Wetting and Dewetting Transitions on Submerged Superhydrophobic

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Wetting and Dewetting Transitions on Submerged Superhydrophobic Surfaces with Hierarchical Structures Huaping Wu, Zhe Yang, Binbin Cao, Zheng Zhang, Kai Zhu, Bingbing Wu, Shaofei Jiang, and Guozhong Chai Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b03752 • Publication Date (Web): 17 Dec 2016 Downloaded from http://pubs.acs.org on December 20, 2016

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Wetting and Dewetting Transitions on Submerged Superhydrophobic Surfaces with Hierarchical Structures Huaping Wu,1,2,* Zhe Yang1, Binbin Cao1, Zheng Zhang, Kai Zhu1, Bingbing Wu1, Shaofei Jiang1, Guozhong Chai1*

1

Key Laboratory of E&M (Zhejiang University of Technology), Ministry of Education & Zhejiang Province, Hangzhou 310014, China

2

Jiangsu Key Laboratory of Engineering Mechanics, Southeast University, Nanjing 210096, China *Corresponding author. Email: [email protected]; [email protected].

Abstract: The wetting transition on submersed superhydrophobic surfaces with hierarchical structures and the influence of trapped air on superhydrophobic stability are predicted based on the thermodynamics and mechanical analyses. The dewetting transition on the hierarchically structured surfaces is investigated, and two necessary thermodynamic conditions and a mechanical balance condition for dewetting transition are proposed. The corresponding thermodynamic phase diagram of reversible transition and the critical reversed pressure well explain the experimental results reported previously. Our theory provides a useful guideline for precise controlling of breaking down and recovering of superhydrophobicity by designing superhydrophobic surfaces with hierarchical structures under water.

Keywords：：Submerged superhydrophobic surfaces, Hierarchical structures, Trapped air, Wetting 1

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transition, Dewetting transition

1. Introduction

Natural superhydrophobic surfaces, such as lotus leaves 1 and water strider legs 2, are known for their ability to completely repel water. These superhydrophobic surfaces with hierarchical structures (micro- and nano-structures) and low surface free energy make water drops rest on top of the structures and trap air underneath, that is Cassie−Baxter (CB) state3. The stability of the air-water interface under pressure is critically important and influenced by various external stimuli, such as external pressure 4, critical hydrostatic pressure surface

5,6

, high Laplace pressure

7,8

, impact against the

9-14

, or mechanical vibration15,16, leading to a partial or complete wetting of the surface

(Wenzel state 17) by forcing the water into the air-filled holes or posts of the surface. Further increase in the amplitude of the external stimuli will lead to the evolution of the drop toward the final wetting state, which is the Cassie impregnating18,19. The process of wetting transition can be divided into two situations, one is quick transition taken place under a constant apparent contact angle, the other is slow transition in which the change of the apparent contact angle should be taken into account20. Recently, much attention has been focused on air-liquid interface on the submersed superhydrophobic surfaces with anticipated applications in drag reduction21,22, antibiofouling

23

and

submersed bodies24,25. The air is trapped in the microstructures of superhydrophobic surfaces and the trapped air pressure is found to be great importance for superhydrophobic stability enhancement in this closed system. For example, Forsberg et al.

26

experimentally confirmed the important role of

trapped air pressure in enhanced superhydrophobic stability. Lv et al. 2


27

examined the dynamical

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evolution of liquid-air interfaces on a submerged surface with cylindrical micropore patterns, and established a theoretical formulation to predict the lifetime of trapped air. Hemeda et al. Lobaton et al.

29

28

and

numerically simulated the improved collapse pressure of three-phase contact line

(TCL) depinning induced by trapped air pressure. However, the quantitative explanation and understanding from thermodynamic aspects on how trapped air pressure affects the superhydrophobic stability of hierarchically structured surfaces are still scarce. Moreover, dewetting transition is key feature to the robust superhydrophobicity, though it is usually ignored. Recent several reports have convincingly demonstrated the reversibility of CB-to-Wenzel (CB→W) wetting transition by several methods, such as the coalescence of Wenzel and CB droplets 30, the application of short-pulse electrical current through a conductive substrate 31, magnetically induced reversible transition

32

or vibration-induced transition 33, when the CB droplet

is in a thermodynamically favored state and the Wenzel droplet in a metastable one. The Wenzel-to-CB (W→CB) transition is also observed when the Wenzel droplet is at the global minimum-energy state via high-temperature heating environment induced

35

34

, as well as other ways, such as low-pressure

and humidity-controlled reversible transitions

36

. However, these methods

are impractical for the submersed superhydrophobic system. Additionally, dewetting transition (partial Wenzel state to the CB one) is also confirmed in microfabricated hierarchical structures immersed in water via negative pressure

38

, and generated a stable gas layer

39

or electrolysis

37

40

.

However, how to precisely design the hierarchical structures to achieve dewetting transitions and control the wetting-dewetting transition on submersed superhydrophobic surfaces is still hard. In this work, we establish a theoretical framework to understand how superhydrophobicity 3


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breaks down and recovers on submerged structured surfaces from thermodynamic aspects. In the first section, we investigate wetting transition on hierarchically structured surfaces and the influence of trapped air on superhydrophobic stability. The collapse pressure is predicted and compared with experimental results reported previously. In the second section, the dewetting transition on hierarchically structured surfaces is analyzed. Two necessary thermodynamic conditions are presented, and the corresponding thermodynamic phase diagram of reversible transition is depicted. The mechanical balance condition for dewetting transition and the critical reversed pressure are also given, which is helpful for precise controlling of wetting-dewetting transition on hierarchically structured surfaces under water.

2. Formulation

In our theoretical model, the size of microstructures under consideration is much smaller than that of the capillary length a, defined by a = (γ LV / ρ g )1/2 , where γLV is the liquid surface tension,

ρ is the liquid density, and g is the gravity acceleration. Thus, a meniscus with constant curvature could be assumed, and the effect of liquid gravity on the asperities and line tension is usually negligible 41,42. We first set up a formulation in terms of a model from CB-CB state to CB-W one on the hierarchically structured surfaces with cylindrical pores (radius r1, depth h1) on a periodic square lattice as shown in Figure 1 a. On the hydrophilic substrate, liquid will more likely to penetrate into pores to release surface energy18,19. While on the hydrophobic substrate, liquid will not penetrate into pores and spread along the sidewall spontaneously. And the resistance against wetting of nanostructure is much higher than that of microstructure. Thus, the nanostructures are assumed no 4


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infiltration, that is in CB state38,43. The front and back parts of the labels for the CB-CB and CB-W states represent the wetting states of nanostructures and microstructures, respectively. f 1 and f 2 are the solid fractions of the microstructures and nanostructures. Meanwhile, this formulation based on pore structures will be extended to investigate pillared structures with radius rp1 (Figure 1 b) according to the principle of equivalence proposed by Xue et al, 43 and the equivalent geometric parameters of pillars are obtained (see Part S1 in Supporting Information).

Figure 1. (a) Hierarchically structured superhydrophobic surface with cylindrical pores (radius r1, depth h1) under water and partially enlarged top and side views; (b) hierarchically structured superhydrophobic surface with pillars under water.

The whole closed interface system is assumed to be an isothermal ensemble. The ambient air is at constant temperature T 0 and pressure P0 . Taking the energy of trapped air G air into consideration, the free energy of whole system G can be expressed as 44 G = γLV S LV + γSL S SL + γSV SSV + G air

,

(1)

where γLV , γSL and γSV are the liquid-air, solid-liquid and solid-air interfacial tensions, respectively; S LV , SSL and SSV are liquid-air, solid-liquid and solid-air interfacial areas, respectively. The G air

can be expressed as 45

5


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P1

Gair = G P0 + ∫ VdP P0

(2) .

Here GP0 is the free energy of the trapped air in initial state. V and P represent volume and pressure, respectively. The subscripts 0 and 1 denote the reference and final states of the trapped air, respectively. For submersed superhydrophobic surfaces, the energy barrier which needs to be overcame during infiltrating the microstructure is analyzed and the change of apparent contact angle is not included27. Assume that the energy of initial state is G 0 , and regardless of air dissolution in the water 43,45, then the free energy of other state (except for CB-W state) can be expressed as G = G0 + γLV ∆S LV + γSL ∆ SSL + γSV ∆ SSV + nkT ln( P1 / P0 )

,

(3)

where ∆S LV , ∆SSL and ∆SSV are the change of liquid-air, solid-liquid and solid-air interfacial areas, respectively; n is the gas moles number; k is the ideal gas constant; T is the temperature; the ideal gas law PV = nkT is assumed. Consider a pore as analysis unit, the process of depinning transition on hierarchically structured surfaces can be divided into three processes46,47,48,49 (Figure 2). The first process is TCL pinning of CB-CB state (Figure 2i-iii), in which the meniscus is expanded continuously as the sag angle of liquid−air interface ( α 1 ). The free energy of the system GCB-CB in this process can be expressed as

  2 h GCB-CB = G0 + Iπ r12 γLV ( − 1) + Λ1 ln( 1 )  . h1 − hA1   1 + sin α1

(4)

Here I is the number of pores submersed under the liquid; Λ 1 ( Λ 1 = P0 h1 / γ LV ) is the ratio of free energy of trapped air and liquid-air interface tension in initial state; hA1 is the average protrusion depth (Eq.S3 in Supporting Information).

6


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Figure 2. Schematic illustrating the transition process of CB-CB state to CB-W one: (i)-(iii) three-phase contact line (TCL) pinning; (iii)-(v) TCL moving; (v)-(vii) wetting of substrate bottom. The ambient air is at constant pressure P0, α1 is the sag angle of the liquid−air interface, θadv is the apparent advancing contact angle in sidewalls of the microstructures, x1 is the depth of TCL downward movement, and ∆h1 is the sum of the height of meniscus hm1 and the depth of TCL downward movement x1.

When the liquid pressure reaches a collapse pressure ( Pcr ) A , that is, the sag angle α 1 reaches apparent advancing contact angle θ adv on hierarchically structured sidewalls, the TCL will move downward, as shown in Figure 2(iii-v). θ adv can be expressed by a modified CB equation 3, 43,50 cos θ adv = f 2 cos θ adv0 + f 2 − 1 ,

(5)

where θ adv0 is the advancing contact angle of smooth sidewalls. The free energy of the system GCB-CB → CB-W in this process can be expressed as

  2 2x h1 GCB-CB→CB-W = G0 + Iπ r12 γLV ( − 1 − 1 ( f 2 cos θ0 + f 2 − 1)) + Λ1 ln( ) , r1 h − (hA1 )max − x1   1 + sin θadv

(6)

where θ 0 is the intrinsic contact angle, x1 is the depth of TCL downward movement, ( hA1 ) max is the maximum value of hA1 . The expression of ( hA1 ) max can be obtained by replacing α 1 in Eq. (S3) by apparent advancing contact angle θ adv . In the third process (Figure 2v-vii)，the meniscus contacts substrate and the bottom of substrate is wetted. This process will be analyzed in detail in Part 2. The 7


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completely wetted bottom of substrate will lead to the CB-W state (Figure 2 vii), and the free energy in the CB-W state GCB-W is expressed as

GCB-W = G0 − π r12γLV − (π r12 + 2π rh 1 1 )( f 2 cos θ0 + f 2 −1)γLV − Gp0 . If introduce a nondimensionalized total free energy G * ( G* =

(7)

(G − G0 )(1 − f1 ) ), the total free energy IγLVπ r12

from CB-CB to CB-W transition G * can be expressed as

h1 2  (1 − f1 )(1 + sin α − 1) + (1 − f1 )Λ1 ln( h − h ), (π / 2 ≤ α1＜θadv ) (CB-CB state) 1 1 A1  h1 2 x1 2  ( f 2 cos θ0 + f 2 − 1)) + (1 − f1 )Λ1 ln( ) −1 − (1 − f1 )( * 1 + sin θadv r1 h − (hA1 ) max − x1 . G =  (α1 = θ adv ,0 ≤ x1＜h1 − hm1 ) (CB-CB → CB-W state)  2h1  * (CB-W state) −(1 − f1 ) − (1 − f1 )(1 + r )( f 2 cos θ 0 + f 2 − 1) − GP0 , ( x1 = h1 ) 1 

(8)

Here, hm1 is the height of meniscus and GP*0 is nondimensionalized free energy of the trapped air in initial state. When f 2 = 1, Eq.(8) will be suitable for single-scale rough surfaces. The above process of transition also exists on hierarchically structured surfaces with pillars, and the corresponding expression (Eq. S4) is obtained (see Part S2 in Supporting Information). From Eq.(6), we can find that the free energy will be infinite with increasing x1 . It means that the transition from CB-CB state to CB-W one is hardly complete. It's worth noting that the size of microstructures is often on the order of one micrometer, even one nanometer. Therefore, the volume of trapped air is limited. Previous researches

27,51

showed that the air would dissolve gradually in the water or escape out

under compression, and then the energy of interface could achieve a balance. That is, the increment of energy (caused by the TCL moving downward and compression of trapped air) is equal to the dissipation of energy (caused by the dissolution and escaping out of air), at which the energy will reach the maximum value. After that, the meniscus moves downward sequentially and transfers to 8


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CB-W state. In the following discussion, we assume that trapped air will not escape out or diffuse into the liquid until the meniscus contacts the bottom of substrate or the trapped air is compressed completely26.

3. Results and discussion

3.1 Wetting Transition from CB-CB State to CB-W State Consider a hierarchically structured surface with cylindrical pores, e.g. θ 0 =95°and θ adv0 =103.5°(polyethylene surface 27) and the geometrical sizes of h1 = r1 = 1µm , f1 = 0.25 and f 2 = 0.5 . A single-scale rough surface with the geometrical sizes of f1 = 0.25 and f 2 = 1 is also chosen as a comparison. The nondimensionalized free energy G * as a function of relative protrusion depth ∆ h1 / h1 on hierarchically structured surface (or single-scale rough surface) is shown in Figure 3(a)

(or

Figure

3b),

in

which

∆h1

is

the

sum

of

the

height

hm1

of

the

meniscus

( hm1 / h1 = (sin α1 − 1) / cos α1 ) and the depth of TCL downward movement x1 . The dotted lines in Figure 3(a) and (b) indicate the open interface system as a comparison, namely the impact of trapped gas is ignored. It can be seen that the whole transition process ranges from point A to point D. The free energy difference between point A and point C is the total energy barrier ∆G* . In order to analyze the influence of f 2 and trapped air on wetting stability more clearly, we introduce three kinds of energy barriers according to wetting transition process. The first process is the TCL pinning (Figure 2 (i-iii)) ranging from point A to point B ( B ' ) in Figure 3, and the free energy increases with increasing sag angle α 1 in the process. The free energy difference between point A and point B ' is 9


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* energy barrier of meniscus ∆Gmeniscus , and can be expressed as

2 * ∆Gmeniscus = (1 − f1 )( − 1) , 1 + sin θadv

(9)

where θ adv can be obtained from Eq.(5). The second one is the process of TCL moving downward (Figure 2 (iii)-(v)) as shown from point B ( B ' ) to point C ( C ' ). The free energy difference between * point B ' to point C ' is capillary energy barrier ∆Gcapil lary , and can be expressed as

* ∆Gcapilla ry =

−2 h1 (1 − f1 ) ( f 2 cos θ 0 + f 2 − 1) . r1

(10)

The third one is the wetting process of substrate bottom (from point C ( C ' ) to point D). With the expanding of the liquid-air interface and the downward movement of TCL, the trapped air is compressed which could resist the infiltration of water. Since the resistance exists in the whole process of transition, the free energy difference between the solid line and the dotted line is energy * barrier of the trapped air ∆Gair , and can be expressed as

h1  (π / 2 ≤ α1 ≤ θ adv ) (1 − f1 ) Λ1 ln( h − h )  1 A1 * ∆Gair =  . h1 (1 − f ) Λ ln( ) (α1 = θ adv , 0 ≤ x ≤ h1 − hm1 ) 1 1  h1 − ( hA1 ) max − x1

10


(11)

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3.0

G

∗

2.5

(a)

2.0

CB-CB

1.5

α1→θadv

C

CB-CB→CB-W x1→h1 *

∆Gair

1.0 0.5

*

∆Gcapillary

B

0.0

B'

A

C' D

0.1

G

∗

-0.5

0.0

∆G*meniscus

B'

-1.0

0.32 0.33 0.34 0.35 CB-W ∆h /h -1.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1

1

∆h1/h1

3.2 2.8 2.4 2.0 CB α →θ 1.6 1 adv0 1.2 0.8 0.4 B 0.0 A B′ -0.4 -0.8

*

(b)

G

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C

CB→W x1→h1

C′ W D

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

∆h1/h1

Figure 3. Normalized minimum free energy on submersed superhydrophobic surfaces with cylindrical pores as a function of relative protrusion depth: (a) hierarchically structured surface, (b) single-scale rough surface. The solid and dotted lines indicate the closed and open interface systems, respectively.

* * As shown in Figure 3, three kinds of nondimensionalized energy barriers ∆Gmeniscus , ∆Gcapillary * and ∆Gair are 0.089, 0.538 and 1.89 for a hierarchically structured surface and 0.011, 0.115 and

2.89 for a single-scale rough surface, respectively. The hierarchical structures can increase the * * * and ∆Gcapillary , while decrease ∆Gair . The fact is that larger apparent contact angle θ (can ∆Gmeniscus

be easily obtained by CB equation) and advancing contact angle θ adv can be provided on 11


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hierarchically structured surface than the original θ 0 and θ adv0 on single-scale rough surface. For * , however, hierarchically structured surface can increase the height of meniscus, makes the ∆Gair

meniscus quicker contact with the substrate and achieve a balance of free energy. Therefore, the role of trapped air to resist the infiltration of water is weakened. For pillared surfaces, a similar conclusion can be obtained, as shown in Figure S1 in Supporting Information. Based on above theoretical analysis, three kinds of energy barriers are all influenced by the * * * solid fraction f 2 . The ∆Gmeniscus and ∆Gcapillary increase, while ∆Gair decreases, as the f 2

decreasing (Figure 4a). The total energy barrier slight decreases in the whole transition process as the * * f 2 decreasing (Figure 4b). Additionally, the total energy barrier (includes ∆Gmeniscus and ∆Gair ) in

the first process of transition increases as the f 2 decreasing, as shown in Figure 4c. Therefore, hierarchical structures can improve the superhydrophobic stability of submersed hierarchical surfaces, but no contribution to the total energy barrier in the whole transition process.

12


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3.0 *

∆Gmeniscus

2.5

*

∆Gcapillary *

∆Gair

∆G

*

2.0 1.5 1.0 0.5 0.0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

f2 3.2

α1 =θadv ,x1=h1

2.8 2.4

*

*

∆G

Closed system Open system

∆Gair

2.0 1.6 1.2 0.8 0.4 0.0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

f2 0.9 Closed system Open system

α1 =θadv

0.8 0.7 0.6 *

0.5 0.4

*

∆Gair

∆G

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0.3 0.2 0.1 0.0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

f2

Figure 4. Nondimensionalized energy barrier as a function of solid fraction f2 with given microstructures: (a) three kinds of energy barriers; (b) the whole process of transition; (c) the first process of transition (TCL pinning). Though energy barriers must be overcome for wetting transition from thermodynamic aspect, a 13


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sufficient condition is still scarce, namely how much pressure can force this transition in a closed interface system? Previous prediction either did not take into account the increasing air pressure as the meniscus curvature enlarging in the closed system, or ignored that the balancing upward force caused by surface tension was the largest when α 1 reached the θ adv , rather than the θ . A simple way to estimate the collapse pressure of the TCL depinning is to balance forces from pressure and surface tension for unit cell with symmetric boundary conditions. We denote the cross-sectional area and perimeter of the pore by A1 and L1 . For the hierarchically structured surfaces with pores, the equilibrium of the water-air interface requires (( PA ) cr − PV ) A1 = −γ LV L1 cos θ adv of trapped air PV = P0 ln

52

, and the pressure

h1 . Combining with the Eq.(5), the collapse pressure ( PA ) cr on h1 − (hA1 ) max

hierarchically structured surfaces with pores can be expressed as

( PA )cr =

−2( f 2 cos θadv0 + f 2 − 1) h1 γ LV + P0 ln . r1 h1 − (hA1 )max

(12)

In a similar way, the collapse pressure ( PA ) cr for hierarchically structured surfaces with pillars can be expressed as ( PA ) cr =

− f1 ( f 2 cos θ adv0 + f 2 − 1) h1 γ LV + P0 ln , eff (1 − f1 ) rp1 h1 − ( hA1 ) max

(13)

eff where (hA1 ) max is the equivalent value of ( hA1 ) max for hierarchically structured surfaces with pillars.

The Eq. (12) and Eq. (13) are valid for various pore or pillar cross sections. The second terms in Eq. (12) and Eq. (13) are the contribution of trapped air on superhydrophobic stability. When f 2 = 1, Eq. (12) and Eq. (13) will be suitable for single-scale rough surfaces. When the θ adv0 is replaced by θ 0 , Eq. (13) can degrade into the equation for an open interface system proposed by Zheng et al 52. In order to confirm the validity of our prediction equations, the calculated collapse pressure is 14


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compared with those experimental results and theoretical predictions reported in previous literatures. Here we consider circular pillars with h1 = 9.5µm , radius r1 = 3.0µm , and two different advancing contact angles of θ adv0 =103.5°and θ adv0 =122.1°so as to compare with the experimental data of Ref.26. The collapse pressure ( PA ) cr as a function of the solid fraction f1 (i.e., 0.1-0.45) (together with the experimental results from Forsberg et al.26 and other theoretical predictions from Xue et al. 43

and Hemeda et al.

28

) is plotted in Figure 5. It can be seen that the ( PA ) cr is in good agreement

with the numerical results of Forsberg et al.

26

, Xue et al.

43

and Hemeda et al.

28

in f2 = 1 (a

single-scale rough surface). Our theoretical predictions (and other numerical results) show good agreement with the experimental data for a low f1 (0.1-0.3), while some difference when f1 >0.3. The differences between theoretical predictions and the experimental data are widened as increasing

f1 . While for a special f2 ( f 2 = 0.8 for example), the good agreement between the experimental data and our theoretical predictions can be obtained in the whole f1 range. Since microstructures or defects on substrates can significantly increase the wetting hysteresis due to TCL pinning, the imperfect and unsmooth sidewalls with certain f 2 perhaps enlarge the collapse pressure (as shown in Forsberg’s experiment26). Additionally, the θ adv0 can affect collapse pressure according to Eq. (12) and Eq.(13) and results in a significant increase of ( PA ) cr (Figure S2) whatever for an open system or a closed system. Therefore, an important way for enhancing the superhydrophobic stability is to improve the θ adv0 .

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30

Numerical (Forsberg et al.,2011) Numerical ( Hemeda A. A.et al.,2014) Numerical (Xue et al.,2012) Experiment (Forsberg et al.,2011) θadv=103.5°

25

(PA)cr(kpa)

20 15 10 5 0 0.1

0.2

f2=1

f2=0.8

f2=0.4

f2=1(open system)

0.3

f2=0.6

0.4

0.5

f1 35

Numerical (Forsberg et al.,2011) Numerical (A. A. Hemeda etal.,2014) Numerical (Xue et al.,2012) Experiment (Forsberg et al.,2011) θadv0=122.1°

30

(PA)cr(kpa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 31

25 20 15 10 5 0.1

0.2

f2=1

f2=0.8

f2=0.4

f2=1(open system)

0.3

0.4

f2=0.6

0.5

f1 Figure 5. Collapse pressure of TCL depinning ( PA )cr (calculated by Eq. 13) as a function of the solid fraction f1 of microstructures on the pillared substrates ( r1 = 3µm and h1 = 9µm ): (a) θ adv0 =103.5°, (b) θ

adv0

=122.1°.

3.2 Dewetting Transition from CB-W State to CB-CB State Two thermodynamic conditions must be satisfied for dewetting transition from CB-W state to CB-CB one. The first condition is that CB-CB state is the thermodynamically favored state, that is, CB-CB state is located at global energy minimization, and the nondimensionalized free energy difference between CB-W state and CB-CB one ( ∆G1* ) must be more than zero for hierarchically 16


Page 17 of 31

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Langmuir

structured surfaces with pores. Thus we can get * * ∆G1* = GCB-W − GCB-CB = −(1 − f1 ) − (1 − f1 )(1 +

2 h1 )( f 2 cos θ 0 + f 2 − 1) − GP*0 > 0 , r1

(14)

* * where GCB-W and GCB-CB are nondimensionalized free energies of W-CB and CB-CB states,

respectively. ∆G1* > 0 means that the CB-CB state has the minimum global thermodynamic energy. c If the radius of hole r1 in Eq. (14) is replaced by the effective capillary radius reff1 (Eq. S2), we can

get similar expression suitable for hierarchically structured surfaces with pillars. The second condition for dewetting transition is that the wetting of substrates (Figure 2 v-vii) is a non-spontaneous process. In this stage, it still has to overcome the rest of capillary energy barrier on sidewalls, meanwhile, solid-air interface and liquid-air interface are replaced by solid-liquid interface, accompanied by energy release. The change of nondimensionalized free energy ∆G2* can be expressed as

∆G2* = −(1 − f1 )

2(sin θadv − 1) 2 − (1 − f1 )(1 + )( f 2 cos θ0 + f 2 − 1) . 1 + sin θadv cos θadv

(15)

Detailed derivation of Eq.(15) is available in Part S3 in Supporting Information. For hierarchically structured surfaces with pillars, ∆G2* can be expressed as ∆G2* = − (1 − f1 )

r g 2(sin θ adv − 1) 2 − (1 − f1 )(1 + eff1 )( f 2 cos θ 0 + f 2 − 1) . c 1 + sin θ adv reff1 cos θ adv

(16)

If the released energy in the wetting process of substrate bottom is larger than the rest of capillary energy barrier, that is ∆G2*＜0 , the wetting of substrate bottom is a spontaneous process which would be completed within a few milliseconds. On the contrary, if ∆G2*＞0 , the substrate bottom wetting is a non-spontaneous process with ring-shaped air pockets remaining around the bases of pores or pillars 38, which is beneficial for TCL receding. A reverse pressure can be applied to realize 17


Langmuir

reverse transition. Based on above theoretical analysis, we deduce that the nondimensionalized free energies ∆G1* and ∆G2* are influenced by the solid fraction f1 and f 2 on hierarchically structured surfaces. Assume constant pore and pillar sizes (e.g., r1=h1=1µm, rp1=h1=1µm), and θ adv0 =103.5°, θ 0 =95°, the contours of color fill plots of ∆G1* and ∆G2* as a function of f1 and f 2 are shown in Figure 6. The critical conditions of nondimensionalized free energies ∆G1* and ∆G2* are independent of f1 on hierarchically structured surfaces with pores (Figures 6 (a)-(b)), while a synergistic effect of f1 and f 2 on ∆G1* and ∆G2* exist on hierarchically structured surfaces with pillars (Figures 6

(c)-(d)).

0.9

0.9

-0.6700

0.8

0.1738 *

*

∆G10

0.4

0.5956

0.5

*

∆G2>0

*

∆G20

0.5

0.5400

0.4

0.9900

0.3

*

∆G10

-0.3800 -0.1550

0.5 0.4

*

∆G2

Wetting and Dewetting Transitions on Submerged Superhydrophobic

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