Following or Against Topographic Wettability Gradient: Movements of

Article pubs.acs.org/Langmuir

Following or Against Topographic Wettability Gradient: Movements of Droplets on a Micropatterned Surface Jiayi Zhao† and Shuo Chen*,†,‡ †

School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China Shanghai Key Laboratory of Vehicle Aerodynamics and Vehicle Thermal Management Systems, Tongji University, Shanghai 201804, China

‡

ABSTRACT: Two kinds of possible movements of droplets (i.e., following or against the topographic wettability gradient) on a micropatterned surface are investigated by using a particle-based method: many-body dissipative particle dynamics (MDPD). The displacement along the wettability gradient and contact angles on both sides of the droplet are analyzed. The results show that the migration trajectory of the droplet is determined by the coexistence of Cassie and Wenzel states and the unbalanced Young’s force, which are related to the impact velocity, pillar height, and surface tension. The droplet remains in the Cassie state and advances spontaneously following the wettability gradient under a small impact velocity, high pillar height, and large surface tension. On the contrary, when the coexistence of Cassie and Wenzel states appears, the contact line on the side of the Cassie state retracts and that on the side of the Wenzel state pins, inducing droplet movement against the wettability gradient. Additionally, the critical impinging velocity, which determines the migration direction of the droplet, also depends on the pillar height and surface tension. The outcomes are helpful in designing surfaces with topographical wettability gradients for droplet transportation.

1. INTRODUCTION

Do droplets always move following the wettability gradient? The answer was experimentally explored by Wu et al.25 via depicting a counterintuitive phenomenon in which droplets could migrate with and against a wettability gradient. They hypothesized that local wetting dynamics (i.e., Cassie26 and Wenzel27 states), including the Cassie-to-Wenzel transition (Ec‑w),28 played a critical role in the lateral displacement of the droplet on a nonuniform texture. Hence it arouses controversy for droplet dynamical behavior on textured surfaces with a wettability gradient, suggesting that our understanding is still superficial. Meanwhile, the mechanism of droplet impingement, serving as a highly transient process, is difficult to probe merely by experimental observations.29,30 Therefore, a wide spectrum of numerical simulations is implemented to investigate the process.31−34 In particular, Zhang35 successfully employed the lattice Boltzmann method to model the counterintuitive situation reported by Wu et al.25 He explained different droplet trajectories via analyzing the competition between penetration and capillary emptying from the textures during the lateral recoiling stage. However, there are still some deficiencies existing in Zhang’s work. For example, the Cassie and Wenzel state, including the transition between them, is not able to be derived in Zhang’s model in which there is no basal substrate, whereas the wetting states are considered to be important for nonbouncing, partial bouncing, and complete bouncing.30,36,37 Additionally, the simulation based on a two-dimensional and Eulerian framework is not always able to display series of

Because of the large practical demand in industrial fields involving spray coating,1,2 anti-icing,3 self-cleaning4 and others,5−7 the complex process of droplet impingement on surfaces decorated with micropillars has obtained widespread attention. The scenarios after droplet impacting solid surfaces are generally defined as six situations: deposition, splash promotion, corona splashing, receding break up, partial rebound, and complete rebound.8 This primarily relies on the topography of surfaces9,10 and related similarity criterion numbers.11,12 More details of droplet impingement can be found in recent reviews.13−15 Compared to homogeneous textured surfaces, heterogeneous surfaces with a roughness gradient, employed to manipulate droplet transport, are increasingly becoming an active topic and are often adopted as an effective tool for specific applications.16−22 In particular, droplets usually undergo asymmetrical movement due to a wettability gradient, which is distinguished with symmetrical responses on uniform textures. For instance, Vaikuntanathan et al.23 carried out an experimental analysis, concerning droplet impingement on a dual-textured substrate consisting of both smooth and textured regions. The droplet, forming an asymmetrical shape and with the bulk movement of the droplet away from the impact point, is related to the droplet’s mass and impact velocity. Similarly, directed rebounding of the droplet on a nonuniform surface was investigated by Malouin24 experimentally, and it showed that the droplet motion follows the wettability gradient, and the unbalanced Young’s force was considered to be a key role in the rebound trajectory of a droplet. © 2017 American Chemical Society

Received: February 8, 2017 Revised: May 9, 2017 Published: May 9, 2017 5328

DOI: 10.1021/acs.langmuir.7b00438 Langmuir 2017, 33, 5328−5335

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when the distance between particle pairs exceeds cutoff radii rc and rd. A and B are attractive and repulsive coefficients, respectively, in controlling the interaction strength between particle pairs. In particular, the repulsive term also depends on its local particle density ρ̅i, which is accumulated by weight function wρ(rij), as shown in eqs 5 and 6. Additionally, the dissipative and random forces are identical to DPD, which satisfies the fluctuation−dissipation theorem for working as the thermostat of a constant-temperature system.44 The two key innovations in DPD/MDPD are soft repulsions, pairwise dissipation, and random forces, in which the former allows large time steps to be taken in simulations and the later basically works as a thermostat.45 Until now, a wide range of researches have adopted MDPD to study capillarity46,47 and droplet dynamics.48−51 In our article, the principal MDPD parameters for fluid particles are chosen according to the literature43,52 and are listed in Table 1.

phenomena such as the coexistence of Cassie and Wenzel states,38 the triaxial unbalancing of Young’s force,28 and the tracking of tiny deformations in the liquid−gas interface,40 and these phenomena are crucial to determining the droplet trajectory according to Wu et al.’s assumption25 from experimental observation. Therefore, to overcome the limitations of experimental observation and two-dimensional simulation, a particle-based method called many-body dissipative particle dynamics (MDPD) is utilized to investigate droplet impingement on surfaces with a topographic wettability gradient. The important influences of pillar height and surface tension for impalement36,37,39 are taken into consideration. Our results suggest that a droplet can migrate along both directions of the wettability gradient, and this primarily depends on the wetting state (i.e., Cassie and Wenzel states) of droplets. Also, the relationship between wetting states and parameters, such as the impact velocity, pillar height, and surface tension, is discussed. The article is organized as follows. A brief introduction of the algorithms of the MDPD approach as well as the modeling of the wettability gradient is given in Section 2.1. The simulated results are discussed in Section 3.1. In Section 4, the article closes with a conclusion.

Table 1. Primary Simulation Parameters (in MDPD Units)

2. NUMERICAL METHOD 2.1. Algorithms of MDPD. As a modified method for standard dissipative particle dynamics (DPD),41,42 the time evolution of each MDPD particle is analogous to DPD, following Newton’s second law as mi

dvi = Fi = dt

∑ (FCij + FijD + FijR ) j≠i

where the interaction between particle pairs can be described by three terms: the conservative force, dissipative force, and random force. In the MDPD method, it adds a van der Waals loop to the equation of state (EOS)43 to model the sharp liquid/vapor interface. Hence, the conservative force is modified as (2)

⎧⎛ rij ⎞ ⎪ ⎪ ⎜1 − ⎟ rij ≤ rc rc ⎠ wC(rij) = ⎨ ⎝ ⎪ ⎪ 0 rij > rc ⎩

(3)

⎧⎛ rij ⎞ ⎪ ⎪ ⎜1 − ⎟ rij ≤ rd rd ⎠ wd(rij) = ⎨ ⎝ ⎪ ⎪ 0 rij > rd ⎩

(4)

ρi (rij) =

∑ wρ(rij) i≠j

2 rij ⎞ 15 ⎛ wρ(rij) = ⎜1 − ⎟ rd ⎠ 2πrd 3 ⎝

symbol

value

All Bll Δt rc rd kBT X×Y×Z

−60 25 0.01 1.0 0.75 1.0 50 × 50 × 50

In Table 1, All and Bll represent the attractive and repulsive coefficients between liquid particles, respectively. For MDPD, if All and Bll are determined, then we are able to obtain the number density and surface tension of droplets, where the measurement of surface tension is often based on the Laplace pressure after reaching equilibrium.52 The relationship between the physics unit and numerical unit can be established by the physical unit ratio48 or coarse-graining level.53 According to experimental work25 and simulation,48 the length of the MDPD model is normalized by LMDPD = 10 μm, and hence the unit mass of the MDPD system can be achieved as MMDPD = 1.18 × 10−13 kg by matching the density in simulations with the physical water density. Additionally, the unit time is determined to be TMDPD = 6.33 μs based on the surface tension calculation. Furthermore, the impact velocity ranges from 2.37 to 7.11 m/s when the MDPD dimensionless velocity is from 1.5 to 4.5. 2.2. Topographic Wettability Gradient. In the present study, the micropatterned surface is established by frozen MDPD particles interacting with liquid particles according to eq 2. The frozen particles are arranged in a face-centered-cubic (fcc) lattice. Because of the soft interaction between particles, the bounce-back boundary condition is implemented in order to avoid penetration. The bounce-back boundary condition is realized by forcing liquid particles back to the position of the last time step and simultaneously reversing the current velocity if they penetrate the solid surface. The no-slip condition can be satisfied in the implementation of this boundary condition.54 A textured surface with a topographic wettability gradient is created by arraying fixed size pillars with varying center-tocenter spacing. The gradient direction is parallel to the X direction with uniform spacing along the Y direction. Hence, roughness fraction f can be calculated according to Cassie and Wenzel states as follows55

(1)

FCij = [AwC(rij) + B(ρi̅ + ρj̅ )wd(rij)]eij

parameter attractive coefficient repulsive coefficient time step cutoff radius of attractive force cutoff radius of repulsive force temperature computational domain

(5)

(6)

where rij = |ri − rj| is the norm of the distance vector between particle pairs and acts as a variable in weight functions wc(rij) and wd(rij). eij = rij/rij is the unit vector, determining the direction of conservative force. The weight functions vanish 5329

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w2 g2

fCassie = fWenzel =

(7)

4wh +1 g2

(8)

where g can be divided into two types (i.e., gx and gy). For example, gx is the gap length between pillars that varies to create a gradient along the X direction. w is the side length of square pillars, and h is the height of pillars. Besides, the basal substrate is also established at the bottom of pillars. The parameter values for the surface are listed in Table 2. Table 2. Primary Parameters for the Surface (in MDPD Units) parameter

symbol

value

attractive coefficient repulsive coefficient pillar width pillar gap in the X direction pillar gap in the Y direction pillar height

Alw Blw w gx gy h

−25 10 0.75 1.0−2.5 1.3 0.75−2.25

The attractive and repulsive coefficients (i.e., Alw and Blw) are utilized to determine the wettability of the surface, and the intrinsic contact angle is approximately 105° for a flat surface with given parameters. Additionally, we define that the direction following the wettability gradient is identical to the direction of decrease of the contact angle. The schematic diagram is shown in Figure 1.

Figure 2. Time evolution of impingement droplets under two different impact velocities: (a) V = 1.5 and (b) V = 4.5. Figure 1. Schematic diagram of a micropatterned surface.

contacting stage, spreading stage, recoiling stage, and equilibrium stage). When a droplet touches the surface, the increase in the droplet Laplace pressure causes liquid impalement.37 Therefore, at the contacting stage the droplet penetrates the gap between pillars and then turns into a spreading stage with the release of the Laplace pressure. Having reached the balance between inertia and the capillary effect,39 the maximal spreading diameter, associated with We,56 is acquired and the droplet starts to recoil. Then the droplet retracts, followed by the merging of two gibbous frontiers on both edges, and it moves toward the equilibrium state. The simulated results show that a droplet under small V migrates along the wettability gradient (Figure 2a), in contrast to that under large V (Figure 2b).

3. RESULTS AND DISCUSSION 3.1. Influence of Impact Velocity. In this article, we initially locate the droplet above the center of the surface, where there is a pillar with gx = 1.7 and 1.8, respectively, on both sides of it. In this section, we discuss the influence of the impact velocity under constant pillar height and droplet surface tension, which are fixed at 1.5 and 24.47, respectively. The complete processes of droplet impingement for two impact velocities V of 1.5 and 4.5 (i.e., We = ρV2d/σ = 9.32 and 83.91) are qualitatively depicted in Figure 2. According to the status of droplets, the process is divided into four substages (i.e., 5330

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Figure 3. History of left and right contact angles along the wettability gradient: (a) V = 1.5 and (b) V = 4.5.

Figure 4. History of left and right contact angles orthogonal to the wettability gradient: (a) V = 1.5 and (b) V = 4.5.

According to Deng’s thoery,57 the states of droplet impingement on a textured surface are segregated into the total wetting state, partial wetting state, and total nonwetting state. The formation of the three states relies on the relationship among the wetting pressure (PD = ρV2/2), antiwetting pressure (PC = −2(21/2)γ cos θA/B, where B is the length of the gap between pillars, B = gx, in the present simulation), and effective water hammer pressure (PEWH = kρcV). γ is the surface tension of a droplet, and θA represents the advancing contact angle. In additional, k is a fitting parameter, and c stands for the speed of sound in water. From Figure 2a, only the center of a droplet penetrates the space due to small V. After the recoiling stage, we obtain the total nonwetting state (i.e., PC > PEWH > PD),57 meaning that the Cassie state survives finally. In this situation, the droplet spontaneously moves along the wettiability gradient, which does not generally happen without an incentive.54 For the Cassie state, which indicates low contact angle hysteresis, the droplet is easily impacted by the unbalanced Young’s force. On the basis of Wu et al’s assumption,25 the receding contact angles on both sides generate a driving force for vectoring the droplet in the direction of decreasing contact angle as dF = γ(cos θ1 − cos θ2)ds

calculated by the tangent line at the three-phase junction and averaged within every 50 time steps. During the recoiling stage, the right contact angle, which is in a dilute pillar area, is continuously larger than the left one (θL < θR) for smaller V, generating the unbalanced Young’s force to drive the droplet to migrate following the wettability gradient. Additionally, both left and right contact angles exceed 100°, and the droplet maintains a Cassie state with a low contact angle hystersis,58 which is crucial for spontaneous motion. On the other hand, Figure 2b shows that the droplet undergoes opposite direction motion against wettiability gradient, which is counter intuition. Under large V, the impalement of droplet becomes evident because of larger PEWH.57 In stark contrast to Figure 2a, the partial wetting state (i.e., PEWH > PC > PD),56 which is the coexistence of the Cassie and Wenzel states, is captured. In Deng’s theory, the partial wetting state generally means that only the center of the droplet fills the space on uniform textures. However, in the present study parameter gx, being the spacing between pillars, increases from left to right as shown in Figure 1. Hence, the partial wetting state on a surface with a topographic wettability gradient is quite different from that in previous studies. Through snapshots in Figure 2b, we find that the right part of the droplet undergoes a transition from the Cassie to Wenzel state and then is pinned by pillars whereas the left part of the droplet remains in the Cassie state as a result of large PC (small gx). The asymmetric coexistence of Cassie and Wenzel states facilitates different contact angles between the left and right sides parallel to the wettability gradient (θL > θR), as shown in

(9)

where F is the unbalanced Young’s force, γ is the surface tension, θ1 and θ2 are contact angles on both sides, respectively, and ds is the contact line.28 Figure 3a displays the history of left and right receding contact angles along the wettability gradient under small V. The advancing and receding contact angles are 5331

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experimental observation of Wu et al.25 and verify the assumption of the coexistence of Cassie and Wenzel states. 3.2. Influence of the Pillar Height and Surface Tension of Droplets. Next, we focus on the influence of ratio ϕh of pillar height h to pillar side length w (i.e., ϕh = h/w) and surface tension σ of droplets on movement with or against the wettability gradient, respectively. As we have mentioned above, the coexistence of Cassie and Wenzel states plays a critical role in the direction of droplet migration. The surface tension, which hinders wetting, is the core for the transition of the Cassie-to-Wenzel state under a given impact velocity.37 According to Bartolo’s proposal,37 two impalement scenarios (i.e., touching down and sliding) are depicted in the present study as the touching down pressure and sliding pressure

Figure 3b. In other words, it leads to different mobility between the left and right contact lines. Specifically, the left contact line depins easily, but the right contact line sticks, and the unbalanced Young’s force causes the droplet to advance against instead of follow the wettability gradient. Figure 3b also indicates that the right contact angle exceeds the left one (θL < θR) during the later recoiling stage, which is mainly caused by the swing in the droplet without resulting in moving, following the wettability gradient. Therefore, it is not enough to determine the direction of droplet movement only through the unbalanced Young’s force, and the wetting state should also be taken into consideration. Because the right part of the droplet stays in the Wenzel state, which means low mobility and large wetting hystersis, the droplet is arrested by the pillars and fails to move further. Even though an unbalanced Young’s force acts on the droplet, the droplet does not move. Additionally, we analyzed left and right contact angles orthogonal to the wettability gradient as shown in Figure 4. It apparently indicates that contact angles on two sides of the droplet are approximately symmetric, owing to uniform textures along the Y direction, and Young’s force between the two sides is also balanced. Therefore, almost zero lateral displacement is observed along the Y direction regardless of the impact velocity and wetting state. On the surface with a topographic wettability gradient, PC and the wetting state vary with the positions of droples. The lateral displacement of a droplet under various V values is shown in Figure 5, which indicates that the lateral displacement

T ≈ Pimp

S Pimp =

σh g2

2fCassie 1 − fCassie

(10)

|cos θa|

σ w

(11)

where θa is the local advancing contact angle and the other symbols are the same as in our previous definition. The droplet undergoes an impalement transition (i.e., a Cassie-to-Wenzel transition) if the wetting pressure is larger than the minimums in PTimp and PSimp. Therefore, pillar height h and surface tension σ are two non-negligible factors in droplet impingment. In brief, a larger pillar height and surface tension enhance the robustness against liquid impalement, to some extent, for the surviving Cassie state.38,39 At first, the influence of pillar height h on the lateral displacement of the droplet is investigated by varying ϕh while keeping We fixed. Figure 6 shows the comparison between lateral displacements of droplets for various pillar heights of micropatterned surfaces (i.e., ϕh = 1.0, 2.0, and 3.0). In Figure 6a, the droplet moves against the wettability gradient under smaller ϕh (i.e., ϕh = 1.0) while motions following the wettability gradient occur for higher ϕh values of 2.0 and 3.0. And the Cassie state is observed for larger ϕh (i.e., ϕh= 2.0 and 3.0) with a Wenzel state for smaller ϕh (i.e., ϕh= 1.0), in which a larger ϕh increases the PTimp in eq 10. Because of different ϕh, the droplet undergoes a Cassie or Wenzel state and shows a different direction of movement for the same We. The pillar height is a significant paramenter in determining the trajectory of a droplet after impacting the nonuniform texture. However, in Figure 6b all droplets move against the wettability gradient for larger impact velocities (i.e., We = 83.91). For larger impact velocities (i.e., We = 83.91), the displacement of a droplet, as a function of ϕh, is larger than that under a small impact velocity (i.e., We = 9.32). For small ϕh with w fixed at 0.75 in the present study (as shown in Table 2), PTimp is small, tending to form a complete Wenzel state. If more parts of the droplet are in the Wenzel state, then this will lead to a larger bulk displacement for a droplet against the wettability gradient compared to the impalement process under large ϕh. Second, the influence of droplet surface tension is discussed with ϕh = 2.0. In the MDPD simulation, the surface tension can be manipulated by choosing a different value of attractive coefficient All in Table 1. The surface tension of a droplet is σ = 13.56, 24.47, 32.09 for All = −50, −60, and −70, respectively. As shown in Figure 7, the surface tension of the droplet also determines the transition between the Cassie and Wenzel states as predicted in eqs 10 and 11. Because there is a linear

Figure 5. Lateral displacement of droplet versus time under different impact velocities.

is a function of V and the droplet migrates much farther from its initial impacting position under larger V. Larger V, which means larger PEWH, results in more parts of the droplet undergoing the sag transition39 (i.e., Wenzel state), and the droplet is pinned. Meanwhile, a droplet under a smaller V trends to move spontaneously owning to low contact angle hystersis for the Cassie state. The critical impact velocity, where the droplet spreads symmetrically without lateral displacement, is approximately 3.0, and it is also influenced by the ratio of pillar height h to pillar width w and by the surface tension of the droplet, which will be discussed in the next section. In summary, besides the unbalanced Young’s force caused by the wettability gradient, the coexistence of Cassie and Wenzel states also affects the direction of droplet migration under various impact velocities. The results are in agreement with the 5332

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Figure 6. Lateral displacement of a droplet with different ϕh: (a) We = 9.32 and (b) We = 83.91.

Figure 7. Lateral displacement of a droplet with different σ: (a) V = 1.5 and (b) V = 4.5.

relationship between the surface tension and wetting pressures under constant pillar width w, pillar height h, and pillar gap g (i.e., PTimp and PSimp), the droplet with small surface tension tends to fill the texture and be pinned by pillars, indicating displacement plateaus in Figure 7a, and it also shows that larger surface tension (i.e., σ = 24.47 and 32.09) helps a drop stay in the Cassie state and advance spontaneously along the wettability gradient. As for V = 4.5, as shown in Figure 7b, the droplet with the largest surface tension, in which the droplet is in the Cassie state, still moves along the wettability gradient compared to the others. It again demonstrates that the surface tension plays an important role in the direction of movement of a droplet on micropatterned surfaces. The comprehensive influence of the pillar height ratio ϕh and Weber number on the direction of movement of the droplet is summarized as shown in Figure 8. In Figure 8, the direction of movement of droplets is represented by different symbols. Small We and large ϕh (top left portion with blue triangles, labeled as the Following Region) tend to make droplets move following wettability, in contrast to large We and small ϕh (bottom right portion with black squares, labeled as the Against Region). Because the antiwetting pressure PC is increased for large h and σ, the droplet needs a greater PD value to complete the Cassie−Wenzle transition and move against wettability. The middle part, labelled as the Symmetry Region, is symmetrical spreading, where the red circle indicates a droplet without evident movement. It suggests that the critical velocity

Figure 8. Diagram for the movement direction of droplet as a function of pillar height and Weber number.

needed to determine the movement direction relies on the relationship between PC and PD. And pillar height h and the Weber number are two important factors that may influence the critical velocity. Especially deserving of mention is that the red circle in the top left corner is found to appear under the smallest We and largest ϕh, where the droplet is not able to move with very small impact velocity because more energy is needed to overcome the contact angle hysteresis. 5333

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4. CONCLUSIONS Two kinds of movement (i.e., with or against the wettability gradient) after droplets impact a micropatterned surface are simulated by using many-body dissipative particle dynamics. The influence of pillar height and surface tension on droplet impingment is investigated. The droplet can migrate either with or against the wettability gradient, depending on the pillar height and surface tension. The simulation results demonstrate that the coexistence of Cassie and Wenzel states and the unbalanced Young’s force plays an important role in the displacement of the impacting droplet. The droplet under a large pillar height and surface tension, meaning a larger antiwetting pressure, tends to generate a Cassie state for selfmotion following the wettability gradient. A comparatively small antiwetting force due to a small pillar height and surface tension of the dropet leads to a Wenzel state after impact. Meanwhile, the nonuniform spacing between pillars dominates the coexistence of the Cassie and Wenzel states in which depinning and pinning exist on either side of the droplet, respectivly, and the coexistence of Cassie and Wenzel states causes the droplet to migrate against the wettability gradient. The critical velocity needed to determine the movement direction of the droplet is also related to the pillar height and surface tension.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Shuo Chen: 0000-0002-0580-5292 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (grant no. 51276130). The grant is gratefully acknowledged.

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