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Mechanism Study on Transition of Cassie Droplets to Wenzel State after Meniscus Touching Substrate of Pillars Tianqing Liu, Yanjie Li, Xiangqin Li, and Wei Sun J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b12042 • Publication Date (Web): 17 Apr 2017 Downloaded from http://pubs.acs.org on April 23, 2017

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The Journal of Physical Chemistry

Mechanism Study on Transition of Cassie Droplets to Wenzel State after Meniscus Touching Substrate of Pillars

Tianqing Liu*1, Yanjie Li1, Xiangqin Li1, Wei Sun2

1 School of Chemical Engineering, Dalian University of Technology, Dalian, 116024,

Liaoning Province, P.R.China.

2 Guangming Research & Design Institute of Chemical Industry, Dalian, 116000, Liaoning Province, P.R.China.

*Corresponding author.Tel.:+86 0411 84706360. E-mail address: [email protected] (T.Q. Liu). [email protected] [email protected] [email protected]

ACS Paragon Plus Environment

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Abstract: To understand the conditions and mechanism of droplet wetting transition from Cassie to Wenzel state (C-W) and furthermore to prohibit this transition are the key research contents about super-hydrophobic materials. In this study, the C-W transition process after meniscus touching substrate (MTS) was divided into different stages. Then, the changes of droplet interfacial free energy (IFE) after MTS were analyzed. And the resistance on three-phase contact line (TPCL) was also investigated. Furthermore, based on the droplet IFE always changing from high to low, a criterion formula for C-W transition was derived so that the mathematical model was founded. The calculation results show that the smaller the pitch and/or diameter of pillars, the more difficult the MTS and C-W transition for an initial sessile Cassie droplet. Therefore, nano textures can efficiently prevent the C-W transition. Besides, the MTS of droplets on short and high pillars can be realized with sag and TPCL depinning impalement respectively. Additionally, the greater the intrinsic contact angle, the more unfavorable the droplet C-W transition. Moreover, micro-nano two-tier textures can effectively inhibit the C-W transition. Finally, the model results are in good agreement with the experimental measurements reported in literatures, with 92% accuracy.


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1 Introduction Cassie and Wenzel states are two typical wetting statuses of droplets on rough surfaces. Droplets show large contact angles and small contact angle hysteresis, i.e. superhydrophobic properties, only when they exist in Cassie state. However, when droplets appear in Wenzel state, contact angle hysteresis is substantial, and the surface loses its superhydrophobicity. Therefore, the conditions and mechanism of droplet transition from Cassie to Wenzel state (C-W) on textured surfaces are the key research contents of superhydrophobic materials1. It is generally considered that there are two types of mechanism of droplet C-W transition, i.e. the threephase contact line (TPCL) below a droplet depinning impalement on high pillars, and the sag impalement pathway on short posts, respectively2-6. In fact, not only the liquid-gas interface beneath a droplet on short pillars is curved during the sag process, but the liquid-gas interface on high posts also appears bent while the TPCL depins and moves downwards3. Therefore, no matter the pillars are high or low, the mechanism is TPCL depinning impalement or sag impalement, the front of the meniscus must contact the substrate of pillars before the droplet C-W transition completes. That is, the meniscus touching substrate (MTS) is the necessary step for any a C-W transition. Then the further wetting whole structures beneath a droplet after MTS is the second and last step of the C-W transition. Although there have been lots of research reports about the spontaneous C-W transition of droplets on textured surfaces1-25, most of them were not related to the process after MTS. These studies included experimental observation3,7-12 of droplet wetting states and C-W transition conditions, the C-W transition driving forces6,11-21, and transition energy barrier5,7,10,22-25 etc. There are only a very few reports related to the wetting process after MTS below droplets26-30. Tsai et al26 and Sbragaglia et al27 recorded the wetting evolution process of droplets on transparent substrates with microscope and high-speed camera, showing the shape of initial wetting point and its expanding area beneath a droplet, as well as the moving speed of TPCL. However, these two reports were all limited to the description of experimental phenomena, and they did not present how the bottom of a Cassie state droplet was initially wetted, and did not involve the driving force of C-W transition. Most researchers believe that the wetting of pillar side and substrate under a droplet will be spontaneously


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completed once the front of meniscus touches the substrate1,3-5,7,8,18,31-33. However, the experimental results of Luo et al showed that the sagging triggered C-W transition could not be completed on micro/nano two-tier surfaces even if the meniscus beneath a droplet contacts the micro structural substrate28-30. Therefore, the C-W transition through MTS is conditional, i.e. may not be able to complete spontaneously. It is necessary to investigate in depth the mechanism and conditions of droplet C-W transition on various textured surfaces after MTS takes place. The C-W transition process of droplets after MTS was divided into different stages in this study. Then, the changes of interfacial free energy (IFE) of a small droplet after MTS were analyzed in the different stages. And the resistance of TPCL moving during the wetting process on the substrate of pillars was also investigated. Furthermore, based on the fundamental of droplet IFE always changing from high to low during the spontaneous transition of their wetting states, the criterion formula of whether droplet C-W transition through MTS was derived so that the physical and mathematical models of droplet C-W transition through MTS were founded. To our knowledge, this is the first time that the theoretical work based on force analysis to describe the C-W transition process after MTS is proposed.

2 Physical and mathematical model As shown in Figure 1, the liquid-air interface beneath a small composite droplet sitting on a structural surface tends to bend under the action of IFE gradient1. When the front of the meniscus contacts the substrate of pillars, MTS triggered C-W transition starts. In this study, the IFE gradient of a small droplet was taken as the force to drive both the liquid-air interface below the drop to bend and the MTS triggered wetting process to complete. The whole C-W transition process of a droplet on textured surfaces involves the initial and continuous bending of the liquid-air interface beneath the drop, the TPCL on high pillars depinning, and the MTS caused final C-W transition both on high and low pillars. The driving forces and resistances were describe blow for these different processes.


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A2

A1

A3 R1

hx2 H

A5

A4 x=H-hx1

x=H-hx2

A6 x=H-hx1+h

hx1 2l p

B1

B2

B3

B4’ hx2=H

φ

B6

hx1=H

d

B5

B4

x hx1=H-x

B4’’ hx1-h=H

Figure 1. Schematics of partial meniscus beneath a composite droplet and its C-W transition via MTS. x: the distance of the TPCL moving downward; d, H and p: diameter, height and pitch of pillars respectively; R1 and R2: the radiuses of two arcs on the meniscus, respectively locating between the two adjacent and diagonal pillars with a square array; hx1 and hx2: the respective bending depth of the above two arcs; h: the part of hx1 already below the base; l: the characteristic length of wetted solid-liquid interface on the substrate between two pillars; φ =π-θ≥π-θA; θ and θA are the contact angle and advancing contact angle on the side of pillars respectively. (A1)~(A6): meniscus declines by TPCL depinning impalement on tall pillars. (A1): meniscus bends before the TPCL depins; (A2): TPCL starts to move downwards; (A3): Front of meniscus just contacts the substrate of pillars with arc R2 just touching the base while arc R1 still not; (A4): The first stage of C-W transition via MTS finishes with arc R1 just contacting the base. The shape of arc R1 keeps unchanging during the TPCL moving; (A5): The C-W transition via MTS proceeds during the second stage with arc R1 continuously moving downwards and the solid-liquid area on substrate continuously increasing; (A6): The second stage of C-W transition via MTS finishes with all substrate wetted, i.e. the TPCL has moved to the foot of pillars. (B1)~(B6): meniscus declines by sag mechanism on short pillars. (B1) and (B2): meniscus bends before MTS; (B3): Front of meniscus just contacts the substrate of pillars with arc R2 just touching the base while arc R1 still not; (B4’): Case 1: The first stage of C-W transition via MTS finishes with arc R1 just contacting the base withφ> π-θA and the TPCL still pinned; (B4”): Case 1: The C-W transition via MTS proceeds during the second stage withφcontinuously declining until equal to π-θA, during which R1 keeps decreasing and hx1 increases continuously with the TPCL pinned; (B4): Case 2: The first stage of C-W transition via MTS finishes with arc R1 just contacting the base withφ= π-θA and the TPCL already depinning; (B5): The C-W transition via MTS proceeds during the second stage withφ=π-θA during whichφ, R1 and hx1 keep unchanging while the TPCL moves downward; (B6): The second stage of C-W transition via MTS finishes with all substrate wetted, i.e. the TPCL has moved to the foot of pillars.


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2.1 Bending of liquid-air interface beneath a droplet and the depinning of TPCL Since the bending of the liquid-air interface beneath a composite droplet under the action of IFE gradient was already described detailed in our early study1, the relative derivation is only illustrated briefly below. The formula of IFE gradient acted on a composite droplet on textured surfaces shown in figure 1 is1:

Fd = -

2 dEx  ∂Ex ∂Ex ∂θd  r  = 4πσ LG f s cosθI + 2πσ LG rs (1 − f ) sin θd = - + dx d  ∂x ∂θd ∂x 

(1)

where Ex is the IFE of the droplet, x is the distance of the TPCL moving downward, θd and θI are the apparent and intrinsic contact angle of the droplet, σLG is surface tension, f and d are the solid area fraction and pillar diameter respectively on textured surfaces, and rs is the droplet base radius.

The physical meaning of equation (1) can be explained as follows. The TPCL below a composite droplet spontaneously moving a short distance dx is resulted from its IFE decrease of dEx from high to low, i.e. the IFE change of a small droplet from high to low is the driving force of its wetting state spontaneous variation, which is in agreement with the fundamental that any an object will always change its position from high to low energy state spontaneously. If each term in equation (1) is divided by droplet base area, the C-W transition pressure acted on a composite droplet on textured surfaces can be obtained1:

Pt =

4σ LG f cos θ I 2σ LG (1 − f ) sin θ d + d rs

(2)

The liquid-gas interface beneath a composite droplet on textured surfaces with intrinsic contact angle greater than 90° tends to bend firstly under the action of above C-W transition pressure while the TPCL keeps pinned1, 34

. Meanwhile, a resistance to inhibit the bending of the liquid-gas interface, or to prevent the enlargement of

the liquid-gas interface area, will form at the same time once the liquid-gas interface curves. This resistance is expressed as1:

FR = πrs

2

(r − 1) σ H

LG

cos θ

(3)

in which r and H are Wenzel roughness factor and pillar height of textured surfaces respectively, θ is the contact angle between the liquid-air interface and the side wall of pillars below the droplet. As the IFE gradient


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shown in equation (1) or the C-W transition pressure shown in equation (2) increases, the bending degree of the liquid-gas interface is enhanced, the resistance shown in equation (3) also increases with the θ in the equation rising. Especially when θ reaches the advancing contact angle θA, the TPCL on pillars will start to depin. At this time, the resistance to prevent the bending of liquid-gas interface becomes maximum, which is the wetting force of the TPCL on pillar side walls1:

FW = πrs

2

(r − 1) σ H

LG

cosθ A = 4πrs

f σ LG cosθ A d

2

(4)

For cylindrical posts, the Wenzel roughness factor r and solid area fraction f in the above equations are given as:

f = π (d / p) 2 / 4

(5)

r = 1 + πdH / p 2

(6)

where d, H and p are diameter, height and pitch of pillars respectively. In summary, the criterion whether the liquid-gas interface bends beneath a composite droplet on textured surfaces, the calculation of the bending degree of the meniscus, as well as the criterion when the TPCL starts to depin, all can be expressed as follows: a) Pt ≤ 0, in this case there is no C-W transition pressure acting on the droplet, and the droplet stays at Cassie state without its bottom liquid-air interface bending. b) 0 < (1 − f ) Pt = FR2 ≤ FW2 , in this case the droplet is affected by the pressure which is balanced by the πrs πrs resistance, and its bottom liquid-air interface is bending while the contact angle on the side wall of posts is less than the advancing contact angle, so that the TPCL is still pinned. c) (1 − f ) Pt > FW2 , the bending degree of meniscus reaches maximum, the contact angle on pillars sides πrs becomes advancing contact angle, and the TPCL depins in this situation. The parameters describing the meniscus can be calculated with the following relations1:

(1 − f )Pt =

FR (r − 1) = σ LG cos θ H πrs 2

(7)


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cos θ =

(p − d ) = ( 2 R1

2p−d 2 R2

)

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（8）

hx1 = R1 (1− sinθ )

(9)

hx2 = R2 (1 − sinθ )

(10)

Namely, the contact angle of curved liquid-air interface can be firstly found according to equations (7) and (2), and then the other parameters can be figured out with equations (8)-(10), in which, R1 and R2 are the two arc radiuses on the curved liquid-air interface beneath a droplet, respectively locating between the two adjacent and diagonal pillars with a square array. And hx1 and hx2 mean the bending depth of the above two arcs respectively, as shown in Figure 1. The shape of the liquid-air interface below a droplet is a very complex one. The contact angle here is used to determine the local meniscus shape around pillars. And the two arcs, one between the two adjacent pillars with a square array, the other one between the two diagonal pillars, are applied approximately to represent the two curves on the meniscus, in view of the fact that the measured shape of these two curves on the meniscus is close to arc as reported by Papadopoulos et al3. Moreover, the calculations related meniscus area below a droplet after MTS are given in the supporting information. 2.2 C-W transition of droplets triggered by MTS on high pillars by TPCL depinning mechanism 1) Qualitative description The TPCL on pillars beneath a droplet will start to move downwards when the liquid-gas interface below the drop bends to maximum, i.e. the contact angle on the side of pillars reaches to the advancing contact angle. The parameters of φ or (π- θ ), R1, R2, hx1 and hx2 etc. all keep unchanging during the downward moving of the TPCL until the top of the meniscus touches the substrate of pillars, after which the C-W transition via MTS begins. Therefore, the critical condition of the C-W transition occurrence via MTS is x+hx2=H

(11)

As shown in figure 1, the whole process of C-W transition via MTS is divided into two stages. Stage 1 is the sub-process during which the bending depth of meniscus between four pillars changes from hx2 to hx1, namely, the TPCL on pillars moves downwards from x=H-hx2 to x=H-hx1 (figures 1 A3 to A4). Stage 2 means the sub-


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process during which the TPCL on pillars moves downwards continuously till x=H (figures 1 A4 to A6). Because the contact angle of the TPCL on posts keeps unchanged advancing contact angle during the two stages, R1 and hx1 will also keep unchanging according to the equations (8) and (9). But the solid-liquid interface below the droplet increases continuously until the entire substrate below the drop is completely wetted. 2) Analysis of the relationship between forces and corresponding energy and/or work during the C-W transition via MTS Because the direction of the wetting force at the TPCL on the substrate below a droplet is parallel to the substrate while the wetting force direction on pillar sides is parallel to the pillar sides, the directions of the two wetting forces at the two places are different and perpendicular. Therefore, the directions of resistances of TPCL to move forward (to wet solid surface) at different positions beneath a droplet are diverse during wetting transition process. On the other hand, the forces with different directions could not be plus simply. Therefore, the criterion of driving force equal to the sum of resistances cannot be applied here to calculate the C-W transition via MTS. In order to consider comprehensively the effect of forces in different directions on the droplet C-W transition via MTS, the forces in different directions were correlated in this model with the corresponding energy changes and/or the word done by the forces, so that the criterion equation to describe the droplet C-W transition after MTS can be established according to the energy and work conservation. The driving force of droplet C-W transition via MTS originates from the IFE decrease of the drop, i.e. Fd = −

dE x , which can be understood as the IFE keeps declining as the TPCL of the drop moves. This formula dx

can be changed to

− dE x = Fd dx , namely, the decrease of IFE equals to the work done by the driving force of

IFE gradient. Meanwhile, according to energy conservation, this reduction of IFE also equals to the increase of other energy and/or the work done to overcome certain resistance. For the problem now, the work covers two parts, one is the work to overcome the resistance preventing TPCL moving, capillary force35-37 or called wetting force Fw1, on pillar sides, i.e. FW dx ; the other one is the work to overcome the wetting force Fwb on the substrate of pillars, i.e. FWb ds . The s here represents the distance of the TPCL to move on the substrate. Finally, in consideration of Fd (1-f) being as the effective driving force to cause the C-W transition11,12,24, the energy


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conservation relation during the wetting transition process via MTS can be written as:

− (1 − f )dE x = FW dx + FWb ds

(12)

Furthermore, if Lp and Lb are respectively to represent the TPCL length on pillar sides and on substrate of pillars, the wetting forces on these two locations can be expressed as:

FW = Lpσ LG cos θ A

(13)

FWb = Lbσ LG cos θ A

(14)

Then, the following equation can be obtained after substituting equations (13) and (14) into equation (12):

− (1 − f )dEx = σ LG cos θ A ( Lp dx + Lb ds ) = σ LG cos θ A (dAp + dAb )

(15)

where dAp and dAb mean the wetting area formed after the TPCL moves dx and ds, respectively on the pillar sides and on substrate. Moreover, the equation (15) can be integrated for any a sub-process during the wetting transition via MTS:

∫

E xL

E xH

− (1 − f )dE x = σ LG cos θ A

(∫ dA + ∫ dA ) p

(16)

b

And the following equation can be found:

(1 − f )( ExH − ExL ) = σ LG cos θ A (∆Ap + ∆Ab )

(17)

in which ExH and ExL represent the initial high and the terminate low IFE of a droplet in the sub-process after MTS, while ∆Ap and ∆Ab denote the wetted solid areas on pillar sides and on substrate respectively after the end of the sub-process. The equation (17) is to be used as the criterion whether the C-W transition can be

(

completed through MTS. Namely, (1 − f )( ExH − ExL ) ≥ σ LG cos θA ∆Ap + ∆Ab

)

is the sufficient and

necessary condition for the C-W transition to be completed after MTS. It should be mentioned that both the gravity potential of a droplet and the energy loss cause by viscous dissipation during the wetting state changing of the droplet were neglected in the above energy and work derivation because droplets considered here are very small (their size is less than the capillary length) and the moving speed of the TPCL beneath droplets is slow3. 3) Formulas of related parameters during the first stage of wetting transition triggered by MTS For a droplet within a selected area Atotal on a textured surface, its IFE covers the liquid-gas, solid-gas and


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solid-liquid interfacial energy1,10:

Ex = (Ad + AbLG)σLG + (rAtotal − AbSL)σSG + AbSLσSL = (Ad + AbLG)σLG − AbSLcosθIσLG + rAtotalσSG

(18)

in which Ad, and AbLG are the liquid-gas interfacial areas of the drop cap and beneath the droplet respectively, while AbSL is the solid-liquid contact areas below the drop; σ LG , σSG and σ SL are respectively the liquid-gas, solid-gas and solid-liquid tension. The formula of Ad is:

2πrs 1 + cos θ d 2

Ad =

(19)

which can be substituted into equation (18), then we can get:

 2πrs 2  Ex =  + AbLG − AbSL cosθI σ LG + rAtotalσ SG 1 + cosθd 

(20)

For Cassie droplets, the above AbLG and AbSL equal to (1 − f )πrs and fπrs respectively, therefore, the IFE 2

2

expression of Cassie droplets is:

 2   2  2 2 EC = πrs σLG − f cosθI − f +1 + rAtotalσSG = πrs σLG − cosθEC + rAtotalσSG 1+ cosθd  1+ cosθd  where

θEC

(21)

denotes the equilibrium contact angle of Cassie drops.

Moreover, it is considered that the whole C-W transition process initiates from a Cassie droplet in this model. Namely, the whole wetting state transition of C-W includes the liquid-gas interface continuous bending beneath a Cassie drop, the TPCL depinning, MTS and its triggered next wetting process. Therefore, the initial IFE of any a sub-process during the C-W transition triggered by MTS, ExH in equation (17), can be found by the following integral of equation (15):

∫

E xH

EC

− (1 − f )dE x = σ LG cos θ A

(∫ dA + ∫ dA ) p

b

(22)

that is:

(1 − f )( EC − ExH ) = σ LG cos θ A (∆ApA + ∆AbA )

(23)

Thus the expression of ExH is:


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ExH = EC − σ LG cos θ A (∆ApA + ∆AbA ) /(1 − f )

Page 12 of 37

(24)

where ∆ApA and ∆AbA represent the already wetted solid areas on pillar sides and on substrate, respectively, below a droplet when a sub-process during the wetting transition via MTS begins. Meanwhile the ExL in equation (17) can be calculated according to equation (20). Then, whether the MST triggered C-W transition is able to complete can be determined by equation (17) after the ExH and ExL are found. The detailed formulas of all the area parameters in equations (17), (20) and (24) are shown in the supporting information. 2.3 C-W transition of droplets triggered by MTS on short pillars by sag mechanism 1) Qualitative description For short pillars, when the meniscus below a droplet touches the substrate of pillars the contact angle on the pillar sides is still less than the advancing one, namely, the MST triggered process begins while the TPCL still pins. Similar to the method used in the above high pillars, here the whole process of C-W transition via MTS is also divided into two stages. Stage 1 is the sub-process during which the bending depth of meniscus between four pillars changes from hx2 to hx1. During the first stage the solid-liquid area increases and the contact angle on the pillar side also rises (figure 1 B). Stage 2 means the sub-process during which the meniscus moves downwards continuously until the entire substrate below the drop is completely wetted. Since the contact angle on pillar side may either reach advancing contact angle or not reach when the above first stage of C-W transition via MTS finishes, the related parameters of the meniscus below a droplet, R1, hx1 and AbSL etc, have to be calculated separately according to the two cases whether the contact angle on pillar side reaches advancing one at the end of the first stage. As shown in figures 1 B4’ to B6, case one means that the contact angle does not reach the advancing one, and the TPCL there does not depin yet (figure 1 B4’) after the completion of the first stage. In this case, the contact angle on pillar side will firstly rise till to the advancing one during the next period of the second stage, in which R1 continuously decreases, hx1 continuously increases (figure 1 B4”), and the TPCL keeps pined. After the contact angle reaches the advancing one, the TPCL will depin until it reaches the foot of pillars (figure 1 B6), during which R1 and hx1 keep unchanging (figure 1 B5). Case two is shown in figures 1 B4 to B6. In this case, the contact angle on


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pillar side already reaches the advancing one, and the TPCL already moves downward (figure 1 B4) at the end of the first stage, after which R1 and hx1 will keep unchanging, and the TPCL will continuously move downward (figure 1 B5) until it reaches the foot of pillars (figure 1 B6). 2) Formulas of related parameters during the first stage of wetting transition triggered by MTS For short posts, equation (17) is still used as criterion to determine whether the C-W transition induced by MTS is able to complete. But the relative area parameters will be different in this case. And the detailed calculation methods are provided in the supporting information. 2.4 Calculation of droplet apparent contact angle during C-W transition via MTS Finally, the apparent contact angle, θ d , of a droplet in different shape during its C-W transition via MTS has to be calculated based on the upper segment volume of the drop above posts Vup . Their relation is:

1 3 (2 − 3 cosθ d + cos3 θ d ) Vup = πrs 3 sin 3 θ d

(25)

Meanwhile, Vup equals the whole volume of the droplet, V, minus the part into structures, Vin, namely:

Vup = V − Vin

(26)

Therefore, it is necessary to find the expression of Vin (see the supporting information):

(

2

 πr  Vin = πrs (1− f )H −  s   p 2

)

 p2 − d 2 (hx1 − h) (hx1 − h)3  2 − + R1 hx1 − h2 − hx1h(hx1 − h)   4 3     − p R 2  arcsin b − arcsin l  − b (R − h ) + l (R − h)  1 x1 1  2  1  2R1 R1  2  

(

)

(27)

2.5 Calculation of C-W transition triggered by MTS on micro-nano two tier surfaces All above equations are valid for the calculation of micro-nano hierarchical textures under the conditions of micro-structures able to be wetted while the nano-textures unable and by using the structural parameters of f, r, d, p, H etc. in above equations with those of the micro-textures and using the equilibrium contact angle of Cassie droplet on nano-structures to replace the intrinsic contact angle in all formulas. 2.6 Progress and notation of droplet C-W transition triggered by MTS The calculation results of droplet C-W transition progresses via MTS are expressed with notations as below: （1）The meniscus of a droplet is unable to touch substrate, i.e. the C-W transition via MTS could not be


13


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Page 14 of 37

implemented, expressed with letter C. （2）The first stage of MTS triggered transition is unable to complete although the meniscus of a droplet already touches textured substrate. TS and TD are used to express the wetting states of droplet touching structural substrate with sag and TPCL depinning impalement respectively. （3）The first stage of C-W transition after MTS already finishes, but the second stage could not be implemented, which is noted with PW1S and PW1D respectively through sag and TPCL depinning mechanism. （4）The second stage of C-W transition after MTS already starts, but could not complete, which is noted with PW2S and PW2D respectively via sag and TPCL depinning mechanism. （5）The second stage of C-W transition via MTS is able to accomplish, expressed with WS and WD respectively via sag and TPCL depinning mechanism. Table 1 summarizes all these progresses and the corresponding notations. These symbol letters will be used in the next figures of calculation results to show the different progresses of C-W transition after MTS. Table 1. Progresses and notations of droplet C-W transition after MTS on textured surfaces

Short pillars High pillars

meniscus is unable to touch substrate C C

meniscus is able to touch substrate

first stage of transition is able to complete

during the second stage of transition

second stage of transition is able to complete

TS TD

PW1S PW1D

PW2S PW2D

WS WD

The calculation block diagram and related program are in the supporting information.

3 Results and discussions 3.1 Effect of structural parameters on C-W transition triggered by MTS Figure 2 shows the calculation results of the effect of micro/nano structural parameters on the C-W transition after MTS. It is clear from figure 2A to 2E, which represent the effect of pillar diameter, that the blue area increases with the pillar diameter reducing from micrometers to nano-meters. Namely, droplets on nanotextured surfaces are easier to keep in Cassie state, or the meniscus of these drops is difficult to touch the substrate, which sufficiently illustrates that nano-textured surfaces could prevent the C-W transition. For instance, the pillar diameter in figure 2A is 100 nm, it can be seen from this figure that droplets of 2µL will


14

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appear in Cassie state on this nano-structural surface once d/p is greater than 0.011, i.e. p less than 9.1µm. Overall, the larger the pitch of micro/nano structures, or the smaller the d/p value, the easier the C-W transition for an initial sessile Cassie droplet triggered by MTS even though the larger spacing between posts gave rise to an easier Wenzel-to-Cassie transition for impinging drops38. Moreover, it can also be seen from figure 2 that the region of WS is small for droplets on short pillar textured surfaces to complete their C-W transition after MTS since WS area requires not only relatively lower pillars but also suitable d/p values. 3.2 Effect of droplet volume on C-W transition triggered by MTS

C

WD

(C)

H, µm

WD

(B)

H, µm

H, µm

(A)

C WS

(E)

C

H, µm

(D)

WD

WS

d/p

d/p

d/p

C

WD

PW1S

PW1S

H, µm

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


C

WD

PW1S

WS PW1S d/p

d/p

Figure 2. Effect of structural parameters on C-W transition after MTS. (A) d=0.1µm; (B) d=0.3µm; (C) d=1µm; (D) d=5µm; (E) d=10µm. θI=110°, θA=115°, V=2µL.

Figure 2 is only the results under the condition of fixed droplet volume. However, drop volume is also one of the most important parameters influencing the C-W transition. Thus, the effects of both drop volume and the textured parameters on the C-W transition were calculated, which is shown in figure 3 and 4. It is clear from the two figures that the influence of two parameters of d and p on C-W transition is similar to that in figure 2. But as the droplet volume decreases, the droplet C-W transition becomes easier, as it shown in figure 4 that the C region becomes smaller and smaller while the completely wetted area gets larger and larger. Furthermore,


15


the C-W transition after MTS of droplets on short pillars (figure 4) is more complicated than that on high pillars (figure 3), with more wetting states existing on short pillar surfaces, such as PW1S and WS. 3.3 Effects of intrinsic and advancing contact angles on C-W transition triggered by MTS The intrinsic contact angle reflects the hydrophobicity of a material surface while the advancing contact angle reveals the difficult degree of the TPCL depinning. For the droplet C-W transition, the increase of the two contact angles is beneficial to prevent the C-W transition, which is calculated as shown in figure 5. It is clear that droplets are easier to maintain Cassie state as the increase of θ I , which is in agreement with the result of Luo et al28. θ I was calculated only up to 130° in this study because the maximum of the intrinsic contact angle of hydrophobic materials to date is 120°13. Besides, it can also been seen from figure 5 that the effect of advancing contact angle on the droplet C-W transition is not as obvious as that of θ I and structural parameters, d and p, although the raise of advancing contact angle can also inhibit the C-W transition. Furthermore, advancing contact angle is not an independent parameter, which depends on both the chemical and physical

V, µL

(A)

(B)

C

WD

d/p

WD

(E) C

C

d/p

d/p

d/p (D)

WD

(C)

C

C

WD

V, µL

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

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WD

d/p

Figure 3. Effects of droplet volume and structural parameters on C-W transition after MTS. (A) d=0.1µm; (B) d=0.3µm; (C) d=1µm; (D) d=5µm; (E) d=10µm; θI=110°, θA=115°, H=40µm.


16

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(A)

(B)

(C)

V, µL

WS

C

C

C WD

WD

WD

d/p (D)

d/p C

PW1S

d/p

(E) C PW1S

V, µL

WS WD

WD

WS d/p

d/p


(A)

(B)

(C)

PW1H

θA－θI, °

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


TH C WH

WH

C

C WH PW2H

,°

,°

,°

Figure 5. Effects of intrinsic and advancing contact angle on C-W transition via MTS. (A) p=75µm; (B) p=100µm; (C) p=125µm; d=10µm, H=40µm, V=2µL.

properties of a material surface, so that it is difficult to change this parameter freely. Figure 6 shows the effects of both droplet volume and θ I on the C-W transition via MTS. It can been seen that droplets on textured surfaces mainly appear in two states, Cassie state of large drops and Wenzel state of small droplets, when θ I is in the normal range (100° < θ I < 120°). But several more complicated wetting


17


modes may exist after θ I is greater than 120°. In any case, droplets on micro-structural surfaces will finally appear in Wenzel state as long as the size of droplets is small enough. (A)

(C)

(B)

C

C

C

TH

V, µL

PW1H TH

PW1H

PW2H

WH

WH

WH

PW2H

TH PW2H

,°

,°

,°

Figure 6. Effects of intrinsic contact angle and drop volume on C-W transition via MTS. (A) p=75µm; (B) p=100µm; (C) p=125µm; d=10µm, H=40µm, V=2µL, θA=θI. 3.4 Effects of micro-nano hierarchical textures on C-W transition triggered by MTS

(A)

(C)

(B)

C

C C

PW1S

WS

V, µL

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 18 of 37

PW1D

WS

WD WD

PW1D

WD

PW1D

fn

fn

fn

Figure 7. Effects of hierarchical textures and drop volume on C-W transition after MTS. (A) p=75µm; (B) p=100µm; (C) p=125µm; d=10µm, H=40µm, V=2µL, θI=110°, θA equals to the equilibrium contact angle of a Cassie droplet on nano-textured surfaces. “WD” and “WS” here mean “Wenzel” state of droplets which only wet the micro-structures but do not wet the nano-textures of the two-tier architectures.

Micro-nano hierarchical textures have been proved to be able to effectively prevent the C-W transition by experiments16,22,28-30,39,40. But detailed theoretical analysis is very limited. Here figure 7 shows the calculation results of whether the C-W transition of droplets with different volume on two-tier textures could be completed


18

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after MTS, in which “WD” and “WS” etc. mean “Wenzel” state of droplets which only wet the micro-structures but do not wet the nano-textures of the two-tier architectures40. In fact that is a kind of partially wetted state of droplets41, i.e. not all the solid surfaces are wetted. It can been seen from figure 7 that the area to keep droplets in Cassie state increases with the decrease of the solid area fraction of nano-structures, fn (fn=1 means single micro-textures without nano-structures). Namely, as the nano-textures on micro-posts become more and more prominent, more droplets will appear in Cassie state, which clearly represents the fact that two-tier surfaces can effectively inhibit the C-W transition. And our results are well accord with the experimental results16,22,2830,39,40

. In addition, our results also indicate that even very small droplets are unable to complete the C-W

transition after MTS as long as the fn of nano-textures is small enough, such as less than 0.01. In this case, small drops stay in “PW1D” state, namely only partial micro-structures are wetted while the entire microstructures could not be wetted, and the nano-textures are even more impossible to be wetted. Therefore, the contact angle of even very small droplets are very large on hierarchical surfaces, showing strong superhydrophobic performance, which is also in agreement with the experimental observation of drop evaporation39. Meanwhile, it can be found by comparing figure 7 with figure 6 that suitable nano-textures are more efficient than the increase of intrinsic contact angle in order to prevent the C-W transition. The hierarchical textures can be manufactured with photolithography and deep reactive-ion etching technology39. 3.5 Comparison of model calculations with experimental results Table 2 represents the comparison of the calculation results of droplet C-W transition of this model with those of experiments. It can been seen that the calculation results of the droplet wetting states or the droplet CW transition parameters are well accord with experimental observations on the firstly 26 textured surfaces out of total 38 surfaces, with the accuracy of 68.4%. Besides, the calculation results of C-W transition is also in agreement with experiments in regularity on the 9 surfaces numbered 27-35, but with a certain error in the detailed C-W transition parameters. For instance, on the surface of No. 27, the experimental C-W transition parameter of pillar diameter d is 3-5µm, while the calculated d is 2.2µm. The model accuracy will be 92% if the calculation on these 9 surfaces is also considered to be in agreement with experimental results. In fact, only on the 3 surfaces of No. 36-38, the calculated C-W transition parameters are not accord with experimental results.


19


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Page 20 of 37

The experimental conditions in table 2 include both sessile droplets and drops in evaporation on both micro and mciro/nano hierarchical surfaces. Overall, the model calculations are well accord with the experimental results, i.e. our model is able to predict the wetting states of droplets, the textured parameters and the critical droplet volume when the C-W transition takes place on different textured surfaces. Therefore, the model is valuable to guide the design of superhydrophobic surfaces with the ability of maintaining droplets in Cassie state and preventing them transiting to Wenzel state. Besides the C-W transition model after droplet MTS in this study, there are several other theoretical investigations related to the wetting transition with different methods including molecular dynamics simulation (MDS)42-45. And the results of MDS is in consistence with the experiments in many aspects42-44. For instance, there is an energy barrier whether for C-W or for W-C transition, and the barrier of C-W transition usually is much greater than that of W-C transition42; the W-C transition can be realized under the conditions of vibration or impinging velocity42,43. On the other hand, the MDS results by Wang et al45 indicated that the nano-droplets in Wenzel state could transit to Cassie state simultaneously, which is different from most experimental results with micro or larger droplets, although Zhang et al46 considered that the dewetting W-C transition became spontaneous on cone-shaped textures. Overall, there is no MDS study reported to date about the C-W transition process after droplet touching substrate of structures similar to this model. Several other studies illustrated that droplets in different wetting states might coexist on a same textured surface42,47-49, which resulted from the different initial location of the droplets42. For example, during a condensation process, Cassie drops formed on the top of rough structures while Wenzel droplets appeared at the base of the same textured surface, and the so called partially wetted droplets also formed on the surface48,49. In addition, the wetting states of drops on a same surface also depend on the size of droplets. Smaller drops appears in Wenzel state easily while larger droplets tend to be in Cassie state, which is the reason why an initial Cassie droplet will change to Wenzel state in the later stage during its evaporation process on a textured surface. The coexistence of droplet states on a same structure results from the fact that these drops exist in either stable or metastable state. And the transition between different wetting states needs to overcome corresponding energy barrier. Therefore, the drops in different wetting states can exist independently and stably.


20

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Finally, it should be pointed out that our model is only limited to determine the final wetting state of a sessile droplet initially on the top of structures with fixed volume. Namely, this model cannot be used for the analysis of droplet state in the processes of condensation and evaporation etc. Table 2. Comparison of model calculations of droplet C-W transition conditions on textured surfaces with experimental results No 1 2 3

4

Experimental parameters and conditions V=5µL, d=5µm, H=10µm, θ I = 109 o ,

Droplet wetting states and/or C-W transition parameters Experimental results Calculation results

Literatures

Droplets changed to Wenzel state at p=45-75µm

Droplets changed to Wenzel state at p=74µm

8

H=30µm,


Droplets changed to Wenzel state at p=125µm

8

H=40µm,


V=3µL, d=3µm, H=6 µm, θ I = 110 o ,

Droplets transited to partially wetted state at p/d=12-20

Droplets changed to Wenzel state at p=106µm Droplets transited to partially wetted state at p/d=18.7, the first stage of C-W transition after MTS has finished.

θ A = 109 o V=5µL, d=14µm, θ I = 109 o , θ A = 109 o V=3µL, d=10µm, θ I = 110 o , θ A = 116 o

θ A = 116 o V=3µL, d=5µm, H=10µm, θ I = 110 o ,

10

10

Droplets changed to Wenzel state at p/d=12.5-15

Droplets changed to Wenzel state at p/d=14.8

10

V=3µL, d=10µm, H=20µm, θ I = 110 o , θ A = 116 o V=3µL, d=14µm, H=28µm, θ I = 110 o , θ A = 116 o V=3µL, d=20µm, H=40µm, θ I = 110 o , θ A = 116 o V=4.3µL, d=105µm, H=150µm, θ I = 107 .5 o , θ A = 107 .5 o

Droplets changed to Wenzel state at p/d=9-12

Droplets changed to Wenzel state at p/d=10.6

10

Droplets changed to Wenzel state at p/d=9-12 Droplets changed to Wenzel state at p/d >7 Droplets changed to Wenzel state at (p-d)/d=1.8-2

10

d=3µm, H=4.8µm, θ I = 110 o , θ A = 116o

p=20µm,

Droplets changed to Wenzel state when their base area reduced to As=7×10-3 mm2

11

d=5µm, H=10µm, p=7µm, θ I = 109 o , θ A = 109 o ,

12

d=5µm, H=10µm, θ I = 109 o , θ A = 109 o ,

p=7.5µm,

13

d=14µm, H=30µm, θ I = 109 o , θ A = 109 o ,

p=21µm,

14

d=14µm, H=30µm, θ I = 109 o , θ A = 109 o ,

p=23µm,

15

d=14µm, H=30µm, θ I = 109 o , θ A = 109 o ,

p=26µm,

Droplets did not change to Wenzel state during the evaporation process Droplets did not change to Wenzel state during the evaporation process Droplets did not change to Wenzel state during the evaporation process Droplets did not change to Wenzel state during the evaporation process Droplets did not change to Wenzel state during the evaporation process

Droplets changed to Wenzel state at p/d=8.9 Droplets changed to Wenzel state at p/d=7. Droplets changed to Wenzel state at (p-d)/d=2.18 Droplets changed to Wenzel state when their base area reduced to As=6.69×10-3 mm2 Droplets did not change to Wenzel state during the evaporation process Droplets did not change to Wenzel state during the evaporation process Droplets did not change to Wenzel state during the evaporation process Droplets did not change to Wenzel state during the evaporation process Droplets did not change to Wenzel state during the evaporation process

Droplets were in Wenzel state

Droplets are in Wenzel state

29


Droplets are in Wenzel state

29

5 6 7 8 9

16 17

θ A = 116 o

V=3-6µL, d=140µm, H=100µm, p=530µm, θ I = 115 o , θ A = 120 o V=3-6µL, d=140µm, H=100µm, p=630µm, θ I = 115 o , θ A = 120 o

10 10 9

7

8

8

8

8

8


21


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

18

V=3-6µL, d=140µm, H=100µm, p=730µm, θ I = 115 o , θ A = 120 o


19

Hierarchical surface, V=3µL, dm=140µm, Hm=100µm, pm=630µm, θ n = 160 o , θ A = 167 o

MTS happened, but droplets did not change to Wenzel state

20



21



22

d=20µm, H=80µm, θ I = 100 o , θ A = 120 o

p=40µm,

23

d=20µm, H=80µm, θ I = 100 o , θ A = 120 o

p=60µm,

Droplets changed to Wenzel state after being evaporated to very small drops Droplets changed to Wenzel state after being evaporated to very small drops Drops usually appeared in Cassie state. Droplets wetted the micro-structures but were unable to wet the nanotextures in the last period of evaporation Drops usually appeared in Cassie state. Droplets wetted the micro-structures but were unable to wet the nanotextures in the last period of evaporation

24

Hierarchical surface, dm=20µm, Hm=80µm, pm=30µm, dn=0.4µm, Hn=5µm, pn=0.7µm, θ n = 143o ,

θ A = 143

25

26

o

Hierarchical surface, dm=20µm, Hm=80µm, pm=40µm, dn=0.4µm, Hn=5µm, pn=0.7µm, θ n = 143o , θ A = 143 o Hierarchical surface, dm=20µm, Hm=80µm, pm=60µm, dn=0.4µm, Hn=5µm, pn=0.7µm, θ n = 143o ,

θ A = 143

o

27

V=3µL, p=50µm, θ I = 110 o , θ A = 116 o

H=20µm,

28

d=5µm, H=10µm, θ I = 109 o , θ A = 109 o ,

p=25µm,

29

d=5µm, H=10µm, θ I = 109 o , θ A = 109 o ,

p=37.5µm,

30

d=14µm, H=30µm, θ I = 109 o , θ A = 109 o ,

p=35µm,

31

d=14µm, H=30µm, θ I = 109 o , θ A = 109 o ,

p=70µm,

32

d=14µm, H=30µm, θ I = 109 o , θ A = 109 o ,

p=105µm,

33

Hierarchical surface, V=3µL, dm=140µm, Hm=100µm, pm=530µm,

Droplets wetted the microstructures but were unable to wet the nano-textures in the last period of evaporation Droplets changed to Wenzel state at d=3-5µm Droplets changed to Wenzel state when their radius reduced to Rd=252µm during evaporation Droplets changed to Wenzel state when their radius reduced to Rd=435µm during evaporation Droplets changed to Wenzel state when their radius reduced to Rd=150µm during evaporation Droplets changed to Wenzel state when their radius reduced to Rd=250µm during evaporation Droplets changed to Wenzel state when their radius reduced to Rd=425µm during evaporation MTS happened, but droplets did not change to Wenzel

Droplets are in Wenzel state MTS happened, but droplets are unable to finish the first stage of C-W transition The first stage of C-W transition after MTS finishes, but the second stage of C-W transition is unable to proceed MTS happened, but droplets are unable to finish the first stage of C-W transition Droplets changed to Wenzel state after being evaporated to Rd=54µm Droplets changed to Wenzel state after being evaporated to Rd=136µm

Page 22 of 37

29

29

29

29

39

39

Droplets are in Cassie state when their radius reduced to Rd=30µm in the last period of evaporation

39

Droplets are in Cassie state when their radius reduced to Rd=30µm in the last period of evaporation

39

Droplets wetted the microstructures but were unable to wet the nano-textures when their radius reduced to Rd=29µm in evaporation Droplets changed to Wenzel state at d=2.2µm Droplets changed to Wenzel state when their radius reduced to Rd=117µm during evaporation Droplets changed to Wenzel state when their radius reduced to Rd=270µm during evaporation Droplets changed to Wenzel state when their radius reduced to Rd=72µm during evaporation Droplets changed to Wenzel state when their radius reduced to Rd=328µm during evaporation Droplets changed to Wenzel state when their radius reduced to Rd=630µm during evaporation Droplets are in Cassie state

39

10

8

8

8

8

8

29


22

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34

35

θ n = 160 o , θ A = 167 o

state

Hierarchical surface, V=6µL, dm=140µm, Hm=100µm, pm=530µm, θ n = 160 o , θ A = 167 o Hierarchical surface, V=6µL, dm=140µm, Hm=100µm, pm=630µm, θ n = 160 o , θ A = 167 o


36

d=5µm, H=10µm, θ I = 109 o , θ A = 109 o ,

p=10µm,

37

d=5µm, H=10µm, θ I = 109 o , θ A = 109 o ,

p=12.5µm,

38

d=20µm, H=80µm, θ I = 100 o , θ A = 120 o

p=30µm,

Droplets are in Cassie state

29

MTS happened, but droplets did not change to Wenzel state Droplets changed to Wenzel state when their radius reduced to Rd=80µm during evaporation Droplets changed to Wenzel state when their radius reduced to Rd=92µm during evaporation

Droplets are in Cassie state

29

Droplets do not transit to Wenzel state during the evaporation

8

Droplets do not transit to Wenzel state during the evaporation

8

Droplets changed to Wenzel state in the last period of evaporation

Droplets appear in Cassie state when their radius reduced to Rd=30µm during evaporation

39

4 Conclusions (1)

The driving force of droplet C-W transition is the IFE gradient, and the resistance is the wetting force

on TPCL. Only when the driving force is greater than the resistance, is it possible for the C-W transition after MTS to complete. (2)

Depending on the relative height of pillars, there are two cases when C-W transition after MTS takes

place. One case is that the TPCL on pillar sides already depins, the other case is the TPCL is still pinned. The simulation methods and calculation formulas are different for the two cases. (3)

The droplet C-W transition after MTS is significantly influenced by pillar diameter and pitches

between pillars. The larger the diameter and the pitch, the easier the C-W transition. Conversely, the small sized posts, nano-pillars for instance, are able to prevent the C-W transition effectively. (4)

Because the C-W transition driving force increases with the decrease of droplet size, the C-W

transition becomes easier as the droplet volume decreases. (5)

Although the difficulty of C-W transition can be enhanced by rising the intrinsic contact angle of

materials, the hierarchical surfaces are able to inhibit the C-W transition after MTS more efficiently.

Associated content


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The detailed derivation program about mathematical model is shown in the supporting information. This material is available free of charge via the Internet at http://pubs.acs.org.

Acknowledgements The authors acknowledge financial support from the National Natural Science Foundation of China (No. 21676041)

References (1) Liu, T. Q.; Li, Y. J.; Li, X. Q.; Sun, W. Theoretical Analysis of Droplet Transition from Cassie to Wenzel State. Chinese Phys. B 2015, 24, 367-378. (2) Bormashenko, E. Progress in Understanding Wetting Transitions on Rough Surfaces. Adv. Colloid Interface 2015, 222, 92-103. (3) Papadopoulos, P.; Mammen, L.; Deng, X.; Vollmer, D.; Butt, H. J. How Superhydrophobicity Breaks Down. P. Natl. Acad. Sci. USA 2013, 110, 3254-3258. (4) Extrand, C. W. Criteria for Ultralyophobic Surfaces. Langmuir 2004, 20, 5013-5018. (5) Kusumaatmaja, H.; Blow, M. L.; Dupuis, A.; Yeomans, J. M. The Collapse Transition on Superhydrophobic Surfaces. Epl (Europhysics Letters) 2008, 81, 36003. (6) He, Y.; Jiang, C. Y.; Wang, S. K.; Yin, H. X.; Yuan, W. Z. Control Wetting State Transition by Micro-Rod Geometry. Appl. Surf. Sci. 2013, 285, 682-687. (7) Reyssat, M.; Yeomans, J. M.; Quere, D. Impalement of Fakir Drops. Epl(Europhysics Letters) 2008, 81, 26006. (8) Jung, Y. C.; Bhushan, B. Wetting Behaviour During Evaporation and Condensation of Water Microdroplets on Superhydrophobic Patterned Surfaces. J. Microsc.-Oxford 2008, 229, 127-140. (9) Lee, J. B.; Gwon, H. R.; Lee, S. H.; Cho, M. Wetting Transition Characteristics on Microstructured Hydrophobic Surfaces. Mater. Trans. 2010, 51, 1709-1711. (10) Barbieri, L.; Wagner, E.; Hoffmann, P. Water Wetting Transition Parameters of Perfluorinated Substrates with Periodically Distributed Flat-Top Microscale Obstacles. Langmuir 2007, 23, 1723-1734. (11) Moulinet, S.; Bartolo, D. Life and Death of a Fakir Droplet: Impalement Transitions on


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Table of Contents (TOC) Graphic

Cassie state

Wenzel state

The detailed wetting progress of meniscus after MTS


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Table of Contents (TOC) Graphic 70x44mm (300 x 300 DPI)


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Figure 1. Schematics of partial meniscus beneath a composite droplet and its C-W transition via MTS. 152x248mm (300 x 300 DPI)



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Figure 2. Effect of structural parameters on C-W transition after MTS. (A) d=0.1µm; (B) d=0.3µm; (C) d=1µm; (D) d=5µm; (E) d=10µm. θI=110°, θA=115°, V=2µL. 166x133mm (300 x 300 DPI)


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Figure 3. Effects of droplet volume and structural parameters on C-W transition after MTS. (A) d=0.1µm; (B) d=0.3µm; (C) d=1µm; (D) d=5µm; (E) d=10µm; θI=110°, θA=115°, H=40µm. 166x129mm (300 x 300 DPI)



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166x130mm (300 x 300 DPI)


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Figure 5. Effects of intrinsic and advancing contact angle on C-W transition via MTS. (A) p=75µm; (B) p=100µm; (C) p=125µm; d=10µm, H=40µm, V=2µL. 159x77mm (300 x 300 DPI)



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Figure 6. Effects of intrinsic contact angle and drop volume on C-W transition via MTS. (A) p=75µm; (B) p=100µm; (C) p=125µm; d=10µm, H=40µm, V=2µL, θA=θI. 150x90mm (300 x 300 DPI)


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Figure 7. Effects of hierarchical textures and drop volume on C-W transition after MTS. (A) p=75µm; (B) p=100µm; (C) p=125µm; d=10µm, H=40µm, V=2µL, θI=110°, θA equals to the equilibrium contact angle of a Cassie droplet on nano-textured surfaces. “WD” and “WS” here mean “Wenzel” state of droplets which only wet the micro-structures but do not wet the nano-textures of the two-tier architectures. 168x94mm (300 x 300 DPI)


Mechanism Study on Transition of Cassie Droplets ... - ACS Publications

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