Electronic Supplementary Materials 2
Adhesion energy of air bubbles on a flat surface.
As shown in the figure below, the energy balance within the area of interface
(𝑆air ) between attachment (figure a) and detachment (figure b) is
𝛾S 𝑆air + 𝐸ad = (𝛾L + 𝛾SL )𝑆air
(1)
where 𝛾S , 𝛾L and 𝛾SL are surface energy per unit surface area (or surface tension) of
the gas–solid, gas–liquid and solid–liquid interfaces, respectively. Therefore, the
adhesion energy per unit area of interface (𝐸ad /𝑆air ) of the bubble on the flat surface is
𝐸ad /𝑆air = 𝛾L + 𝛾SL − 𝛾S ,
(2)
where the right side is determined by the combination of liquid and solid, and 𝑆air is a
function of the contact angle of bubble on the surface as [11]
𝑆air = π[4⁄(2 − 3 cos 𝜃air + cos 3 𝜃air )]2/3 𝑟02 sin2 𝜃air .
(3)
Here, 𝑟0 is the radius of the air bubble in water (figure b).
The contact angle of air bubbles (𝜃air ) is determined by the wettability
(hydrophilicity and hydrophobicity) of the surface. From the geometry in figure c, the
balance between the surface tensions along the horizontal axis is
𝛾S = 𝛾SL + 𝛾L cos𝜃water .
(4)
From eq. (4) and 𝜃air + 𝜃water = 𝜋, we get the following equation:
cos 𝜃air = cos(𝜋 − 𝜃water ) = − cos 𝜃water = (𝛾SL − 𝛾S )⁄𝛾L .
(5)
Therefore, the relationship between 𝐸ad /𝑆air and cos 𝜃air from eqs. (2) and (5) is
𝐸ad /𝑆air = 𝛾L (1 + cos𝜃air ).
(6)
Energy balance between the energies of adhesion and interface energies. (a) Attachment.
(b) Detachment. (c) The balance among surface tensions to obtain Young’s relation. 𝛾S ,
𝛾L and 𝛾SL are surface energy per unit surface area (or surface tension) of the gas–
solid, gas–liquid and solid–liquid interfaces, respectively. 𝐸ad , adhesion energy of the
bubble; 𝑟0 , radius of the air bubble; 𝑆air , area of interface; 𝜃air , contact angle of the
bubble.