Supporting Information
Preparation and hydrophobicity of solid–liquid bulk composite
using porous glass and fluorinated oil
Yasuhiro Takada, Munetoshi Sakai, Toshihiro Isobe, Sachiko Matsushita, Akira Nakajima*
1. PIV system for single samples
A schematic illustration of our PIV system for single samples is depicted as Fig. S-1(a). For PIV analysis, the
water droplet contained 0.06 mass% fluorescent particles (test fluid). Using a micro-syringe, a 30-L test fluid droplet
was prepared on the sample surface inclined at 35°. A sheet-shaped laser was emitted vertically to the droplet and
was set to the center of droplet during sliding. A dynamic image of the fluorescent light emitted from the particles
was recorded using a high-speed camera. For this study, we estimated the velocity vectors on each pixel region of
about 114  114 m2 – 136  136 m2 in the entire dynamic image of sliding droplets using time separation of ca.
0.33 ms. Pattern matching for PIV analysis was performed with a correlation factor of 0.6. A schematic illustration
of our PIV system for sandwiched samples is presented as Fig. S-1(b).
Measurement
Sheet-shaped laser
High-speed camera
Water droplet
Slit
High-speed camera
Colored glass filter
(< 540 nm)
Fluorescent particles
Ar laser
l= 488 and 514 nm
Intensity: 1000 mW
Emitted light
Pipette
Analysis
Droplet
Tracking same pixel
Tilted at 35°
Spacer
Samples
Sliding direction
Fig. S-1(a)
Fig. S-1(b)
2. Interface energy values and spreading coefficients
Interfacial energy obtained from each contact angle measurement and pendant drop method [1] is listed in Table
S-1. The spreading coefficients calculated from the values are listed in Table S-2. A schematic illustration of the
relation between the wetting state and the sign of the spreading coefficient is displayed in Fig. S-2.
Table S-1
Table S-2
FC-43 [mJ/m2]
S-20 [mJ/m2]
FC-43 [mJ/m2]
S-20 [mJ/m2]
56
56
0.25
- 1.45
16
17.7
- 1.60
- 3.47
0.6
0.77
- 13.9
- 14.1
72.25
15
42.7
① Sow (a) ≡ γwa − γwo − γoa
①
or
Sow (a) > 0
air
water
③
②
② Sos (a) ≡ γsa − γos − γoa
oil
Sow (a) < 0
or
Sos (w) < 0
solid
・air (a)
・solid (s)
・water (w)
・oil (o)
③ Sos (w) ≡ γsw − γos − γow
Sos (a) > 0
or
Sos (w) < 0
Sos (w) > 0
Fig. S-2
3. Calculation of droplet volume
We assumed the droplet shape as an ovoidal solid (long axis, a; short axis, b) tilted  from the origin, and the
contact area shape between the droplet and SLBC as a circle (Fig. 9 in the text). The coordinate (x, y) for the point
on the oval sphere can be described using the following equation.
(𝑥cos𝜑+𝑦sin𝜑)2
𝑎2
+
(−𝑥sin𝜑+𝑦cos𝜑)2
𝑏2
=1
(S2-1)
The expansion of eq. (S2-1) provides the following equation (S2-2).
(𝑏2 cos2 𝜑 + 𝑎2 sin2 𝜑 )𝑥 2 + 2cos𝜑sin𝜑(𝑏2 − 𝑎2 )𝑦𝑥 + (𝑏2 sin2 𝜑 + 𝑎2 cos 2 𝜑)𝑦 2 − 𝑎2 𝑏2 = 0
(S2-2)
The solution of eq. (S2-2) by assuming constant y is the following.
𝑥=
−cos𝜑sin𝜑(𝑏 2 −𝑎2 )𝑦±√cos2 𝜑 sin2 𝜑(𝑏 2 −𝑎2 )2 𝑦 2 −(𝑏2 cos2 𝜑+𝑎2 sin2 𝜑){(𝑏 2 sin2 𝜑+𝑎2 cos2 𝜑)𝑦 2 −𝑎2 𝑏2 }
𝑏 2 cos2 𝜑+𝑎2 sin2 𝜑
(S2-3)
The horizontal length (d) of the droplet at y is equal to the difference of two solutions in eq. (S2-3).
𝑑=
2√cos2 𝜑 sin2 𝜑(𝑏 2 −a2 )2 y2 −(𝑏2 cos2 𝜑+a2 sin2 𝜑){(𝑏 2 sin2 𝜑+a2 cos2 𝜑)𝑦 2 −a2 𝑏2 }
𝑏 2 cos2 𝜑+a2 sin2 𝜑
(S2-4)
The contact area shape is assumed as a circle. Therefore, the entire droplet volume is given as shown in eq. (S2-5).
𝛽
𝑑 2
𝛺droplet = ∫ 𝜋 ( ) 𝑑𝑦
2
𝛼
𝛽
= π∫ [
𝛼
cos 2 𝜑 sin2 𝜑(𝑏 2 − 𝑎2 )2 𝑦 2 − (𝑏 2 cos 2 𝜑 + 𝑎2 sin2 𝜑){(𝑏 2 sin2 𝜑 + a2 cos 2 𝜑)𝑦 2 − 𝑎2 𝑏 2 }
] 𝑑𝑦
(𝑏 2 cos 2 𝜑 + 𝑎2 sin2 𝜑 )2
𝛽 𝑎2 𝑏2 {(𝑏2 cos2 𝜑+𝑎2 sin2 𝜑)−𝑦 2 }
]
(𝑏 2 cos2 𝜑+𝑎2 sin2 𝜑 )2
= π ∫𝛼 [
𝑑𝑦
(S2-5)
Here,  and  are y-coordinate at the top and the bottom of the droplet. The line parallel to x-axis and contact with
the top of the droplet is the following.
y=
(S2-6)
Both eq. (S2-2) and eq. (S2-6) are mutually contacted. The discriminant of the simultaneous equation is expected to
be 0. Consequently, the following equation can be obtained.
𝛽 = √𝑏2 cos2 𝜑 + 𝑎2 sin2 𝜑
(S2-7)
The value of  can be found from droplet height (H) and .
𝛼 = 𝛽 − 𝐻 = √𝑏2 cos2 𝜑 + 𝑎2 sin2 𝜑 − 𝐻
(S2-8)
Using eqs. (S2-5), (S2-7), and (S2-8), the droplet volume can be calculated as shown below.
𝛽
𝛺droplet =
=
𝛽
π𝑎2 𝑏2
π𝑎2 𝑏2 2
1
2
2
2
2
2}
{(𝑏
∫
cos
𝜑
+
𝑎
sin
𝜑)
−
𝑦
𝑑𝑦
=
[𝛽 𝑦 − 𝑦 3 ]
2
2
2
2
2
4
(𝑏 cos 𝜑 + 𝑎 sin 𝜑 ) 𝛼
𝛽
3
𝛼
π𝑎2 𝑏2 2
1
π𝑎2 𝑏2 2
1
2
2
{𝛽
(𝛽
−
𝛼)
−
(𝛽
−
𝛼)(𝛽
+
𝛼𝛽
+
𝛼
)}
=
[𝛽 𝐻 − 𝐻{𝛽 2 + 𝛽(𝛽 − 𝐻) + (𝛽 − 𝐻)2 }]
4
4
𝛽
3
𝛽
3
π𝐻2 𝑎2 𝑏2
= 3(𝑏2 cos2 𝜑+𝑎2 sin2 𝜑 )2 (3√𝑏2 cos2 𝜑 + 𝑎2 sin2 𝜑 − 𝐻)
(S2-9)
4. Effect of droplet volume on the wetting ridge size
Figure S-3 depicts the width and height of the wetting ridge on sample A for water droplets having various
volumes: 1, 2, 3, 5, and 10 L. The wetting ridge size does not depend strongly on the droplet volume on sample A.
hr
b
Wetting ridge
Height and width of wetting ridge [mm]
Droplet
0.1
0.08
0.06
Height
0.04
Width
0.02
0
0
2
4
6
8
10
Water droplet volume [μL]
12
Fig. S-3
5. Calculation of solid–liquid contact area ratio at the interface
When fine roughness is provided onto the solid part with a large roughness, Cassie’s equation is modified as
follows [2].
cos’ = rf cos + (1f ) cos 180o = rf cos + f1 (S4-1)
Here, r stands for the Wenzel’s roughness factor (surface area ratio between rough surface and smooth surface under
the same projected area)[3], and f represents the solid–liquid contact area ratio. In this calculation, we assume that
the surface state of sample C is attained even onto the surface of porous glass or porous glass with fine roughness by
colloidal silica. Hereinafter, 0, 1, 2, and 3 are denoted respectively as the water contact angles for sample C, porous
glass after FAS17 modification, colloidal silica coating onto flat substrate and subsequent FAS17 modification, and
the sample before oil impregnation. Then r1, r2, f1, and f2 respectively denote the roughness factors of porous glass
and that of colloidal silica coating with FAS17 modification, and the solid–liquid contact area ratio for these two
samples. The contact angles of porous glass (with FAS17 modification), colloidal silica (with FAS17 modification),
and the sample before oil impregnation can be described by the following equations.
cos1 = r1f1 cos0 + f1
(S4-2)
cos2 = r2f2 cos0 + f1
(S4-3)
cos3 = r1f1 cos2 + f1 = r1r2f1f2 cos0 + r1f1f2 r1f1 + f1 1
(S4-4)
Eqs. (S4-2) and (S4-4) provide the following.
r1 = (cos1 f1 +1) / f1cos0 = (cos3 f1 +1) / f1cos2
𝑓1 =
1
cos𝜃2 −cos𝜃0
(S4-5)
(cos𝜃2 cos 𝜃1 − cos𝜃0 cos𝜃3 + cos𝜃2 − cos𝜃0 ) (S4-6)
From eq. (S4-3), the following can be inferred.
f2 = (cos2 +1) / (r2cos0 +1)
(S4-7)
The respective practical values of 0, 1, 2,3, and r2 are 110.6°±0.1°, 136.9°±1.4°, 133.1°±0.8°147.6°±1.6°, and r2
= 1.53±0.36 which were obtained from AFM measurements. Also, r1r2f1f2 and r1f1f2 r1f1 + f1 1 are calculable
respectively as 0.3756 and 0.7157 from these values, which correspond to the practical solid–liquid contact area
ratio and the air–liquid contact area ratio on the surface with two roughness values. These values are converted to
0.344 and 0.656 when the total value is equal to unity.
6. Sliding resistance on the SLBC surface of water droplets with various volumes
Sliding velocities of water droplets with various volumes were evaluated on samples A and B tilted at 35°. The
motion equation of the droplet with mass m and contact radius r can be described by the following equation [4].
𝑚𝑎 = 𝑚g sin 35° − 2𝜋r𝑘
(S5-1)
𝑚𝑎 = 𝑚g sin 35° − πr2 𝑘′ (S5-2)
Therein, a is the sliding acceleration of the droplet; k and k’ are constants. Equation (S5-1) shows that the sliding
resistance is governed by the outer perimeter of the contact area. The meaning of equation (S5-2) is that the sliding
resistance is determined by the contact area itself. The equations are converted, respectively, to equations (S5-3) and
(S5-4).
𝑎 = g sin 35° − 2𝜋𝑘
𝑎 = gsin35° − πk ′
𝑟
𝑚
𝑟2
𝑚
= 5.62 − 2𝜋𝑘
= 5.62 − πk ′
𝑟
𝑚
𝑟2
𝑚
(S5-3)
(S5-4)
Figure S-4 shows plots for the relation between a and r/m (Fig. S-4(a)) or r2/m (Fig. S-4(b)). The y-intercept of
sample became 5.85 against r/m; a linear relation was obtained. Although the data points were scattered, the yintercept was close to the expected value (5.62) for sample A also when the x-axis is r/m. These results imply that the
outer perimeter of the contact area plays a more important role for the sliding resistance than the contact area itself.
8
(a)
6
Sample A
4
Sample B
y = -0.0405x + 5.854
R² = 0.9792
2
0
0
50
r/m [m/kg]
100
Sliding acceleration [mm/s2]
Sliding acceleration [mm/s2]
Because of the high viscosity of the impregnated oil, this trend is more remarkable in sample B.
8
(b)
Sample A
6
4
Sample B
y = -57.864x + 10.969
R² = 0.8436
2
0
0
r2/m
Fig. S-4
0.1
[m2/kg]
0.2
7. Laplacian pressure on the sandwiched configuration of SLBC
A schematic illustration of the liquid droplet sandwiched between two SLBCs is depicted in Fig. S-5. Laplacian
pressure for the droplet and wetting ridge is calculable as shown below.
1
1
∆𝑃 = 𝛾(𝑅 + 𝑅 )
1
2
(S7-1)
Here,  denotes the surface tension of the liquid; R1 and R2 respectively stand for the corresponding curvatures in Fig.
S-5. The respective practical values of these curvatures and calculated Laplacian pressure are listed in Tables S-3.
Table S-3
(a)
R2
R1
R2’
R1’
R1, R2 : radius curvature of droplet
R1’, R2’ : radius curvature of wetting ridge
Fig. S-5
wetting ridge (R1’: mm)
sample
droplet
(R1: mm)
top side
bottom side
A–A
2.39
0.09
0.11
B–B
2.43
0.14
0.11
(b)
wetting ridge (R2’: mm)
sample
droplet
(R2: mm)
top side
bottom side
A–A
2.04
1.54
1.79
B–B
2.11
1.68
1.94
(c)
wetting ridge (N/m2)
sample
droplet
(N/m2)
top side
bottom side
A–A
65.7
845
856
B–B
64.0
540
548
8. Practical distribution of velocity vector in the test-fluid droplet sliding between two samples
Fig. S-6
References
[1] Y. Tsuruki, M. Sakai, T. Isobe, S. Matsushita, A. Nakajima, Static and dynamic hydrophobicity of alumina-
based porous ceramics impregnated with fluorinated oil, J. Mater. Res. 29(14) (2014) 1546–1555.
[2] M. Miwa, A. Nakajima, A. Fujishima, K. Hashimoto, T. Watanabe, Effects of the Surface Roughness on Sliding
Angles of Water Droplets on Super-Hydrophobic Surfaces, Langmuir 16(13) (2000) 5754–5760.
[3] R.N. Wenzel, Resistance of solid surfaces to wetting by water, Ind. Eng. Chem. 28 (1936) 988–994.
[4] A. Nakajima, S. Suzuki, Y. Kameshima, N. Yoshida, T. Watanabe, K. Okada, Sliding Mode Transition of
Water Droplet on the Silicon Surface Coated with Octadecyltrichlorosilane, Chem．Lett. 32(12) (2003) 1148–
1149．