Numerical Study on Droplet Sliding across Micropillars - Langmuir

Apr 10, 2015 - Langmuir , 2015, 31 (16), pp 4673–4677 .... Sahba Salarian , Reza Kamali. Journal of Industrial and Engineering Chemistry 2016 42, 1-...
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Numerical Study on Droplet Sliding across Micropillars Yuxiang Wang and Shuo Chen* School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China S Supporting Information *

ABSTRACT: Droplet sliding on surfaces is an important phenomenon since it widely happens in microfluidic industry. In this article, we simulate droplets sliding across micropillars on smooth substrates to test how the pillars with different intrinsic wettability influence the movement of droplets. The simulation is performed using a particle-based numerical method, many-body dissipative particle dynamics (MDPD). The simulated results show that the heterogeneous area (built by arranged micropillars) can influence the dynamical contact angles significantly. Both the advancing and receding contact angles increase when the droplet front slides on the heterogeneous area, and their difference is also enlarged, thus the contact line may be pinned. The droplet shows a creeping motion style when its front climbs over each pillar. We also find when the droplet enwraps all pillars, the composite liquid/ solid surfaces have no effect on the advancing and receding contact angles. The outcomes support the viewpoint that the wettability is a contact-line-based problem instead of a contact-area-based one.

1. INTRODUCTION Because of the promising applications of superhydrophobic surfaces such as anti-icing of airplanes and power lines and selfcleaning living facilities, research on droplet wetting dynamics has witnessed rapid development in the last few decades.1−4 In general, roughening the surfaces is a workable and economical method for making hydrophobic surfaces superhydrophobic compared to thermal or chemical methods. Two main theories for droplet wetting on rough surfaces were proposed: Wenzel’s theory5 and Cassie−Baxter’s theory.6 Many experiments and theoretical analyses have proven that the droplets adhere more strongly to the rough surfaces in the Wenzel state7−9 than in the Cassie state, thus the Wenzel state can cause more contact angle hysteresis. Because these two theories just take the area underneath the droplet into account, the wettability can be treated as a contact-area-based problem, i.e., a 2D problem. However, the two theories have been open to discussion.10−18 Pease10 questioned the 2D viewpoint first and suggested that “all the equilibrium, advancing and receding contact angles are related directly to the mean work of adhesion that the threephase contact line can assume”, implying that wettability is a contact-area-based problem, or a 1D problem. However, this 1D viewpoint has almost been ignored ever since. Recently, on the basis of many experimental data and phenomena, Gao and McCarthy11 directly challenged the 2D theories in a paper titled “How Wenzel and Cassie Were Wrong”, and their experimental data showed that the surface properties around the contact line, instead of the whole contact area under the droplet, are important in determining wettability. This bold idea soon aroused a heated discussion.12−16 McHale15 thought both the 1D and 2D viewpoints are appropriate in a certain scope of applicability: if the rough surfaces are “everywhere similar and isotropic”, then the Wenzel and Cassie equations © XXXX American Chemical Society

are available, while when the surfaces are equipped with a single defect, the local form of the modified Wenzel and Cassie equations should be used. Patankar and co-workers16 pointed out that the overall trend in the Cassie−Baxter equation is in agreement with their experimental and numerical data for advancing and receding contact angles. Marmur et al.18 suggested that the contact-line-based and the contact-areabased approaches can be deduced from each other. This debate is ongoing. The dynamic properties of droplet sliding on heterogeneous surfaces has been studied intensively because it is a major scientific challenge and is particularly useful in droplet-based microfluidics.19−25 Sbragaglia and co-workers20,21 carried out experiments on droplets sliding down inclined heterogeneous surfaces with parallel hydrophilic and hydrophobic stripes. Their results demonstrated a very a notable stick−slip moving style, and they also explained that a large contrast in wettability between the hydrophilic and hydrophobic stripes could contribute to this visible phenomenon. Together with a twodimensional numerical simulation using lattice Boltzmann models, they observed the interesting moving style. Also, by using a diffuse-interface model with a generalized Navier boundary condition, Wang and co-workers22 found that the wettability contrast between two solid strips affects the droplet behavior significantly. Nakajima and partners23−25 did a series of experiments to study the sliding behavior of droplets on surfaces, and using a particle tracing velocimetry method, they observed the flow pattern in the sliding droplet directly. They found that on a superhydrophobic surface there are almost no Received: January 28, 2015 Revised: April 1, 2015

A

DOI: 10.1021/acs.langmuir.5b00353 Langmuir XXXX, XXX, XXX−XXX

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Langmuir rolling moving styles but for a hydrophobic surface with fixed velocity the rolling of indicator particles can be observed easily. However, most of the previous research fabricated or set the heterogeneous property on all of the concerned surfaces. From a macroprospective, droplets always move on surfaces with a “homogeneous” property if we treat the adjacent two microstripes with different wettabilities as a single unit. How will the droplet behave if it moves from a homogeneous area to a heterogeneous area and passes it? In this article, by using many-body dissipative particle dynamics, we focused on droplets sliding across pillars with varied wettability on a smooth substrate. The shape change of the droplet could be observed, and the influences on the dynamic contact angles in different stages due to heterogeneous areas were analyzed. Furthermore, we used the simulated data to illustrate why the wettability is a 1D issue.

Table 1. Parameters of the MDPD Scheme (All in MDPD Units)

ωc(rij) = 1 −

ωd(rij) = 1 −

ωρ(rij) =

value

attractive coefficient repulsive coefficient time step temperature of the system random coefficient dissipative coefficient particle number of the droplet computational domain empirical velocity Verlet coefficient

A B ΔT kBT σ γ NL X×Y λ

−300 10 0.01 1.0 2.0 σ2/(2kBT) 14 400 200 × 100 0.65

(1)

rij rc

(2)

rij rd

(3)

In these equations, A and B are the maximum attraction and repulsion between two fluid particles i and j, respectively. eij = rij/rij determines the direction of the force. rij = rj − ri and rij = | rij|, in which ri and rj are the positions of ith and jth particles in a Cartesian coordinate system. ωc(rij) and ωd(rij) are the weight functions which both depend on distance rij and will decrease to zero when rij ≥ rc for ωc(rij) and rij ≥ rd for ωd(rij). Here rc = 1.0 and rd = 0.75. Another important modification of MDPD is the introduction of local density ρ̅ for the repulsive part, here taking the ith particle for example: ρi (rij) =

symbol of parameter

Also, we have explained why the surface tension is very high and advantageous to our previous study.33 2.2. Boundary Condition. The solid substrate and pillars are built by freezing-arranged solid particles, and the bounceback reflection boundary condition is employed to force back the penetrating liquid particles. Figure 1 shows that no penetration occurs and the droplet front intrudes into the space between two pillars naturally.

2. NUMERICAL METHOD 2.1. MDPD. As a modified scheme of classical DPD,26,27 Warren28 introduced the van der Waals loop into the equation of state (EOS) to make it suitable for simulating the free liquid/ vapor interface. Since then, MDPD has been used increasingly to investigate fluid systems with free fluid interfaces, such as capillarity,29,30 flow in a microtube,31,32 droplet dynamics,33 and so on. In our simulation, Warren’s MDPD scheme is adopted, in which the only difference from classical DPD is the conservative force as eq 1. The random and dissipative forces are kept the same as those in classical DPD. FCij = Aωc(rij)e ij + B(ρi + ρj ) ωd(rij)e ij

name of parameter

Figure 1. Droplet sliding over pillars.

To generate different wettability, the solid/fluid conservative force is modeled by combining short-range repulsive and longrange attractive forces. Here, three linear weight functions are linked together as follows ⎧ ⎛ r ⎞ ⎪ Fb⎜1 − ij ⎟ , rij ≤ rb rb ⎠ ⎪ ⎝ ⎪ ⎪ rb − rij FslC = ⎨ Fa , rb < rij ≤ ra ⎪ ra − rb ⎪ r −r ij ⎪ F sl ⎪ a r − r , ra < rij ≤ rsl ⎩ a sl

∑ ωρ(rij) i≠j

(4)

2 rij ⎞ 15 ⎛ 1 − ⎜ ⎟ rd ⎠ 2πrd 3 ⎝

(5)

In this work, the simulation was performed in a 2D rectangular computational domain, and the modified velocity Verlet algorithm was used to run the iterative calculation. Table 1 lists all of the necessary parameters in the simulation. Some important fluid attributes of the droplet can be deduced from these parameters, such as a number density of 24.5 (in MDPD units), a surface tension of 510 (in MDPD units), and a kinematic viscosity of 0.2686 (in MDPD units).

(6)

where is the conservative force between solid and fluid particles and depends on their distance rij. Fa and Fb determine the strength of the attractive and repulsive parts. rsl is the cutoff of the solid/fluid interaction, and the two subranges (ra and rb) are the positions of the maximum point of Fa and the vanishing point of Fb. To simplify the simulation, Fb = 200, rsl = 0.3, ra = FCsl

B

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Langmuir 0.15, and rb = 0.1 are fixed; only Fa is varied to generate various wettabilities. Figure 2 illustrates eq 6, and the inlet shows the relationship between Fa and θE when other parameters are fixed.

Figure 3. Advancing and receding contact angle evolution versus time. The pair of curves is divided into five stages.

substrate in this stage, respectively. In stage 3, the droplet front is beginning to cross the pillars when the rear is still on the substrate. It shows a creeping motion style when the droplet front climbs over each pillar, as illustrated in the inset of Figure 6. In this stage, θA3 and θR3 are both enlarged compared to those in stage 2. The droplet liquid can totally wet the substrate between two pillars in all cases. In other words, the droplet is always in the Wenzel state. In Table 2, the duration of this stage is presented, and more analysis will be given later. When the droplet front crosses over the last pillar and begins to slide on the substrate and at the same time the rear has not come into contact with the first pillar, all pillars are wrapped by the droplet and the values of θA4 and θR4 approximate those of θA2 and θR2, respectively. In the final stage, the droplet rear gets in touch with the first pillar and is pinned by it. The wettability is more sensitive to the contact line than to the contact area, and it could be explained as follows: (1) In Table 2 we can see that the duration time of the droplet in stage 3 decreases when the intrinsic wettability is higher whereas it increases with the increasing number of pillars. Furthermore, Figure 5 shows the details of a droplet sliding across a pillar. One can see that the droplet will spend more time crossing an MA pillar than crossing an MC pillar because material A is far more hydrophobic than material C. Both the table and figure can verify that the heterogeneous areas do have a significant effect when the droplets slide across the pillars in stage 3. As a rational conjecture, people who treat Wenzel’s theory or Cassie−Baxter’s theory as law may expect the same effect in stage 4, in which the whole heterogeneous area is involved beneath the droplet. Unfortunately, in stage 4, θA4 and θR4 almost equal θA2 and θR2, though the middle of liquid/solid contact area, which is a little far away from the contact line, is composed of a smooth substrate and pillars with two different materials. Only the front and rear of the droplet are on the substrate as in stage 2. Under this condition, θA4 and θR4 should not equal θA2 and θR2 according to either Wenzel’s theory or Cassie−Baxter’s theory. As mentioned above, “is the Wenzel and Cassie−Baxter formulas relevant?” is still in dispute. The hottest point is whether wetting is contact-line-dependent or contact-area-dependent. McCarthy’s group11−13 used many experimental facts to consolidate their argument that “contact angle behavior (advancing, receding, and hysteresis) is determined by interactions of the liquid and the solid at the

Figure 2. Illustration of eq 6. The inset is the relationship between Fa and θE.

By varying Fa, different static contact angles θE on smooth surfaces (or the intrinsic contact angle of a specified material) can be obtained. We focus on three types of material whose intrinsic contact angles are 135.2° with Fa = 40 (material A, MA), 114° with Fa = 70 (material B, MB), and 84.7° with Fa = 120 (material C, MC), respectively. Here we should state that in nature smooth surfaces usually have intrinsic contact angles smaller than 120° but in simulation it is a very commendable attempt to set such an unphysical situation. The surface consists of a smooth substrate and some pillars which are used to construct a heterogeneous area on the substrate. We studied different surfaces by arranging one, three, or seven pillars periodically on the smooth substrate. Both of the pillars and the substrate can be specified by different materials. The simulation consists of three groups, and each group includes three cases; for example, case MA/MB_3 indicates three MA pillars on the MB substrate. Here we neglect gravity, though it may have an effect on the transition of droplet states.34 The droplet is driven by a horizontal body force G to slide on the surface. The advancing contact angle θA and receding contact angle θR and the displacement of the droplet front are recorded through the whole process.

3. RESULTS AND DISCUSSION For a systematic comparison, we recorded the mentioned data on all nine surfaces. Case MB/MB_3 is used for describing details in Figure 3, and the other eight cases can be found in Supporting Information. Here we should state that case MB/ MB_3 is a typical case and not a special case. All the descriptions below for this case can also be applied to the other eight cases. The whole sliding process can be divided into five stages clearly according to Figure 3. Five snapshots for each stage are shown in Figure 4. Stage 1 shows a droplet being gently deposited on the substrate and relaxing itself for a while to get a static contact angle of θE, here θA1 = θR1 = θE (subscript 1 denotes stage 1 and so on). Then body force G = 0.05 is exerted on the droplet and drives it to slide on the substrate in stage 2. Here θA2 > θE > θR2, and θA2 and θR2 can be treated as the intrinsic advancing and receding contact angles of the C

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Figure 4. Five snapshots in each stage of case MA/MB_7.

Table 2. Duration Time of Stage 3 (in MDPD Time Units), where N Denotes the Number of Pillars and M Denotes the Material of Pillars N↓ 1 3 7

M→

MA

MB

MC

68.0 156.4 257.5

43.4 86.0 137.2

25.5 46.3 63.8

Figure 5. Snapshots of droplet sliding over a MA or MC pillar (in MDPD time units). For both MA and MC rows, the first snapshot is the initial position of the droplet front, the second is the first instant of the droplet front contacting the left side of the pillar top, the third is the first instant contacting the right side of the pillar top, and the last snapshot is the last instance of the droplet front still on the right side of the pillar top.

Figure 6. Advancing contact angle and displacement of the droplet front evolution versus time. Inset: enlarged views of the green dashed frame for advancing contact angle and the red for displacement.

surface just around the contact line, which is why there exist different slopes between stages 3 and 4 and why the slopes are similar in stages 2 and 4. It is also of interest to notice that in the inset of Figure 6 there exist three relatively flat periods on the displacement curve which implies that the droplet front is pinned intermittently. This phenomenon can be well explained in Figure 5. Take the MA row for example. When the droplet front wets the whole top of a pillar (MA, t = 23.4), it will be pinned and stop moving until enough liquid is accumulated in the upper part of the droplet, and then it will squash the droplet front to wet the side part of the pillar or jump to the next pillar top (MA, t = 66.3). This moving style looks like a worm creeping, and it is also referred to as stick−slip motion in the literature. A more insightful description can be found in refs 20−22.

three-phase contact line alone and that the interfacial area within the contact perimeter is irrelevant.”11 The droplet state in stage 4 is similar to that in McCarthy’s experiments, in which they measured contact angles on surfaces containing spots of different wettability and size within the contact lines of the droplets. In their results, Wenzel’s and Cassie−Baxter’s theories were consistent with none of them. Furthermore, in their later paper,12 they clearly proved this viewpoint again with four simple demonstrations. Whatever the intrinsic wettability or geometrical configuration in the regions remote from the threephase contact line, it has no direct effect on the apparent contact angle. One can also compare our simulation with Wang’s.22 By using a diffuse-interface numerical model, they also performed a droplet sliding simulation on a patterned surface with a contrast wettability (109.4 and 70.6° to be precise). The same phenomenon can be found in their Figure 13e,h, the counterparts of our stages 2 and 4. (2) In stage 3, as shown in Figure 3, both θA3 and θR3 are enlarged, but the gap between them is also larger than that in stage 2. As a result, it is more difficult for the droplet to slide. In Figure 6, the slope of the displacement curve represents the magnitude of the droplet front moving velocity. Indeed, the moving velocity is lower in stage 3 than in stages 2 and 4. Moreover, in stages 2 and 4 the moving velocities are almost equal. This fact indicates that in stage 4 the pillars do not influence the sliding velocity as they do in stage 3. It can be explained by the liquid particle moving pattern35 along the contact line. The interfacial fluid particles which are far away from the three-phase contact line do not move with the droplet. Only the liquid particles which participate in wetting and dewetting the solid surface near the contact line move. This kind of movement is referred to as “tank tread” in ref 32. In other words, most of the interfacial liquid particles are stationary and only particles in the front and rear ends move. The sliding style is determined only by solid

4. CONCLUSIONS In this article, we perform a simple droplet sliding simulation to argue that wettability is a 1D problem. In our simulation, the droplet moving characteristics are well captured, such as the advancing and receding contact angles when sliding, the pinning effect by the heterogeneous area, and the creeping motion style. The analysis of stages 3 and 4 shows that the wetting process is determined by the spreading of the contact line. Once the contact line spreads over a given solid area, whatever the properties (geometrical configuration or intrinsic wettability) the solid area has, it will contribute nothing to the wetting process. Thus we can say that the wettability is a contact-line-based problem and not a contact-area-based problem. Our numerical results can be seen as convincing proof of this viewpoint. This viewpoint may lead to many great potential applications. As mentioned in ref 13, more complex solid surface patterns can be designed to control the droplet shape, which is particularly useful in plastic or metal modeling. In addition, this simulation can provide a new approach for designing and testing droplet-based devices. D

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(19) Yao, X.; Hu, Y.; Grinthal, A.; Wong, T. S.; Mahadevan, L.; Aizenberg, J. Adaptive fluid-infused porous films with tunable transparency and wettability. Nat. Mater. 2013, 12, 529−534. (20) Varagnolo, S.; Ferraro, D.; Fantinel, P.; Pierno, M.; Mistura, G.; Amati, G.; Biferale, L.; Sbragaglia, M. Stick-slip sliding of water drops on chemically heterogeneous surfaces. Phys. Rev. Lett. 2013, 111, 066101. (21) Sbragaglia, M.; Biferale, L.; Amati, G.; Varagnolo, S.; Ferraro, D.; Mistura, G.; Pierno, M. Sliding drops across alternating hydrophobic and hydrophilic stripes. Phys. Rev. E 2014, 89, 012406. (22) Wang, X. P.; Qian, T.; Sheng, P. Moving contact line on chemically patterned surfaces. J. Fluid Mech. 2008, 605, 59−78. (23) Song, J. H.; Sakai, M.; Yoshida, N.; Suzuki, S.; Kameshima, Y.; Nakajima, A. Dynamic hydrophobicity of water droplets on the linepatterned hydrophobic surfaces. Surf. Sci. 2006, 600, 2711−2717. (24) Suzuki, S.; Nakajima, A.; Sakai, M.; Song, J. H.; Yoshida, N.; Kameshima, Y.; Okada, K. Sliding acceleration of water droplets on a surface coated with fluoroalkylsilane and octadecyltrimethoxysilane. Surf. Sci. 2006, 600, 2214−2219. (25) Suzuki, S.; Nakajima, A.; Tanaka, K.; Sakai, M.; Hashimoto, A.; Yoshida, N.; Kameshima, Y.; Okada, K. Sliding behavior of water droplets on line-patterned hydrophobic surfaces. Appl. Surf. Sci. 2008, 254, 1797. (26) Hoogerbrugge, P. J.; Koelman, J. M. V. A. Simulating Microscopic Hydrodynamic Phenomena with Dissipative Particle Dynamics. Europhys. Lett. 1992, 19, 155−160. (27) Groot, R. D.; Warren, P. B. Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation. J. Chem. Phys. 1997, 107, 4423−4435. (28) Warren, P. B. Vapor-liquid coexistence in many-body dissipative particle dynamics. Phys. Rev. E 2003, 68, 066702. (29) Cupelli, C.; Henrich, B.; Glatzel, T.; Zengerle, R.; Moseler, M.; Santer, M. Dynamic capillary wetting studied with dissipative particle dynamics. New J. Phys. 2008, 10, 043009. (30) Chen, C.; Gao, C.; Zhuang, L.; Li, X.; Wu, P.; Dong, J.; Lu, J. A Many-Body Dissipative Particle Dynamics Study of Spontaneous Capillary Imbibition and Drainage. Langmuir 2010, 26, 9533−9538. (31) Chen, C.; Zhuang, L.; Li, X.; Dong, J.; Lu, J. A Many-Body Dissipative Particle Dynamics Study of Forced Water Oil Displacement in Capillary. Langmuir 2012, 28, 1330−1336. (32) Arienti, M.; Pan, W.; Li, X.; Karniadakis, G. Many-body dissipative particle dynamics simulation of liquid/vapor and liquid/ solid interactions. J. Chem. Phys. 2011, 134, 204114. (33) Wang, Y.; Chen, S. Droplets impact on textured surfaces: Mesoscopic simulation of spreading dynamics. Appl. Surf. Sci. 2015, 327, 159−167. (34) Daisiane, M.; Lopes, D. M.; Ramos, S. M. M.; de Oliveira, L. R.; Mombach, J. C. Cassie−Baxter to Wenzel state wetting transition: a 2D numerical simulation. RSC Adv. 2013, 3, 24530−24534. (35) Gao, L.; McCarthy, T. J. Contact Angle Hysteresis Explained. Langmuir 2006, 22, 6234−6237.

ASSOCIATED CONTENT

S Supporting Information *

The advancing contact angle θA and receding contact angle θR of all nine cases. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (grant nos. 51276130 and 10872152). The grants are gratefully acknowledged.

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REFERENCES

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DOI: 10.1021/acs.langmuir.5b00353 Langmuir XXXX, XXX, XXX−XXX