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Droplets can rebound toward both directions on textured surfaces with a wettability gradient Bo Zhang, Qing Lei, Zuankai Wang, and Xianren Zhang Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.5b04365 • Publication Date (Web): 15 Dec 2015 Downloaded from http://pubs.acs.org on December 16, 2015
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Droplets can rebound toward both directions on textured surfaces with a wettability gradient
Bo Zhang,1 Qing Lei,1 Zuankai Wang2,* and Xianren Zhang1,* 1
State Key Laboratory of Organic-Inorganic Composites, Beijing University of Chemical Technology, Beijing 100029, China 2
Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong 999077, China
Email:
[email protected];
[email protected] 1
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Abstract: The impact of water droplets on superhydrophobic surfaces with a wettability gradient is studied using the lattice Boltzmann simulation. Droplets impacting on such kind of textured surfaces have been previously reported to rebound obliquely following the wettability gradient due to the unbalanced interfacial forces created by the heterogeneous architectures. Here we demonstrate that droplets can rebound toward both directions on textured surfaces with a wettability gradient. Our simulation results indicate that the rebound trajectory of droplets is determined by the competition between the lateral recoil of the liquid and the penetration and capillary emptying of the penetrated liquid from the textures in the vertical direction. When the timescale for the droplet penetration and capillary emptying process is smaller than the time for the lateral spreading, the droplet will rebound following the wettability gradient. By contrast, the droplet will display a bouncing against the wettability gradient direction because of the significant capillary penetration and emptying in the transverse direction. We believe that our study provides important insight for the design of micro/nanotextured surfaces for controlled droplet manipulation.
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Introduction
The study of liquid droplets impacting on solid surfaces has been an active topic in both experimental and theoretical investigations for more than a century,1-12 as they play an important role for a variety of technical applications, including ink-jet printing, spray coating, surface cooling, and anti-icing.13-15 For superhydrophobic surfaces with uniform structures,16-20 it has been shown that impacting droplets exhibit various patterns from complete rebounding to sticking, without net lateral move.5, 15, 21-26 For the substrates with nonuniform structures, however, the impact of water droplets behaves rather different, and directed motion was observed following the wettability gradient.27,28 Malouin et al27 reported that droplets rebound toward the more wettable side of the substrate with a wettability gradient that gives rise to a resultant force, which drives the impacting droplets toward the less hydrophobic area. Vaikuntanathan et al28 showed experimentally that when water drops impacting onto the junction line between hydrophobic texture and hydrophilic smooth portions of a dual-textured substrate, it will migrate toward the hydrophilic smooth portions. A counter-intuitive observation has been reported by Wu et al for the droplet migrations on substrates having a wettability gradient.8 The authors showed that droplets can migrate both along and against the wettability gradient as a result of the competition between the capillary pressure and the effective water hammer pressure. This poses a similar question on the bouncing of droplets impacting on a substrate 3
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with roughness gradient: Do droplets rebound following the wettability gradient of the substrate? In the present work, a lattice Boltzmann (LB) method was used to investigate the droplet rebound on textured surfaces with a wettability gradient. Depending on droplet size and impact velocity as well as substrate wettability gradient, we find the droplets can rebound either along or against the gradient direction, instead of directed rebounding solely toward the less hydrophobic side.
Model description LB method is a numerically robust technique for simulating various wetting phenomena of microdroplets that have a size beyond the scope of atomistic computer simulations.29 In this work, the two dimensional, multiphase Shan-Chen (SC) type LB method30,31 based on D2Q9 lattice32 were implemented.33,34 SC model has been successfully used to study droplets impacting on pillared substrates,35 the effects of the entrapped bubbles on flow,36 impingement of liquid drops on dry surfaces,37 drag reduction of superhydrophobic surfaces,38 and surface roughness-hydrophobicity coupling in channel flows.39 In the LB model, both space and time are discrete, and particles move on a regular lattice by applying consecutive propagation and collision processes. The density distribution functions are calculated by solving the lattice Boltzmann equation (LBE), the general form of LBE with the BGK approximation can be written as40: fα ( x + eα ∆t , t + ∆t ) − fα ( x, t ) = −
1
τf
[ fα ( x, t ) − fαeq ( x, t )]
(1) 4
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where the left side is the streaming part, the right side is the collision term, f is the density distribution function, x refers to space position, t time, α the velocity direction, ∆t the time step, and τf the dimensionless relaxation time (τf = 1.0 in this work), In Eq. (1), the equilibrium distribution function is calculated as:
fαeq ( x, t ) = ωα ρ [1 + 3
eα ⋅ u eq (eα ⋅ u eq )2 (u eq )2 9 3 ] + − c2 2c 4 2c 2
For a D2Q9 model, the weighting factor, w˛, is chosen to be 4/9 for α = 0,
α= 1, 2,3, 4,
(2) 1/9 for
and 1/36 forα= 5, 6, 7, 8. eα can be expressed as
0 1 0 − 1 0 1 −1 −1 1 eα = 0 0 1 0 −1 1 1 −1 −1 8
ρ = ∑ fα is the fluid density,
u eq = u +
α =0
the computation of f˛ with
1
τf is the macroscopic velocity used in ρ
8
∑f e . ρα α α
u=
(3)
F contains two types of force, namely,
=0
that from the nearby fluid Fc and that from the solid Fs, which are determined, respectively, with 8
Fc ( x, t ) = −Gcψ ( x, t )∑ ωαψ ( x + eα ∆t , t )eα
(4)
α =0 8
Fs ( x, t ) = −Gsψ ( x, t ) ∑ ωα s ( x + eα ∆t , t )eα
(5)
α =0
In above equations Gc is the interaction strength between fluids, and Gs is the adsorption coefficient between the fluid (liquid or vapor) and the solid surface determined by the interaction potential of ψ ( ρ ) = ψ 0 exp(
− ρ0
ρ
) , with ψ 0 =4 and
ρ 0 =200. s is an indicator function with the value of 1 if the neighbor lattice of the fluid lattice is a solid boundary and 0 for a neighbor fluid lattice.41,42 In this work, all 5
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the quantities used are dimensionless, particular lengths are expressed in the unit of the lattice spacing. GC is fixed to -160, and GS is -130. A two-dimensional simulation box of 400 × 600 was adopted in this work, and a textured surface with a wettability gradient was placed at the bottom of the box. A number of pillars with a size of 3×150 were introduced to represent the substrate roughness, and the spacing between adjacent pillars varies to set a roughness gradient. The distances between pillars decrease evenly, from left to right, with a constant difference of 1 between two successive spacings. In all of our configurations, the middlemost of the substrate was denoted by a specified size of the spacing, B0. A rectangle water droplet was initially placed in the middle of the box (along x direction) above the pillars, and exactly over the middlemost of the substrate. In the initial configurations, liquid density was set to be 887.06, and vapor density was 60.87. The initial configuration was then equilibrated in a short simulation of 8000 ts (time step), and with the obtained final configuration (see Figure 1) a downward vertical velocity was imposed to the liquid droplet. For the following discussion, R represents the radius of the droplet, B0 the spacing for the middlemost interval, ΔB the spacing difference between two nearest intervals, A is the width of the post.
Results and discussion
Droplet impacting with various velocities
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Figure 2A and B show the snapshots of a droplet impacting on a nonuniform solid surface featured with B0=16, A=3, R=60, spacing gradient ξ =VB / A = 1/ 3 ,
and different initial velocities of 0.15 and 0.24, respectively. When impacting on a textured surface, the droplet is found to deform first and spread rapidly owing to a rapid increase in pressure at the point of impact,43 and part of the liquid is driven into the space between posts because of the downward momentum of the droplet. After reaching its maximum spreading and deepest penetration, the droplet undergoes viscous dissipation and moves back toward the center, and finally bounces off. The analysis of images reveals that at low impact velocity, the droplet bounces off following the wettability gradient as expected (Figure 2A). However, the droplet at a higher impact velocity would bounce off against the wettability gradient (Figure 2B) which has not been observed before. For a droplet impacting on a surface, liquid will penetrate into the space between pillars44 when the capillary pressure
Pc = −2 2γ LV cos θ A / B can be balanced by
the water hammer pressure PWH = k ρ cV , with γ LV the surface tension, θ A the advancing contact angle on the flat surface, c the speed of sound in water, and B the space between two adjacent posts. For droplets with higher velocity (see Figure 2B), a larger dynamic pressure and water hammer pressure will be generated, and as a result, more liquid can penetrate into the space between pillars. This behavior can be found from Figure 3, which illustrates the maximum ratio of liquid penetrated into the gaps as a function of Weber number. When the liquid on the interface recoils, the part penetrated rises up under the 7
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capillary pressure at the same time. If liquid penetrates into the gap between pillars more deeply, the recoil of liquid on the interface would be significantly slowed down due to the unfinished capillary rise of the penetrated liquid, as shown in Figure 2B(c), especially on the left side of the droplet. In this case, the capillary pressure that relates to interval radius is the dominating factor to slow down the droplet receding on the left side. On the right, however, the liquid in the right gap will rise up faster owing to the higher capillary pressure, resulting in a fast receding velocity on the right side, and so that the droplet will rebound toward the left side. The molecular mechanism can be confirmed by measuring the time evolution of capillary rise and liquid receding processes. Here the contact area (length of contact line in 2D) between liquid above interface and the substrate, Sup, is used to represent the liquid receding process, and contact area (again length of contact line) between the substrate and penetrated liquid, Sdown, is used to represent the capillary rise of penetrated liquid (see Fig 4). For different dynamics of droplet rebounding shown in Figure 2A and 2B, the evolution of Sup and Sdown is given in Fig 5. The figure shows that upon impacting, both of them spread until reaching their maximum length in a short time, and then begin to recede until the droplet bouncing off. For the case with droplet bouncing along the roughness gradient (Figure 2A), the ratio of penetrated liquid is relatively small, and the penetrated liquid rises up rather quickly. Therefore, Sdown reaches 0 before the receding process finishes, and then the liquid recoils sequentially. At this time, the wettability gradient of the substrate is the dominating factor that influences the recoil, resulting in a net 8
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unbalanced Young force in the direction of the wetting gradient16 that forces the drop bouncing off toward the hydrophilic side. For the case with droplet rebounding against the wettability gradient (Figure 2B), different trends for Sup and Sdown are found (Figure 5(b)). In this case, Sup and Sdown reach the same value shortly after the receding process, and then Sup and Sdown decrease sequentially until bouncing off. This observation suggests that the following receding process is limited by the capillary rise. Thus, the uneven distribution of the required time for capillary rise results in the rebounding against the wettability gradient. Three timescales are compared in Figure 6, in which tmax is the time for the droplet reaching its maximum spreading, te the time for capillary penetration, and liquid emptying from the interval, tc the time for the droplet bounding off. In our simulations, a droplet bounces off right when V
