2.1. Experimental set-up

Figure 2(a) shows the schematic diagram for the experimental set-up, similar to Ruth et al. (Reference Ruth, Aiyer, Rivière, Perrard and Deike2022). It consists of a polycarbonate tank filled with approximately 150 l of solution, and a hollow hemispherical metal cup is positioned underwater at the tank centre, equidistant from the four corners of the tank where four pumps (Rule 20 DA 800 GPH Bilge Pump) are mounted. The cup initially faces downwards and is filled with a constant volume $V_0 = 20$ ml of air. At the beginning of an experiment, the cup is rotated to face upwards while a fast camera (Phantom VEO4K-990-L, 500 fps) is simultaneously triggered.

Figure 2.

(a) Schematic of the experimental set-up: air is injected into an upside down cup that is flipped at the beginning of the experiment, and a large air cavity is released into the turbulent flow. (b) Temporal evolution of the total bubble count $N$ for different SDS concentrations in deionised (DI) water. (c) Example of images comparing clean (light blue, bottom), and contaminated cases (dark blue, top, 2000 µmol l⁻¹ of SDS). The shaded area ( $2.5T_{\textit{int}}\sim 6T_{\textit{int}}$ ) represents the time range used to compute distributions in the following.

The air cavity is released into the turbulent flow and experiences a sequence of breakups which results in a broad distribution of bubble sizes. Such experimental configuration is designed to characterise the transient fragmentation process of a large air cavity into a distribution of bubbles of various sizes. The high-speed camera is coupled with two different lenses to improve the resolution using multiple views: a large-scale view (Nikon Micro 200 mm leading to 35 µm/pixel) and a zoomed-in view (K2 DistaMax Infinity Photo-Optical Co. lens leading to 9 µm/pixel). The field of view is illuminated by a coaxial LED panel from the opposite side. To ensure statistical convergence of the size distribution, each experimental condition is repeated 15 times using the 200 mm lens and 25 times using the K2 lens. The depth of field $\delta (d)$ was calibrated by translating a target containing 14 chrome-sputtered dots with diameters ranging from 30 to 3000 $\unicode{x03BC}$ m along the optical axis, similarly to Erinin et al. (Reference Erinin, Néel, Mazzatenta, Duncan and Deike2023b ). The large-scale field of view is processed with a two step algorithm: thresholding for larger bubbles then edge detection on an image where large bubbles have been removed to detect the smaller ones. The small-scale field of view is processed using thresholding followed by a filter on the sharpness of the detected objects, to only keep well focused bubbles. We therefore obtain the bubble size distribution $\mathcal{N}(d)$ , defined as the number of bubbles per unit bin size $[d, d+\Delta d]$ , normalised by the cross-sectional area $A$ of the camera and the depth of field at that given size bin $\delta (d)$ , and averaged over all realisations. The distributions from the two camera views are combined into a single distribution (see Mazzatenta et al. Reference Mazzatenta, Erinin, Néel and Deike2025). The uncertainties in the distribution are estimated by a Monte Carlo approach in which data are randomly sampled from the repeated experiments using bootstrapping (100 realisations).

2.2. Surfactant and salt conditions

There are five different sets of solutions in this study. We used three different surfactants: SDS, CTAB and Tween 20 used as purchased and mixed with DI water. We also repeated experiments with SDS and Tween 20, adding a mixture of salts representative of salts present in the ocean (ASTM D1141-98 Artificial Sea Salt, the same as in Mazzatenta et al. Reference Mazzatenta, Erinin, Néel and Deike2025). The case with SDS was repeated a second time to ensure robustness. For each experimental case, the solution’s surface tension isotherm was measured at the beginning and end of the experiment using a Langmuir trough (KSV NIMA 1003), and the static surface tension prior to surface compression was recorded as $\gamma _s$ . Surfactants concentrations were chosen such that a given set of experiments covers the whole range from clean interfaces to fully contaminated (concentration close to the critical micelle concentration (CMC) or surface tension close to 40 mN m⁻¹). During all experiments, the tank was closed with an acrylic lid to minimise contamination from ambient air. Experimental conditions are summed up in table 1.

Table 1.

Summary of surfactant and solvent conditions tested in the study, with the corresponding evolution of the measured static surface tension $\gamma _s$ , the Hinze scale $d_H$ computed using (1.1) and the Sauter mean diameter of the measured distributions $d_{32}$ . Concentrations are given as a fraction of the CMC: 8.2 mmol ^l−1 for SDS, and 59 µmol l⁻¹ for Tween 20 and 1 mmol l⁻¹ for CTAB.

Table 1. Long description

A table with five rows and seven columns comparing different surfactant and solvent conditions. The columns are labeled Surfactant, Sea salt, Concentration (% CMC), γs (mN m-1), dH (mm), and d32 (mm). The rows present data for different combinations of surfactants (SDS, Tween-20, CTAB) with and without sea salt. Each row lists the surfactant type, presence of sea salt, concentration range as a percentage of the critical micelle concentration (CMC), static surface tension (γs) range in mN m-1, Hinze scale (dH) range in millimeters, and Sauter mean diameter (d32) range in millimeters. The table provides specific values for each combination, showing how these parameters vary with different experimental conditions.

Figure 2(b) presents the temporal evolution of the bubble count, $N = \int \mathcal{N}(d)\mathrm{d}d$ , for experiments conducted with varying concentrations of SDS in DI water. Across all experimental conditions, $N$ exhibits a sharp increase immediately after the cup is flipped ( $t=0$ ), followed by a quasi-steady plateau over the interval $2.5T_{\textit{int}} \leqslant t \leqslant 6T_{\textit{int}}$ . The total number of recorded bubbles increases with the concentration of SDS by almost an order of magnitude. Figure 2(c) compares images of the clean case (no surfactants, bottom) and a contaminated case (SDS = 2000 µmol l⁻¹, top) at $t = 4T_{\textit{int}}$ . The scale bar represents 1 cm. A substantial increase in the number of small bubbles is observed in the contaminated case, whereas the population of large bubbles remains nearly unchanged. Similar observations are made when considering the other types of solutions in terms of time evolution (shown in Appendix). Bubble coalescence in the present experiment is expected to be negligible given the low volume fraction of air, as discussed in experiments (Ruth et al. Reference Ruth, Aiyer, Rivière, Perrard and Deike2022) and in simulations (Rivière et al. Reference Rivière, Mostert, Perrard and Deike2021, Reference Rivière, Ruth, Mostert, Deike and Perrard2022).

We note that the experiment is transient by design as a large cavity is released and breaks due to turbulence. In our experiment, we observe bubbles generated over a relatively short window after the release (approximately 1 s, or $10 T_{\textit{int}}$ ). During that time, bubbles rise to the surface and break: we observe an initial increase in the number of bubbles which then plateaus for some time. This number would then decrease at much longer times if we were continuing to record as all bubbles move out of the experimental field of view. As we analyse the influence of surfactant on bubble breakup, we will compare the adsorption time with the time in between the breakups in the sequence, as new area is constantly being created through successive breakups.

2.3. Turbulent properties

We measured the turbulent properties of the flow using particle image velocimetry (PIV) in the central plane of the experiment, just over the cup with a field of view slightly larger than the one used in the experiment. The set-up, processing and analyses are very similar to the ones described in Ruth et al. (Reference Ruth, Vernet, Perrard and Deike2021). We recorded the flow seeded with tracer particles for 2.5 s with a sampling rate of 1000 frames per second. The data are processed using a standard library for PIV analysis. The mean flow is similar to the one used in Ruth et al. (Reference Ruth, Aiyer, Rivière, Perrard and Deike2022): a small downwards flow just above the cup due to the jets of the pumps meeting in the centre (figure 3 a). We estimated the turbulent properties of the flow using the scaling argument in homogeneous and isotropic turbulence (De Jong et al. Reference De Jong, Cao, Woodward, Salazar, Collins and Meng2009): $\varepsilon =AU_{\textit{rms}}^3/L_{\textit{int}}$ with $U_{\textit{rms}}$ the root-mean-square of the velocity fluctuations and $L_{\textit{int}}$ the integral length scale, using $A=0.7$ (Sreenivasan Reference Sreenivasan1998). We compute the integral length scale as the maximum of the integral of the autocorrelation of the velocity fluctuations, and average between both directions. Since there is some inhomogeneity in the flow structure, we computed the integral length scale and root-mean-square velocity locally, giving us a local dissipation rate. The resulting integral length scale (cyan lines) and Hinze scale (black circles, using (1.1)) are plotted in figure 3(b) at several locations in space. The integral length scale is $L_{\textit{int}} =2.5$ cm (10 %: 2.2, 90 %: 2.8), the root-mean-square velocity is $U_{\textit{rms}}= 34$ cm s⁻¹ (10 %: 28, 90 %: 42), giving a dissipation rate $\varepsilon =0.39$ m $^2$ s $^{-3}$ (10 %: 0.22, 90 %: 0.75) where the value reported is the median of the spatial distribution and the values in parentheses are the quantiles, showing some inhomogeneity in our set-up. The associated Kolmogorov scale is 40 µm and the integral time scale is $T_{\textit{int}}=101$ ms. In figure 3(c), we show the second-order longitudinal structure function $D_{LL}(r)=\langle (\Delta u(r))^2\rangle _{\boldsymbol{x}, t}$ , in both the $x$ and $y$ directions. In addition we plot the inertial range scaling, with the dissipation computed from the approach above, and the prefactor $C_2=2.0$ (Pope Reference Pope2000). Despite the limited inertial range in our set-up, we see that both approaches are consistent. Overall, the flow field in the set-up presents strong turbulent fluctuations, required to obtain the breakup of millimetric bubbles and keep them within the field of view long enough so that the size distribution can be recorded. In exchange for this large intensity, the turbulent flow field is not homogeneous or isotropic, but we do not expect those features to affect our conclusions.

Figure 3.

Properties of the turbulent flow field in the set-up. Panel (a) shows th mean flow $\langle U \rangle$ intensity and direction. Panel (b) shows the root-mean-square velocity in colour. The dashed rectangle represents the field of view, the cyan lines show local estimations of the integral length scale $L_{\textit{int}}$ and the circles show the local Hinze scale $d_H$ . Panel (c) shows the second-order longitudinal structure functions in both directions. The dashed line is the inertial range scaling, with the prefactor $C_2=2.0$ .

3.1. Representative breakup dynamics with and without surfactant

We qualitatively discuss examples of large bubble breakup in our experiments, leading to populations of sub-Hinze-scale bubbles, in the presence and absence of surfactant. Figure 4 presents representative examples of the fragmentation process with configurations showing filament formation or shedding behind large bubbles (without surfactants: (a) and (b) and with $4000\,\unicode{x03BC} \text{mol}\,\text{l}^{-1}$ of SDS: (c) and (d)).

Figure 4.

Representative image sequences illustrating bubble breakup with filament fragmentation and subsequent sub-Hinze bubble generation in the presence and absence of surfactant. The initial parent bubble has a size of the order of the Hinze scale. Each row presents four time-resolved snapshots, with a temporal spacing of 1 ms (a,c) or 6 ms (b,d) between consecutive frames: (a) and (b) DI water without surfactant, showing the formation of no or a single sub-Hinze bubble; (c) and (d): 4000 µmol l⁻¹ of SDS, where several sub-Hinze bubbles are produced.

Figure 4. Long description

Panel A: Four time-resolved snapshots with a temporal spacing of 6 milliseconds between consecutive frames, showing bubble breakup in DI water without surfactant. The initial parent bubble has a size of the order of the Hinze scale, and the formation of no or a single sub-Hinze bubble is observed. Panel B: Four time-resolved snapshots with a temporal spacing of 1 millisecond between consecutive frames, showing bubble breakup in DI water without surfactant. The initial parent bubble has a size of the order of the Hinze scale, and the formation of no or a single sub-Hinze bubble is observed. Panel C: Four time-resolved snapshots with a temporal spacing of 6 milliseconds between consecutive frames, showing bubble breakup in the presence of 4000 mol/l of SDS. The initial parent bubble has a size of the order of the Hinze scale, and several sub-Hinze bubbles are produced. Panel D: Four time-resolved snapshots with a temporal spacing of 1 millisecond between consecutive frames, showing bubble breakup in the presence of 4000 mol/l of SDS. The initial parent bubble has a size of the order of the Hinze scale, and several sub-Hinze bubbles are produced.

Rivière et al. (Reference Rivière, Mostert, Perrard and Deike2021, Reference Rivière, Ruth, Mostert, Deike and Perrard2022), Ruth et al. (Reference Ruth, Aiyer, Rivière, Perrard and Deike2022) and Qi et al. (Reference Qi, Tan, Corbitt, Urbanik, Salibindla and Ni2022) discuss large bubble breakup in turbulence as happening through two successive stages. First, the parent bubble is deformed by a turbulent eddy with a characteristic length scale comparable to the bubble diameter, over a time scale controlled by the eddy turnover time ( $T_{\textit{eddy}} \sim 25 \text{ ms}$ in figure 4 b,d). This interaction stretches the bubble into an unstable configuration consisting of two lobes connected by a thinning neck, as illustrated in figure 4(b,d). Subsequently, the neck undergoes further thinning under the action of capillary forces, sometimes forming a filament, which ultimately breaks via a Rayleigh–Plateau-like instability, over the capillary time scale ( $T_{\textit{cap}}\sim 0.1-1$ ms based on the filament thickness in figure 4 b,d), producing small capillary bubbles.

The representations of the sub-Hinze bubble production in figures 4(b) and 4(d) could suggest that surfactants influence this mode of production, by increasing the number of child bubbles formed by a single air filament. This configuration is, however, an idealisation of the multiple concurrent fragmentations occurring with the large bubbles used in this study (the injection scale is $d_{\textit{inj}} \approx 40 d_H$ ) visible in figures 4(a) and 4(c). In those case a single large bubble can be seen fragmenting into a large number of smaller bubbles, with many fragmentation events occurring between subsequent frames. One key difference is the resulting distribution of bubbles: in the clean case a few intermediate sized bubbles (around $d_H$ ) are formed as well as a similar number of sub-Hinze bubbles. In the contaminated case, however, a similar number of intermediate sized bubbles are formed, but the number of sub-Hinze bubbles is drastically larger.

Different mechanisms may account for the enhanced production of small bubbles in the surfactant-laden case. Surfactants may increase the number of small child bubbles produced during the deformation and breakup of large bubbles in a tearing- or shedding-like process or during the rupture of elongated individual ligaments. Distinguishing between these mechanisms would require dynamically resolved experiments or simulations focusing on these different scenarios. We comment, however, that our time resolution and the use of a single camera is not sufficient to access the detailed dynamics in a reliable way. Our study focuses on the statistical analysis of the influence of surfactant on the fragmentation of a relatively large air cavity in turbulence. Detailed observation of the breakup dynamics would require zoomed in views at higher temporal resolution, with a dedicated experimental set-up with multiple cameras, such as some past studies (without surfactant) on pinch-off in turbulence (Ruth et al. Reference Ruth, Mostert, Perrard and Deike2019) or breakup by a single vortex (Qi et al. Reference Qi, Tan, Corbitt, Urbanik, Salibindla and Ni2022).

3.2. Bubble size distribution

Figure 5 presents the bubble size distribution $\mathcal{N}(d)$ for the tested solutions: SDS (a), SDS with sea salt (b), CTAB (c), Tween 20 (d) and Tween 20 with sea salt (e) at increasing surfactant concentrations. The dashed lines are shown as references for the $-10/3$ scaling in the super-Hinze regime and the $-3/2$ scaling in the sub-Hinze regime. The observed transition between these two regimes in the bubble size distribution occurs at $d \approx 2.5$ mm, which is close to the calculated Hinze scale, and will be used as the effective Hinze scale $d_{H,\textit{eff}}$ .

Figure 5.

Comparison between the bubble size distributions of different solution compositions at varying surfactant concentration. Bubble size distribution for (a) SDS, (b) SDS and sea salt, (c) CTAB, (d) Tween20, (e) Tween 20 and sea salt. Colours indicate the surfactant concentration. Dashed lines indicate the $d^{-3/2}$ and $d^{10/3}$ power laws. The shaded regions represent uncertainties in the distributions, estimated using a Monte Carlo approach, while the solid lines indicate the mean values in the distributions.

For a given set of experiments, the distribution in the super-Hinze-scale range ( $d \gt d_{H,\textit{eff}}$ ) does not change when adding surfactants, and is very similar across the different solutions tested. As the concentration of SDS is increased, with or without sea salt (figures 5 a and 5 b), the total number of bubbles in the sub-Hinze range increases while that in the super-Hinze regime remains unchanged. The increase in the number of bubbles in the sub-Hinze range is associated with an increase in the slope of the distribution. In contrast, when Tween20 is used (figures 5 d and 5 e), the bubble size distribution remains unchanged both in the super-Hinze and sub-Hinze ranges. This conclusion is not altered by the presence of sea salt. Using CTAB leads to an intermediate behaviour between that of SDS and Tween 20, where the number of sub-Hinze bubbles increases significantly once the concentration is large enough. Details of the solution (salt, surfactant) only affect sub-Hinze-scale bubbles, and the effect is strongly dependent on the type of surfactant used, while the super-Hinze size distribution remains unchanged.

For all conditions, the power-law distribution in the sub-Hinze regime at the lowest contamination is compatible with the $d^{-3/2}$ distribution due to capillary breakup. For the cases that show a increase in the number of sub-Hinze bubble count, this increase is linked to a steeper distribution as surfactant concentration is increased (i.e. a more negative power-law exponent). This modification of the shape of the distribution, together with the fact that we do not observe any changes in the effective Hinze scale when adding surfactant, rules out the possibility that the trends observed in this study are merely a result of increasing the Weber number through a change of the static surface tension.

3.3. Counts of super-Hinze and sub-Hinze bubbles

We now quantify the increase of sub-Hinze bubbles ( $N_{d\lt d_H}$ ) for the various solutions used.

Figure 6.

(a) Variation in the counts of super-Hinze (open triangles) and sub-Hinze (solid squares) bubbles with surface tension. The solid black edge denotes the second realisation of the SDS experiment. (b) Same data as a function of the bulk concentration of surfactant, represented as a fraction of the CMC.

Figure 6. Long description

Panel A: A scatter plot shows the variation in the counts of super-Hinze (open triangles) and sub-Hinze (solid squares) bubbles with surface tension. The x-axis represents surface tension in mN/m, and the y-axis represents the ratio of bubble counts (N/N0). Different colors and symbols represent different surfactants and conditions, including SDS, SDS with sea salt, Tween20, Tween20 with sea salt, and CTAB. Panel B: Another scatter plot shows the same data as a function of the bulk concentration of surfactant, represented as a fraction of the critical micelle concentration (CMC). The x-axis is logarithmic and represents c/CMC, while the y-axis is the same as in Panel A. The data points show how bubble counts vary with different surfactant concentrations and conditions.

Figure 6(a) shows the counts of super-Hinze bubbles $N_{d\gt d_H}$ (open triangles) and sub-Hinze bubbles $N_{d\lt d_H}$ (full squares) as contamination increases (represented by the static surface tension, note the inverted x-axis) for the different types of contamination (colours). We define the reference bubble count, $N_0$ , as the bubble count measured in pure water without surfactant or salt. With this definition, the normalised bubble count, $N/N_0$ , collapses to unity at zero surfactant concentration (corresponding to the maximum static surface tension, $\gamma _s$ ) for SDS, Tween 20 and CTAB. For surfactant solutions including sea salt, however, the initial normalised counts are slightly greater than unity, possibly due to trace amounts of surfactants present in the sea salt that we use. Figure 6(b) shows the same data as a function of the bulk concentration of surfactant, represented as a fraction of the CMC.

As contamination increases with the addition of surfactant (surface tension is reduced), the super-Hinze bubble count remains unchanged in all cases (triangles). For sub-Hinze bubbles, all solutions with a fast surfactant (for instance SDS, blue squares) show an increase in counts: for very low surface tensions, the counts of sub-Hinze bubbles are increased a factor 5. In contrast, for cases where surface tension is modified by adding a slow surfactant (Tween 20, green squares), the sub-Hinze bubble counts remain unchanged. The addition of salt is associated with a similar trend: the number of sub-Hinze bubbles increases with increasing SDS concentration (dark blue squares), while remaining essentially unchanged with increasing Tween 20 concentration (dark green squares). The intermediate surfactant (CTAB, yellow squares) exhibits a transitional behaviour: it behaves like a slow surfactant at higher static surface tension $\gamma _s$ (low concentration), and increasingly resembles a fast surfactant as $\gamma _s$ further decreases (high concentration). In particular, as the CTAB concentration is increased, the sub-Hinze bubble count remains unchanged for $\gamma _s \gtrsim 60\,\text{mN}\,\text{m}^{-1}$ , before increasing as $\gamma _s$ is further reduced. The concentration of surfactants required to obtain a similar increase in counts can be vastly different depending the the surfactant used or on whether sea salt is present (figure 6 b), suggesting a reinforcement of the effect of SDS (anionic) upon the addition of ions. Another way to view the same effect is is the effective reduction of the CMC of charged surfactants in the presence of ions: indeed the CMC of SDS is reduced by almost two orders of magnitude when adding NaCl at seawater concentration (Schick Reference Schick1964; Srinivasan & Blankschtein Reference Srinivasan and Blankschtein2003), comparable to the differences seen in figure 6(b). This effect is also relatively well be captured by plotting the counts as a function of the static surface tension (figure 6 a).

As stated above, the various cases with Tween 20 show no differences in counts when varying the concentration despite a reduction in surface tension comparable to the reduction when using SDS, highlighting that the increase in the number of sub-Hinze bubbles and the change in the power law observed here is not simply due to a reduction in surface tension and associated increase in Weber number. Tween 20 is non-ionic and interactions between the surfactant and sea salt are therefore very limited.

As hinted at in the introduction and now demonstrated experimentally, the influence of surfactant on the breakup process is tightly linked to their adsorption time. Surfactants will only affect the breakup process if a significant amount can adsorb to the interface of a bubble between two successive fragmentation events. Indeed, during the stretching preceding fragmentation (through tearing or through the formation of a ligament), a large amount of surface area is created and therefore the surfactant surface concentration on a newly formed bubble is less than the value at equilibrium with the bulk (see sketch in figure 1 b). If the time scale for adsorption of surfactants to the interface $\tau _{\textit{ads}}$ is much longer than the typical time between fragmentation events, then after a few breakups, the bubbles will essentially behave as clean bubbles by successive dilutions. We consider here a large-scale turbulent time: the eddy turnover time at the integral length scale $T_{\textit{eddy}}(L_{\textit{int}})$ (equal to the integral time scale $T_{\textit{int}}$ ) as the representative time for the turbulence breakup sequence to be compared with the adsorption process. To quantify the competition between turbulent transport and adsorption kinetics, we introduce a Biot number (Tasoglu et al. Reference Tasoglu, Demirci and Muradoglu2008; Pico et al. Reference Pico, Kahouadji, Shin, Chergui, Juric and Matar2024), comparing the turbulence time scale with the adsorption time scale

(3.1) \begin{align} Bi = \frac {T_{\textit{eddy}}(L_{\textit{int}})}{\tau _{\textit{ads}}}. \end{align}

Figure 7.

Variation in the number of sub-Hinze bubbles as a function of the Biot number $ \textit{Bi}$ for SDS (blue), CTAB (yellow) and Tween 20 (green). The second SDS realisation is denoted with a black edge. Inset: adsorption time $\tau _{\textit{ads}}$ as a function of bulk concentration of surfactant in solution. Data points are from the literature, shaded areas represent the ranges used in this study and dashed lines are fits used to extend estimated adsorption times to lower concentrations. Yellow for CTAB (Qazi et al. Reference Qazi, Schlegel, Backus, Bonn, Bonn and Shahidzadeh2020), blue for SDS (Fainerman et al. Reference Fainerman, Lylyk, Aksenenko, Petkov, Yorke and Miller2010) and green for Tween 80 (Qazi et al. Reference Qazi, Schlegel, Backus, Bonn, Bonn and Shahidzadeh2020).

We therefore need to estimate the adsorption time scale for the various contamination cases considered. The inset of figure 7 shows data acquired using a maximum bubble pressure tensiometer for SDS (extracted from Fainerman et al. Reference Fainerman, Lylyk, Aksenenko, Petkov, Yorke and Miller2010), Tween 80 (a surfactant similar to Tween 20 but with more oxyethylene groups, extracted from Qazi et al. Reference Qazi, Schlegel, Backus, Bonn, Bonn and Shahidzadeh2020) and CTAB (also extracted from Qazi et al. Reference Qazi, Schlegel, Backus, Bonn, Bonn and Shahidzadeh2020). The adsorption time scale differs markedly across surfactants, with Tween 80 ( $\sim 10\,\text{s}$ ) being much slower than SDS ( $\sim 10\,\text{ms}$ ), while CTAB lies in between ( $\sim 1\,\text{s}$ ). In all cases, the adsorption time scales become much faster as concentration is increased, with a variation close to $\tau _{\textit{ads}} \propto c^{-2}$ . Note, however, that the datasets cover concentrations that are larger than the ones we use (highlighted with the shaded areas). We therefore need to extend $\tau _{\textit{ads}}$ to lower concentrations via modelling.

We follow the approach detailed by Ferri & Stebe (Reference Ferri and Stebe2000): under the assumption that the adsorption is diffusion limited, the adsorption time scale can be expressed as $\tau _{\textit{ads}} \sim h^2 / D$ , where $D$ is the diffusivity of the surfactant and $h$ is the depletion depth, the thickness of the layer depleted of surfactants below the interface. This depth is then given by $h = \varGamma / c$ , with $c$ the bulk concentration and $\varGamma$ the surface concentration at equilibrium, which is not directly accessible but given by the isotherm (or surfactant equation of state) that the surfactant follows. We make the hypothesis that both surfactants follow a Langmuir isotherm, in which case $\varGamma (c) = \varGamma _\infty K c / (1 + K c)$ , with $\varGamma _\infty$ and $K$ two constants. Bringing these results together, we obtain finally that the adsorption time scale should follow $\tau _{\textit{ads}} = t_0 / (1 + c / c_0)^2$ , with $t_0$ and $c_0$ two fitting parameters. This functional form fits well the available literature data, in particular the $c^{-2}$ scaling observed in all datasets (dashed lines). Note, however, that the approach that we use here is only done to extract a typical adsorption time and has several limitations: first, we have no proof that the adsorption is indeed diffusion limited and that kinetics do not start to play a role for some of the concentrations (although if diffusion is the limiting factor, one expects the $c^{-2}$ scaling observed in the data; Ward & Tordai Reference Ward and Tordai1946). Second, both datasets were recorded in quiescent conditions, whereas in our case turbulence constantly mixes the fluid and brings surfactant close to the interface. As a consequence, diffusion may be significantly faster than expected (or the depletion depth thinner) and the adsorption kinetics may play a role (Chang & Franses Reference Chang and Franses1995).

We can now plot the normalised count of sub-Hinze bubbles as a function of the Biot number, computed using the fitted adsorption time scale in the inset of figure 7. As expected, we now very clearly see two different regimes: at low $ \textit{Bi}$ , the surfactant is too slow for a significant amount to be adsorbed between breakup events and as a consequence the dynamics is identical to the clean case ( $N/N_0\approx 1$ ). At higher $ \textit{Bi}$ , however, the number of sub-Hinze bubbles progressively increases as a significant amount of surfactants can adsorb to bubbles before they fragment through a capillary instability, during which important surface tension gradients can then form and influence the resulting size distribution. We note that using another turbulent time would lead to a similar plot, simply shifting the transitional Biot number. With the present choice of the slowest turbulent time, the transition is for $ \textit{Bi}$ between 0.1 and 1.