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Directional migration of single droplet on multi-wetting gradient surface by 3D lattice Boltzmann method
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Directional migration of single droplet on multi-wetting gradient surface by 3D lattice Boltzmann method Xin Wang , Bo Xu , Yi Wang , Zhenqian Chen PII: DOI: Reference:
S0045-7930(19)30350-0 https://doi.org/10.1016/j.compfluid.2019.104392 CAF 104392
To appear in:
Computers and Fluids
Received date: Revised date: Accepted date:
21 June 2019 6 September 2019 17 November 2019
Please cite this article as: Xin Wang , Bo Xu , Yi Wang , Zhenqian Chen , Directional migration of single droplet on multi-wetting gradient surface by 3D lattice Boltzmann method, Computers and Fluids (2019), doi: https://doi.org/10.1016/j.compfluid.2019.104392
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Highlights
Spontaneous movement of droplet on multi-wetting gradient surface was simulated.
Both hydrophobicity and hydrophilicity can be strengthened by micropillar arrays.
Wetting gradient and surface morphology had a great effect on droplet movement.
The droplet state transition from partial Cassie to Wenzel state can be observed.
Gravitational coefficient should be considered during climbing-upward movement.
Directional migration of single droplet on multi-wetting gradient surface by 3D lattice Boltzmann method Xin Wang a, Bo Xu a, b, c, Yi Wang a, Zhenqian Chen a, b, c *
(a School of Energy and Environment, Southeast University, Nanjing, P. R. China; b
Jiangsu Provincial Key Laboratory of Solar Energy Science and Technology, School of Energy and Environment, Southeast University, Nanjing, P. R. China;
c
Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of Energy and Environment, Southeast University, Nanjing, P. R. China)
* Corresponding author, E-mail address: [email protected], Tel/Fax: +86-25-83790626.
Submitted to (Computers and Fluids)
Abstract: Directional migration of single microdroplet on a microstructured surface with multi wetting gradients plays a significant role in many industrial applications. To understand the mechanism of spontaneous movement of microdroplet on wetting gradient surfaces under different gravities, the model of microdroplet dynamics is built by 3D lattice Boltzmann method. Effects of wetting gradient, surface morphology, surface orientation and gravitational force on spontaneous movement are investigated. The results demonstrate that static contact angles are greatly affected by micropillars. Both hydrophilicity and hydrophobicity can be strengthened by micropillar arrays. As a droplet moves from hydrophobic to hydrophilic driven by net capillary force, a transition from partial Cassie state to Wenzel state can be found. A larger wetting gradient and larger solid fraction promote the spontaneous movement of microdroplet. In the process of movement, surface free energy shows an increasing tendency due to part of kinetic energy converted to surface free energy. In spite of two-layer surface capable of maintaining a droplet in partial Cassie state, it will decrease the movement velocity. For Bo≤0.0126, surface orientation has little effect on climbing-upward movement. When the Bo number ranges from 0.0126 to 0.063, gravitational coefficient should be considered in the process of microdroplet (2μm) climbing- upward movement. The surface with multi wetting gradients can be applied to remove droplets during condensation to enhance heat transfer. Keywords: 3D LBM; Microdroplet; Wetting gradient; Micropillar; Gravity
1. Introduction Surface wettability is quite significant for condensation heat transfer, affecting the condensation mode: filmwise condensation or dropwise condensation. The wettability of liquid on the solid substrate is distinguished by the equilibrium contact angle (θe) of three-phase contact line. θe>90° is considered to be a hydrophobic surface, while θe<90° is referred to a hydrophilic surface. Inspired by biological surfaces [1-5], preparation of superhydrophobic (SH) surface has attracted extensive attention due to the higher heat transfer coefficient during dropwise condensation.
Since the current fabrication technology of preparing SH surface is unsatisfactory, condensate droplets on SH substrates will change from Cassie state to Wenzel state after operating several tens of hours [6]. Droplets pinned on the SH surface are difficult to remove, resulting in the failure of SH surface. Hence, the removal of condensate droplets has now emerged as a crucial engineering problem. In recent years, self-removal of droplets during condensation has become possible. Spontaneous movement of droplet on spider silk [7, 8], conical cactus spines [9, 10] and desert beetle [11] was experimentally observed. Those organisms possess surface structures with wetting gradient and unique shapes, which can drive the droplet directional migration. Spontaneous movement of droplets plays a vital role in a great number of industrial applications, such as fuel lines, flexible electronics, solar cells as well as condensers. In the past few decades, a great deal of experimental work on droplet dynamics on the surface with multi wetting gradients has been investigated. Whitesides and Chaudhury [12, 13] first reported the experiment on droplet motion on wetting gradient surfaces. It was found that self-propelled movement of droplet along the direction of stronger wettability was observed driven by gradient surface free energy. Subsequently, several experiments of droplet directional migration on multi wetting gradient surfaces were conducted. Lai et al. [14] fabricated a gradient wettability surface for transporting droplets from SH region to hydrophilic region by combining a structure gradient and a self-assembled-monolayer gradient. Besides, a bidirectional gradient surface that changes the droplet moving direction was achieved. Paradisanos et al. [15] and Ghosh et al. [16] used the laser technology to create a wetting gradient surface and obtained the droplet moving speed of several hundred millimeters per second. Deng et al. [17] prepared a wetting gradient surface with wedge-shaped pattern using the improved anodic oxidation method. Displacement of droplet movement can be controlled by adjusting the wedge angle and droplet volume. Khoo and Tseng [18] carried out an experiment of droplet movement on a wetting gradient surface with nano-textured pattern, which can deliver a wide range of droplets at speeds up to 0.5m/s. However, due to experimental uncertainties including complex interface and visual observation, it is difficult to understand the mechanism
of microdroplet movement on wetting gradient surfaces in depth. There also has been a few numerical work devoted to the droplet movement on wetting gradient surfaces. Li et al. [19] simulated droplet movement on a linear gradient wettability surface by dissipative particle dynamics. And effects of wetting gradient and droplet size on the movement velocity were studied. Chakraborty et al. [20] used molecular dynamics to simulate droplet migration on the surface with chemically induced wettability gradient and their results agreed well with the model of droplet self-propelled movement on wetting gradient surfaces. Deng et al. [21] simulated the droplet dynamics on wetting gradient surfaces during condensation using the free energy lattice Boltzmann model. Their results indicated that condensate droplets can be removed by wetting gradient surface, thus enhancing condensation heat transfer. On account of its advantages in computational efficiency and dealing with complex boundaries, lattice Boltzmann method (LBM) has become an effective and essential tool in multiphase flows. Pseudopotential model, as one of the lattice Boltzmann models, has been widely used to simulate the microdroplet dynamics. Pravinraj and Patrikar [22] investigated the droplet splitting and directional migration on the surface with chemical gradient and structural gradient. In addition, the change of surface free energy and contact angle hysteresis were analyzed. Lee et al. [23] reported the microdroplet movement on wetting gradient and structural gradient surfaces by 3D pseudopotential model. Droplet movement was also analyzed in terms of kinetic, gravitational and surface free energy. Although a little numerical work has been carried out of the spontaneous motion on wetting gradient surfaces, effect of surface structure on the droplet dynamics are neglected. Moreover, the above experimental and numerical investigations are on horizontal surfaces without considering the influence of gravitational coefficient and surface orientation on droplet dynamics. It has been demonstrated that micropillars, surface orientation and gravity play a significant role on the dynamic characteristics of droplets in the process of condensation [24-28]. Here, 3D pseudopotential model is adopted to simulate the spontaneous movement of single droplet on rough wetting gradient surfaces. Effects
of surface orientation and gravitational coefficient on droplet dynamics are investigated. In addition, displacement, penetrated depth and surface free energy are also analyzed to provide insight into dynamic behaviors of droplet migration on wetting gradient surfaces. It is meaningful and significant for the design of multi wetting gradient surface with micropillar arrays. The surface with multi wetting gradients can be applied to getting rid of droplets during condensation, thus enhancing condensation heat transfer.
2. Model description 2.1 3D lattice Boltzmann model In the single-component multi-phase model, migration and collision of fluids can be described by specific particle distribution functions. Recently, Xiong and Cheng [29] proposed a new piecewise distribution function with two different relaxation times. A larger relaxation time is first applied to maintain the numerical stability in early time steps. Then, a smaller relaxation time is applied to obtain a lower viscous droplet. In this paper, the evolution function with BGK collision operator can be given by: (
t
where
𝛿)
( t)
{
[ ( t)
( t)]
( t) t
[ ( t)
( t)]
( t) t
( t) denotes the density distribution function with velocity
lattice x and time t. 𝛿 is the time step.
(1) at the
( t) is the force term, including
fluid-fluid force, fluid-solid force as well as body force.
is the corresponding
equilibrium distribution function [30], written as: 𝜔 𝜌[
𝛼 ∙𝐮
𝑐s
( 𝛼 ∙𝐮)𝟐 2𝑐s4
𝐮 2𝑐s
]
(2)
where ωα are the weighting factors. For D3Q19 model, ωα=1-3, ωα=1/18 and ωα=1/36. 𝑐𝑠
𝑐/√3 denotes the lattice sound speed (𝑐
D3Q19 model is written as:
𝛿x /𝛿 ). The discrete velocity
in
( ) )( ± )( {(± ± ) (± ± ) ( ± ± ) (± ± )
𝛼 𝛼 𝛼
~6 7~ 8
(3)
The macroscopic density and velocity can be obtained by the density distribution function: 𝜌
∑
( t)
∑
𝜌𝐮
( t)
(4)
Through the multi-scale analysis of Chapman-Enskog expansion, the macroscopic Navier-Stokes equations can be restored and the kinematic viscosity of fluid is given by: 𝜐
𝑐s2 (𝜏2
.5)𝛿
(5)
The total force F involved in this paper is mainly composed of fluid-fluid force, fluid-solid force and gravitational force. In the past few decades, numerous schemes have been proposed to incorporate the force term in the LBM. Here, exact difference method (EDM) [31] is adopted to incorporate the force term due to the better numerical accuracy and independent of relaxation time, written as: ( t) where
[𝜌( t) 𝐮
𝐮
𝐮]
[𝜌( t) 𝐮]
(6)
𝐅𝛿 /𝜌 is the velocity change caused by the total force F.
The fluid-fluid force in the form of combined approximation proposed by Kupershtokh et al. [32, 33] and Gong and Cheng [34] is adopted, written as: 𝐅in
𝛽𝜓 ( ) ∑ G( −𝛽 2
𝛿 )𝜓( ∑ G(
𝛿) 𝛿 )𝜓 2 (
𝛿)
(7)
where β is a constant parameter. G denotes the interaction strength between particles, expressed as: G(
𝛿)
g {g 2
| | | | √2 otherwise
(8)
where g1=g0 and g2=g0/2 in D3Q19 model. And the pseudopotential function 𝜓( ) in Eq. (7) is specially defined as: 𝜓( )
√2(𝑝
𝜌𝑐𝑠2 )/𝑐0 g
(9)
where the pressure p can be obtained by actual equation of state (EOS). Here, PengRobinson EOS is used, written as:
𝑝
𝜌𝑅𝑇 −𝑏𝜌
(𝑇)
𝑎𝜌
(10)
+2𝑏𝜌−𝑏 𝜌
where R=1, 𝛼(𝑇)
[
.87324(
2
√𝑇/𝑇𝑐𝑟 )] , a=2/49, b=2/21. Since P-R EOS
is adopted in this paper, the value of parameter β is equal to 1.16. The fluid-solid force can be used to characterize the wettability between the fluid and solid wall, defined as: 𝐅s
Gs 𝜓( ) ∑ 𝜔 S(
𝛿)
(11)
where the parameter Gs denotes the interaction strength between the fluid and solid wall. Surface wettability can be controlled by changing the value of parameter Gs. S is an indicator function with S=1 for solid and S=0 for fluid. In addition, the gravitational force is calculated by: 𝐅g
(𝜌( )
𝜌𝑣 )g
(12)
where g is the gravitational acceleration coefficient and ρv is the density of vapor phase. The real fluid velocity U is defined as the average of velocity before and after collision, written as: 𝜌𝐔
𝜌𝐮
.5𝛿 𝐅
(13)
2.2 Model validation Laplace’s law is considered to validate 3D pseudopotential lattice Boltzmann model. According to Laplace’s law, the pressure difference inside and outside the droplet is linear with the inverse radius (∆p=2σ/r, where σ denotes the surface tension). Initially, a droplet is placed at the center of computational domain (151×151×151). Periodic boundary conditions are applied for all boundaries. The fluid density is initialized as: 𝜌 (x y z)
𝜌l
𝜌v 2
𝜌l
𝜌v 2
[tanh(2(√(x
x 0 )2
(y
y0 )2
(z
z 0 )2
r)/W)]
(14)
where W is the thickness of two-phase interface (W=3). As shown in Fig.1, the inverse radius increases linearly with the pressure difference at different temperatures (T/Tcr=0.9, T/Tcr=0.85, T/Tcr=0.8), and the corresponding surface tensions are 0.0903,
0.1521 and 0.2287, respectively. In addition, our LB results are compared with Maxwell construction. It can be seen from Table.1 that vapor and liquid densities in equilibrium are in good agreement with the Maxwell construction. 0.05
T/Tcr=0.90, =0.0903 T/Tcr=0.85, =0.1521
0.04
T/Tcr=0.80, =0.2287
p
0.03
0.02
0.01
0 0
0.02
0.04
0.06
0.08
0.10
1/r
Fig.1 Laplace’s law at different temperatures
Table.1 Comparison of LB results with Maxwell analytical solution Analytical solution
T/Tcr 0.90 0.85 0.80
LB results
ρl
ρg
ρl
ρg
ρl/ρg
σ
5.907 6.627 7.205
0.580 0.342 0.197
5.904 6.641 7.198
0.580 0.333 0.185
10.18 19.94 38.89
0.0903 0.1521 0.2287
3. Results and discussion In this section, comparison of static contact angle on smooth and microstructured surface will be discussed. Then, Effects of wetting gradient and surface morphology on
horizontal
movement
of
microdroplet
will
be
investigated.
Finally,
climbing-upward movement on inclined surfaces will be explored. In this simulation, the initial liquid and gas densities are 7.2 and 0.197, respectively. 3.1 Contact angle As mentioned earlier, surface wettability is greatly significant for the droplet dynamics. Therefore, single droplet on smooth and microstructured surfaces with
different wettability is simulated. Bounce-back scheme is applied to the bottom surface and periodic boundary conditions are applied to other borders. Initially, a droplet with r=20 lattice unit is located at the center of bottom surface, corresponding to 2μm in real units. The calculation won’t stop until the relative error is less than 10-6, indicating that a droplet is static on solid substrate. In pseudopotential model, wettability can be adjusted by the interaction strength between the fluid and solid wall to change static contact angles. Fig.2 shows the relationship between static contact angles (θe) and interaction strength Gs on smooth and microstructured surfaces. It can be seen that static contact angle increases with the greater interaction strength G s. For smooth surface, static contact angles increase linearly with G s. When Gs ranges from -19.0 to -4.0, θe increases from 5.2º to 155.7º. The droplet morphology changes from a thin liquid film to a spherical droplet and the liquid-solid contact area decreases significantly, as shown in Fig.3. For microstructured surface (a=2μm, s=5μm, h=5μm), static contact angles (0º~177.8º) increase nonlinearly with Gs, which is different from smooth surface. When Gs is less than -9.0, the droplets are in Wenzel state and a portion of droplets penetrate into micropillar arrays. In general, as a droplet is in Wenzel state, the wetting of chemically homogenous surface is governed by the Wenzel model [35]: cos w rf cos e
(15)
where rf is defined as the ratio of real surface in contact with liquid to its projection onto the substrate. Due to the real contact area larger than the projected area, hydrophilic property is strengthened by surface textures. Therefore, liquid-solid contact area on microstructured surface is larger than that on smooth surface for the same Gs, resulting in a greater viscous force and a smaller contact angle. When Gs is more than -8.0, droplets are in Cassie or partially Cassie state. There exists some air between the droplet and substrate. The Cassie-Baxter equation [36] can be applied to a droplet in Cassie state, given by:
cos c 1 s (1 cos e )
(16)
where s denotes the friction of solid in contact with liquid. Compared with the smooth surface, contact area between liquid and solid on microstructured surface is
smaller for the same Gs (Fig.3). In this case, viscous force decreases and leads to a larger contact angle. Hence, hydrophobic property is strengthened by micropillar arrays.
180
150
Smooth surface Microstructured surface Linear fit Wenzel =10.18×Gs+197.71
e/°
120
90
60
30
a=0.2μm, s=0.5μm, h=0.5μm 0 -18
-16
-14
-12
-10
-8
-6
-4
Gs
Fig.2 Variation of contact angle with G s
0°
5.2°
0°
(a) Gs=-19.0 19.6°
(b) Gs=-17.0 45.1°
(c) Gs=-15.0 85.0°
24.8°
Liquid film
Liquid film
41.6°
67.5°
(d) Gs=-13.0 84.0°
97.7°
105.5°
(e) Gs=-11.0 138.2°
(g) Gs=-7.0
(f) Gs=-9.0 129.0°
165.9°
146.1°
(h) Gs=-5.0
Fig.3 Comparison of droplet morphologies on smooth and microstructured surfaces
3.2 Directional migration on horizontal wetting gradient surface
Fig.4 shows the computational model of spontaneous movement of droplet on wetting gradient surface with micropillars. In this simulation, the computational domain is 181×81×81 (18μm×8μm×8μm). Bounce-back scheme is applied to the bottom surface and periodic boundary conditions are applied to other borders. Initially, a droplet is located at the position (30, 40, 25), corresponding to 2μm in real units. Twelve different chemical gradients are used on the bottom surface, thereby obtaining a wetting gradient surface. As a droplet contacts the bottom substrate, it will move from a weaker wetting region to a stronger wetting region. The enlarged view in Fig.4 shows a schematic of surface with micropillars. In this simulation, the letters a, s and h represent width, spacing and height of micropillars, respectively. In order to be related to actual physical properties, lattice units should be converted to real physical units. Ohnesorge number (Oh
𝜇𝑙 /√𝜌𝑙 𝜎𝑙𝑣 r) is a dimensionless number, indicating
the relationship between viscous force and inertial and surface tension. Due to [Oh]lu=[Oh]real, Oh number is used to convert the lattice unit to real unit, calculated by: 𝑙0 [𝑡]real [𝑡]lu
[𝑙]real
[Oh]real
[𝜇𝑙 /(𝜌𝑙𝜎𝑙𝑣 )]real
[𝑙]lu
[Oh]lu
[𝜇𝑙 /(𝜌𝑙𝜎𝑙𝑣 )]lu
𝑙0.5
(17)
[√𝜌𝑙/𝜎𝑙𝑣 ]real
(18)
[√𝜌𝑙 /𝜎𝑙𝑣 ]lu
Square micropillar a s h g Gs2 Gs1 w w r0
Weaker wettability
stronger wettability
Fig.4 Computational model of droplet movement on microstructured surface
3.2.1 Wetting gradient Fig.5 shows the directional migration of single droplet (r=2μm) on different
wetting gradient surfaces (a=0.2μm, s=0.5μm, h=0.5μm). Wetting gradients are set to Case 1: 0.25Gs/1.5μm, Case 2: 0.375Gs/1.5μm and Case 3: 0.5Gs/1.5μm, respectively. “0.25Gs/1.5μm” represents that the value of Gs decreases by 0.25 per 1.5μm along the x direction. As shown in Fig.4, Gs decreases from -7.0 (x=1~15) to -9.75 (x=166~180) along the x direction from the red region to the blue region.
Similarly,
“0.375Gs/1.5μm” and “0.5Gs/1.5μm” represent that Gs decrease by 0.375 and 0.5 per 1.5μm along the x direction. Initially, the trailing portion of droplet is located at the weaker wetting area and the leading portion is located on the stronger wetting area. The shape of droplet with θe=138.2° is symmetrical in XZ plane. Due to the action of unbalanced capillary force, a droplet begins to move along the gradient direction (from hydrophobic to hydrophilic). At t=2.088ms, the droplet is deformed with the penetrated volume increased. Moreover, contact area between droplet and solid wall is increased. However, a small amount of air is reserved between the micropillars, indicating that the droplet still maintains partial Cassie state. It can be found that the penetrated volume and displacement of droplet increase with a greater wetting gradient. At t=4.175ms, the state of droplet in Case 2 and Case 3 changes from the originally partial Cassie to Wenzel state. It should be noted that the droplet state is dependent on the depth penetrated into micropillars. In contrast, the droplet in Case 1 remains in the partial Cassie state.
0ms
2.088ms
4.175ms
5.219ms
4.175ms
5.219ms
top view side view
(a) Case 1: 0.25Gs/1.5μm 0ms
2.088ms
top view Wenzel
side view
(b) Case 2: 0.375 Gs/1.5μm
Wenzel
0ms
2.088ms
4.175ms
5.219ms
top view Wenzel
side view
Wenzel
(c) Case 3: 0.5 Gs/1.5μm
Fig.5 Directional migration of single droplet on different wetting gradient surface (top view and side view)
The efficiency of condensate droplet removal is directly influenced by displacement of droplet movement. To analyze this transport mechanism, variation of leading and trailing edge position and penetrated depth with time is plotted, as shown in Fig.6. As can be seen from the schematic diagram in Fig. 6(a), the leading and trailing edge points of droplet are defined as three-phase points where the front and rear portions are in contact with micropillars from side view. Fig. 6(a) shows variation of leading edge position with time. For Case 2 and Case 3, the droplet with partial Cassie state is transformed into Wenzel state in the process of movement at 2.818ms and 4.071ms, respectively (See critical transition point marked in red). In contrast, the droplet in Case 1 always maintains the partial Cassie state in 5.219ms due to moving from hydrophobic area to weaker hydrophobic area, which is consistent with the picture in Fig.5. Displacement of the leading and trailing edge point moves faster with the larger wetting gradient because the droplet is subjected by a driving force in the opposite direction produced by a larger unbalanced Laplace pressure. Displacement of trailing edge point tends to increase with time, which is similar to leading edge point, as shown in Fig. 6(b). Furthermore, a larger wetting gradient has a positive effect on displacement velocity. It is worth noting that displacement of trailing edge point is almost the same before 1.5μm, independent of the wetting gradient. This is due to the trailing edge point located at the weaker wetting area and dragged by the front portion of droplet instead of driven by capillary force. When displacement exceeds 1.5μm, directional migration is dominated by unbalanced capillary force. Displacement of trailing edge point increases more rapidly with a larger wetting gradient. During the
spontaneous movement, part of the droplet penetrated into micropillars has to overcome the resistance caused by micropillars. Therefore, repeating from a gentle rise to a rapid raise can be seen from displacement of the leading and trailing edge point. Penetrated depth is a significant parameter for judging the state of droplet (Wenzel or Cassie), defined as the distance between the bottom of droplet and solid, as can be seen from enlarged view in Fig. 6(c). Variation of the penetrated depth of droplet with time can be divided into two stages: (I) spreading on textured surface dominated by viscous force; (II) Directional migration of droplet affected by unbalanced capillary force. In stage (I), the droplet contacts and spreads on the microstructured surface dominated by viscous force, resulting in a small portion of droplet penetrating into micropillars. Moreover, penetrated depth decreases rapidly with time, independent of wetting gradient. In stage (II), penetrated depth decreases slowly with time. Affected by surface wettability, it decreases faster with a larger wetting gradient. Here, penetrated depth equal to 0.1μm is defined as the critical transition point from Partial Cassie state to Wenzel state. Due to locating at the stronger wetting region, the transition from Partial Cassie state to Wenzel state can be seen in Case 2 and Case 3. Case 1 Case 2 Case 3
8
7
Critical transition point
The trailing edge position/μm
The leading edge position/m
10
6 Trailing edge
Leading adge
4
2
Case 1 Case 2 Case 3
6
Critical transition point
5 4 3 2 1
(b) Displacement of the trailing edge
(a) Displacement of the leading edge 0
0 0
1
2
3
t/ms
4
5
0
1
2
3
t/ms
4
5
0.30
Case 1 Case 2 Case 3
Penetrated depth/μm
0.25
0.20
0.15
0.10
I II Wenzel State
(c) Penetrated depth 0.05 0
1
2
3
4
5
t/ms
Fig.6 Variation of droplet position and penetrated depth with time
Surface free energy characteristics are also investigated in the process of droplet spontaneous movement. To obtain normalized surface free energy, difference method is used to calculate the surface area of gas-liquid interface and solid-liquid interface at different time steps. The initial surface free energy before spontaneous movement is written as: SE0
Aal0 𝛾al
Asa0 𝛾sa
(19)
where Aal0 and Asa0 are initial surface area of gas-liquid interface and gas-solid interface. When a droplet moves along gradient direction, surface free energy is changed as: SE
Aal 𝛾al
(Asa
Als )𝛾sa
Als 𝛾ls
(20)
where Aal, Asa and Als are the surface area of gas-liquid interface, gas-solid interface and liquid-solid interface, and 𝛾al , γsa and 𝛾ls are the corresponding surface tension respectively. The change of surface free energy is the difference between SE 1 and SE0, expressed as: SE
SE
SE0
(Aal
Aal0 )𝛾al
Als (𝛾ls
𝛾sa )
(21)
Normalized surface free energy can be obtained by dividing 𝛾al : SE 𝛾al
(Aal
Aal0 )
Als (𝛾ls −𝛾sa ) 𝛾al
(22)
The unknown terms of 𝛾al , γsa and 𝛾ls should be removed. According to Young equation: 𝛾ls
𝛾sa
𝛾al cos𝜃e
(23)
Here, θe denotes static contact angles on smooth surface. Substituting Eq. (23) into Eq. (22): SE 𝛾al
(Aal
Aal0 )
Als cos𝜃e
(24)
Due to the chemically heterogeneous surface, different wetting gradients correspond to a variety of static contact angles, Eq. (24) is rewritten as: SE 𝛾al
(Aal
Aal0 )
∑ Anls cos𝜃e n
(25)
Fig. 7(a) shows time evolution of the droplet superficial area during movement. As the droplet migrates along the stronger wetting region, contact area between liquid and solid wall increases and the shape of droplet is spread from spherical to semi-ellipsoidal. The droplet superficial area tends to increase with time. And the droplet spreads more flatly caused by a larger wetting gradient, resulting in a larger superficial area. In the process of movement, superficial area is reduced locally for several times. Fig. 7(b) shows time evolution of normalized surface free energy. According to Eq. (25), normalized surface free energy is composed of gas-liquid surface area increment and the product of solid-liquid surface area and the cosine of static contact angle. Lee et al. [22] and Pravinraj and Patrikar [21] analyzed the surface free energy characteristics during migration on smooth wetting gradient surface. It was found that normalized surface free energy decreases with time. However, our results indicate that normalized surface free energy tends to increase with time, which is quite different from the results from Lee and Pravinraj. This is due n to the different second term in Eq. (25), ∑ Cls cos𝜃e n . In Lee’s and Pravinraj’s model,
the value of cos𝜃e is positive (θe<90º), while the droplet moves on textured surface with static contact angle ranging from 138.2º to 95.9º in our model (cos𝜃e <0).
16
Case 1 Case 2 Case 3
60
Surface free energy/μm2
Superficial area/μm2
58
56
54
52
Case 1 Case 2 Case 3
12
8
4
(a) 50 0
1
2
t/ms
3
4
(b) 0 0
1
2
3
4
t/ms
Fig.7 Evolution of superficial area and normalized surface free energy over time
3.2.2 Surface morphology Surface structure is also a non-negligible parameter during spontaneous movement. Here, solid fraction is used to represent the roughness of microstructured surface, defined as the ratio of width and spacing of micropillars (η=a/s). Fig. 8(a) droplet dynamics on different microstructured surfaces at 5.219ms from side view. Displacement of leading and trailing edge point increases with the increase in solid fraction because the volume of droplet penetrates into micropillars increases, resulting in a smaller net driving force. For η≤0.25, the droplet stays in place without moving, indicating that unbalanced capillary force is dissipated completely by viscous force and resistance caused by micropillars. Fig. 8(b) shows displacement of the leading and trailing edge point on different microstructured surfaces. As the solid fraction increases, displacement of leading and trailing edge point increase from 4.63μm to 14.31μm and from 1.26μm to 10.61μm, respectively. Fig. 8(c) shows superficial area and surface free energy as a function of solid fraction. For η≤0.286, superficial area and surface free energy tends to decrease, mainly caused by deformation. And the droplets are all penetrated into four micropillars. On account of the smaller η corresponding to a larger spacing of micropillars, the droplet extends and a larger portion of droplet is immersed in micropillars. For 0.286<η≤0.5, an increasing tendency of superficial area and surface free energy with time is shown, which is opposite for η>0.5. To sum up, a larger solid fraction is available for the microdroplet spontaneous movement.
(a)
16
a=0.2μm, s=0.3μm, h=0.5μm
The trailing edge position The leading edge position
14
a=0.2μm, s=0.4μm, h=0.5μm
a=0.2μm, s=0.5μm, h=0.5μm
Displacement/μm
12 10 8 6 4 2
(b)
0
a=0.2μm, s=0.6μm, h=0.5μm
0.2
0.3
0.4
0.5
0.6
0.7
Solid fraction 62 16
58
12
56 8 54
52
4
a=0.2μm, s=0.9μm, h=0.5μm
Surface free energy/μm2
Superficial area/μm2
a=0.2μm, s=0.7μm, h=0.5μm
a=0.2μm, s=0.8μm, h=0.5μm
Superficial area Surface free energy
60
(c)
50
0 0.2
0.3
0.4
0.5
0.6
0.7
Solid fraction
Fig.8 (a) Droplet dynamics on different microstructured surfaces at 5.219ms; (b) Displacement of the leading and trailing edge point; (c) Superficial area and Surface free energy.
Hydrophobic materials are capable of inhibiting droplets from entering micropillars to maintain the Cassie or partial Cassie state. To prevent the droplet from being pinned on textured surface, spontaneous migration of single droplet on single-layer and two-layer microstructured surface are compared and investigated. Schematic diagram of single-layer and a two-layer wetting gradient surface is depicted in Fig.9. Single-layer surface is the same as Case 3 in Fig.5. Two-layer surface is composed of bottom hydrophobic surface (h2=0.4μm) and upper wetting gradient surface (h1=0.1μm), as shown in Fig. 9(b). The bottom layer is to prevent droplet from entering into micropillars and the upper one is to provide a driving force. Fig.10 shows comparison of spontaneous movement of single droplet on single-layer and two-layer surface from side view. Wetting gradients of two different surfaces are consistent with Case 3. As shown in Fig.10, the hydrophobic layer of two-layer surface is capable of inhibiting the droplet from entering into micropillars, thereby
keeping the droplet in partial Cassie state. However, velocity of movement is smaller compared with single-layer surface due to a reduction in net capillary force. Wetting gradient surface
h
(a) Single-layer wetting gradient surface
Wetting gradient surface
h1
Hydrophobic surface
h2
(b) Two-layer wetting gradient surface
Fig.9 Schematic diagram of single-layer and two-layer wetting gradient surfaces
0ms
2.088ms
5.219ms
(a) Single-layer wetting gradient surface 0ms
2.088ms
5.219ms
(b) Two-layer wetting gradient surface Fig.10 Droplet dynamics on single-layer and two-layer wetting gradient surfaces (Side view)
Fig.11 shows the relationship between penetrated depth and time on single-layer and two-layer surfaces. As mentioned in Fig.6, it can be divided into two stages: (I) spreading process dominated by viscous force, and (II) directional migration driven by unbalanced capillary force. In stage (I), penetrated depth decreases rapidly after contacting with micropillars. As a droplet moves along stronger wetting area, penetrated depth decreases slowly, corresponding to stage (II). Penetrated depth of droplet on single-layer surface decreases below 0.1μm, causing the state transition from partial Cassie to Wenzel. In contrast, the droplet on two-layer surface always maintains a partial Cassie state. The results indicate that although two-layer surface is capable of prevent droplets from entering into micropillars, velocity of directional migration is decreased.
0.30
Two-layer surface Singe-layer surface
Penetrated depth/μm
0.25
0.20
0.15
I
II
0.10
Wenzel state 0.05 0
1
2
3
4
5
6
t/ms
Fig.11 Variation of penetrated depth with time on single-layer and two-layer surfaces
3.3 Climbing-upward movement on inclined wetting gradient surface 3.3.1 Surface orientation Spontaneous movement of single droplet on horizontal surface has been studied in Section 3.2. However, solid surface is inclined instead of horizontal in many engineering applications. Therefore, it is of great significance to simulate climbingupward movement on inclined surface. Fig.12 shows droplet migration on wetting gradient surface with different orientations, where inclined angles are equal to 15º, 30º and 45º, respectively. Gravitational coefficient is set as 10-6 and surface structure and wetting gradient are consistent with Case 2 in Fig.4. Different from horizontal surface, tangential component of gravity (Fgsinα) should be taken into consideration in addition to viscous and resistance caused by micropillars. Although surface orientations are different, displacement of droplet is roughly the same at 4.076ms, as shown in Table.2. It is concluded that gravitational force (g=10-6) is smaller relative to unbalanced capillary force. Penetrated depth is less than 0.1μm at 4.076ms, indicating a transition from partial Cassie to Wenzel state. The numerical results demonstrate that g≤10-6 has little influence on droplet climbing-upward movement. 0ms
0.941ms
1.986ms
(a) α=15º
4.076ms
0ms
0.941ms
1.986ms
4.076ms
(a) α=30º 0ms
0.941ms
1.986ms
4.076ms
(a) α=45º Fig.12 Droplet motion on wetting gradient surface with different orientations
Table.2 Value of parameters at 4.076ms Surface orientation 15º 30º 45º
Leading edge position/μm 5.25 5.14 4.57
Trailing edge position/μm
Penetrated depth/μm
Surface free energy/μm2
3.97 3.88 3.85
0.082 0.082 0.087
6.05 5.92 6.02
3.3.2 Gravitational coefficient Various gravitational coefficients are applied to investigate the droplet dynamic characteristics. The surface inclination is 30º and micropillars and wetting gradient are consistent with Case 2 in Fig.5. Gravitational coefficients are 5×10-7, 7.5×10-7, 10×10-7, 25×10-7 and 50×10-7, respectively. Bond number (Bo) is a dimensionless number, denoting the relative importance of gravity and surface tension: Bo
𝜌𝑙 gr 2 /𝜎
(24)
Here, Bo number is used to represent the effect of gravity on droplet climbing-upward movement. Effect of Bo number on displacement and penetrated depth is depicted in Fig.13. Displacement of leading and trailing edge point decreases with an increasing Bo. For Bo≤0.0126, displacement of droplet decreases slightly with the larger Bo number, in which case tangential component of gravity is negligible compared with unbalanced surface tension. For Bo>0.0126, it decreases rapidly with an increasing Bo number. When Bo number is increased to 0.063, the droplet remains in its original
position without displacement, where net capillary force is less than tangential component of gravity. On the contrary, penetrated depth shows an increasing trend with Bo number because the droplet moves to a stronger wetting region. Fig.14 shows effect of Bo number on superficial area and surface free energy at 4.076ms. In the process of movement, part of the kinetic energy generated by net capillary force is converted into surface free energy. Superficial area and surface free energy decrease with the increase in Bo number. The results indicate that a smaller gravitational coefficient (Bo) is beneficial for climbing-upward movement, increasing kinetic energy of droplet. The Bo number ranging from 0.0126 to 0.063 has a great effect on climbing-upward movement of microdroplet (20μm). 7
0.16
The trailing edge position The leading edge position Penetrated depth
6
0.15
Displacement/μm
0.14 4 0.13
3 2
0.12
1
0.11
Penetrated depth/μm
5
0 0.10 -1 0.01
0.1
Bo
Fig.13 Variation of Displacement and Penetrated depth with Bo 5.0
Superficial area Surface free energy
4.5
Superficial area/μm2
55 4.0
3.5 54 3.0 53
2.5
Surface free energy/μm2
56
2.0 52 0.01
0.1
Bo
Fig.14 Effects of Bo on superficial area and surface free energy
Acknowledgment This work was supported by ESA-CMSA International Cooperation of Space Experiment Project (Study on Condensation and Enhancement Methods under Microgravity).
4. Conclusion The model of spontaneous movement of single droplet on microstructured surface with multi wetting gradients is established and investigated by 3D pseudopotential lattice Boltzmann method. Effects of wetting gradient, surface morphology, surface orientation and gravitational coefficient on droplet dynamics are analyzed. Several conclusions can be drawn as follows. (1) Static contact angles on microstructured surface increase non-linearly with Gs, which is different from smooth surface. When a droplet is in Wenzel state (Gs≤-9.0), micropillars have a negative influence on static contact angles. In contrast, it can greatly increase static contact angle of droplet in Cassie or partial Cassie state (Gs≥-8.0). (2) Wetting gradient and surface morphology are of great significance for spontaneous movement. A larger wetting gradient and solid friction is beneficial to increasing movement velocity and surface free energy. Although two-layer wetting gradient surface is capable of prevent the droplet entering micropillars, it has a negative effect on movement velocity. (3) When the Bo number ranges from 0.0126 to 0.063, gravitational coefficient should be considered in the process of microdroplet (20μm) climbing-upward movement. For Bo≤0.0126, surface orientation has little effect on climbing-upward movement.
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Directional migration of single droplet on multi-wetting gradient surface by 3D lattice Boltzmann method Xin Wang , Bo Xu , Yi Wang , Zhenqian Chen PII: DOI: Reference:
S0045-7930(19)30350-0 https://doi.org/10.1016/j.compfluid.2019.104392 CAF 104392
To appear in:
Computers and Fluids
Received date: Revised date: Accepted date:
21 June 2019 6 September 2019 17 November 2019
Please cite this article as: Xin Wang , Bo Xu , Yi Wang , Zhenqian Chen , Directional migration of single droplet on multi-wetting gradient surface by 3D lattice Boltzmann method, Computers and Fluids (2019), doi: https://doi.org/10.1016/j.compfluid.2019.104392
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Highlights
Spontaneous movement of droplet on multi-wetting gradient surface was simulated.
Both hydrophobicity and hydrophilicity can be strengthened by micropillar arrays.
Wetting gradient and surface morphology had a great effect on droplet movement.
The droplet state transition from partial Cassie to Wenzel state can be observed.
Gravitational coefficient should be considered during climbing-upward movement.
Directional migration of single droplet on multi-wetting gradient surface by 3D lattice Boltzmann method Xin Wang a, Bo Xu a, b, c, Yi Wang a, Zhenqian Chen a, b, c *
(a School of Energy and Environment, Southeast University, Nanjing, P. R. China; b
Jiangsu Provincial Key Laboratory of Solar Energy Science and Technology, School of Energy and Environment, Southeast University, Nanjing, P. R. China;
c
Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education, School of Energy and Environment, Southeast University, Nanjing, P. R. China)
* Corresponding author, E-mail address: [email protected], Tel/Fax: +86-25-83790626.
Submitted to (Computers and Fluids)
Abstract: Directional migration of single microdroplet on a microstructured surface with multi wetting gradients plays a significant role in many industrial applications. To understand the mechanism of spontaneous movement of microdroplet on wetting gradient surfaces under different gravities, the model of microdroplet dynamics is built by 3D lattice Boltzmann method. Effects of wetting gradient, surface morphology, surface orientation and gravitational force on spontaneous movement are investigated. The results demonstrate that static contact angles are greatly affected by micropillars. Both hydrophilicity and hydrophobicity can be strengthened by micropillar arrays. As a droplet moves from hydrophobic to hydrophilic driven by net capillary force, a transition from partial Cassie state to Wenzel state can be found. A larger wetting gradient and larger solid fraction promote the spontaneous movement of microdroplet. In the process of movement, surface free energy shows an increasing tendency due to part of kinetic energy converted to surface free energy. In spite of two-layer surface capable of maintaining a droplet in partial Cassie state, it will decrease the movement velocity. For Bo≤0.0126, surface orientation has little effect on climbing-upward movement. When the Bo number ranges from 0.0126 to 0.063, gravitational coefficient should be considered in the process of microdroplet (2μm) climbing- upward movement. The surface with multi wetting gradients can be applied to remove droplets during condensation to enhance heat transfer. Keywords: 3D LBM; Microdroplet; Wetting gradient; Micropillar; Gravity
1. Introduction Surface wettability is quite significant for condensation heat transfer, affecting the condensation mode: filmwise condensation or dropwise condensation. The wettability of liquid on the solid substrate is distinguished by the equilibrium contact angle (θe) of three-phase contact line. θe>90° is considered to be a hydrophobic surface, while θe<90° is referred to a hydrophilic surface. Inspired by biological surfaces [1-5], preparation of superhydrophobic (SH) surface has attracted extensive attention due to the higher heat transfer coefficient during dropwise condensation.
Since the current fabrication technology of preparing SH surface is unsatisfactory, condensate droplets on SH substrates will change from Cassie state to Wenzel state after operating several tens of hours [6]. Droplets pinned on the SH surface are difficult to remove, resulting in the failure of SH surface. Hence, the removal of condensate droplets has now emerged as a crucial engineering problem. In recent years, self-removal of droplets during condensation has become possible. Spontaneous movement of droplet on spider silk [7, 8], conical cactus spines [9, 10] and desert beetle [11] was experimentally observed. Those organisms possess surface structures with wetting gradient and unique shapes, which can drive the droplet directional migration. Spontaneous movement of droplets plays a vital role in a great number of industrial applications, such as fuel lines, flexible electronics, solar cells as well as condensers. In the past few decades, a great deal of experimental work on droplet dynamics on the surface with multi wetting gradients has been investigated. Whitesides and Chaudhury [12, 13] first reported the experiment on droplet motion on wetting gradient surfaces. It was found that self-propelled movement of droplet along the direction of stronger wettability was observed driven by gradient surface free energy. Subsequently, several experiments of droplet directional migration on multi wetting gradient surfaces were conducted. Lai et al. [14] fabricated a gradient wettability surface for transporting droplets from SH region to hydrophilic region by combining a structure gradient and a self-assembled-monolayer gradient. Besides, a bidirectional gradient surface that changes the droplet moving direction was achieved. Paradisanos et al. [15] and Ghosh et al. [16] used the laser technology to create a wetting gradient surface and obtained the droplet moving speed of several hundred millimeters per second. Deng et al. [17] prepared a wetting gradient surface with wedge-shaped pattern using the improved anodic oxidation method. Displacement of droplet movement can be controlled by adjusting the wedge angle and droplet volume. Khoo and Tseng [18] carried out an experiment of droplet movement on a wetting gradient surface with nano-textured pattern, which can deliver a wide range of droplets at speeds up to 0.5m/s. However, due to experimental uncertainties including complex interface and visual observation, it is difficult to understand the mechanism
of microdroplet movement on wetting gradient surfaces in depth. There also has been a few numerical work devoted to the droplet movement on wetting gradient surfaces. Li et al. [19] simulated droplet movement on a linear gradient wettability surface by dissipative particle dynamics. And effects of wetting gradient and droplet size on the movement velocity were studied. Chakraborty et al. [20] used molecular dynamics to simulate droplet migration on the surface with chemically induced wettability gradient and their results agreed well with the model of droplet self-propelled movement on wetting gradient surfaces. Deng et al. [21] simulated the droplet dynamics on wetting gradient surfaces during condensation using the free energy lattice Boltzmann model. Their results indicated that condensate droplets can be removed by wetting gradient surface, thus enhancing condensation heat transfer. On account of its advantages in computational efficiency and dealing with complex boundaries, lattice Boltzmann method (LBM) has become an effective and essential tool in multiphase flows. Pseudopotential model, as one of the lattice Boltzmann models, has been widely used to simulate the microdroplet dynamics. Pravinraj and Patrikar [22] investigated the droplet splitting and directional migration on the surface with chemical gradient and structural gradient. In addition, the change of surface free energy and contact angle hysteresis were analyzed. Lee et al. [23] reported the microdroplet movement on wetting gradient and structural gradient surfaces by 3D pseudopotential model. Droplet movement was also analyzed in terms of kinetic, gravitational and surface free energy. Although a little numerical work has been carried out of the spontaneous motion on wetting gradient surfaces, effect of surface structure on the droplet dynamics are neglected. Moreover, the above experimental and numerical investigations are on horizontal surfaces without considering the influence of gravitational coefficient and surface orientation on droplet dynamics. It has been demonstrated that micropillars, surface orientation and gravity play a significant role on the dynamic characteristics of droplets in the process of condensation [24-28]. Here, 3D pseudopotential model is adopted to simulate the spontaneous movement of single droplet on rough wetting gradient surfaces. Effects
of surface orientation and gravitational coefficient on droplet dynamics are investigated. In addition, displacement, penetrated depth and surface free energy are also analyzed to provide insight into dynamic behaviors of droplet migration on wetting gradient surfaces. It is meaningful and significant for the design of multi wetting gradient surface with micropillar arrays. The surface with multi wetting gradients can be applied to getting rid of droplets during condensation, thus enhancing condensation heat transfer.
2. Model description 2.1 3D lattice Boltzmann model In the single-component multi-phase model, migration and collision of fluids can be described by specific particle distribution functions. Recently, Xiong and Cheng [29] proposed a new piecewise distribution function with two different relaxation times. A larger relaxation time is first applied to maintain the numerical stability in early time steps. Then, a smaller relaxation time is applied to obtain a lower viscous droplet. In this paper, the evolution function with BGK collision operator can be given by: (
t
where
𝛿)
( t)
{
[ ( t)
( t)]
( t) t
[ ( t)
( t)]
( t) t
( t) denotes the density distribution function with velocity
lattice x and time t. 𝛿 is the time step.
(1) at the
( t) is the force term, including
fluid-fluid force, fluid-solid force as well as body force.
is the corresponding
equilibrium distribution function [30], written as: 𝜔 𝜌[
𝛼 ∙𝐮
𝑐s
( 𝛼 ∙𝐮)𝟐 2𝑐s4
𝐮 2𝑐s
]
(2)
where ωα are the weighting factors. For D3Q19 model, ωα=1-3, ωα=1/18 and ωα=1/36. 𝑐𝑠
𝑐/√3 denotes the lattice sound speed (𝑐
D3Q19 model is written as:
𝛿x /𝛿 ). The discrete velocity
in
( ) )( ± )( {(± ± ) (± ± ) ( ± ± ) (± ± )
𝛼 𝛼 𝛼
~6 7~ 8
(3)
The macroscopic density and velocity can be obtained by the density distribution function: 𝜌
∑
( t)
∑
𝜌𝐮
( t)
(4)
Through the multi-scale analysis of Chapman-Enskog expansion, the macroscopic Navier-Stokes equations can be restored and the kinematic viscosity of fluid is given by: 𝜐
𝑐s2 (𝜏2
.5)𝛿
(5)
The total force F involved in this paper is mainly composed of fluid-fluid force, fluid-solid force and gravitational force. In the past few decades, numerous schemes have been proposed to incorporate the force term in the LBM. Here, exact difference method (EDM) [31] is adopted to incorporate the force term due to the better numerical accuracy and independent of relaxation time, written as: ( t) where
[𝜌( t) 𝐮
𝐮
𝐮]
[𝜌( t) 𝐮]
(6)
𝐅𝛿 /𝜌 is the velocity change caused by the total force F.
The fluid-fluid force in the form of combined approximation proposed by Kupershtokh et al. [32, 33] and Gong and Cheng [34] is adopted, written as: 𝐅in
𝛽𝜓 ( ) ∑ G( −𝛽 2
𝛿 )𝜓( ∑ G(
𝛿) 𝛿 )𝜓 2 (
𝛿)
(7)
where β is a constant parameter. G denotes the interaction strength between particles, expressed as: G(
𝛿)
g {g 2
| | | | √2 otherwise
(8)
where g1=g0 and g2=g0/2 in D3Q19 model. And the pseudopotential function 𝜓( ) in Eq. (7) is specially defined as: 𝜓( )
√2(𝑝
𝜌𝑐𝑠2 )/𝑐0 g
(9)
where the pressure p can be obtained by actual equation of state (EOS). Here, PengRobinson EOS is used, written as:
𝑝
𝜌𝑅𝑇 −𝑏𝜌
(𝑇)
𝑎𝜌
(10)
+2𝑏𝜌−𝑏 𝜌
where R=1, 𝛼(𝑇)
[
.87324(
2
√𝑇/𝑇𝑐𝑟 )] , a=2/49, b=2/21. Since P-R EOS
is adopted in this paper, the value of parameter β is equal to 1.16. The fluid-solid force can be used to characterize the wettability between the fluid and solid wall, defined as: 𝐅s
Gs 𝜓( ) ∑ 𝜔 S(
𝛿)
(11)
where the parameter Gs denotes the interaction strength between the fluid and solid wall. Surface wettability can be controlled by changing the value of parameter Gs. S is an indicator function with S=1 for solid and S=0 for fluid. In addition, the gravitational force is calculated by: 𝐅g
(𝜌( )
𝜌𝑣 )g
(12)
where g is the gravitational acceleration coefficient and ρv is the density of vapor phase. The real fluid velocity U is defined as the average of velocity before and after collision, written as: 𝜌𝐔
𝜌𝐮
.5𝛿 𝐅
(13)
2.2 Model validation Laplace’s law is considered to validate 3D pseudopotential lattice Boltzmann model. According to Laplace’s law, the pressure difference inside and outside the droplet is linear with the inverse radius (∆p=2σ/r, where σ denotes the surface tension). Initially, a droplet is placed at the center of computational domain (151×151×151). Periodic boundary conditions are applied for all boundaries. The fluid density is initialized as: 𝜌 (x y z)
𝜌l
𝜌v 2
𝜌l
𝜌v 2
[tanh(2(√(x
x 0 )2
(y
y0 )2
(z
z 0 )2
r)/W)]
(14)
where W is the thickness of two-phase interface (W=3). As shown in Fig.1, the inverse radius increases linearly with the pressure difference at different temperatures (T/Tcr=0.9, T/Tcr=0.85, T/Tcr=0.8), and the corresponding surface tensions are 0.0903,
0.1521 and 0.2287, respectively. In addition, our LB results are compared with Maxwell construction. It can be seen from Table.1 that vapor and liquid densities in equilibrium are in good agreement with the Maxwell construction. 0.05
T/Tcr=0.90, =0.0903 T/Tcr=0.85, =0.1521
0.04
T/Tcr=0.80, =0.2287
p
0.03
0.02
0.01
0 0
0.02
0.04
0.06
0.08
0.10
1/r
Fig.1 Laplace’s law at different temperatures
Table.1 Comparison of LB results with Maxwell analytical solution Analytical solution
T/Tcr 0.90 0.85 0.80
LB results
ρl
ρg
ρl
ρg
ρl/ρg
σ
5.907 6.627 7.205
0.580 0.342 0.197
5.904 6.641 7.198
0.580 0.333 0.185
10.18 19.94 38.89
0.0903 0.1521 0.2287
3. Results and discussion In this section, comparison of static contact angle on smooth and microstructured surface will be discussed. Then, Effects of wetting gradient and surface morphology on
horizontal
movement
of
microdroplet
will
be
investigated.
Finally,
climbing-upward movement on inclined surfaces will be explored. In this simulation, the initial liquid and gas densities are 7.2 and 0.197, respectively. 3.1 Contact angle As mentioned earlier, surface wettability is greatly significant for the droplet dynamics. Therefore, single droplet on smooth and microstructured surfaces with
different wettability is simulated. Bounce-back scheme is applied to the bottom surface and periodic boundary conditions are applied to other borders. Initially, a droplet with r=20 lattice unit is located at the center of bottom surface, corresponding to 2μm in real units. The calculation won’t stop until the relative error is less than 10-6, indicating that a droplet is static on solid substrate. In pseudopotential model, wettability can be adjusted by the interaction strength between the fluid and solid wall to change static contact angles. Fig.2 shows the relationship between static contact angles (θe) and interaction strength Gs on smooth and microstructured surfaces. It can be seen that static contact angle increases with the greater interaction strength G s. For smooth surface, static contact angles increase linearly with G s. When Gs ranges from -19.0 to -4.0, θe increases from 5.2º to 155.7º. The droplet morphology changes from a thin liquid film to a spherical droplet and the liquid-solid contact area decreases significantly, as shown in Fig.3. For microstructured surface (a=2μm, s=5μm, h=5μm), static contact angles (0º~177.8º) increase nonlinearly with Gs, which is different from smooth surface. When Gs is less than -9.0, the droplets are in Wenzel state and a portion of droplets penetrate into micropillar arrays. In general, as a droplet is in Wenzel state, the wetting of chemically homogenous surface is governed by the Wenzel model [35]: cos w rf cos e
(15)
where rf is defined as the ratio of real surface in contact with liquid to its projection onto the substrate. Due to the real contact area larger than the projected area, hydrophilic property is strengthened by surface textures. Therefore, liquid-solid contact area on microstructured surface is larger than that on smooth surface for the same Gs, resulting in a greater viscous force and a smaller contact angle. When Gs is more than -8.0, droplets are in Cassie or partially Cassie state. There exists some air between the droplet and substrate. The Cassie-Baxter equation [36] can be applied to a droplet in Cassie state, given by:
cos c 1 s (1 cos e )
(16)
where s denotes the friction of solid in contact with liquid. Compared with the smooth surface, contact area between liquid and solid on microstructured surface is
smaller for the same Gs (Fig.3). In this case, viscous force decreases and leads to a larger contact angle. Hence, hydrophobic property is strengthened by micropillar arrays.
180
150
Smooth surface Microstructured surface Linear fit Wenzel =10.18×Gs+197.71
e/°
120
90
60
30
a=0.2μm, s=0.5μm, h=0.5μm 0 -18
-16
-14
-12
-10
-8
-6
-4
Gs
Fig.2 Variation of contact angle with G s
0°
5.2°
0°
(a) Gs=-19.0 19.6°
(b) Gs=-17.0 45.1°
(c) Gs=-15.0 85.0°
24.8°
Liquid film
Liquid film
41.6°
67.5°
(d) Gs=-13.0 84.0°
97.7°
105.5°
(e) Gs=-11.0 138.2°
(g) Gs=-7.0
(f) Gs=-9.0 129.0°
165.9°
146.1°
(h) Gs=-5.0
Fig.3 Comparison of droplet morphologies on smooth and microstructured surfaces
3.2 Directional migration on horizontal wetting gradient surface
Fig.4 shows the computational model of spontaneous movement of droplet on wetting gradient surface with micropillars. In this simulation, the computational domain is 181×81×81 (18μm×8μm×8μm). Bounce-back scheme is applied to the bottom surface and periodic boundary conditions are applied to other borders. Initially, a droplet is located at the position (30, 40, 25), corresponding to 2μm in real units. Twelve different chemical gradients are used on the bottom surface, thereby obtaining a wetting gradient surface. As a droplet contacts the bottom substrate, it will move from a weaker wetting region to a stronger wetting region. The enlarged view in Fig.4 shows a schematic of surface with micropillars. In this simulation, the letters a, s and h represent width, spacing and height of micropillars, respectively. In order to be related to actual physical properties, lattice units should be converted to real physical units. Ohnesorge number (Oh
𝜇𝑙 /√𝜌𝑙 𝜎𝑙𝑣 r) is a dimensionless number, indicating
the relationship between viscous force and inertial and surface tension. Due to [Oh]lu=[Oh]real, Oh number is used to convert the lattice unit to real unit, calculated by: 𝑙0 [𝑡]real [𝑡]lu
[𝑙]real
[Oh]real
[𝜇𝑙 /(𝜌𝑙𝜎𝑙𝑣 )]real
[𝑙]lu
[Oh]lu
[𝜇𝑙 /(𝜌𝑙𝜎𝑙𝑣 )]lu
𝑙0.5
(17)
[√𝜌𝑙/𝜎𝑙𝑣 ]real
(18)
[√𝜌𝑙 /𝜎𝑙𝑣 ]lu
Square micropillar a s h g Gs2 Gs1 w w r0
Weaker wettability
stronger wettability
Fig.4 Computational model of droplet movement on microstructured surface
3.2.1 Wetting gradient Fig.5 shows the directional migration of single droplet (r=2μm) on different
wetting gradient surfaces (a=0.2μm, s=0.5μm, h=0.5μm). Wetting gradients are set to Case 1: 0.25Gs/1.5μm, Case 2: 0.375Gs/1.5μm and Case 3: 0.5Gs/1.5μm, respectively. “0.25Gs/1.5μm” represents that the value of Gs decreases by 0.25 per 1.5μm along the x direction. As shown in Fig.4, Gs decreases from -7.0 (x=1~15) to -9.75 (x=166~180) along the x direction from the red region to the blue region.
Similarly,
“0.375Gs/1.5μm” and “0.5Gs/1.5μm” represent that Gs decrease by 0.375 and 0.5 per 1.5μm along the x direction. Initially, the trailing portion of droplet is located at the weaker wetting area and the leading portion is located on the stronger wetting area. The shape of droplet with θe=138.2° is symmetrical in XZ plane. Due to the action of unbalanced capillary force, a droplet begins to move along the gradient direction (from hydrophobic to hydrophilic). At t=2.088ms, the droplet is deformed with the penetrated volume increased. Moreover, contact area between droplet and solid wall is increased. However, a small amount of air is reserved between the micropillars, indicating that the droplet still maintains partial Cassie state. It can be found that the penetrated volume and displacement of droplet increase with a greater wetting gradient. At t=4.175ms, the state of droplet in Case 2 and Case 3 changes from the originally partial Cassie to Wenzel state. It should be noted that the droplet state is dependent on the depth penetrated into micropillars. In contrast, the droplet in Case 1 remains in the partial Cassie state.
0ms
2.088ms
4.175ms
5.219ms
4.175ms
5.219ms
top view side view
(a) Case 1: 0.25Gs/1.5μm 0ms
2.088ms
top view Wenzel
side view
(b) Case 2: 0.375 Gs/1.5μm
Wenzel
0ms
2.088ms
4.175ms
5.219ms
top view Wenzel
side view
Wenzel
(c) Case 3: 0.5 Gs/1.5μm
Fig.5 Directional migration of single droplet on different wetting gradient surface (top view and side view)
The efficiency of condensate droplet removal is directly influenced by displacement of droplet movement. To analyze this transport mechanism, variation of leading and trailing edge position and penetrated depth with time is plotted, as shown in Fig.6. As can be seen from the schematic diagram in Fig. 6(a), the leading and trailing edge points of droplet are defined as three-phase points where the front and rear portions are in contact with micropillars from side view. Fig. 6(a) shows variation of leading edge position with time. For Case 2 and Case 3, the droplet with partial Cassie state is transformed into Wenzel state in the process of movement at 2.818ms and 4.071ms, respectively (See critical transition point marked in red). In contrast, the droplet in Case 1 always maintains the partial Cassie state in 5.219ms due to moving from hydrophobic area to weaker hydrophobic area, which is consistent with the picture in Fig.5. Displacement of the leading and trailing edge point moves faster with the larger wetting gradient because the droplet is subjected by a driving force in the opposite direction produced by a larger unbalanced Laplace pressure. Displacement of trailing edge point tends to increase with time, which is similar to leading edge point, as shown in Fig. 6(b). Furthermore, a larger wetting gradient has a positive effect on displacement velocity. It is worth noting that displacement of trailing edge point is almost the same before 1.5μm, independent of the wetting gradient. This is due to the trailing edge point located at the weaker wetting area and dragged by the front portion of droplet instead of driven by capillary force. When displacement exceeds 1.5μm, directional migration is dominated by unbalanced capillary force. Displacement of trailing edge point increases more rapidly with a larger wetting gradient. During the
spontaneous movement, part of the droplet penetrated into micropillars has to overcome the resistance caused by micropillars. Therefore, repeating from a gentle rise to a rapid raise can be seen from displacement of the leading and trailing edge point. Penetrated depth is a significant parameter for judging the state of droplet (Wenzel or Cassie), defined as the distance between the bottom of droplet and solid, as can be seen from enlarged view in Fig. 6(c). Variation of the penetrated depth of droplet with time can be divided into two stages: (I) spreading on textured surface dominated by viscous force; (II) Directional migration of droplet affected by unbalanced capillary force. In stage (I), the droplet contacts and spreads on the microstructured surface dominated by viscous force, resulting in a small portion of droplet penetrating into micropillars. Moreover, penetrated depth decreases rapidly with time, independent of wetting gradient. In stage (II), penetrated depth decreases slowly with time. Affected by surface wettability, it decreases faster with a larger wetting gradient. Here, penetrated depth equal to 0.1μm is defined as the critical transition point from Partial Cassie state to Wenzel state. Due to locating at the stronger wetting region, the transition from Partial Cassie state to Wenzel state can be seen in Case 2 and Case 3. Case 1 Case 2 Case 3
8
7
Critical transition point
The trailing edge position/μm
The leading edge position/m
10
6 Trailing edge
Leading adge
4
2
Case 1 Case 2 Case 3
6
Critical transition point
5 4 3 2 1
(b) Displacement of the trailing edge
(a) Displacement of the leading edge 0
0 0
1
2
3
t/ms
4
5
0
1
2
3
t/ms
4
5
0.30
Case 1 Case 2 Case 3
Penetrated depth/μm
0.25
0.20
0.15
0.10
I II Wenzel State
(c) Penetrated depth 0.05 0
1
2
3
4
5
t/ms
Fig.6 Variation of droplet position and penetrated depth with time
Surface free energy characteristics are also investigated in the process of droplet spontaneous movement. To obtain normalized surface free energy, difference method is used to calculate the surface area of gas-liquid interface and solid-liquid interface at different time steps. The initial surface free energy before spontaneous movement is written as: SE0
Aal0 𝛾al
Asa0 𝛾sa
(19)
where Aal0 and Asa0 are initial surface area of gas-liquid interface and gas-solid interface. When a droplet moves along gradient direction, surface free energy is changed as: SE
Aal 𝛾al
(Asa
Als )𝛾sa
Als 𝛾ls
(20)
where Aal, Asa and Als are the surface area of gas-liquid interface, gas-solid interface and liquid-solid interface, and 𝛾al , γsa and 𝛾ls are the corresponding surface tension respectively. The change of surface free energy is the difference between SE 1 and SE0, expressed as: SE
SE
SE0
(Aal
Aal0 )𝛾al
Als (𝛾ls
𝛾sa )
(21)
Normalized surface free energy can be obtained by dividing 𝛾al : SE 𝛾al
(Aal
Aal0 )
Als (𝛾ls −𝛾sa ) 𝛾al
(22)
The unknown terms of 𝛾al , γsa and 𝛾ls should be removed. According to Young equation: 𝛾ls
𝛾sa
𝛾al cos𝜃e
(23)
Here, θe denotes static contact angles on smooth surface. Substituting Eq. (23) into Eq. (22): SE 𝛾al
(Aal
Aal0 )
Als cos𝜃e
(24)
Due to the chemically heterogeneous surface, different wetting gradients correspond to a variety of static contact angles, Eq. (24) is rewritten as: SE 𝛾al
(Aal
Aal0 )
∑ Anls cos𝜃e n
(25)
Fig. 7(a) shows time evolution of the droplet superficial area during movement. As the droplet migrates along the stronger wetting region, contact area between liquid and solid wall increases and the shape of droplet is spread from spherical to semi-ellipsoidal. The droplet superficial area tends to increase with time. And the droplet spreads more flatly caused by a larger wetting gradient, resulting in a larger superficial area. In the process of movement, superficial area is reduced locally for several times. Fig. 7(b) shows time evolution of normalized surface free energy. According to Eq. (25), normalized surface free energy is composed of gas-liquid surface area increment and the product of solid-liquid surface area and the cosine of static contact angle. Lee et al. [22] and Pravinraj and Patrikar [21] analyzed the surface free energy characteristics during migration on smooth wetting gradient surface. It was found that normalized surface free energy decreases with time. However, our results indicate that normalized surface free energy tends to increase with time, which is quite different from the results from Lee and Pravinraj. This is due n to the different second term in Eq. (25), ∑ Cls cos𝜃e n . In Lee’s and Pravinraj’s model,
the value of cos𝜃e is positive (θe<90º), while the droplet moves on textured surface with static contact angle ranging from 138.2º to 95.9º in our model (cos𝜃e <0).
16
Case 1 Case 2 Case 3
60
Surface free energy/μm2
Superficial area/μm2
58
56
54
52
Case 1 Case 2 Case 3
12
8
4
(a) 50 0
1
2
t/ms
3
4
(b) 0 0
1
2
3
4
t/ms
Fig.7 Evolution of superficial area and normalized surface free energy over time
3.2.2 Surface morphology Surface structure is also a non-negligible parameter during spontaneous movement. Here, solid fraction is used to represent the roughness of microstructured surface, defined as the ratio of width and spacing of micropillars (η=a/s). Fig. 8(a) droplet dynamics on different microstructured surfaces at 5.219ms from side view. Displacement of leading and trailing edge point increases with the increase in solid fraction because the volume of droplet penetrates into micropillars increases, resulting in a smaller net driving force. For η≤0.25, the droplet stays in place without moving, indicating that unbalanced capillary force is dissipated completely by viscous force and resistance caused by micropillars. Fig. 8(b) shows displacement of the leading and trailing edge point on different microstructured surfaces. As the solid fraction increases, displacement of leading and trailing edge point increase from 4.63μm to 14.31μm and from 1.26μm to 10.61μm, respectively. Fig. 8(c) shows superficial area and surface free energy as a function of solid fraction. For η≤0.286, superficial area and surface free energy tends to decrease, mainly caused by deformation. And the droplets are all penetrated into four micropillars. On account of the smaller η corresponding to a larger spacing of micropillars, the droplet extends and a larger portion of droplet is immersed in micropillars. For 0.286<η≤0.5, an increasing tendency of superficial area and surface free energy with time is shown, which is opposite for η>0.5. To sum up, a larger solid fraction is available for the microdroplet spontaneous movement.
(a)
16
a=0.2μm, s=0.3μm, h=0.5μm
The trailing edge position The leading edge position
14
a=0.2μm, s=0.4μm, h=0.5μm
a=0.2μm, s=0.5μm, h=0.5μm
Displacement/μm
12 10 8 6 4 2
(b)
0
a=0.2μm, s=0.6μm, h=0.5μm
0.2
0.3
0.4
0.5
0.6
0.7
Solid fraction 62 16
58
12
56 8 54
52
4
a=0.2μm, s=0.9μm, h=0.5μm
Surface free energy/μm2
Superficial area/μm2
a=0.2μm, s=0.7μm, h=0.5μm
a=0.2μm, s=0.8μm, h=0.5μm
Superficial area Surface free energy
60
(c)
50
0 0.2
0.3
0.4
0.5
0.6
0.7
Solid fraction
Fig.8 (a) Droplet dynamics on different microstructured surfaces at 5.219ms; (b) Displacement of the leading and trailing edge point; (c) Superficial area and Surface free energy.
Hydrophobic materials are capable of inhibiting droplets from entering micropillars to maintain the Cassie or partial Cassie state. To prevent the droplet from being pinned on textured surface, spontaneous migration of single droplet on single-layer and two-layer microstructured surface are compared and investigated. Schematic diagram of single-layer and a two-layer wetting gradient surface is depicted in Fig.9. Single-layer surface is the same as Case 3 in Fig.5. Two-layer surface is composed of bottom hydrophobic surface (h2=0.4μm) and upper wetting gradient surface (h1=0.1μm), as shown in Fig. 9(b). The bottom layer is to prevent droplet from entering into micropillars and the upper one is to provide a driving force. Fig.10 shows comparison of spontaneous movement of single droplet on single-layer and two-layer surface from side view. Wetting gradients of two different surfaces are consistent with Case 3. As shown in Fig.10, the hydrophobic layer of two-layer surface is capable of inhibiting the droplet from entering into micropillars, thereby
keeping the droplet in partial Cassie state. However, velocity of movement is smaller compared with single-layer surface due to a reduction in net capillary force. Wetting gradient surface
h
(a) Single-layer wetting gradient surface
Wetting gradient surface
h1
Hydrophobic surface
h2
(b) Two-layer wetting gradient surface
Fig.9 Schematic diagram of single-layer and two-layer wetting gradient surfaces
0ms
2.088ms
5.219ms
(a) Single-layer wetting gradient surface 0ms
2.088ms
5.219ms
(b) Two-layer wetting gradient surface Fig.10 Droplet dynamics on single-layer and two-layer wetting gradient surfaces (Side view)
Fig.11 shows the relationship between penetrated depth and time on single-layer and two-layer surfaces. As mentioned in Fig.6, it can be divided into two stages: (I) spreading process dominated by viscous force, and (II) directional migration driven by unbalanced capillary force. In stage (I), penetrated depth decreases rapidly after contacting with micropillars. As a droplet moves along stronger wetting area, penetrated depth decreases slowly, corresponding to stage (II). Penetrated depth of droplet on single-layer surface decreases below 0.1μm, causing the state transition from partial Cassie to Wenzel. In contrast, the droplet on two-layer surface always maintains a partial Cassie state. The results indicate that although two-layer surface is capable of prevent droplets from entering into micropillars, velocity of directional migration is decreased.
0.30
Two-layer surface Singe-layer surface
Penetrated depth/μm
0.25
0.20
0.15
I
II
0.10
Wenzel state 0.05 0
1
2
3
4
5
6
t/ms
Fig.11 Variation of penetrated depth with time on single-layer and two-layer surfaces
3.3 Climbing-upward movement on inclined wetting gradient surface 3.3.1 Surface orientation Spontaneous movement of single droplet on horizontal surface has been studied in Section 3.2. However, solid surface is inclined instead of horizontal in many engineering applications. Therefore, it is of great significance to simulate climbingupward movement on inclined surface. Fig.12 shows droplet migration on wetting gradient surface with different orientations, where inclined angles are equal to 15º, 30º and 45º, respectively. Gravitational coefficient is set as 10-6 and surface structure and wetting gradient are consistent with Case 2 in Fig.4. Different from horizontal surface, tangential component of gravity (Fgsinα) should be taken into consideration in addition to viscous and resistance caused by micropillars. Although surface orientations are different, displacement of droplet is roughly the same at 4.076ms, as shown in Table.2. It is concluded that gravitational force (g=10-6) is smaller relative to unbalanced capillary force. Penetrated depth is less than 0.1μm at 4.076ms, indicating a transition from partial Cassie to Wenzel state. The numerical results demonstrate that g≤10-6 has little influence on droplet climbing-upward movement. 0ms
0.941ms
1.986ms
(a) α=15º
4.076ms
0ms
0.941ms
1.986ms
4.076ms
(a) α=30º 0ms
0.941ms
1.986ms
4.076ms
(a) α=45º Fig.12 Droplet motion on wetting gradient surface with different orientations
Table.2 Value of parameters at 4.076ms Surface orientation 15º 30º 45º
Leading edge position/μm 5.25 5.14 4.57
Trailing edge position/μm
Penetrated depth/μm
Surface free energy/μm2
3.97 3.88 3.85
0.082 0.082 0.087
6.05 5.92 6.02
3.3.2 Gravitational coefficient Various gravitational coefficients are applied to investigate the droplet dynamic characteristics. The surface inclination is 30º and micropillars and wetting gradient are consistent with Case 2 in Fig.5. Gravitational coefficients are 5×10-7, 7.5×10-7, 10×10-7, 25×10-7 and 50×10-7, respectively. Bond number (Bo) is a dimensionless number, denoting the relative importance of gravity and surface tension: Bo
𝜌𝑙 gr 2 /𝜎
(24)
Here, Bo number is used to represent the effect of gravity on droplet climbing-upward movement. Effect of Bo number on displacement and penetrated depth is depicted in Fig.13. Displacement of leading and trailing edge point decreases with an increasing Bo. For Bo≤0.0126, displacement of droplet decreases slightly with the larger Bo number, in which case tangential component of gravity is negligible compared with unbalanced surface tension. For Bo>0.0126, it decreases rapidly with an increasing Bo number. When Bo number is increased to 0.063, the droplet remains in its original
position without displacement, where net capillary force is less than tangential component of gravity. On the contrary, penetrated depth shows an increasing trend with Bo number because the droplet moves to a stronger wetting region. Fig.14 shows effect of Bo number on superficial area and surface free energy at 4.076ms. In the process of movement, part of the kinetic energy generated by net capillary force is converted into surface free energy. Superficial area and surface free energy decrease with the increase in Bo number. The results indicate that a smaller gravitational coefficient (Bo) is beneficial for climbing-upward movement, increasing kinetic energy of droplet. The Bo number ranging from 0.0126 to 0.063 has a great effect on climbing-upward movement of microdroplet (20μm). 7
0.16
The trailing edge position The leading edge position Penetrated depth
6
0.15
Displacement/μm
0.14 4 0.13
3 2
0.12
1
0.11
Penetrated depth/μm
5
0 0.10 -1 0.01
0.1
Bo
Fig.13 Variation of Displacement and Penetrated depth with Bo 5.0
Superficial area Surface free energy
4.5
Superficial area/μm2
55 4.0
3.5 54 3.0 53
2.5
Surface free energy/μm2
56
2.0 52 0.01
0.1
Bo
Fig.14 Effects of Bo on superficial area and surface free energy
Acknowledgment This work was supported by ESA-CMSA International Cooperation of Space Experiment Project (Study on Condensation and Enhancement Methods under Microgravity).
4. Conclusion The model of spontaneous movement of single droplet on microstructured surface with multi wetting gradients is established and investigated by 3D pseudopotential lattice Boltzmann method. Effects of wetting gradient, surface morphology, surface orientation and gravitational coefficient on droplet dynamics are analyzed. Several conclusions can be drawn as follows. (1) Static contact angles on microstructured surface increase non-linearly with Gs, which is different from smooth surface. When a droplet is in Wenzel state (Gs≤-9.0), micropillars have a negative influence on static contact angles. In contrast, it can greatly increase static contact angle of droplet in Cassie or partial Cassie state (Gs≥-8.0). (2) Wetting gradient and surface morphology are of great significance for spontaneous movement. A larger wetting gradient and solid friction is beneficial to increasing movement velocity and surface free energy. Although two-layer wetting gradient surface is capable of prevent the droplet entering micropillars, it has a negative effect on movement velocity. (3) When the Bo number ranges from 0.0126 to 0.063, gravitational coefficient should be considered in the process of microdroplet (20μm) climbing-upward movement. For Bo≤0.0126, surface orientation has little effect on climbing-upward movement.
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