pubs.acs.org/Langmuir © 2010 American Chemical Society
A New Wetting Mechanism Based upon Triple Contact Line Pinning Jianlin Liu,*,† Yue Mei,† and Re Xia*,‡ †
Department of Engineering Mechanics, China University of Petroleum, Qingdao 266555, P. R. China, and ‡ School of Power and Mechanical Engineering, Wuhan University Wuhan, 430072, P. R. China Received September 12, 2010. Revised Manuscript Received November 14, 2010
The classical Wenzel and Cassie models fail to give a physical explanation of such phenomenon as the macroscopic contact angle actually being equal to the Young’s contact angle if there is a spot (surface defect) inside the droplet. Here, we derive the expression of the macroscopic contact angle for this special substrate in use of the principle of least potential energy, and our analytical results are in good agreement with the experimental data. Our findings also suggest that it is the triple contact line (TCL) rather than the contact area that dominates the contact angle. Therefore a new model based upon the TCL pinning is developed to explain the different wetting properties of the Wenzel and Cassie models for hydrophilic and hydrophobic cases. Moreover, the new model predicts the macroscopic contact angle in a broader range accurately, which is consistent with the existing experimental findings. This study revisits the fundamentals of wetting on rough substrates. The new model derived will help to design better superhydrophobic materials and provide the prediction required to engineer novel microfluidic devices.
1. Introduction Most wetting and dewetting properties of solid materials can be dramatically affected by the micro- and/or nanostructures on their surfaces,1,2 such as the leaves of lotus and Lady’s Mantle, which can keep off rain drops and dust as a result of the micro/ nanomorphologies of their surfaces, and this phenomenon is called the “self-cleaning effect” or “lotus effect”.3,4 Similar microtextures are also found on the surfaces of some animals. For example, water striders can stand, walk, and jump on still or even flowing water because their legs can provide a very high superhydrophobic force to propel the body without being wetted.5 This extremely large driving force also benefits from the special microstructures on their legs.6-8 Recently, Wu et al. also reported the superhydrophobicity and superior floating behavior of the mosquito leg, which was ascribed to micrometer-size scales with alternating nanoridge and microgroove pattern.9 In another case, the Stenocara beetle in the Namib desert has special surface microstructures on its carapace that enable its collection of dew.10 Inspired by these functional surface properties in nature, various ultrahydrophobic materials with fractal or hierarchical surface structures have been designed *Corresponding author. E-mail:
[email protected] (J.L. Liu);
[email protected] (R.X.).
(1) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827–863. (2) Quere, D. Physica A 2002, 313, 32–46. (3) Neinhuis, C.; Barthlott, W. Ann. Bot. 1997, 79, 667–677. (4) Otten, A.; Herminghaus, S. Langmuir 2004, 20, 2405–2408. (5) Hu, D. L.; Chan, B.; Bush, J. W. Nature 2004, 424, 663–666. (6) Feng, X. Q.; Gao, X. F.; Wu, Z. N.; Jiang, L.; Zheng, Q. S. Langmuir 2007, 23, 4892–4896. (7) Liu, J. L.; Feng, X. Q.; Wang, G. F. Phys. Rev. E 2007, 76, 066103. (8) Shi, F.; Niu, J.; Liu, J. L.; Liu, F.; Wang, Z. Q.; Feng, X. Q.; Zhang, X. Adv. Mater. 2007, 19, 2257–2261. (9) Wu, C. W.; Kong, X. Q.; Wu, D. Phys. Rev. E 2007, 76, 017301. (10) Parker, A. R.; Lawrence, C. R. Nature 2001, 414, 33–34. (11) Zhai, L.; Cebeci, F. C.; Cohen, R. E.; Rubner, M. F. Nano Lett. 2004, 4, 1349–1353. (12) Lau, K. K. S.; Bico, J.; Teo, K. B. K.; Chhowalla, M.; Amaratunga, G. A. J.; Milne, W. I.; McKinley, G. H.; Gleason, K. K. Nano Lett. 2003, 3, 1701– 1705. (13) Hosono, E.; Fujihara, S.; Honma, I.; Zhou, H. J. Am. Chem. Soc. 2005, 127, 13458–13459. (14) Yoshimitsu, Z.; Nakajima, A.; Watanabe, T.; Hashimoto, K. Langmuir 2002, 18, 5818–5822.
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and prepared to mimic the microstructures of biological materials.11-15 Furthermore, rough surfaces with appropriate wetting properties have been widely used in industry and our daily life, e.g., porous media,16 microfluidic devices,17-19 self-cleaning paints, and glass windows.20 The dependence relationship between the macroscopic contact angle and microstructure of solid surfaces has attracted considerable attention in the past decades (see, e.g., the excellent reviews1,2). The macroscopic contact angle of a liquid drop on a rough surface can be approximately predicted by the classical Wenzel model, the Cassie-Baxter model, or some other established models. The Wenzel model indicates that part of the liquid pierces into the microstructures on the solid substrate completely, and this theory shows that the surface roughness amplifies the hydrophobicity of hydrophobic surfaces as well as enhances the hydrophilicity of hydrophilic surfaces.21 In the case of the Cassie state, the liquid does not go into the microstructures, as there is plenty of trapped air inside it.22 Moreover, Zheng et al. also proposed a mixed model to illustrate their observations, in which the model part of the substrate is in the Cassie state, and the other is in the Wenzel state.23 The Wenzel and Cassie models were so popular and successful to elucidate the superhydrophobic phenomenon of rough substrate, that they were even cited as hot topics in publications with an exponentially increasing during the recent years,24 but there were still some different views and questions about them. As early as in 1945, Pease proposed that the wettability measured by contact angle is a one-dimensional issue and not related with the contact zone.25 Until about 60 years later, Gao et al. also argued (15) (16) (17) (18) (19) 3213. (20) (21) (22) (23) (24) (25)
Onda, T.; Shibuichi, S.; Satoh, N.; Tsujii, K. Langmuir 1996, 12, 2125–2127. Bico, J.; Thiele, U.; Quere, D. Colloids Surf., A 2002, 206, 41–46. Patankar, N. A. Langmuir 2003, 19, 1249–1253. Lee, J.; He, B.; Patankar, N. A. J. Micromech. Microeng. 2005, 15, 591–600. Liu, J. L.; Xia, R.; Li, B. W.; Feng, X. Q. Chin. Phys. Lett. 2007, 24, 3210– Blossey, R. Nat. Mater. 2003, 2, 301–306. Wenzel, R. N. Ind. Eng. Chem. 1936, 28, 988–994. Cassie, A. B. D.; Baxter, S. Trans. Faraday Soc. 1944, 40, 546–551. Zheng, Q. S.; Yu, Y.; Zhao, Z. H. Langmuir 2005, 21, 12207–12212. Gao, L.; McCarthy, T. J. Langmuir 2007, 23, 3762–3765. Pease, D. C. J. Phys. Chem. 1945, 49, 107–110.
Published on Web 11/30/2010
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Figure 1. The classical wetting models: (a) Wenzel model; (b) Cassie model.
that it is not the contact areas but the contact lines that determine the wettability.24 In their study, three types of two-component surfaces containing “spots (surface defects)” in a surrounding field were prepared, and the corresponding macroscopic contact angles were measured. The results showed that the contact angle behavior (advancing, receding, and hysteresis) depends only on the interactions of liquid and solid at the triple contact line (TCL), and is irrelevant to the interfacial area inside the contact perimeter. Although Gao’s experimental results are elegant and faithful, there are still some problems that deserve further attention. For example, what are the limitations and applicable ranges of the Wenzel and Cassie models? When there is a spot inside a droplet, how does one explain the paradox between the experimental result and the Wenzel result? How does one predict the macroscopic contact angle if the TCL is pinned on the rough substrates? Therefore, to address these critical questions, this study aims to revisit the fundamental mechanism of a droplet wetting a rough substrate by analyzing the TCL.
2. Limitations of the Wenzel and Cassie Models Let us first consider how the Wenzel and Cassie models are derived in theory. As shown in Figure 1a, the Wenzel model means that the liquid penetrates into the microstructures on the solid substrate completely. However, in the Cassie model (Figure 1b), there is a composite interface between the liquid and the substrate, i.e., there is vapor in the microstructures of the substrate. Hence, no liquid could go into the defects of the solid. For the Wenzel model, given an increment dx of the contact line, the potential energy change can be normally expressed as2 dΠ ¼ rðγSL - γSV Þdx þ γ dx cos θW
ð1Þ
where γSV, γSL, and γ denote the interfacial tensions of solid/ vapor, solid/liquid and liquid/vapor interfaces, respectively, which satisfy the classical Young’s equation γSV - γSL ¼ γ cos θY
ð2Þ
where θY is Young’s contact angle on a smooth substrate. The symbol θW in eq 1 is the macroscopic contact angle of the Wenzel Langmuir 2011, 27(1), 196–200
Figure 2. A spot (surface defect) in a smooth field that is inside a droplet: (a) Wenzel state; (b) Cassie state.
state, and r is the roughness factor, which is the ratio of the actual area of a rough surface to the geometric projected area. Then according to the condition dΠ/dx = 0, one can deduce the Wenzel model: cos θW ¼ r cos θY
ð3Þ
Similarly, for the Cassie model, the potential energy change caused by dx is dΠ ¼ φs ðγSL - γSV Þdx þ ð1 - φs Þγ dx þ γ dx cos θC
ð4Þ
and the corresponding wetting model is cos θC ¼ φs cos θY þ φs - 1
ð5Þ
where θC is the macroscopic contact angle of the Cassie state, and φs is the percentage of the solid/liquid interface area to the total composite interface area. While these deductions and derivations seemingly satisfy the energy theory, they fail to recognize some critical underlying limitations of the two models. First, these models are applicable only when the dimensions of the microstructures are much smaller than that of the liquid droplet. In the energy expression, only the energy change of the solid/liquid interface is considered, and so the variation of the liquid/vapor interface is neglected and excluded. Second, these two models hold if and only if the microstructures are uniformly distributed on the substrate. It is clearly seen that just under these conditions, the energy change in the range of the increment dx can be written as in eqs 1 and 4. The roughness factor r and the percentage φs are the characteristic parameters for the whole rough properties of the substrate, and DOI: 10.1021/la103652s
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they are both functions of the position of a selected area; henthforth, only when the microstructures are uniformly distributed on the substrate are the two parameters constants. More importantly, the results predicted by the two models are not consistent with some important experimental findings. For a rough spot or some aligned pillars in a smooth field, as is included in a droplet, the macroscopic contact angle is surprisingly observed to be θY,24 but according to the Wenzel model of eq 3, the value of the macroscopic angle is θW > θY for the roughness factor r > 1. Thus we can see that there is clearly a discrepancy between experiments and the prediction of the wetting models.
3. Macroscopic Contact Angle for a Spot Inside a Smooth Field Now we will derive the macroscopic contact angle for the case of a spot or a cave in a smooth field, which are included inside a droplet, as shown in Figure 2. The morphology of the droplet is assumed to be a spherical cap for symmetry, as p itsffiffiffiffiffiffiffiffiffiffiffiffiffiffi characteristic size is much less than the capillary length K - 1 ¼ γ=ðFgÞ, where F is the mass density of the liquid, and g is the gravitational acceleration. For the Wenzel and Cassie models, the radius of the projected surface area of the solid/liquid interface is denoted as a, and the maximum height of the droplet is h. Therefore, the areas of the liquid/vapor interface ALV and the projected surface of the solid/liquid interface ASL are expressed respectively as ALV ¼ πða2 þ h2 Þ
ð6Þ
ASL ¼ πa2
ð7Þ
Here, the roughness factor for the Wenzel model is a function of a, which reads rðaÞ ¼ 1 þ
As - Ap πa2
ð8Þ
where As is the solid/liquid interfacial area of the rough spot, and Ap is its projected area. For the Cassie model, the percentage of the solid/liquid interface is φs ðaÞ ¼ 1 -
Ap πa2
ð9Þ
The total potential energy of the droplet can be written as Π ¼ γALV þ ðγSL - γSV Þeff ASL þ λF
ð10Þ
where (γSL - γSV)eff is equal to r(γSL - γSV) for the Wenzel model, and φs(γSL - γSV)þ(1 - φs)γ for the Cassie model. The parameter λ is a Lagrange multiplier, and F is the constraint condition of the droplet’s volume, which reads F ¼
πhðh2 þ 3a2 Þ - V0 - V ¼ 0 6
ð11Þ
for the Wenzel model, and F ¼
πhðh2 þ 3a2 Þ -V ¼ 0 6
ð12Þ
for the Cassie model, where V is the total volume of the liquid, and V0 is the volume of the liquid penetrating into the microstructures. Using the principle of least potential energy, the condition of ∂Π/∂a = 0 yields 2πγa - 2πγa cos θY þ πλha ¼ 0 198 DOI: 10.1021/la103652s
ð13Þ
that is, λ ¼
2γðcos θY - 1Þ h
ð14Þ
The condition of ∂Π/∂h = 0 can give the result 4hγ þ λðh2 þ a2 Þ ¼ 0
ð15Þ
Substituting eq 14 into 15, one has a2 ¼
1 þ cos θY 2 h 1 - cos θY
ð16Þ
In view of eqs 14 and 16, one can obtain the macroscopic contact angle via the spherical cap assumption cos θW ¼
h=ð1 - cos θY Þ - h ¼ cos θY h=ð1 - cos θY Þ
ð17Þ
i.e., θW = θY. By the same process of derivation, for the Cassie model, we also have θC = θY. Once again, this theoretical analysis agrees with the experimental observation,24 showing that the Wenzel model is not closely appropriate in this case. When the TCL is far away from the position of the spot, the value of the macroscopic contact angle is always the same as that of the Young’s contact angle. This result indicates that the macroscopic contact angle is not dependent on the size of the spot inside a droplet, but is related to the TCL pinning phenomenon when the TCL is located on some microstructures of the substrate.
4. New Model Based upon TCL Pinning When the three-phase contact line is pinned at the roughness, the contact angle is actually the advancing or receding angle because of the hysteresis, in which case the droplet is in a metastable state. In this state, the interfacial area within the contact perimeter is not related to the macroscopic contact angle. Thus, predicting the real contact angle with the pinned TCL becomes a critical problem. When the TCL is pinned at the rough microstructures, i.e., the radius a is a constant, the effective volume of the droplet is a monotonic function of the macroscopic contact angle, namely,14 3Ve ð1 - cos θÞð2 þ cos θÞ ¼ 3 sin θð1 þ cos θÞ πa
ð18Þ
where Ve denotes the liquid volume above the microstructure tops, and Ve = V - V0 for the Wenzel model, Ve = V for the Cassie model. The dependence relationship between Ve and θ may be depicted as in Figure 3. From the figure we can see that, for a given radius a, with the increase of the effective volume, the macroscopic contact angle increases too. This means that when the TCL is pinned, the macroscopic contact angle is dominated merely by the effective volume of the liquid, whether or not there are some microstructures on the substrate. The new model can illustrate some properties predicted by the Wenzel and Cassie models. The comparisons of Wenzel and Cassie models with Young’s equation are shown in Figure 4. In the Wenzel model, when the substrate is hydrophilic (θY < 90°), the macroscopic contact angle θW is less than θY; when hydrophobic (θY > 90°), one has θW > θY. For the Wenzel model, with a hydrophilic substrate, some liquid penetrates into the roughness, then the remaining volume is Ve = V - V0 < V (the total volume), and henceforth the contact angle θW is less than θY. Langmuir 2011, 27(1), 196–200
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Figure 3. The dependence relationship between the effective volume of the droplet and the macroscopic contact angle.
Figure 5. Pillarlike microstructural surfaces: (a) platform of the square structure used in the experiment of Yoshimitsu et al.;14 (b) side view of part a.
Figure 4. The wetting properties of the Wenzel and Cassie models compared with Young’s equation.
For the hydrophobic case, when a droplet is deposited on a rough substrate, the contact area between the droplet and the solid is nearly a point resulting from the repulsion of the solid to liquid. Thus the radius a is a very small variable, and the volume of the liquid piercing into the microstructures is very small, as Ve ≈ V, so 3Ve 3V the nondimensional variable has the relation πa 3 > πa3 . As a e result, the macroscopic angle is bigger than the Young’s contact angle, i.e., θW > θY in the hydrophobic state. For the Cassie model, one always has θC > θY whether the substrate is hydrophilic or hydrophobic. This is because for Cassie model the contact surface is a composite interface including solid/ liquid and liquid/vapor phases. For the vapor phase, the contact angle of liquid is 180°, i.e., vapor can be regarded as a completely superhydrophobic material. Consequently, the TCL can not come across from one solid microstructure to the adjacent one easily, since there is some vapor as a barrier between them. It is different from the Wenzel model, where the TCL totally contacts the solid substrate. In the Cassie state, the effective volume is equal to the total volume of the liquid, but the radius ae is less than a as the TCL is pinned by the solid roughness, then the 3Ve 3V nondimensional variable has πa 3 > πa3 no matter what the sube strate is hydrophilic or hydrophobic. Therefore, it also leads to the macroscopic contact angle θC > θY. Langmuir 2011, 27(1), 196–200
Moreover, our new model is also consistent with some experimental results quantitatively. For instance, Yoshimitsu et al.14 prepared some surfaces with various pillarlike structures by wafer dicing. In their study, the cross-section of the aligned pillar is square, with the side width d = 50 μm, height c, the groove width b = 100 μm, and volume of the liquid drop is V=1 mm3. The detailed microstructures and physical parameters of the substrate are shown in Figure 5. In such conditions, we can obtain the macroscopic contact angle for a given a by eq 18. For the Wenzel state, the effective volume is Ve ¼ V - ðπa2 - nd 2 Þc
ð19Þ
where n is the number of the pillars inside the droplet. The roughness factor is r ¼
ðd þ bÞ2 þ 4dc ðd þ bÞ2
ð20Þ
For the Cassie state, the effective volume is Ve = V = 1 mm3, and the percentage parameter is φs ¼
nd 2 πa2
ð21Þ
Taking the height of the pillar c as 0, 10, 36, 148, and 282 μm, the roughness factor r is calculated as 1, 1.1, 1.2, 2.0, and 3.1 according to eq 20, respectively. The corresponding radius of a is 650, 400, 250, 250, and 325 μm taken from the micrographs, and the effective volume Ve is 1 (smooth substrate), 0.995 (Wenzel state), 1 (Cassie state), 1 (Cassie state), and 1 (Cassie state) mm3, respectivley. All results from the Wenzel model, the Cassie model, DOI: 10.1021/la103652s
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Figure 6. The macroscopic contact angles predicted by the Wenzel and Cassie models, the experiments, and the new model.
the experiments, and the new model are compared in Figure 6. Clearly, the result predicted by our model is consistent with the experiments, which widens the effective range of the Wenzel and Cassie model. Especially for r = 1.1, the macroscopic contact angle calculated from the Wenzel and Cassie model is 117° and 152°, respectively, while the experimental result is 138°. The relative large error can be compensated by the new model, where the predicted result is 140°. In conclusion, the new model does well in explaining the existing experimental results both qualitatively and quantitatively.
5. Conclusion In the present paper, we first discussed the limitations of the classical Wenzel and Cassie model, which are listed as follows: (1)
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The two models are only applicable in cases when the dimensions of the microstructures are much smaller than that of the liquid droplet. (2) The two models hold only when the microstructures are uniformly distributed on the substrate. (3) The results predicted by the two models are not consistent with the experimental results if a rough spot or some aligned pillars or caves are in a smooth field that is included in a droplet. Next, we derived the macroscopic contact angle according to the principle of least potential energy for the case of a spot inside the droplet, and the theoretical analysis result supports the existing experimental data. Furthermore, we proposed that it is the TCL and not the contact area that dominates the macroscopic contact angle. When the three-phase contact line is pinned at the roughness, the contact angle is actually the advancing or receding angle due to hysteresis, in which case the droplet is in a metastable state. In this condition, the interfacial area within the contact perimeter is irrelevant to the macroscopic contact angle. We then constructed a new model based upon the TCL pinning, which can illustrate the existing properties for Wenzel and Cassie model qualitatively. The new model can predict the macroscopic contact angle in a wider range more accurately, which is quantitatively consistent with the experimental results. Our analysis can help in better understanding the wetting mechanism and to design more accurate superhydrophobic materials or new microfluidic devices. Also, our findings may bring inspirations for devising novel superfloating and drag-reducing sinks of aquatic miniature robots, and so forth. Acknowledgment. The project is supported by National Natural Science Foundation of China (10802099), the Doctoral Fund of the Ministry of Education of China (200804251520), and the Natural Science Foundation of Shandong Province (2009ZRA05008).
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