Size Amplification of Jet Drops due to Insoluble Surfactants

TL;DR

This study investigates how insoluble surfactants influence jet drop formation during bubble bursting at a liquid–air interface. By combining experiments with 2D axisymmetric two-phase Navier–Stokes simulations that incorporate measured surface-tension isotherms as the equation of state and Marangoni stresses, the authors reveal a reversal in surfactant impact: at low Laplace numbers ($La\approx 10^3$) surfactants enlarge the first ejected drop radius $R_d$ by smoothing the collapsing cavity, while at high Laplace numbers ($La\approx 10^4$–$10^5$) they shrink $R_d$ by damping precursor capillary waves. The results show quantitative agreement between experiment and simulation when the isotherm is used as the EOS, and demonstrate a data collapse of key jetting metrics when parameterized by the surfactant strength $\beta$ and excess $E$. The findings advance understanding of aerosol emissions from contaminated surfaces and provide a robust framework for linking surface chemistry measurements to dynamic jetting behavior in complex fluids.

Abstract

Surface bubbles in the environment or engineering configurations, such as the ocean-atmosphere interface, sparkling wine, or during volcanic eruptions typically live on contaminated surfaces. A particularly common type of contamination is surface active agents (surfactants). We consider the effect of insoluble surfactant on jet drop formation by bubble bursting. Contrary to the observed trend that surfactants decrease the ejected drop radius for bubbles with precursor capillary waves, we find that surfactants increase the ejected drop radius for bubbles without precursor capillary waves - a regime characteristic of small bubbles. Consequently, the results have fundamental implications for understanding aerosol distributions in contaminated conditions. We find that the trend reversal is due to the effect of Marangoni stresses on the focusing of the collapsing cavity. We demonstrate quantitative agreement on the jet velocity and drop size between laboratory experiments and numerical simulations by using the measured surface tension dependence on surfactant concentration as the equation of state for the simulations. *Jun Eshima and Tristan Aurégan contributed equally to this work.

Figures (4)

• Figure 1: Jet drop formation without surfactants (a,b) and with surfactants (c,d), showing the effects of the Laplace number $\textit{La}=\rho \gamma_c R_0/\mu^2$. At high $\textit{La}$, the addition of surfactants increases the ejected drop radius $R_d$. At low $\textit{La}$, the addition of surfactants decreases $R_d$. Results shown are numerical. The axes are given by the interface profile $h$ and the radial coordinate is denoted $r$. The colors in (c,d) show the surfactant surface concentration $\Gamma$, nondimensionalized by the initial surfactant concentration $\Gamma_0$. (a) Spatial and temporal evolution shown for $\textit{La}=75000$, showing precursor capillary waves (ripples) as the jet is formed. Times shown are $t/t_{\text{ic}}=0,0.2,0.35, 0.7$ (colored light gray to black), where $t_{ic}=\sqrt{\rho R_0^3/\gamma_c}$ is the inertia-capillary timescale. (b) Spatial and temporal evolution shown for $\textit{La}=2400$, where the lack of precursor capillary waves allows the cavity to focus more efficiently. Times shown are $t/t_{\text{ic}}=0,0.2,0.39, 0.5$. (c) Spatial and temporal evolution shown for a typical case with surfactants for $\textit{La}=75000$. Times shown are $t/t_{\text{ic}}=0,0.2,0.43, 0.6$. For reference (to be defined later in the text), the surfactant parameters shown in (c) are given by $(\beta, \Delta \gamma_{\infty}, E)=(0.3, 0.52, 0.5)$. (d) Spatial and temporal evolution shown for a typical case with surfactants for $\textit{La} = 2400$. Times shown are $t/t_{\text{ic}}=0,0.2,0.45, 0.8$. The surfactant parameters shown in (c) are given by $(\beta, \Delta \gamma_{\infty}, E)=(0.3, 0.55, 0.5)$. The Bond number shown in (a,c) is $\textit{Bo}= \rho g R_0^2/\gamma_c=0.13$ and (b,d) is $\textit{Bo}= \rho g R_0^2/\gamma_c=0.16$. The initial bubble shape is given by the Young-Laplace equations with the bubble cap removed at the foot of the cap $r= r_{\text{cap}}$.
• Figure 2: Surfactant effects on jet drops. (a) Experimentally measured surfactant equation of state (markers) for several bulk surfactant concentrations as a function of the surface surfactant concentration $\Gamma$ (see legend in (b)), with the corresponding fits (dashed curves) according to Eq. \ref{['eq:non_dim_isotherm']}. Glycerol weight fraction is 50%. (b) Laplace number $La_{d}=\rho \gamma_c R_d/\mu^2$ of the first ejected drop radius $R_d$ as a function of $\textit{La}$. Experimental points (filled) are colored by the bulk Triton X-100 concentration. Numerical points (open) are colored by the surfactant parameter $\beta$ (Eq. \ref{['eq:non_dim_isotherm']}). The black line and shaded area represent the average and typical spread found in the literature data (without surfactant) berny_role_2020brasz_minimum_2018. Triangles represent experimental pierre_influence_2022vega_influence_2024 (filled) and numerical constante_dynamics_2021pico_surfactantladen_2024 (open) data available in the literature on jet drops with surfactant. Arrows indicate the effect of adding surfactant at a given $\textit{La}$. (c,d) Illustrations of the jet drop dynamics and comparisons between experiments and simulations ((c,d): $(\textit{La},\textit{Bo}, \Delta \gamma_{\infty}, E) = (2000, 0.16, 0.55, 0.5)$, (c): $\beta=0.24$, (d): $\beta=0.37$). The background images are experimental snapshots and the color Crameri2018 overlays show the interface profile and the surfactant concentration from the simulations. Supplementary videos are available.
• Figure 3: Effect of the surfactant parameter $\beta$ on the first ejected drop radius $R_d$, velocity $V_d$, and pinch-off time $t_d$. (a,b,c) Relationships between $(R_d, V_d, t_d)$ and $\beta$, nondimensionalized by the bubble radius $R_0$, capillary velocity $\gamma_c/\mu$, and inertiocapillary timescale $\sqrt{\rho R_0^3/\gamma_c}$ respectively. (d,e,f) Relationships between non-dimensionalized $(R_d, V_d, t_d)$. The same numerical and experimental data points are used throughout (a-f). For $\textit{Bo}=0.16$ as considered, $r_{\text{cap}}/R_0\approx 0.5$ and hence results with $E = 0-1$ are shown.
• Figure 4: Self-similar profile comparison between a clean ($\textit{La} = 2400, \textit{Bo} = 0.16$) and correspondingly contaminated ($\textit{La} = 2400, \textit{Bo} = 0.16, \beta = 0.3, \Delta \gamma_{\infty}=0.55, E = 0.5$) bubble. The clean case is analogous to Figs. 3,4 of lai_bubble_2018. The interface is given by $z=h(r,t)$. (a) The time evolution of the cavity collapse in time $(t < t_0)$ for the clean (gray) and contaminated bubble (colors). The times shown are $(t_0-t)/t_{ic}=0.018, 0.024, 0.03$, where $t_{ic}=\sqrt{\rho R_0^3/\gamma_c}$ is the inertia-capillary timescale. (b) The time evolution of the cavity collapse in self-similar coordinates, corresponding to curves in (a). (c) The time evolution of jet production ($t>t_0$) (same convention as (a)), at times $(t-t_0)/t_{ic}=0.008, 0.011, 0.015, 0.019$. (d) The time evolution of jet production in self-similar coordinates corresponding to curves in (c). In (b,d), $\min$ is taken over the profile of the cavity or jet. The colors for (a,b,c,d) are given by the surfactant concentration $\Gamma$ and Marangoni stress $d\gamma/ds$ for surface tension $\gamma$ and arclength $s$ measured from $r=0$. The particular choice of times plotted in (a,b,c,d) is the same as lai_bubble_2018, which captures the self-similarity.