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Title: Energy and Hydrodynamic Analyses of Coalescence-Induced Jumping Droplets
Experimental Method
We used commercially available copper foils (99.8 % purity, 0.8mm thickness, and 2 cm × 2
cm size) as starting substrates. The substrate was cleaned in an ultrasonic bath with acetone for
10 min and rinsed with deionized (DI) water. The substrate was then dipped into a 2.0 M
hydrochloric acid solution for 30 s to remove pre-existing oxide layers, rinsed with DI water, and
dried in a nitrogen blow. Then the Cu substrate was nanostructured with copper oxides by
immerging the substrate into a hot alkaline solution composed of NaClO2, NaOH, Na3PO4 
12H2O and DI water (3.75:5:10:100 wt %). Then the nanostructured CuO is functionalized with
TFTS (trichloro(1H,1H,2H,2H-perfluorooctyl)silane, Sigma) through the vapor deposition
Global condensation behaviors were investigated with high speed imaging in a temperature
and humidity controlled environment. As in Figure S5, samples were mounted to a custom-made
temperature controlled cold plate located in the temperature/humidity controlled stage. The
samples were first cooled to the target temperature by circulating cooling flow, and then watersaturated N2 flow was supplied to the test chamber. The superstation level was determined by
𝑆 = 𝑃𝑣 /𝑃𝑤 , where Pv is the vapor saturation pressure and Pw is the saturation pressure
corresponding to the sample temperature. The microscopic condensation behaviors are measured
from the top using a high speed microscopy (Phantom M110 connected to Olympus BX53). The
a) & b)
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droplet merging and jumping behaviors are also observed from the side with a high speed camera
(Phantom M110) connected to multiple extension tubes.
Figure S1. Schematic of the condensation experimental setup.
Numerical Method
We use the level contour reconstruction method, a hybrid scheme that combines the advantage
of the front tracking and level set methods. The method explicitly tracks implicitly connected
individual interface elements as in the front tracking method, which is ideal for capturing precise
location of the contact points. Since the Lagrangian interface can be reconstructed using Eulerian
function field (e.g. distance function) that can be computed directly from given interface location,
merging and pinch- off of interfaces can also be dealt automatically as in the level set method.
Using the method, the continuity and momentum equations were solved with the properties
described by the Heaviside function I(x,t) that varies from zero to one near the interface.
We use a simple Navier-slip model to account for the contact line behavior at the boundary
wall which allows contact line movement proportional to shear stress at the contact point. The
contact line velocity is determined as follows:
U cl  
n wall
where, Ucl denotes the contact line speed, u / n wall is the shear strain rate at the wall, and  is a
proportionality constant. In this study, the size of a grid cell has been used for . With a slip
condition at the wall, we can alleviate the infinite shear stress condition near the contact line.
The model should also account for contact angle hysteresis. We put the advancing (a) and
receding (r) contact angles measured by experiments. If a contact angle is less than the
prescribed receding angle r or larger than advancing angle a, the interface are free to move
with the given receding or advancing contact angle. If the contact angle is between the advancing
and receding angles, i.e. during hysteresis, no fixed contact angle is imposed and the contact
angle is free to change between the advancing and receding angles without any interfacial
Using the developed model, the droplets merged on the superhydrophobic CuO surface are
simulated in a three-dimensional frame work. Initially, two near-spherical droplets with diameter
of 30 µm are placed on the bottom of the domain as in Figure S2. The wall and open boundary
conditions were applied to the bottom and top surfaces, respectively. The periodic boundary
condition has been applied to x and y direction.
Figure S2. Schematic of the calculated domain with two droplets placed on a superhydrophobic
surface. The diameter of each droplet is 30 µm and the advancing and receding contact angles
are determined as 165.1° and 159.8°, respectively, based on the experimental characterization.
Movies of Simulated Droplet Coalescence on the Superhydrophobic Surface
Figure S3. Evolution of liquid-vapor interface of droplets coalesced on the superhdyrophobic
surface. (θa = 165.1° and θr = 159.8°), (enhanced online).
Figure S4. Evolution of liquid-solid interface (base area) of droplets coalesced on the
superhdyrophobic surface. (θa = 165.1° and θr = 159.8°), (enhanced online).
Figure S5. Evolution of velocity vectors of droplets coalesced on the superhdyrophobic surface.
(θa = 165.1° and θr = 159.8°), (enhanced online).
Figure S6. Evolution of pressure distribution of droplets coalesced on the superhdyrophobic
surface. (θa = 165.1° and θr = 159.8°), (enhanced online).