Liquid-liquid-driven spreading process based on Marangoni effect

Drop dynamics at liquid surfaces is existent in nature and industry, which is of great value in studying droplet self-propulsion, surface coating, and drug delivery, and possesses great potential applications in microfluidics and biological process. Here, we analyze the role of Marangoni effect in the spontaneously driving system by studying the driving effect of a low surface tension liquid at the liquid substrate on another liquid. A three-phase liquid system is established to explore the liquid-driven spreading process, including non-volatile silicone oil as driving solvent, n-hexadecane as driven solvent, and sodium dodecyl sulfate (SDS) solution with different concentrations as aqueous substrates. The spreading process of n-hexadecane driven by silicone oil can be divided into two stages. N-hexadecane is first driven to form a thin rim, and then the rim breaks up into small liquid beads. Afterwards, the driving mechanism, spreading scaling laws and instability characteristic parameters of the liquid-driven spreading process are analyzed theoretically. The analysis of driving mechanism indicates that the differences in surface tension among silicone oil, n-hexadecane and SDS solution cause surface tension gradient at the liquid-liquid interface, which plays a crucial role in spreading the n-hexadecane. The results also demonstrate that the maximum spreading radius of n-hexadecane is affected by the concentration of the aqueous substrate. When the concentration of SDS solution is lower than the critical micelle concentration, the maximum spreading radius of n-hexadecane is proportional to the concentration of SDS solution. Meanwhile, the scaling law between the spreading radius <i>R</i> and time <i>t</i> driven by silicone oil conforms to the classical theoretical <inline-formula><tex-math id="M1">\begin{document}$ \mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\;R\left(t\right)\propto {t}^{3/4} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210485_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210485_M1.png"/></alternatives></inline-formula>. In addition, the classical analysis model is used to explain the instability pattern of n-hexadecane breaking into small beads from rim in the liquid-driven spreading process, which is called Rayleigh-Plateau instability. The fastest instability wavelength <inline-formula><tex-math id="M2">\begin{document}$ {\lambda }_{\mathrm{s}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210485_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210485_M2.png"/></alternatives></inline-formula> and the constant radius <inline-formula><tex-math id="M3">\begin{document}$ {r}_{\mathrm{c}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210485_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210485_M3.png"/></alternatives></inline-formula> of the n-hexadecane liquid rim are related by <inline-formula><tex-math id="M4">\begin{document}$ {\lambda }_{\mathrm{s}}\approx 9{r}_{\mathrm{c}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210485_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20210485_M4.png"/></alternatives></inline-formula>. Our results prove the applicability of the spreading scaling law to the liquid-driven spreading process, and also help to understand in depth the mechanism of the liquid-driven spreading and the instability pattern in the spreading process.