Asymmetric coalescence of two droplets with different surface tensions is caused by capillary waves

When two droplets with different surface tensions collide, the shape evolution of the merging droplets is asymmetric. Using experimental and numerical techniques, we reveal that this asymmetry is caused by asymmetric capillary waves, which are the result of the different surface tensions of the droplets. We show that the asymmetry is enhanced by increasing the surface tension difference, and suppressed by increasing the inertia of the colliding droplets. Furthermore, we study capillary waves in the limit of no collisional inertia. We reveal that the asymmetry is not directly caused by Marangoni forces. In fact, somehow counterintuitive, asymmetry is strongly reduced by the Marangoni effect. Rather, the different intrinsic capillary wave amplitudes and velocities associated with the different surface tensions of the droplets lie at the origin of the asymmetry during droplet coalescence.

Figure 1

(a) Schematic of the experimental setup (top view). (b) Definitions of the collision parameters (side view).

Figure 2

Temporal evolution of the neck width $w$ for two coalescing water droplets ($\gamma =72.5\mathrm{mN}/\mathrm{m}$, $\rho =997\mathrm{kg}/{\mathrm{m}}^3$, and $\eta =0.92\mathrm{mPa}\mathrm{s}$). The experimental data are an average of five measurements with $\mathrm{We}=0.17$. The error bars indicate the standard deviation between the measurements at each time step.

Figure 3

Direct comparison of experiments (images) and numerical simulations (colored lines). The experimental parameters $v_i$, $R_i$, ${\gamma }_i$, ${\eta }_i$, and ${\rho }_i$ ($i\in [1,2]$) were used as input for the numerical simulations. Collisions between (a) two water droplets with $\tilde{\gamma }=0$, $\mathrm{We}=0.17$, $\mathrm{Bo}=1.6\times {10}^{-4}$, $\mathrm{Oh}=1.8\times {10}^{-2}$; (b) two water droplets with $\tilde{\gamma }=0$, $\mathrm{We}=22.88$, $\mathrm{Bo}=2.2\times {10}^{-4}$, $\mathrm{Oh}=1.7\times {10}^{-2}$; (c) a water droplet and an ethanol droplet with $\tilde{\gamma }=2.17$, $\mathrm{We}=0.40$, $\mathrm{Bo}=1.3\times {10}^{-4}$, $\mathrm{Oh}=2.0\times {10}^{-2}$; (d) a water droplet and an ethanol droplet (no simulations available, see the discussion in Sec. 2c) with $\tilde{\gamma }=2.17$, $\mathrm{We}=11.80$, $\mathrm{Bo}=1.7\times {10}^{-4}$, $\mathrm{Oh}=1.9\times {10}^{-2}$; and (e) a water droplet and a 34.5 wt % ethanol droplet with $\tilde{\gamma }=1.27$, $\mathrm{We}=2.27$, $\mathrm{Bo}=1.5\times {10}^{-4}$, $\mathrm{Oh}=1.9\times {10}^{-2}$. See the Supplemental Material [35] for the movies of these collisions.

Figure 4

The maximum asymmetry $\max [L_2\left(t\right)/L_1\left(t\right)]$ as a function of ${\mathrm{We}}^{1/2}$ for several $\tilde{\gamma }$. The schematic inset shows the definitions of $L_1\left(t\right)$ and $L_2\left(t\right)$. In these experiments (circles) and simulations (squares) ${\gamma }_1$ was varied and ${\gamma }_2=72.5\mathrm{mN}/\mathrm{m}$ was fixed, except for the green symbols, where ${\gamma }_1={\gamma }_2=22.9\mathrm{mN}/\mathrm{m}$. The snapshots (i)–(vi) correspond to the indicated data points. The scalebar corresponds to $100\mu \mathrm{m}$ and applies to all snapshots.

Figure 5

Snapshots from the numerical simulations with $\tilde{\gamma }=1$ and various $\mathrm{We}$. Here, $\rho =1000\mathrm{kg}/{\mathrm{m}}^3$, $\eta =1\mathrm{mPa}\mathrm{s}$, ${\gamma }_1=40\mathrm{mN}/\mathrm{m}$, and ${\gamma }_2=80\mathrm{mN}/\mathrm{m}$. The numbers above the snapshots indicate the nondimensional time $\tilde{t}=t/{\tau }_2$, with $\tilde{t}=0$ being the moment of first contact between the droplets. The zoom provides a closer look at the engulfing film. The protrusion amplitude (and the collision time scale) are smaller for higher $\mathrm{We}$. See the Supplemental Material [35] for the corresponding movies.

Figure 6

Snapshots from the numerical simulations with $\mathrm{We}=0$ and various $\tilde{\gamma }$. Here, $\rho =1000\mathrm{kg}/{\mathrm{m}}^3$, $\eta =1\mathrm{mPa}\mathrm{s}$, ${\gamma }_1=20\mathrm{mN}/\mathrm{m}$, and ${\gamma }_2$ is varied between 20 and 60 mN/m, resulting in the shown values of $\tilde{\gamma }$. The numbers above the snapshots indicate $\tilde{t}=t/{\tau }_2$. The protrusion amplitude increases with increasing $\tilde{\gamma }$, resulting in highly asymmetric droplet shapes. See the Supplemental Material [35] for the corresponding movies.

Figure 7

Schematic of the capillary waves on the droplets. The amplitude $A$ of the capillary waves is determined with respect to the shape of the droplet at $t=0$ as a function of $\theta$, which is the angle between an arbitrary point on the droplet interface and the center of the droplet at $t=0$. In this case we neglect the small initial connection between the droplets, and assume that the droplets are perfect circles. The zoom shows the definition of $A$.

Figure 8

Capillary waves on coalescing droplets for $\tilde{\gamma }=0$ and $\mathrm{We}=0$, from our numerical simulations. (a) Capillary waves on a droplet with $\gamma =60\mathrm{mN}/\mathrm{m}$, where $A$ is the amplitude and $\theta$ the angular coordinate as defined in Fig. 7. (b) Similarity collapse of the capillary waves, following Ref. [46]. (c) The location of the capillary wave maximum as a function of time. (d) The amplitude of the capillary wave as a function of time.

Figure 9

Capillary waves for various $\tilde{\gamma }$ at ${\tilde{t}}_1=0.15$ and ${\tilde{t}}_2=0.50$, from our numerical simulations. Here, ${\gamma }_1=20\mathrm{mN}/\mathrm{m}$, and ${\gamma }_2=40,60,80\mathrm{mN}/\mathrm{m}$, resulting in $\tilde{\gamma }=1,2,3$. (a) Capillary wave amplitude on the low surface tension droplet ($\tilde{t}=t/{\tau }_1$). (b) Capillary wave amplitude on the high surface tension droplet ($\tilde{t}=t/{\tau }_2$).

Figure 10

Capillary waves for $\tilde{\gamma }=0$ and 2 at ${\tilde{t}}_1=0.15$ and ${\tilde{t}}_2=0.50$, from our numerical simulations. (a) Capillary wave amplitude on the low surface tension droplet ($\tilde{t}=t/{\tau }_1$), with ${\gamma }_1=20\mathrm{mN}/\mathrm{m}$. The black line indicates capillary waves on a droplet with $\gamma =20\mathrm{mN}/\mathrm{m}$. (b) Capillary wave amplitude on the high surface tension droplet ($\tilde{t}=t/{\tau }_2$), with ${\gamma }_2=40\mathrm{mN}/\mathrm{m}$. The black line indicates capillary waves on a droplet with $\gamma =40\mathrm{mN}/\mathrm{m}$.

Figure 11

Surface tension and relative capillary wave amplitude on droplet 2 from our numerical simulations for (a) $\tilde{\gamma }=1$ and (b) $\tilde{\gamma }=4$. In both cases ${\gamma }_1=20\mathrm{mN}/\mathrm{m}$. The vertical dashed lines indicate the engulfing front locations.

Figure 12

Rescaled location of the engulfing front as a function of rescaled time, from our numerical simulations. The schematic inset shows the definition of ${\theta }_{\mathrm{front}}$, which is the angle between the location of the engulfing front on the droplet interface with respect to the center of the initial droplet position at $t=0$. The data are cut off when ${\theta }_{\mathrm{front}}=\pi$. The front dynamics converge to $\widehat{L}\propto t^{3/4}$, in accordance with the results of Koldeweij et al. [27].