4.1. Impact of the relative humidity of air

To rationalise the increase in film stability induced by the presence of salt, we make 10 films at the four relative humidities studied and measure their thicknesses. We observe that all thinning curves collapse, proving good reproducibility. This motivates the use of only the longest film from each experimental condition for detailed analysis.

Figure 3( $a$ ) shows the evolution of the film thickness corresponding to the most stable films measured at each relative humidity studied in the salt-free system. We observe that, aside from a slight horizontal shift of the curve at $\mathcal{R}_h = 40\,\%$ (blue) likely due to an offset in the zero reference of the translation stage, all curves closely follow the same dynamics down to a thickness of 400 nm. As shown in the inset of figure 3( $a$ ) (log-lin scale), the curves start to deviate significantly from one another below 400 nm. We observe that, at such small thicknesses, a higher relative humidity leads to a slower thinning rate of the film. As a consequence, at a given generation time $t$ , films formed at higher humidity are systematically thicker.

This observation is in good agreement with the previous work by Champougny et al. (Reference Champougny, Henaff, Restagno, Boulogne and Rio2018). The authors considered that drainage and evaporation are decorrelated. Under saturated vapour, where thinning is governed exclusively by drainage, Champougny et al. (Reference Champougny, Henaff, Restagno, Boulogne and Rio2018) observed that the drainage rate decreases over time. Under non-saturated conditions ( $\mathcal{R}_h \lt$ 100 $\%$ ), evaporation also contributes to thinning, but its effect becomes significant only for film thicknesses under $h=400$ nm. This does not necessarily mean that evaporation dominates over drainage in this regime, but that we need to consider both. Eventually, the films rupture as soon as their thickness reaches 10 nm, indicating that no stable CBF or NBF form under these conditions.

Figure 3( $b$ ) shows four thinning curves corresponding to the most stable films at each tested relative humidity, for the system with added salt. As in the salt-free case, all four curves collapse onto a common curve down to 400 nm. Below this thickness, the curves begin to differ significantly. Thus, the drainage-dominated regime above 400 is followed by a regime in which evaporation plays an important role. The inset in figure 3( $b$ ) (log-lin scale) highlights the [ $0, 400$ ] nm range, where we observe local thickness variations. These are likely due to the emergence of localised thick patches – an effect previously reported by Aurégan & Deike (Reference Aurégan and Deike2025). In figure 3 ( $b$ ), we also observe the presence of a thickness plateau at all the studied humidities. The corresponding thickness on this plateau is constant at approximately 5 nm independent of the relative humidity. The higher the humidity, the longer the plateau. We identify this plateau as corresponding to the formation of a stable NBF. Indeed, in the literature, it is demonstrated that such films, stabilised by a steric repulsion between the surfactant heads at the interfaces, have a thickness between 5 and 10 nm (Schulze-Schlarmann et al. Reference Schulze-Schlarmann, Buchavzov and Stubenrauch2006).

The thickness at the plateau $h_p = 5$ nm is measured by microinterferometry using a single-layer model with two reflecting interfaces, analogous to a Fabry–Pérot cavity (Ziapkoff et al. Reference Ziapkoff, Boulogne, Salonen and Rio2026). This model does not account for the surfactant layer. However, at such thicknesses, the contribution of the surfactant layer cannot be neglected. We can use multilayer models instead to refine the description of the film layer (Mysels et al. Reference Mysels, Shinoda and Frankel1959). The three-layer model introduced by Duyvis (Reference Duyvis1962) considers an aqueous core of thickness $h^\dagger$ including an ammonium TTAB head sandwiched between tetradecane tails of thickness $h_1$ and yields a total layer thickness $h_{\textit{tot}}$ . According to this model, the calculation of $h_{\textit{tot}}$ and $h^\dagger$ are given by

(4.1) \begin{equation} h_{\textit{tot}} = 2h_1 + h^\dagger ; \quad h^\dagger = h_p - 2h_1 \frac {n_1^2 - 1}{n_2^2 - 1}, \end{equation}

where $n_1 = 1.4290$ is the refractive index of pure tetradecane (Lide Reference Lide1993) and $n_2$ the refractive index of the aqueous core, assumed here as $n_2 = ({1}/{m}) \sum\nolimits _{i=1}^{m} n(\lambda _i,C_s^\circ )$ , with $m = 1106$ the number of discrete wavelength values in the experimental spectrum where $\lambda \in [450,800]$ nm. The linear tetradecane chains oriented perpendicular to the interface yield a thickness $h_1$ of 1.7 nm (Simister et al. Reference Simister, Lee, Thomas and Penfold1992). Using these values, the actual thickness of the core at any given relative humidity $\mathcal{R}_h$ is $h^\dagger \approx 0.52$ nm, consistent with neutron scattering measurements reported by Simister et al. (Reference Simister, Lee, Thomas and Penfold1992). Note that we have neglected the variation of the refractive index with the salt concentration. We estimate an increase of $n_2$ of approximately 3 % at saturation (from Li et al. (Reference Li, Liu, Zhao and Tan2015)), in which experiments are performed on salty water, in the absence of surfactant. Taking this into account would lead to $h^\dagger =1.13$ nm. Note that this error, which increases over time, will also impact the values of $h$ in figure 3(b) by at most 3 %.

4.2. Impact of salt

From the thinning curves presented in figure 3, a direct comparison can be made for the liquid composition at fixed relative humidity $\mathcal{R}_h$ . Figure 4 shows the thinning dynamics with and without salt at $\mathcal{R}_h = 40\,\%$ , 60 %, 80 % and 100 % in subfigures ( $a$ ), ( $b$ ), ( $c$ ) and ( $d$ ), respectively. Each figure includes an inset representing the data in log-lin scale.

In figure 4, for each tested humidity, the thinning curves with or without salt follow the same dynamics down to 100 nm, in agreement with results reported previously by Aurégan & Deike (Reference Aurégan and Deike2025). In contrast, when $h \lt 100$ nm, the presence of salt tends to slow down thinning and to stabilise the NBF.

Figure 4.

Thinning curves of films with ( $\square$ ) and without ( $\blacktriangle$ ) 32.5 g L $^{-1}$ NaCl for the four fixed relative humidities: ( $a$ ) $\mathcal{R}_h = 40\,\%$ , ( $b$ ) $\mathcal{R}_h =60\,\%$ , ( $c$ ) $\mathcal{R}_h =80\,\%$ and ( $d$ ) $\mathcal{R}_h =100\,\%$ . The $\blacktriangle$ markers for films without salt are outlined in black for clarity. Insets show the data in log-lin scale.

5.1. Film rupture stochasticity

The stochastic nature of film rupture has been explored in different contexts, including public studies of horizontal film lifetimes sealed in bamboo cylinders by Tobin et al. (Reference Tobin, Meagher, Bulfin, Möbius and Hutzler2011), experimental studies on two-dimensional foams by Forel et al. (Reference Forel, Dollet, Langevin and Rio2019), droplet coalescence simulations by Perumanath et al. (Reference Perumanath, Borg, Chubynsky, Sprittles and Reese2019) and more recently, thin-film pressure balance (TFPB) measurements by Chatzigiannakis & Vermant (Reference Chatzigiannakis and Vermant2020). However, the dynamics of vertical film rupture remains underexplored.

Given the data shown in § 3, we quantify the degree of stochasticity in vertical film rupture by following the approach of Shaw (Reference Shaw1984). Specifically, we represent the difference of lifetime of consecutive films through return maps, plotting the interval

(5.1) \begin{equation} T_{i+1} = \tau _{f}^{i+2} - \tau _{f}^{i+1} \end{equation}

between the lifetimes of films $i+2$ and $i+1$ as a function of the interval $T_{i}$ between the lifetimes of films $i+1$ and $i$ . If rupture events, considered here as independent, are strongly correlated, characteristic patterns should emerge in these maps. Conversely, if rupture events are independent, the resulting cloud of points should be homogeneously distributed and centred, reflecting a stochastic dynamics.

Figures 5( $a$ ) and 5( $b$ ) display return maps for films with or without salt at four different relative humidities, $\mathcal{R}_h \in [40,60,80,100] \ \%$ , using the same colour code as in figure 2. To quantify any privileged angular directions from the cloud data points, we construct a new variable $z_i(\theta )$ defined as

(5.2) \begin{equation} z_i(\theta ) = \big (T_{i} - \tilde {T}\big )\cos \theta + \big (T_{i+1} - \tilde {T}\big )\sin \theta , \end{equation}

with $i\in [0, N-1]$ and $\theta \in [0,2\pi ]$ and where the mean value is defined as

(5.3) \begin{equation} \tilde {T} = \frac {1}{N}\sum _{i=1}^{N} T_{i}. \end{equation}

Figures 5( $c$ ) and 5( $d$ ) then present the variance

(5.4) \begin{equation} \langle z(\theta )^2\rangle = \frac {1}{N}\sum _{i=1}^{N} z_i(\theta )^2, \end{equation}

plotted in polar coordinates as a function of $\theta$ . For clarity, angles are expressed in degrees in these polar representations.

The results in figures 5(c) and 5(d) indicate that, for both systems and at all relative humidities, the variance is consistently more pronounced along the $315^{\circ }$ – $135^{\circ }$ direction, with the amplitude of the variance lobes increasing with humidity. This trend reflects a negative correlation between rupture events. When $T_{i}$ is larger than average, $T_{i+1}$ tends to be smaller, and $\textit {vice versa}$ . In other words, long films are statistically more likely to be followed by shorter ones. Salt addition does not significantly alter this behaviour. This robust feature suggests that film rupture is neither purely stochastic nor fully deterministic, but rather exhibits an intermediate dynamics.

Figure 5.

( $a$ ) Experimental return maps for films without salt at four relative humidities $\mathcal{R}_h \in [40, 60, 80, 100]\ \%$ . ( $b$ ) Experimental return maps for films with 32.5 g $.$ L $^{-1}$ sodium chloride addition under identical humidity conditions. Polar representation of the variance $\langle z_i(\theta )^2\rangle$ as a function of the angle $\theta$ for films (c) without salt and (d) with salt addition, at relative humidities $\mathcal{R}_h \in [40, 60, 80, 100]\ \%$ . All figures use the same colour code as in figure 2.

5.2. Film evaporation without salt

In this section, we provide a quantitative description of the film thinning dynamics over the entire film lifetime, under all humidity conditions, to rationalise the results reported in § 3. We consider here only salt-free films. As shown in figure 3 $(a)$ , small offsets are present in the experimental data, which we attribute to slight differences in the initial positioning of the translation stage. We correct this artefact by shifting each dataset in time, as explained in the following.

We observe that the early stages of film thinning are mostly governed by drainage and do not depend on humidity. This leads us to take the experimental data obtained in a saturated environment, displayed as red triangles, as a reference. We minimise the least-squares error between the $t(h)$ curves and the reference curve, restricted to the interval $h \in [1000, 3000]$ nm, in the early stage of the film dynamics and with a sufficient number of data points to determine the optimal temporal shifts $t_0$ . The corresponding temporal corrections are smaller than one second. The resulting shifted datasets are shown in figure 6, using triangular symbols and the same colour code as in the previous sections.

Figure 6.

Time shifted evolution of film thickness for the TTAB film shown in figure 3 $(a)$ at relative humidity $\mathcal{R}_h = 40\,\%, 60\,\%, 80\,\%, 100\,\%$ without salt. The respective time shifts $t_0$ at $\mathcal{R}_h = 40\,\%, 60\,\%, 80\,\%$ are 0.764, 0.047 and 0.166 s. The black line represents the computation of (5.5) at saturation, whereas dashed lines represent the computation at relative humidity $\mathcal{R}_h = 40\,\%, 60\,\%, 80\,\%$ . The fitting values of the prefactor $a$ are $\{0.74,0.67,0.55\}$ and the evaporation rate $j_{\textit{e}}$ is $[21, 13, 5]$ nm s $^{-1}$ , for the three different humidities. Inset shows the models and experimental data in log-lin scale.

To determine theoretically the evolution of the thickness in time $h(t)$ , we use the approach by Pasquet et al. (Reference Pasquet, Boulogne, Restagno and Rio2024) and by Champougny et al. (Reference Champougny, Henaff, Restagno, Boulogne and Rio2018). In the following, we make the hypothesis that the film thickness is uniform. The slope of each curve in figure 6 writes

(5.5) \begin{equation} \left .\frac {\mathrm{d} h}{\mathrm{d} t}\right |_{\mathcal{R}_h,\, \mathcal{T}} = -\,2 j_{\textit{e}} - j_{\textit{d}}, \end{equation}

where $\mathcal{T}$ is the film temperature, assumed to be equal to the room temperature, $j_{\textit{e}}$ is the water evaporation rate from the film, the factor 2 comes from the presence of two interfaces and $j_{\textit{d}}$ is the drainage rate inside the film, supposed independent of the atmospheric humidity. Both contributions are defined as positive. By definition, in the saturated experimental data, the evaporation rate $j_{\textit{e}} = 0$ and $\left . ({\mathrm{d} h}/{\mathrm{d} t})\right |_{\mathcal{R}_h = 100\ \%,\, \mathcal{T}} = - j_{\textit{d}}$ .

To extract the drainage curves $j_{\textit{d}}(h)$ , we first smooth the experimental data obtained in a saturated atmosphere with a third-order Savitzky–Golay filter (Savitzky & Golay Reference Savitzky and Golay1964) with a window size of 11 points and derive the temporal data $h(t)$ . The derivation, noted $j_{\textit{d}}(t)$ , is smoothed with the same procedure, and an interpolation gives us the drainage rate $j_{\textit{d}}(h)$ . We numerically solve (5.5) with the function odeint from the scipy.integrate package and the result is displayed in figure 6 as a black line. We restrain our computation to $h\gt 5$ nm because we do not take into account the disjoining pressure, which is important at these length scales. As expected, the numerical solution accurately follows the experimental data.

To perform this analysis for data at any relative humidity $\mathcal{R}_h$ , following (5.5), we need to determine the evaporation rate $j_{\textit{e}}(h)$ for any thickness. We use the framework established by Schmidt & Beckmann (Reference Schmidt and Beckmann1930), Dollet & Boulogne (Reference Dollet and Boulogne2017), Boulogne & Dollet (Reference Boulogne and Dollet2018) and Pasquet et al. (Reference Pasquet, Boulogne, Restagno and Rio2024) to obtain an expression for $j_{\textit{e}}(h)$ . In particular, Boulogne & Dollet (Reference Boulogne and Dollet2018) demonstrated that natural convection evaporation significantly affects vertical aqueous films of a few centimetres or more, a criterion that we meet (§ 3). This observation is also supported by the direct calculation of the Grashof number

(5.6) \begin{equation} \textit{Gr} = \frac {g}{\nu ^2} \frac {\rho _\infty - \rho _s}{\rho _\infty } \mathcal{L}^3, \end{equation}

where $g = 9.81$ ms $^{-2}$ is the gravitational acceleration, $\nu = 1.5 \times 10^{-5}$ m $^{2}$ s $^{-1}$ the kinematic viscosity of water vapour in air, $\rho _s$ and $\rho _\infty = \rho _s\mathcal{R}_h$ , the air densities at saturation and far from the evaporating surface, respectively, and $\mathcal{L}$ the characteristic vertical length scale of the evaporating film, which we take equal to $20$ mm for the calculation. The value of $\textit{Gr}$ quantifies the relative importance of buoyancy-driven flow. Large $\textit{Gr}$ values indicate that natural convection dominates the evaporative transport. Following Jones (Reference Jones1978) and Boulogne & Dollet (Reference Boulogne and Dollet2018), we compute $\rho _s$ and $\rho _\infty$ at any relative humidity. Across all experimental conditions, we consistently find $\textit{Gr} \gg 1$ (typically $\textit{Gr} \gt 200$ ), confirming that buoyancy-driven convection is the dominant mechanism.

Accordingly, the evaporative rate of a film can be approximated as

(5.7) \begin{equation} j_{\textit{e}} = a \frac { \hat {c}_0 \mathcal{D}\left (c_{\textit{sat}}-c_{\infty }\right )}{\rho _w} \frac {\textit{Gr}^{1 / 4}}{\mathcal{L}}, \end{equation}

where $\hat {c}_0 = 0.478$ is a prefactor arising from a numerical solution (Boulogne & Dollet Reference Boulogne and Dollet2018), $\mathcal{D} = 2.44 \times 10^{-5}$ m $^{2}$ s $^{-1}$ is the diffusion coefficient of water vapour in air, $c_{\textit{sat}}$ and $c_\infty =c_{\textit{sat}}\mathcal{R}_h$ are the mass concentrations of saturated vapour and ambient water vapour, respectively, and $\rho _w = 1000$ kgm $^{-3}$ is the water density. At $\mathcal{T} = 20^{\ \circ }$ C, $c_{\textit{sat}} \approx 17$ gm $^{-3}$ for water. The prefactor $a$ will be used as a fit parameter.

We solve equation (5.5) (Pasquet et al. Reference Pasquet, Boulogne, Restagno and Rio2024) using the explicit expression of $j_e$ (Boulogne & Dollet Reference Boulogne and Dollet2018) given by (5.7). We start all computations at $t=t_0$ and $h=3.2\ {\unicode{x03BC}}$ m, which is the first thickness detected in the reference data. We numerically solve equation (5.5) with the function odeint from the scipy.integrate package. The result is displayed in figure 6 as dashed black lines for relative humidities $\mathcal{R}_h \in [40, 60, 80]\,\%$ and captures the experimental thinning dynamics. Indeed, the values of $j_{\textit{e}} \in [21, 13, 5]\,\mathrm{nm\,s^{-1}}$ are comparable to the values $j_{\textit{e}} \in [16, 8, 3.5]\,\mathrm{nm \, s^{-1}}$ reported by Champougny et al. (Reference Champougny, Henaff, Restagno, Boulogne and Rio2018).

The fitting parameter $a$ accounts for deviations from the idealised model, such as convection, thermal exchange, complex geometry (including the presence of the bath), the variation of $\mathcal{L}$ and Gr in time and the presence of concentration and temperature gradients. Notably, $a$ (see table 2) remains of order 1, indicating that our expression of $j_e$ captures the dominant physical mechanisms.

Table 2.

Summary of the fitted values of parameters $a$ (5.7) and $a'$ (5.10).

5.3. Film evaporation with salt

To rationalise the results presented in figure 3 $(b)$ , we follow the same approach as in § 5.2. As observed in figure 4, salt has no significant effect on drainage. We also previously argued that drainage also does not depend on humidity. Consequently, to mitigate potential offset on experimental data, we shift by $t_0$ all datasets in time with respect to the same reference–the red triangle displayed in figure 6–using the same least-squares error minimisation than in § 5.2. The corresponding temporal corrections for all humidity conditions tested are smaller than one second. Figure 7 $(a)$ shows the shifted datasets using square symbols and the same colour as in the previous sections. The inset is the log-lin representation.

Figure 7.

$(a)$ Time shifted evolution of the film thicknesses for TTAB films with salt shown in figure 3 $(b)$ . The dashed lines represent the computation of $h(t)$ at the four relative humidities with the function odeint for (5.5) and (5.8). The respective time shifts $t_0$ are 0.672, 0.669, 0.105 and 0.672 s and the values of the prefactor $a'$ are $\{1,0.7,0.7,0.7\}$ . Insets show the models and experimental data in log-lin scale. $(b)$ Numerical solutions of the salt concentration from (5.5) and (5.8) for salt concentration $C_s$ with relative humidity $\mathcal{R}_h = 40\,\%, 60\,\%, 80\,\%, 100\,\%$ . Coloured crosses correspond to $t^\star$ such as $h^\star (t^\star ) = 100$ nm, where the corresponding salt concentrations are $C_s(t^\star )$ . The black straight line represents the saturation of salt in water.

The difficulty in applying the model developed in § 5.2 arises because, as the film thins by evaporation, the salt concentration $C_s$ increases and, in turn, affects the evaporation described in (5.7). This coupled effect results from the dependence of the parameters $c_{\textit{sat}}(C_s)$ and $\textit{Gr}(C_s)$ on the salt concentration. Additionally, it is well established in the literature (O’Brien Reference O’Brien1948; Kou Reference Kou1999; Generous et al. Reference Generous, Qasem, Qureshi and Zubair2020; López-Borrell et al. Reference López-Borrell, Lora-García, Cardona, López-Pérez and Fombuena2024) that the presence of salt affects the humidity fixed point. For instance, at $C_s = 32.5\ \mathrm{g\,L^{-1}}$ , the liquid–vapour equilibrium is reached at $\mathcal{R}_h = 98\ \%$ , while at saturation ( $C_s = 360\ \mathrm{g\,L^{-1}}$ at $\mathcal{T} = 20\,^\circ \mathrm{C}$ ), the equilibrium humidity drops to $\mathcal{R}_h \approx 78\ \%$ . A full modelling of the film dynamics requires an explicit description of the temporal evolution of the salt concentration over time.

Applying a mass balance between a time $t$ and a time $t + {\mathrm{d}}t$ over an infinitesimal film element of length ${\mathrm{d}} z$ both in the absence of evaporation and with evaporation leads us to the equation for the evolution of the salt concentration $C_s$ in the film

(5.8) \begin{equation} \left .\frac {\text{d} C_s}{\mathrm{d} t}\right |_{\mathcal{R}_h,\, \mathcal{T}} = 2j_{\textit{e}}\frac {C_s}{h}. \end{equation}

Using the ideal gas law for water vapour in air, we express the saturated vapour mass concentration as

(5.9) \begin{equation} c_{\textit{sat}}(C_s) = c_{\textit{sat}}\frac {P_{\textit{sat}}(C_s)}{P_{\textit{sat}}(0)}, \end{equation}

where $P_{\textit{sat}}(C_s)$ is the saturated vapour pressure at salt concentration $C_s$ , and $P_{\textit{sat}}(0) = 2340$ Pa the saturated vapour pressure of pure water. To obtain $P_{\textit{sat}}(C_s)$ at any given $C_s$ , we interpolate the data of López-Borrell et al. (Reference López-Borrell, Lora-García, Cardona, López-Pérez and Fombuena2024). As shown by (5.6), $\mathcal{L}$ is the only variable of the Grashoff number, and we take it equal to 20 mm as in § 5.2. The resulting new expression for $j_{\textit{e}}$ is

(5.10) \begin{equation} j_{\textit{e}} = a' \hat {c}_0 \frac {\mathcal{D}c_{\textit{sat}}\left (\frac {P_{\textit{sat}}(C_s)}{P_{\textit{sat}}(0)}-\mathcal{R}_h\right )}{\rho _w} \frac {\textit{Gr}^{1 / 4}}{\mathcal{L}}, \end{equation}

with $\mathcal{D}$ and $\rho _w$ the same values as in equation (5.7) and $a'$ the new fit parameter. As in the previous section, we numerically solve equations (5.5) and (5.8) down to $h\gt 5$ nm, with the same expression of $j_{\textit{d}}$ but with the expression of $j_{\textit{e}}$ given by equation (5.10) using the function odeint from the scipy.integrate package. The results for $h(t)$ are displayed as dashed black lines for the different relative humidities in figure 7 $(a)$ and the values of the fitting parameter $a'$ are given in table 2.

This new model accurately captures the experimental data for most of its duration, down to $h =100$ nm in all humidity conditions. Note that the thinning curves with or without salt are very similar, which means that the effect of salt on evaporation is negligible down to such thicknesses. Nevertheless, it allows us to estimate the salt concentration over time. The variation $C_s(t)$ for each humidity condition is displayed in figure 7 $(b)$ as coloured lines for the different relative humidities. The black horizontal line represents the saturation of salt in water. We identify the time $t^\star$ at which the thickness reaches $h^\star =100 \ \mathrm{nm}$ . The corresponding salt concentrations for relative humidities $\mathcal{R}_h \in [40, 60, 80, 100]\,\%$ are $C_s(t^\star ) \in \{150,73,58,31\}\ \mathrm{g\,L^{-1}}$ . These points are displayed as coloured crosses in figure 7 $(b)$ . Since the model is no longer valid below these thicknesses, we cannot predict what happens in these concentrated thin films.

Below a thickness of $100$ nm, the model predicts a more rapid thinning than observed in the experiments. At such high salt concentrations, Aurégan & Deike (Reference Aurégan and Deike2025) observed instabilities in the film thickness. The thinning slowdown noticed in experimental data below $h^{\star } =100$ nm could be attributed to those instabilities, or to a complex coupling between drainage and evaporation.

Our conclusion is that neither the drainage nor the evaporation rates are measurably affected by the addition of salt, despite the strong increase in salt concentration. The impact of salt only appears at smaller thicknesses and is threefold: (i) the thinning rate is smaller than predicted by the model, (ii) some thickening regions are observed and (iii) the NBF is more stable. Among these three effects, the one that controls the film rupture seems to be the stabilisation of the NBF, which we quantify in the following. From our model, we extract the salt concentration over time and we show that it cannot explain the evolution of the thinning curves.

5.4. Film rupture

As illustrated in figure 8 $(a)$ , in addition to the film lifetime $\tau _f$ determined in figure 2, two characteristic times can be extracted from each thinning curve: the thinning lifetime $\tau _t$ , and the plateau lifetime $\tau _p$ . Here, $\tau _t$ is determined by plotting the thickness $1/h$ against $t$ as shown in the inset of figure 8 $(a)$ in log-lin scale. In this example, $\tau _t = 22.5$ s corresponds to the jump in $1/h$ . If the identification of this time is not as obvious as in this example, we take the intersection between a horizontal line at the mean plateau $1/h_p$ and an affine regression, $1/h = pt+q$ with $p$ and $q$ constants, fitted to the data in the range $h \in [50, 200]$ nm. The plateau lifetime $\tau _p$ is defined as the difference $\tau _f - \tau _t$ .

By extracting these two characteristic times for each set of 10 films, together with the film lifetime $\tau _f$ (figure 2), we plot in figure 8 $(b)$ the mean film lifetime $\langle \tau _f\rangle$ as a function of the mean thinning time $\langle \tau _t\rangle$ . The circle’s colour code is the same as in figure 2 for relative humidity. The blue and red crosses present salt-free and salt containing systems, respectively. The $\langle \tau _f\rangle = \langle \tau _t\rangle$ line is included in black. The collapse of all blue points onto this line indicates that film rupture occurs immediately after thinning ceases, implying that no stable plateau forms without salt. On the other hand, the red crosses are above the $\langle \tau _f\rangle = \langle \tau _t\rangle$ line, suggesting that film rupture occurs during the plateau phase in salted systems.

Figure 8.

( $a$ ) Illustration of the three characteristic times, the film lifetime $\tau _f$ , the thinning lifetime $\tau _t$ and the plateau lifetime $\tau _p$ for a thinning curve obtained with $C_s^\circ =$ 32.5 g L $^{-1}$ NaCl at a relative humidity of $\mathcal{R}_h = 80\ \%$ . The inset shows the determination of $\tau _t$ from the plot of $1/h$ versus $t$ (log-lin scale). ( $b$ ) Mean film lifetime, $\langle \tau _f \rangle$ , as a function of the mean thinning time, $\langle \tau _t \rangle$ , for all relative humidity conditions tested using the same colour code as previous sections, either with ( ) or without ( ) 32.5 g L $^{-1}$ NaCl. The solid black line corresponds to $\langle \tau _f\rangle = \langle \tau _t\rangle$ .

Previous TFPB studies (Exerowa et al. Reference Exerowa, Nikolov and Zacharieva1981) have shown that a NBF can be stabilised above a critical salt concentration $C_{\textit{el,cr}}$ . For films made with TTAB, Schulze-Schlarmann et al. (Reference Schulze-Schlarmann, Buchavzov and Stubenrauch2006) reported a critical sodium chloride concentration $C_{\textit{el,cr}} = 23.4$ g L $^{-1}$ that is lower than our initial salt concentration of $C_s^\circ = 32.5$ g L $^{-1}$ . The plateau is thus probably stabilised by the same process. A surprising feature is that no CBF is stabilised in the absence of salt. This is unexpected as, in the absence of salt, the screening is much smaller and an electrostatic repulsion between the interfaces should appear (Bergeron Reference Bergeron1997). The absence of CBF stabilisation could be attributed to the dynamics of the process. However, further investigation would be necessary to clarify the mechanism. For example, systematic measurements in a thin-film pressure balance, controlling the pressure jump applied on the thin film could give insights on the transition between CBF and NBF.

Figure 9.

Thinning curves of films in figure 4 $(c)$ with the addition of the thinning curve at $\mathcal{R}_h = 80\,\%$ with initial concentration $C_s^\circ \in [3.25,65,97.5]$ g L $^{-1}$ represented using a blue-scale gradient indicating concentration. Inset shows the data in log-lin scale.

Figure 10.

Mean film thickness $\langle h_p \rangle$ averaged over the plateau once local thickening is no longer observed in the thinning curves in figures 4 and 9. Here, $\sigma = 1$ nm is the uncertainty estimated on each measurement. The blue-scale gradient indicates the salt concentration.

We performed complementary measurements at $\mathcal{R}_h = 80\,\%$ with different initial concentrations $C_s^\circ$ . Figure 9 shows the results of figure 4 $(c)$ together with initial concentration $C_s^\circ \in [3.25,65,97.5]$ g L $^{-1}$ represented using a blue-scale gradient indicating concentration. We note that for $C_s^\circ \in [65,97.5]$ g L $^{-1}$ , most films attain the 81 mm limit of the translation stage. Thus, in figure 9 the thinning curves do not correspond to the longer film as in § 4, but rather to the longest film attained within this limit.

The observed thickness, about 5 nm corresponds to the typical thickness of NBF as observed in the TFPB Exerowa et al. (Reference Exerowa, Nikolov and Zacharieva1981). Except without added salt, an NBF forms at all concentrations, even below $C_{\textit{el,cr}}$ . This could be attributed to the increase of salt concentration over time, leading to a concentration higher than $C_{\textit{el,cr}}$ on the plateau. In figure 10, the thickness on the plateau $\left \langle h_p \right \rangle$ measured in each experiment is plotted as a function of atmospheric humidity. $\left \langle h_p \right \rangle$ is obtained by averaging the thickness on the plateau, at the later times, once local thickening is no longer observed. The uncertainty $\sigma$ is determined the error bar measured on each measure, before averaging. We note that the thickness plateau is constant in all experiments at approximately 5 $ \pm\, 1$ nm.