Diffusion-Dominated Pinch-Off of Ultralow Surface Tension Fluids

We study the breakup of a liquid thread inside another liquid at different surface tensions. In general, the pinch-off of a liquid thread is governed by the dynamics of fluid flow. However, when the interfacial tension is ultralow (2–3 orders lower than normal liquids), we find that the pinch-off dynamics can be governed by bulk diffusion. By studying the velocity and the profile of the pinch-off, we explain why the diffusion-dominated pinch-off takes over the conventional breakup at ultralow surface tensions.

Figure 1

Neck radius $r$ as a function of the time to pinch-off $t$. (a) Two example curves in which there is a 100 times difference in surface tension $\gamma$. The solid curves are the best fit. For $\gamma =10.7\mathrm{mN}/\mathrm{m}$, the curve is linear. For $\gamma =0.1\mathrm{mN}/\mathrm{m}$, the curve follows a power law with an exponent 0.33. Both samples are ATPS-A samples. The upper inset shows a fluorescence image of liquid thread before pinch-off. The outer phase (polymer rich) is brighter than the inner phase (salt rich). The scale bar is $10\mu \mathrm{m}$. The lower inset shows a sketch of free energy [see Eq. (5)], and the dashed curve indicates a near-critical situation with a very small ${ϵ}_0$. (b) Log-log plots of typical measurements for samples with various surface tensions. The solid lines are the best fit. (c) Replotting the low surface tension data in (b) as a function of $t/\sqrt{N}$, where $N$ are the molecular weights of polymers. Panels (b) and (c) share the same legend.

Figure 2

Snapshots of Movie 1 of SM [40] showing the self-similarity of the interface. Top: Raw images of the pinch-off process at different time. Bottom: Corresponding rescaled images, which are resacled by a length scale $L\equiv t^n$ in both radial and axial directions. The exponent $n$ is obtained from fitting the $r\text{−}t$ curves as shown in Fig. 1. The scale bars in all images are $10\mu \mathrm{m}$.

Figure 3

Mean exponents $n$ for different surface tensions $\gamma$ in log-log scale. The error bars include standard deviations of multiple measurements and instrumental uncertainties. The red line indicates the characteristic exponent of Stokes regime, which is 1. The blue line indicates the characteristic exponent of bulk diffusion regime, which is $1/3$. The measured exponents are consistent with these two corresponding characteristic exponents. The green dashed line indicates the characteristic exponent of thermal fluctuation regime ($\sim 0.42$). Apparently it does not agree with the data.

Figure 4

Overlaying the theoretical cone angle on the captured images at different surface tensions at $t=1\mathrm{ms}$. The black dashed lines represent the theoretical asymptotic cone angle of $\theta =76°$. The brightness of the captured images is displayed in pseudocolor to enhance contrast. The blue region is the inner phase, the red region is the outer phase. The scale bar is $10\mu \mathrm{m}$.

Figure 5

The ratio between measured velocity $v_{\mathrm{expt}}$ and theoretical velocity of Stoke regime $v_{\text{St}}$. The theoretical velocity $v_{\text{St}}$ is calculated without any fitting parameter. The velocities are measured or calculated at $r=5\mu \mathrm{m}$. All data points correspond to ATPS-A samples. The large values in the diffusion regime quantitatively verify that $v_{\mathrm{diff}}\gg v_{\text{St}}$ and the faster mechanism dominates the final outcome.