validity of the equations for the contact angle on real ... · 2.3. surface energy and surface...

© 2013 The Korean Society of Rheology and Springer 175

Korea-Australia Rheology Journal, Vol.25, No.3, pp.175-180 (August 2013)DOI: 10.1007/s13367-013-0018-5

www.springer.com/13367

Validity of the equations for the contact angle on real surfaces

Kwangseok Seo, Minyoung Kim and Do Hyun Kim*Department of Chemical and Biomolecular Engineering, Korea Advanced Institute of Science and Technology,

Daejeon, 305-701, Republic of Korea

(Received July 31, 2013; accepted August 6, 2013)

The wetting property between liquid and solid is very important in many industries besides natural sys-tems. The simplest method to determine the wetting property is just dropping a liquid drop on a solidsurface and measuring a contact angle from the shape of the drop. Since the Young's equation has beenused as a basic equation to relate a contact angle and interfacial tensions over 200 years, it is importantto understand a derivation and limits of the Young's equation. We derived the Young's equation followingenergy minimization with simple mathematics. By expanding the derivation, the modified forms of theCassie-Baxter equation and the Wenzel equation were also derived. From analyses of the derivations, itwas deduced that a contact angle on an ideal surface is only related to the infinitesimal region in thevicinity of contact line, not internal area surrounded by the contact line. Although the Cassie-Baxtermodel and the Wenzel model were not rigorously built, they have been widely used for a supe-rhydrophobic surface, because the apparent forms are similar to those of rigorously derived models whenthe contact line can easily move on the surface.

Keywords: Young’s equation, Wenzel equation, Cassie-Baxter equation, contact angle, energy minimization.

1. Introduction

Wetting typically means spreading of liquid on a solidsurface. Wetting plays an essential role in many industries,such as painting (Song et al., 2011), printing (Wang et al.,2004; Zahner et al., 2011; Li et al., 2012), photolithog-raphy (Bauer et al., 1996), cosmetic (Lodge and Bhushan,2006; Wagner et al., 2011; Murdan et al., 2012), microf-luidics (Gau et al., 1999), agriculture (Xu et al., 2011),fabric industry (Leng et al., 2009), besides natural systems(Sun et al., 2009; Byun et al., 2009; Prakash et al., 2008).

In many instances including the real life issues, both thewetting property and the rheological property are important.For example, when painting on a plastic substrate, we usu-ally spread paint with a brush. Although the spreading ofpaint is related to rheology, the stable maintenance of painton the surface is related to wetting. If watercolor with highsurface tension is used on the hydrophobic plastic substrate,dewetting will occur. Both the wetting properties and therheological properties are important for comprehensiveunderstanding of complex flow systems with solid walls.

Dropping a liquid drop on a solid surface is the simplestmethod to measure the degree of wetting. Depending onthe wetting property between them, the drop shows a spe-cific contact angle. The Young's equation has been used as

a basic theoretical equation to quantify the wetting prop-erty in terms of interfacial tensions without sufficient ver-ifications. In spite of misunderstanding involved in theYoung’s equation, it has been accepted for a long time(Gao and McCarthy, 2007).

In order to interpret the meaning of a contact angle inmacroscopic view, the Young's equation has been derivedin detail by energy minimization with simple mathemat-ics. By expanding the derivation, the Cassie-Baxter equa-tion and the Wenzel equation has been also derived. Somecautions for practical application and limitations of theseequations followed the derivations.

2. Scope and Terminology

2.1. Surface tension as bulk property in macro-scopic view

Surface tension is the physical property directly relatedto the force acting between molecules. The so-called sur-face tension is the value measured in macroscopic system,such as pendant drop method, du Nouy ring method andpressure measurement method. It should be mentionedthat although the surface tension is originated from molec-ular interaction, the surface tension measured in macro-scopic system is meaningful only in macroscale. Analysisof the microscopic system with the bulk property is likelyto cause an error. In order to analyze surface tension-related phenomena exactly in microscopic system ofnanometer scale, additional correction factors such as linetension and disjoining pressure are required. However,

# This paper is based on an invited lecture presented by the corre-sponding author at the 13th International Symposium on Applied Rhe-ology (ISAR), held May 23, 2013, Seoul.*Corresponding author: [email protected]

Kwangseok Seo, Minyoung Kim and Do Hyun Kim

176 Korea-Australia Rheology J., Vol. 25, No. 3 (2013)

these factors can be neglected in a system with a scale ofmore than micrometers. In this paper, these factors areneglected assuming the length scale of the system isgreater than a micrometer.

2.2. Interfacial tension and thermodynamicallyunstable energy state

The molecules in bulk liquid pull each other in everydirection. Thus, net force acting on a molecule is zero.There exists no extra tension between molecules. However,when two kinds of immiscible molecules are separated bythe interface, the intermolecular force at the interface is dif-ferent from the force acting on the molecules in the bulk.As a result, net force acting on the molecule at the interfaceis not zero and there exists a tension. The interfacial ten-sion will be determined depending on the degree of forcemismatching acting on the molecules at the interface.

The interface between two immiscible phases is knownto be at an unstable state with a higher energy. In order tostabilize the high energy state thermodynamically, thenumber of molecules at the interface, i.e. interface area,should be reduced as much as possible. A shape that sat-isfies this condition is a sphere in three-dimensional sys-tem. It is well known that the water droplets can form aperfect sphere in space.

2.3. Surface energy and surface tensionSurface energy and surface tension have been used inter-

changeably in a field related to wetting phenomena. Sur-face energy (J/m2) is basically a scalar representing theenergy, having no direction but only a magnitude. But,surface tension (N/m) is basically a vector representing theforce. Thus, surface tension has both magnitude and direc-tion. Although these units are used interchangeably witheach other (J/m2 [=] N/m), the difference in physicalmeaning between these is very important. The origins ofthe surface tension and surface energy are the same. Bothof them are originated from force mismatching on themolecules at the interface. Surface energy and surface ten-sion are generally used for solids and liquids, respectively.They are distinguished by the difference in liquidity ofmaterials, schematically shown in Fig. 1.

Solid below the melting point is at a state where atomsor molecules are tied firmly by the collective cohesionbetween them, showing no liquidity. As previously men-tioned, intermolecular force is mismatched at the interfaceand the force acting on the molecules at the interface is avector that has both magnitude and direction. However, ifthe movement of the molecules is locked as in a solid, thedirectionality on the solid surface is meaningless. Thus, onthe solid surface, surface energy is used that reflects onlythe magnitude of mismatched force at the interface.

Liquidity is a characteristic of a liquid, where the mol-ecules can freely move. Thus, in the case of liquid, surface

tension is used that reflects both the magnitude and thedirection of mismatched force at the interface.

For example, when water is at room temperature, sur-face tension will be used. But, when the temperature fallsdown below zero and the water becomes a solid ice, sur-face energy will be used.

3. Detailed derivation of the Young’s equationon an ideal surface

3.1. Previous description of the Young’s equationTo quantify a contact angle in terms of interfacial ten-

sions, the Young’s equation has been widely used (Albertiand Desimone, 2005; Quéré, 2005; Bachmann et al.,2006; Bachmann and McHale, 2009). As shown in Fig. 2,direction and magnitude of each force are represented bythe direction and the length of the arrow, respectively.From vector calculation, it can be written as follows.

, (1)

where , , and are liquid/gas surface tension, solid/liquid interfacial energy, solid/gas surface energy, respec-tively. Eq. (1) is the conventionally used Young's equation.

3.2. Assumptions for derivation of the Young’sequation

It is necessary to derive the Young’s equation in a morerigorous way (Whyman et al., 2008; Bormashenko, 2009;Xu and Wang, 2010). Young's equation has been derivedin detail from a thermodynamic point of view. For the der-ivation, the following assumption is used: First, becausethe surface is ideal (i.e. rigid, flat, insoluble, chemicallyhomogenous, and smooth), there is no contact angle hys-teresis. It means that the contact line can freely move

sl cos+ so=

sl so

Fig. 1. (Color online) Surface tension as a vector and surfaceenergy as a scalar.

Fig. 2. (Color online) General scheme for the description of theYoung’s equation.


Korea-Australia Rheology J., Vol. 25, No. 3 (2013) 177

around. Second, in zero gravity, the drop shape on the sur-face is part of the sphere, i.e. spherical cap, because unsta-ble interface minimizes its area for stability.

3.3. Process for the static contact angleAs shown in Fig. 3, when the drop spreads or contracts,

the wetting surface will vary with a contact angle givingthe contact angle as a one-to-one function of the wettingsurface area. Because the contact line can move freely onan ideal surface, the drop can also change freely its shapeto minimize the energy of the system. Thus, at the moststable state, the shape of the drop will be always a sectionof sphere and no residual force will exist at the contactline.

3.4. Derivation of the Young’s equationBy employing a thermodynamic approach, the equation

can be derived only with energy terms without consid-ering the size and direction of the force (Whyman et al.,2008). As shown in Fig. 4, when a drop is on the surface,the volume of the cap is

or and the

surface area of the cap is or .The total energy of the system is as follows.

(2)

The variation of the energy is as follows.

(3)

The variation of the energy is zero at equilibrium, i.e. dE= 0. Dividing both sides by dR,

(4)

means that when the shape of the drop is changed

infinitesimally, there is no energy variation. At this time,the drop shape corresponds to the state of minimum total

energy. Here, in order to obtain , the condition that dV

= 0 are used, because the volume of the drop is constant.The volume of the drop is as follows,

(5)

can be obtained from the conditions that dV = 0.

can be written as,

(6)

Substitution Eq. (6) into Eq. (4) gives rise to

(7)

By rearranging the above equation, the Young's equationis derived.

(8)

During the derivation of the equation, only energy termswere used. Contact angle was determined when the droptakes the minimum energy.

4. Detailed Cassie-Baxter equation and Wenzelequation

4.1. Previous Wenzel model and Cassie-Baxtermodel

There are two models to describe the contact angle on

Vh6

------ 3r2 h2+ = VR3

3--------- 1 cos– 2 2 cos+ =

A 2Rh= A 2R2 1 cos– =

E r2 sl so– 2Rh + R sin 2 sl so– = =

2R2 1 cos– +

dE 2R sl so– dR R dcossin+2sin =

2 1 cos– dR R dsin+ +

dEdR------ 2R sl so– R d

dR------cossin+

2sin⎝ ⎠⎛ ⎞=

2 1 cos– R ddR------sin+⎝ ⎠

⎛ ⎞+ 0=

dEdR------ 0=

ddR------

VR3

3--------- 1 cos– 2 2 cos+ =

ddR------ d

dR------

ddR------ 1 cos– 2 cos+

R 1 cos+ sin-----------------------------------------------–=

sl so– 2 1 cos– 2 cos+ 1 cos+

-----------------------------------------------–⎝ ⎠⎛ ⎞cos+sin⎝ ⎠

⎛ ⎞

2 1 cos– 1 cos– 2 cos+ 1 cos+

-----------------------------------------------–⎝ ⎠⎛ ⎞+ 0=

sl cos+ so=

Fig. 3. (Color online) Process of shape change of a drop to givea static contact angle. The drop will change its shape to minimizethe energy of the system.

Fig. 4. (Color online) Cross-section of a liquid drop on an idealsurface.



a real surface. The significant differences between thereal surface and the ideal surface are heterogeneity androughness. One of two models is the Wenzel model onthe chemically homogenous surface but with roughness(Wenzel, 1936). The other one is the Cassie-Baxter modelon the chemically heterogeneous surface without rough-ness, i.e. perfect flat and smooth (Cassie and Baxter,1944). In the Wenzel model, the surface roughness r isdefined as the ratio of the actual area over the apparentsurface area of the substrate and apparent contact anglecan be written as:

, (9)

where is the apparent contact angle and is the equi-librium contact angles on a ideal solid.

In the Cassie-Baxter model, f1 and f2 are the area frac-tions of solid and air in composite materials and apparentcontact angle can be written as:

, (10)

where is the apparent contact angle and is the equi-librium contact angles on the solid.

In the case of the Wenzel model, the surface roughnessenhances the wettability property of the original surface.Surface roughness makes hydrophilic surface more hydro-philic and hydrophobic surface more hydrophobic. In thecase of the Cassie-Baxter model, the area fraction of thesolid under the drop is an important factor.

These equations have been used over half a century.Recently, with active superhydrophobic surface research,these equations are more widely used. However, theseequations should be rigorously derived, as Gao andMcCarthy pointed out the erroneousness of these modelsexperimentally (Gao and McCarthy, 2007).

4.2. Modified derivation of Wenzel modelFor the Wenzel model in Fig. 5, the radius of smooth

region under the drop is b and the radius of the drop is a.When b=0, the surface become a uniform rough surface.It is assumed that the contact line of drop is positioned onthe rough region of the surface.

As shown in Fig. 5, the total energy of the system is asfollows.

(11)

where As is the interfacial area of the drop contacting withair and K is the roughness factor as defined above. Thevariation of the energy is as follows.

(12)

Rearranging above equation gives rise to

(13)

It should be mentioned that the terms related to b2 were

deleted, because they are constant. The internal area is notrelated at all in obtaining the contact angle. It is totally dif-ferent from previous Wenzel model where the internal sur-face area is considered to be important. From dE=0 atequilibrium and Eq. (6), we have

, (14)

where is the equilibrium contact angle on solid and is the apparent contact angle.

Previous Wenzel model and this equation look similar.However, there is an important difference. It is the def-inition on the roughness factor K. While the roughnessfactor r in Wenzel model was obtained from the total areaunder the drop, roughness factor K in Eq. (14) is relatedto only local region near contact line. Also, the property ofinternal surface under the drop does not affect the finalcontact angle. Fig. 6 shows the infinitesimal region da todefine roughness factor K in the vicinity of contact lineand any crack, hump, or defect under a drop does notaffect a contact angle.

4.3. Modified derivation of the Cassie-Baxter modelIn the Cassie-Baxter model in Fig. 7, the composite sur-

face consists of two kinds of ideal surfaces. The area frac-tions of red region and yellow region are f1 and f2,respectively. As shown in Fig. 7, the total energy of thesystem is as follows

(15)

Here, As is the interfacial area of the drop contactingwith air. The variation of the energy is as follows.

(16)

cos r Ycos=

Y

cos f1 Y f2–cos=

Y

E b2 sl so– K a2 b2– sl so– As+ +=

dE d b2 sl so– d K a2 b2– sl so– d As + +=

dE K sl so– d a2 d AS +=

K Ycos cos=

Y

E f1a2 sl f1so f1

– f2a2 sl f2so f2

– As+ +=

dE C1d R2 2sin C2d R2 1 cos– +=

Fig. 5. (Color online) Schematic of the Wenzel model. The radiusof the drop is a and the radius of the smooth region is b.

Fig. 6. (Color online) Local roughness factor K that is defined inthe vicinity of contact line. The state of surface inside the dropdoes not affect the contact angle.


Korea-Australia Rheology J., Vol. 25, No. 3 (2013) 179

Here, ,

From dE=0 at equilibrium and Eq. (6), we have

(17)

This equation and previous Cassie-Baxter model alsolook similar (Assuming f2 as a fraction contacting with airand ). However, like modified Wenzel model, thestate of surface inside the drop does not affect the finalcontact angle. Thus, only the area fractions in the vincinityof contact line are important.

5. Deductions from the derivations

From the derivation of the Young’s equation, it can bededuced that a contact angle on ideal surface is determinedby only the infinitesimal region in the vicinity of contactline. In forming the contact angle from the Young’s equa-tion, only three surface energies near contact line are used.When the contact angle satisfies the Young’s equation, thetotal energy of the system is minimized. Unless the Young’sequation is satisfied at the contact line, the total energy ofthe system is not at a local minimum and there exists resid-ual force at the contact line. Thus, the shape of the drop willbe changed until the residual force disappears, when thecontact line is assumed to move freely on the ideal surface.It implies that the Young’s equation is a necessary conditionat the contact line for the minimization of the total energy.Thus, the contact angle does not depend on surface-energy-independent factors, such as gravity, drop shape and size,and curvature of substrate. This result is also applicable tothe Wenzel model or the Cassie-Baxter model.

Regions inside the contact line can not affect a contactangle, as verified during the derivations. An equilibriumcontact angle is the value which is determined from thelowest point of the total energy of a system. If each drophas a different pattern inside a contact line, the total energyin the vicinity of the contact angle is translated in parallelabout the y-axis, as shown in Fig. 8. At this time, the con-tact angle satisfying the lowest energy is not changed,because the contact angle is independent of the absolutevalue of the total energy.

During the derivations of the Young’s equation, theWenzel equation, and the Cassie-Baxter equation, the con-

tact line was assumed to move freely on the ideal surface,i.e. no contact angle hysteresis. This assumption may beviolated under the actual situation on the real commonsurface with contact angle hysteresis. The contact line onthe real surface cannot move freely on the surface. Thestatic contact angle also has some values between twoextreme values, i.e. a receding angle and an advancingangle. So, it is hard to describe the actual surface with asingle equation such as the Young’s equation, the Wenzelequation or the Cassie-Baxter equation. However, becausesuperhydrophobic surface has a very low contact anglehysteresis, the contact line can easily move on the supe-rhydrophobic surface and the surface has small range ofstatic contact angles. This is why the Cassie-Baxter modeland the Wenzel model have been widely accepted for thesuperhydrophobic surface, even if their derivations arebased on wrong assumptions.

6. Conclusion

For better understanding of a contact angle of a liquiddrop on a solid surface, three basic equations, the Young’sequation, the Wenzel equation, and the Cassie-Baxterequation, were derived by energy minimization. From thederivations, it was deduced that a contact angle on an idealsurface is determined by only the infinitesimal region inthe vicinity of contact line while surface state of regionsinside the contact line does not affect a contact angle. Itwas also explained why the Cassie-Baxter model and theWenzel model have been widely used for a superhydro-phobic surface. These models are not proper models forcontact angle behavior on the real surface because of theviolation of the assumption of no contact angle hysteresis.We hope that this study will provide a guideline to adeeper understanding of a contact angle.

Acknowledgements

This research was supported by Basic Science ResearchProgram through a National Research Foundation ofKorea (NRF) grant funded by the Ministry of Education,Science, and Technology (2012R1A2A2A01047371).

C1 f1 sl f1so f1

– f2 sl f2so f2

– += C2 2=

f1 1 f2 2cos+cos cos=

2 =

Fig. 7. (Color online) Schematic of the Cassie-Baxter model. (a)side-view, (b) top-view.

Fig. 8. (Color online) The area inside the contact line affects thetotal energy of the system, not the contact angle. The contactangle is independent of the absolute value of the total energy.



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