Tunable structures of compound droplets formed by collision of immiscible microdroplets
Xiaodong Chen1,2 · Yingnan Sun3 · Chundong Xue1,2 · Yude Yu3 · Guoqing Hu1,2
1 Validations of three-phase VOF method
We validate the three-phase VOF method by the spreading of a water lens between air and oil, following Boyer et al. (2010). Initially, a spherical water droplet locates at the middle of a straight surface between the air and oil phase. The droplet then relaxes to an equilibrium shape (Figure 2(a)) under the balancing of the interfacial tensions. Since the lens is the intersection of two spherical caps, we can obtain the theoretical solution of the shape according to the geometrical relationship. The contact angles, θw, θo, and θaare angles between the interfacial tensions in the water, oil and air phases, respectively, and are determined by the Neumann’s relation,
The radius of the interface of the two spherical caps r (see Figure 2(a)) is determined from equal volumes of the initial shape and the steady shape as,
where R is the radius of the initial spherical droplet and / (S2)
1 State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics,
Chinese Academy of Sciences, Beijing 100190, China
2 School of Engineering Science, University of Chinese Academy of Sciences, Beijing 100049, China
3 State Key Laboratory on Integrated Optoelectronics, Institute of Semiconductors,
Chinese Academy of Sciences, Beijing 100083, China
For any combination of the three interfacial tensions, the contact angles can be obtained by solving Equation (S1) numerically combining with
θw+ θa+ θo= 2π.(S4)
We then perform axisymmetric numerical simulations with different combinations of interfacial tensions and compare with the theoretical solutions of Equation (S2). Table S1 lists the value of r for a few combinations. The good agreements indicate that the balancing of interfacial tensions at the triple-line is well captured in the present three-phase volume-of-fluid method.
Table S1 Relative error of the radius of the interface between the two spherical caps r for different combinations of interfacial tensions, which are scaled by σoa.(σwa;σoa;σwo) / theoretical value / numerical value / relative error
(0.6; 1.0;0.6) / 1.625 / 1.653 / 1.723%
(1.0; 1.0; 1.0) / 1.276 / 1.291 / 1.194%
(1.8;1.0; 1.8) / 1.143 / 1.147 / 0.350%
(0.8; 1.0; 1.4) / 1.074 / 1.094 / 1.862%
(0.8; 1.0;1.6) / 0.839 / 0.854 / 1.788%
2 Experimental setup
Figure S1 shows the photos of the experimental setup. Mineral oil (M8410, Sigma-Aldrich, USA) and deionized water are deposited using Jetlab 4 inkjet nozzle (1) (MicroFab Technologies Inc., USA) to a silanized silicon dioxide surface (2). Through a long-working-distance Nikon objective (3) (S Plan Fluor ELWD 20×/0.45) and a homemade tube lens (4) with a cemented achromatic doublet, the collision dynamics are recorded with ahigh-speed camera (5) (Phantom v7.3, Vision Research Inc., USA).
Fig. S1 Experimental installation. (1) Jetlab 4 inkjet nozzle; (2) a silanized silicon dioxide surface; (3) a long-workingdistance Nikon objective (S Plan Fluor ELWD 20×/0.45); (4) a homemade tube lens with a cemented achromatic doublet; (5) a high-speed camera (Phantom v7.3, Vision Research Inc., USA).
3 Additional validations
Figure S2 compares the images from the experiment in Figure 3(a) and (b) and numerical simulations under the same physical conditions. Good agreements are observed in terms of both the dynamics of triple-line and the final structure.
Fig. S2 Comparisons of experimental images (top) and numerical results (bottom) for the experimental conditions in Figure3(a) and (b) of the main text, respectively. (a) Ro = 171.0 µm, Rw= 22.5µm, U0 = 0.8 m/s, Wew=ρwU02Rw/σwa = 0.2, Rew=ρwU0Rw/µw = 18.0, Ohw= 0.025; (b) Ro = 171.0 µm, Rw= 22.0 µm, U0 = 2.2 m/s, Wew= 1.5, Rew = 48.3, Ohw= 0.025.
Boyer F, Lapuerta C, Minjeaud S, Piar B, Quintard M (2010) Cahn-Hilliard/Navier-Stokes model for the simulation of three-phase flows. Transport Porous Med 82(3):463–483