Shaping a Surface Microdroplet by Marangoni Forces along a Moving Contact Line of Four Immiscible Phases

Figure 1 shows the typical snapshots of receding contact line over an octanol microdroplet with different v_cl and $r_{o}$. Initially, the octanol microdroplet rests inside the water drop on the substrate. The term v_cl is defined as the speed of the three-phase contact line of gas-water-solid before it reaches the octanol droplet. In the experiments, v_cl was controlled by the flow rate of the injection pump. The $r_{o}$ is radius of contact line of octanol droplet. Here $t$ = 0 ms represents the time when solid-oil-water-gas four phase contact line forms, namely the moment when the three-phase contact line around the water drop just touches the oil droplet.

Figure 1(a) shows the optical images of octanol droplet with radius of 83 $\mu$m and with v_cl of 0.36 mm/s. The contact line quickly traverses the octanol microdroplet once the contact line meets the edge of microdroplet within 53 ms. Then, the contact line passes the droplet and remains fixed until the outer slowly moving parts catch up, leaving an unbroken droplet behind the contact line at t=1148 ms.

A fluorescence dye of Nile red at 5 $\mu$M was added into octanol to trace the flow of octanol indicated by the bright part in the image. Figure 1 (b) shows the fluorescence microscope image of the octanol droplet with radius of 139 $\mu$m and with v_cl of 0.9 mm/s. Similar with Figure 1(a), the octanol droplet is completely left behind the contact line of the water drop. This stable behaviour of oil droplet, i.e., being completely transferred from water to air, is dominated by capillary force, which we will discuss later.

The octanol droplet with radius of 419 $\mu$m exhibits radically contrasting behaviour at v_cl of 0.07 mm/s as shown in Figure 1 (c). The contact line passes the edge of microdroplet, but does not sweep cross the entire droplet, as shown in Figure 1 (c, 6 ms to 20 ms). In 60 ms, octanol in the microdroplet spreads along the contact line and evolves into a crescent shape like a bow. The crescent is darker than water, as shown in Figure 1 (c, 80 ms to 400 ms). Figure 1 (d) shows the similar spreading phenomenon captured by using fluorescence microscopy where the radius of the octanol droplet was 251 $\mu$m and the v_cl is 0.01 mm/s. At 300 ms, a bow shape of octanol microdroplet appears in the image, which reflect the spreading of the microdroplet along the deformed contact line as a result of the Marangoni stress.

Figure 1 (e) show the typical snapshots of a microdroplet with the radius of 145 $\mu$m at v_cl of 0.01 mm/s. In the images taken by the laser scanning confocal microscope, the red, green and black colors indicate octanol, water and air phases, respectively. We see that the contact line moves over the octanol droplet once the contact line meets the edge of the microdroplet, as shown in Figure 1 (e, 0 ms to 20 ms). After that, the octanol spreads laterally along the contact line and gradually evolves to a bow shape, as shown in Figure 1 (e, 40 ms to 160 ms). Simultaneously, a part of water was stranded on the substrate due to the pinning effect.

Tiny twin water droplets were left on lateral sides of the octanol microdroplet, as shown in Figure 1 (e, 160 ms). Finally, octanol fully cloaks the water-air interface and shapes like a circle, as shown in the left image in Figure 1 (f). The spreading behaviour shown in Figure 1 (c-f) is dominated by Marangoni effect, which we will discuss later. The dynamics of contact line before and after the contact with the edge of microdroplet is sketched in Figure 1 (f).

By varying the values of $r_{o}$ and $v_{cl}$, we are able to identify the transition between spreading (Marangoni effect-dominated) and stable (capillarity-dominated) behaviors. In Figure 2 (a), the octanol microdroplet spreads along contact line when $r_{o}$=193 $\mu$m and $v_{cl}$=0.36 mm/s, while the octanol microdroplet remains stable when $r_{o}$=83 $\mu$m and $v_{cl}$=0.36 mm/s. This result indicates the smaller $r_{o}$ may lead to the transition from Marangoni effect-dominated behavior to capillarity-dominated behavior. In Figure 2(b), the octanol microdroplet spreads along the contact line when $r_{o}$=227 $\mu$m and $v_{cl}$=0.32 mm/s, but remains stable when $r_{o}$=213 $\mu$m and $v_{cl}$=3.73 mm/s. This indicates that the increase in $v_{cl}$ may also lead to the transition from Marangoni effect-dominated behavior to capillarity-dominated behavior.

To get more insight into the dynamics of this four-phase system, we conducted systematic experiments by tuning the $v_{cl}$ from 0.01 mm/s to 5 mm/s and $r_{o}$ from 80 $\mu$m and 500 $\mu$m. Figure 2(c) shows the phase diagram of $r_{o}$ versus $v_{cl}$. Clearly, there is a demarcation line between capillarity-dominated and Marangoni effect-dominated behavior. Smaller $r_{o}$ and faster $v_{cl}$ contribute to the transition from Marangoni effect-dominated behavior to capillarity-dominated behaviour.

Next, we seek to confirm that the spreading behaviour is driven by Marangoni effect. After formation of four-phase contact line, the original water-air interface is gradually displaced by water-octanol and octanol-air interfaces. The Marangoni effect driven spreading can be identified by the spreading coefficient $S=\delta_{12}-(\delta_{13}+\delta_{23})$, where $\delta_{ij}$ is interfacial tension between phase $i$ and phase $j$ [38]. The fluid of phase $3$ will spontaneously spread between phase $1$ and phase $2$ if $S>0$. In our case, the phase $1$, $2$, $3$ are water, air and octanol, respectively. The spreading coefficient $S$ can be calculated by

where $\delta_{wa}$, $\delta_{ow}$, and $\delta_{oa}$ are interfacial tensions between water-air interface (72.8 mN/m), water-octanol interface (8.4 mN/m), and octanol-air interface (27.5mN/m), respectively.

To get a better understanding of this Marangoni effect driven spreading behavior, the key is to reveal the coupling between the Marangoni effect and the sliding of water drop contact line acting on the octanol microdroplet. The time scale for the contact line sliding through the oil droplet is:

The time scale for the oil droplet spreading on the water drop can be obtained by

where $\rho_{w}$ is the water density, and $\Delta\sigma$ is the interfacial tension difference between octanol-water interface and water-air interface.

For the oil droplet spreading on the water drop, sliding time scale should be longer than spreading time scale, $\tau_{s}>\tau_{M}$. Figure 2(d) displays the phase diagram replotted with $\tau_{s}$ vs $\tau_{M}$. It clearly shows that two phases are separated by a straight line, below which the octanol droplet always spreads. The slope of the straight line is measured to be about 0.09 in the range of 0 $\lesssim\tau_{M}\lesssim$ 40. Notably that the measured slope is not equal to 1, which is due to the time scales of $\tau_{s}$ and $\tau_{M}$ only standing for the order of magnitude, but not the exact values.

When we use silicone oil or squalane, the magic wand work well with same principle, as shown in Figure 3(a-c). Similar with octanol oil, silicone oil and squalane droplet also transit from Marangoni effect-dominated behaviour to capillarity-dominated behaviour when decreasing $r_{o}$ and increasing $v_{cl}$.

Here we perform the dimensionless analysis for the spreading droplet. The Reynolds number $\rm{Re_{w}}$ is defined as the ratio of the inertial force and viscosity force on the water drop. The Schmidt number $\rm{Sc_{w}}$ is defined as the ratio of kinematic viscosity and mass diffusivity.

The Marangoni number $\rm{Ma_{o}}$ can be used to estimate the significance of the Marangoni effect acting on the octanol microdroplet.

Here $r_{w}/r_{o}$ is the ratio of water drop radius and octanol droplet radius; $\mu_{w}/\mu_{o}$, the ratio of the dynamic viscosity of water and octanol, and $D$, the mass diffusivity of octanol in water. Then $\tau_{M}>\tau_{s}$ can be rewritten as

To demonstrate the robustness of dimensionless criteria, $\mu_{w}$ was tuned within the range of 1.0 mPa$\cdot$s $<\mu_{w}<$ 4.0 mPa$\cdot$s by adding glycerol into the water drop. By substituting the corresponding $\mu_{w}$, $\mu_{o}$, $\Delta\sigma$, $\rho_{w}$, $r_{o}$, $r_{w}$ and $v_{cl}$ depicted in Tables 1 and 2, we obtain the plot of $Re_{w}^{2}Sc_{w}(r_{w}/r_{o})(\mu_{w}/\mu_{o})$ vs $Ma_{o}$, as shown in Figure 3(d). It clearly shows two phases are separated by a straight line and the spreading takes place when $Ma_{o}$ is below the reference line.

Next, we quantitatively investigate the evolution of spreading length of the microdroplet, $L(t)$, with time $t$. Figure 4(a$-$d) show the typical snapshots captured by fluorescence microscope with $r_{o}$ = 251 $\mu$m and $r_{w}$ = 1409 $\mu$m. We observed that octanol phase continues extending to lateral sides with time, and forms an arc shape. Here we define the arc length $L(t)$ to represent the boundary of the drop covered by oil, as shown in Figure 4 (b).

In Figure 4(e), we show the normalized spreading length $L^{\ast}$ as a function the normalized time $t^{\ast}$.

where $\eta_{1}$ = 1.0 mPa$\cdot$s is dynamic viscosity of water, $\Delta\delta$ = 45.3 mN/m is the difference in interfacial tensions between water-air interface and octanol-air interface, and $\rho_{1}$ = 998 kg/m³ is the density of water.

All the data in Figure 4(e) were extracted from snapshots of fluorescence microscope experiments, with $r_{o}$ changing from 220 to 434 $\mu$m, $r_{w}$ changing from 1376 to 1865 $\mu$m and the velocity of water drop contact line $v_{cl}$ less than 0.1 mm/s. We find that $L(t)$ increases with time, and all seven sets of data collapse to one curve and are parallel to the reference solid line, corresponding to a scaling law of $L^{\ast}\sim(t^{\ast})^{3/4}$.

According to literature [39, 40], for Marangoni dominated spreading, the spreading length $L$ obeys a scaling law of $L\sim t^{3/4}S^{1/2}/(\rho_{w}\eta_{w})^{1/4}$, where $\rho_{w}$ is density of water and $\eta_{w}$ is viscosity of water. The consistency of this scaling law with our experimental results, further demonstrates that the spreading behavior of octanol along the water-air interface is governed by the Marangoni stress.

When we changed the oil type from octanol to silicone oil, the viscosity, as well as interfacial tension, were changed. As expected, the phenomenon was different from octanol droplets. As shown in Figure 5 (a-I, a-II) the contact line of the water drop was pinned by the silicone oil droplet, rather than sliding immediately on the oil drop after the water contact line contact with oil drop. At one moment, as shown in Figure 5 (a-III), the contact line of water drop depins and slides quickly on the oil droplet. At the same time, the oil spreads along water-air interface under the Marangoni stress. In this process, the shape of the silicone oil droplet evolves into a shape that is broad, triangular, and have smoothly rounded edges, resembling the shape of the pectoral fins of a stingray. Next, we conducted the experiments with multiple silicone oil droplets encapsulated in a water drop, all the silicone oil droplets are shaped into similar shapes, as shown in Figure 5 (c). Changing the velocity of water-air contact line ($v_{cl}$), we found a negative relationship between vertex angle of the triangle and $v_{cl}$, as shown in Figure 5 (d). This indicates that we can shape viscous oil droplet by changing the moving velocity of water contact line. On the other hand, the behaviour of viscous liquid drop under the Marangoni stress implies that those viscous microdroplets may not be removed from the surface during cleaning.