The role of drop shape in impact and splash

The impact and splash of liquid drops on solid substrates are ubiquitous in many important fields. However, previous studies have mainly focused on spherical drops while the non-spherical situations, such as raindrops, charged drops, oscillating drops, and drops affected by electromagnetic field, remain largely unexplored. Using ferrofluid, we realize various drop shapes and illustrate the fundamental role of shape in impact and splash. Experiments show that different drop shapes produce large variations in spreading dynamics, splash onset, and splash amount. However, underlying all these variations we discover universal mechanisms across various drop shapes: the impact dynamics is governed by the superellipse model, the splash onset is triggered by the Kelvin-Helmholtz instability, and the amount of splash is determined by the energy dissipation before liquid taking off. Our study generalizes the drop impact research beyond the spherical geometry, and reveals the potential of using drop shape to control impact and splash.


Supplementary Table 1 | Viscosity of the ferrofluid measured by rheometer under different magnetic field strength.
The minimum field strength in the experiment region increases from 0 to 3.00 mT, which is 10 times stronger than the field during our impact process (smaller than 0.3 mT). However, no change in the viscosity is observed. Therefore, the weak residue of magnetic field in our experiment does not induce any observable change in viscosity.

Minimum magnetic field in the region (mT)
Maximum magnetic field in the region (mT) The minimum field strength in the experiment region increases from 0 to 2.83 mT, which is over 9 times stronger than the field during our impact process (smaller than 0.3 mT). However, no change in the surface tension is observed. Therefore, the weak residue of magnetic field in our experiment does not induce any observable change in surface tension.

Minimum magnetic field in the region (mT)
Maximum magnetic field in the region (mT)  1 , where ρ is the liquid density, σ is the surface tension, and R is the drop radius. The characteristic time scale of the impact is defined as T=L/2V with V the impact speed (2.9 ± 0.2 m/s), and L the drop length. In our study, L varies between 1.59 mm and 4.35 mm, and thus T ranges from 0.28 ms to 0.76 ms, which is much shorter than the oscillation period of 31 ms. Therefore, the drop shape can be considered as stable throughout the entire impact process.
In addition, we also measure the oscillation velocity, dL/dt, for each drop shape, as shown in Supplementary Fig. 3. Most shapes exhibit small oscillation velocities as shown by the orange region.
To make sure that the oscillation speed does not affect the impact, we eliminate situations with large oscillation speeds (the green region), as represented by the shapes d and e. Only the shapes with small oscillation speeds are selected (the orange region) in our analysis.

Supplementary Fig. 3 | The oscillation velocity of each drop shape.
The blue curve represents all shapes we can achieve, shapes a to m (the red points) are selected as representative examples for demonstration. The drop shapes in green regions are rejected due to their high oscillation velocities, while the shapes in the orange region are selected due to their small oscillation velocities.

Supplementary Note 3 | Derivation of equation (3) and (4) of the main text
After a liquid drop impacts onto a solid substrate (see the drawing Supplementary Fig. 4 below), the liquid at the bottom part (region A) will be replaced by the substrate. We assume that this part of the liquid goes to region B and forms a cylinder 2-4 . By volume conservation, the volume of region A and B must be equal, or equivalently, volume of region A + C and B + C are equal, thus where r is the radius of the wetting front, V is the impact velocity, t is the time after impact, and ( ) x a = − − is the drop shape described by the super-ellipse function (equivalent to equation (2)  by the splash criterion. Note that h is quite discrete due to the limited spatial resolution of our camera, which may explain some large deviations from unity in Fig. 3e in the main text.

Supplementary Note 4 | Derivation of viscous dissipation energy
The viscous dissipation energy of an axisymmetric flow is given by 2 because of the thin thickness of the spreading liquid sheet. It is further approximated as where μ is the viscosity, vr is the radial velocity, h is the thickness of liquid sheet, V is the volume and T is the timescale. Similar approximations can successfully deduce the scaling of maximum spreading diameter of drops by energy conservation [5][6][7][8][9] . In our model we take

Supplementary Note 5 | Experiments on different substrates
We performed our experiments on three different substrates. The first substrate, which is identical to that of the main text, is a glass microscope slide washed by acetone, IPA, and deionized water. The other two substrates are polymethyl methacrylate (PMMA or acrylic glass) surface and piranhacleaned glass, which have lower and higher surface energy than the microscope slide respectively. The contact angles between a water drop and the acrylic glass, microscope slide, and piranha-cleaned glass are 74±4°, 26±4°, and 7±3° respectively. The corresponding literature values of the surface energy of the acrylic glass, microscope slide, and piranha-cleaned glass are 42 mN/m, 68 mN/m and 83 mN/m respectively 11,12 . The ferrofluid, whose surface tension is 19 mN/m, wets all three substrates.
The data of microscope slide (Black) is overlaid with the new data of acrylic glass (Pink) and piranhacleaned glass (Green) in Supplementary Fig. 10 below, and they largely overlap. Therefore, our conclusion holds for substrates with different surface energy. This is also consistent with the previous study of Latka et al. that substrate wetting property has little or no effect on high-speed impact dynamics and splash 13 .