Supplementary Material:
Drop splashing on a rough surface: how surface morphology affects splashing
threshold
Hyungmo Kim†, Usung Park†, Chan Lee, Moo Hwan Kim, Hyungdae Kim and Joonwon Kim*
†
These authors equally contributed to this work.
*
Corresponding author: Prof. Joonwon Kim (joonwon@postech.ac.kr)
1. Derivation of projected area A(tc)
This process starts with the following two equations:
A  Db2 / 4
2
2
2
 Db   D0

D 
 vi t    0  ,

 
 2   2

 2 
then combines them to yield
A(t c )   (vi D0t c  vi2t c2 ) .
For small tc, we ignore
𝑣𝑖2 𝑡𝑐2 and obtain
A(tc )  vi D0tc
FIG. S1. Schematic drawing and derivation of projected area A(tc) of an impacting drop.
2. Fabrication process
FIG. S2. Process of specimen fabrication. All specimens have both microstructures and nanostructures. The
nanostructures were fabricated by the black silicon formation method using deep reactive-ion etching 1, 2. The
microstructures were fabricated using silicon wet etching. A silicon dioxide (SiO2) layer (500 nm) was
thermally grown in a furnace and patterned by conventional lithography using a photoresist. Each photomask
was differently designed for different cavity sizes. Tetramethyl ammonium hydroxide solution was used to
etch the Si surface, and the patterned SiO2 layer functioned as the masking layer during etching. The etching
depth was controlled precisely by the etching time and temperature (i.e. 12 min at 90 °C). Then hierarchical
micro/nano-structures were fabricated sequentially by the same black silicon formation method. Formation of
conformal black silicon nanostructures was possible due to the slope-sided walls formed by silicon wet etching.
Finally, all exposed surfaces were coated with a hydrophobic self-assembled monolayer, which was applied
by coating the surface with a solution of 0.1% heptadecafluoro-1,1,2,2-tetrahydrodecyl trichlorosilane in nhexane 3, 4.
3. Specimen information
Table S1. Specimen information including surface roughness (Ra and H, where H means depth of the each
cavity) of microstructures and contact angles. All specimens had almost the same contact angles, and were
superhydrophobic. a: cavity size; b: gap between cavities (Fig. 2).
a / b (µm / µm)
Designed
Measured
20 / 40
21.4 / 38.3
30 / 30
32.7 / 28.4
40 / 20
42.6 / 18.0
50 / 10
52.2 / 8.3
Specimen
G20
G30
G40
G50
Measured roughness
Ra (µm)
H (µm)
3.06
10.2
3.66
8.9
3.61
8.9
2.58
9.0
Static
163 ± 3
163 ± 3
161 ± 2
161 ± 2
Contact angle (°)
Receding Advancing
161 ± 2
164 ± 1
162 ± 3
165 ± 2
160 ± 2
164 ± 2
160 ± 2
164 ± 1
4. Method of drop impact experiments
We selected three variables: impact velocity vi, drop diameter D0, and four surface morphologies (G20, G30,
G40, and G50 [Fig. 2]; Ra was largest on G30). We conducted drop impact experiments on each specimen
with various vi and D0. Drops of known different volume (from ~1.3 μL to ~5.4 μL) were released at different
heights above a target surface at room temperature and pressure. vi and D0 were automatically obtained from
recorded images by programmed software.
5. Physical parameter selection and derivation of Eq. (S1)
ST  va 

Vc / A
5  Vtot / A 1 / 2 
5

L p vi D01 / 2  
Lp1 / 2 vi D01 / 2

7  Aout / L
7  hn  n Deff  Vc / A


(S1)
<Measured parameters>
Vc/A: cavity volume per unit area (calculated for each specimen)
hn: nano-pillar height = ~5.0 μm (from SEM images)
εn: porosity of nanostructured layer = ~0.7 (from SEM images)
<Selected parameters>
Deff: effective diffusivity in Knudsen diffusion inside nanostructures = selected as 7% based on 5
<Description>
In our cases, capillary pressure Pc,n for nanostructures was over ~1 MPa 6, but the capillary pressures Pc,m for
microstructures were from ~1.7 kPa to ~4 kPa 7. PEWH in our experimental conditions was estimated to be less
than ~0.9 MPa from the relation PEWH ≈ 0.2ρwcvi where ρw is the density of water and c is the speed of sound
in water 6, 8. PEWH was not sufficient to cause the water drop to penetrate between dense nanostructures, but
was sufficient to cause it to penetrate between the microstructures (partial wetting state) 6. Therefore, we can
estimate the cavity volume per unit area Vc/A by considering only microstructures using measurements from
SEM images (i.e., Vc/A = 0 for a smooth surface and Vc/A > 0 for a rough surface). We simply postulate that
Vtot/A = Vc/A. From the same consideration, we obtained the relation Aout/L = Am/L + hnεnDeff, where Am/L is the
opening area of microstructures per unit length, hn is the height of nanostructures, εn is the porosity of
nanostructures, and Deff is the effective diffusivity of nanostructure. Am/L can be replaced by Vc/A in the concept
of mean value when the penetrating drop can reach the bottom of the surface. If the structure is too deep for
the drop to reach the bottom, the Vc/A < Am/L. Finally, we can obtain Eq. S1 with va for ST for our cases.
To apply Eq. (S1) to our experimental results [Fig. 4(a)], and εnDeff was obtained by analyzing SEM images
and Knudsen diffusion regime,5 and hn was measured as ~5 µm in SEM images [insets of Fig. 2]. Vc/A for each
specimen was calculated from geometry conditions.
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