the high frequency discrepancy observed by the authors.

All these pieces of evidence tend to show there is a need for a local model of the flow and surfactant fields near the meniscus. Notably, the length L added by the authors would require a proper description, which is provided by Bussonnière et al. [4] and detailed a bit later in 0.4.5.

### 0.4.4 Recent numerical advances

Simulating elementary foams is a numerical challenge on its own, as it requires to predict flow and surfactant fields in menisci and foam films, whose geometries evolve in time and which have very different aspect ratios. Titta et al. [3] (2018) have performed an exhaustive simulation of an elementary foam-like configuration illustrated in Fig. 26, undergoing a global shear imposed by setting the relative velocity U of the two supporting plates. The aspect ratio of the simulation is the one shown in the figure.

The bulk flow is described using the momentum conservation:

$$ \nabla \cdot \mathbf{u} = 0 \qquad (40) $$

$$ \rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} \right) = \nabla \cdot \bar{\bar{\sigma}} + \gamma \kappa \delta_{\Gamma} \mathbf{n} + (\nabla_s \gamma) \delta_{\Gamma} \qquad (41) $$

Where $\mathbf{u}$ is the velocity, $\bar{\bar{\sigma}} = -p\bar{\bar{I}} + \eta(\bar{\bar{\nabla}}\mathbf{u})^t \bar{\bar{\nabla}}\mathbf{u})$ is the bulk stress tensor, $\gamma$ is the surface tension, $\kappa$ the local curvature of the interface, $\nabla_s$ the del operator projected on the interface, which has a normal vector $\mathbf{n}$ and is localised by the Dirac function $\delta_{\Gamma}$. The first term on the right-hand side accounts for the bulk stress, the second term for the Laplace pressure jump at the interface and the third one for the Marangoni stress at the interface.

That accounts for the flow field, which is directly dependent of $\gamma$ of which we need a law to close the problem. This is solved by giving a conservation law for surfactants (i.e. for bulk
To focus on Besson's paper again, where the description of the surfactant field is more exhaustive, a few points can be raised:

* The model is geometry dependent, and it does not describe the local flow fields in the films and the meniscus, meaning it is hard to predict the robustness of it if $R$ is taken much bigger or much shorter.

* The amplitude of the mechanical solicitations is small, with deformation of the order of $\epsilon \sim 5 \times 10^{-2}$, allowing the authors to neglect the potential exchanges of interface with the two bubble supports. For bigger amplitudes of deformation, which locally exist in a macroscopic foam even for small global deformations (during a T1 event for instance), this assumption cannot be made.

* The ad hoc time $\tau$ hides the variables $L$, which is unpredicted, making the problem free of the local detail (i.e. the flow field and the surfactant field in the vicinity of the meniscus). However, if $L$ starts being of the order of $R$ for instance, it adds some geometry-dependence to the problem. Besides, $L$ might very well be a function of the pulsation $L(\omega)$ in all generality, making it a possible candidate to explain the high frequency discrepancy observed by the authors.

All these pieces of evidence tend to show there is a need for a local model of the flow and surfactant fields near the meniscus. Notably, the length $L$ added by the authors would require a proper description, which is provided by Bussonnière et al. [4] and detailed a bit later in 0.4.5.

### 0.4.4 Recent numerical advances

Simulating elementary foams is a numerical challenge on its own, as it requires to predict flow and surfactant fields in menisci and foam films, whose geometries evolve in time and which have very different aspect ratios. Titta et al. [3] (2018) have performed an exhaustive simulation of an elementary foam-like configuration illustrated in Fig.26, undergoing a global shear imposed by setting the relative velocity $U$ of the two supporting plates. The aspect ratio of the simulation is the one shown in the figure.

The bulk flow is described using the momentum conservation:

$$ \nabla \cdot \boldsymbol{u} = 0 \qquad (40) $$

$$ \rho \left( \frac{\partial \boldsymbol{u}}{\partial t} + (\boldsymbol{u} \cdot \nabla) \boldsymbol{u} \right) = \nabla \cdot \overline{\overline{\boldsymbol{\sigma}}} + \gamma \kappa \delta_{\Gamma} \boldsymbol{n} + (\nabla_s \gamma) \delta_{\Gamma} \qquad (41) $$

Where $\boldsymbol{u}$ is the velocity, $\overline{\overline{\boldsymbol{\sigma}}} = -p\overline{\overline{\boldsymbol{I}}} + \eta(\overline{\overline{\nabla}}\boldsymbol{u})^t \overline{\overline{\nabla}}\boldsymbol{u}$ is the bulk stress tensor, $\gamma$ is the surface tension, $\kappa$ the local curvature of the interface, $\nabla_s$ the del operator projected on the interface, which has a normal vector $\boldsymbol{n}$ and is localised by the Dirac function $\delta_{\Gamma}$. The first term on the right-hand side accounts for the bulk stress, the second term for the Laplace pressure jump at the interface and the third one for the Marangoni stress at the interface.

That accounts for the flow field, which is directly dependent of $\gamma$ of which we need a law to close the problem. This is solved by giving a conservation law for surfactants (i.e. for bulk
<figure><img src="image_31.png" /><figcaption>Figure 26: Notations and geometry of the elementary foam studied in [3] (faithful aspect ratios). (a) Initial state of the foam at the beginning of the simple shear (of imposed strain rate $\dot{\epsilon} = U/H$). (b) The elementary foam under shear between two T1 events. Credits to Titta et al. [3]</figcaption></figure>

concentration $c$ and surface concentration $\Gamma$), associated with an isotherm model giving the relation $\gamma = f(\Gamma)$. The bulk and surface conservations of surfactants, as well as the Langmuir isotherm chosen for the simulation write:

$$ \frac{\partial c}{\partial t} + \nabla \cdot (\boldsymbol{u}c) = D_c \Delta c \qquad (42) $$

$$ \frac{\partial \Gamma}{\partial t} + \nabla_s \cdot (\boldsymbol{u}\Gamma) = D_{\Gamma} \Delta \Gamma + j \qquad (43) $$

$$ \gamma = \gamma_0 + RT\Gamma_{\infty} \ln \left( 1 - \frac{\Gamma}{\Gamma_{\infty}} \right) \qquad (44) $$

Where $D_c$ and $D_{\Gamma}$ are the bulk and surface diffusion coefficients for the surfactant, $j$ is a surfactant flux from the bulk given by the Langmuir isotherm, $RT$ is the molar thermal energy and $\Gamma_{\infty}$ a saturating surface excess. I will elaborate more on the Langmuir isotherm model later in chapter 1.

These equations are solved given the initial geometry shown in Fig.26(a) and using a level-set function locating the interface over time (adding one more numerical transport law to the physical problem). This gives access to the force exerted by the fluid on the plates (Fig.27(a)) in a transient regime and in steady state. Thus, the authors are allowed to compute the initial elastic loading of the system followed by force oscillations. Past the transient regime, it is possible to compute the dissipated power of both the bulk and of the interface, and to see how the injected power is shared between both contributions (Fig.27(b)).

In Fig.27(a), the black line is the sum of all the forces exerted on the plate as a function of time when the velocity $U$ is turned on (orange the capillary forces, red the viscous forces in the liquid, blue in the air). After a transient regime, the oscillations correspond to a series of T1 events occurring in the system. The dissipated energy during this process is computed and plotted in Fig.27(b). The squares correspond to the injected energy of a cycle, which is totally dissipated over one period of oscillations, either by viscous bulk forces (red circles) or surface dissipation (the difference between the two). The latter contribution is due here to the sole
<figure><img src="image_31.png" /><figcaption>Figure 26: Notations and geometry of the elementary foam studied in [3] (faithful aspect ratios). (a) Initial state of the foam at the beginning of the simple shear (of imposed strain rate $\dot{\epsilon} = U/H$). (b) The elementary foam under shear between two T1 events. Credits to Titta et al. [3]</figcaption></figure>

concentration $c$ and surface concentration $\Gamma$), associated with an isotherm model giving the relation $\gamma = f(\Gamma)$. The bulk and surface conservations of surfactants, as well as the Langmuir isotherm chosen for the simulation write:

$$ \frac{\partial c}{\partial t} + \nabla \cdot (\boldsymbol{u}c) = D_c \Delta c \qquad (42) $$

$$ \frac{\partial \Gamma}{\partial t} + \nabla_s \cdot (\boldsymbol{u}\Gamma) = D_{\Gamma} \Delta \Gamma + j \qquad (43) $$

$$ \gamma = \gamma_0 + RT\Gamma_{\infty} \ln \left( 1 - \frac{\Gamma}{\Gamma_{\infty}} \right) \qquad (44) $$

Where $D_c$ and $D_{\Gamma}$ are the bulk and surface diffusion coefficients for the surfactant, $j$ is a surfactant flux from the bulk given by the Langmuir isotherm, $RT$ is the molar thermal energy and $\Gamma_{\infty}$ a saturating surface excess. I will elaborate more on the Langmuir isotherm model later in chapter 1.

These equations are solved given the initial geometry shown in Fig.26(a) and using a level-set function locating the interface over time (adding one more numerical transport law to the physical problem). This gives access to the force exerted by the fluid on the plates (Fig.27(a)) in a transient regime and in steady state. Thus, the authors are allowed to compute the initial elastic loading of the system followed by force oscillations. Past the transient regime, it is possible to compute the dissipated power of both the bulk and of the interface, and to see how the injected power is shared between both contributions (Fig.27(b)).

In Fig.27(a), the black line is the sum of all the forces exerted on the plate as a function of time when the velocity $U$ is turned on (orange the capillary forces, red the viscous forces in the liquid, blue in the air). After a transient regime, the oscillations correspond to a series of T1 events occurring in the system. The dissipated energy during this process is computed and plotted in Fig.27(b). The squares correspond to the injected energy of a cycle, which is totally dissipated over one period of oscillations, either by viscous bulk forces (red circles) or surface dissipation (the difference between the two). The latter contribution is due here to the sole
<figure><img src="image_32.png" /><figcaption>Figure 27: (a) Forces applied by the foam on the plates, Black is the sum of all the others: orange are the capillary forces, red to viscous forces in the liquid and blue to viscous forces in the gas. (b) Time-averaged injected (black) and dissipated (red) powers over a cycle. Credits to Titta et al. [3]</figcaption></figure>

extrinsic viscosity of the interfaces, as no dilatational or shear viscosities have been added to the surface rheology. The abscissa is the Péclet number of the system, which is defined as:

$$Pe = \frac{UH}{D_c} = \frac{UH}{D_\Gamma} \qquad (45)$$

Whose second equality is a hypothesis of the authors.

What we see is that as the velocity of the mechanical solicitation increases, the contribution of the interface for the dissipation increases until it Plateaus. This means that the ratio $\langle \tilde{D}_s \rangle / \langle \tilde{D}_v \rangle$ peaks at around $Pe \simeq 1$, where the extrinsic viscosity yields its maximal dissipation. This is not intuitive, but already predicted by a model from Lucassen et al. [81].

This is the state of the art in terms of numerics for a local description. The problem of dealing with the real aspect ratio of the films and menisci is still a problem today, but these toy 2D foam models seem to already capture a lot of the mechanical response of the elementary liquid foams. In that respect, numerical models for foams and foam films with more complex structures are fed with unphysical local models [82, 83]. Also, building the bridge towards a full model for macroscopic liquid foam rheology is a whole field of research, on which I will not elaborate. Efforts from our community in this direction have been made in particular by Besson et al. [2] and Denkov et al. [84], as well as in Cohen-Addad's and Höhler's groups [85–87].

### 0.4.5 Rheology of an elementary liquid foam: state of the art in Isabelle Cantat's group at the beginning of my Ph.D

Adrien Bussonnière, Emmanuel Schaub and Isabelle Cantat developed an experimental setup [4] allowing the large deformation of an elementary liquid foam which resembles the Besson-Debrégeas setup [78] but with an invariance by translation symmetry instead of the axisymmetry.
<figure><img src="image_32.png" /><figcaption>Figure 27: (a) Forces applied by the foam on the plates, Black is the sum of all the others: orange are the capillary forces, red to viscous forces in the liquid and blue to viscous forces in the gas. (b) Time-averaged injected (black) and dissipated (red) powers over a cycle. Credits to Titta et al. [3]</figcaption></figure>

extrinsic viscosity of the interfaces, as no dilatational or shear viscosities have been added to the surface rheology. The abscissa is the Péclet number of the system, which is defined as:

$$Pe = \frac{UH}{D_c} = \frac{UH}{D_r} \qquad (45)$$

Whose second equality is a hypothesis of the authors.

What we see is that as the velocity of the mechanical solicitation increases, the contribution of the interface for the dissipation increases until it Plateaus. This means that the ratio $\langle \tilde{D}_s \rangle / \langle \tilde{D}_v \rangle$ peaks at around $Pe \simeq 1$, where the extrinsic viscosity yields its maximal dissipation. This is not intuitive, but already predicted by a model from Lucassen et al. [81].

This is the state of the art in terms of numerics for a local description. The problem of dealing with the real aspect ratio of the films and menisci is still a problem today, but these toy 2D foam models seem to already capture a lot of the mechanical response of the elementary liquid foams. In that respect, numerical models for foams and foam films with more complex structures are fed with unphysical local models [82, 83]. Also, building the bridge towards a full model for macroscopic liquid foam rheology is a whole field of research, on which I will not elaborate. Efforts from our community in this direction have been made in particular by Besson et al. [2] and Denkov et al. [84], as well as in Cohen-Addad's and Höhler's groups [85–87].

### 0.4.5 Rheology of an elementary liquid foam: state of the art in Isabelle Cantat's group at the beginning of my Ph.D

Adrien Bussonnière, Emmanuel Schaub and Isabelle Cantat developed an experimental setup [4] allowing the large deformation of an elementary liquid foam which resembles the Besson-Debrégeas setup [78] but with an invariance by translation symmetry instead of the axisymmetry.
This foam is shown schematically in Fig.28(a). It is 4.2 cm wide in the z direction, and the four edges parallel to z are mobile edges imposing a deformation at the four peripheral films.

<figure><img src="image_33.png" /></figure>

Figure 28: Deformation of an elementary liquid foam with 5 films. (a) The mobile sides are the four external edges of the vertical films, they impose deformations $\pm\epsilon_{mot}$ of opposite signs on the right/left sides. (b-d) Schematic view of the time evolution of the films throughout the deformation, where the coloured dots are elementary material systems at the interface which are followed along their trajectories: compressed films thicken and give up interface to the central film, stretched films thin out and extract interface from the central film. (e-f) Local view of the compression/stretching of the films. (g) Local view of the flow fields responsible for the interface exchanges, that is plug flows far from the menisci and local shears near them. Credits to Adrien Bussonnière et al. [4].

Doing so results in the scenario detailed in Fig.28(b-d): The coloured dots represent elementary material systems at the interface which are followed along their trajectories, thus illustrating film compression/extension as well as interface transfers from one film to the other. Compressed films thicken and relax through giving up some interface, while the stretched films get thinner and accept some new extracted film in their plane.

Under deformation, this system remains invariant by translation along z for a time long enough (this is later discussed in chapter 3) to allow us to define everything in terms of 2D deformations in each films. Thus, each film undergoes a certain deformation $\epsilon$ which is defined using the notations in Fig.28(e-f) as $\epsilon = ds/ds_0 - 1 = h_0/h - 1$ which is uniform in the film because of the uniformity of tension far from a meniscus, as discussed in subsection 0.2.4. These deformations are defined for each film by looking at the pieces of films already present in the plane before the deformation (when their size $ds_0$ is defined). Because of the interface transfers, these deformations are thus distinguishable from the imposed deformations $\epsilon_{mot} = L_m/L_m(t=0) - 1$, which are defined for each motor with the distance $L_m$ between the motor and the rest positions of the neighbouring free meniscus.

On the other hand, it is possible to get a measurement of the difference of tensions $\Delta\sigma$ between the deformed films based on the same principle as in the Besson-Debrégeas experiment [78], that is by looking at the angles around the menisci and their deviations from the Plateau's laws. The
This foam is shown schematically in Fig.28(a). It is 4.2 cm wide in the z direction, and the four edges parallel to z are mobile edges imposing a deformation at the four peripheral films.

<figure><img src="image_33.png" /></figure>

Figure 28: Deformation of an elementary liquid foam with 5 films. (a) The mobile sides are the four external edges of the vertical films, they impose deformations $\pm\epsilon_{mot}$ of opposite signs on the right/left sides. (b-d) Schematic view of the time evolution of the films throughout the deformation, where the coloured dots are elementary material systems at the interface which are followed along their trajectories: compressed films thicken and give up interface to the central film, stretched films thin out and extract interface from the central film. (e-f) Local view of the compression/stretching of the films. (g) Local view of the flow fields responsible for the interface exchanges, that is plug flows far from the menisci and local shears near them. Credits to Adrien Bussonnière et al. [4].

Doing so results in the scenario detailed in Fig.28(b-d): The coloured dots represent elementary material systems at the interface which are followed along their trajectories, thus illustrating film compression/extension as well as interface transfers from one film to the other. Compressed films thicken and relax through giving up some interface, while the stretched films get thinner and accept some new extracted film in their plane.

Under deformation, this system remains invariant by translation along z for a time long enough (this is later discussed in chapter 3) to allow us to define everything in terms of 2D deformations in each films. Thus, each film undergoes a certain deformation $\epsilon$ which is defined using the notations in Fig.28(e-f) as $\epsilon = ds/ds_0 - 1 = h_0/h - 1$ which is uniform in the film because of the uniformity of tension far from a meniscus, as discussed in subsection 0.2.4. These deformations are defined for each film by looking at the pieces of films already present in the plane before the deformation (when their size $ds_0$ is defined). Because of the interface transfers, these deformations are thus distinguishable from the imposed deformations $\epsilon_{mot} = L_m/L_m(t=0) - 1$, which are defined for each motor with the distance $L_m$ between the motor and the rest positions of the neighbouring free meniscus.

On the other hand, it is possible to get a measurement of the difference of tensions $\Delta\sigma$ between the deformed films based on the same principle as in the Besson-Debrégeas experiment [78], that is by looking at the angles around the menisci and their deviations from the Plateau's laws. The
principle of this measurement is more detailed in chapter 3. Thus, by monitoring the difference of tensions $\Delta\sigma$ between the films, the deformations $\epsilon$ of the films, and knowing the imposed deformations $\epsilon_{mot}$ to the system, we are able to draw the graphs of Fig. 29.

In **A**, blue is the global deformation imposed to the whole elementary foam, and we show here a ramp experiment imposed in 0.1 s, thus corresponding to a strain rate of 10 Hz. Red is the response of the system with a difference of tension $\Delta\sigma$ normalised by the equilibrium tension $\sigma_0$ known with our pendant drop campaign. **This response is that of a Maxwell fluid: elastic at shorter times, viscous at longer times.**

In **B**, we still plot the difference of tension $\Delta\sigma$, but this time against the deformation $\epsilon$ measured in the films. The blue curve is obtained during the ramp of deformation, and the purple one is obtained during the relaxation process. The collapse of both curves means that the viscous dissipation is not located in the pieces of film far from the meniscus, which react as non-linear springs. This elastic behaviour is well understood and characterised, and is the topic of chapter 1 where measurements are rationalised with a thermochemical model.

<figure><img src="image_34.png" /><figcaption>Figure 29: Rheological curves of an elementary liquid foam. **A** Blue: ramp of deformation imposed by the motors; Red: response of the system in terms of difference of tension between the stretched and compressed films. It is the response of a Maxwell fluid. **B** Stress-strain curve of the system computing the deformation $\epsilon$ of the films (i.e. far from the meniscus), we have a non-linear elastic response from the spring component of our Maxwell fluid.</figcaption></figure>

The more mysterious part of this dynamics is the relaxation through viscous dissipation. If not for the compressed/stretched films, where is it located? How can we describe it? This is what Bussonnière et al. [4] addressed in a paper published right at the beginning of my thesis work: the viscous dissipation is expected to be located in a very localised zone at the junction with the menisci, in the form of a simple shear flow in the thickness of the films. Its existence is driven by a coupling with a difference of Marangoni stresses between both interfaces, arising as the moving interfaces meet at the meniscus.

These interfaces must be supplied in/get rid of surfactants as they are deformed, and the exchange mechanisms between the interfaces and with the meniscus were left in the model with adjustment parameters. The aim of chapter 3, is about quantifying those exchanges in order to close the model. We will do so by looking at an even more elementary liquid foams: 3 films connected by a single free meniscus.
principle of this measurement is more detailed in chapter 3. Thus, by monitoring the difference of tensions $\Delta\sigma$ between the films, the deformations $\epsilon$ of the films, and knowing the imposed deformations $\epsilon_{mot}$ to the system, we are able to draw the graphs of Fig. 29.

In **A**, blue is the global deformation imposed to the whole elementary foam, and we show here a ramp experiment imposed in 0.1 s, thus corresponding to a strain rate of 10 Hz. Red is the response of the system with a difference of tension $\Delta\sigma$ normalised by the equilibrium tension $\sigma_0$ known with our pendant drop campaign. **This response is that of a Maxwell fluid: elastic at shorter times, viscous at longer times.**

In **B**, we still plot the difference of tension $\Delta\sigma$, but this time against the deformation $\epsilon$ measured in the films. The blue curve is obtained during the ramp of deformation, and the purple one is obtained during the relaxation process. The collapse of both curves means that the viscous dissipation is not located in the pieces of film far from the meniscus, which react as non-linear springs. This elastic behaviour is well understood and characterised, and is the topic of chapter 1 where measurements are rationalised with a thermochemical model.

<figure><img src="image_34.png" /><figcaption>Figure 29: Rheological curves of an elementary liquid foam. **A** Blue: ramp of deformation imposed by the motors; Red: response of the system in terms of difference of tension between the stretched and compressed films. It is the response of a Maxwell fluid. **B** Stress-strain curve of the system computing the deformation $\epsilon$ of the films (i.e. far from the meniscus), we have a non-linear elastic response from the spring component of our Maxwell fluid.</figcaption></figure>

The more mysterious part of this dynamics is the relaxation through viscous dissipation. If not for the compressed/stretched films, where is it located? How can we describe it? This is what Bussonnière et al. [4] addressed in a paper published right at the beginning of my thesis work: the viscous dissipation is expected to be located in a very localised zone at the junction with the menisci, in the form of a simple shear flow in the thickness of the films. Its existence is driven by a coupling with a difference of Marangoni stresses between both interfaces, arising as the moving interfaces meet at the meniscus.

These interfaces must be supplied in/get rid of surfactants as they are deformed, and the exchange mechanisms between the interfaces and with the meniscus were left in the model with adjustment parameters. The aim of chapter 3, is about quantifying those exchanges in order to close the model. We will do so by looking at an even more elementary liquid foams: 3 films connected by a single free meniscus.
In the three articles we discussed [1, 4, 78], the amount of surfactant flowing from one film to its neighbour is a crucial parameter, unknown and required to close the model. In the three cases, it is used as an adjustable parameter.

The prediction of this surfactant transfer is made difficult because the role of the meniscus in the process is an open question : does it play, or not, a role of surfactant reservoir for the adjacent films ?

**The main result of chapter 3 will be a direct measure of this transfer, leading to the conclusion that the meniscus does not provide or absorb surfactant during the process.**

## 0.5 Spontaneous evolution of foams and films : some other physical processes

### 0.5.1 Normal motion of the films

When it comes to compare the physics this manuscript is about to macroscopic foam and single bubble physics, it is worth keeping in mind that a first line we draw here is that we do not address the problem of normal motions of foam films. This is because the shape of our elementary foam is always very close to its equilibrium shape. The normal motions are much better known than the tangential ones addressed in this manuscript, but important questions are still investigated.

For instance, these motions are especially important in liquid foams to explain their acoustics [88], as the speed of sound can be resolved only taking into account the normal oscillations to the films. They also are important in the case of film burst, where the film actually retracts with a violent normal flapping motion [89] ! This last peculiar dynamics could even be relevant for the matter of salt microparticles projection from the sea to the atmosphere, which is important for cloud formation.

Back to rheology, unless the strain rate becomes too high [90], the structure stays close to its equilibrium shape most of the time, as in our setup. Just after T1 however, the shape is far from equilibrium and fast normal motions are observed, which can not be reproduced with our set-up.

### 0.5.2 Gravity drainage

Both elementary and disordered macroscopic liquid foams undergo drainage at longer times. That includes gravity drainage, which makes the liquid fraction increase at the bottom of the foam (or equivalently makes the bottom films thicker in elementary foams), as well as capillary drainage (cf. 0.3.3) responsible for the thinning of foam films near their edges.

Gravity drainage leads to a very heterogeneous ageing, as shown in Fig. 30, with thinner films, bigger bubbles on top and thicker films, smaller bubbles at the bottom. This can make the global mechanical properties difficult to predict, and the rheology hard to characterise as it becomes time-dependent. In the frame of this manuscript, with the elementary liquid foams we study here, gravity can also become a problem as it leads to thicker films at the bottom compared to the top ones and induces flows in the meniscus which are not horizontal. However, the time scales
In the three articles we discussed [1, 4, 78], the amount of surfactant flowing from one film to its neighbour is a crucial parameter, unknown and required to close the model. In the three cases, it is used as an adjustable parameter.

The prediction of this surfactant transfer is made difficult because the role of the meniscus in the process is an open question : does it play, or not, a role of surfactant reservoir for the adjacent films ?

**The main result of chapter 3 will be a direct measure of this transfer, leading to the conclusion that the meniscus does not provide or absorb surfactant during the process.**

## 0.5 Spontaneous evolution of foams and films : some other physical processes

### 0.5.1 Normal motion of the films

When it comes to compare the physics this manuscript is about to macroscopic foam and single bubble physics, it is worth keeping in mind that a first line we draw here is that we do not address the problem of normal motions of foam films. This is because the shape of our elementary foam is always very close to its equilibrium shape. The normal motions are much better known than the tangential ones addressed in this manuscript, but important questions are still investigated.

For instance, these motions are especially important in liquid foams to explain their acoustics [88], as the speed of sound can be resolved only taking into account the normal oscillations to the films. They also are important in the case of film burst, where the film actually retracts with a violent normal flapping motion [89] ! This last peculiar dynamics could even be relevant for the matter of salt microparticles projection from the sea to the atmosphere, which is important for cloud formation.

Back to rheology, unless the strain rate becomes too high [90], the structure stays close to its equilibrium shape most of the time, as in our setup. Just after T1 however, the shape is far from equilibrium and fast normal motions are observed, which can not be reproduced with our set-up.

### 0.5.2 Gravity drainage

Both elementary and disordered macroscopic liquid foams undergo drainage at longer times. That includes gravity drainage, which makes the liquid fraction increase at the bottom of the foam (or equivalently makes the bottom films thicker in elementary foams), as well as capillary drainage (cf. 0.3.3) responsible for the thinning of foam films near their edges.

Gravity drainage leads to a very heterogeneous ageing, as shown in Fig. 30, with thinner films, bigger bubbles on top and thicker films, smaller bubbles at the bottom. This can make the global mechanical properties difficult to predict, and the rheology hard to characterise as it becomes time-dependent. In the frame of this manuscript, with the elementary liquid foams we study here, gravity can also become a problem as it leads to thicker films at the bottom compared to the top ones and induces flows in the meniscus which are not horizontal. However, the time scales
we address are shorter than the typical time scale of the gravitational drainage, thus allowing us to neglect its influence.

<figure><img src="image_35.png" /><figcaption>Figure 30: A liquid foam ageing through gravity drainage. Top is dry, bottom is wet. Credits to S. Cohen-Addad, R.M. Guillermic & A. Saint-Jalmes [7].</figcaption></figure>

### 0.5.3 Coarsening of liquid foams

In the frame of foam physics, coarsening is how the foam tends to have a decreasing number of bubbles over time, and an increasing mean size for them. This is due to two main mechanisms: film rupture, leading to the destruction of two bubbles and the creation of a bigger one, and Ostwald ripening, when small bubbles with bigger pressures empty themselves in larger ones through gas diffusion across foam films. Our elementary foam has bubbles opened to the atmosphere, and we consider only cases without ruptures. So these ageing processes do not occur in our situation.

Film ruptures occur mainly due to the drainage mechanisms discussed before. These events lead to bubble rearrangements as the menisci supporting broken films move to a new equilibrium position, and often cause T1 events in their vicinity. Thus, even without exterior solicitations, the study of elastic and dissipative phenomena at stake in the deformation of elementary foams is still relevant.

The same goes for Ostwald ripening, as a change of bubble size can trigger a T1 event. Moreover, a recent study from A.Saint-Jalmes & C.Trégouët [91] has shown that a steady shear slows down the ageing process. The reason behind it was found to be the constant extraction of Frankel films (cf. subsection 0.3.2) out of the menisci, counterbalancing drainage by renewing foam films with thick pieces. Thicker films means slower gas diffusion, and thus slower Ostwald ripening. Here again, the evolution of film thickness induced by bubble deformation, which is at the core of our study, plays an important role.
we address are shorter than the typical time scale of the gravitational drainage, thus allowing us
to neglect its influence.

<figure><img src="image_35.png" /><figcaption>Figure 30: A liquid foam ageing through gravity drainage. Top is dry, bottom is wet. Credits to S. Cohen-Addad, R.M. Guillermic & A. Saint-Jalmes [7].</figcaption></figure>

### 0.5.3 Coarsening of liquid foams

In the frame of foam physics, coarsening is how the foam tends to have a decreasing number of bubbles over time, and an increasing mean size for them. This is due to two main mechanisms: film rupture, leading to the destruction of two bubbles and the creation of a bigger one, and Ostwald ripening, when small bubbles with bigger pressures empty themselves in larger ones through gas diffusion across foam films. Our elementary foam has bubbles opened to the atmosphere, and we consider only cases without ruptures. So these ageing processes do not occur in our situation.

Film ruptures occur mainly due to the drainage mechanisms discussed before. These events lead to bubble rearrangements as the menisci supporting broken films move to a new equilibrium position, and often cause T1 events in their vicinity. Thus, even without exterior solicitations, the study of elastic and dissipative phenomena at stake in the deformation of elementary foams is still relevant.

The same goes for Ostwald ripening, as a change of bubble size can trigger a T1 event. Moreover, a recent study from A.Saint-Jalmes & C.Trégouët [91] has shown that a steady shear slows down the ageing process. The reason behind it was found to be the constant extraction of Frankel films (cf. subsection 0.3.2) out of the menisci, counterbalancing drainage by renewing foam films with thick pieces. Thicker films means slower gas diffusion, and thus slower Ostwald ripening. Here again, the evolution of film thickness induced by bubble deformation, which is at the core of our study, plays an important role.
### 0.5.4 Link to liquid foams in the industry and daily life

This thesis work is about a very simple chemistry in the eye of an engineer. Yet, a mixture of two species, SDS and DOH (or even three as fluorescein counts as salt cf. chapter 1) is already a complex problem for hydrodynamicists. In fact, SDS as a soluble surfactant allows the existence of our foam, while DOH as a poorly soluble one changes drastically the elasticity of the films. The boundary conditions for our flow describing this chemical reality are full of subtleties already.

Then, what to think of scaling up this work not only for macroscopic foams, but rather actual foams used in the industry or encountered in daily life? This is where we must be even more cautious in our approach.

#### Scaling up our elementary foams: Part of the physics is captured

The use of the physics behind liquid foams in the industry is kept at the scale of classical rheology and correlations between the surface properties (Marangoni modulus, surface shear viscosity...) and the effective properties of the macroscopic materials. From there, a wide variety of behaviours exists and the scale of the elementary foam is relevant for part of it.

As discussed earlier, viscoelastic response of dry foams can be already recovered in elementary foams, but it has to be noted that real foams also show many other properties not captured/not rationalised at our local scale: plasticity due to film rupture under shear or repositioning at the edges of the foam, thixotropy as the foam ages... Most of these phenomena depend either on bubble size, liquid fraction, or ageing time. Our elementary foams model huge centimetric bubbles, so that all the typical length scales are decoupled. This serves the physics best, but also mean that we will not be able to probe directly all these coupled dynamics depending on the bubble size or the liquid fraction.

#### More complex chemistries: even more physical ingredients

The simplest surfactants used broadly in industries such as cosmetics, detergent or food industries are quite close to SDS and are grouped under the name "sulfates". They are produced using byproducts of the oil industry, hence their abundance, and are mixes of molecules with a skeleton ranging from 8 to 16 carbon atoms with a sulfate group located on a random carbon. SDS (of which the structure is recalled in Fig. 4) is actually a model surfactant for these systems.

Sulfates are used for daily life applications as a foaming agent (dishwashing liquid, shampoo...), as well as in many industries. Half the global production of surfactants (including sulfates) goes to the mining industry for what is called froth flotation, in which valuable minerals are separated from the rest by being put into contact with foams. This separating process relies on the difference of wetting properties of the materials, and foam are used to exacerbate this differentiation.

For other industries, such as the dairy industry for instance, uncontrolled foaming is a huge problem. Due to the potentially very high air fraction of these materials (up to 99%), piping foamy liquids can represent a loss of energy and storage volume.

In all these cases, the effect of chemistry is already well rationalised, and these industries have many foaming/anti-foaming agents for a wide range of situations. However, some of these
### 0.5.4 Link to liquid foams in the industry and daily life

This thesis work is about a very simple chemistry in the eye of an engineer. Yet, a mixture of two species, SDS and DOH (or even three as fluorescein counts as salt cf. chapter 1) is already a complex problem for hydrodynamicists. In fact, SDS as a soluble surfactant allows the existence of our foam, while DOH as a poorly soluble one changes drastically the elasticity of the films. The boundary conditions for our flow describing this chemical reality are full of subtleties already.

Then, what to think of scaling up this work not only for macroscopic foams, but rather actual foams used in the industry or encountered in daily life? This is where we must be even more cautious in our approach.

#### Scaling up our elementary foams: Part of the physics is captured

The use of the physics behind liquid foams in the industry is kept at the scale of classical rheology and correlations between the surface properties (Marangoni modulus, surface shear viscosity...) and the effective properties of the macroscopic materials. From there, a wide variety of behaviours exists and the scale of the elementary foam is relevant for part of it.

As discussed earlier, viscoelastic response of dry foams can be already recovered in elementary foams, but it has to be noted that real foams also show many other properties not captured/not rationalised at our local scale: plasticity due to film rupture under shear or repositioning at the edges of the foam, thixotropy as the foam ages... Most of these phenomena depend either on bubble size, liquid fraction, or ageing time. Our elementary foams model huge centimetric bubbles, so that all the typical length scales are decoupled. This serves the physics best, but also mean that we will not be able to probe directly all these coupled dynamics depending on the bubble size or the liquid fraction.

#### More complex chemistries: even more physical ingredients

The simplest surfactants used broadly in industries such as cosmetics, detergent or food industries are quite close to SDS and are grouped under the name "sulfates". They are produced using byproducts of the oil industry, hence their abundance, and are mixes of molecules with a skeleton ranging from 8 to 16 carbon atoms with a sulfate group located on a random carbon. SDS (of which the structure is recalled in Fig. 4) is actually a model surfactant for these systems.

Sulfates are used for daily life applications as a foaming agent (dishwashing liquid, shampoo...), as well as in many industries. Half the global production of surfactants (including sulfates) goes to the mining industry for what is called froth flotation, in which valuable minerals are separated from the rest by being put into contact with foams. This separating process relies on the difference of wetting properties of the materials, and foam are used to exacerbate this differentiation.

For other industries, such as the dairy industry for instance, uncontrolled foaming is a huge problem. Due to the potentially very high air fraction of these materials (up to 99%), piping foamy liquids can represent a loss of energy and storage volume.

In all these cases, the effect of chemistry is already well rationalised, and these industries have many foaming/anti-foaming agents for a wide range of situations. However, some of these
chemical species endow liquid foams with other physical ingredients which are still to be better described by physicists!

For example, foam films can become very stiff when made out of protein solutions or gels, as the molecules can go at the interface where they react and form 2D networks. This has been observed with bovine serum albumine (BSA) for instance [92]. This leads to enhanced stability, possibly greater film elasticity... and even the existence of surface shear elasticity [26]!

Another remarkable fact when using such larger molecules is their much bigger diffusion times. This is expected to play a major role at the local scale of elementary foams already, as the adsorption/desorption dynamics sets the shear flow described in chapter 3, thus controlling the mechanical dissipation of the foam.

Knowing all these open questions surrounding surface rheology, elementary and macroscopic foams, how they relate, how different they are, let us now take a step back. Back to our local scale. Back to a simpler chemistry.
chemical species endow liquid foams with other physical ingredients which are still to be better described by physicists!

For example, foam films can become very stiff when made out of protein solutions or gels, as the molecules can go at the interface where they react and form 2D networks. This has been observed with bovine serum albumine (BSA) for instance [92]. This leads to enhanced stability, possibly greater film elasticity... and even the existence of surface shear elasticity [26]!

Another remarkable fact when using such larger molecules is their much bigger diffusion times. This is expected to play a major role at the local scale of elementary foams already, as the adsorption/desorption dynamics sets the shear flow described in chapter 3, thus controlling the mechanical dissipation of the foam.

Knowing all these open questions surrounding surface rheology, elementary and macroscopic foams, how they relate, how different they are, let us now take a step back. Back to our local scale. Back to a simpler chemistry.
# Plan

What follows is the main body of this manuscript, my thesis work.

Chapter 1 is about the elastic response of an elementary liquid foam and how to predict it with a thermochemical model I developed and solved numerically.

Chapter 2 is about a novel measurement of a line tension caused by the presence of thickness heterogeneities in a foam film. I present there the experiments I supervised and the theoretical model I used to describe the in-plane motions in the film in order to validate our force measurement.

Chapter 3 is the core experimental work of my thesis. I first elaborate on the state of the art by presenting the model describing the flow fields in an elementary liquid foam. This model raises a question: does a patch of interface exchange any surfactants with the meniscus when it goes from one film to another ? I answered this question experimentally and present the results in this chapter.
Plan

What follows is the main body of this manuscript, my thesis work.

Chapter 1 is about the elastic response of an elementary liquid foam and how to predict it with a thermochemical model I developed and solved numerically.

Chapter 2 is about a novel measurement of a line tension caused by the presence of thickness heterogeneities in a foam film. I present there the experiments I supervised and the theoretical model I used to describe the in-plane motions in the film in order to validate our force measurement.

Chapter 3 is the core experimental work of my thesis. I first elaborate on the state of the art by presenting the model describing the flow fields in an elementary liquid foam. This model raises a question: does a patch of interface exchange any surfactants with the meniscus when it goes from one film to another ? I answered this question experimentally and present the results in this chapter.
# 1

Elasticity of a single foam film and how to
measure it

In subsection 0.2.3, we introduce the notion of film elasticity, which is directly linked to the Marangoni modulus of the interfaces sandwiching the bulk. This elastic behaviour arises from a finite size reservoir-effect of the bulk in the peculiar geometry of a foam film.

In this chapter I show the thermochemical model I developed to predict the elasticity of foam
films, made of a mixture of SDS and dodecanol, which I will use in the whole manuscript. The
originality of this work is to address the non linear regime, due to geometrical and physico-
chemical effects. My theoretical results are confronted to the experimental data obtained by
Raphaël Poryles and Adrien Bussonnière at the IPR before my arrival. These results are published
in *Soft Matter* [6].
# 1

Elasticity of a single foam film and how to
measure it

In subsection 0.2.3, we introduce the notion of film elasticity, which is directly linked to the Marangoni modulus of the interfaces sandwiching the bulk. This elastic behaviour arises from a finite size reservoir-effect of the bulk in the peculiar geometry of a foam film.

In this chapter I show the thermochemical model I developed to predict the elasticity of foam
films, made of a mixture of SDS and dodecanol, which I will use in the whole manuscript. The
originality of this work is to address the non linear regime, due to geometrical and physico-
chemical effects. My theoretical results are confronted to the experimental data obtained by
Raphaël Poryles and Adrien Bussonnière at the IPR before my arrival. These results are published
in *Soft Matter* [6].
## 1.1 Surface and film elasticity measurements in the literature

### 1.1.1 SDS/DOH mixtures: a few more comments

The classical methods for measuring surface tension presented in subsection 0.1.2 are key characterisation methods both in the academic and industrial worlds, and have yielded a vast literature for SDS/DOH alone. We made our own measurements for the equilibrium surface tensions of our SDS/DOH mixtures, which we presented in Fig.9. The main interest of this surfactant combination is that it allows to tune quite finely the surface rheology of the foam films with tiny amounts of DOH, hence allowing us to explore a lot of different regimes.

Interestingly, SDS is a common surfactant, making it a popular troublemaker! It is well known to host DOH traces, as it is its hydrolysis product, and to be difficult to purify through recrystallisation. Both the amount of these traces in commercial SDS and the kinetics of the hydrolysis reaction have been discussed, from the 80's until quite recently [93, 94]. Besides, as DOH tends to hide in SDS micelles, new parameters appear when trying to solve the thermochemical problem of quantifying population of DOH at the interface. This will be taken into account in our thermochemical model for the mechanical response of foam films made with SDS/DOH mixtures, and will even allow us to estimate the amount of impurities.

Also, the critical micellar concentration of SDS is a sensible parameter, as it varies with temperature, pH, the presence of other ionic species, varying from 5 – 10 mmol/L [5, 41, 95–97] with a convergence towards the value of CMC = 8.1 mmol/L in normal conditions of temperature and pH [98]. This value is the one we retain in the following.

### 1.1.2 Film elasticity and model in the submicellar regime: the Prins experiment

As an introduction to this chapter, we present a historical experiment and model developed by Prins et al. [5] in 1967 where the elastic response of a foam film is probed with an ingenious system. The main problem when it comes to deform foam films are the tiny forces at play, making the system easy to break under disproportionate solicitations. In fact a first order of magnitude for a deformation $ϵ = 1$ (i.e. an extension of 100%) is $F ∼ E_f W ∼ 10^{-5}$ N for $E_f ∼ 1$ mN/m and a typical width of film $W ∼ 1$ cm.

Thus, the *tour de force* introduced by Prins et al. which has become a general idea in this context, is to counteract these tiny tension forces with... the tension of another foam film. The authors have designed the experimental setup illustrated in Fig.1.1 (Left) where they create a vertical film out of a bath of soapy solution. They monitor the vertical thickness profile of the film using monochromatic light interferences, with a sodium lamp seen in reflection on the film. Once the initial film is formed, they stretch it vertically a second time, drawing out a Frankel film in the plane of the preexisting film. The thickness of this new film is also measured as well as the evolution of the profile of the preexisting one. The former allows them to deduce the weight imposed at the bottom of the preexisting film, that is the driving force of the deformation, and
## 1.1 Surface and film elasticity measurements in the literature

### 1.1.1 SDS/DOH mixtures: a few more comments

The classical methods for measuring surface tension presented in subsection 0.1.2 are key characterisation methods both in the academic and industrial worlds, and have yielded a vast literature for SDS/DOH alone. We made our own measurements for the equilibrium surface tensions of our SDS/DOH mixtures, which we presented in Fig.9. The main interest of this surfactant combination is that it allows to tune quite finely the surface rheology of the foam films with tiny amounts of DOH, hence allowing us to explore a lot of different regimes.

Interestingly, SDS is a common surfactant, making it a popular troublemaker! It is well known to host DOH traces, as it is its hydrolysis product, and to be difficult to purify through recrystallisation. Both the amount of these traces in commercial SDS and the kinetics of the hydrolysis reaction have been discussed, from the 80's until quite recently [93, 94]. Besides, as DOH tends to hide in SDS micelles, new parameters appear when trying to solve the thermochemical problem of quantifying population of DOH at the interface. This will be taken into account in our thermochemical model for the mechanical response of foam films made with SDS/DOH mixtures, and will even allow us to estimate the amount of impurities.

Also, the critical micellar concentration of SDS is a sensible parameter, as it varies with temperature, pH, the presence of other ionic species, varying from 5 – 10 mmol/L [5, 41, 95–97] with a convergence towards the value of CMC = 8.1 mmol/L in normal conditions of temperature and pH [98]. This value is the one we retain in the following.

### 1.1.2 Film elasticity and model in the submicellar regime: the Prins experiment

As an introduction to this chapter, we present a historical experiment and model developed by Prins et al. [5] in 1967 where the elastic response of a foam film is probed with an ingenious system. The main problem when it comes to deform foam films are the tiny forces at play, making the system easy to break under disproportionate solicitations. In fact a first order of magnitude for a deformation $ϵ = 1$ (i.e. an extension of 100%) is $F ∼ E_f W ∼ 10^{-5}$ N for $E_f ∼ 1$ mN/m and a typical width of film $W ∼ 1$ cm.

Thus, the *tour de force* introduced by Prins et al. which has become a general idea in this context, is to counteract these tiny tension forces with... the tension of another foam film. The authors have designed the experimental setup illustrated in Fig.1.1 (Left) where they create a vertical film out of a bath of soapy solution. They monitor the vertical thickness profile of the film using monochromatic light interferences, with a sodium lamp seen in reflection on the film. Once the initial film is formed, they stretch it vertically a second time, drawing out a Frankel film in the plane of the preexisting film. The thickness of this new film is also measured as well as the evolution of the profile of the preexisting one. The former allows them to deduce the weight imposed at the bottom of the preexisting film, that is the driving force of the deformation, and
<figure><img src="image_36.png" /><figcaption>TABLE I
ELASTICITY OF FILMS DRAWN FROM SOLUTIONS OF SODIUM DODECYL SULFATE (e.m.c. 5 × 10⁻⁶ moles.em⁻³)</figcaption></figure>

<table><thead><tr><th rowspan="2">Thickness (micron units)</th><th colspan="2">Concentration (moles.cm<sup>-3</sup>)</th></tr><tr><th>4.10<sup>-6</sup></th><th>15.10<sup>-6</sup></th></tr></thead><tbody><tr><td>0.452</td><td></td><td>2.3 ± 0.4</td></tr><tr><td>0.678</td><td>58</td><td>1.7 ± 0.2</td></tr><tr><td>0.904</td><td>47 ± 3</td><td>1.3 ± 0.10</td></tr><tr><td>1.130</td><td>38 ± 1</td><td>1.18 ± 0.06</td></tr><tr><td>1.356</td><td>36 ± 3</td><td>1.07 ± 0.05</td></tr><tr><td>1.582</td><td>33 ± 2</td><td>1.00 ± 0.04</td></tr><tr><td>1.808</td><td>31 ± 2</td><td>0.94 ± 0.07</td></tr><tr><td>2.034</td><td>28 ± 1</td><td>0.91 ± 0.04</td></tr><tr><td>2.260</td><td>27 ± 1</td><td>0.89 ± 0.07</td></tr></tbody></table>

Figure 1.1: **Left:** Schematic view of the experimental setup. **Right:** Computed film elasticities for SDS solutions in sub-micellar (middle column) and micellar (right column) regimes.

the latter allows them to compute the deformation of the film, as it gets thinner to compensate for the extension of its area.

Knowing the factor $\epsilon$ of deformation, so that the area of the preexisting film is $A = A_0(1 + \epsilon)$ and its thickness profile $h = h_0/(1 + \epsilon)$, where the 0 indices account for the initial values, the authors define the film elasticity:

$$E_f = \frac{d\sigma^f}{d \ln A} \qquad (1.1)$$

This definition coincides with the definition we will use later on in this chapter. Also we must notice that for a vertical film, the variation of film tension $d\sigma^f$ must balance at each altitude the weight of the pieces of film below it. This last point means that careful integration of the thickness profile must be performed, as well as a lot of experimental precautions. For instance the necessary presence of a balancing twin frame, with its own foam film, that plunges into the bath as the main frame is drawn out to keep constant the total area of interface present at the surface of the bath.

Also, because the initial film is let to drain for a short while, its thickness distribution is at mechanical equilibrium, meaning that the thinner parts are on top of the thicker ones. By monitoring the different $\epsilon(z)$ with the interferences, and integrating the thickness profiles to yield the $d\sigma^f(z)$ it is possible to measure different elasticities in the same film for different thicknesses, as we expect $E_f = f(h)$ to be a function of the thickness of the film.

This allows the authors to compute $E_f$ for a wide range of thicknesses and chemistries. Regarding SDS/DOH mixtures, the authors first report the Table I of Fig.1.1 (Right) for SDS “alone” at two concentrations corresponding respectively to 0.62 CMC and 1.85 CMC of SDS. For the same thicknesses (the values of the measurable thicknesses are fixed by the wavelength used) a factor 20 is observed between the submicellar and micellar regimes! For a given concentration of SDS, the elasticity of the film decreases with the thickness, with a factor 1/2 for a film 5 times thicker.
<figure><img src="image_36.png" /><figcaption>TABLE I
ELASTICITY OF FILMS DRAWN FROM SOLUTIONS OF SODIUM DODECYL SULFATE (e.m.c. 5 × 10⁻⁶ moles.em⁻³)</figcaption></figure>

<table><thead><tr><th rowspan="2">Thickness (micron units)</th><th colspan="2">Concentration (moles.cm<sup>-3</sup>)</th></tr><tr><th>4.10<sup>-6</sup></th><th>15.10<sup>-6</sup></th></tr></thead><tbody><tr><td>0.452</td><td></td><td>2.3 ± 0.4</td></tr><tr><td>0.678</td><td>58</td><td>1.7 ± 0.2</td></tr><tr><td>0.904</td><td>47 ± 3</td><td>1.3 ± 0.10</td></tr><tr><td>1.130</td><td>38 ± 1</td><td>1.18 ± 0.06</td></tr><tr><td>1.356</td><td>36 ± 3</td><td>1.07 ± 0.05</td></tr><tr><td>1.582</td><td>33 ± 2</td><td>1.00 ± 0.04</td></tr><tr><td>1.808</td><td>31 ± 2</td><td>0.94 ± 0.07</td></tr><tr><td>2.034</td><td>28 ± 1</td><td>0.91 ± 0.04</td></tr><tr><td>2.260</td><td>27 ± 1</td><td>0.89 ± 0.07</td></tr></tbody></table>

Figure 1.1: **Left:** Schematic view of the experimental setup. **Right:** Computed film elasticities for SDS solutions in sub-micellar (middle column) and micellar (right column) regimes.

the latter allows them to compute the deformation of the film, as it gets thinner to compensate for the extension of its area.

Knowing the factor $\epsilon$ of deformation, so that the area of the preexisting film is $A = A_0(1 + \epsilon)$ and its thickness profile $h = h_0/(1 + \epsilon)$, where the 0 indices account for the initial values, the authors define the film elasticity:

$$E_f = \frac{d\sigma^f}{d \ln A} \qquad (1.1)$$

This definition coincides with the definition we will use later on in this chapter. Also we must notice that for a vertical film, the variation of film tension $d\sigma^f$ must balance at each altitude the weight of the pieces of film below it. This last point means that careful integration of the thickness profile must be performed, as well as a lot of experimental precautions. For instance the necessary presence of a balancing twin frame, with its own foam film, that plunges into the bath as the main frame is drawn out to keep constant the total area of interface present at the surface of the bath.

Also, because the initial film is let to drain for a short while, its thickness distribution is at mechanical equilibrium, meaning that the thinner parts are on top of the thicker ones. By monitoring the different $\epsilon(z)$ with the interferences, and integrating the thickness profiles to yield the $d\sigma^f(z)$ it is possible to measure different elasticities in the same film for different thicknesses, as we expect $E_f = f(h)$ to be a function of the thickness of the film.

This allows the authors to compute $E_f$ for a wide range of thicknesses and chemistries. Regarding SDS/DOH mixtures, the authors first report the Table I of Fig.1.1 (Right) for SDS “alone” at two concentrations corresponding respectively to 0.62 CMC and 1.85 CMC of SDS. For the same thicknesses (the values of the measurable thicknesses are fixed by the wavelength used) a factor 20 is observed between the submicellar and micellar regimes! For a given concentration of SDS, the elasticity of the film decreases with the thickness, with a factor 1/2 for a film 5 times thicker.
<figure><img src="image_37.png" /><figcaption>Figure 1.2: Film elasticity as a function of film thickness in the sub-micellar regime. The solid line is a fit by adding a concentration of dodecanol supposedly taking into account the impurities of SDS.</figcaption></figure>

The authors also describe a model for submicellar mixtures giving access to a full expression for the film elasticity. It is based on a Langmuir isotherm model of adsorption/desorption (which is presented in details in section 1.2) combined to mass conservation which allows them to get an analytical formula for $E_f$. This expression depends on the thickness $h_0$ of the film and the adsorption/desorption coefficients for SDS/DOH, which are known thanks to prior Langmuir trough experiments. This model fits the data of Fig.1.2, that is the submicellar column of Table I of Fig.1.1... if a virtual small quantity of dodecanol is added besides SDS, which is here to take into account the impurities present in commercial SDS.

However, this model is not adaptable to the micellar regime in an analytic form, as it would mean taking into account another phase, the micellar phase, which hosts and exchanges both SDS/DOH with the bulk monomeric phase. This is where the numerics of nowadays can help, and we will see in section 1.2 how we implement this to our own experiments to predict the strain-stress curves, both below and above the CMC, and for large deformations in the non-linear regime.

## 1.2 Thermochemical models for foam film elasticities

Prins et al. developed an analytical model for submicellar solutions (with a single surfactant or a mixture), which we will extend to the micellar regime with numerical tools.

The assumptions of our model are (most of them being those of Prins et al. [5]):

* Surface tension is uniform in the foam film, and both its interfaces have the same value (cf. subsection 0.2.4).

* Chemical equilibrium is reached instantaneously whenever a deformation is applied (cf. subsection 0.2.2).
<figure><img src="image_37.png" /><figcaption>Figure 1.2: Film elasticity as a function of film thickness in the sub-micellar regime. The solid line is a fit by adding a concentration of dodecanol supposedly taking into account the impurities of SDS.</figcaption></figure>

The authors also describe a model for submicellar mixtures giving access to a full expression for the film elasticity. It is based on a Langmuir isotherm model of adsorption/desorption (which is presented in details in section 1.2) combined to mass conservation which allows them to get an analytical formula for $E_f$. This expression depends on the thickness $h_0$ of the film and the adsorption/desorption coefficients for SDS/DOH, which are known thanks to prior Langmuir trough experiments. This model fits the data of Fig.1.2, that is the submicellar column of Table I of Fig.1.1... if a virtual small quantity of dodecanol is added besides SDS, which is here to take into account the impurities present in commercial SDS.

However, this model is not adaptable to the micellar regime in an analytic form, as it would mean taking into account another phase, the micellar phase, which hosts and exchanges both SDS/DOH with the bulk monomeric phase. This is where the numerics of nowadays can help, and we will see in section 1.2 how we implement this to our own experiments to predict the strain-stress curves, both below and above the CMC, and for large deformations in the non-linear regime.

## 1.2 Thermochemical models for foam film elasticities

Prins et al. developed an analytical model for submicellar solutions (with a single surfactant or a mixture), which we will extend to the micellar regime with numerical tools.

The assumptions of our model are (most of them being those of Prins et al. [5]):

* Surface tension is uniform in the foam film, and both its interfaces have the same value (cf. subsection 0.2.4).

* Chemical equilibrium is reached instantaneously whenever a deformation is applied (cf. subsection 0.2.2).
* The piece of film we consider is a closed system (cf. subsection 0.2.2): volume is conserved and the total quantities of surfactants (SDS and DOH) contained in the interfaces and the bulk are conserved.

* The bulk monomeric concentration of SDS is capped $c_s^m = CMC$ whenever a micellar phase exists.

* Dodecanol population in the bulk is shared between a monomeric population, and a micellar concentration proportional to the concentration of micellar SDS.

* The chemical equilibrium between the bulk and the interfaces is modelled with Langmuir isotherms and concerns the bulk monomeric concentrations of surfactants only.

* SDS is an anionic surfactant, each molecule comes with a sodium counterion $Na^+$ whose populations will likewise be shared between the bulk, the micelles and the interface.

This is the overview of our model, which will be solved numerically in the end. Let us now write down the proper system of equations.

### 1.2.1 Our model for SDS/DOH mixtures

#### Adsorption laws

We start by writing the population shares of SDS and DOH in the bulk. The former has a surface excess $\Gamma_s$ and a total bulk concentration $c_s = c_s^m + c_s^M$ shared between monomers $c_s^m$ and micelles $c_s^M$ so that:

$$c_s^m = c_s \quad \text{if} \quad c_s < CMC \quad \text{and} \quad c_s^m = CMC \quad \text{otherwise.} \qquad (1.2)$$

The total DOH population has a surface excess $\Gamma_d$ and a bulk population $c_d = c_d^m + c_d^M$ that is shared between a monomeric population $c_d^m$ and a micellar population $c_d^M$. As micelles are composed of SDS for an overwhelming majority, we consider that the micellar population of DOH is simply given by a law of mass action $c_d^M = K c_d^m c_s^M$ writing:

$$c_d = c_d^m (1 + K c_s^M) \qquad (1.3)$$

Also, the sodium counterion population must be taken into account, as bringing a molecule of SDS at the interface means bringing a counterion along with it to ensure the electroneutrality of the interface. This comes with consequences for the statistical presence of SDS at the interface. Thus, let us denote $\Gamma_n = \Gamma_s$ the surface excess of $Na^+$, $c_n^m = c_s^m$ its bulk concentration for the monomeric SDS phase, and $c_n^M = c_s^M$ its bulk concentration associated with the SDS micellar phase.

The chemical equilibria between the interface, the bulk monomeric phase and the bulk micellar phase are written using a Langmuir isotherm model. This model considers: (i) that the interface is a set of fixation sites of maximum surface excess $\Gamma_\infty$, (ii) that the probability of desorption of a species $i$ is proportional to its surface excess $\Gamma_i$, (iii) that its probability of adsorption is proportional to its bulk monomeric concentration $c_i^m$ and to the number of vacant sites $\Gamma_\infty - \sum_k \Gamma_k$. This can be written in terms of flux for any species $i = \{s, d, n\}$:
* The piece of film we consider is a closed system (cf. subsection 0.2.2): volume is conserved and the total quantities of surfactants (SDS and DOH) contained in the interfaces and the bulk are conserved.

* The bulk monomeric concentration of SDS is capped $c_s^m = CMC$ whenever a micellar phase exists.

* Dodecanol population in the bulk is shared between a monomeric population, and a micellar concentration proportional to the concentration of micellar SDS.

* The chemical equilibrium between the bulk and the interfaces is modelled with Langmuir isotherms and concerns the bulk monomeric concentrations of surfactants only.

* SDS is an anionic surfactant, each molecule comes with a sodium counterion $Na^+$ whose populations will likewise be shared between the bulk, the micelles and the interface.

This is the overview of our model, which will be solved numerically in the end. Let us now write down the proper system of equations.

### 1.2.1 Our model for SDS/DOH mixtures

#### Adsorption laws

We start by writing the population shares of SDS and DOH in the bulk. The former has a surface excess $\Gamma_s$ and a total bulk concentration $c_s = c_s^m + c_s^M$ shared between monomers $c_s^m$ and micelles $c_s^M$ so that:

$$c_s^m = c_s \quad \text{if} \quad c_s < CMC \quad \text{and} \quad c_s^m = CMC \quad \text{otherwise.} \qquad (1.2)$$

The total DOH population has a surface excess $\Gamma_d$ and a bulk population $c_d = c_d^m + c_d^M$ that is shared between a monomeric population $c_d^m$ and a micellar population $c_d^M$. As micelles are composed of SDS for an overwhelming majority, we consider that the micellar population of DOH is simply given by a law of mass action $c_d^M = K c_d^m c_s^M$ writing:

$$c_d = c_d^m (1 + K c_s^M) \qquad (1.3)$$

Also, the sodium counterion population must be taken into account, as bringing a molecule of SDS at the interface means bringing a counterion along with it to ensure the electroneutrality of the interface. This comes with consequences for the statistical presence of SDS at the interface. Thus, let us denote $\Gamma_n = \Gamma_s$ the surface excess of $Na^+$, $c_n^m = c_s^m$ its bulk concentration for the monomeric SDS phase, and $c_n^M = c_s^M$ its bulk concentration associated with the SDS micellar phase.

The chemical equilibria between the interface, the bulk monomeric phase and the bulk micellar phase are written using a Langmuir isotherm model. This model considers: (i) that the interface is a set of fixation sites of maximum surface excess $\Gamma_\infty$, (ii) that the probability of desorption of a species $i$ is proportional to its surface excess $\Gamma_i$, (iii) that its probability of adsorption is proportional to its bulk monomeric concentration $c_i^m$ and to the number of vacant sites $\Gamma_\infty - \sum_k \Gamma_k$. This can be written in terms of flux for any species $i = \{s, d, n\}$:
$$j_i^- = k_i^- \Gamma_i \qquad (1.4)$$

$$j_i^+ = k_i^+ c_i^m (\Gamma_\infty - \sum_k \Gamma_k) \qquad (1.5)$$

Writing the chemical equilibrium in this case means equating $j_i^- = j_i^+$ for the three populations of SDS, DOH and sodium ions. We define $K_i = k_i^+/k_i^-$ for each species, and solving these three equations yields the surface excesses:

$$\Gamma_d = \Gamma_\infty \frac{K_d c_d^m}{1 + 2K_s c_s^m + K_d c_d^m} \qquad (1.6)$$

$$\Gamma_s = \Gamma_n = \Gamma_\infty \frac{K_s c_s^m}{1 + 2K_s c_s^m + K_d c_d^m} \qquad (1.7)$$

Note that the factor 2 at the denominators is the result of the electroneutrality imposed to the interface. This has an entropic cost for a counterion to be brought at the interface along with an SDS molecule.

## Mass conservations

As we assume that each piece of deformed film of volume $V = hS$ is a closed system, and that the tension is uniform in the film (meaning the populations of surfactants are uniform), we can write the same law for both SDS and DOH: $2\Gamma_i S + c_i h S = \text{cst}$, with $i = \{s, d\}$. Also, the deformation of this piece of film can be written as $\epsilon = S/S_0 - 1$ with $S_0$ its initial surface. As the volume is preserved, we thus have $S = S_0(1 + \epsilon)$ and $h = h_0/(1 + \epsilon)$ for respectively the surface and thickness of film. Casting this into the conservations of SDS and DOH yields:

$$2\Gamma_d(1 + \epsilon) + c_d h_0 = 2\Gamma_{d,0} + c_{d,0} h_0 \qquad (1.8)$$

$$2\Gamma_s(1 + \epsilon) + c_s h_0 = 2\Gamma_{s,0} + c_{s,0} h_0 \qquad (1.9)$$

These conservation laws, when fed with a given deformation $\epsilon$ and initial conditions ($c_{s,0}$, $c_{d,0}$, $h_0$), are a set of equations which become implicit in the case of the micellar regime, as the total bulk concentrations $c_s$ and $c_d$ are actually each the sum of two variables: a monomeric and a micellar concentration. This is why this model becomes analytically intractable in the micellar regime. For an analytical resolution in the submicellar regime, see Prins et al. [5].

## Thermodynamic identity

We need now to link surface tension to the surface excesses of all the species, in order to couple it with the mass conservation equations. Thus, we first write formally the evolution of surface tension with respect to the chemical potentials of all the surface active species. We do so by expressing the variation of internal energy $dU$ for a piece of film, that is the first fundamental thermodynamic relation:
$$j_i^- = k_i^- \Gamma_i \qquad (1.4)$$

$$j_i^+ = k_i^+ c_i^m (\Gamma_\infty - \sum_k \Gamma_k) \qquad (1.5)$$

Writing the chemical equilibrium in this case means equating $j_i^- = j_i^+$ for the three populations of SDS, DOH and sodium ions. We define $K_i = k_i^+/k_i^-$ for each species, and solving these three equations yields the surface excesses:

$$\Gamma_d = \Gamma_\infty \frac{K_d c_d^m}{1 + 2K_s c_s^m + K_d c_d^m} \qquad (1.6)$$

$$\Gamma_s = \Gamma_n = \Gamma_\infty \frac{K_s c_s^m}{1 + 2K_s c_s^m + K_d c_d^m} \qquad (1.7)$$

Note that the factor 2 at the denominators is the result of the electroneutrality imposed to the interface. This has an entropic cost for a counterion to be brought at the interface along with an SDS molecule.

### Mass conservations

As we assume that each piece of deformed film of volume $V = hS$ is a closed system, and that the tension is uniform in the film (meaning the populations of surfactants are uniform), we can write the same law for both SDS and DOH: $2\Gamma_i S + c_i hS = \text{cst}$, with $i = \{s, d\}$. Also, the deformation of this piece of film can be written as $\epsilon = S/S_0 - 1$ with $S_0$ its initial surface. As the volume is preserved, we thus have $S = S_0(1 + \epsilon)$ and $h = h_0/(1 + \epsilon)$ for respectively the surface and thickness of film. Casting this into the conservations of SDS and DOH yields:

$$2\Gamma_d(1 + \epsilon) + c_d h_0 = 2\Gamma_{d,0} + c_{d,0} h_0 \qquad (1.8)$$

$$2\Gamma_s(1 + \epsilon) + c_s h_0 = 2\Gamma_{s,0} + c_{s,0} h_0 \qquad (1.9)$$

These conservation laws, when fed with a given deformation $\epsilon$ and initial conditions ($c_{s,0}$, $c_{d,0}$, $h_0$), are a set of equations which become implicit in the case of the micellar regime, as the total bulk concentrations $c_s$ and $c_d$ are actually each the sum of two variables: a monomeric and a micellar concentration. This is why this model becomes analytically intractable in the micellar regime. For an analytical resolution in the submicellar regime, see Prins et al. [5].

### Thermodynamic identity

We need now to link surface tension to the surface excesses of all the species, in order to couple it with the mass conservation equations. Thus, we first write formally the evolution of surface tension with respect to the chemical potentials of all the surface active species. We do so by expressing the variation of internal energy $dU$ for a piece of film, that is the first fundamental thermodynamic relation:
$$dU = -pdV + TdS + \sum_k \mu_k dN_k + \gamma dS \quad (1.10)$$

Where $p$ is the pressure in the film, $V$ the volume of the piece of film, $T$ the temperature, $S$ the entropy, $\mu_k$ the chemical potential of each species $k = \{s, d, n\}$ at the interface, $N_k$ their surface populations, $\gamma$ the surface tension and $S$ the surface of the piece of film.

On the other hand, as the internal energy $U(V, S, N_k, S)$ is an extensive function of volume, entropy, surfactant populations and surface, the Euler's theorem for homogeneous functions writes $U = -Vp + ST + \sum_k \mu_k N_k + \gamma S$, which can be differentiated:

$$dU = -pdV - Vdp + TdS + SdT + \sum_k (\mu_k dN_k + N_k d\mu_k) + \gamma dS + Sd\gamma \quad (1.11)$$

This can now be equated with the fundamental thermodynamics identity and yields:

$$d\gamma = hdp - sdT - \sum_k \Gamma_k d\mu_k \quad (1.12)$$

Where $h$ is the thickness of the piece of film, $s$ is its entropy per unit area and $\Gamma_k = N_k/S$ are the surface excesses which eventually appear here. As we are in normal conditions with constant pressure and temperature throughout the whole experiment, we can thus directly write the relation between surface tension and the chemical potentials of the surface active species:

$$d\gamma = - \sum_k \Gamma_k d\mu_k \quad (1.13)$$

### Chemical equilibrium

A thermodynamic consequence of the instantaneous chemical equilibrium we impose in the piece of film is that the chemical potentials of the interface and the bulk must be equal. More precisely, for each species, the equality of the chemical potential between the interface, the bulk monomeric populations and the bulk micellar populations must be respected. These conditions are already enforced with all the equilibrium and adsorption laws written in subsection 1.2.1. What we need to write here besides all this is the form of the chemical potential of the monomeric phase:

$$\mu_{k,interface} = \mu_{k,bulk,mono} = \mu_{k,0} + RT \ln(c_k^m) \quad (1.14)$$

Meaning we can now express the surface tension variation as:

$$d\gamma = - \sum_k RT\Gamma_k d(\ln c_k^m) \quad (1.15)$$

This can be integrated using the expressions of the $\Gamma_k$'s 1.6 and yields the full expression:

$$\gamma = \gamma_w - RT\Gamma_{\infty} \ln (1 + 2K_s c_s^m + K_d c_d^m) \quad (1.16)$$

Where $\gamma_w = 72.8 \text{ mN/m}$ is the surface tension of pure water, or the glycerol/water mix free of any surfactant. This is the relation between the surface tension and the monomeric concentrations of SDS and DOH.
$$dU = -pdV + TdS + \sum_k \mu_k dN_k + \gamma dS \quad (1.10)$$

Where $p$ is the pressure in the film, $V$ the volume of the piece of film, $T$ the temperature, $S$ the entropy, $\mu_k$ the chemical potential of each species $k = \{s, d, n\}$ at the interface, $N_k$ their surface populations, $\gamma$ the surface tension and $S$ the surface of the piece of film.

On the other hand, as the internal energy $U(V, S, N_k, S)$ is an extensive function of volume, entropy, surfactant populations and surface, the Euler's theorem for homogeneous functions writes $U = -Vp + ST + \sum_k \mu_k N_k + \gamma S$, which can be differentiated:

$$dU = -pdV - Vdp + TdS + SdT + \sum_k (\mu_k dN_k + N_k d\mu_k) + \gamma dS + Sd\gamma \quad (1.11)$$

This can now be equated with the fundamental thermodynamics identity and yields:

$$d\gamma = hdp - sdT - \sum_k \Gamma_k d\mu_k \quad (1.12)$$

Where $h$ is the thickness of the piece of film, $s$ is its entropy per unit area and $\Gamma_k = N_k/S$ are the surface excesses which eventually appear here. As we are in normal conditions with constant pressure and temperature throughout the whole experiment, we can thus directly write the relation between surface tension and the chemical potentials of the surface active species:

$$d\gamma = -\sum_k \Gamma_k d\mu_k \quad (1.13)$$

### Chemical equilibrium

A thermodynamic consequence of the instantaneous chemical equilibrium we impose in the piece of film is that the chemical potentials of the interface and the bulk must be equal. More precisely, for each species, the equality of the chemical potential between the interface, the bulk monomeric populations and the bulk micellar populations must be respected. These conditions are already enforced with all the equilibrium and adsorption laws written in subsection 1.2.1. What we need to write here besides all this is the form of the chemical potential of the monomeric phase:

$$\mu_{k,interface} = \mu_{k,bulk,mono} = \mu_{k,0} + RT \ln(c_k^m) \quad (1.14)$$

Meaning we can now express the surface tension variation as:

$$d\gamma = -\sum_k RT\Gamma_k d(\ln c_k^m) \quad (1.15)$$

This can be integrated using the expressions of the $\Gamma_k$'s 1.6 and yields the full expression:

$$\gamma = \gamma_w - RT\Gamma_{\infty} \ln (1 + 2K_s c_s^m + K_d c_d^m) \quad (1.16)$$

Where $\gamma_w = 72.8 \text{ mN/m}$ is the surface tension of pure water, or the glycerol/water mix free of any surfactant. This is the relation between the surface tension and the monomeric concentrations of SDS and DOH.
<table><caption>Table 1.1: Table of the co-adsorption Langmuir parameters found in the literature, by chronological order. Most parameters are deduced by the authors from comparison with their experimental data, but some are taken from the previous articles ('from lit.') and some are not given in the paper but can be deduced from the data ('from data').</caption><thead><tr><th></th><th>Γ<sub>∞</sub> (mol/m<sup>2</sup>)</th><th>K<sub>s</sub> (m<sup>3</sup>/mol)</th><th>K<sub>d</sub> (m<sup>3</sup>/mol)</th><th>K (m<sup>3</sup>/mol)</th></tr></thead><tbody><tr><td>van den Tempel et al. [31]</td><td>2.5 – 5 · 10<sup>-6</sup></td><td>5</td><td></td><td></td></tr><tr><td>Prins et al. [5]</td><td>7.10<sup>-6</sup></td><td>0.38</td><td>233</td><td></td></tr><tr><td>Joos et al. [95]</td><td>7.10<sup>-6</sup> (from lit.)</td><td></td><td></td><td></td></tr><tr><td>Vollhardt et al. [96]</td><td>14.10<sup>-6</sup> (from data)</td><td>0.13</td><td>98</td><td></td></tr><tr><td>Fang et al. [41]</td><td>6.10<sup>-6</sup> (from lit.)</td><td>0.78 ou 0.48</td><td>1600</td><td>4.67 or 5.83</td></tr><tr><td>Vollhardt et al. [97]</td><td>6.5 · 10<sup>-6</sup></td><td>0.13 (from lit.)</td><td>98 (from lit.)</td><td></td></tr></tbody></table>

## Numerical resolution and chemical parameters

Equation 1.16 giving $\gamma = f(c_s^m, c_d^m)$ is coupled to the mass conservations 1.8-1.9 we recall:

$$2\Gamma_d(1 + \epsilon) + c_d h_0 = 2\Gamma_{d,0} + c_{d,0} h_0$$

$$2\Gamma_s(1 + \epsilon) + c_s h_0 = 2\Gamma_{s,0} + c_{s,0} h_0$$

And the chemical equilibrium laws between the monomeric and micellar concentrations:

$$c_s^m = c_s \quad \text{if} \quad c_s < CMC \quad \text{and} \quad c_s^m = CMC \quad \text{otherwise.}$$

$$c_d = c_d^m (1 + K c_s^M)$$

This set of equations must be fed a deformation $\epsilon$, initial concentrations $c_{s,0}$ and $c_{d,0}$, an initial thickness $h_0$ and the chemical parameters $K$, $K_s$, $K_d$, $\Gamma_\infty$. Then, a numerical resolution can be performed to determine $\gamma$. It is related to the film tension $\sigma^f$ defined in eq. 0.2.4 by $\sigma^f = 2\gamma$.

This is done with a MATLAB code using symbolic computation to approach a solution where mass conservation laws are enforced as close as possible. Performing the minimisation starting with the trivial case $\epsilon = 0$ and increasing (resp. decreasing) gradually searching in the vicinity of the previous solution yields very good results and essentially no numerical error on the total solution $\sigma^f(\epsilon)$.

Regarding the chemical parameters, a wide literature gives access to them, although certain values took time to converge and some are still open to debate [5, 31, 40, 41, 95, 97]. I report Table 1.1 extracted from our paper Poryles et al. [6], which summarises all the parameters we found. Notably, the set of parameters we retain for our numerical resolution is: $K = 4.67 \text{ m}^3/\text{mol}$, $K_s = 0.14 \text{ m}^3/\text{mol}$, $K_d = 98 \text{ m}^3/\text{mol}$ and $\Gamma_\infty = 6.5 \times 10^{-6} \text{ mol/m}^2$.
<table><caption>Table 1.1: Table of the co-adsorption Langmuir parameters found in the literature, by chronological order. Most parameters are deduced by the authors from comparison with their experimental data, but some are taken from the previous articles ('from lit.') and some are not given in the paper but can be deduced from the data ('from data').</caption><thead><tr><th></th><th>Γ<sub>∞</sub> (mol/m<sup>2</sup>)</th><th>K<sub>s</sub> (m<sup>3</sup>/mol)</th><th>K<sub>d</sub> (m<sup>3</sup>/mol)</th><th>K (m<sup>3</sup>/mol)</th></tr></thead><tbody><tr><td>van den Tempel et al. [31]</td><td>2.5 – 5 · 10<sup>-6</sup></td><td>5</td><td></td><td></td></tr><tr><td>Prins et al. [5]</td><td>7.10<sup>-6</sup></td><td>0.38</td><td>233</td><td></td></tr><tr><td>Joos et al. [95]</td><td>7.10<sup>-6</sup> (from lit.)</td><td></td><td></td><td></td></tr><tr><td>Vollhardt et al. [96]</td><td>14.10<sup>-6</sup> (from data)</td><td>0.13</td><td>98</td><td></td></tr><tr><td>Fang et al. [41]</td><td>6.10<sup>-6</sup> (from lit.)</td><td>0.78 ou 0.48</td><td>1600</td><td>4.67 or 5.83</td></tr><tr><td>Vollhardt et al. [97]</td><td>6.5 · 10<sup>-6</sup></td><td>0.13 (from lit.)</td><td>98 (from lit.)</td><td></td></tr></tbody></table>

## Numerical resolution and chemical parameters

Equation 1.16 giving $\gamma = f(c_s^m, c_d^m)$ is coupled to the mass conservations 1.8-1.9 we recall:

$$2\Gamma_d(1 + \epsilon) + c_d h_0 = 2\Gamma_{d,0} + c_{d,0} h_0$$

$$2\Gamma_s(1 + \epsilon) + c_s h_0 = 2\Gamma_{s,0} + c_{s,0} h_0$$

And the chemical equilibrium laws between the monomeric and micellar concentrations:

$$c_s^m = c_s \quad \text{if} \quad c_s < \text{CMC} \quad \text{and} \quad c_s^m = \text{CMC} \quad \text{otherwise.}$$

$$c_d = c_d^m (1 + K c_s^M)$$

This set of equations must be fed a deformation $\epsilon$, initial concentrations $c_{s,0}$ and $c_{d,0}$, an initial thickness $h_0$ and the chemical parameters $K$, $K_s$, $K_d$, $\Gamma_\infty$. Then, a numerical resolution can be performed to determine $\gamma$. It is related to the film tension $\sigma^f$ defined in eq. 0.2.4 by $\sigma^f = 2\gamma$.

This is done with a MATLAB code using symbolic computation to approach a solution where mass conservation laws are enforced as close as possible. Performing the minimisation starting with the trivial case $\epsilon = 0$ and increasing (resp. decreasing) gradually searching in the vicinity of the previous solution yields very good results and essentially no numerical error on the total solution $\sigma^f(\epsilon)$.

Regarding the chemical parameters, a wide literature gives access to them, although certain values took time to converge and some are still open to debate [5, 31, 40, 41, 95, 97]. I report Table 1.1 extracted from our paper Poryles et al. [6], which summarises all the parameters we found. Notably, the set of parameters we retain for our numerical resolution is: $K = 4.67 \text{ m}^3/\text{mol}$, $K_s = 0.14 \text{ m}^3/\text{mol}$, $K_d = 98 \text{ m}^3/\text{mol}$ and $\Gamma_\infty = 6.5 \times 10^{-6} \text{ mol/m}^2$.
### 1.2.2 Numerical results

In Fig.1.3, we report our numerical resolution of the system of subsection 1.2.1, in micellar regime (A) and submicellar regime (B).

<figure><img src="image_38.png" /><figcaption>Figure 1.3: Numerical resolution of the strain-stress curves (centred and normalised with the equilibrium tension $\sigma_0$). A: with $c_s = 2.4$ CMC and $c_d = 153$ mg/L. The different colours account for different thickness of films $h_0$ (which is a parameter of the model) from blue to red: $h_0 = 0.3, 0.5, 1, 2$ µm. The black solid line, which is the full numerical resolution with $h = 1$ µm, is fitted with the black dotted line with a function $E_f \frac{\epsilon}{1+\epsilon}$ (fitting on $E_f$). B: without dodecanol with an SDS concentration $c_s = 0.9$ CMC close to the micellar regime. The solid line is the numerical resolution: for some negative threshold value of $\epsilon$, the micellar regime is attained, and the contribution of SDS to film elasticity vanishes. The dashed line is the best fit with the same function $E_f \frac{\epsilon}{1+\epsilon}$ as previously.</figcaption></figure>

Fig.1.3 A shows the strain-stress curves predictions for various thicknesses of film (solid lines) varying over an order of magnitude $h_0 = 0.3, 0.5, 1, 2$ µm. We define the film elasticity $E_f$ as the slope in origin of these curves, and we see that varying $h$ by an order of magnitude leads to a change in elasticity by a factor 2 at best. **Film elasticity is a slowly varying function of film thickness.**

As eq.16 already suggests despite its crude model, these curves are non-linear in $\epsilon$, Also, we choose to fit our data with the following function:

$$ \Delta\sigma = E_f \frac{\epsilon}{1 + \epsilon} \qquad (1.17) $$

Which defines a film elasticity $E_f$ as its slope in origin.

**Important Remark:** This function is **NOT** an application of the model of subsection 0.2.3 with $h_\Gamma \gg h_0$ and $E_f = 2E_M$ which we will not risk identifying. It only captures the non-linearity of the curves and allows us to compare the relative stiffness of the different solutions.

The dashed lines in Fig.1.3 are fits with eq.1.17 for $h = 1$ µm. We see that they indeed capture the non-linearity, but there is also a small discrepancy at large $\epsilon$. As we anticipate that the thicknesses of our films are varying around $h = 0.5 - 1.5$ µm and are not always known, the
### 1.2.2 Numerical results

In Fig.1.3, we report our numerical resolution of the system of subsection 1.2.1, in micellar regime (A) and submicellar regime (B).

<figure><img src="image_38.png" /><figcaption>Figure 1.3: Numerical resolution of the strain-stress curves (centred and normalised with the equilibrium tension $\sigma_0$). A: with $c_s = 2.4$ CMC and $c_d = 153$ mg/L. The different colours account for different thickness of films $h_0$ (which is a parameter of the model) from blue to red: $h_0 = 0.3, 0.5, 1, 2$ µm. The black solid line, which is the full numerical resolution with $h = 1$ µm, is fitted with the black dotted line with a function $E_f \frac{\epsilon}{1+\epsilon}$ (fitting on $E_f$). B: without dodecanol with an SDS concentration $c_s = 0.9$ CMC close to the micellar regime. The solid line is the numerical resolution: for some negative threshold value of $\epsilon$, the micellar regime is attained, and the contribution of SDS to film elasticity vanishes. The dashed line is the best fit with the same function $E_f \frac{\epsilon}{1+\epsilon}$ as previously.</figcaption></figure>

Fig.1.3 A shows the strain-stress curves predictions for various thicknesses of film (solid lines) varying over an order of magnitude $h_0 = 0.3, 0.5, 1, 2$ µm. We define the film elasticity $E_f$ as the slope in origin of these curves, and we see that varying $h$ by an order of magnitude leads to a change in elasticity by a factor 2 at best. **Film elasticity is a slowly varying function of film thickness.**

As eq.16 already suggests despite its crude model, these curves are non-linear in $\epsilon$, Also, we choose to fit our data with the following function:

$$ \Delta\sigma = E_f \frac{\epsilon}{1 + \epsilon} \qquad (1.17) $$

Which defines a film elasticity $E_f$ as its slope in origin.

**Important Remark:** This function is **NOT** an application of the model of subsection 0.2.3 with $h_\Gamma \gg h_0$ and $E_f = 2E_M$ which we will not risk identifying. It only captures the non-linearity of the curves and allows us to compare the relative stiffness of the different solutions.

The dashed lines in Fig.1.3 are fits with eq.1.17 for $h = 1$ µm. We see that they indeed capture the non-linearity, but there is also a small discrepancy at large $\epsilon$. As we anticipate that the thicknesses of our films are varying around $h = 0.5 - 1.5$ µm and are not always known, the
small discrepancy of the fit will remain under our error bars and eq.1.17 remains a solid tool to
characterise our film elasticities.

Fig.1.3 B shows the stress-strain curve for a submicellar regime of SDS without dodecanol. In this regime, an interesting feature that we highlight is the passage from the submicellar to the micellar regime occurring for $\epsilon$ going below $\sim -0.2$. When the micellar regime is attained, the film elasticity completely vanishes, meaning that micellar SDS alone has no contribution to it. This can be rationalised by the fact that above the CMC, the bulk monomeric concentration of SDS is always the same, capped at the CMC. Thus, the surface population in equilibrium with this bulk concentration is the same whatever the $\epsilon$, and $\gamma$ is constant.

**Important Remark 2:** The other consequence is that the elasticity shown in Fig.1.3 **A** is due to the contribution of dodecanol! In this latter case, the SDS concentration above the CMC still matters, as dodecanol is shared between the surface, the bulk and the SDS micelles.

<figure><img src="image_39.png" /><figcaption>Figure 1.4: Film elasticities $E_f$ fitted on our numerical model as a function of dodecanol concentration for: (red) $c_s = 1.2$ CMC, (blue) $c_s = 2.4$ CMC.</figcaption></figure>

Back to the micellar regime, Fig.1.4 shows how $E_f$ (computed through our fits on the numerical resolutions of stress-strain curves) depends on the dodecanol concentration $c_d$. **Dodecanol is expected to be the main parameter to tune the elasticity of our solutions, as a factor 2 on its concentration can lead to one order of magnitude on $E_f$!**

This is all the predictions we can make on these stress-strain curves, extending the pioneering work of Prins *et al.* [5] to micellar regimes of SDS/DOH mixtures. We shall now look at the experiments which have allowed us to measure those curves, and compare them to our model.
small discrepancy of the fit will remain under our error bars and eq.1.17 remains a solid tool to
characterise our film elasticities.

Fig.1.3 B shows the stress-strain curve for a submicellar regime of SDS without dodecanol. In this regime, an interesting feature that we highlight is the passage from the submicellar to the micellar regime occurring for $\epsilon$ going below $\sim -0.2$. When the micellar regime is attained, the film elasticity completely vanishes, meaning that micellar SDS alone has no contribution to it. This can be rationalised by the fact that above the CMC, the bulk monomeric concentration of SDS is always the same, capped at the CMC. Thus, the surface population in equilibrium with this bulk concentration is the same whatever the $\epsilon$, and $\gamma$ is constant.

**Important Remark 2:** The other consequence is that the elasticity shown in Fig.1.3 **A** is due to the contribution of dodecanol! In this latter case, the SDS concentration above the CMC still matters, as dodecanol is shared between the surface, the bulk and the SDS micelles.

<figure><img src="image_39.png" /><figcaption>Figure 1.4: Film elasticities $E_f$ fitted on our numerical model as a function of dodecanol concentration for: (red) $c_s = 1.2$ CMC, (blue) $c_s = 2.4$ CMC.</figcaption></figure>

Back to the micellar regime, Fig.1.4 shows how $E_f$ (computed through our fits on the numerical resolutions of stress-strain curves) depends on the dodecanol concentration $c_d$. **Dodecanol is expected to be the main parameter to tune the elasticity of our solutions, as a factor 2 on its concentration can lead to one order of magnitude on $E_f$!**

This is all the predictions we can make on these stress-strain curves, extending the pioneering work of Prins *et al.* [5] to micellar regimes of SDS/DOH mixtures. We shall now look at the experiments which have allowed us to measure those curves, and compare them to our model.
1.3 A foam film rheometer

The experimental values of the film elasticity we discuss below have been obtained by Raphaël Poryles, using a set up very similar to the set up I used during my PhD to obtain the experimental results of Chapter 3. Here the elementary foam has 5 films, whereas I used a 3 films set-up.

In this section, I briefly present the 5 films set-up, but the precise data analysis will not be discussed here, as it is not part of my work. **I will elaborate much more about both the technological aspects and the data processing in chapter 3 for this reason.**

1.3.1 Experimental setup

The experimental setup is about creating a deformable centimetric elementary foam as depicted
in Fig.1.5 and presented more thoroughly by Bussonnière et al. [4]. It is composed of five films
linked by two free menisci, embedding a central film of length *d*<sub>c</sub> = 6 mm. The edges at min
*z* = 0 and max *z* = 42 mm are supported by a rigid metallic frame along which four motors are
free to slide at imposed velocities *U*<sub>m</sub>, supporting the four edges with the red velocity arrows.
The rigid frame respects the 120° of the rest angles between the films.

As it will be shown later, even during rather fast deformations, the angle between the films
stay close to 120 °, as the relative tension variation remains small. In that context, a great
advantage of this system is that the films are thus flat rectangles that stay in the same plane
whatever the position of the motors. This makes the image acquisition much easier, which is at
the origin of all our measurements.

<figure><img src="image_40.png" /><figcaption>Figure 1.5: Notations and geometry for the elementary foam with 5 films used in this chapter.</figcaption></figure>

This geometry is very close to that of the Besson & Debrégeas setup [78], with an invariance
by translation along *z* instead of an axisymmetry. The gain here is that we have more degrees
of freedom for the deformation of the elementary foam, as each motor can have its own ve-
locity/displacement instructions, and also allows us to probe greater amplitudes of deformations
(ε ~ 1 here vs. ε ~ 0.05 in the Besson & Debrégeas setup). The main shortcoming of our setup
1.3 A foam film rheometer

The experimental values of the film elasticity we discuss below have been obtained by Raphaël Poryles, using a set up very similar to the set up I used during my PhD to obtain the experimental results of Chapter 3. Here the elementary foam has 5 films, whereas I used a 3 films set-up.

In this section, I briefly present the 5 films set-up, but the precise data analysis will not be discussed here, as it is not part of my work. **I will elaborate much more about both the technological aspects and the data processing in chapter 3 for this reason.**

1.3.1 Experimental setup

The experimental setup is about creating a deformable centimetric elementary foam as depicted
in Fig.1.5 and presented more thoroughly by Bussonnière et al. [4]. It is composed of five films
linked by two free menisci, embedding a central film of length d_c = 6 mm. The edges at min
z = 0 and max z = 42 mm are supported by a rigid metallic frame along which four motors are
free to slide at imposed velocities U_m, supporting the four edges with the red velocity arrows.
The rigid frame respects the 120° of the rest angles between the films.

As it will be shown later, even during rather fast deformations, the angle between the films stay close to 120 °, as the relative tension variation remains small. In that context, a great advantage of this system is that the films are thus flat rectangles that stay in the same plane whatever the position of the motors. This makes the image acquisition much easier, which is at the origin of all our measurements.

<figure><img src="image_40.png" /><figcaption>Figure 1.5: Notations and geometry for the elementary foam with 5 films used in this chapter.</figcaption></figure>

This geometry is very close to that of the Besson & Debrégeas setup [78], with an invariance
by translation along *z* instead of an axisymmetry. The gain here is that we have more degrees
of freedom for the deformation of the elementary foam, as each motor can have its own ve-
locity/displacement instructions, and also allows us to probe greater amplitudes of deformations
(ε ~ 1 here vs. ε ~ 0.05 in the Besson & Debrégeas setup). The main shortcoming of our setup
however is that we will not be able to probe as wide of a range of deformation rates because of the camera resolution and the limitations of the motors (we will have $\dot{\epsilon} \sim 1 - 10$ Hz vs. $\omega \sim 0.01 - 100$ rad/s).

The foam films are made by bathing the whole deformable frame in soapy solutions which are SDS/DOH aqueous mixtures with 15% of glycerol in volume and fluorescein (at [Fluo] = 0.8 g/L). This action is automatically performed in a repeatable way using a step-by-step motor rising a bath towards the frame.

The system is monitored from the top using fluorescence cameras with the relevant colour filter for fluorescein. This allows us to track the displacements of the menisci (which are saturated given their thicknesses ~ 300 µm compared to the films ~ 1 - 10 µm).

The applied deformation has a top-down symmetry, compressing two sides, stretching the two others as depicted in Fig.1.6A. The situation is also symmetrical in amplitude, as the compressed films go from $d^-(0) = d_0+A/2$ to $d^-(t_{end}) = d_0-A/2$, the stretched films from $d^+(0) = d_0-A/2$ to $d^+(t_{end}) = d_0+A/2$. $d_0 = 12.2$ mm and A varies in the range [4, 14] mm. A first experimental observation is that the central film has a deformation $\epsilon$ that remains small ~ 0.1 compared to the other films ~ 1. This leads to consider that the central film remains at its equilibrium tension $\sigma_0$.

<figure><img src="image_41.png" /><figcaption>Figure 1.6: Notations for the stress and deformation measurements in the elementary foam. **A**: Compressing and stretching the films result in decreased/increased tensions that induce changes of angles between the films. **B**: Extraction/Withdrawal of lengths of films in the planes of the films must be taken into account to compute the actual deformations of the films.</figcaption></figure>

Fig.1.6A also illustrates the stress measurement in the system. It is based on the same principle as in Besson et al. [78], that is a measure of the dynamics of the angles $\theta^{\pm}$, whose deviation from a static value (here given by the second Plateau's law $2\theta_0 = 120^{\circ}$) is directly linked to a difference of film tension between the three neighbouring films. Writing the tension balance projected on the horizontal direction at all times (neglecting inertia) gives:

$$ \frac{\Delta\sigma^{\pm}}{\sigma_0} = \frac{\sigma^{\pm} - \sigma_0}{\sigma_0} = \frac{1}{2\cos\theta^{\pm}} - 1 \qquad (1.18) $$
however is that we will not be able to probe as wide of a range of deformation rates because of the camera resolution and the limitations of the motors (we will have $\dot{\epsilon} \sim 1 - 10$ Hz vs. $\omega \sim 0.01 - 100$ rad/s).

The foam films are made by bathing the whole deformable frame in soapy solutions which are SDS/DOH aqueous mixtures with 15% of glycerol in volume and fluorescein (at [Fluo] = 0.8 g/L). This action is automatically performed in a repeatable way using a step-by-step motor rising a bath towards the frame.

The system is monitored from the top using fluorescence cameras with the relevant colour filter for fluorescein. This allows us to track the displacements of the menisci (which are saturated given their thicknesses ~ 300 µm compared to the films ~ 1 - 10 µm).

The applied deformation has a top-down symmetry, compressing two sides, stretching the two others as depicted in Fig.1.6A. The situation is also symmetrical in amplitude, as the compressed films go from $d^-(0) = d_0+A/2$ to $d^-(t_{end}) = d_0-A/2$, the stretched films from $d^+(0) = d_0-A/2$ to $d^+(t_{end}) = d_0+A/2$. $d_0 = 12.2$ mm and A varies in the range [4, 14] mm. A first experimental observation is that the central film has a deformation $\epsilon$ that remains small ~ 0.1 compared to the other films ~ 1. This leads to consider that the central film remains at its equilibrium tension $\sigma_0$.

<figure><img src="image_41.png" /><figcaption>Figure 1.6: Notations for the stress and deformation measurements in the elementary foam. **A:** Compressing and stretching the films result in decreased/increased tensions that induce changes of angles between the films. **B:** Extraction/Withdrawal of lengths of films in the planes of the films must be taken into account to compute the actual deformations of the films.</figcaption></figure>

Fig.1.6A also illustrates the stress measurement in the system. It is based on the same principle as in Besson et al. [78], that is a measure of the dynamics of the angles $\theta^{\pm}$, whose deviation from a static value (here given by the second Plateau's law $2\theta_0 = 120^{\circ}$) is directly linked to a difference of film tension between the three neighbouring films. Writing the tension balance projected on the horizontal direction at all times (neglecting inertia) gives:

$$ \frac{\Delta\sigma^{\pm}}{\sigma_0} = \frac{\sigma^{\pm} - \sigma_0}{\sigma_0} = \frac{1}{2\cos\theta^{\pm}} - 1 \qquad (1.18) $$
The angles $\theta^{\pm}$ can be deduced from the displacements $\delta^{\pm}$ of the menisci, and the known positions of the motors at all times. This gives experimental access to the dynamics of film tension.

On the other hand, in Fig.1.6B, we illustrate how the deformations $\epsilon$ are obtained. As discussed in 0.4.5 the film deformation can not be directly deduced from the motor and mensicus motion. Fortunately, we have a clear signature of the film creation at the boundary between the streched films and the meniscus : these new pieces of film obey the Frankel law discussed in subsection 0.3.2. They are much thicker (usually 10 times), and thus appear much brighter in the fluorescence images, than the film initially present on the frame, before the deformation. This allows to track these preexisting pieces of film and to define their deformations as:

$$ \epsilon^{\pm}(t) = \frac{L^{\pm}(t)}{L^{\pm}(0)} - 1 \qquad (1.19) $$

Where $L^{\pm}(0) = d_0 \mp A/2$ is the initial length of preexisting film, and $L^{\pm}(t) = d^{\pm} \mp 2L^{Fr}(t)$ is the current value estimated by taking into account the presence of Frankel films. The underlying assumption of these expressions is that the same quantity of film $L_{Fr}(t)$ (measured only in the central film) goes out/in of all the menisci. **This a rather bold assumption that is needed to further advance with these experiments. Notably, chapter 3, which is about a similar setup, aims at going beyond this approximation.**

### 1.3.2 Experimental results and comparison with the predictions

The two measures of $\Delta\sigma^{\pm}$ and $\epsilon^{\pm}$ give experimental access to stress-strain curves for the system. Both can be put in the same graph, as $\Delta\sigma^{+}, \epsilon^{+} > 0$ and $\Delta\sigma^{-}, \epsilon^{-} > 0$ are respectively the stress-strain curves in extension and in compression.

<figure><img src="image_42.png" /><figcaption>Figure 1.7: Stress strain curves: **A:** For a given chemistry ($c_s = 2.4$ CMC for SDS, $c_d = 35$ mg/L for DOH) with different velocities of deformations (red $U_m = 5$ mm/s, black $U_m = 10$ mm/s). **B:** For different concentration of dodecanol with the same deformation $U_m = 10$ mm/s and SDS concentration $c_s = 2.4$ CMC (from red to black: $c_d = 0, 15, 35, 50$ mg/L).</figcaption></figure>

In Fig.1.7A, we show two typical curves for a micellar SDS/DOH mixture ($c_s = 2.4$ CMC for SDS, $c_d = 35$ mg/L for DOH) for two different velocities of deformation $U_m = 5$ mm/s (red)
The angles $\theta^{\pm}$ can be deduced from the displacements $\delta^{\pm}$ of the menisci, and the known positions of the motors at all times. This gives experimental access to the dynamics of film tension.

On the other hand, in Fig.1.6B, we illustrate how the deformations $\epsilon$ are obtained. As discussed in 0.4.5 the film deformation can not be directly deduced from the motor and mensicus motion. Fortunately, we have a clear signature of the film creation at the boundary between the streched films and the meniscus : these new pieces of film obey the Frankel law discussed in subsection 0.3.2. They are much thicker (usually 10 times), and thus appear much brighter in the fluorescence images, than the film initially present on the frame, before the deformation. This allows to track these preexisting pieces of film and to define their deformations as:

$$ \epsilon^{\pm}(t) = \frac{L^{\pm}(t)}{L^{\pm}(0)} - 1 \qquad (1.19) $$

Where $L^{\pm}(0) = d_0 \mp A/2$ is the initial length of preexisting film, and $L^{\pm}(t) = d^{\pm} \mp 2L^{Fr}(t)$ is the current value estimated by taking into account the presence of Frankel films. The underlying assumption of these expressions is that the same quantity of film $L_{Fr}(t)$ (measured only in the central film) goes out/in of all the menisci. **This a rather bold assumption that is needed to further advance with these experiments. Notably, chapter 3, which is about a similar setup, aims at going beyond this approximation.**

### 1.3.2 Experimental results and comparison with the predictions

The two measures of $\Delta\sigma^{\pm}$ and $\epsilon^{\pm}$ give experimental access to stress-strain curves for the system. Both can be put in the same graph, as $\Delta\sigma^{+}, \epsilon^{+} > 0$ and $\Delta\sigma^{-}, \epsilon^{-} > 0$ are respectively the stress-strain curves in extension and in compression.

<figure><img src="image_42.png" /><figcaption>Figure 1.7: Stress strain curves: **A:** For a given chemistry ($c_s = 2.4$ CMC for SDS, $c_d = 35$ mg/L for DOH) with different velocities of deformations (red $U_m = 5$ mm/s, black $U_m = 10$ mm/s). **B:** For different concentration of dodecanol with the same deformation $U_m = 10$ mm/s and SDS concentration $c_s = 2.4$ CMC (from red to black: $c_d = 0, 15, 35, 50$ mg/L).</figcaption></figure>

In Fig.1.7A, we show two typical curves for a micellar SDS/DOH mixture ($c_s = 2.4$ CMC for SDS, $c_d = 35$ mg/L for DOH) for two different velocities of deformation $U_m = 5$ mm/s (red)
and $U_m = 10 \text{ mm/s}$ (black) at fixed $A = 6 \text{ mm}$. Their superimposition is a piece of evidence that the contributions of the intrinsic/extrinsic viscosities of the interfaces can be neglected.

**NB 1:** The origin is a slightly angular point, this may be due to an initial repositioning of the films when we start the deformation, which is also encountered in chapter 3 and more thoroughly investigated in subsection 3.8.1.

In Fig.1.7B, we fix $U_m = 10 \text{ mm/s}$ and the dodecanol concentration is varied in the range $c_d = 0, 15, 35, 50 \text{ mg/L}$ from red to black (the green of Fig.1.7B corresponds to the black of Fig.1.7A). **We thus see that the more we add dodecanol (a poorly soluble surfactant), the stiffer the foam films get.** The full lines are the experimental data with errors remaining on average below ~5%.

We have now given ourselves everything to draw these stress-strain curves for deformed foam films, let us compare them to the thermochemical model of section 1.2. Fig.1.8 is a reboot of Fig.1.3 with now the experimental data (open circles).

<figure><img src="image_43.png" /></figure>

Figure 1.8: Predicted stress strain curves taken from Fig.1.3 with experimental data as the black circles: **A:** with $c_s = 2.4 \text{ CMC}$ and $c_d = 153 \text{ mg/L}$. The different colours account for different thickness of films $h_0$ (which is a parameter of the model) from blue to red: $h_0 = 0.3, 0.5, 1, 2 \text{ µm}$. **B:** without dodecanol with an SDS concentration $c_s = 0.9 \text{ CMC}$ close to the micellar regime. The solid line is the numerical resolution: for some negative threshold value of $\epsilon$, the micellar regime is attained, and the contribution of SDS to film elasticity vanishes.

In the first case **A**, we have $c_s = 2.4 \text{ CMC}$ and $c_d^{nom} = 35 \text{ mg/L}$ the nominal concentration in DOH. The black circles are the experimental data corresponding to the green curve of Fig.1.7B. We first tried to plot the numerical prediction using the nominal concentration of DOH $c_d^{nom}$, resulting in the black dotted line. The slope remained too small, even when we start sweeping realistic values for the film thickness which has been measured as $h_0 = (1.0 \pm 0.5) \text{ µm}$. We thus took the intermediate value $h_0 = 1 \text{ µm}$ and fitted on an effective dodecanol concentration $c_d^{fit}$, with in mind to take into account the impurities present in the solution besides the nominal concentration of dodecanol added by us. The black solid line is the main fit with $h_0 = 1 \text{ µm}$ and $c_d^{fit} = 153 \text{ mg/L}$.

In the second case **B** with $c_{s,0} = 0.9 \text{ CMC}$, the circles are the experimental data, the solid line is our model, and the slope agreement is rather good without any fitting parameter. The
and $U_m = 10 \text{ mm/s}$ (black) at fixed $A = 6 \text{ mm}$. Their superimposition is a piece of evidence
that the contributions of the intrinsic/extrinsic viscosities of the interfaces can be neglected.

**NB 1:** The origin is a slightly angular point, this may be due to an initial repositioning of the films when we start the deformation, which is also encountered in chapter 3 and more thoroughly investigated in subsection 3.8.1.

In Fig.1.7B, we fix $U_m = 10 \text{ mm/s}$ and the dodecanol concentration is varied in the range $c_d = 0, 15, 35, 50 \text{ mg/L}$ from red to black (the green of Fig.1.7B corresponds to the black of Fig.1.7A). **We thus see that the more we add dodecanol (a poorly soluble surfactant), the stiffer the foam films get.** The full lines are the experimental data with errors remaining on average below ~5%.

We have now given ourselves everything to draw these stress-strain curves for deformed foam films, let us compare them to the thermochemical model of section 1.2. Fig.1.8 is a reboot of Fig.1.3 with now the experimental data (open circles).

<figure><img src="image_43.png" /></figure>

Figure 1.8: Predicted stress strain curves taken from Fig.1.3 with experimental data as the black circles: **A:** with $c_s = 2.4 \text{ CMC}$ and $c_d = 153 \text{ mg/L}$. The different colours account for different thickness of films $h_0$ (which is a parameter of the model) from blue to red: $h_0 = 0.3, 0.5, 1, 2 \text{ µm}$. **B:** without dodecanol with an SDS concentration $c_s = 0.9 \text{ CMC}$ close to the micellar regime. The solid line is the numerical resolution: for some negative threshold value of $\epsilon$, the micellar regime is attained, and the contribution of SDS to film elasticity vanishes.

In the first case **A**, we have $c_s = 2.4 \text{ CMC}$ and $c_d^{\text{nom}} = 35 \text{ mg/L}$ the nominal concentration in DOH. The black circles are the experimental data corresponding to the green curve of Fig.1.7B. We first tried to plot the numerical prediction using the nominal concentration of DOH $c_d^{\text{nom}}$, resulting in the black dotted line. The slope remained too small, even when we start sweeping realistic values for the film thickness which has been measured as $h_0 = (1.0 \pm 0.5) \text{ µm}$. We thus took the intermediate value $h_0 = 1 \text{ µm}$ and fitted on an effective dodecanol concentration $c_d^{\text{fit}}$, with in mind to take into account the impurities present in the solution besides the nominal concentration of dodecanol added by us. The black solid line is the main fit with $h_0 = 1 \text{ µm}$ and $c_d^{\text{fit}} = 153 \text{ mg/L}$.

In the second case **B** with $c_{s,0} = 0.9 \text{ CMC}$, the circles are the experimental data, the solid line is our model, and the slope agreement is rather good without any fitting parameter. The
only physical ingredient we do not capture is the elasticity of the film as it is compressed enough for it to reach $c_s = CMC$ when the interfaces evacuate surfactants towards the bulk. As said above, in our model, without the presence of any other surfactants, the elastic contribution of SDS vanishes, which we do not observe experimentally. One reason might be what we suppose for our initial conditions (given by the foam films we extract from the bath), that is we have a piece of bulk with a total concentration $c_{s,0}$, which is the nominal concentration of surfactant added to the solution, sandwiched between two interfaces whose surfactant populations $\Gamma_{s,0}$ are at equilibrium with it. Given the short amount of time it takes to create these interfaces when they are extracted, we could expect them to tap into the already-isolated piece of bulk being extracted, thus lowering the initial concentration $c_{s,0} < c_s^{nom}$ below the nominal concentration. This problem of film extraction, and to what extent the initial extracted film shares the same concentrations than the bath, is still open to debate. This was discussed by de Gennes [43] in 2001, in which a model keeping mechanical and thermodynamic equilibrium instantaneous, even at the short time scales of a film pullout, can lead to predict unrealistic shapes for foam films.

Now, it is also possible to extract a film elasticity $E_f$ out of the experimental data using the same fit of eq.1.17, just as we did with our numerical predictions. These fits are represented as the dotted lines in Fig.1.7B and describe well our experimental data as well.

1.3.3 Theoretical and experimental film elasticities

Experimental film elasticities are reported in Fig.1.9: **A** where SDS is at $c_s = 1.2 CMC$ (red) and $c_s = 2.4 CMC$ (blue, corresponds to the four curves of Fig.1.7B) and **B** where no dodecanol is added to the solution and $c_s$ varies from 0.75 *CMC* to 6 *CMC* (the red and blue points match the points of **A** at $c_d^{nom} = 0$). The dotted lines in **A** are only guides to the eye. The denomination $c_d^{nom}$ in **A** is the nominal concentration of dodecanol, that is the quantity effectively added to the solutions.

A first observation, without dodecanol on Fig.1.9B, is that film elasticity is a decreasing function of $c_s$. The extreme values tell us that an increase by a factor 10 in SDS concentration can yield a decrease by a factor 50 in film elasticity. This behaviour is not easy to rationalise, as a pure micellar SDS solution is expected to have a vanishing film elasticity under the assumption that the chemical equilibrium is met at all times. This very last point of chemical equilibrium hypothesis is not to be called into question again however, since if it was the problem, we would see a velocity dependency of our elasticity, which we do not. **As discussed earlier, SDS is known to come with residual dodecanol, and its micelles to host them in the bulk [42,93], which is a first unwanted source of surface active impurities contributing to the elasticity as the main surfactant contribution vanishes.** Also, the exchanges of these impurities between the bulk and the micellar phase (more micelles means more soluble impurities) make the dependency of $E_f$ with $c_s > CMC$ non trivial, even at fixed quantities of impurities, making it even harder to rationalise.

Also, when we start adding DOH ourselves (cf. Fig.1.9A), the elasticity of foam films increases quite rapidly, over an order of magnitude by adding up to $c_d^{nom} = 50 \text{ mg/L} = 2.7 \times 10^{-4} \text{ mol/L}$ making it comparable to the elasticity of SDS in its submicellar regime. This increase in elasticity is all the more important as the SDS concentration is close to the CMC. This can be qualitatively rationalise by the fact that more micelles means a greater share of the population of DOH in the
only physical ingredient we do not capture is the elasticity of the film as it is compressed enough for it to reach $c_s = CMC$ when the interfaces evacuate surfactants towards the bulk. As said above, in our model, without the presence of any other surfactants, the elastic contribution of SDS vanishes, which we do not observe experimentally. One reason might be what we suppose for our initial conditions (given by the foam films we extract from the bath), that is we have a piece of bulk with a total concentration $c_{s,0}$, which is the nominal concentration of surfactant added to the solution, sandwiched between two interfaces whose surfactant populations $\Gamma_{s,0}$ are at equilibrium with it. Given the short amount of time it takes to create these interfaces when they are extracted, we could expect them to tap into the already-isolated piece of bulk being extracted, thus lowering the initial concentration $c_{s,0} < c_s^{nom}$ below the nominal concentration. This problem of film extraction, and to what extent the initial extracted film shares the same concentrations than the bath, is still open to debate. This was discussed by de Gennes [43] in 2001, in which a model keeping mechanical and thermodynamic equilibrium instantaneous, even at the short time scales of a film pullout, can lead to predict unrealistic shapes for foam films.

Now, it is also possible to extract a film elasticity $E_f$ out of the experimental data using the same fit of eq.1.17, just as we did with our numerical predictions. These fits are represented as the dotted lines in Fig.1.7B and describe well our experimental data as well.

1.3.3 Theoretical and experimental film elasticities

Experimental film elasticities are reported in Fig.1.9: **A** where SDS is at $c_s = 1.2 CMC$ (red) and $c_s = 2.4 CMC$ (blue, corresponds to the four curves of Fig.1.7B) and **B** where no dodecanol is added to the solution and $c_s$ varies from 0.75 *CMC* to 6 *CMC* (the red and blue points match the points of **A** at $c_d^{nom} = 0$). The dotted lines in **A** are only guides to the eye. The denomination $c_d^{nom}$ in **A** is the nominal concentration of dodecanol, that is the quantity effectively added to the solutions.

A first observation, without dodecanol on Fig.1.9B, is that film elasticity is a decreasing function of $c_s$. The extreme values tell us that an increase by a factor 10 in SDS concentration can yield a decrease by a factor 50 in film elasticity. This behaviour is not easy to rationalise, as a pure micellar SDS solution is expected to have a vanishing film elasticity under the assumption that the chemical equilibrium is met at all times. This very last point of chemical equilibrium hypothesis is not to be called into question again however, since if it was the problem, we would see a velocity dependency of our elasticity, which we do not. **As discussed earlier, SDS is known to come with residual dodecanol, and its micelles to host them in the bulk [42,93], which is a first unwanted source of surface active impurities contributing to the elasticity as the main surfactant contribution vanishes.** Also, the exchanges of these impurities between the bulk and the micellar phase (more micelles means more soluble impurities) make the dependency of $E_f$ with $c_s > CMC$ non trivial, even at fixed quantities of impurities, making it even harder to rationalise.

Also, when we start adding DOH ourselves (cf. Fig.1.9A), the elasticity of foam films increases quite rapidly, over an order of magnitude by adding up to $c_d^{nom} = 50 \text{ mg/L} = 2.7 \times 10^{-4} \text{ mol/L}$ making it comparable to the elasticity of SDS in its submicellar regime. This increase in elasticity is all the more important as the SDS concentration is close to the CMC. This can be qualitatively rationalise by the fact that more micelles means a greater share of the population of DOH in the
<figure><img src="image_44.png" /><figcaption>Figure 1.9: Film elasticities: **A:** as a function of the dodecanol concentration $c_d$ (red $c_s = 1.2$ CMC, blue $c_s = 2.4$ CMC); **B:** as a function of SDS concentration without any added dodecanol.</figcaption></figure>

micellar phase, making its equilibrium surface population variations lower when the interface is deformed.

Effective concentration of dodecanol

These fits over the dodecanol concentration can be repeated so that we match the films elasticities $E^{f,th} = E^{f,exp}$ measured earlier, and we estimate the effective dodecanol concentrations $c_d^{fit} = f(c_d^{nom})$ in Fig.1.10. In red we have the data for $c_s = 1.2$ CMC, in blue $c_s = 2.4$ CMC. The black dotted line is the first bisector, where the data would be in the case of a flawless model applied to ideal solutions.

<figure><img src="image_45.png" /><figcaption>Figure 1.10: Fitted concentration of dodecanol $c_d^{fit}$ as a function of the nominal concentration $c_d^{nom}$ added to the solutions. Blue is with $c_s = 2.4$ CMC, red is for $c_s = 1.2$ CMC. The black dotted line is the function $y = x$.</figcaption></figure>

The first observation is that the global quantity of dodecanol virtually added to the solution to get a quantitative match with the data increases with the concentration of SDS. This is expected, as more SDS means more impurities in the solution. However, the slopes are larger
<figure><img src="image_44.png" /><figcaption>Figure 1.9: Film elasticities: **A:** as a function of the dodecanol concentration $c_d$ (red $c_s = 1.2$ CMC, blue $c_s = 2.4$ CMC); **B:** as a function of SDS concentration without any added dodecanol.</figcaption></figure>

micellar phase, making its equilibrium surface population variations lower when the interface is deformed.

## Effective concentration of dodecanol

These fits over the dodecanol concentration can be repeated so that we match the films elasticities $E^{f,th} = E^{f,exp}$ measured earlier, and we estimate the effective dodecanol concentrations $c_d^{fit} = f(c_d^{nom})$ in Fig. 1.10. In red we have the data for $c_s = 1.2$ CMC, in blue $c_s = 2.4$ CMC. The black dotted line is the first bisector, where the data would be in the case of a flawless model applied to ideal solutions.

<figure><img src="image_45.png" /><figcaption>Figure 1.10: Fitted concentration of dodecanol $c_d^{fit}$ as a function of the nominal concentration $c_d^{nom}$ added to the solutions. Blue is with $c_s = 2.4$ CMC, red is for $c_s = 1.2$ CMC. The black dotted line is the function $y = x$.</figcaption></figure>

The first observation is that the global quantity of dodecanol virtually added to the solution to get a quantitative match with the data increases with the concentration of SDS. This is expected, as more SDS means more impurities in the solution. However, the slopes are larger
than 1, with the red one, close to the CMC being almost 1 while the blue one is closer to 3. This
is the signature that SDS impurities alone cannot explain the quantitative mismatch between
our data and our model fed with the nominal concentrations of DOH. Seeing the dependency of
this slope with the concentration of SDS, this might be a problem caused by a flawed value for
*K*, the exchange coefficient between the micellar and monomeric bulk phases. Another possible
explanation could be direct DOH exchanges between the SDS micelles and the interface.
than 1, with the red one, close to the CMC being almost 1 while the blue one is closer to 3. This
is the signature that SDS impurities alone cannot explain the quantitative mismatch between
our data and our model fed with the nominal concentrations of DOH. Seeing the dependency of
this slope with the concentration of SDS, this might be a problem caused by a flawed value for
*K*, the exchange coefficient between the micellar and monomeric bulk phases. Another possible
explanation could be direct DOH exchanges between the SDS micelles and the interface.
1.4 Conclusions of Chapter 1

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

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

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

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

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

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