Equilibrium Contact Angles of Liquid Droplets on Ideal Rough Solids

ARTICLE pubs.acs.org/Langmuir

Equilibrium Contact Angles of Liquid Droplets on Ideal Rough Solids Hie Chan Kang*,† and Anthony M. Jacobi‡ † ‡

School of Mechanical and Automotive Engineering, Kunsan National University, Daehangro 558, Gunsan 573-701, Republic of Korea Department of Mechanical Science and Engineering, University of Illinois at Urbana
Champaign, 1206 West Green Street, Urbana, Illinois 61801, United States ABSTRACT: This work proposes a theoretical model for predicting the apparent equilibrium contact angle of a liquid on an ideal rough surface that is homogeneous and has a negligible body force, line tension, or contact angle hysteresis between solid and liquid. The model is derived from the conservation equations and the free-energy minimization theory for the changes of state of liquid droplets. The work of adhesion is expressed as the contact angles in the wetting process of the liquid droplets. Equilibrium contact angles of liquid droplets for rough surfaces are expressed as functions of the area ratios for the solid, liquid, and surrounding gas and the roughness ratio and wetting ratio of the liquid on the solid for the partially and fully wet states. It is found that the ideal critical angle for accentuating the contact angles by the surface roughness is 48°. The present model is compared with existing experimental data and the classical Wenzel and Cassie
Baxter models and agrees with most of the experimental data for various surfaces and liquids better than does the Wenzel model and accounts for trends that the Wenzel model cannot explain.

1. INTRODUCTION During the last two centuries, the wetting of solids by liquids was studied by many researchers. Young1 treated the contact angle of a liquid as the result of the mechanical equilibrium of a drop resting on a plane solid surface subject to three surface tensions, one for each interface (solid/liquid, solid/gas, and liquid/gas). Following Young, Dupre2 introduced the concepts of the work of cohesion and the work of adhesion. His approach employed thermodynamic notation to express the surface tension in terms of free energy, namely, the free energy per unit area as well as the force per unit length. Sumner3 derived Young’s equation thermodynamically for the ideal plane solid surface. Wenzel4 proposed a model for the contact angle as a function of the macroscopic roughness of the solid surface. Later, Cassie and Baxter5 made Wenzel’s model more generally applicable to structures. For 75 years, the Wenzel and Cassie
Baxter models have contributed greatly to the understanding of interfacial energies, and many researchers have reviewed the models theoretically and empirically. The validity of the Wenzel model for modern usage was discussed in detail.6
8 Dettre and Johnson9 conducted experiments and found that the contact angles of rough surfaces followed Wenzel’s model qualitatively within their experimental range. Onda et al.10 and Shibuichi et al.11 reported experimental data and empirical correlations for surfaces that were super-waterrepellent owing to their fractal structures. Many reseachers reported the contact angles of various liquids on various rough surfaces and compared their experimental data with the Wenzel model and/or the Cassie
Baxter model.12
20 Their data showed that the wetting behavior for θ > 90° correlated well with the predictions made according to the Cassie
Baxter heterogeneous wetting model. However, the wetting behavior for θ < 90° could not be described r 2011 American Chemical Society

by the Wenzel, Cassie
Baxter, or Onda
Shibuichi model. Experimentation found that the contact angles for rough surfaces were greater than 90° when those for smooth surfaces were in the range of 60° < θ < 90°. This evidence was a direct contradiction of the Wenzel model. Chung and Bhushan20 studied the wetting behavior of water and oil droplets for hydrophobic, hydrophilic, oleophobic, and oleophilic surfaces with three-phase interfaces and reported extensive experimental data on the contact angles of water, hexadecane, and hexadecane in water for pillar material having different heights. The results contradicted the Wenzel model for the fully wet condition qualitatively and quantitatively. Therefore, an improved theoretical model is needed to understand wettability on rough surfaces. The objective of this study is to investigate the relationship between morphology and wettability for liquid droplets on solids. A model is proposed on the basis of the mass, force, and energy conservation equations and free-energy minimization. The effects of morphology on the wetting process are modeled theoretically and compared to experimental data.

2. EQUILIBRIUM CONTACT ANGLE THEORY 2.1. Work of Adhesion of a Droplet on a Smooth Surface. Consider the process of putting a droplet slowly on an ideal surface that is wetted as shown in Figure 1. State 1 is the initial state of the liquid droplet, and state 2 is the final state in which the liquid is located on the plane surface in the form of a spherical cap with an apparent contact angle θ. Assumptions for wetting the Received: August 11, 2011 Revised: November 3, 2011 Published: November 04, 2011 14910

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Using eqs 3, 7, 8, and 9, we transform eq 6 to

Figure 1. Surface wetting process of a liquid droplet.

ideal surfaces are as follows: surface energies are dominant in the process; the surface is homogeneous; there is no contact angle hysteresis; the line tension is negligible;21 all body forces are negligible; electromagnetic forces are negligible; the liquid does not diffuse into the solid; and all properties are constant. Mass conservation is expressed as FV1 ¼ FV2

ð1Þ

From Young’s equation, the force balance at the contact line is as given below σ sg
σsl ¼ σ lg cos θ

1
αlg w ¼ϕ þ σ lg αsl

ð10Þ

δ ¼ ϕαsl
αlg þ 1

ð11Þ

Parameter w/σlg in eq 10 is the work of adhesion for the liquid
gas surface tension, termed the “specific work of adhesion” in this work. w/σlg and δ are functions of surface energy ratio ϕ (i.e., Young’s contact angle θY and area ratios αlg and αsl). Young’s contact angle is a physical property pertaining to the solid
liquid
gas combination and is usually different from the apparent contact angle. Young’s contact angle is the same as the apparent contact angle for the ideal smooth surface discussed above. Helmholtz free-energy minimization prescribes that the liquid
gas surface is spherical to minimize the surface area in the final state. The specific work of adhesion and the normalized work of adhesion are as below from the geometrical relation between a liquid droplet and a spherical cap in Table 1:

ð2Þ 4

where σ is the surface tension (or surface free energy) and subscripts sg, sl, and lg refer to the solid
gas, solid
liquid, and liquid
gas interfaces, respectively. If the surface is ideal and there is no hysteresis, then Young’s contact angle θY is the same as the apparent contact angle θ. The surface energy ratio ϕ is the same as cos θY. σ sg
σsl ¼ ϕ ¼ cos θY ð3Þ σlg

w ¼ϕ þ σ lg

σsg
σ sl Asl2 Alg2 wAsl2 ¼ þ σlg Alg1 Alg1 σlg Alg1

ð5Þ

ð6Þ

Asl2 αsl ¼ Alg1

ð7Þ

Alg2 Alg1

ð8Þ

wAsl2 w ¼ αsl ¼ δ σ lg σlg Alg1

!2=3

!2=3 þ 1

ð13Þ

2.2. Contact Angle on a Rough Surface with Full Wetting. We can consider four cases in which a liquid droplet is put on a solid surface. Case a in Figure 2 is a baseline smooth surface. Case b is fully wet and the case of the Wenzel state. Contact area ratio fsl is the ratio of the solid
liquid contact area to the solid
liquid interfacial projected area on the solid plane:

fsl ¼

ð9Þ

δ is the normalized work of adhesion (i.e., the ratio of the total work of adhesion to the total surface energy of the initial liquid sphere).

Asl, c A0sl2

ð14Þ

The ratio of the liquid
gas contact area on the bottom of the liquid droplet to the solid
liquid interfacial projected area on the solid plane is

We define area ratio α as

αlg ¼

ð12Þ

θ

1
cos θ 4
2 ð2 þ cos θÞð1
cos θÞ2

Equation 4 can be rearranged to 1 þ


2ð1
cos θÞ

ð4Þ

The parameter w is the work of adhesion per solid
liquid contact area (J/m2), described in the same units as the surface tension (N/m). The work of adhesion depends on characteristics of the solid
liquid pair and the condition of the solid. The solid
gas surface area in the initial state is the same as the solid
liquid surface area on the bottom of the liquid droplet in the final state: Asg1 ¼ Asl2

sin2

!2=3

sin2 θ cos θ 4 δ¼ϕ 4 ð2 þ cos θÞð1
cos θÞ2

Energy conservation for a rough surface for the initial and final states yields σlg Alg1 þ σsg Asg1 ¼ ðσ lg Alg2 þ σ sl Asl2 Þ þ wAsl2

ð2 þ cos θÞð1
cos θÞ2 4

flg ¼

Alg, c A0sl2

ð15Þ

Figure 2c is the case of the Cassie state; the sum of the solid
liquid and liquid
gas contact areas is the same as the area at the bottom of the spherical cap. Practical cases are represented by Figure 2d in which the sum of the solid
liquid and liquid
gas contact areas is greater than bottom area of the spherical cap and the liquid
gas contact area at the bottom of the spherical cap is greater than zero. It is assumed that the liquid droplet completely penetrates into the rough grooves in the case of Figure 2b and is sufficiently large compared to the roughness scale. On a rough surface, the apparent 14911

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Table 1. Geometric Relationship between the Initial Sphere and the Final Spherical Caps

a

The radius ratio satisfies the mass conservation for the initial and final states.

contact angle is related to the ideal contact angle by the Wenzel equation as given below: 0

cos θ ¼ fsl cos θ

1
α0lg α0sl

¼ fsl

1
αlg αsl

ð20Þ

By minimizing the Helmholz free energy and using Table 1, the apparent contact angle on the rough surface can be expressed as below. 1
cos θ0
2

ð2 þ cos θ0 Þð1
cos θ0 Þ2 4

!2=3

sin2 θ0 1
cos θ
2 ¼ fsl

ð2 þ cos θÞð1
cos θÞ2 4

!2=3

sin2 θ

ð21Þ

2.3. Contact Angle on a Rough Surface with Partial Wetting. The Cassie
Baxter model is applicable to rough solid sur-

faces in partially wet states in the case of Figure 2c. cos θ0 ¼ fsl cos θ þ flg

ð18Þ

The specific work of adhesion w/σlg is the same in the processes for the smooth and rough surfaces; eq 18 can be combined with

ð19Þ

After rearranging terms, we have

ð17Þ

Primed and nonprimed parameters are denoted for rough and smooth surfaces, respectively. The initial solid
gas area A0 sg1 and the final solid
liquid contact area A0 sl2 are the same because both areas are the projected area of the solid
liquid side interface on the solid plane. In this work, the total work of adhesion is assumed to be proportional to the wetted solid surface area; thus the work of adhesion on a rough surface per area is the same as that on smooth surface, w0 = w, and eq 17 becomes A0lg2 σsg
σ sl A0sl2 w A0sl2 1 þ fsl ¼ þ fsl σ lg Alg1 σ lg Alg1 Alg1

  1
αlg 1 þ ϕfsl α0sl ¼ α0lg þ fsl α0sl ϕ þ αsl

ð16Þ

In the equation, θ0 is the contact angle of the rough surface and θ is the contact angle of the smooth surface. Consider the process of putting the same droplet slowly on a rough surface instead of the smooth surface in the previous section. Only the surface roughness has been changed, so the wetted solid surface area is increased, as shown by fsl. The surface tension ratio ϕ is not changed. The apparent contact angle and interfacial areas in the final state changed with the increase in the solid surface area. Energy conservation for the rough surface in the described wetting process is σ lg Alg1 þ σ sg fsl A0sg1 ¼ ðσ lg A0lg2 þ σ sl fsl A0sl2 Þ þ w0 fsl A0sl2

eq 10 as given below:

ð22Þ

Consider liquid droplets placed slowly on rough surfaces that become partially wet as shown in Figure 2c,d. The surface energies are the same as that of the smooth surface. The energy conservation 14912

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Figure 3. Variations in the normalized and specific works of adhesion along the contact angle of a spherical droplet wetting an ideal solid surface.

Figure 2. Schematic diagram of the wetting states of a liquid on solid surfaces.

for a rough surface in such a wetting process is shown below. σ lg Alg1 þ σ sg fsl A0sg1 ¼ ðσ lg A0lg2 þ σ lg flg A0sl2 þ σ sl fsl A0sl2 Þ þ w0 fsl A0sl2

ð23Þ Figure 4. Comparison of the present model with the Wenzel model for the contact area ratio of solid
liquid fsl for the fully wet condition of flg = 0.

By rearranging eq 23 and using eqs 3 and 10, we have 1
α0lg α0sl

¼ fsl

1
αlg þ flg αsl

ð24Þ

If the solid surface is fully wet (flg = 0), then eq 24 becomes eq 20. Using Table 1, eq 24 can be written as a function of the apparent angles of the smooth and rough surfaces: 1
cos θ0
2

ð2 þ cos θ0 Þð1
cos θ0 Þ2 4

!2=3

sin2 θ0 2

ð2 þ cos θÞð1
cos θÞ2 6 61
cos θ
2 6 4 ¼ fsl 6 6 sin2 θ 6 4

!2=3 3 7 7 7 7 þ f lg 7 7 5

ð25Þ

Parameter (1
αlg)/(αsl) in eqs 20 and 24 of the present model corresponds to cos θ in eqs 16 and 22 of the Wenzel and Cassie
Baxter models.

3. RESULTS AND DISCUSSION 3.1. Work of Adhesion. The normalized work of adhesion δ (or wAsl2/σlgAlg1) is the ratio of the total work of adhesion used by the solid and liquid to the total initial free surface energy during the process of a droplet contacting a solid surface. If Young’s contact angle is the same as the apparent contact angle, then eqs 12 and 13 apply, as drawn in Figure 3. The normalized work of adhesion is a function of the contact angle and decreases from 1 to 0 as the contact angle increases from 0 to 180°. At a 0° contact angle, the initial surface free energy of the spherical droplet is completely used for adhesion. At 180°, the final state is the same as the initial state and the surface free energy of the initial state is conserved. The specific work of adhesion w/σlg is the ratio of the work of adhesion per unit area to the liquid
gas surface tension. Near zero contact angle, the specific work of adhesion becomes zero because of the infinite contact area ratio αsl = Asl2/Alg1, and it is also zero at an 180° contact angle because the work of adhesion w is zero. The value reaches a maximum at 48°; this angle is named 14913

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Langmuir the critical contact angle for an ideal surface, is denoted by θcr in this work, and is based on the assumptions described before. The Wenzel model assumes that the critical angle needed to accentuate the apparent contact angle for the rough surface is 90°. The contact angle of a solid surface decreases for an increasing solid surface area if the solid
liquid is hydrophilic, θ < θcr. The contact angle of a solid surface increases for an increasing surface area if the solid surface is hydrophobic, θ > θcr. 3.2. Characteristics of Full and Partial Wetting. Figure 4 shows the variations of the contact angles of the present and Wenzel models for changes in the contact area ratio of solid
liquid for the fully wet condition. The horizontal and vertical axes indicate the cosine of the contact angle at the baseline surface having a smooth

Figure 5. Contact angle variation along the contact area combinations (fsl and flg) of solid
liquid and liquid
gas of a solid surface area for the partially wet condition.

ARTICLE

surface and the cosine of the contact angle at the rough surface, respectively. The hydrophilic and hydrophobic characteristics are enhanced as the solid surface area increases. The trend is similar in the two models; however, the critical angle θcr is 90° in the Wenzel model and 48° (cos θcr = 0.669) in the present model. The Wenzel model cannot explain the behavior of the contact angle in quadrant IV and in the region below the diagonal line in quadrant I in Figure 4. The contact angle of the present model increases as the roughness ratio increases while showing the reverse trend as the Wenzel model in the range of 48° < θ < 90°. As an example, for a smooth surface of 63° (cos θ = 0.454) and a rough surface of fsl = 2, the Wenzel model predicts a contact angle of 28° (cos θ0 = 0.883), but the present model predicts 85° (cos θ0 = 0.087). Figure 5 shows the characteristics of the apparent contact angles on a rough surface for the solid
liquid contact area ratio fsl and the liquid
gas contact area ratio flg for the projected solid surface area of a spherical cap of liquid. In the case of fsl + flg = 1 as in case c in Figure 2, the present model shows almost the same results as the Cassie
Baxter model at cos θ < 0 but predicts a smaller contact angle on the rough surface than the Cassie
Baxter model at cos θ > 0. If the liquid
gas contact area ratio flg is greater than zero and the contact angle on the rough surface θ0 is greater than that of the smooth surface θ, then the contact angle on the rough surface is increased as the solid
liquid contact area ratio fsl is decreased. When fsl + flg > 1 and flg > 0 (case d in Figure 2), near 180° the contact angle on a rough surface (superhydrophobic condition) can be obtained from the less-hydrophobic baseline surface even though θ < 90°, which is known to be a hydrophilic surface. A typical example is the case of fsl = 0.2 and flg = 0.9 in Figure 5. The partially wet and excess contact area ratios of flg > 0 and fsl + flg > 1 may be obtained by the distortion of the liquid interface at the bottom of the spherical cap. The hierarchical

Figure 6. Contact angle changes with increasing surface roughness for the fully wet condition (flg = 0). (a) Comparison with Jung and Bhushan’s experimental data20 for cylindrical pillars and with Yoshimitsu et al’s.19 data for rectangular pillars at high contact angles on smooth surfaces. (b) Comparison with Jung and Bhushan’s experimental data at low contact angles on smooth surfaces. The roughness ratios of Jung and Bhushan were reproduced, and experimental data were selected for the fully wet state for fsl < 1.3. The detailed test conditions are listed in Table 2. 14914

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14915

coating with FAS-17

and 150 μm pillar width, 100 μm groove width, and varying

14 μm diameter,

30 μm height, and varying pitch

active vibration of the surface in the contact angle measurement

interfaces

830 μL (11.6 mm)

520 μL (10 mm),

minimum

global energy

fsl = 1
1.25, flg = 0b

solid
water
oil

1 μL (1.2 mm)

sessile drop method

fsl = 1
1.3, flg = 0b

water, ethylene glycol

melted beeswax

1000) dipped into

(grits #240, 320, and

abrasive papers

beeswax

Meiron et al.14

solid
air
water and

5 μL (2.1 mm)

method

sessile drop

fsl = 1
1.3, flg = 0d

hexadecane in water

water

pillar height

square pillars of 50

cylindrical pillars with

water, hexadecane,

micropatterned

micropatterned

coated with C20F42

silicon wafer

the same surface

Yoshimitsu et al.19

epoxy resin and

Jung and Bhushan20

fsl = 0.2, flg = 0.8e

∼4 μL (∼2.1 mm)

not specified

d

fsl = 2.6, flg = 0,

water, mercury, etc.

8 liquids,

30 μm pitch

pillars with

diameter; cylindrical

square lattice of 15 μm

SU-8 photoresist

McHale et al.13

contact angles

advancing and receding

10 μL (2.7 mm)

sessile drop method

d

fsl = 0.11, flg = 0.89, fsl = 6, flg = 0

c

b

e

fsl = 0.2, flg = 0.8

measurement

the contact angle

gentle vibration in

0.52 μL (1 mm)

sessile drop method

d

fsl = 10, flg = 0,

not specified,

water, glycerol, etc.

fsl = 1.7, flg = 0,

16 liquids,

water, n-alkanes, etc.

in an orderly surface configuration

2 μL (1.6 mm)

fsl = 6, flg = 0d sessile drop method

flg = 0.78,c

flg = 0,b fsl = 0.22,

fsl = 1.8,

water, octane, etc.

9 liquids,

and a 2 μm average fiber diameter

having 80% porosity

fiber membranes and 40 μm in length

PCL/Teflon coaxial

5 μm in diameter

Teflon

Han and Steckl18

pillars approximately

poly (alkylpyrrole) film

Kurogi et al.15

13 liquids,

silica particles

0.2-μm-diameter

from FSI-coated,

fractal surface prepared

FSI

Synytska et al.17

Diameter of the initial spherical droplet used in the measurement of the contact angle. b Roughness factor for the Wenzel state described in the reference. c Roughness factor for the Cassie
Baxter state described in the reference. d Reproduced or estimated roughness factor for the Wenzel state in the present work from the reference data. e Reproduced or estimated roughness factor for the Cassie
Baxter state in the present work from the reference data.

a

remarks

(diametera)

droplet volume

measuring method

(area ratios)

roughness factors

liquid

solid surface shape

solid material

author

Table 2. Experimental Conditions for the Contact Angle Measurements on the Rough Surface Compared with Theoretical Models

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Figure 7. Comparison of the contact angle models with the experimental data for the rough surfaces of (a) McHale et al.13 for regular cylindrical pillar, (b) Synytska et al.17 for fractal surface of particles, (c) Kurogi et al.15 for arbitrary pillars, and (d) Han and Steckl18 for fiber membrane. The experimental conditions are summarized in Table 2.

microstructure of lotus leaves seems to use this principle. Empirical evidence is needed in the future to prove this phenomenon. 3.3. Comparison to Experimental Data. Figure 6a,b shows that the contact angle changes with increasing surface roughness of the pillar surfaces for the fully wet condition. Jung and Bhushan20 measured the contact angles of water, oil, and oil in water for cylindrical pillar surfaces of epoxy and a C20F42 coating on epoxy having different surface area ratios. The area ratios were calculated to be fsl = 1 + (πDpH)/(Pp2) from their test conditions. Yoshimitsu et al.19 measured the contact angles of water for rectangular pillar surfaces of a silicon wafer having two pillar pitches and heights. Jung and Bhushan and Yoshimitsu et al. tested for the ranges of 1 < fsl < 4 and 1 < fsl < 3.1, respectively. The data shown in Figure 6 express only the range 1 < fsl < 1.3, which is for the fully wet condition. The detailed test conditions for the solid, liquid, and measuring method are listed in Table 2. The contact angle changes almost linearly as the solid surface area ratio fsl increases near the critical contact angle θcr but increases dramatically at θ > 90°. The contact angles of water on epoxy and oil on C20F42 were both 76° (solid and hollow circles in Figure 6a), increasing as the surface area ratio increased. The present model predicts the trend in the experimental data. However, the Wenzel model shows the reverse trend: the contact angle should decrease as the roughness increases. The experimental data for the oil in water on an epoxy surface follow Wenzel’s model for θ = 109°; however, by considering the data scatter for 1 < fsl < 1.05, the present theory for θ = 100° is also reasonable. Yoshimitsu’s

experimental data also agree better with the present model than with the Wenzel model (Figure 6a). As shown in Figure 6b, the contact angle of oil was 13° on the smooth surface and decreased as the contact area ratio fsl increased in the range of 0 < fsl < 1.3. The present model traces the experimental data very well, but the Wenzel model falls to 0° at fsl = 1.03. The present model predicts the apparent contact angle on the rough surface for the fully wet condition better than does the Wenzel model. However, more precise consideration and experimentation are needed for the water
oil interface with the solid. Meiron et al.14 reported experimental data for contact angles on rough surfaces as listed in Table 2. They used 520 and 830 μL of ethylene glycol and water, assuming that the ratios between the drop base diameters and surface roughness parameters would be sufficiently large for the Wenzel equation to hold. They also used a global energy minimum (GEM) on real surfaces, vibrated the surfaces, and calculated the contact angles from the drop diameters and weights. Their data are almost identical to those obtained via the Wenzel model but differ from those obtained via the present model. The conventional sessile drop method uses about 0.5 to 10 μL of liquid to reduce the effect of gravity. It remains for future research to disclose whether this discrepancy pertains to the experimental method or the present theory. Figure 7a
d shows comparisons of the present theory with the experimental data of McHale et al.,13 Synyska et al.,17 Kurogi et al.,15 and Han and Steckl18 and the Wenzel and Cassie
Baxter 14916

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Langmuir models for the solid
liquid combinations for the contact angles of various liquids on various rough surfaces such as regular cylindrical pillars, arbitrary pillars, spherical particles, and fiber membranes. The test conditions for the solid material and the shape, liquid, measuring method, volume of liquid, and contact area ratios fsl and flg are listed in Table 2. The contact area ratios for the Wenzel and Cassie
Baxter states are taken from the reference or reproduced using the geometric data. The roughness factor of Kurogi is estimated from the geometrical data of the pillar in the reference. One and a half layers of silica particles were assumed to be wetted for the case of Synyska et al., and multiple layers (about 10 layers of 2-μm-thick fiber) of fiber membranes were assumed to be wetted for the case of Han and Steckl. In Figure 7a
d, the states of wetting change from the Wenzel to Cassie
Baxter states in the range of about 0.3 < cos θ < 0.75 for increasing contact angle (or decreasing cos θ) on the smooth surface. The Cassie
Baxter and present models agree well with the experimental data, although the theoretical models are different for 0.3 < cos θ. For the Wenzel state, the present model predicts the experimental data better than does the classical Wenzel model. There has been significant controversy concerning experiments for the hatched region of quadrants I and IV in Figure 4. This comparison clarifies the situation. The transition from the partial to fully wet states in the present model is close to that in the experimental data. Further study is needed to understand the transition phenomena. 3.4. Limits of the Present Model and Future Work. The present theory is based on the equilibrium contact angle of liquid droplet on an ideal solid surface. In practical applications, there is contact angle hysteresis between the solid and liquid; furthermore, the critical contact angle can vary with the conditions of the solid and the particular solid
liquid combination. This work assumes that the work of adhesion is proportional to the ideal solid
liquid contact area and does not account for the size of the droplet or the shape of the solid surface. The experimental data for the oil in water on the rough surface partially agreed with the present model. These should be verified by experimental data in the future. The present work could contribute to our understanding of the ideal contact angle of solid surface, the surface energies, and the hierarchical microstructure of plant and animal surfaces.

4. CONCLUDING REMARKS A model has been proposed to predict the apparent equilibrium contact angle of a liquid droplet wetting an ideal solid surface. Equilibrium contact angles for rough surfaces are expressed as a function of solid
liquid
gas area ratios, contact area ratios of the liquid and gas on the solid, and smooth surface contact angle for full and partial wettings. The critical angle for accentuating the contact angles by the surface roughness is 48° for the ideal solid surface, whose apparent contact angle is the same as the Young’s contact angle. The present model provides a reasonable basis for predicting and controlling the wettability of a solid by the surface morphology for the full and partial wetting cases. Also, the present model explains trends in the experimental data that the classical Wenzel model cannot. ’ AUTHOR INFORMATION Corresponding Author

*Tel: 82 + 63-469-4722. Fax: 82 + 63-469-4727. E-mail: hckang@ kunsan.ac.kr.

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’ ACKNOWLEDGMENT This work was financially supported by the IT R&D program (10033910) of MKE, and the authors thank Dr. Jung Hoon Yeom and Sung Chul Bae (University of Illinois at Urbana-Champaign) for helpful discussions. ’ NOMENCLATURE a spherical cap diameter on the contact plane, m liquid
gas interfacial area of initial liquid droplet, 4πr12, m2 Alg1 Alg2 final liquid
gas interfacial area on the top of the liquid droplet, m2 Asg1 initial projected area to be covered with liquid on the solid plane, m2 Asl2 final solid
liquid interfacial projected area on the solid plane, equal to Asg1, m2 Alg,c contact area of liquid and gas on the bottom of the liquid droplet, m2 Asl,c contact area of the solid and liquid on the bottom of the liquid droplet, m2 Dp pillar diameter, m ratio of liquid
gas contact area on the bottom of the liquid flg droplet to the solid
liquid interfacial projected area on the solid plane, Alg,c/A0 sl2 fsl ratio of the solid
liquid contact area to the solid
liquid interfacial projected area on the solid plane, Asl,c/A0 sl2 h spherical cap height, m pillar height, m Hp pillar pitch, m Pp r radius of a liquid droplet or spherical cap, m V volume of a liquid droplet or spherical cap, m3 w work of adhesion per solid
liquid contact area, N m
1 ’ GREEK SYMBOLS αlg ratio of liquid
gas interfacial area on the top of liquid droplet to the initial liquid
gas interfacial area, Alg2/Alg1 αsl ratio of the solid
liquid interfacial projected area on the solid plane to the initial liquid
gas interfacial area, Asl2/Alg1 δ normalized work of adhesion ϕ surface tension ratio in Young’s equation θ contact angle, deg or rad critical contact angle, deg or rad θcr Young’s contact angle, deg or rad θY F density of liquid, kg m
3 σ surface tension, N m
1 ’ SUBSCRIPTS AND SUPERSCRIPTS 1 initial state 2 final state c contact cr critical lg liquid
gas sg solid
gas sl solid
liquid 0 rough surface ’ REFERENCES (1) Young, T. Philos. Trans. R. Soc. London 1805, 95, 65. (2) Dupre, A. Theorie Mecanique de la Chaleur; Gauthier-Villars: Paris, 1869. (3) Sumner, C. G. Symposium on Detergency; Chemical Publ. Co.: New York, 1937. 14917

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dx.doi.org/10.1021/la2031413 |Langmuir 2011, 27, 14910–14918