1.School of Mechanical and Automotive Engineering, Qingdao University of Technology, Qingdao 266520, China 2.National Microgravity Laboratory, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China 3.School of Engineering Science, University of Chinese Academy of Sciences, Beijing 100049, China
Fund Project:Project supported by the Key R&D Program of Shandong Province, China (Grant No. 2019GGX102023) and the National Natural Science Foundation of China (Grant Nos. U1738118, 11472275)
Received Date:12 March 2021
Accepted Date:11 April 2021
Available Online:07 June 2021
Published Online:20 September 2021
Abstract: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 R and time t driven by silicone oil conforms to the classical theoretical $ \mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\;R\left(t\right)\propto {t}^{3/4} $. 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 $ {\lambda }_{\mathrm{s}} $ and the constant radius $ {r}_{\mathrm{c}} $ of the n-hexadecane liquid rim are related by $ {\lambda }_{\mathrm{s}}\approx 9{r}_{\mathrm{c}} $. Our results prove the applicability of the spreading scaling law to the liquid-driven spreading process, and also help to understand in depth the mechanism of the liquid-driven spreading and the instability pattern in the spreading process. Keywords:driven spreading/ drop dynamics/ interface instability/ surface tension
从表2可以看出, 正十六烷在浓度为5, 6和7 mmol/L的SDS溶液表面铺展系数$ S < 0 $, 表现为正十六烷局部铺展, 以透镜的形式存在(如图2(a)—(c)所示); 而浓度为8 mmol/L时铺展系数$ S > 0 $, 在实验中表现为正十六烷在溶液表面未完全覆盖整个表面, 此时正十六烷处于赝不完全浸润状态(如图2(d)所示). 图 2 不同浓度SDS溶液表面正十六烷初始铺展状态(比例尺为2 mm) Figure2. Initial spreading state of n-hexadecane on the surface of SDS solutions with different concentrations (scale bar = 2 mm).
图3显示了在浓度为8 mmol/L的SDS溶液表面, 正十六烷的部分铺展状态图像, 其中, 正十六烷内侧的红色线条表示硅油-正十六烷界面, 正十六烷外侧的黑色线条表示正十六烷-SDS溶液界面. 我们注意到, 硅油滴入后, 正十六烷液滴迅速受驱动由中心向外铺展, 由透镜状(图3(a))变为环形带状(图3(b)). 由于正十六烷带的内边缘直接受硅油驱动, 内外边缘的铺展速度不同, 正十六烷带的宽度逐渐减小, 直到变为液柱状(图3(c)). 之后, 正十六烷液柱继续向外铺展(图3(d)), 由于表面张力会通过减小半径来使表面积尽可能的小, 液柱自发扰动形成波浪形(图3(e)), 最终破裂形成一圈小液珠(图3(f)). 图 3 SDS溶液(8 mmol/L)表面正十六烷受驱动铺展过程(比例尺为10 mm) Figure3. Spreading process of n-hexadecane on the surface of SDS solution (8 mmol/L) (scale bar = 10 mm).
在实验中, 由于驱动液体(硅油, 表面张力为15.9 mN·m–1)表面张力较小, 且不易挥发, 与正十六烷和基底溶液之间会形成稳定的界面张力差, 这一不同液体之间的张力差异使得硅油对正十六烷产生驱动作用. 图4是硅油的驱动作用示意图. 首先, 硅油滴入后, 浸入正十六烷(图4I阶段); 之后, 硅油接触基底SDS溶液(图4II阶段), 在形成稳定界面张力差$ \Delta \sigma $的条件下驱动正十六烷向外铺展, 此时的界面张力差$ \Delta \sigma $可以近似表示为 图 4 硅油的驱动作用示意图. 1, 2和3分别为驱动溶剂硅油、受驱动溶剂正十六烷和基底SDS溶液, $ {\sigma }_{23} $和$ {\sigma }_{12} $分别表示正十六烷与SDS基底溶液之间和硅油与正十六烷之间的界面张力, $ \nabla \sigma $表示界面张力梯度, ${\Delta }P$表示压力梯度 Figure4. Schematic diagram of the driving effect of silicone oil. 1, 2 and 3 are driving solvent (silicone oil), driven solvent (n-hexadecane) and aqueous substrate (SDS solution), $ {\sigma }_{23} $ and $ {\sigma }_{12} $ represent the interfacial tension between n-hexadecane and SDS solution and between silicone oil and n-hexadecane, respectively. $ \nabla \sigma $ represents the interfacial tension gradient, and $ {\Delta }P $ represents the pressure gradient.
因此, 受硅油驱动的正十六烷失稳前的铺展半径R应遵循$ R\left(t\right)\propto {t}^{3/4} $, 即“3/4”标度律. 图6显示了在铺展的第一阶段, 正十六烷在不同浓度的SDS溶液表面受硅油驱动的铺展半径R随时间t的演变过程. 从图6可以看出: 50 ms内铺展各浓度铺展半径变化几乎一致; 50 ms后, 随着基底溶液浓度的增加, 铺展半径的最值逐渐增大. 将不同浓度的铺展标度率与“3/4”进行比较, 发现浓度为5 mmol/L时, 铺展标度率略小于3/4, 而浓度为6—8 mmol/L时, 铺展标度率略大于3/4. 总之, 受硅油驱动的正十六烷失稳前的铺展半径与时间的标度关系符合理论$ R\left(t\right)\propto {t}^{3/4} $. 图 6 失稳前不同浓度的SDS溶液表面正十六烷铺展半径R随时间的变化(在5, 6, 7和8 mmol/L的SDS溶液表面, 正十六烷受驱动铺展的标度率分别为0.73, 0.80, 0.82, 0.83, 黄线为理论值0.75) Figure6. Variations of the spreading radius R of n-hexadecane on the surface of SDS solutions with different concentrations before instability occurs. The driven-spreading scale rate of n-hexadecane is 0.73, 0.80, 0.82, and 0.83 on the surface of 5, 6, 7 and 8 mmol/L SDS solution, respectively. The yellow line is the theoretical rate of 0.75.
图 1 实验装置示意图
图 2 不同浓度SDS溶液表面正十六烷初始铺展状态(比例尺为2 mm)
图 3 SDS溶液(8 mmol/L)表面正十六烷受驱动铺展过程(比例尺为10 mm)
图 4 硅油的驱动作用示意图. 1, 2和3分别为驱动溶剂硅油、受驱动溶剂正十六烷和基底SDS溶液, 
图 5 正十六烷液滴受驱动铺展阶段示意图
图 6 失稳前不同浓度的SDS溶液表面正十六烷铺展半径R随时间的变化(在5, 6, 7和8 mmol/L的SDS溶液表面, 正十六烷受驱动铺展的标度率分别为0.73, 0.80, 0.82, 0.83, 黄线为理论值0.75)
