DOI: 10.1021/la102981e ALangmuir XXXX, XXX(XX), XXX–XXX

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©XXXX American Chemical Society

Is the Cassie-Baxter Formula Relevant?

Y. Kwon,† S. Choi,† N. Anantharaju,‡ J. Lee,† M. V. Panchagnula,‡ and N. A. Patankar*,§

†School of Mechanical and Aerospace Engineering, Seoul National University, Seoul, Korea, ‡Departmentof Mechanical Engineering, Tennessee Tech University, Cookeville, Tennessee 38505, United States, and

§Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208, United States

Received July 27, 2010. Revised Manuscript Received September 2, 2010

The utility of the Cassie-Baxter formula to predict the apparent contact angle of a drop on rough hydrophobicsurfaces has been questioned recently. To resolve this issue, experimental and numerical data for advancing and recedingcontact angles are reported. In all cases considered it is seen that contact angles follow the overall trend of theCassie-Baxter formula, except for the advancing front on pillar type roughness. It is shown that deviations fromthe Cassie-Baxter angle have a one-to-one correlation withmicroscopic distortions of the contact line with respect to itsconfiguration in the Cassie-Baxter state.

1. Introduction

Two approaches to understand the state of a drop on roughsurfaces, in particular hydrophobic surfaces, have been discussedin the literature. The first approach is considered to be based onthe interfacial energy minimization technique that, after homo-genization approximation, leads to Cassie-Baxter1 and Wenzel2

formulas (depending on the wetting configuration) for the ap-parent contact angle of a drop.3 The second approach is con-sidered to be the one that analyzes the shape of the contact lineitself.4,5 Provided equivalent approximations are used, the energy-based and contact-line-based approaches are in fact equivalent.6,7

Minimization of the interfacial energy leads to the equation offorce balance at the contact line.

The apparent contact angles, according to Cassie-Baxter(C-B) and Wenzel formulas, represent an average (or homo-genized) measure of the minimum-energy state experienced by acontact line.6,7 While these formulas provide useful guidance todesign rough superhydrophobic surfaces,6,8 they do not capturehysteresis.9 It has been proposed that the degree of deviation ofthe advancing or receding contact angles from the C-B formula(in this work the focus is only on theC-B case) is ameasure of theadditional energy associatedwith pinning/depinning events of thecontact line.9,10Motivated by some experimental data, one view isto propose that the deviation of contact angles, from the predic-tion based on the C-B formula, is so substantial that using such

homogenized interfacial-energy-based models is not practicallymeaningful.5,11 There have been many insightful discussions inthe literature but with incomplete resolution of this criticallyimportant and fundamental issue in this field.5-7,11

This paper addresses the above issue by enquiring if the C-Bformula is practically relevant. Two sets of experimental data andone set of numerical data for advancing and receding contactangles are reported and compared to predictions from the C-Bformula. Pillar and hole type rough surfaces are considered in theexperiments, whereas a flat checkerboard pattern with differentcontact angles is considered in numerical simulations. Section 2briefly describes the experimental and numerical methods andpresents the key results. Discussion are presented in section 3, andconclusions are summarized in section 4.

2. Methods and Results

2.1. Experimental Section. Square (24 μm � 24 μm) pillarand hole arrays (Figure 1) were fabricated on a silicon surface byusing deep reactive ion etching (RIE). A range of roughnessgeometries were fabricated by varying the spacing between thepillars and holes in the respective cases. The height and depth ofthe pillars and holes were typically 56 μm. The silicon surface wascoated with HDFS (heptadecafluoro-1,1,2,2-tetrahydrodecyltri-chlorosilane, Gelest #SIH5841.0). The advancing and recedingangles, on a flat surface of this silanizedmonolayer,were 111� and97�, respectively. A water droplet of 10 μL was gently placed onthe fabricated sample. The droplet, in a C-B state, was thenforced to slide on the rough surface byusing a probe tip (Figure 2).Contact angles at advancing and receding fronts in the state ofimpending motion were measured from video images (Figure 2)and comparedwith predictions from theC-B formula (Figure 3).TheC-B angles were calculated as follows. The area fractionφ ofthe solid at the liquid-solid contact underneath the droplet wascalculated according to the following equations:

φ ¼ a2

ðaþ bÞ2, for pillar geometry

¼ 1-a2

ðaþ bÞ2, for hole geometry

)ð1Þ

where a and b are as defined in Figure 2. For geometriesconsidered here, a= 24 μm and b is varied as shown in Figure 1.

*Corresponding author. E-mail: n-patankar@northwestern.edu.(1) Cassie, A. B. D.; Baxter, S. Wettability of porous surfaces. Trans. Faraday

Soc. 1944, 40, 546–551.(2) Wenzel, R. N. Surface roughness and contact angle. J. Phys. Colloid Chem.

1949, 53(9), 1466–1467.(3) Alberti, G.; DeSimone, A. Wetting of rough surfaces: a homogenization

approach. Proc. R. Soc. London, Ser. A 2005, 461(2053), 79–97.(4) Extrand, C. W. Criteria for ultralyophobic surfaces. Langmuir 2004, 20(12),

5013–5018.(5) Gao, L. C.; McCarthy, T. J. HowWenzel and Cassie were wrong. Langmuir

2007, 23(7), 3762–3765.(6) Marmur, A.; Bittoun, E. When Wenzel and Cassie Are Right: Reconciling

Local and Global Considerations. Langmuir 2009, 25(3), 1277–1281.(7) Panchagnula, M. V.; Vedantam, S. Comment on how Wenzel and Cassie

were wrong by Gao and McCarthy. Langmuir 2007, 23(26), 13242–13242.(8) Patankar, N. A. Mimicking the lotus effect: Influence of double roughness

structures and slender pillars. Langmuir 2004, 20(19), 8209–8213.(9) Patankar, N. A. Hysteresis with Regard to Cassie and Wenzel States on

Superhydrophobic Surfaces. Langmuir 2010, 26(10), 7498–7503.(10) Joanny, J. F.; Degennes, P. G. A Model for Contact-Angle Hysteresis.

J. Chem. Phys. 1984, 81(1), 552–562.(11) Gao, L. C.; McCarthy, T. J. An Attempt to Correct the Faulty Intuition

Perpetuated by theWenzel and Cassie “Laws”. Langmuir 2009, 25(13), 7249–7255.

B DOI: 10.1021/la102981e Langmuir XXXX, XXX(XX), XXX–XXX

Article Kwon et al.

The values of φ are listed in Figure 2. The C-B angles arepredicted and plotted in Figure 3 according to the followingequation:

cos θCass ¼ φ cos θðadv=recÞ þφ- 1 ð2Þwhere θCass is the predicted C-B angle. θadv=111� is theadvancing angle on the flat surface that is used if it is an advancingfront, andθrec=97� is the receding angle on the flat surface that isused if it is a receding front.

2.2. Numerical. Numerical simulations of advancing andreceding fronts on a flat checkerboard patterned surface weredone (Figure 4). Each square in the pattern was 1 cm � 1 cm.The advancing/receding angles were 145�/125� and 135�/115� forthe two domains on the patterned surface. The simulations were

based on a novel technique that couples Surface Evolver12 with aphase-fieldmodel for contact linemotion.13 Themethod is brieflydescribed below.

Figure 1. SEM images of surfaces with pillar and hole geometries. The pillar and hole spacings are in increasing order from left to right.

Figure 2. Snapshots of droplets being dragged to the left or to theright on surfaces with pillar and hole geometries. The inset in eachsnapshot shows the measured angles at the left and right fronts ofthe droplet, respectively. Depending on which way the droplet isbeing dragged, these angles are the advancing or receding angles.

Figure 3. Comparison between experimentally measured advan-cing and receding contact angles, for the pillar (a) and hole (b) typesurface roughness, with predictions based on the C-B formula.Lines joining the experimental data points are for visual aid only.

Figure 4. Schematic of the numerical simulation setup.The liquid-vapor interface is moved as shown by the arrow. One edge of theinterface moves on a flat surface with checkerboard pattern ofsurface wettability (darker shade is more hydrophobic).

(12) Brakke, K. A. The Surface Evolver, 1992 (http://www.susqu.edu/brakke/evolver/evolver.html).

DOI: 10.1021/la102981e CLangmuir XXXX, XXX(XX), XXX–XXX

Kwon et al. Article

Surface Evolver (SE)12 is an interactive program for the studyof surfaces shaped by surface tension and other energies. The userspecifies an initial surface, the constraints that the surface shouldsatisfy throughout the evolution, and an energy function thatdepends on the surface. SE then modifies the surface, subject tothe given constraints, so as to minimize the energy. This mini-mization of energy is performed through a gradient descentmethod. The numerical results are based on an approach thatconstitutively incorporates contact angle hysteresis. Therefore,the advancing and receding angles of the twomaterials that makeup the chessboard are all inputs to the model.

In the current problem, the geometry is as shown in Figure 4.For a given volume V of the liquid, the liquid-air surface isevolved until the total energy, with surface tension and gravita-tional energies being two components, is minimized. On thecontact line (see Figure 4) the local contact angle is satisfied ateach point. This is ensured by using a phase-field model forcontact line motion.13 The advancing front scenario is emulatedby increasing the liquid volume V in steps and calculatingthe minimum-energy state at each volume. Similarly, the reced-ing front is emulated by decreasing the liquid volume V insteps. More details of the simulation technique can be foundelsewhere.13,14 The apparent contact angle is calculated byaveraging the local contact angle (which is calculated from thelocal meniscus shape obtained from the simulation solution)along the contact line on the checkerboard surface. This is doneat each liquid volume during advancing and receding simula-tions. The calculated contact angles are plotted in Figure 5where the inset shows the shapes of the contact line. Thecalculated apparent contact angle is compared with the C-Bformula in Figure 5.

3. Discussion

Two issues are discussed. First is whether the C-B formula isrelevant at all to model the apparent contact angles of drops onrough hydrophobic surfaces (section 3.1). Second pertains to the

deviation of the observed contact angle with respect to the C-Bformula (section 3.2).3.1. Evidence of Relevance of the Cassie-Baxter For-

mula.We compare the predictions for the apparent contact anglewith respect to the area fraction of the solid to experimentalobservations. Two fundamentally different kinds of roughnessfeatures, a pillar type geometry, which results in a discontin-uous contact line, and a hole type roughness, which results in acontinuous contact line, are considered. Figure 3 shows that inall cases considered, except for the advancing front on pillartype roughness which has strong pinning and therefore a con-stant contact angle, the apparent contact angles follow theoverall trend of the C-B formula. Replotting past data forpillar type geometry in the same way leads to similar conclu-sions.9 (Note: it was not always possible to get the entire rangeof area fraction from prior data;9 no systematic data wereavailable for hole type roughness.) Deviations from the C-Bformula do not significantly compromise the primary trend,except for one case among those considered by us. Thus,disregarding the C-B formula as irrelevant is not warrantedin general.3.2. ContactLineDistortionand theCassie-BaxterAngle.

The goal of this section is twofold. First is to emphasize that theC-B angle in general corresponds to a “microscopic” state, to bereferred to as theC-B state of the contact line. The secondgoal is tocorrelate deviations of the observed angle, with respect to the C-Bangle, to distortions of the contact line away from its C-B state. Tothis end, numerical simulations were performed, as explained insection 2.2 (Figure 4). A checkerboard pattern was chosen becauseit is an ideal case to illustrate the two points listed above.

It is seen from Figure 5 that in each of the advancing andreceding cases the contact line configurations A andE correspondto theC-B state where the apparent contact angle is in reasonableagreement with the C-B angle. When the advancing front pins atthe edge of the more hydrophobic patch on the checkerboardpattern, the microscopic state of the contact line changes (seecontact lines B andC inFigure 5a) from theC-B state to a pinnedstate. Correspondingly, the apparent angle deviates away from theC-Bangle.When the contact line depins (transition of the contactline from C to D to E in Figure 5a), the C-B state of the contactline is recovered and the apparent contact angle drops back to theC-B angle within the margin of the numerical error. Similarbehavior is observed in case of the receding front in Figure 5b. Thepresence of the “ideal” C-B state of the contact line, which resultsin an apparent angle nearly equal to the C-B angle, is also seen inprior experimental results (see Figure 2 in ref 15). Thus, the C-Bformula is not divorced from the configuration of the contact line.

The C-B state of the contact line can, thus, be described as astate where the microscopic deformation of the contact lineremains unchanged during the advancing or receding process inthe absence of pinning. In this scenario the contact line onlyundergoes a general translational transformation. However,when pinning/depinning occurs, the shape of the contact lineundergoes a change. This causes energy to be stored during thepinning process and dissipated during the depinning process.These pinning/depinning processes are the root cause of devia-tions from the C-B state and therefore hysteresis. As long as suchpinning/depinning processes can be avoided, it is expected that theC-B formula will be obeyed. This argument is also consistentwith that set forth originally by Cassie and Baxter themselves.1

Figure 5. Deviation of the numerically calculated values of advan-cing (a) and receding (b) angles compared to the C-B anglescorrelate with pinning/depinning events as shown in the inset.The simulation setup is as shown in Figure 2. In this case theC-B formula is cos θCass = φ1 cos θ1(adv/rec) þ φ2 cos θ2(adv/rec),where subscripts 1 and 2 denote the respective wetting domains onthe checkerboard pattern.

(13) Vedantam, S.; Panchagnula, M. V. Phase field modeling of hysteresis insessile drops Phys. Rev. Lett. 2007, 99 (17): p Artn 176102.(14) Anantharaju, N. Wetting by moving triple lines. In Mechanical Engineer-

ing; Tennessee Technological University: Cookeville, TN, 2010.

(15) Choi, W.; Tuteja, A.; Mabry, J. M.; Cohen, R. E.; McKinley, G. H. Amodified Cassie-Baxter relationship to explain contact angle hysteresis andanisotropy on non-wetting textured surfaces. J. Colloid Interface Sci. 2009, 339(1), 208–216.

D DOI: 10.1021/la102981e Langmuir XXXX, XXX(XX), XXX–XXX

Article Kwon et al.

The corresponding mathematical derivation relies on the assump-tion that each infinitesimal (advancing or receding) motion of thetriple line causes the same area fraction to be wetted or dewetted.This is possible if the microscopic shape of the contact line ispreserved.Thus, a surfacematerial that is devoidofpinning siteswillshow no hysteresis and will be in agreement with the C-B formula.Joanny anddeGennes10 show that surfaceswill be devoid of pinningif the heterogeneities are weak/small for pinning to happen. It hasalso been demonstrated computationally and theoretically that ifthe size of a droplet is large relative to the heterogeneity size onthe surface, then the agreement with Cassie-Baxter and Wenzelformulas is improved due to weak pinning.16,17

4. Conclusions

In summary, theC-B formula is found to be relevant in generalto model the apparent contact angle on rough surfaces. The actual

static contact line configuration is one of the states that locallyminimizes the interfacial energy of the system.Themethodbased onminimization of interfacial energy is not incorrect but is in factformally equivalent to force-balance-based approach. The C-Bformula uses an average measure of the interfacial energy of thesurface. Therefore, it is an approximation anddoes not represent theactual contact line configuration during advancing or receding ifpinning occurs. However, it is improper to consider the C-Bformula to be unrelated to contact line configuration. It is shownthat the C-B formula approximately represents a microscopicconfiguration of the contact line.Deviations of the apparent contactangle from the C-B angle directly correlate with distortions of theactual contact line, due to pinning, from its C-B state.

Acknowledgment. NAP acknowledges support from the Ini-tiative for Sustainability and Energy at Northwestern University.YK, SC, and JL acknowledge support from the Pioneer ResearchCenter Program (Grant No.: 2010-0002217) and the NationalResearch Laboratory Program (Grant No. 20100018904), boththrough the National Research Foundation of Korea funded bythe Ministry of Education, Science and Technology.

(16) Brandon, S.; Haimovich, N.; Yeger, E.; Marmur, A. Partial wetting ofchemically patterned surfaces: The effect of drop size. J. Colloid Interface Sci.2003, 263(1), 237–243.(17) Wolansky, G.; Marmur, A. Apparent contact angles on rough surfaces: the

Wenzel equation revisited. Colloids Surf., A 1999, 156(1-3), 381–388.