Article pubs.acs.org/Langmuir
Indirect Methods to Measure Wetting and Contact Angles on Spherical Convex and Concave Surfaces C. W. Extrand* and Sung In Moon Entegris, Inc., 101 Peavey Road, Chaska, Minnesota 55318, United States ABSTRACT: In this work, a method was developed for indirectly estimating contact angles of sessile liquid drops on convex and concave surfaces. Assuming that drops were sufficiently small that no gravitational distortion occurred, equations were derived to compute intrinsic contact angles from the radius of curvature of the solid surface, the volume of the liquid drop, and its contact diameter. These expressions were tested against experimental data for various liquids on polytetrafluoroethylene (PTFE) and polycarbonate (PC) in the form of flat surfaces, spheres, and concave cavities. Intrinsic contact angles estimated indirectly using dimensions and volumes generally agreed with the values measured directly from flat surfaces using the traditional tangent method.
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INTRODUCTION Sessile liquid drops are widely used to assess wettability.1,2 Usually, a small drop is gently deposited on a horizontal solid surface. A small amount of liquid may be added to or withdrawn from the drop to advance or retract its contact line; then, a contact angle, depicted in Figure 1a, is measured. In the tangent method, a horizontal baseline is drawn that passes through the triple point on the side of the drop where the liquid, solid, and surrounding gas meet. Another line, also
originating from the triple point, is drawn tangent to the drop. The angle between the baseline and tangent line defines the intrinsic contact angle (θ0) of a flat, horizontal surface. The tangent method described above lends itself to flat surfaces with unobstructed side views. However, there are many scenarios where it is desirable to measure wettability of more complex surfaces. Surfaces that are rough may hamper establishment of a well-defined baseline. Alternatively, geometry may preclude an unobstructed side view. In these cases, contact angles can be measured indirectly.3−7 For example, if the drop is sufficiently small that no gravitational distortion occurs such that the drop has the proportions of a spherical segment, θ0 can be estimated from the base diameter (2a) and drop volume (V)8 ⎧⎡ ⎛ ⎪ ⎢ 48V 48V ⎪ ⎢ π(2a)3 + ⎜⎝4 + π(2a)3 ⎪⎣ θ0 = 2·arctan⎨ ⎡ ⎪ ⎛ ⎪ 21/3⎢ 48V 3 + ⎜4 + ⎪ ⎝ ⎢⎣ π(2a) ⎩
2/3 2 ⎞1/2 ⎤
( ) ⎟⎠
⎥ ⎥⎦
−2
1/3 2 ⎞1/2 ⎤
( ) ⎟⎠ 48V π (2a)3
⎫
2/3 ⎪
⎥ ⎥⎦
⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭
(1)
This expression is accurate for flat, horizontal surfaces. However, if a surface has curvature, it is not relevant. Therefore, in this study, we develop and experimentally test a method to indirectly measure contact angles on spherical convex and concave surfaces.
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THEORY While the shape of drops on curved surfaces has been studied previously,9 to the best of our knowledge, adaptations of eq 1 for curved surfaces do not exist.10 Therefore in this section, we Figure 1. Illustrations of small, sessile liquid drops of volume V on various types of surfaces: (a) on a flat, horizontal surface, (b) on a solid sphere, or (c) inside of a hemispherical cavity. © 2012 American Chemical Society
Received: March 29, 2012 Revised: April 23, 2012 Published: April 24, 2012 7775
dx.doi.org/10.1021/la301312v | Langmuir 2012, 28, 7775−7779
Langmuir
Article
⎧⎡ ⎛ 48Vt ⎪ ⎢ 48Vt ⎪ ⎢ π(2a)3 + ⎜⎝4 + π(2a)3 ⎪⎣ θ0 = 2·arctan⎨ ⎡ ⎪ ⎛ ⎪ 21/3⎢ 48Vt 3 + ⎜4 + π (2a) ⎪ ⎝ ⎢ ⎣ ⎩
derive working equations for indirectly estimating contact angles from drops on convex and concave surfaces. Convex Case. Consider a sessile liquid drop of volume V on a solid sphere, as depicted in Figure 1b. Assume that the drop was deposited such that the liquid is symmetrically centered, i.e., the vertical z axis of the sphere passes through the apex of the drop. Also, assume that the drop is not distorted by gravity and has spread along the curved surface to produce an intrinsic contact angle, θ0, which is the angle that one would expect to measure if the liquid were deposited on a flat, horizontal surface of the same composition.11 Due to the curvature of the sphere, the apparent contact angle, θ, measured relative to the horizon is greater than the intrinsic value, θ0. The deviation of this convex geometry from a flat, horizontal surface is given by the curvature angle (ϕb). From Figure 1b, the intrinsic contact angle is the difference between the apparent value and the curvature angle θ0 = θ − ϕb
( ) ⎟⎠
Vt = V +
1/2 ⎡ ⎛ 2a ⎞⎤ ⎡ ⎛ 2a ⎞2 ⎤ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ = 1− cos⎢arcsin ⎝ 2R ⎠⎥⎦ ⎣ ⎝ 2R ⎠ ⎦ ⎣
π (2a)3
⎥ ⎥⎦
⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭
(8)
1/2 ⎧ ⎡ ⎪ ⎛ 2a ⎞2 ⎤ 1 π ·R3⎨2 − 3⎢1 − ⎜ ⎟ ⎥ ⎪ ⎝ 2R ⎠ ⎦ 3 ⎣ ⎩
(9)
Concave Case. Consider a sessile liquid drop of volume V in a spherical cavity of a solid (Figure 1c). As before, assume that the drop is symmetrically centered, is not distorted by gravity, and has spread to produce an intrinsic contact angle, θ0. The analysis of the concave case differs from that of the convex case in two aspects. First, the intrinsic contact angle (θ0) is the sum of the apparent value (θ) and the curvature angle (ϕb) θ0 = θ + ϕb
(10)
Second, the apparent drop volume on the concave surface (Vc) is less than V
Vc = V − Vb
(11)
By following the approach given above, we arrive at an expression that allows for indirect contact angle estimation on concave surfaces in terms of dimensions and volume ⎧⎡ ⎛ 48Vc ⎪ ⎢ 48Vc ⎪ ⎢ π(2a)3 + ⎜⎝4 + π(2a)3 ⎪⎣ θ0 = 2·arctan⎨ ⎡ ⎪ ⎛ ⎪ 21/3⎢ 48Vc 3 + ⎜4 + ⎪ ⎝ ⎢⎣ π(2a) ⎩
2/3 2 ⎞1/2 ⎤
( ) ⎟⎠
(4)
The volume of the wetted spherical segment is related to the curvature angle12
By using the following trigonometric relation
1/3 2 ⎞1/2 ⎤
3/2 ⎫ ⎡ ⎛ 2a ⎞2 ⎤ ⎪ + ⎢1 − ⎜ ⎟ ⎥ ⎬ ⎝ 2R ⎠ ⎦ ⎪ ⎣ ⎭
(3)
1 π ·R3(2 − 3 cos ϕb + cos3 ϕb) 3
−2
where
A value of θ can be estimated using a modified form of eq 1, where the volume of the liquid drop (V) must be replaced by apparent drop volume (Vt), which on a convex surface includes an additional volume associated with the wetted portion of the sphere (Vb)
Vb =
48Vt
⎫
2/3 ⎪
⎛ 2a ⎞ − arcsin⎜ ⎟ ⎝ 2R ⎠
In order to indirectly solve for θ0, the angles θ and ϕb must be given in terms of dimensions and drop volume. The sphere has a diameter of 2R. The chord that intersects the contact line and passes orthogonally through the vertical axis of the sphere has a length of 2a. The curvature angle (ϕb) can be estimated from the sphere and drop dimensions as
Vt = V + Vb
⎥ ⎥⎦
( ) ⎟⎠
(2)
⎛ 2a ⎞ ϕb = arcsin⎜ ⎟ ⎝ 2R ⎠
2/3 2 ⎞1/2 ⎤
⎥ ⎥⎦
π (2a)3
⎛ 2a ⎞ + arcsin⎜ ⎟ ⎝ 2R ⎠
(5)
−2
1/3 2 ⎞1/2 ⎤
( ) ⎟⎠ 48Vc
⎫
2/3 ⎪
⎥ ⎥⎦
⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭
(12)
where
12
Vc = V − (6)
the curvature angle can be eliminated and Vb can be written solely in terms of 2a and R
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1/2 3/2 ⎫ ⎧ ⎡ ⎡ ⎪ ⎛ 2a ⎞2 ⎤ ⎛ 2a ⎞2 ⎤ ⎪ 1 3 ⎜ ⎟ ⎜ ⎟ ⎥ + ⎢1 − ⎥ ⎬ Vb = π ·R ⎨2 − 3⎢1 − ⎪ ⎝ 2R ⎠ ⎦ ⎝ 2R ⎠ ⎦ ⎪ 3 ⎣ ⎣ ⎩ ⎭
1/2 ⎧ ⎡ ⎪ ⎛ 2a ⎞2 ⎤ 1 π ·R3⎨2 − 3⎢1 − ⎜ ⎟ ⎥ ⎪ ⎝ 2R ⎠ ⎦ 3 ⎣ ⎩
3/2 ⎫ ⎡ ⎛ 2a ⎞2 ⎤ ⎪ + ⎢1 − ⎜ ⎟ ⎥ ⎬ ⎝ 2R ⎠ ⎦ ⎪ ⎣ ⎭
(13)
EXPERIMENTAL DETAILS
The liquids and solids were chosen to produce a wide range of contact angles, which exhibited both acute and obtuse values. The liquids used in the experiments were 18 MΩ·cm deionized (DI) water, ethylene glycol (EG, Sigma-Aldrich, anhydrous 99.8%), and hexadecane (HX, Alfa Æsar, 99%). Surface tension (γ) and density (ρ) of the liquids are summarized in Table 1.1,13,14 The solid surfaces were polytetrafluoroethylene (PTFE) and polycarbonate (PC).
(7)
By combining eqs 1−4 and 7, we arrive at an expression that allows for indirect contact angle estimation on convex surfaces in terms of dimensions and volume 7776
dx.doi.org/10.1021/la301312v | Langmuir 2012, 28, 7775−7779
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RESULTS AND DISCUSSION Wettability of the Flat Surfaces. Figure 2a shows an 8 μL drop of water on a flat PTFE surface. The drop is sufficiently
Table 1. Properties of the Liquids liquid
γ (mN/m)
ρ (kg/m )
water ethylene glycol hexadecane
72 48 28
998 1110 773
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For convex surfaces, solid PTFE and PC spheres were purchased from McMaster Carr. Their diameters (2R) ranged from 3.2 mm (1/8 in.) to 25.4 mm (1 in.). For concave surfaces, PTFE (Entegris Hy-Q) and PC (McMaster Carr) stock were first machined to rectangular blocks with fly cutters and then hemispherical cavities were cut with ball end mills. The cavity diameters (2R) ranged from 6.4 mm (1/4 in.) to 25.4 mm (1 in.). To create a smooth optical finish after machining, the PC cavity blocks were vapor-polished. Direct measurements of intrinsic contact angles (θ0) were made on a flat portion of each solid. For spheres, a flat PTFE surface was made by machining away a section from a 25.4 mm sphere and then polishing with 600 mesh sand paper. PC was flattened by placing a 12.7 mm PC sphere between Al sheets and slowly applying 4500 kg of force via a platen press (PHI Bench Design Hydraulic Compression Press). Prior to use, the polymer surfaces were rinsed with isopropanol (Brenntag Co.) and 18 MΩ·cm DI water, then blown dry with clean, filtered air. Spheres were immobilized using double-sided adhesive tape to keep them from rolling. Small liquid drops were gently deposited from a one-milliliter, glass syringe (M-S, Tokyo, Japan) or a micropipet (Eppendorf Reference Series 2000, 2−20 μL). To minimize gravitational distortion, drop volumes (Vi) were selected according to the following criterion8
Figure 2. 8 μL water drops on PTFE. (a) Side view on a flat surface. (b) Side view on top of a sphere, 2R = 6.4 mm. (c) Plane view at the bottom of a spherical cavity, 2R = 6.4 mm.
small that gravity does not distort its shape. Thus, it is spherically proportioned and exhibits an advancing contact angle of 108°. The flattened spheres and flat areas of concave surfaces showed almost identical contact angles for each liquid−solid combination we tried. Values are listed in Table 2. They ranged from 41° for hexadecane on PTFE to 108° for water on PTFE. These values generally agreed with those reported in the literature.1,14,15 Table 2. Advancing Intrinsic Contact Angles (θ0) from Flat, Convex, and Concave Surfaces θ0 (°) Flat
Convex
Concave
solid
measured directly by tangent method
estimated indirectly from eqs 8 and 9
estimated indirectly from eqs 12 and 13
PTFE PC PTFE PC PTFE
108 89 89 60 41
107 90 91 ---
106 88 -56 36
3/2 1 ⎛γ ⎞ θ⎛ θ⎞ Vi < π ⎜ ⎟ tan ⎜3 + tan 2 ⎟ 48 ⎝ ρg ⎠ 2⎝ 2⎠
⎡⎛ ⎤3 2 ⎞1/2 ⎢⎜1 + 8 sin θ ⎟ − 1⎥ ⎢⎣⎝ ⎥⎦ 1 − cos θ ⎠
liquid water
(14)
ethylene glycol
where i = t for drops on convex surfaces, i = c for drops on concave surfaces, γ is the surface tension of the liquid, ρ its density, and g is the acceleration due to gravity (9.81 m/s2). For the flat surfaces, images were captured with a drop shape analyzer (Krüss DSA10), and intrinsic contact angles (θ0) were measured by the tangent method using the resident software. For spheres, each experiment began by depositing a 1 μL drop at the apex of a sphere and then immediately adding another 1 μL to advance the contact line. Additional liquid was sequentially deposited in 2 μL increments. At each 2 μL increment (2 μL, 4 μL, 6 μL, ...), an image was captured. Apparent contact angles (θ) were measured using the drop shape analyzer. Contact line chords (2a) and sphere diameters (2R) were estimated from images using Image-Pro Plus software. For the concave cavities, plan view images of the drops were captured from above with a stereo optical microscope (Nikon SMZ 1500 with a DSFi1 digital camera) at a magnification of 10×. The contact diameter (2a) was measured using resident software. Unless stated otherwise, contact angles were advancing values. All measurements were performed at 25 ± 1 °C. The standard deviation of the dispensed liquid volume from the syringe or pipet was around 1%. Standard deviation of the directly measured θ0 values was ±1−2°. For the curved surfaces, the standard deviation in 2R was
