Needle-free drop deposition technique for contact angle measurements of superhydrophobic surfaces
Prashant R. Waghmare2 and Sushanta K. Mitra1
1
Micro and Nanoscale Transport Laboratory, Department of Mechanical Engineering, Lassonde School of Engineering, York
University, Toronto, M3J1P3
2
Department of Mechanical Engineering, University of Alberta, Edmonton, T6G2G8
SUPPORTING MATERIAL:
Impact Weber number (Wet) on the top low energy surface (St):
For this analysis, we have considered the energy conservation between two stages during the drop deposition process(1–3). The
first stage (labelled as subscript 1) corresponds to the drop before impact (Fig. 3 (a)) and second stage (labelled as subscript 2)
corresponds to the drop in contact with St (Fig. 3(b)). One can write energy conservation between state 1 and state 2 as(1–3)
𝐾. 𝐸.1 + 𝑆. 𝐸.1 = 𝐾. 𝐸.2 + 𝑆. 𝐸.2 + 𝐸𝑑𝑖𝑠𝑠
(S1)
In Eq. (S1), the left hand side is the sum of the kinetic energy (K.E.) and surface energy (S.E.) of the drop before the impact, and
individually, each contribution can be explicitly written as (1–3)
𝜋𝐷𝑜2 𝛾
𝐾. 𝐸.1 = (
) Wet
12
(S2)
𝑆. 𝐸.1 = 𝜋𝐷𝑜2 𝛾
(S3).
Here, γ is the liquid- air surface tension, Do is the diameter of the drop and Wet =
ρV2 Do
γ
. The right hand side of Eq. (S1) repre-
sents the total energy in the drop in contact with the top surface St, i.e., at maximum spread. It is to be noted that in this configuration, the drop is at rest therefore K. E.2 = 0. During the attainment of maximum spreading configuration, the drop gets deformed and attains a non-spherical configuration. Assuming near elastic collision with the surface St, one can neglect the viscous
dissipation Ediss in Eq. (S1) for Wet << 1 (4),(5). It is observed that in the present case, the drop remains almost spherical with
D
minimal deformation in the shape i.e., x ≪ o⁄2 as shown in Fig. 3 (b). For such configuration, the surface energy of the drop
with maximum spread diameter dm can be written as;
𝑆. 𝐸.2 = (𝜋𝐷𝑜2 +
𝜋 2
𝑑 cos 𝜃) 𝛾
4 𝑚
(S4)
Here, θ is the equilibrium contact angle corresponding to the top surface S t. As explained in the main manuscript, this “needlefree” drop deposition technique only works for significant superhydrophobicity of S t, which suggests that θ > 900. Hence, using
equations (S1) – (S4), we obtain
𝑑𝑚 2
𝑊𝑒𝑡 = 3 (
𝐷𝑜
) cos(180𝑜 − 𝜃) for 𝜃 > 900
(S5)
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