Numerical investigation of dynamic effects for sliding drops on wetting defects

The ability to trap or deflect sliding drops is of great interest in microfluidics, as it has several technological applications, ranging from self-cleaning and fog harvesting surfaces to laboratory-on-a-chip devices. We present a three-dimensional numerical model that describes sliding droplets interacting with wetting defects of variable strength and size. This approach provides relevant insight if compared to simplified analytic models, as it allows us to assess the relevance of the internal degrees of freedom of the droplet. We observe that the deformation of the drop enhances the effective strength and range of the defect, and we quantify this effect by comparison to a point-mass model. We also analyze the role of the steepness and strength of the defect on the drop motion, observing that small, strong defects are more effective at trapping than large, shallow traps of same excess surface energy. Finally, our results show quantitative agreement with previously reported electrowetting experiments, suggesting a universal behavior in droplet trapping that does not depend strongly on the nature of the defect.

Figure 1

Sketch of the computational domain.

Figure 2

(a) Values of terminal velocities for water droplets of volume $V=20,40,60\mu \mathrm{L}$ (red, green, and blue sets, with increasing line thickness) as a function of the sine of the slope angle $sin\alpha$. The circles represents numerical results, while the crosses are experimental measurements from Ref. [16] for water drops on a glass slide covered with an ITO and oil layer, as discussed in the main text. The lines are linear fits to the numerical data. (b) Dynamic contact angles for increasing ${\theta }_Y$, according to the different models described in the main text.

Figure 3

(a) Sketch showing the relevant geometric parameters for the energy and force calculations under the assumption that the drop retains a spherical cap shape. (b) Energy landscape for a 40-$\mu \mathrm{L}$ water drop (spherical cap) sliding over a stripe of width $a=0.5$ mm of contact angle ${\theta }_{\mathrm{trap}}={45}^{\mathrm{o}}$ as a function of the distance $x$ between the center of the drop and the midpoint of the stripe. (c) Corresponding force acting on the drop as a function of $x$.

Figure 4

Selected snapshot and corresponding base profile for $\alpha =5^{\circ }$, ${\theta }_{\mathrm{trap}}={30}^{\circ }$ (top), and $\alpha =7^{\circ }$, ${\theta }_{\mathrm{trap}}={45}^{\circ }$ (bottom). The wetting defect is represented by the red stripe.

Figure 5

Diagram showing numerical simulations resulting in trapping (red circles) and escaping (green crosses) for a 40-$\mu \mathrm{L}$ water droplet, as a function of the slope $\alpha$ and the strength of the trap, which is proportional to $\Delta {\theta }_{\mathrm{trap}}={\theta }_0-{\theta }_{\mathrm{trap}}$. The darker shade of red represent the trapping region according to the analytic “spherical cap” approximation.

Figure 6

(a) The energy landscape experienced by the sliding drop in the numerical simulation (circles, red fit) is compared to the prediction from the spherical cap approximation (blue solid line). (b) Phase-space plot for $\alpha =5^{\circ }$ and ${\theta }_{\mathrm{trap}}={45}^{\circ }$: Both the spherical cap model (blue full lines) and the simulation (red thick dashed lines) predict trapping. (c) Phase-space plot for $\alpha =6^{\circ }$ and ${\theta }_{\mathrm{trap}}={45}^{\circ }$: Thanks to stretching in the sliding direction, the drop is retained in the numerical simulation, while the spherical cap approximation predicts escaping. In (b) and (c), the green dashed line shows the range and magnitude of the force acting on ther drop in the spherical cap approximation (arbitrary units).

Figure 7

A comparison of defects of different size and shapes. Contact angle profiles for wetting defects of (a) fixed width $a$ and different contact angle ${\theta }_{\mathrm{trap}}$. (c) fixed contact angle ${\theta }_{\mathrm{trap}}$ and variable trap width $a$. (e) fixed excess energy, $acos{\theta }_{\mathrm{trap}}=\mathrm{const}$. The shape of the trapped droplet for a wide and thin trap are also shown as an insets. In all pictures, red solid lines represent simulations in which the droplet is captured, while for green dashed lines the droplet is only slowed down. [(b) and (d)–(f)] Corresponding energy landscapes in the spherical cap approximation, including the gravity contribution for $\alpha =5^{\circ }$. The profiles in (f) are shifted to ease the visualization.

Figure 8

Phase-space plots for traps of increasing width $a$ and fixed ${\theta }_{\mathrm{trap}}$ [(a), (c), and (e)] and traps of fixed width $a$ and different ${\theta }_{\mathrm{trap}}$ [(b), (d), and (f)]. The colors are the same as in Figs. 6–6. As discussed in the main text, deeper energy wells (see Fig. 7) result in wider oscillations of the droplet velocity. Notice how in (c), once again, the numerical simulation shows trapping, while the analytic model predicts escaping.

Figure 9

Normalized trapping and escaping landscape for (a) the experiments in Ref. [16] and (b) the numerical simulations in this work. As usual, (light) green represents sliding droplets and (dark) red trapped ones. In (b), full dots represent simulations with linearly varying contact angle, as in Fig. 11, and triangles are traps with a constant static contact angle ${\theta }_{\mathrm{trap}}$. In both cases, the black line represents the transition between trapping and escaping in the spherical cap approximation, which underestimates the trapping at large $Q$.

Figure 10

Convergence plot for freely sliding drops. The blue circles are obtained for homogeneous resolution meshes (as in the sketch in the top right corner). The green crosses are obtained for locally refined meshes (sketched in the lower left corner). The red square represents a reduced domain size, still showing good agreement with the high-resolution simulation.

Figure 11

A comparison of defects of different size and shapes for $acos{\theta }_{\mathrm{trap}}=\mathrm{const}$ (see also Fig. 7). In both pictures, red solid lines represents simulations in which the droplet is captured, while for green dashed lines the droplet is only slowed down. (a) Contact angle profiles. (b) Corresponding energy landscapes in the spherical cap approximation for $\alpha =5^{\circ }$.