Supplementary materials for Control of a bouncing magnitude on a heated substrate via ellipsoidal drop shape

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1 Supplementary materials for Control of a bouncing magnitude on a heated substrate via ellipsoidal drop shape Shape oscillation of bouncing ellipsoidal drops Shape oscillation of drops before impacting can be approximately represented by Laplace s spherical harmonics [1,2], providing a radius: r(t, φ) = r0[1 + ζ cos(wt)p2(cos φ)], where r0 is the radius of an unperturbed sphere, ζ the amplitude of the mode, w the time period of the mode, P2(x) = (3x 2-1)/2 the Legendre polynomial of degree 2, and φ the azimuthal angle. The modal shape is axisymmetric with respect to the y-axis as described in Fig. S1, which alternates between prolate and oblate ellipsoidal shapes. In non-axisymmetric spreading and retraction, a drop in prolate or oblate ellipsoidal shape becomes aligned with a certain horizontal axis (x-axis or y-axis), which is reproduced numerically as shown in Fig. S2. The aligned liquid is released and squeezed outward along the axis and initiates alternate expansion and contraction of the drop along the horizontal axes. In other words, the alignment of liquid triggers the shape oscillation due to the capillary retraction. The symmetry axis of the oscillation is closely related to the axis with which the liquid was aligned, as described in Fig. S2. Note that, before the (prolate or oblate) ellipsoidal drop collides with the surface, the shape oscillation is axisymmetric with respect to the y-axis [3]. For the prolate ellipsoidal drop impacting on the surface, it completes the alignment with the x-axis and then bounces off the surface, having the symmetry axis for oscillation as the same axis (Fig. S2a). For the oblate ellipsoidal case, it completes the alignment with the y-axis and then bounces off the surface, having the symmetry axis for oscillation as the same axis (Fig. S2b). Shape oscillation of bouncing drops can also be approximately represented by Laplace s spherical harmonics. The theoretical aspect ratio in the horizontal cross section (xy-plain) of the drop is given as α = r(t, 0)/r(t, π/2) = [1 + ζ cos(wt)]/[1 - (ζ /2)cos(wt)], which implies that the amplitude of one horizontal axis (x-axis or y-axis) is greater than that of the other axis (yaxis or x-axis). 1

2 Fig. S1. The oscillation phase of the ellipsoidal drop alternating between prolate ellipsoid and oblate ellipsoid before impacting. Fig. S2. The impact behavior of (a) a drop in prolate ellipsoidal shape and (b) a drop in oblate ellipsoidal shape obtained numerically for the controlled parameters of AR = b/a = d/c = 1.48 and We = 13; the dashed boundaries in the bottom view denote the outermost liquid-vapor interfaces at the moment of impact. The symmetry axis of the oscillation is closely related to the axis with which the liquid was aligned. 2

3 Impact dynamics of drops in prolate and oblate ellipsoids We examine the effects of the oscillation phase on drop impact by using the simulation. For the prolate ellipsoidal drop impacting on the surface, it spreads and completes the alignment with the x-axis and then bounces off the surface at t ~ 7 ms (Fig. S2a). For the oblate ellipsoidal case, likewise, it spreads and completes the alignment with the y-axis and then bounces off the surface at t ~ 7 ms (Fig. S2b). It should be noted that the values of AR in the prolate and oblate are identical (AR = 1.48 & We = 13). Once ARs are identical, the impact behavior of the prolate ellipsoidal drop is only slightly different from that of the oblate one, which is proven based on the horizontal widths, the vertical width, and bounce height of the drop in Fig. S3. There is no major difference in bouncing efficiency. This implies that the impact behavior is highly dependent on AR rather than the oscillating phase at We = 13. In addition, we deduce that the impact behavior is weakly dependent on drop oscillation at the given We. At the impact point, the prolate ellipsoidal drop is contracting along the y-axis, the phase of which corresponds to π/2 < ϕ < π in Fig. S1; in contrast, the oblate ellipsoidal drop is expanding along the y-axis, the phase of which corresponds to 3π/2 < ϕ < 2π in Fig. S1. As described earlier, the impact behaviors are found to be only slightly different from each other at We = 13, once ARs are identical. If the drop has higher impact velocity (higher We), the shape could be frozen like an ellipsoidal solid at the moment of impact, so the impact dynamics would be independent of the shape oscillation. The effect of the shape oscillation on the impact dynamics can be simply estimated by the ratio of the oscillating time scale [τosc ~ (ρd0 3 /σ) 1/2 ] to the crashing time scale (τcra ~ D0/U0), which gives us that τosc/τcra ~ We 1/2. Because the shorter time scale is dominant, the shape oscillation only slightly affects the impact behavior at high We. 3

4 Fig. S3. Similarity of the impact behavior between a drop in prolate ellipsoidal shape and a drop in oblate ellipsoidal shape, based on temporal variations in (a) the horizontal width in the major axis, (b) the horizontal width in the minor axis, (c) the vertical width, and (d) the bounce height, using simulation for AR = 1.48 and We = 13. The major axis denotes the longitudinal direction of the liquid alignment during retraction. 4

5 Discussion on discrepancies between experiment and simulation In this work, a slight discrepancy in maximum bounce height and contact time is observed for experimental and numerical results. We present some findings to account for the discrepancy. First, the hole formation occurred due to the rupture of the lamella in the bottomview experiment of the ellipsoidal drop impact (Fig. S4a), which cannot be reproduced by the simulation. We found that the hole formation did not always appear in our impact experiment. In addition, it did not lead to liquid fragmentation and disappeared due to the retraction along the y-axis at t ~ 6 ms (Fig. S4b). We do not currently have a clear explanation for the origin of the hole formation; this phenomenon could be triggered by single defects, which has been studied in literatures [4,5]. The hydrodynamics due to the rupture might affect the aligning process and bouncing height. Second, we compare the drop impact dynamics quantitatively for the experiment and simulation in terms of the normalized contact width D/D0 for several ARs. In Fig. S5, the spreading dynamics show that the maximum spreading diameter of the y-axis Dmy is found earlier and is smaller than that of the x-axis Dmx. In addition, Dmx increases with AR, whereas Dmy rarely depends on AR. The ratio between maximum spreading diameters between of the x- and y-axes Dmx/Dmy increases with AR. The overall temporal evolution in contact width of the experiment is roughly comparable to that of the simulation, in spite of a difference in maximum spreading diameter between the experiment and simulation. The difference can affect the magnitude of liquid alignment during retraction and the consequent shape oscillation of bouncing drops, which can make a difference in bounce height and contact time. In the simulation, we did not employ a vapor film model [6,7], but considered a free slip condition at the contact area and a contact angle of 180 to simply represent the effects of the vapor layer [8,9]. 5

6 Fig. S4. (a) Bottom view of impact behaviors for AR = 1.43 and We = 28: the hole formation occurs during retraction; the experimental setup is similar to the work of Tran et al. [10]. (b) Illustration of the impact behavior and the hole formation. Fig. S5. Temporal evolution of the normalized contact width D/D 0 of drops for We = 28 and several ARs. (a) AR = 1.08 and (b) AR = 1.50 and (c) AR = Symbols represent experimental data in the x-axis (circle) and the y-axis (triangle) as shown in the inset of (a). Solid lines represent numerical data in the x-axis (pink line) and the y-axis (blue line). 6

7 Reference [1] Rayleigh, L. (1879). On the capillary phenomena of jets. Proceedings of the Royal Society of London, 29( ), [2] Lamb, H. Hydrodynamics (6th edn.)(1932). [3] Yun, S., & Lim, G. (2014). Ellipsoidal drop impact on a solid surface for rebound suppression. Journal of Fluid Mechanics, 752, [4] Biance, A. L., Pirat, C., & Ybert, C. (2011). Drop fragmentation due to hole formation during Leidenfrost impact. Physics of Fluids (1994-present), 23(2), [5] Lastakowski, H., Boyer, F., Biance, A. L., Pirat, C., & Ybert, C. (2014). Bridging local to global dynamics of drop impact onto solid substrates. Journal of Fluid Mechanics, 747, [6] Harvie, D. J., & Fletcher, D. F. (2001). A hydrodynamic and thermodynamic simulation of droplet impacts on hot surfaces, Part I: theoretical model. International Journal of Heat and Mass Transfer, 44(14), [7] Harvie, D. J., & Fletcher, D. F. (2001). A hydrodynamic and thermodynamic simulation of droplet impacts on hot surfaces, Part II: validation and applications. International Journal of Heat and Mass Transfer, 44(14), [8] Karl, A., Anders, K., Rieber, M., & Frohn, A. (1996). Deformation of liquid droplets during collisions with hot walls: experimental and numerical results. Particle & particle systems characterization, 13(3), [9] Mitra, S., Sathe, M. J., Doroodchi, E., Utikar, R., Shah, M. K., Pareek, V.,... & Evans, G. M. (2013). Droplet impact dynamics on a spherical particle. Chemical Engineering Science, 100, [10] Tran, T., Staat, H. J., Prosperetti, A., Sun, C., & Lohse, D. (2012). Drop impact on superheated surfaces. Physical review letters, 108(3),