Article pubs.acs.org/Langmuir
Direction Dependence of Adhesion Force for Droplets on Rough Substrates Shan Chen,† Bo Zhang,†,‡ Xiangyu Gao,† Zhiping Liu,*,† and Xianren Zhang*,† †
State Key Laboratory of Organic−Inorganic Composites, Beijing University of Chemical Technology, Beijing 100029, China Key Laboratory of Bioinspired Smart Interfacial Science, Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Beijing 100190, China
‡
ABSTRACT: Determination of solid surface free energy is still an open problem. At present, there are two leading theories on how to determine the adhesion of droplets on rough substrates: one theory stresses that the droplet adhesion force lies in the areas of contact and interaction energy between liquid and solid molecules, whereas the other holds that the length of the edge of drops is essential. In this work, we unify the two theories through lattice Boltzmann simulations and demonstrate that the adhesion force could depend on either the contact area or the contact line, depending on the direction of the adhesion force measured, that is, by vertically separating the two materials or laterally sliding the droplet on the substrate. We reveal that for separating droplets away from rough substrates, the vertical adhesion (pull-off) force depends more significantly on the contact area rather than on the contact line. However, for sliding a droplet on substrates, the lateral adhesion force depends on the contact line while being independent of the contact area.
1. INTRODUCTION Adhesion of droplets on various substrates has been a hot topic in both experimental and theoretical investigations because of its wide applications, including spraying of pesticides, printing, and liquid transportation.1−9 For a sessile liquid drop on a solid substrate, the work of adhesion is usually evaluated indirectly using the contact angle of droplets by assuming that a lower contact angle corresponds to a higher surface energy. The relation between contact angle and work of adhesion for droplets on flat homogeneous substrates can be theoretically given by the Young−Dupre equation10 W = γLV + γSV − γSL = γLV(1 + cos θ )
Alternatively, the adhesion of droplets is also directly related to contact angle hysteresis (CAH), as the CAH is a measure of energy dissipation during droplet movement.17−19 For instance, droplets under external forces (e.g., gravity) can stay on horizontal or tilted surfaces that are balanced by the adhesion force exerted by substrate roughness or heterogeneity.20−22 As a result, the adhesion force is often related to surface tension and the advancing and receding contact angles θA and θR as follows23−30 Fa /R = KγLV(cos θR − cos θA)
where Fa is the adhesion force, R is a characteristic length representing the size of the droplet contour, and K is the proportionality constant. The adhesion force models based on CAH predict the sliding angle as a function of the length of the periphery of the contact area, that is, the contact line, without considering its relationship with the liquid−solid contact area. Experimentally, CAH can be determined using various methods.31−37 The tilted plate method and the centrifugal force balance method measure the contact angle of the droplet by placing a droplet on an inclined plane, and its contact angles are measured when it starts sliding down. The sessile drop method measures the advancing and the receding angles by pumping liquid into and out of a droplet. As a modification of the sessile drop method, the evaporation method38−40 measures the receding angle by evaporating droplets.
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where θ is the droplet contact angle and γLV, γSV, and γSL represent the interfacial tensions for the vapor−liquid, vapor− solid, and liquid−solid interfaces, respectively. For rough or chemically heterogeneous substrates, the contact angle θ in eq 1 should be determined using Cassie11 or Wenzel12 equation depending on the wetting state of sessile droplets13 because equilibrium contact angles appear as the transversality conditions of the appropriate variational problem of wetting.14,15 Equation 1 indicates that the adhesion strength for sessile droplets depends on the contact area over which droplets and solid surfaces contact. In other words, the equation predicts that adhesion will be affected by surface chemistry or roughness away from the contact line. Bormashenko and Bormashenko16 also demonstrated that the energy of adhesion depends on the physical and chemical properties of the entire area underneath the droplet. © XXXX American Chemical Society
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Received: December 30, 2016 Revised: February 9, 2017 Published: February 10, 2017 A
DOI: 10.1021/acs.langmuir.6b04668 Langmuir XXXX, XXX, XXX−XXX
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correspond to 21.13, 60.87, and 887.06, respectively. The details of the applied method can be found in our previous work.48−50 In this work, lengths were expressed in the unit of lattice spacing, and all other quantities used were dimensionless. A simulation box of 100 × 100 × 90 in three dimensions was adopted, with a textured surface placed at the bottom of the box. Many pillars with a size of 3 × 3 × 10 were introduced to coat the substrate surface to represent the substrate roughness (see Figure 1). Initially, a spherical droplet was placed above
In general, there are two leading theories for determining the adhesion of droplets: one theory stresses that the contact areas and liquid−solid interaction energy are both essential, whereas the other emphasizes that the length of the edge of drops is critical. The different models pose a question as to which of these theories can work well in determining the adhesion of droplets on rough or chemically heterogeneous substrates. Agreeing with the latter, for example, Dussan et al.26 found that the adhesion force depends on the shape of the contact line. Paxson and Varanasi38 measured the dynamic behavior of three-phase contact lines at micron-length scales and predicted that the adhesion force is under the influence of the morphology of the contact line. Gao et al.39 suggested that the CAH is related to the three-phase contact line between droplets and solid surfaces alone and that it is irrelevant to the interfacial area. On the other hand, through force analyses, Yong et al.40 found that the adhesion force considers the contact area between the solid and the droplet rather than the contour of the contact line. Escobar and Castillo41 found that a small but finite force of adhesion still exists even if the macroscopic CAH vanished. Note that Tadmor et al.42 managed to measure the lateral adhesion force that acts between a droplet and substrates, by decoupling it from the normal forces. They demonstrated the normal force effect,42 namely, the normal acceleration influencing the measured lateral force, and the effect, as pointed out, obscured the interpretation of the experiments that were typically performed with the tile stage changing both normal and lateral accelerations at the same time.43 In summary, there exist two different adhesion models that are related to either the contact area or the contact line, and this poses a question as to how to determine the adhesion of droplets on a rough or chemically heterogeneous substrate. In the present work, the lattice Boltzmann (LB) method is used to investigate which factor is the droplet adhesion force that lies on: the contact area or the contact line. Our simulation results unify the two existing theories, and the results show that the adhesion force in fact depends on either the contact area or the contact line, depending on the direction of the adhesion force measured, that is, it is to vertically separate the two materials or to laterally slide the droplet on the substrate.
Figure 1. (a) Structure of rough substrate and (b) typical initial configuration with a droplet of a radius of 30 lattice units that was placed above the substrate. W, L, S, and H represent the width, length, spacing, and height of the pillars, respectively. (c,d) After the droplet impacted the substrate and reached its stable state, the hydrophobicity of a chosen patch of the substrate beneath the sessile droplet was changed, the corresponding wetting state was stabilized for a while, and then, the forces of different directions were uniformly applied on the lattices of the sessile droplet to measure the adhesion force: (c) exerting a vertical force or (d) tilting force of 10°.
the center of the substrate. It then moved to the substrate by a small impacting velocity. After the sessile droplet quickly achieved its stable wetting state, a local patch of the substrate beneath the sessile droplet was chosen and its hydrophobicity was changed. An LB simulation run was performed until the corresponding wetting state of the droplet was stabilized. Finally, a force in a given direction was applied on every occupied lattice of the droplet to measure the adhesion force.
2. LATTICE BOLTZMANN (LB) METHOD AND THE SHAN−CHEN MODEL The LB method is a numerically robust technique for simulating various wetting phenomena of microdroplets for which their size is well beyond the scope of atomistic computer simulations. In this work, the three-dimensional, multiphase Shan−Chen-type LB method44,45 based on D3Q19 lattice46,47 was implemented. In this method, a pseudopotential was employed to represent different interactions. The corresponding potential function can be expressed as Vkk ̅(x , x′) = Gkk ̅ (|x − x′|)ψk(x)ψ (x′), in which ψk represents the effective density of component k and the Green function Gkk̅ determines the interaction strength between components k and k̅ for neighboring lattice sites x and x′. In our simulations, the fluid−fluid interaction was kept unchanged to fixed liquid− vapor interfacial tension, whereas the solid−fluid interaction was varied to model different intrinsic contact angles of solid surfaces from 70° to 130°. The bounce-back boundary condition was adopted for the solid boundary. The dimensionless surface tension, gas density, and liquid density
3. RESULTS AND DISCUSSION 3.1. Pull-Out Force Depends on the Contact Area between Droplets and Substrates. To prepare the initial configuration for a sessile droplet, first, we placed a microdroplet on a pillared substrate that had a Young contact angle of 90°. The droplet moved downward the substrate and spread and then retracted until reaching its equilibrium configuration in the Wenzel state. Then, we changed the given size of the central patch of the hydrophilic substrate exactly beneath the droplet into a hydrophobic one that has a contact angle of 130° and observed how the droplet in the Wenzel state responded to the variation of substrate hydrophobicity. As shown in Figure 2, changing the patch of the substrate into a hydrophobic one caused the liquid to be expelled from the patch. We also changed the size of the central hydrophobic patch. However, B
DOI: 10.1021/acs.langmuir.6b04668 Langmuir XXXX, XXX, XXX−XXX
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Figure 2. (a−e) Final configuration of a sessile droplet on the substrate with a central hydrophobic patch of various sizes (with a Young contact angle of 130°) confined within the contact line. From left to right, the patch size and contact angles are (a) 1 × 1 and the apparent contact angle θ = 127.77°; (b) 3 × 3 and θ = 128.08°; (c) 5 × 5 and θ = 128.44°; (d) 7 × 7 and θ = 129.49°; and (e) 9 × 9 and θ = 133.7°. (f) Typical dynamic process for wetting transition from the Wenzel to the Cassie state when we changed the Young contact angle of the 9 × 9 central patch to 130°. Figure 3. (a) Obtained vertical adhesion force as a function of the size of the central hydrophobic patch. (b) Typical snapshot of the droplets under the critical applied force right before detachment. (c) Panel shows how the final droplet morphology changes as the applied force increases.
the droplet contact angle remained unchanged until the transition of wetting state occurred when the patch size was nearly equal to the contact area, or in other words, when the hydrophobic patch reached the contact line region (see Figure 2). This observation demonstrates the critical role played by the contact line on the apparent contact angle.51 When the hydrophobic patch reaches the contact line region, the contact angle changed significantly because of the occurrence of wetting transition. Figure 2f shows the dynamic process for wetting transition from the Wenzel to the Cassie state as a result of the increase in surface energy. The penetrated liquid at the central patch began to recede as the capillary force renders receding of the penetrated liquid in the emptying process. Compared with the adjacent grooves at the boundary between patches of different hydrophobicities, the central area is more hydrophobic, and as a result, the liquid in the middle of the substrate was emptied first, and then, the adjacent groove was emptied. We then used the final configurations obtained above as the initial configurations to study how the adhesion force of the droplet depends on the contact area over which the droplet and the substrate meet. To explore the effect of patch size on the adhesion force, we exerted a variety of pulling forces on the droplet along the Z direction to separate the droplet from the substrate (see Figure 1c). The minimal pulling force required to cause droplet detachment that balances the adhesion force is shown in Figure 3. The figure clearly indicates that the fraction of the surface heterogeneity featured with the size of hydrophobic patch is important, crucially affecting the required pull-off force. As the area of the central hydrophobic patch increases while keeping the contact line for the droplet outer edge unchanged, the required pulling force for droplet detachment decreases significantly. Meanwhile, we also found that the adhesion force for separating Wenzel droplets from substrates should be much stronger than that for droplets in the Cassie state (Figure 3c). For the Wenzel state, parts of the droplet completely penetrated into the pillar intervals and formed a much larger contact area, and thus, the droplet exhibits a much higher adhesive property than that in the Cassie state, as demonstrated experimentally.52,53 We also determined cos θ for the droplets that experienced the critical applied force (see Figure 3a). The figure clearly indicates that the pulling force and cos θ do not follow the same trend for droplets at the Wenzel state. This is because for surfaces with large chemical heterogeneities, the contact-line
pinning effect would result in the deviation of contact angle from the Young equation.48 This again demonstrates that the adhesion force depends on the properties of the entire contact area underneath the droplet, as discussed in another paper.16 In summary, we reveal in this part that if we want to detach sessile droplets from substrates, the force of adhesion is critically influenced by the interfacial free energy that depends on the contact area over which the droplets and the surfaces contact. Moreover, we show that the adhesion force is related to the surface chemistry or roughness away from the contact line. In other words, the force of adhesion that corresponds to the pull-off force required to detach the liquid from the surface, which is experimentally measured by slowly separating the liquid droplet from the contacting surface, would relate to the surface chemistry or the roughness away from the contact line. 3.2. Lateral Adhesion Force Depends on the Contact Line Rather than on the Contact Area. In this part, the Young contact angle of the material making the rough substrate was set to 75°. Then, we changed a patch of the substrate in the central region to be hydrophobic, generating a contact angle of 120°. After the droplet reached equilibrium, we exerted a force with a tilting angle of 10° to make the droplet slide (see Figure 1d). By varying the applied force, we can find out the minimal force that is needed to move the droplet. Figure 4a gives the effect of the variation in the area of the hydrophobic patch and shows the corresponding critical force needed to let the droplet move. The critical applied force for moving the droplet, which balances the adhesion force (defined in eq 2), is found to be located between 0.038 and 0.039. More importantly, the measured adhesion force does not change if we change only the size of the hydrophobic patch, as long as it is far from the contact line. Because all situations in Figure 4a are under the same conditions for the triple contact line, it is reasonable to assume that in this case the hydrophobic area within the contact perimeter does not influence the critical adhesion force to move the droplet. We also investigated the variation in the adhesion force against sliding a droplet as we changed the hydrophobicity near the contact line region by varying the contact angle in the C
DOI: 10.1021/acs.langmuir.6b04668 Langmuir XXXX, XXX, XXX−XXX
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length of the edge of a drop on a surface and the CAH. In this model, the adhesion force depends on surface tension and CAH, predicting adhesion as a function of the length of the periphery of the liquid−solid contact rather than as a function of the contact area. In this work, the LB method was used to investigate whether the adhesion force between droplets and the substrate it attached depends on the contact area or the contact line. Our simulation results unified the two existing theories and demonstrated that the adhesion force could be dependent on either the contact area or the contact line, depending on the direction of the adhesion force measured, that is, to vertically cleave the two materials or to laterally slide the droplet on the substrate. We reveal that for separating a droplet away from rough substrates, the vertical adhesion (pull-off) force depends on the contact area rather than on the contact line. However, for sliding a droplet on substrates, the lateral adhesion force depends on the contact line while being independent of the contact area.
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AUTHOR INFORMATION
Corresponding Authors
*E-mail:
[email protected] (Z.L.). *E-mail:
[email protected] (X.Z.).
Figure 4. (a) Influence of the central area on lateral adhesion force and (b) typical snapshots before droplet sliding. The Young contact angle of the boundary region was set to 75°, whereas that for the central region was 120°. (c) Influence of the contact line region on the lateral adhesion force and (d) Final state of the droplet that exerted a critical force. The Young contact angle for the central patch was 75°, whereas that for contact line region was set to 70°, 80°, 90°, 100°, and 110°.
ORCID
Xianren Zhang: 0000-0002-8026-9012 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work is supported by National Natural Science Foundation of China (Nos. 21276007 and 91434204).
vicinity of the contact line (the region apart from the central patch of 3 × 3) from 70° to 110°, while keeping the central patch with the fixed Young contact angle of 75°. The measured lateral adhesion force is given in Figure 4c, which demonstrates a significant change in the adhesion force as the Young contact angle of the contact line region changes. Therefore, our simulation results show the critical dependence of lateral adhesion force on the contact line region rather than on the contact area far from the contact line. We also determined cos θR − cos θA for the droplets that experienced the critical applied force, and the data are shown in Figure 4c. The figure clearly indicates that the lateral adhesion force and cos θR − cos θA follows the same trend as expected from eq 2. This again demonstrates that the adhesion force depends on the CAH as shown in eq 2. In general, we show here that if we want to slide sessile droplets from rough substrates, the force of adhesion corresponding to the lateral sliding force depends on the CAH. We demonstrate further that the critical lateral adhesion force against the droplet sliding is determined by the interactions of the liquid and the solid at the three-phase contact line alone and that it is nearly irrelevant to the interfacial area within the contact perimeter.
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4. CONCLUSIONS Although the adhesion of droplets on rough substrates is a hot topic, determination of surface adhesion is still an open problem. There are two leading theories on how to calculate the adhesion of droplets. For a sessile liquid drop, the first theory stresses that the adhesion depends on the contact angle of the droplet and the contact area over which the droplet and the solid surface contact. The second theory focuses on the D
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