- Email: [email protected]
Effect of pillar height on the wettability of micro-textured surface: Volume-of-fluid simulations
Author’s Accepted Manuscript Effect of pillar height on the wettability of microtextured surface：Volume-of-Fluid simulations Wei Zhang, Run-run Zhang, Cheng-gang Jiang, Cheng-wei Wu www.elsevier.com/locate/ijadhadh
PII: DOI: Reference:
S0143-7496(16)30258-5 http://dx.doi.org/10.1016/j.ijadhadh.2016.12.011 JAAD1944
To appear in: International Journal of Adhesion and Adhesives Accepted date: 21 December 2016 Cite this article as: Wei Zhang, Run-run Zhang, Cheng-gang Jiang and Chengwei Wu, Effect of pillar height on the wettability of micro-textured surface： Volume-of-Fluid simulations, International Journal of Adhesion and Adhesives, http://dx.doi.org/10.1016/j.ijadhadh.2016.12.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Effect of pillar height on the wettability of micro-textured surface： Volume-of-Fluid simulations
Wei Zhang, Run-run Zhang, Cheng-gang Jiang, Cheng-wei Wu State Key Laboratory of Structure Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, China *
Corresponding author. Email: [email protected], Tel: +86 411 84706353
Abstract: The wetting and spreading characteristics of a water drop on a surface is highly dependent on the geometric parameters of the micro-texture of the surface. To obtain a stable Cassie drop, the condition that the pillar height must be higher than the sag height of the meniscus is a necessary, yet not a sufficient, condition. From the viewpoint of energy minimization, a new criterion was proposed to design the height of the pillar for achieving a stable and robust anti-adhesive drop state. To ensure such a composite interface, the height of the pillars should be taller than the critical height where the critical contact angle is equal to the intrinsic contact angle. If this height requirement is not met, the drop exists in either a Wenzel or a metastable Cassie state. The numerical simulations using the Volume-of-Fluid method in Fluent support the above proposals. Keyword: Volume of fluid, superhydrophobic, anti-adhesive, pillar height, simulation
1
1. Introduction On rough hydrophobic surfaces, a water drop can either reside on the top of the roughness grooves, i.e. in Cassie states or impale the roughness grooves, i.e. in Wenzel states. Owing to the low resistance from the air pocket, a Cassie drop shows considerably less hysteresis in contrast to a Wenzel drop; this means the drop can move with less adhesion and friction and more easily relative to the surface [1]. Hence, in applications such as surface-tension-induced motion in microfluidic devices [2], self-cleaning surfaces [3, 4], magnetic storage devices, and micro/nano electromechanical systems [5, 6], it is imperative that a composite contact is achieved, rather than with the surface grooves being wet with liquid.
Then it comes to the question: in practice, how to design and fabricate a superhydrophobic surface with robust anti-adhesive properties? For any given low surface energy materials, the drop state is then largely determined by the geometric parameters of the surface roughness [7, 8]. Consider a small water drop suspended on a superhydrophobic surface consisting of a regular array of pillars, which is one of most popular structures used currently and can be made easily by lithographic techniques [9]. Based on the geometric consideration, Patankar et al. proposed that the depth of the cavity must be greater than the droop; otherwise the meniscus will eventually touch the relief bottom, leading to the collapse of the Cassie wetting and to the Cassie-Wenzel transition [10, 11]. On the other hand, Nosonovsky et al. proposed that the surface energy of a water drop on surface with nanoscale imperfections is a function of the vertical position of the liquid-vapor interface under the droplet [12].
2
Then, we wonder, from the surface energy point of view, how to design the height of the pillars of the surface that can ensure the drop exists in a Cassie state? And this is the motivation of the simulation work in this communication.
2. Effect of pillar height To date, the creation of a micro-structured surface with superhydrophobicity and anti-adhesive property remains a challenge to researchers. Apart from fulfilling the surface force equilibriums, the height of the pillar, h, should be so tall that the curved liquid-air interface cannot touch the bottom of the roughness groove, as expressed in Inequality 1. h hi 2 (l a)
1 s i n 2c o s
(1)
Where hi is the maximum sag distance of the curved interface allowed by the micro structure, see Fig. 1.
Fig. 1 Schematic of a liquid-gas meniscus suspended on top of pillars
3
2
a Substituting area fraction of solid-liquid interface in the horizontal plane , , l the height criterion, based on geometric consideration, for achieving a Cassie drop state can be obtained, as expressed in Inequality 2. To remove the effect of size, the dimensionless quantity, i.e. h/l, was used here. h hi 1 s i n 2 (1 ) l l 2c o s
(2)
On the other hand, if a surface is superhydrophobic, the drop must remain in the Cassie state and cannot transit to a Wenzel one. Hence, from the energy viewpoint, the surface energy of the structure with air entrapped must be smaller than the wetted state. Suppose a small displacement dx of the triple-phase line along the parallel direction of the micro-pillar structured surface, as shown in Fig. 2, when the droplet spreads on the micro-textured surface, the change of the surface free energy dE can be calculated using either Equation (3) for a Wenzel state or Equation (4) if a Cassie state is reached [13].
Fig. 2 Scheme of the triple-phase line between the droplet and the rough surface
dE r ( SL SG )dx GLdx cos * dE ( SL SG )dx (1 ) GLdx GLdx cos *
4
(3) (4)
where SL, SG, GL are surface tension of solid-liquid, solid-gas, and gas-liquid interfaces, respectively. is the apparent angle on the rough solid surface. r is surface roughness. Since an equilibrium drop shape with a lower value of the apparent contact angle will have a lower the surface energy [14], the droplet will stabilize with the smallest possible apparent contact angle. When the system reaches a thermodynamic equilibrium, dE=0, Inequality 5 should be satisfied.
( SL SG ) (1 ) GL r ( SL SG )
(5)
Substituting into Young’s equation, SG SL GL cos , we obtain Inequality 6.
cr ， cos cr
1 r
(6)
It has been shown that the critical contact angle can be realized by designing a geometric structure such that the Wenzel and the Cassie states have the same energy at a given intrinsic contact angle. In this case, the Wenzel and Cassie states have the same chance of existence, and thus this approach is not desirable if one wants to form a Cassie surface. For convenience, we define hcr as the critical height, i.e. the height of the pillar when the critical contact angle,
, is equal to the intrinsic contact angle,
h a
. Substituting r (4 1) , another height criterion (Inequality 7), based on energy minimization, is proposed here.
h hcr
a( 1)(cos 1) 4 cos
(7)
Analogously, to remove the effect of size, the dimensionless quantity, i.e. h/l was used.
5
h hcr ( 1)(cos 1) l l 4 cos
(8)
As discussed above, the height of the pillar is a crucial parameter that affects the wettability of the surface. To obtain a stable composite state, the height must be not only higher than the sag height of the meniscus caused by the external pressure, but also needs to meet the requirement of energy minimization. Then the question comes out: how about the compatibility of these two height criteria, hi/l and hcr/l?
Based on geometric consideration and energy minimization discussed above, the relationship of the stabilized state of a droplet and h/l, can be expressed using Fig. 3. Clearly, no matter what the values of h/l, and arehcr/l is always larger than hi/l. This means to obtain a stabilized Cassie droplet, h/l must be larger than hcr/l, i.e. meeting the energy criteria. If hi/l < h/l < hcr/l, the droplet may exist in a Cassie state but the application of any external disturbance may incur the transition from Cassie state to Wenzel sate, a metastable Cassie state. If the geometric criteria is not met, h/l < hi/l, a Wenzel droplet will be obtained.
6
Fig. 3 The ratio of the height (h) to the spacing (l) as a function of surface property for micro-texture with different intrinsic contact angle ( ) and area fraction of solid-liquid interface (). A red surface represents the geometric criteria obtained using Eq. (2). A green surface represents the energy criteria obtained using Eq. (8).
3. Computation model The Volume-Of-Fluid (VOF) method originally developed by Hirt and Nichols in 1981 [15] has become the most frequently used approach to simulate free-surface flows and study the phase change phenomena. And Fluent is the world's most widely used CFD software. To verify the effect of pillar height on the stabilized droplet state that we proposed above, the VOF method in Fluent was adopted to simulate the gas-liquid two-phase flow with a free surface.
7
Periodic structure consisting of a regular array of square pillars with equal side length a, height h and spacing l is widely adopted in the manufacture of super hydrophobic surfaces and this surface was used for the simulations, as shown in Fig. 4.
Fig. 4 Schematic of the periodic micro-structure.: a is the length of pillar, l is the spacing between pillars, h is the height of pillar
Due to symmetry, 1/4 of the model was used for simulation and the computed space is a rectangular structure (Fig. 5). In the set of boundary conditions, the surface of the pillar and the bottom surface of the rectangular space are set as the wall and can be given different contact angles as required; the front and left surface of rectangular space are set as symmetric; the top, back and right surface of rectangular space are set as constant pressure inlet. In the set of initial field, air is considered as the primary phase and water is the secondary phase.
8
Fig. 5 The set of simulation model and initial field
One-fourth of the sphere adjacent to the pillar is endowed the properties of the secondary phase and represents the morphology of the simulated droplet, as illustrated as blue in Fig.5. To ensure that the surface tension governs the deformation of a droplet, the radius of the droplet must be smaller than the capillary length. As the water capillary length was reported to be 2.7 mm [16], the initial radius of the droplet, R0, is set as 0.25 mm.
3.1 Quasi-static contact To alleviate the added dynamic effects, the velocity of the droplet is set as 1 mm/s. The Weber number is much less than 1 and the whole process is thought of being
9
quasi-static. The geometric parameter and property of the micro-pillar structure are given in Table 1.
Table 1 The geometric parameter and property of the micro-pillar structure
a / μm
l / μm
h / μm
θ/°
θcr / °
50
100
100
120
105.8
50
100
100
95
105.8
When the size of the micro structure of a surface is given, the wettability of the surface is mainly determined by the surface property. If the intrinsic contact angle, θ, is larger than the critical contact angle θcr, the droplet will be of a Cassie state; if θ < θcr, the droplet will be of a Wenzel state. As θcr is 105.8° in the model, two intrinsic angles, i.e. 120° and 95° are adopted for the simulation.
Fig. 6 diagrammatically demonstrates the simulation results; and it is apparent that for the intrinsic contact angle of 1200 the droplet sits on the top of the peaks of the rough surface and air exits between the droplet and pillars, a typical Cassie state with the apparent contact angle being around 153.4° (the top row of Fig. 6). This agrees with the value predicted by the Cassie formula [17], 151°. In terms of a contact angle of 95°, the droplet replaces the air layer and wets the grooves of the rough surface, a typical Wenzel state, see the bottom row of Fig. 6. The apparent contact angle measured is 109°, in good agreement with the one predicted by the Wenzel formula
10
[18], 105.20. Clearly, these models can reflect the effect of intrinsic contact angle on the wettability of the surface.
Fig. 6 The simulation result with intrinsic contact angle of 120° (top row) and 95° (bottom row)
By fixing the side length of pillar, the spacing between the pillars, and the intrinsic contact angle, we examined the effect of height of pillar on the wettability of the surface. Table 2 lists the geometric parameters and properties of different pillar surfaces. We assume the three surfaces have the same intrinsic contact angle θ (1100). Apparently, the area fraction of the solid-liquid interface φ (0.25), critical height hcr (72.1 μm), and sag height hi (6.2 μm), and the mere difference exists in the height of pillars, being 100 μm, 50 μm and 18 μm.
11
Table 2 The geometric parameters and properties of the micro-pillar structure
a / μm
l / μm
h / μm
θ/°
φ
θcr / °
hi / μm
hcr / μm
50
100
100
110
0.25
105.8
6.2
72.1
50
100
50
110
0.25
115.4
6.2
72.1
50
100
18
110
0.25
132.5
6.2
72.1
Herein, the height of the pillar was chosen in reference to hcr and hi. In all three cases, the height of pillar h is higher than hi. According to the well documented geometric criterion, the droplet should settle in a Cassie state and give a contact angle of about 146.7°. To alleviate the additional dynamic effects, the initial velocity of the droplet was kept small and constant at 1mm/s. The final simulated states of the droplet are illustrated in Fig.7.
Strikingly, not all the droplets settle in the same state. For pillars with heights of 100 μm and 50 μm, the droplet settles in the Cassie state and an air layer exists between the droplet and the surface, with the apparent contact angle being 143° and 144.2°, respectively, fairly close to the predicted values. When the height of the pillar is 18μm, however, the droplet wets the pillar and the apparent contact angle is only 112.3°, which is closer to the value predicted by the Wenzel model. It is apparent that at the pillar height of 100 μm (h = 1.39 hcr), θcr < θ, and a Cassie drop is expected; whereas at the pillar height of 18μm (h = 0.25 hcr), θcr > θ and a Wenzel drop is obtained. These results indicate that in quasi-static contact, h > hi is not a sufficient condition that can guarantee that the droplet will not make contact with the bottom of the
12
micro-textured structure, i.e. a Cassie state. On multi-groups of simulations by varying θ, a, and l, we found only when h > 0.25 hcr, the drop is prone to be in a Cassie state.
Fig. 7 The spreading of a droplet and its apparent contact angle on three microstructure surfaces with different pillar heights. Left: 3D front view, right: measure of apparent contact angle
It is worth mentioning that even though the initial velocity of the droplet was set at 1 µm/s, the droplet will continue to wet the pillar with the height of 18µm. When the pillar height was set at 50 μm, h =0.69 hcr, although θcr is slightly larger than θ, under quasi-static conditions, the droplet exists in a Cassie state. The plausible explanation for this is that, upon quasi-static conditions, the kinetic energy obtained by the droplet
13
is not high enough to break the pressure balance of the gas/liquid surface and thus the droplet cannot wet the space of the micro-structure. This is in good agreement with experimental observations by Li et al. [19] and Yoshimitsu et al. [20]. Their experimental results show that, with the increase of the ratio of h to l, the hydrophobicity of the micro-texture surface is improved.
3.2 Impacting contact In the aforementioned simulations, the collision velocity is set at 1 mm/s and thus the whole process can be considered as quasi-static. In many practical cases, however, the drop may collide with surface with much higher velocity. For a raindrop with a diameter of 1 mm, the average falling velocity can reach 4 m/s [21]. In ink-jet printing, the typical drop speed is 6-9 m/s [22]. In these cases, the dynamic effect of the liquid drop is obvious and cannot be ignored. Extensive studies have been carried out with regards to the dynamic impact behavior of a water drop. Bird et al. [23] and Liu et al. [24] investigated the effects of surface structure on the contact time of a bouncing drop. They reported the contact time does not depend on the impact velocity and that a water drop can behave as a harmonic spring. Jung et al. [25] and Bartolo et al. [26] have shown that the transition from a Cassie to a Wenzel state can occur when a water drop collides with a surface which has specific micro-patterns. As such, we further investigated the effect of collision velocity on the wettability of a water drop on the micro pillar-textured surface. The geometric parameter of the pillar and the intrinsic contact angle are given in Table 3. The quasi-static simulations above reveal that for both of these two surfaces air remains trapped below the water drop and the drop sits
14
on a composite surface made of pillar and air. Then, we wonder what will happen if we increase the collision velocity, and whether this external actuation will incur any change in the state of the drop? Table 3 The geometric parameters and properties of the micro-pillar structures
a / μm
l / μm
h / μm
θ/°
1st surface
50
100
50
110
2nd surface
50
100
100
120
Fig. 8 demonstrates the wetting and spreading of the water drop after collision with the first surface, where the cross-sections of the drop are selected. It was found that, with the increase of the initial collision velocity, the Cassie droplet under quasi-static conditions becomes unstable. After initiating contact with the surface, the droplet enters the space between micro-structures, replaces the air and eventually contacts the bottom of the groove. After oscillations, a Wenzel droplet is obtained.
Fig. 8 The wetting and spreading of water drop after colliding with the first kind of surface with a velocity of 1m/s
This indicates that the hydrophobicity of the first surface is not stable and may transit from a Cassie to a Wenzel state upon external actuation, impacting here. This is in agreement with the observation that the Cassie regime can be obtained for θ <θcr in
15
spite of a higher energy [27]. Once such a transition happens, the system becomes more stable due to the drop in energy and a reverse transition is hard to be obtained. The cross sections of the drop after the collision with the second surface are illustrated in Fig. 9, where it was found that when contacting the surface, the drop deforms, wets the sides of the pillars and fills in the groove due to the increase of collision velocity, and simultaneously a certain volume of air is embedded between the bottom center of the drop and the surface, forming an air pocket inside the drop. This is in good agreement with experimental observations by Reyssat et al. [28]. After reaching the maximum spreading radius at about 0.5 ms, the spreading liquid starts to retract and finally bounced off the surface at about 1.5 ms. Owing to the effect of gravity, the speed of the drop reduces gradually, and becomes zero at 11 ms, yielding a maximum rebounding height of 0.8 mm. The drop then falls back and contacts the surface again. After small amplitude oscillations, the drop reaches a static state and sits on the top of the surface. It should be noted that the rebounding of a drop can take place only when the collision speed is sufficiently high since both spreading and retracting of the drop will consume energy. For a drop with a radius of R0, the stored surface energy in spreading and retracting is estimated to fall in the order of R02. Only when the kinetic energy (in the order of R03V02) of the droplet is larger than the surface energy, is it possible for the droplet to rebound. This means the initial collision speed must be higher than ( /R0)
1/2
[29, 30]. For the model used, the minimum speed for
rebounding is about 0.2 m/s. This explains why in quasi-static contact (collision speed: 1 mm/s) and the second contact of the drop with the surface (collision speed: 1 m/s), no further rebounding phenomena were observed.
16
Fig. 9 The deformation and rebounding of a water drop after colliding with the second kind of surface with a velocity of 1m/s
These results imply that when h < hcr, although a Cassie drop can be obtained, it is metastable and will transit to a Wenzel state under the external actuations to minimize the energy of the system. To ensure a stable composite contact, the height of the pillar should be chosen such that the Cassie drop has a lower energy than the Wenzel drop.
4. Conclusions The height of pillar, h, is a crucial parameter that affects the wettability of the surface with a regular array of pillars. To obtain a stable composite state, surmounting the sag height of the meniscus caused by the external pressure is a necessary but not sufficient condition. This height geometric criterion must be resolved in terms of an energy minimization approach, where it is proposed that the pillar should be taller than the critical height, hcr. When h < 0.25 hcr. a Wenzel state can be obtained. When 0.25 hcr
17
< h < hcr, a Cassie state can be obtained on quasi-static conditions; however, this state is metastable. On impacting contact with a velocity of 1 m/s, the water drop wets the surface after deformation, demonstrating a transition from a Cassie to a Wenzel state. Under the same impacting velocity, when the water drop hits the surface with the pillar height being over hcr, it deforms and then retracts and bounces off the surface. Finally, the droplet sits on the surface and has a high contact angle, which suggests the formation of a solid-air-liquid interface. Based on these, a conservative criterion is proposed for achieving a stable Cassie drop with respect to the pillar height, i.e. .
Acknowledgements The NSFC (11272080,11572080), the NSFC of Liaoning Province (2015020198), the Fundamental Research Funds for the Central Universities of China (DUT15JJG06, DUT14LK36), National Basic Research Program of China (2015CB057306) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars (State Education Ministry) were acknowledged for the financial support.
References: [1] Cottin-Bizonne C, Barrat JL, Bocquet L, Charlaix E (2003) Low-friction flows of liquid at nanopatterned interfaces. Nat Mater 2(4): 237-240. [2] Patankar NA (2004) Transition between superhydrophobic states on rough surfaces. Langmuir 20(17): 7097-7102.
18
[3] Yao L, He J (2014) Recent progress in antireflection and self-cleaning technology–From surface engineering to functional surfaces. Prog Mater Sci 61: 94-143. [4] Nishimoto S, Bhushan B (2013) Bioinspired self-cleaning surfaces with superhydrophobicity, superoleophobicity, and superhydrophilicity. RSC Advances 3(3): 671-690. [5] Jung YC, Bhushan B (2008) Dynamic effects of bouncing water droplets on superhydrophobic surfaces. Langmuir 24(12): 6262-6269. [6] Jung YC, Bhushan B (2007) Wetting transition of water droplets on superhydrophobic patterned surfaces. Scripta Mater 57(12): 1057-1060. [7] Reyssat M, Yeomans JM, Quéré D (2008) Impalement of fakir drops. EPL 81(2): 26006. [8] Yoshimitsu Z, Nakajima A, Watanabe T et al (2002) Effects of surface structure on the hydrophobicity and sliding behavior of water droplets. Langmuir 18(15): 5818-5822. [9] Roach P, Shirtcliffe NJ, Newton MI (2008) Progress in superhydrophobic surface development. Soft Matter 4(2): 224-240. [10] Patankar NA (2010) Consolidation of hydrophobic transition criteria by using an approximate energy minimization approach. Langmuir 26(11): 8941-8945. [11] Whyman G Bormashenko E (2011) How to make the Cassie wetting state stable?
19
Langmuir 27(13): 8171-8176. [12] Nosonovsky M, Bhushan B (2008) Patterned nonadhesive surfaces: superhydrophobicity and wetting regime transitions. Langmuir 24(4): 1525-1533. [13] Wang S, Liu K, Yao X et al (2015) Bioinspired surfaces with superwettability: new insight on theory, design, and applications. Chem Rev 115(16): 8230-8293. [14] Patankar NA (2003) On the modeling of hydrophobic contact angles on rough surfaces. Langmuir 19(4): 1249-1253. [15] Hirt CW, Nichols BD (1981) Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 39(1): 201-225. [16] Jung YC, Bhushan B (2007) Wetting transition of water droplets on superhydrophobic patterned surfaces. Scripta Mater 57(12), 1057-1060. [17] Cassie ABD, Baxter S (1944) Wettability of porous surfaces. Transactions of the Faraday Society 40: 546-551. [18] Wenzel RN (1936) Resistance of solid surfaces to wetting by water. Industrial & Engineering Chemistry 28(8): 988-994. [19] Li X, Mao L, Ma X (2013) Dynamic behavior of water droplet impact on microtextured surfaces: the effect of geometrical parameters on anisotropic wetting and the maximum spreading diameter. Langmuir 29: 1129-1138. [20] Yoshimitsu Z, Nakajima A, Watanabe T et al (2002) Effects of surface structure on the hydrophobicity and sliding behavior of water droplets. Langmuir 18: 5818-5822
20
[21] Dingle N, Lee Y (1972) Terminal fall speeds of raindrops. J Appl Meteorol 11(5): 877-879. [22] Hutchings I, Tuladhar T, Mackley M et al (2009) Links between ink rheology, drop-on-demand jet formation, and printability. J Imaging Sci Techn 53(4): 41208-1-41208-8. [23] Bird JC, Dhiman R, Kwon HM et al (2013) Reducing the contact time of a bouncing drop. Nature 503(7476): 385-388. [24] Liu Y, Moevius L, Xu X et al (2014) Pancake bouncing on superhydrophobic surfaces. Nature Physics 10(7): 515-519. [25] Jung YC, Bhushan B (2008) Dynamic effects of bouncing water droplets on superhydrophobic surfaces. Langmuir 24(12): 6262-6269. [26] Bartolo D, Bouamrirene F, Verneuil É et al (2006) Bouncing or sticky droplets: Impalement
transitions
on
superhydrophobic
micropatterned
surfaces.
EPL
(Europhysics Letters) 74(2): 299. [27] Lafuma A, Quéré D (2003) Superhydrophobic states. Nat Mater 2(7): 457-460. [28] Reyssat M, Richard D, Clanet C et al (2010) Dynamical superhydrophobicity. Faraday Discussions 146: 19-33. [29] Reyssat M, Pépin A, Marty F et al (2006) Bouncing transitions on microtextured materials. EPL(Europhysics Letters) 74(2): 306.
21
[30] Rioboo R, Voué M, Vaillant A et al (2008) Drop impact on porous superhydrophobic polymer surfaces. Langmuir 24(24): 14074-14077.
22
PII: DOI: Reference:
S0143-7496(16)30258-5 http://dx.doi.org/10.1016/j.ijadhadh.2016.12.011 JAAD1944
To appear in: International Journal of Adhesion and Adhesives Accepted date: 21 December 2016 Cite this article as: Wei Zhang, Run-run Zhang, Cheng-gang Jiang and Chengwei Wu, Effect of pillar height on the wettability of micro-textured surface： Volume-of-Fluid simulations, International Journal of Adhesion and Adhesives, http://dx.doi.org/10.1016/j.ijadhadh.2016.12.011 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Effect of pillar height on the wettability of micro-textured surface： Volume-of-Fluid simulations
Wei Zhang, Run-run Zhang, Cheng-gang Jiang, Cheng-wei Wu State Key Laboratory of Structure Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, China *
Corresponding author. Email: [email protected], Tel: +86 411 84706353
Abstract: The wetting and spreading characteristics of a water drop on a surface is highly dependent on the geometric parameters of the micro-texture of the surface. To obtain a stable Cassie drop, the condition that the pillar height must be higher than the sag height of the meniscus is a necessary, yet not a sufficient, condition. From the viewpoint of energy minimization, a new criterion was proposed to design the height of the pillar for achieving a stable and robust anti-adhesive drop state. To ensure such a composite interface, the height of the pillars should be taller than the critical height where the critical contact angle is equal to the intrinsic contact angle. If this height requirement is not met, the drop exists in either a Wenzel or a metastable Cassie state. The numerical simulations using the Volume-of-Fluid method in Fluent support the above proposals. Keyword: Volume of fluid, superhydrophobic, anti-adhesive, pillar height, simulation
1
1. Introduction On rough hydrophobic surfaces, a water drop can either reside on the top of the roughness grooves, i.e. in Cassie states or impale the roughness grooves, i.e. in Wenzel states. Owing to the low resistance from the air pocket, a Cassie drop shows considerably less hysteresis in contrast to a Wenzel drop; this means the drop can move with less adhesion and friction and more easily relative to the surface [1]. Hence, in applications such as surface-tension-induced motion in microfluidic devices [2], self-cleaning surfaces [3, 4], magnetic storage devices, and micro/nano electromechanical systems [5, 6], it is imperative that a composite contact is achieved, rather than with the surface grooves being wet with liquid.
Then it comes to the question: in practice, how to design and fabricate a superhydrophobic surface with robust anti-adhesive properties? For any given low surface energy materials, the drop state is then largely determined by the geometric parameters of the surface roughness [7, 8]. Consider a small water drop suspended on a superhydrophobic surface consisting of a regular array of pillars, which is one of most popular structures used currently and can be made easily by lithographic techniques [9]. Based on the geometric consideration, Patankar et al. proposed that the depth of the cavity must be greater than the droop; otherwise the meniscus will eventually touch the relief bottom, leading to the collapse of the Cassie wetting and to the Cassie-Wenzel transition [10, 11]. On the other hand, Nosonovsky et al. proposed that the surface energy of a water drop on surface with nanoscale imperfections is a function of the vertical position of the liquid-vapor interface under the droplet [12].
2
Then, we wonder, from the surface energy point of view, how to design the height of the pillars of the surface that can ensure the drop exists in a Cassie state? And this is the motivation of the simulation work in this communication.
2. Effect of pillar height To date, the creation of a micro-structured surface with superhydrophobicity and anti-adhesive property remains a challenge to researchers. Apart from fulfilling the surface force equilibriums, the height of the pillar, h, should be so tall that the curved liquid-air interface cannot touch the bottom of the roughness groove, as expressed in Inequality 1. h hi 2 (l a)
1 s i n 2c o s
(1)
Where hi is the maximum sag distance of the curved interface allowed by the micro structure, see Fig. 1.
Fig. 1 Schematic of a liquid-gas meniscus suspended on top of pillars
3
2
a Substituting area fraction of solid-liquid interface in the horizontal plane , , l the height criterion, based on geometric consideration, for achieving a Cassie drop state can be obtained, as expressed in Inequality 2. To remove the effect of size, the dimensionless quantity, i.e. h/l, was used here. h hi 1 s i n 2 (1 ) l l 2c o s
(2)
On the other hand, if a surface is superhydrophobic, the drop must remain in the Cassie state and cannot transit to a Wenzel one. Hence, from the energy viewpoint, the surface energy of the structure with air entrapped must be smaller than the wetted state. Suppose a small displacement dx of the triple-phase line along the parallel direction of the micro-pillar structured surface, as shown in Fig. 2, when the droplet spreads on the micro-textured surface, the change of the surface free energy dE can be calculated using either Equation (3) for a Wenzel state or Equation (4) if a Cassie state is reached [13].
Fig. 2 Scheme of the triple-phase line between the droplet and the rough surface
dE r ( SL SG )dx GLdx cos * dE ( SL SG )dx (1 ) GLdx GLdx cos *
4
(3) (4)
where SL, SG, GL are surface tension of solid-liquid, solid-gas, and gas-liquid interfaces, respectively. is the apparent angle on the rough solid surface. r is surface roughness. Since an equilibrium drop shape with a lower value of the apparent contact angle will have a lower the surface energy [14], the droplet will stabilize with the smallest possible apparent contact angle. When the system reaches a thermodynamic equilibrium, dE=0, Inequality 5 should be satisfied.
( SL SG ) (1 ) GL r ( SL SG )
(5)
Substituting into Young’s equation, SG SL GL cos , we obtain Inequality 6.
cr ， cos cr
1 r
(6)
It has been shown that the critical contact angle can be realized by designing a geometric structure such that the Wenzel and the Cassie states have the same energy at a given intrinsic contact angle. In this case, the Wenzel and Cassie states have the same chance of existence, and thus this approach is not desirable if one wants to form a Cassie surface. For convenience, we define hcr as the critical height, i.e. the height of the pillar when the critical contact angle,
, is equal to the intrinsic contact angle,
h a
. Substituting r (4 1) , another height criterion (Inequality 7), based on energy minimization, is proposed here.
h hcr
a( 1)(cos 1) 4 cos
(7)
Analogously, to remove the effect of size, the dimensionless quantity, i.e. h/l was used.
5
h hcr ( 1)(cos 1) l l 4 cos
(8)
As discussed above, the height of the pillar is a crucial parameter that affects the wettability of the surface. To obtain a stable composite state, the height must be not only higher than the sag height of the meniscus caused by the external pressure, but also needs to meet the requirement of energy minimization. Then the question comes out: how about the compatibility of these two height criteria, hi/l and hcr/l?
Based on geometric consideration and energy minimization discussed above, the relationship of the stabilized state of a droplet and h/l, can be expressed using Fig. 3. Clearly, no matter what the values of h/l, and arehcr/l is always larger than hi/l. This means to obtain a stabilized Cassie droplet, h/l must be larger than hcr/l, i.e. meeting the energy criteria. If hi/l < h/l < hcr/l, the droplet may exist in a Cassie state but the application of any external disturbance may incur the transition from Cassie state to Wenzel sate, a metastable Cassie state. If the geometric criteria is not met, h/l < hi/l, a Wenzel droplet will be obtained.
6
Fig. 3 The ratio of the height (h) to the spacing (l) as a function of surface property for micro-texture with different intrinsic contact angle ( ) and area fraction of solid-liquid interface (). A red surface represents the geometric criteria obtained using Eq. (2). A green surface represents the energy criteria obtained using Eq. (8).
3. Computation model The Volume-Of-Fluid (VOF) method originally developed by Hirt and Nichols in 1981 [15] has become the most frequently used approach to simulate free-surface flows and study the phase change phenomena. And Fluent is the world's most widely used CFD software. To verify the effect of pillar height on the stabilized droplet state that we proposed above, the VOF method in Fluent was adopted to simulate the gas-liquid two-phase flow with a free surface.
7
Periodic structure consisting of a regular array of square pillars with equal side length a, height h and spacing l is widely adopted in the manufacture of super hydrophobic surfaces and this surface was used for the simulations, as shown in Fig. 4.
Fig. 4 Schematic of the periodic micro-structure.: a is the length of pillar, l is the spacing between pillars, h is the height of pillar
Due to symmetry, 1/4 of the model was used for simulation and the computed space is a rectangular structure (Fig. 5). In the set of boundary conditions, the surface of the pillar and the bottom surface of the rectangular space are set as the wall and can be given different contact angles as required; the front and left surface of rectangular space are set as symmetric; the top, back and right surface of rectangular space are set as constant pressure inlet. In the set of initial field, air is considered as the primary phase and water is the secondary phase.
8
Fig. 5 The set of simulation model and initial field
One-fourth of the sphere adjacent to the pillar is endowed the properties of the secondary phase and represents the morphology of the simulated droplet, as illustrated as blue in Fig.5. To ensure that the surface tension governs the deformation of a droplet, the radius of the droplet must be smaller than the capillary length. As the water capillary length was reported to be 2.7 mm [16], the initial radius of the droplet, R0, is set as 0.25 mm.
3.1 Quasi-static contact To alleviate the added dynamic effects, the velocity of the droplet is set as 1 mm/s. The Weber number is much less than 1 and the whole process is thought of being
9
quasi-static. The geometric parameter and property of the micro-pillar structure are given in Table 1.
Table 1 The geometric parameter and property of the micro-pillar structure
a / μm
l / μm
h / μm
θ/°
θcr / °
50
100
100
120
105.8
50
100
100
95
105.8
When the size of the micro structure of a surface is given, the wettability of the surface is mainly determined by the surface property. If the intrinsic contact angle, θ, is larger than the critical contact angle θcr, the droplet will be of a Cassie state; if θ < θcr, the droplet will be of a Wenzel state. As θcr is 105.8° in the model, two intrinsic angles, i.e. 120° and 95° are adopted for the simulation.
Fig. 6 diagrammatically demonstrates the simulation results; and it is apparent that for the intrinsic contact angle of 1200 the droplet sits on the top of the peaks of the rough surface and air exits between the droplet and pillars, a typical Cassie state with the apparent contact angle being around 153.4° (the top row of Fig. 6). This agrees with the value predicted by the Cassie formula [17], 151°. In terms of a contact angle of 95°, the droplet replaces the air layer and wets the grooves of the rough surface, a typical Wenzel state, see the bottom row of Fig. 6. The apparent contact angle measured is 109°, in good agreement with the one predicted by the Wenzel formula
10
[18], 105.20. Clearly, these models can reflect the effect of intrinsic contact angle on the wettability of the surface.
Fig. 6 The simulation result with intrinsic contact angle of 120° (top row) and 95° (bottom row)
By fixing the side length of pillar, the spacing between the pillars, and the intrinsic contact angle, we examined the effect of height of pillar on the wettability of the surface. Table 2 lists the geometric parameters and properties of different pillar surfaces. We assume the three surfaces have the same intrinsic contact angle θ (1100). Apparently, the area fraction of the solid-liquid interface φ (0.25), critical height hcr (72.1 μm), and sag height hi (6.2 μm), and the mere difference exists in the height of pillars, being 100 μm, 50 μm and 18 μm.
11
Table 2 The geometric parameters and properties of the micro-pillar structure
a / μm
l / μm
h / μm
θ/°
φ
θcr / °
hi / μm
hcr / μm
50
100
100
110
0.25
105.8
6.2
72.1
50
100
50
110
0.25
115.4
6.2
72.1
50
100
18
110
0.25
132.5
6.2
72.1
Herein, the height of the pillar was chosen in reference to hcr and hi. In all three cases, the height of pillar h is higher than hi. According to the well documented geometric criterion, the droplet should settle in a Cassie state and give a contact angle of about 146.7°. To alleviate the additional dynamic effects, the initial velocity of the droplet was kept small and constant at 1mm/s. The final simulated states of the droplet are illustrated in Fig.7.
Strikingly, not all the droplets settle in the same state. For pillars with heights of 100 μm and 50 μm, the droplet settles in the Cassie state and an air layer exists between the droplet and the surface, with the apparent contact angle being 143° and 144.2°, respectively, fairly close to the predicted values. When the height of the pillar is 18μm, however, the droplet wets the pillar and the apparent contact angle is only 112.3°, which is closer to the value predicted by the Wenzel model. It is apparent that at the pillar height of 100 μm (h = 1.39 hcr), θcr < θ, and a Cassie drop is expected; whereas at the pillar height of 18μm (h = 0.25 hcr), θcr > θ and a Wenzel drop is obtained. These results indicate that in quasi-static contact, h > hi is not a sufficient condition that can guarantee that the droplet will not make contact with the bottom of the
12
micro-textured structure, i.e. a Cassie state. On multi-groups of simulations by varying θ, a, and l, we found only when h > 0.25 hcr, the drop is prone to be in a Cassie state.
Fig. 7 The spreading of a droplet and its apparent contact angle on three microstructure surfaces with different pillar heights. Left: 3D front view, right: measure of apparent contact angle
It is worth mentioning that even though the initial velocity of the droplet was set at 1 µm/s, the droplet will continue to wet the pillar with the height of 18µm. When the pillar height was set at 50 μm, h =0.69 hcr, although θcr is slightly larger than θ, under quasi-static conditions, the droplet exists in a Cassie state. The plausible explanation for this is that, upon quasi-static conditions, the kinetic energy obtained by the droplet
13
is not high enough to break the pressure balance of the gas/liquid surface and thus the droplet cannot wet the space of the micro-structure. This is in good agreement with experimental observations by Li et al. [19] and Yoshimitsu et al. [20]. Their experimental results show that, with the increase of the ratio of h to l, the hydrophobicity of the micro-texture surface is improved.
3.2 Impacting contact In the aforementioned simulations, the collision velocity is set at 1 mm/s and thus the whole process can be considered as quasi-static. In many practical cases, however, the drop may collide with surface with much higher velocity. For a raindrop with a diameter of 1 mm, the average falling velocity can reach 4 m/s [21]. In ink-jet printing, the typical drop speed is 6-9 m/s [22]. In these cases, the dynamic effect of the liquid drop is obvious and cannot be ignored. Extensive studies have been carried out with regards to the dynamic impact behavior of a water drop. Bird et al. [23] and Liu et al. [24] investigated the effects of surface structure on the contact time of a bouncing drop. They reported the contact time does not depend on the impact velocity and that a water drop can behave as a harmonic spring. Jung et al. [25] and Bartolo et al. [26] have shown that the transition from a Cassie to a Wenzel state can occur when a water drop collides with a surface which has specific micro-patterns. As such, we further investigated the effect of collision velocity on the wettability of a water drop on the micro pillar-textured surface. The geometric parameter of the pillar and the intrinsic contact angle are given in Table 3. The quasi-static simulations above reveal that for both of these two surfaces air remains trapped below the water drop and the drop sits
14
on a composite surface made of pillar and air. Then, we wonder what will happen if we increase the collision velocity, and whether this external actuation will incur any change in the state of the drop? Table 3 The geometric parameters and properties of the micro-pillar structures
a / μm
l / μm
h / μm
θ/°
1st surface
50
100
50
110
2nd surface
50
100
100
120
Fig. 8 demonstrates the wetting and spreading of the water drop after collision with the first surface, where the cross-sections of the drop are selected. It was found that, with the increase of the initial collision velocity, the Cassie droplet under quasi-static conditions becomes unstable. After initiating contact with the surface, the droplet enters the space between micro-structures, replaces the air and eventually contacts the bottom of the groove. After oscillations, a Wenzel droplet is obtained.
Fig. 8 The wetting and spreading of water drop after colliding with the first kind of surface with a velocity of 1m/s
This indicates that the hydrophobicity of the first surface is not stable and may transit from a Cassie to a Wenzel state upon external actuation, impacting here. This is in agreement with the observation that the Cassie regime can be obtained for θ <θcr in
15
spite of a higher energy [27]. Once such a transition happens, the system becomes more stable due to the drop in energy and a reverse transition is hard to be obtained. The cross sections of the drop after the collision with the second surface are illustrated in Fig. 9, where it was found that when contacting the surface, the drop deforms, wets the sides of the pillars and fills in the groove due to the increase of collision velocity, and simultaneously a certain volume of air is embedded between the bottom center of the drop and the surface, forming an air pocket inside the drop. This is in good agreement with experimental observations by Reyssat et al. [28]. After reaching the maximum spreading radius at about 0.5 ms, the spreading liquid starts to retract and finally bounced off the surface at about 1.5 ms. Owing to the effect of gravity, the speed of the drop reduces gradually, and becomes zero at 11 ms, yielding a maximum rebounding height of 0.8 mm. The drop then falls back and contacts the surface again. After small amplitude oscillations, the drop reaches a static state and sits on the top of the surface. It should be noted that the rebounding of a drop can take place only when the collision speed is sufficiently high since both spreading and retracting of the drop will consume energy. For a drop with a radius of R0, the stored surface energy in spreading and retracting is estimated to fall in the order of R02. Only when the kinetic energy (in the order of R03V02) of the droplet is larger than the surface energy, is it possible for the droplet to rebound. This means the initial collision speed must be higher than ( /R0)
1/2
[29, 30]. For the model used, the minimum speed for
rebounding is about 0.2 m/s. This explains why in quasi-static contact (collision speed: 1 mm/s) and the second contact of the drop with the surface (collision speed: 1 m/s), no further rebounding phenomena were observed.
16
Fig. 9 The deformation and rebounding of a water drop after colliding with the second kind of surface with a velocity of 1m/s
These results imply that when h < hcr, although a Cassie drop can be obtained, it is metastable and will transit to a Wenzel state under the external actuations to minimize the energy of the system. To ensure a stable composite contact, the height of the pillar should be chosen such that the Cassie drop has a lower energy than the Wenzel drop.
4. Conclusions The height of pillar, h, is a crucial parameter that affects the wettability of the surface with a regular array of pillars. To obtain a stable composite state, surmounting the sag height of the meniscus caused by the external pressure is a necessary but not sufficient condition. This height geometric criterion must be resolved in terms of an energy minimization approach, where it is proposed that the pillar should be taller than the critical height, hcr. When h < 0.25 hcr. a Wenzel state can be obtained. When 0.25 hcr
17
< h < hcr, a Cassie state can be obtained on quasi-static conditions; however, this state is metastable. On impacting contact with a velocity of 1 m/s, the water drop wets the surface after deformation, demonstrating a transition from a Cassie to a Wenzel state. Under the same impacting velocity, when the water drop hits the surface with the pillar height being over hcr, it deforms and then retracts and bounces off the surface. Finally, the droplet sits on the surface and has a high contact angle, which suggests the formation of a solid-air-liquid interface. Based on these, a conservative criterion is proposed for achieving a stable Cassie drop with respect to the pillar height, i.e. .
Acknowledgements The NSFC (11272080,11572080), the NSFC of Liaoning Province (2015020198), the Fundamental Research Funds for the Central Universities of China (DUT15JJG06, DUT14LK36), National Basic Research Program of China (2015CB057306) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars (State Education Ministry) were acknowledged for the financial support.
References: [1] Cottin-Bizonne C, Barrat JL, Bocquet L, Charlaix E (2003) Low-friction flows of liquid at nanopatterned interfaces. Nat Mater 2(4): 237-240. [2] Patankar NA (2004) Transition between superhydrophobic states on rough surfaces. Langmuir 20(17): 7097-7102.
18
[3] Yao L, He J (2014) Recent progress in antireflection and self-cleaning technology–From surface engineering to functional surfaces. Prog Mater Sci 61: 94-143. [4] Nishimoto S, Bhushan B (2013) Bioinspired self-cleaning surfaces with superhydrophobicity, superoleophobicity, and superhydrophilicity. RSC Advances 3(3): 671-690. [5] Jung YC, Bhushan B (2008) Dynamic effects of bouncing water droplets on superhydrophobic surfaces. Langmuir 24(12): 6262-6269. [6] Jung YC, Bhushan B (2007) Wetting transition of water droplets on superhydrophobic patterned surfaces. Scripta Mater 57(12): 1057-1060. [7] Reyssat M, Yeomans JM, Quéré D (2008) Impalement of fakir drops. EPL 81(2): 26006. [8] Yoshimitsu Z, Nakajima A, Watanabe T et al (2002) Effects of surface structure on the hydrophobicity and sliding behavior of water droplets. Langmuir 18(15): 5818-5822. [9] Roach P, Shirtcliffe NJ, Newton MI (2008) Progress in superhydrophobic surface development. Soft Matter 4(2): 224-240. [10] Patankar NA (2010) Consolidation of hydrophobic transition criteria by using an approximate energy minimization approach. Langmuir 26(11): 8941-8945. [11] Whyman G Bormashenko E (2011) How to make the Cassie wetting state stable?
19
Langmuir 27(13): 8171-8176. [12] Nosonovsky M, Bhushan B (2008) Patterned nonadhesive surfaces: superhydrophobicity and wetting regime transitions. Langmuir 24(4): 1525-1533. [13] Wang S, Liu K, Yao X et al (2015) Bioinspired surfaces with superwettability: new insight on theory, design, and applications. Chem Rev 115(16): 8230-8293. [14] Patankar NA (2003) On the modeling of hydrophobic contact angles on rough surfaces. Langmuir 19(4): 1249-1253. [15] Hirt CW, Nichols BD (1981) Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 39(1): 201-225. [16] Jung YC, Bhushan B (2007) Wetting transition of water droplets on superhydrophobic patterned surfaces. Scripta Mater 57(12), 1057-1060. [17] Cassie ABD, Baxter S (1944) Wettability of porous surfaces. Transactions of the Faraday Society 40: 546-551. [18] Wenzel RN (1936) Resistance of solid surfaces to wetting by water. Industrial & Engineering Chemistry 28(8): 988-994. [19] Li X, Mao L, Ma X (2013) Dynamic behavior of water droplet impact on microtextured surfaces: the effect of geometrical parameters on anisotropic wetting and the maximum spreading diameter. Langmuir 29: 1129-1138. [20] Yoshimitsu Z, Nakajima A, Watanabe T et al (2002) Effects of surface structure on the hydrophobicity and sliding behavior of water droplets. Langmuir 18: 5818-5822
20
[21] Dingle N, Lee Y (1972) Terminal fall speeds of raindrops. J Appl Meteorol 11(5): 877-879. [22] Hutchings I, Tuladhar T, Mackley M et al (2009) Links between ink rheology, drop-on-demand jet formation, and printability. J Imaging Sci Techn 53(4): 41208-1-41208-8. [23] Bird JC, Dhiman R, Kwon HM et al (2013) Reducing the contact time of a bouncing drop. Nature 503(7476): 385-388. [24] Liu Y, Moevius L, Xu X et al (2014) Pancake bouncing on superhydrophobic surfaces. Nature Physics 10(7): 515-519. [25] Jung YC, Bhushan B (2008) Dynamic effects of bouncing water droplets on superhydrophobic surfaces. Langmuir 24(12): 6262-6269. [26] Bartolo D, Bouamrirene F, Verneuil É et al (2006) Bouncing or sticky droplets: Impalement
transitions
on
superhydrophobic
micropatterned
surfaces.
EPL
(Europhysics Letters) 74(2): 299. [27] Lafuma A, Quéré D (2003) Superhydrophobic states. Nat Mater 2(7): 457-460. [28] Reyssat M, Richard D, Clanet C et al (2010) Dynamical superhydrophobicity. Faraday Discussions 146: 19-33. [29] Reyssat M, Pépin A, Marty F et al (2006) Bouncing transitions on microtextured materials. EPL(Europhysics Letters) 74(2): 306.
21
[30] Rioboo R, Voué M, Vaillant A et al (2008) Drop impact on porous superhydrophobic polymer surfaces. Langmuir 24(24): 14074-14077.
22