A simple method for measuring the superhydrophobic contact angle ...

REVIEW OF SCIENTIFIC INSTRUMENTS 81, 065105 共2010兲

A simple method for measuring the superhydrophobic contact angle with high accuracy Yi-Lin Hung, Yao-Yuan Chang, Meng-Jiy Wang, and Shi-Yow Lina兲 Department of Chemical Engineering, National Taiwan University of Science and Technology, 43, Keelung Rd., Sec. 4, Taipei 106, Taiwan

共Received 17 March 2010; accepted 13 May 2010; published online 24 June 2010兲 A modified selected-plane method for contact angle 共␪兲 measurement is proposed in this study that avoids the difficulty of finding the real contact point and image-distortion effects adjacent to the contact point. This method is particularly suitable for superhydrophobic surfaces. The sessile-drop method coupled with the tangent line is the most popular method to find the contact angle in literature, but it entails unavoidable errors in determining the air-solid base line due to the smoothness problem and substrate tilting. In addition, the tangent-line technique requires finding the actual contact point. The measurement error due to the base line problem becomes more profound for superhydrophobic surfaces. A larger ␪ deviation results from a more superhydrophobic surface with a fixed base line error. The proposed modified selected-plane method requires only four data points 共droplet apex, droplet height, and two interfacial loci close to the air-solid interface兲, avoiding the problem of the sessile-drop-tangent method in finding the contact point and saving the trouble of the sessile-drop-fitting method for best fitting of the numerous edge points with the theoretical profile. A careful error analysis was performed, and a user-friendly program was provided in this work. This method resulted in an accurate ␪ measurement and a method that was much improved over the classical selected plane and the sessile-drop-tangent methods. The ␪ difference between this method and the sessile-drop-fitting method was found to be less than three degrees. © 2010 American Institute of Physics. 关doi:10.1063/1.3449325兴

I. INTRODUCTION

Superhydrophobic surfaces have drawn extensive attention due to their nearly complete nonwetting properties, which can avoid liquid adhesion. Due to these notable properties, the industrial applications of superhydrophobicity include antibiofouling or self-cleaning paints for vehicles, and snow-antistick coatings for windows and antennas.1–4 A superhydrophobic surface is usually defined as a material with an advancing water-contact angle 共WCA兲 exceeding 150° with low hysteresis. Superhydrophobicity can be generated either by nanostructures or microstructures on the solid surface or by chemical composition. The primary method applied to measure superhydrophobic WCA is the sessile-drop technique.5–8 The Wilhelmy plate method and capillary-rise method have also been employed.9–11 A water-shedding angle method has also been developed to characterize the wetting properties of a superhydrophobic textile.12,13 A sliding-drop method 共by moving the solid substrate兲 has been reported to obtain a WCA up to 159° on porous polypropylene 共PP兲.7,14 The superhydrophobicity of a 169° WCA has also been identified by the popular Wilhelmy plate method for a raspberrylike particulate film.15,16 A 176° WCA was found for a ZnO-nanowire structured surface by using the sessile-drop-fitting method.6 WCA larger than 160° or even 170° have also been reported for various surfaces in Author to whom correspondence should be addressed. Tel.: ⫹886-2-27376648. FAX: ⫹886-2-2737-6644. Electronic mail: [email protected].

a兲

0034-6748/2010/81共6兲/065105/9/$30.00

literature.17–19 However, the accuracy of superhydrophobic WCA measurements has rarely been discussed. Visualization of a sessile drop coupled with a tangent line through the contact point 共sessile-drop-tangent technique兲 is the most widely employed method for superhydrophobic WCA measurement: This technique has yielded WCA values of 171.2° for a surface of amphiphilic poly共vinyl alcohol兲 nanofibers,20 168° for a polytetrafluoroethylene 共PTFE兲 surface covered with carbon nanotubes,5 168° for a silicone nanofilament surface,8 160° for a multilayer polyelectrolyte surface 共mimicking the surface structure of the Namib Desert beetle兲,21 and 160° for a transparent, porous thin film.22 However, every measurement technique is fraught with some potential errors. The Wilhelmy plate method 共F = ␥ P cos ␪兲 requires surfaces with a symmetrical coating on both sides and an accurate peripheral determination.7 The sliding method suffers from the difficulty of approaching the thermodynamic equilibrium state for a moving solid substrate.7 The sessile-drop-tangent method bears the difficulty of the determination of the three-phase contact point and of the edge coordinates for the drop interface adjacent to the contact point. The sessile-drop-fitting method saves the trouble of finding a contact point but suffers from a troublesome best-fitting algorithm. It should also be noted that all three above methods involving the imaging process also suffer from the surface tilting problem in the light passing direction when determining the liquid-solid interface. For a superhydrophobic surface, an

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0 0

I dy dx

P1 : X1/Z1=0.50

0

s'

z'

P2 : X2/Z2=0.30

B=100 30 10

0.5

Xe g

X

Z/R0

s

ds Ze

x'

I R2  Z



R1 3

1

b

1 FIG. 1. Coordinate system of a sessile drop.

1.5 0.3

extra difficulty arises from an image distortion due to the narrow slit near the contact point, such that the determination of an accurate edge location becomes a crucial problem. Moreover, a larger error arises in the determination of the contact point, and a thus large error in the WCA measurement may result. In this study, the accuracy of the superhydrophobic WCA measurement using the sessile-drop method was examined, and a simple method with higher accuracy was proposed. This method was modified from the classic selectedplane method in which there is no need to find the threephase contact point. The role of the contact point was replaced by the measurement of the droplet height, which can be acquired much more easily. Only the positions of three edge points on a drop surface 共the droplet apex and two interfacial loci near the three-phase contact point兲 and one edge point at the air-solid base line are needed in this method. By inputting the four edge point data, the capillary constant and the radius of curvature at the apex can be easily estimated from a third-order polynomial equation, which correlates the two newly defined parameters 共a drop-shape function and a drop-shape factor兲. The use of the two interfacial loci near the three-phase contact point yields an accurate WCA determination for a superhydrophobic surface via a user-friendly program provided in this work.

The theoretical shape of a sessile drop is governed by the Young–Laplace 共YL兲 equation, which describes the pressure difference across the curved fluid interface23–25

冊

1 1 + = ⌬P, R1 R2

0.01 0

共1兲

where ␥ is the surface tension, R1 and R2 are the two principal radii of curvature of the surface, and ⌬P is the pressure difference across the interface. Equation 共1兲 can be recast as a set of three first-order differential equations for the spatial positions x and z and turning angle ␾ of the interface as a function of the arc length s 共Fig. 1兲 and then integrated with the boundary conditions x共0兲 = z共0兲 = ␾共0兲 = 0. The first-order differential equations can be written in the following form by applying the dimensionless variables xⴱ = x / R0, zⴱ = z / R0, and sⴱ = s / R0:

0.5

X/R 0

1

FIG. 2. 共Color online兲 Sessile-drop profiles derived from the YL equation at various capillary constants.

sin ␾ d␾ ⴱ , ⴱ = 2 + Bz − ds xⴱ

共2a兲

dxⴱ = cos ␾ , dsⴱ

共2b兲

dzⴱ = sin ␾ , dsⴱ

共2c兲

where B is the capillary constant 共⌬␳gR20 / ␥兲, ⌬␳ is the density difference between the fluid phases, g is the gravitational acceleration, and R0 is the radius of curvature at the apex. The equations are subject to the boundary conditions xⴱ共0兲 = zⴱ共0兲 = ␾共0兲 = 0. Equation 共2兲 was integrated by using a fourth-order Runge–Kutta scheme initialized with the approximate solution

zⴱ =

A. Drop profile

冉

2

xⴱ = sⴱ ,

II. CONTACT ANGLE FROM THE SESSILE-DROP METHOD

␥

0.1

␾=

共3a兲

2关1 − I0共冑Bxⴱ兲兴 , −B 2I1共冑Bxⴱ兲

冑B

.

共3b兲

共3c兲

Equation 共3兲 is valid near the apex, where ␾ Ⰶ 1.23 Here, In共xⴱ兲 is the modified Bessel function of the first kind. The profile of a sessile drop 关solution of Eq. 共2兲兴 depends only on the capillary constant B and R0, i.e., the theoretical drop profile is fixed once B and R0 are given. In other words, the drop profile is uniquely determined by any three specific points at the drop interface. Figure 2 demonstrates some drop profiles for various B with R0 = 1. Note that the contact angle ␪ is set to be equal to ␾s, where ␾s is the turning angle of the intercept point between the theoretical profile of the sessile-drop and air-solid interface 共i.e., the three-phase contact point兲.25

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TABLE I. The coefficients of the polynomial equations for the selected-plane method: B = a0 + a1Q + a2Q2 + a3Q3 and Xe / R0 = b0 + b1Q + b2Q2 共Q = Xe / Ze − 1兲. B

a0

a1

a2

a3

b0

b1

b2

0.01–1 0.6–10 9–100

0.000 383 9 ⫺0.1945 ⫺65.32

4.330 7.391 277.8

11.17 ⫺4.137 ⫺385.1

17.37 42.80 226.5

1.0001 1.0041 1.256

0.7217 0.6813 ⫺0.066 18

0.7567 0.8440 1.411

B. Selected-plane method

When a water droplet is put on a superhydrophobic surface, the drop has a clear equator. In the early 20th century, the selected-plane method24 was commonly applied to identify the profile of a sessile drop with an equator, i.e., to determine R0 and B of the drop. In the selected-plane method, Xe 共the radius of the equator, Fig. 1兲 and Ze 共the vertical distance between the equator and the apex兲 were used to estimate B and R0. By integrating Eq. 共2兲 for different values of B, we constructed the relationships between B and Ze / Xe and between Xe / R0 and Ze / Xe. These relationships can be well described by polynomial equations. Once Xe and Ze were obtained from a drop image via the image-processing technique, values of B and R0 were then calculated from the following equations: B = a 0 + a 1Q + a 2Q 2 + a 3Q 3 ,

共4a兲

Xe/R0 = b0 + b1Q + b2Q2 ,

共4b兲

where Q = Xe / Ze − 1. The coefficients 共ai and bi兲 of the polynomial equations are listed in Table I for several ranges of B values. C. Modified selected-plane method

A method similar to the selected-plane method was proposed in this work. According to the theoretical YL drop profiles, the corresponding loci of two sessile drops with different B values depart farther when the locus is farther away from the apex. In other words, for a sessile drop on a solid surface, the locus near the three-phase contact circle can be used to identify the drop profile more precisely than the data at the equator. This advantage becomes more significant for the case of a liquid droplet on superhydrophobic surface because its contact angle is high and there is a large arc length between the contact point and the droplet equator.

I P1 P2

XA2/R0 = b0 + b1 S + b2 S2 + b3 S3 ,

共5b兲

where S = XA1 / XA2 − 1. Parts of these relationships are summarized in Table II. The suitable working range of B for a specific set of 共A1 , A2兲 is also listed. When the values of Al and A2 are selected, P1, P2, X1, X2, Z1, and Z2 are known from the raw data, and therefore the initial guesses of B and R0 can be obtained from Eq. 共5兲.

0.75 X0.47/R0

0.8

Z1 g

共5a兲

1.2 B

s

I dy dx

B = a0 + a1 S + a2 S2 + a3 S3 ,

X

0

ds

Two interfacial loci on the drop profile, P1 = 共X1 , Z1兲 and P2 = 共X2 , Z2兲, as shown in Fig. 3, were selected from the raw data of the digitized drop image to replace the roles of Ze and Xe. The parameters Xl, X2 and Z1, Z2 are the distances of P1 and P2 away from the droplet apex in the x and z directions, respectively. Interfacial loci on the droplet interface close to the contact circle are recommended to obtain a more accurate estimate. A drop-shape function XAi and a drop-shape factor XA1 / XA2 were then defined. Here, XAi is the value of Xi / R0 at the interfacial locus Pi, where the ratio of Xi / R0 and Zi / R0 is equal to Ai共=Xi / Zi兲. The role of the traditional shape factor ds / de was replaced by XA1 / XA2. As the contact angle for a sessile drop on a superhydrophobic surface is high, only the case of ␪ ⬎ 120° was considered in this study. An example set of loci P1 and P2 for XAi = 0.5 and 0.3 is shown in Fig. 2 for three drop profiles with different B values. Equation 共2兲 was integrated for different values of B to obtain the drop-shape function XAi. The relationships between B and XA1 / XA2 and between XA2 / R0 and XA1 / XA2 were then constructed. An example illustrating the typical relationships was shown in Fig. 4, which can be described by the following polynomial equations:

Z2

0.65

H 0.4

X1

0.55 0 1.17

X2

Z FIG. 3. Selected interfacial loci P1 = 共X1 , Z1兲 and P2 = 共X2 , Z2兲 from the drop profile.

1.22 X 1.27 0.63/X0.47

FIG. 4. 共Color online兲 Representative relationships of the capillary constant B vs XA1 / XA2 and XA2 / R0 vs XA2 / XA1. A1 = 0.63 and A2 = 0.47 for 0.01⬍ B ⬍ 1.5.

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TABLE II. The coefficients of the polynomial equations for the modified selected-plane method: B = a0 + a1 S + a2 S2 + a3 S3 and XA2 / R0 = b0 + b1 S + b2 S2 + b3 S3 共S = XA1 / XA2 − 1兲. A1

A2

B

a0

a1

a2

a3

b0

b1

b2

b3

1.55 1.45 1.35 1.25 1.1 0.95 0.85 0.8 0.7 0.63 0.57 0.52 0.5

1.45 1.3 1.2 1.1 0.95 0.82 0.75 0.65 0.55 0.47 0.4 0.35 0.3

40–100 25–80 15–50 10–30 5–20 2–10 0.85–7 0.5–5 0.1–2.5 0.01–1.5 0.01–1 0.01–0.85 0.01–0.5

34.60 26.98 17.19 12.09 9.148 5.137 3.436 3.105 1.800 1.720 2.500 3.925 3.574

⫺3259 ⫺1573 ⫺915.8 ⫺586.5 ⫺381.99 ⫺215.9 ⫺176.0 ⫺90.52 ⫺45.67 ⫺32.47 ⫺32.02 ⫺38.416 ⫺25.147

2.304⫻ 105 5.521⫻ 104 2.837⫻ 104 1.507⫻ 104 6.759⫻ 103 3346 2980 810.6 307.0 154.79 110.25 107.99 50.32

−2.130⫻ 106 −3.093⫻ 105 −1.372⫻ 105 −6.224⫻ 104 −2.275⫻ 104 ⫺9804 ⫺9240 ⫺1465 ⫺369.3 ⫺139.0 ⫺87.60 ⫺80.82 ⫺27.00

0.671 0.756 0.8448 0.9269 1.079 1.143 1.160 1.194 1.136 1.072 0.9982 0.9410 1.341

⫺18.0 ⫺11.03 ⫺10.85 ⫺9.937 ⫺9.500 ⫺7.137 ⫺6.065 ⫺3.555 ⫺1.083 0.0315 0.5152 0.6621 –2.296

190 63.0 54.36 36.82 31.75 ⫺1.376 ⫺34.21 ⫺11.41 ⫺19.13 ⫺15.49 ⫺10.92 ⫺8.457 1.22

¯ ¯ ¯ 44.42 ¯ 113.8 300.8 55.60 51.58 29.5 15.38 9.900 ¯

III. MEASUREMENTS

To demonstrate the advantages and accuracy of the modified selected-plane method, we measured the contact angles for water drops on two superhydrophobic surfaces: a lotus leaf and a CF4 plasma-modified PP film. A. Materials

Water was purified with a Barnstead Nanopure waterpurification system, with the output water having a specific conductance less than 0.057 ␮S / cm. The value of the surface tension of the air-water interface was measured using the pendant-bubble technique26,27 as 72.0 mN/m at 25.0⫾ 0.2 ° C. The PP film was purchased from Celgard® 共NC, USA兲 and ultrasonically cleaned with ethanol, acetone, and pure water sequentially. After drying under nitrogen gas, the PP film was treated with oxygen plasma for 5 min 关100 W, with a 10 SCCM 共SCCM denotes cubic centimeter per minute at STP兴 flow rate at 100 mTorr pressure兴 followed by CF4 plasma treatment for 5 min 共20 W, with a 10 SCCM flow rate at 100 mTorr pressure兲. A fresh lotus leaf was taken from a plant on the campus and washed extensively with pure water. B. Apparatus

The equipment shown in Fig. 5 was used to create a silhouette of a sessile drop, to video image the silhouette, M PC

S DA

A

B C D C

E

F

G

FIG. 5. Sessile-drop apparatus and video-image digitization equipment: A is the light source; B is the filter; C is the biconvex lens; D is the pin hole; DA, D/A data translation card; E is the chamber, suspending needle, solid substrate; F is the objective lens; G is the CCD camera; M is the monitor; PC is a personal computer; S is the syringe and syringe pump.

and to digitize the image. The equipment consists of an image-forming and recording system, a drop-forming system, a video-image profile digitizer, and a solid substrate. The image-forming and recording system consisted of a light source, a planoconvex lens system for producing a collimated beam, an objective lens, and a solid-state video camera. A halogen lamp with a constant light intensity was used. The water drop and solid substrate were placed inside a thermostatically controlled chamber at T = 25.0⫾ 0.2 ° C. The drop on the active area of the camera was magnified by approximately 1.5⫻. A video-image frame grabber digitized the picture into 480 lines⫻ 512 pixels and assigned to each one a gray level with an 8-bit resolution.

C. Drop edge

An edge-detection routine was performed as follows. The change in the gray level ranged from black 共0兲 inside to bright light 共255兲 outside across a few pixels. The change was not a stepwise increase from 0 to 255. The variation was nearly symmetric and the change was almost linear around a gray level of 127.5. Therefore, the edge was defined as the x or z position that, for the interpolated line, corresponded to an intensity of 127.5. A different approach was presented by Liggier and Passerone28 which an acquisition module was applied. Note that the edge location procedure was performed in the coordinate direction in which the normal to the surface has the larger component. Thus, near the apex the edge location was performed along the z direction, while near the sides of the drop, it was carried out along the x direction. For the drop interface near the three-phase contact point, the edge location was performed along both x and z directions. The air-solid surface was also located by finding the edge along the vertical direction. The image-forming system was calibrated by digitizing a stainless steel sphere with a known diameter of 1.581 mm. The coordinates of the digitized steel ball were processed to calculate the average length between pixels along a row and along a column, i.e., the pixels per centimeter horizontally and vertically.

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There were approximately 800⫾ 200 measured coordinates of the drop edge for each sessile-drop image. The contact angle was computed from the drop-edge data and either the modified selected-plane method or the best-fit method in the following ways. To determine the best fit with the theoretical drop profile, an objective function E was defined as the sum of the squares of the normal distance dn between the measured points un and the calculated curve v obtained from the integration of N 关dn共un , v兲兴2, where N is the total number Eq. 共2兲 共i.e., E = 兺n=1 of edge points兲. The objective function depends on four unknown variables, X0 and Z0 共the actual location of the apex兲, R0, and the capillary constant B. To obtain dn for each point, a theoretical drop profile consisting of 12 000⫾ 2000 points was first generated, and then dn was computed as the distance from the experimental edge point to the closest of the discrete points of the theoretical drop profile. To obtain the optimum congruence between the theoretical profile and the data points, the objective function was minimized with respect to the four parameters 共⳵E / ⳵qi = 0, i = 1 – 4兲. Minimization equations were solved by directly applying the Newton–Raphson method. The contact angle ␪ was then computed from the optimum values 共R, B, and Z0兲 and H 共the distance between the apex and the air-solid base line兲. To determine the contact angle ␪ using the modified selected-plane method, the digitized edge coordinates of the sessile drop were used. First, the drop-shape function XAi was calculated for the last measured drop edge, which was very close to the three-phase contact point. Two interfacial loci, P1 and P2, were then selected based on a set of XA1 and XA2, selecting from the available set with the smallest XAi in Table II. In other words, we chose the set of XA1 and XA2 for which P1 and P2 were closest to the contact point. Once P1 and P2 were selected, we obtained the drop-shape factor XA1 / XA2. The capillary constant B and the radius of curvature at apex R0 were then calculated from Eq. 共5兲. When the classical selected-plane method was applied, the equator and the drop apex were obtained from the measured edge points. Xe and Ze were then calculated 共Fig. 1兲, after which B and R0 were determined from Eq. 共4兲. The contact angle was computed from the theoretical profile of the sessile drop 共fixed by the B and R0 values obtained above兲 and the droplet height H. Because it is not straightforward to calculate the theoretical drop curve, a compiled program is provided in the supplementary material. To use the program, one simply inputs the three items of droplet information 共B, R0, and H兲 listed in a window and the program returns the contact angle. IV. ILLUSTRATIONS

In the following, we first show the image analysis for a stainless steel ball to demonstrate the potential hardware errors in the determination of the contact angle ␪ and of the edge coordinates at the region adjacent to the contact point. Note that it is a key issue in the contact-angle measurement to understand the possible errors in the measurement system

100

a

200

Z (pixel)

D. Contact angle

300

130 348

Z (pixel)

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349 350

230 X (pixel)

330

b

351 100

200 X (pixel)

300

400

FIG. 6. 共Color online兲 共a兲 Digitized image 关inset in 共a兲兴 and edge coordinates 共O兲 of a stainless steel ball on a glass slide surface and 共b兲 the air-glass interface with 共+兲 and without 共O兲 a stainless steel ball on the slide from the video-image digitization equipment.

especially for the superhydrophobic surfaces. Therefore, two superhydrophobic surfaces 共a lotus leaf and a CF4-modified PP film兲 were used to analyze the measurement error of the widely used sessile-drop method. The advantages of and an accuracy analysis of ␪, found using the modified selectedplane method, are also demonstrated at the end of this section. A. Difficulty in finding the contact point

To demonstrate the system error inherent in determining the contact angle, the video-enhanced sessile-drop apparatus detailed above was used. Figure 6共a兲 shows the images and edge coordinates of a stainless steel ball on a microscope slide. The edge coordinates were fitted to a circle nearly perfectly except at the region adjacent to the three-phase contact point. The stainless steel ball is equivalent to a liquid drop with infinite surface tension and presents a system with an 180° contact angle on a solid substrate. Figure 6共b兲 shows both the air-glass interfaces 共base line兲 with and without a steel ball on the glass. The base line shows nearly no change 共⬍0.05 pixels兲 except for the region adjacent to the ball-glass contact point. The data clearly demonstrate that the flatness variation of the base line in the x direction is only around 0.2 pixels 共about 2.5 ␮m兲 in the 5 mm window region. It is reasonable to assume that the flatness variation is similar in both the x and y directions. Based on the information for the flatness variation, it is believed that the stainless steel ball always stands on a concave position on the glass surface. Although the light source employed in this apparatus is nearly perfectly parallel, an unavoidable tilting of the glass substrate in the y direction still exists. Therefore, a small part of the bottom of the spherical ball was unavoidably shielded. In other words, a small part of the contour of the spherical interface in the vicinity of threephase contact point became undetectable. This undetectable

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Z

343 344 345 346 347 348 349 350 351 352 353

a

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 9 24 50 0 0 0 0 0 0 0 0 0 0 2 10 24 48 85 136 0 0 0 0 0 0 0 0 7 17 34 59 95 140 188 231 0 0 0 0 0 0 9 24 49 81 121 164 205 239 255 255 0 0 0 0 10 31 63 106 158 209 254 255 255 255 255 255 0 0 9 29 57 96 142 192 233 255 255 255 255 255 255 255 7 19 40 67 98 127 152 173 185 190 191 192 192 192 192 193 19 32 46 61 74 83 88 92 94 94 93 97 99 97 95 96 16 23 28 34 38 40 42 43 44 43 44 46 46 45 45 46 10 13 16 19 22 24 25 25 24 24 24 24 24 24 24 23 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 X

350

b

(225.9, 350.2)

(229.2, 350.8)

θ = 170o

Z (pixel)

351 (232.6, 351.4)

174o

(236.0, 351.7)

352

225

172o

(253.1, 351.4)

176o (249.7, 351.7) 180o

(239.4, 351.9) 178o (246.3, 351.9) (242.8, 352.0)

235

245

X (pixel)

255

350 255

(259.8, 350.2)

(256.5, 350.8)

340 Z (pixel)

065105-6

265

FIG. 7. 共Color online兲 共a兲 The gray level for the region around the detected contact point and 共b兲 imaginary edge coordinates 共O兲 and contact angle around the contact point. Open circles and pluses in 共a兲 indicates the detected edge points.

interface region for a spherical ball with a diameter of 2 mm covered about 50 pixels 共⬃0.63 mm兲 in the x direction in our sessile-drop system. It is highly possible that the three-phase contact point for any liquid sessile drop is hardly to be the real contact point unless the solid substrate is perfectly flat and there is no tilting. This prompts the following question: How far is the distance between the real contact point and the one detected for a liquid sessile drop on a solid substrate? In this illustration, a spherical ball mimics a liquid drop with infinite surface tension and presents an 180° contact angle on a solid substrate. Note that this example demonstrated the largest possible error for contact-angle measurement. This demonstrates that a more superhydrophobic surface results in a more serious deviation in the ␪ measurement under the condition that some lower part of the liquid drop was shielded, as demonstrated for the case shown in Figs. 6 and 7 共a ball sitting on a concave site with a certain surface tilting兲. This phenomenon will be discussed in further detail later in the text. B. Difficulty in finding the liquid-solid interface

Figure 7共a兲 shows the gray level for the region in the vicinity of the detected right contact point for the image shown in Fig. 6共a兲. Figure 7共b兲 presents the spherical contour 共solid curve兲 and parts of the imaginary edge coordinates 共open circles, predicted from the best-fit spherical profile兲 at the lowest part of the ball. Figure 7共b兲 clearly demonstrates that the lowest edge point was located at x = 242.8 共pixel兲 but the detected right contact point 关Fig. 7共a兲兴 was found at x = 267.5 共pixel兲, which departs nearly 25 pixels 共about 0.32 mm兲 from the real contact point. This result demonstrated the difficulty in finding the real contact point. This 25-pixel

265 X (pixel) 275

285

FIG. 8. 共Color online兲 Best-fit circle 共curve兲, base line, and the edge points 共O兲 near the right detected contact point for the stainless steel ball on the slide. The solid and dashed lines represent the tangent lines using the edge points of different regions near detected contact point. Both tangent lines resulted in a value of ␪ = 152°.

variation was a result of the mentioned concave surface site, the glass tilting in the y direction, and possibly also the image distortion of the light passing the narrow slit between the spherical ball and flat glass surfaces. The liquid-solid interface is not available from the image-processing technique; therefore, it is common to determine the liquid-solid surface from the air-solid interface nearby, as demonstrated in the regions between the dashed lines in Fig. 6共b兲. The edge points obtained from Fig. 6 are shown in Fig. 7共a兲 as open circles. The data points obtained from the best-fit spherical profile 关Fig. 7共b兲兴 clearly demonstrated that the lowest point of the spherical ball was located at y = 352.0 共pixel兲, but the base line determined from Fig. 6 was located at y = 350.6 共pixel兲. An unavoidable discrepancy of 1.4 pixels 共0.018 mm兲 resulted from this measurement. A contact angle of 171.5° was obtained from the measured base line 关Fig. 7共b兲兴, which is in disagreement with the theoretical 180° for the ideal spherical ball. An 8° difference in the contact-angle determination thus resulted from this small 1.4 pixels of error in base line judgment. This is a particularly important issue in the contact-angle determination of superhydrophobic surfaces. C. Image distortion

When parallel light passes through a narrow slit, there commonly exists an image distortion detected from the video camera. Figure 8 shows the deviation between the best-fit spherical profile and the edge points adjacent to the contact point. A 0.6 pixel deviation clearly occurred for the 20 points close to the detected contact point 共at x = 268 pixel兲, which was significantly large compared with the standard deviation 共⬍0.07 pixels兲 for the best-fit between a spherical profile and all of the edge points of the spherical ball. In addition to the previously mentioned 25 pixels 共due to the 1.4 pixels in the z direction shielded from the concave surface site and surface tilting兲, these 20 edge points should be neglected in the contact-angle determination. Two measurements were performed by applying the tangent method for contact-angle determination using the edge points immediately next to the detected contact point. A contact angle of 152° was obtained 共shown in the black solid and dashed lines in Fig. 8兲 for both tangent lines. Note that a 171.5° contact angle resulted from

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TABLE III. Comparison of the imaginary left-side edge coordinates of a water drop on a lotus leaf predicted from the best-fit YL curve and modified selected-plane 共P1 , P2兲 method at ␾ = 152° – 180°.

a

YL best-fit

Z (pixel)

80

180

280 100

252

276

Z (pixel)

262 272

200 X (pixel) 300

b

280 150

282 292 133

153 173 X (pixel)

160

193

FIG. 9. 共Color online兲 共a兲 Digitized image 关inset in 共a兲兴, edge coordinates of a sessile water drop on a lotus leaf, and the best-fit YL curve 共solid curve, ␪ = 159°兲 and 共b兲 the curves predicted from the modified selected-plane method 共dashed curve, ␪ = 156°兲 and from the classic selected-plane method 共dotted curve兲, edge coordinates 共small O兲 and imaginary edge points 共large O and +兲 around the left three-phase contact point. The inset in 共b兲 shows all of the edge points from the imaging process. The symbol “big ┼” shows the selected interfacial loci P1 and P2 used in the modified selected-plane method. B = 0.332 79, R0 = 122.92 pixels, and H = 189.47 pixels.

the best-fit analysis. It is suggested that the first several available edge points should not be used when a superhydrophobic surface is considered.

P1, P2 method

␾

Xleft

Z

Xleft

Z

180 176 172 168 164 160 156 152

191.6 188.1 184.3 180.3 176.2 171.9 167.5 163.1

288.61 288.49 288.08 287.38 286.34 284.95 283.18 281.04

191.6 188.0 184.3 180.3 176.2 171.9 167.6 163.2

288.21 288.09 287.69 286.98 285.96 284.57 282.80 280.66

line in the inset of Fig. 9共b兲兴 resulted from this 3° of difference in ␪ for the modified selected-plane method. Table III shows an illustrative example of how the base line misjudgment can affect the determination of the contact angle. A 3° contact-angle difference can result from a 1.3pixel variation in the z direction at ␪ = 156– 159°. Note that for a more superhydrophobic surface, a larger contact-angle deviation would result from the same variation in the z direction. For example, 0.70 and 0.41 pixel variations in the base line can result in a 4° difference at ␪ = 168° – 172° and at ␪ = 172° – 176°, respectively. Recall that there exists a 1.4 pixel base line uncertainty in our sessile-drop system. Therefore, a 3° ␪ variation is inherent in the system uncertainty at ␪ = 159° for a water drop on a lotus leaf. The classic selected-plane method 共using the Xe / Ze information兲 was also applied 关shown as the dotted curve in Fig. 9共b兲兴 for a water drop on a lotus leaf. Drop parameters B = 0.210 01 and R0 = 113.32 pixels were obtained, yielding a contact angle of 150°. A larger deviation was clearly found for the classic selected-plane method.

D. Example 1: Lotus leaf

Figure 9共a兲 shows the images and edge coordinates of a sessile water drop on a lotus leaf. The edge coordinates can be fitted by the theoretical profile of the sessile drop obtained from the YL equation, except within a small region 共five pixels in the z direction兲 adjacent to the three-phase contact points. A region adjacent to the left three-phase contact point is shown in the inset of Fig. 9共b兲. A contact angle of 159° was obtained from the best-fit YL curve and the air-solid base line. The modified selected-plane method proposed above was applied. The red cross symbols shown in Fig. 9共a兲 are the selected points P1 and P2. Equation 共5兲 and Table II were applied and B = 0.3328 and R0 = 122.92 pixels were obtained. With the information of droplet-height 共H = 189.47 pixel兲, a contact angle of 156° was obtained. The YL best-fit profile resulted from all of the edge points 共several hundred兲 above the base line 关excluding the data for the last five pixels above the base line, shown in Fig. 9共b兲兴. The modified selected-plane method employed only five data points 共apex, P1, P2, and the two points determined for the base line兲. It is presumed that a more accurate contact angle can be determined with the YL best-fit method. A slightly different sessile-drop profile 关shown as the dashed

E. Example 2: Modified PP

Figure 10共a兲 shows the images and edge coordinates of a sessile water drop on a CF4 plasma-treated PP film. A theoretical sessile-drop profile fits the edge points nearly perfectly. The insect of Fig. 10共b兲 shows the data near the left three-phase contact point. In general, the edge coordinates can be fitted by a theoretical sessile-drop profile derived from the YL equation except at a small region 共2–5 pixels in the z-direction, case by case兲 adjacent to the three-phase contact points. A contact angle of 163° was obtained from the best-fit YL profile and the air-solid base line. The red cross symbols shown in Fig. 10共a兲 represent the selected points P1 and P2 for the modified selected-plane method. Equation 共5兲 and Table II were applied, and B = 0.3074 and R0 = 116.5 pixel were obtained. With the droplet-height information 共H = 184.75 pixels兲, a contact angle of 162° was obtained. In this case, only one degree of deviation was found. When the classic selected-plane method was applied 关shown as the dotted curve in Fig. 10共b兲兴 for a water drop on PP film, drop parameters B = 0.3253 and R0 = 106.02 pixels were obtained. However, the drop profile did not reach the

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80 Z (pixel)

a

180

280 100 240

265

b

269

Z (pixel)

250

FIG. 11. 共Color online兲 The window panel of the program for calculating the contact angle by inputting the capillary constant, radius of curvature at the drop apex, and the droplet height.

200 X (pixel) 300

260

180

190

270 165

185 205 X (pixel)

225

FIG. 10. 共Color online兲 共a兲 Digitized image 共inset in a兲, edge coordinates of a sessile water drop on CF4 plasma-modified PP film, and the best-fit YL curve 共solid curve, ␪ = 163°兲 and 共b兲 the curves predicted from the modified selected-plane method 共dashed curve, ␪ = 162°兲 and from the selected-Xe/ Ze-plane method 共dotted curve兲, and imaginary edge points 共large O and +兲 around the left three-phase contact point and 共b兲, edge coordinates 共small O兲. The inset in 共b兲 shows all of the edge points from the imaging process. The symbol big ┼ shows the selected interfacial loci P1 and P2 used in the modified selected-plane method. B = 0.307 36, R0 = 116.50 pixels, and H = 184.75 pixels.

base line in this case. It was concluded that for superhydrophobic surfaces 共i兲 the selected Xe / Ze plane method is not always suitable and 共ii兲 the modified selected-plane method works well for contact-angle measurement. F. User-friendly program

For the users’ convenience, this work supplies a userfriendly program 共as the supplementary material for this manuscript兲 for calculating the contact angle by simply inputting the drop information, i.e., the capillary constant B, the radius of curvature at the drop apex R0, and the droplet height H. Fig. 11 shows the window panel of this program. One can test-run the program using the two examples discussed above for water drops on a lotus leaf and on modified PP to become familiar with the program. The drop information B, R0, and H are listed in the captions of Figs. 9 and 10. By pressing the “run” button, the contact angle result is returned. Moreover, by selecting the “output” box on the panel, one can obtain an output file that contains the sessile-drop profile derived from the YL equation. V. DISCUSSION AND CONCLUSIONS

The sessile-drop technique is the method most widely used for contact-angle measurement. However, due to the difficulties in locating the real contact point and finding the real air-solid base line, this method may result in significant error. This work proposes that it is better to use only the drop information which is a couple of pixels above the base line to

avoid the problem of contact point determination. The classic selected-plane method is a well-known technique that requires only three data inputs 共coordinate of the apex, diameter of the equator, and the droplet height or the base line location兲. The modified selected-plane method in this work employs the drop information close to the contact point instead of the diameter of the equator. Therefore, a more accurate contact angle is obtained. A user-friendly program is also provided to calculate ␪ by simply inputting B, R0, and H. The above analysis on contact-angle measurement accuracy is only a representative illustration. The experimental error using a sessile-drop goniometer depends also on the resolution of the video camera, the light-source quality, and the drop volume. Of course, the parameters such as material uniformity, surface flatness, and surface tilting are also important. The most popular WCA measurement method, the sessile-drop with tangent-line technique, suffers from difficulties in finding the real contact point and from shape distortion due to the light passing through a narrow slit in the region near the contact point. The modified selected-plane method proposed in this study is capable of avoiding the above-mentioned difficulties associated with the sessiledrop-tangent method. Additionally, the modified selectedplane method can save the trouble of best-fitting the several hundreds of edge points with the theoretical YL drop profile to find the drop-shape factors 共capillary constant B and radius curvature at apex R0兲. In this work, several tasks were performed to improve the experimental accuracy. Although the resolution of the video camera used in this work was only 512⫻ 512, a highquality parallel light was produced by using several lenses with a pinhole and a filter. The tilting of the solid substrate was carefully adjusted to reduce the variation to ⬍1 pixel in both the x and y directions. A laboratory-made edgedetermination program was applied to reduce the error in edge location to 0.1 pixels. During the experiment, the temperature variation of the system 共water drop and solid substrate兲 was no more than ⫾0.1 K. The effect of droplet volume on the measurement error is currently under investigation in our laboratory. ACKNOWLEDGMENTS

This work was supported by the National Science Council of Taiwan, Republic of China 共NSC 97-2221-E-011-047MY3兲.

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