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C: Surfaces, Interfaces, Porous Materials, and Catalysis

A General Way to Compute the Intrinsic Contact Angle at Tubes Chengjie Xiang, and Lidong Sun J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b08100 • Publication Date (Web): 27 Nov 2018 Downloaded from http://pubs.acs.org on November 28, 2018

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The Journal of Physical Chemistry

A General Way to Compute the Intrinsic Contact Angle at Tubes Chengjie Xiang,1 Lidong Sun1,2, 1State

Key Laboratory of Mechanical Transmission, School of Materials Science and Engineering,

Chongqing University, Chongqing 400044, PR China 2National

Engineering Laboratory of Highway Maintenance Technology, Changsha University of

Science & Technology, Changsha 410114, PR China AUTHOR INFORMATION Corresponding Author * E-mail: [email protected]

ACS Paragon Plus Environment

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ABSTRACT: Nanostructured coatings of superwettability, especially at the outer and inner surfaces of large-scale tubes, have generated broad interests for anti-biofouling, drag reduction, condensation heat transfer, and so on. The contact angle, a major parameter to evaluate the wettability, is derived and gradually modified mainly for flat surfaces after Thomas Young (1773—1829). However, in the case of tubular surfaces, the curvature affects and biases the measurement. In this article, a tube-in-tube coaxial anodization approach is developed to generate coatings of nanotube arrays at the outer surface of Ti tubes. The wetting property of the resulting nanotubes is tailored; the apparent contact angle (measured in the radial plane) deviates substantially from its intrinsic value (measured in the axial plane) as the tube diameter decreases or droplet size increases. A general equation, suitable for tubular surfaces of different wetting states, is hence derived to compute the intrinsic contact angle in terms of apparent contact angle, droplet radius and tube radius. Intrinsic contact angles calculated using the equation agree well with the values measured directly from the corresponding flat surface, validating its effectiveness. This study provides a theoretical basis to assess the wetting property of tubular or curved surfaces by eliminating the curvature effect.


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INTRODUCTION Wettability is a key factor concerned in surface science that involves multidisciplinary fundamentals of physics, chemistry and materials science. Engineering materials surface at micro and/or nanoscale promises well-textured surfaces of superwettability. This has been found important applications in self-cleaning,1 anti-biofouling,2 anti-icing,3-6 enhanced heat transfer,7-12 drag reduction,13-15 and oil-water separation.16 Theoretical and experimental studies on the longterm stability of superhydrophobic state accelerate their development toward real-world application.14-15, 17-19 In this regard, such surfaces of superwettability are of particular interest when prepared at either inner or outer surfaces of large-scale tubes.20-22 As a prerequisite for practical application, the wettability should be well evaluated in a suitable manner. Contact angle is the main index used to quantify the wettability. A number of methods have been developed to determine the contact angle. As early as 1805, Thomas Young described the contact angle on an ideal surface.23 In 1936, Wenzel modified Young’s equation by considering surface roughness.24 Cassie and Baxter further derived an equation suitable for surfaces of high roughness or high porosity in 1944.25 Bigelow and coworkers first measured the contact angle directly by using a telescope equipped with a goniometer eyepiece in 1946.26 This develops into the sessile drop method that has been widely used today. All of these major developments are focused on the equilibrium contact angle on flat surface, unsuitable for curved ones which are, however, very common in many practical applications. In this case, the surface curvature biases the apparent contact angle from its intrinsic value. A few studies are devoted to exploring the relationship between the apparent contact angle θ and the intrinsic contact angle 𝜃0, with known geometric parameters on spherical surfaces of rotational


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symmetry.27-32 Nonetheless, it is inapplicable to tubular substrates, which are of asymmetric feature in radial and axial directions, and thus more complicated than the spherical counterparts. To evaluate the contact angle on tubular surfaces, Zhu and co-workers developed a method to measure the contact angle in opaque cylindrical tubes by magnetic resonance imaging.33 It is effective but of high cost on the other hand. It is also important to note that a great difficulty arose when determining the contact point and the liquid boundaries; moreover, the signal was insufficient for thick-walled tubes. Carroll reported a method for determining the contact angle on fibers, which was analogous to the sessile drop method for flat surfaces.34 The method is applicable to microscopic fibers (e.g., 200 m diameter) and is restricted to contact angles below about 60. It is noteworthy that the method is limited to drops that are symmetrically wrapped on the fibers, whereas inapplicable to “clam-shell” type drops that are adhered to the side of the fibers. Darzi and Park corrected the contact angle inside a transparent tube by studying light refraction.35 Nevertheless, it is only applicable to tubes of high transparency, limiting its utilization scope. As such, a facile and feasible approach is highly favored to assess the intrinsic contact angle at tubes, which would boost the researches on surface science and superwettability coatings toward practical applications. In this study, a general equation is derived to calculate the intrinsic contact angle at either outer or inner surfaces of tubular substrates of different wetting properties. It is applicable to both opaque and transparent tubes, regardless of the geometrical configuration. The equation is verified by systematic studies on different tubes and water droplets, where the tube radius and droplet size are the two major variables in the equation. The computed results agree well with experimental measurements on corresponding flat surfaces.


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EXPERIMENTAL SECTION Coaxial Anodization. Titanium tubes (outer diameter: 10 mm, wall thickness: 0.5 mm, length: 50 mm, purity: 99.6%) were used as substrates. Prior to anodization, the tubes were ultrasonically cleaned in detergent, isopropanol, ethanol, and deionized water for 10 min in sequence, and dried with a nitrogen stream. Electrochemical anodization was carried out using a Keithley 2450 SourceMeter as the power supply and to record the current transients during the anodization. A tube-in-tube coaxial deployment was adopted, with a stainless steel tube (=18 mm) as the cathode and the titanium tube as the anode. The electrolyte solution was ethylene glycol consisting of 0.3 wt% ammonium fluoride and 2 vol % deionized water. Square wave voltage was employed to obtain well-exposed nanotubes. The configuration of square wave was given in Figure S1, which was performed by alternating the voltage between low (E1, t1) and high (E2, t2) voltage domain. The total anodization duration was 3 h. For superhydrophobic tubes rendering small sliding angle, a two-step anodization was used by: (1) etching in 0.1 M NaCl aqueous solution at 15 V for 1 h to obtain microstructure-patterned titanium tubes, (2) anodizing under alternating voltage at 10 V for 120 s and then 30 V for 20 s with a total duration of 3 h to produce titania nanotubes on the above microstructures. All of the anodizations were conducted at room temperature. After anodization, the samples were rinsed and ultrasonically cleaned in deionized water for 10 min and then dried in air. Hydrophobic Treatment. The obtained tubes were then immersed in n-hexane (99%, Adamas) containing 0.5 vol% Trichloro(1H,1H,2H,2H-Perfluorooctyl) Silane (TPFS, 97%, Sigma-Aldrich) for 30 min, followed by washing with n-hexane and treating under nitrogen atmosphere at 110 °C for 1 h.


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Contact Angle Measurement. Two types of opaque cylindrical tubes were employed in this study to verify the derived equations. For hydrophobic systems, polytetrafluoroethylene (PTFE) tubes were used; for hydrophilic systems, polyurethane (PU) tubes were adopted. The tubes of various radius were cut into small pieces with the length of 15 mm, and ultrasonically cleaned in detergent, isopropanol, ethanol, and deionized water for 10 min each. The water droplet was dispensed onto the small tubules by a micromachined peristaltic pump, and the volume was carefully controlled from 3 to 11 μL. Apparent contact angles of water drops were measured in air using the sessile drop method (Attension® Theta optical tensiometer from KSV Instruments LTD). The static contact angles on tubes were determined by three steps: horizontal baseline determination, circle fitting, and contact angle measurement. The contact angles on the left- and right-hand sides were separately measured for three times each, and the average value was used as the corresponding contact angle. More details of the measurements are described in the Supporting Information. Surface and crosssectional morphologies were characterized by field-emission scanning electron microscopy (FESEM, Zeiss Auriga).

RESULTS AND DISCUSSION Curvature Effect from Hydrophilic to Hydrophobic Surface. To demonstrate the curvature effect on contact angle measurement, planar and tubular surfaces of different wetting properties are used, i.e., from hydrophilic titanium metals to hydrophobic PTFE, and to superhydrophobic coatings of TiO2 nanotube arrays on titanium metals. The first batch of control experiments were conducted on bare titanium metals. The original surface of commercial titanium foils exhibits some cracks induced by manufacturing process, and


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presents a water contact angle of about 62, as shown in Figure S2 (Supporting Information). To reduce the influence of surface morphology on contact angle measurement, electrochemical polishing was applied to titanium foils and tubes, i.e., anodizing at 60 V for 3 h to produce nanotube coatings on the titanium metals and subsequently stripping off the coatings by ultrasonication to expose the fresh metal surfaces. Figure 1a-c displays the resulting surfaces on titanium foils and tubes, which exhibit uniform concave patterns originated from the nanotube bottoms that are stripped off.36 Figure S3 reveals the corresponding surfaces over a large scale, which are of similar morphology that would show negligible influence on contact angle measurement. The water contact angle on titanium foils increases from 62 (Figure S2) to 80 (Figure 1d). Very interestingly, the apparent contact angle, determined using the conventional tangent line methods, shows substantial differences when measured on planar and tubular surfaces, under almost the same surface morphology and roughness, as presented in Figure 1d-f. The apparent contact angle is overestimated at outer tube surface (87 vs. 80), whereas underestimated at inner tube surface (63 vs. 80), because of the curvature effect. The same findings are also demonstrated on hydrophobic surfaces, on which the contact angle is enlarged on the outer PTFE tube surface (115 vs. 108) while reduced on the corresponding inner surface (94 vs. 108), as shown in Figure S4 (Supporting Information).


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Figure 1. Wetting property on bare metal surfaces. Top-view FESEM images of fresh titanium surfaces after stripping off nanotube coatings on planar foils (a), at the outer (b) and inner (c) tube surfaces. Scale bar = 300 nm. The corresponding apparent contact angles determined by the conventional method are shown in (d), (e) and (f), respectively.

The superhydrophobic coatings of TiO2 nanotube arrays on titanium metals are employed to further verify the curvature effect in the range showing large contact angles (150). A tube-intube coaxial deployment was used to produce TiO2 nanotube arrays at the outer surface of Ti tubes, as shown in Figure 2a. Based on our previous studies,20-21, 37 such a coaxial scheme results in uniform distribution of the electric field along both radial and axial directions. This in turn generates uniform coatings of nanotubes at the outer surface, realizing the uniformity control. Figure 2b shows a typical cross-sectional FESEM image of titania nanotubes fabricated under potentiostatic anodization at 10 V for 3 h. The nanotubes exhibit highly oriented feature, with length of ~680 nm and outer diameter of ~40 nm. Figure 2c is the corresponding nanotube surface, which is yet fully opened and renders as nanopores. This is due to the compact oxide layer formed


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at the very beginning of anodization.38-39 To make full use of the nanotube feature for better wettability, well exposed surfaces are more desirable. In general, an effective way is to prolong the anodization duration; in this case, nanowires are gradually developed on the surface and easy to be removed via ultrasonic cleaning.39 Nonetheless, this increases the nanotube length and meanwhile weakens the adhesion strength of nanotube coatings. As such, the anodization was modified to be conducted under alternating voltage in the form of square wave, as shown in Figures S1 and S5 (Supporting Information). The applied potential (E) and anodization duration (t) were varied between low (E1, t1) and high (E2, t2) voltage domain. Figures 2d and 2e reveals that, under the alternating voltage condition, the nanotubes are well exposed with tube length of ~600 nm and outer diameter of ~85 nm. The outer diameter is enlarged because of the anodization implemented at high voltage domain (i.e., 30 V). In principle, the nanotube feature is closely related to the applied potential, which provides the driving force for nanotube growth.38, 40 Therefore, sitting at high voltage for long duration usually leads to larger and longer nanotubes. Figure 2f shows the effect of high voltage (i.e., E2) on the resulting nanotubes, while keeping the other parameters unchanged (E1=10 V, t1=120 s, t2= 20 s). It indicates that the nanotube length, outer diameter and wall thickness increase almost linearly with the E2, uncovering the key role of high voltage. On the other hand, when extending the anodization duration at low voltage (i.e., t1) and leaving the other parameters constant (E1=10 V, E2=30 V, t2=20 s), the total driving force is actually reduced, giving rise to smaller and shorter nanotubes. Figure 2g shows the varying trend of nanotube length, outer diameter and wall thickness, in line with the above discussion, with the wall thickness being slightly increased. Consequently, the nanotube features can be well tailored by designing the square wave


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configuration for different wetting states, besides the conventional parameters such as distance,38 temperature,41 electrolyte pretreatment,42 etc.

Figure 2. Nanotube features at outer tube surface. (a) Schematic illustration of the experimental setup for tube-in-tube coaxial anodization. Typical cross-sectional (b, d) and surface (c, e) FESEM images of the nanotubes prepared under potentiostatic condition at 10 V for 3 h (b, c) and that under alternating voltage at 10 V for 120 s and then 30 V for 20 s with a total duration of 3 h (d, e). The alternating voltage was employed in the manner of square wave at E1 for t1 (low voltage domain) and E2 for t2 (high voltage domain), as shown in Figure S5. Nanotube length, outer diameter and wall thickness as a function of E2 (f) and t1 (g) under square wave anodization. Scale bar = 300 nm.

According to the Cassie-Baxter equation,25 the contact angle on surfaces with high porosity or high roughness is closely associated with the area fraction of solid-liquid interface (f). It can be expressed as cos𝜃 = 𝑓(cos 𝜃0 + 1) ― 1

(1)


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where θ and θ0 are the contact angle on porous (i.e., the apparent contact angle) and flat surface (i.e., the intrinsic contact angle), respectively. Equation (1) indicates that a small f contributes to a large contact angle. For superhydrophobic TiO2 nanotubes, the area fraction f can be computed using outer diameter (d) and wall thickness (), as follows (see Figures 3a and 3b for FESEM image and geometric configuration)

𝑓=

𝑑 2 2

2

[𝜋( ) ― 𝜋( ― 𝜏) ]

1 2

𝑑 2

3 2 4𝑑

=

2𝜋(𝑑𝜏 ― 𝜏2) 3𝑑2

=

[

2π 𝜏 3 𝑑

―

2

(𝑑𝜏) ]

(2)

Equation (2) demonstrates that the f is determined by /d. Accordingly, a small wall thickness and large outer diameter gives rise to a small area fraction and thus a large apparent contact angle. Figure 3c shows that the f reaches the maximum under high voltage of E2=40 V, with the others being comparable. Figure 3d displays that the f endows the smallest values at t1=100 or 120 s. Considering the air capture capacity and coating adhesion strength, the optimal square wave condition is E1=10 V, t1=120 s, E2=30 V, t2=20 s for a total duration of 3 h. Therefore, such a condition was subsequently employed to fabricate nanotube coatings at outer Ti tube surfaces for enhanced hydrophobicity.


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Figure 3. Nanotube feature related area fraction. An FESEM image (a) and corresponding illustration (b) showing the geometric configuration of hexagonal close-packed nanotubes; Dependence of area fraction on square wave voltage (c) and time (d), calculated using Equation (2) and the data in Figures 2f and 2g, respectively.

To achieve the superhydrophobic state, surfaces with high porosity or high roughness is usually required. Figure 4 compares three kinds of morphologies with surface roughness at microscale only (Figure 4a), micro plus nanoscale (Figure 4b), and nanoscale only (Figure 4c). All of the surfaces exhibit superhydrophobic feature (contact angle over 150) upon treatment with low surface energy molecules (i.e., Trichloro(1H,1H,2H,2H-Perfluorooctyl) Silane, TPFS). Nevertheless, for surfaces showing roughness at only one scale, it endows sticky superhydrophobicity, as displayed in Figures 4a and 4c. The droplet was firmly attached to the tube surface even with 90o rotation. In contrast, the droplet was difficult to stay on Ti tube surfaces with two-scale roughness, and immediately slid off when being tilted slightly (Figure 4b). More importantly, the apparent contact angle also shows obvious differences at the outer and inner surfaces of Ti tubes as compared to that on a flat surface, though the same technique was applied to fabricate the two-scale surface roughness, as displayed in Figure 4d-f. This further indicates the curvature effect on contact angle measurement, and suggests that the conventional method is unsuitable for evaluating the true wetting behavior of tubular or curved surfaces, in view of the asymmetric geometry along the radial and axial directions.


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Figure 4. Wetting property at different tube surfaces. FESEM images of the microstructure obtained by anodization in NaCl solution (a), the nanotubes formed on the above microstructure (b), and the nanotubes alone (c). Insets are water contact angles measured at corresponding tube surfaces upon hydrophobic treatment. Apparent contact angles determined by the conventional method on outer (d) and inner (f) tube surfaces, as compared with that on a flat surface (e).

Hydrophobic Outer Surface. Figure 5a and 5b illustrates a spherical droplet residing on top of a tube in the radial and axial plane, respectively. Figure 5a shows two dominant ways that have been widely used to determine the apparent contact angle: 𝜃′0 and θ. The 𝜃′0 is the angle between the tangent line to the tube and that to the droplet, while the θ is computed using the horizontal line and the tangent line to the droplet. These two angles are easily measured either at the outer or inner tube surface, but affected by tube curvature. Figure 5b determines in fact the intrinsic contact angle 𝜃0; the droplet spreads freely along the axial direction without any influence from the curvature, which can be attained by direct measurement at the outer surface but is impossible for inner surface. Obviously, the three angles show large difference between each other, as the spherical droplet on the asymmetric tube yields a three-dimensional solid-liquid contact line that exhibits an elliptical shape in projected plane (Figure 5c). Figure 5d-5f discloses the geometric


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relationship between the droplet and the substrate; a regular sphere and tube with radius of r and R is assumed, respectively. Figure 5d shows that the two intersections produce a line segment (i.e., BB) with a length of 2a in the radial plane. Similarly, Figure 5f reveals a line segment (i.e., CC) with a length of 2b in the axial plane. The shape parameters a and b are the semi-minor and -major axes of the elliptical projection in Figure 5c. Consequently, the following relations stand 𝑎

(3)

𝜃 = 𝜋 ― 𝛼 = 𝜋 ― arcsin𝑟

𝑏

(4)

𝜃0 = 𝜋 ― 𝛽 = 𝜋 ― arcsin𝑟

The distance between the two horizontal planes where the segments BB and CC are involved separately, i.e., AA in Figure 5e, can be computed as (5)

𝑙AA′ = 𝑟(cos 𝛼 ― cos 𝛽) It can also be calculated using the tube geometry in the radial plane (Figure 5d)

(6)

𝑙AA′ = 𝑅(1 ― cos 𝛾)

Combining Equations (36), the following relation is derived using the trigonometric function

(

𝑅 1―

𝑅2 ― 𝑎2 𝑅

) = 𝑟(

𝑟2 ― 𝑎2 𝑟

𝑟2 ― 𝑏2 𝑟

)

―

(7)

It can be rearranged as 𝑏 𝑟

= 1―

(― 𝑅 𝑟

1―

𝑎 2 𝑟

()

―

2 𝑎 2 𝑟

( ) ―( ) ) 𝑅 2 𝑟

(8)

Combining Equations (3), (4) and (8), the intrinsic contact angle 𝜃0 is thus obtained in terms of θ, r and R 𝜃0 = arcsin 1 ―

(― 𝑅 𝑟

2

1 ― (sin 𝜃) ―

𝑅 2 𝑟

()

2

)

2

― (sin 𝜃)

(9)


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Equation (9) correlates the intrinsic contact angle with three variables: tube radius, droplet radius, and apparent contact angle (see Figure 5a for determining method). To verify its effectiveness, control experiments were carried out using hydrophobic tubes of different diameters (varying R) and water droplets of different sizes (varying r). Considering that the purity, shaping technique, post treatment and also other factors may affect the quality of Ti substrates and hence the resulting TiO2 coatings, polytetrafluoroethylene (PTFE) tubes were employed instead. Figure 5g shows that the apparent contact angle, determined by either method in Figure 5a, exhibits enhanced deviations from its intrinsic value, as the tube diameter decreases. In contrast, the intrinsic contact angle calculated using Equation (9) agrees well with the measurement on corresponding flat surface. This is further verified by changing the droplet size, as shown in Figure 5h. The apparent contact angle displays substantial deviations as the droplet increases, whereas falls back to its intrinsic level when corrected using Equation (9). It is obvious that the θ overestimates the contact angle at hydrophobic outer surface, whereas the 𝜃′0 underestimates it. Therefore, Equation (9) could eliminate the curvature effect by considering the tube and droplet geometry. However, it is only derived based on a droplet sitting at the outer surface of a hydrophobic tube. To obtain a general equation, another three scenarios are further taken into account, i.e., the hydrophilic outer surface, the hydrophilic inner surface, and the hydrophobic inner surface, as is discussed below.


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Figure 5. Intrinsic contact angle at hydrophobic outer tube surface. Side view of a droplet in the radial plane (a) and axial plane (b), and the corresponding solid-liquid contact line with elliptical projection (c). Geometric configuration drawn in the radial (d) and axial planes (f), where (e) is the combined drawing of (d) and (f) to demonstrate the relationship in Equations (5) and (6). Water contact angles measured at PTFE tubes of different diameters (g) and by droplets of different sizes (h). The droplet size and tube diameter used in (g) and (h) are 5 μL and 10 mm, respectively.


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Each of the data points is based on at least 5 measurements. (See Figures S6 and S7 in the Supporting Information for typical contact angle data)

Hydrophilic Outer Surface. Figure 6a and 6b illustrates a droplet residing on top of a tube in the radial and axial plane, respectively. Figure 6c depicts the three-dimensional solid-liquid contact line that exhibits an elliptical shape in projected plane. Figure 6d-6f shows the geometric relationship between the droplet and the substrate, where a regular sphere and tube with radius of r and R is assumed, respectively. Figure 6d discloses that the two intersections yield a line segment (i.e., BB) with a length of 2a in the radial plane. Similarly, Figure 6f reveals a line segment (i.e., CC) with a length of 2b in the axial plane. The shape parameters a and b are the semi-major and -minor axes of the elliptical projection in Figure 6c. As a result, the following relations are attained 𝜃 = 𝛼 = arcsin𝑟

𝑎

(10)

𝑏

(11)

𝜃0 = 𝛽 = arcsin𝑟

The distance between the two horizontal planes where the segments BB and CC are contained separately, i.e., AA in Figure 6e, can be expressed as (12)

𝑙AA′ = 𝑟(cos 𝛽 ― cos 𝛼) It can also be computed from the radial plane (Figure 6d)

(13)


Combining Equations (1013), the following relation is obtained based on the trigonometric function

(

𝑅 1―

𝑅2 ― 𝑎2 𝑅

) ( =𝑟

𝑟2 ― 𝑏2 𝑟

―

𝑟2 ― 𝑎2 𝑟

)

(14)

Equation (14) can be reorganized as


17


𝑏 𝑟

= 1―

(

𝑅 𝑟

+ 1―

Page 18 of 32

𝑎 2 𝑟

()

―

2 𝑎 2 𝑟

() ()) 𝑅 2 𝑟

―

(15)

Combining Equations (10), (11) and (15), the intrinsic contact angle 𝜃0 is derived. 𝜃0 = arcsin 1 ―

(

𝑅 𝑟

+ 1 ― (sin 𝜃)2 ―

𝑅 2 𝑟

()

2

)

― (sin 𝜃)2

(16)


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Figure 6. Intrinsic contact angle at hydrophilic outer tube surface. Side view of a droplet in the radial plane (a) and axial plane (b), and the corresponding solid-liquid contact line with elliptical projection (c). Geometric configuration drawn in the radial (d) and axial planes (f), where (e) is the combined drawing of (d) and (f) to demonstrate the relationship in Equations (12) and (13). Water contact angles measured at PU tubes of different diameters (g) and by droplets of different sizes (h). The droplet size and tube diameter used in (g) and (h) are 5 μL and 8 mm, respectively. Each of the data points is based on at least 5 measurements. (See Figures S8 and S9 in the Supporting Information for typical contact angle data)

The hydrophilic polyurethane (PU) tubes of different diameters were used to verify Equation (16). Figures 6g and 6h show that both θ and 𝜃′0 decrease with enlarged tube diameter, whereas increase with droplet size. This reveals that, when the difference between R and r becomes larger, the deviation tends to be suppressed; in this case, the curvature affects less. It is evident that the intrinsic contact angle calculated using Equation (16) is in line with the corresponding measurement on flat surface. In contrast, both of the conventional methods overestimate the contact angle at hydrophilic outer surface. Hydrophilic Inner Surface. Using a similar approach as above, based on the illustration in Figure 7a-7f, the apparent and intrinsic contact angles can be calculated as 𝑎

𝜃 = 𝛼 = arcsin𝑟

𝑏

𝜃0 = 𝛽 = arcsin𝑟

(17) (18)

The distance AA in Figure 7e is computed as 𝑙AA′ = 𝑟(cos 𝛼 ― cos 𝛽)

(19)

It can also be obtained from Figure 7d


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(20)

𝑙AA′ = 𝑅(1 ― cos 𝛾) Combining Equations (1720), the following expression is attainable 𝑅 1―

𝑅2 ― 𝑎2 𝑅

= 1―

(―

(

) ( =𝑟

𝑟2 ― 𝑎2 𝑟

𝑟2 ― 𝑏2 𝑟

―

)

(21)

2 𝑎 2 𝑟

(22)

Equation (21) is reorganized as 𝑏 𝑟

𝑅 𝑟

1―

𝑎 2 𝑟

()

―

( ) ―( ) ) 𝑅 2 𝑟

Combining Equations (17), (18) and (22), the intrinsic contact angle 𝜃0 is expressed. 𝜃0 = arcsin 1 ―

(― 𝑅 𝑟

2

1 ― (sin 𝜃) ―

𝑅 2 𝑟

()

2

)

2

― (sin 𝜃)

(23)

It is noteworthy that Equation (23) is formulated the same form as Equation (9). In other words, a hydrophilic inner tube surface shares the same equation with a hydrophobic outer surface. Nonetheless, the θ underestimates the contact angle at hydrophilic inner surface, whereas the 𝜃′0 overestimates it, rendering an opposite rule. Again, the intrinsic contact angle calculated using Equation (23) is consistent with the measurement on corresponding flat surface.


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Figure 7. Intrinsic contact angle at hydrophilic inner tube surface. Side view of a droplet in the radial plane (a) and axial plane (b), and the corresponding solid-liquid contact line with elliptical projection (c). Geometric configuration drawn in the radial (d) and axial planes (f), where (e) is the combined drawing of (d) and (f) to demonstrate the relationship in Equations (19) and (20). Water contact angles measured at PU tubes of different diameters (g) and by droplets of different sizes (h). The droplet size and tube diameter used in (g) and (h) are 5 μL and 8 mm,


21


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respectively. Each of the data points is based on at least 5 measurements. (See Figures S10 and S11 in the Supporting Information for typical contact angle data)

Hydrophobic Inner Surface. Similarly, the following relations stand based on Figure 8a-8f 𝑎

(24)

𝜃 = 𝜋 ― 𝛼 = 𝜋 ― arcsin𝑟

𝑏

(25)

𝜃0 = 𝜋 ― 𝛽 = 𝜋 ― arcsin𝑟 The distance AA in Figure 8e can be calculated as

𝑙AA′ = 𝑟(cos 𝛽 ― cos 𝛼)

(26)


(27)

It can also be computed in Figure 8d

Combining Equations (2427), the following relations are therefore attained

(

𝑅 1― 𝑏 𝑟

= 1―

(

𝑅 𝑟

𝑅2 ― 𝑎2 𝑅

) = 𝑟(

+ 1―

𝑎 2 𝑟

()

𝑟2 ― 𝑏2 𝑟

―

𝑟2 ― 𝑎2 𝑟

―

)

(28)

2 𝑎 2 𝑟

(29)

() ()) 𝑅 2 𝑟

―

Combining Equations (24), (25) and (29), the intrinsic contact angle 𝜃0 is expressed in terms of θ, r and R. 𝜃0 = arcsin 1 ―

(

𝑅 𝑟

+ 1 ― (sin 𝜃)2 ―

𝑅 2 𝑟

()

2

)

― (sin 𝜃)2

(30)


22

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Figure 8. Intrinsic contact angle at hydrophobic inner tube surface. Side view of a droplet in the radial plane (a) and axial plane (b), and the corresponding solid-liquid contact line with elliptical projection (c). Geometric configuration drawn in the radial (d) and axial planes (f), where (e) is the combined drawing of (d) and (f) to demonstrate the relationship in Equations (26) and (27). Water contact angles measured at PTFE tubes of different diameters (g) and by droplets of different sizes (h). The droplet size and tube diameter used in (g) and (h) are 5 μL and 8 mm, respectively. Each of the data points is based on at least 5 measurements (See Figures S12 and S13 in the Supporting Information for typical contact angle data).


23


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Similarly, Equation (30) is presented the same form as Equation (16). Figure 8g shows that the apparent contact angle, measured by either method in Figure 8a, biases considerably from the intrinsic value as the tube diameter decreases, in line with the above discussion. Figure 8h displays that the apparent contact angle decreases with the enlarged droplet. Besides, the 𝜃′0 overestimates the contact angle at hydrophobic inner surface, whereas the θ underestimates it. More importantly, the calculation using Equation (30) shows good agreement with the measurement on flat surface. Equations (9, 16, 23, 30) are combined to form a general equation for determining the intrinsic contact angle at tubes, as follows 𝜃0 = arcsin 1 ―

(± 𝑅 𝑟

2

1 ― (sin 𝜃) ―

𝑅 2 𝑟

()

2

)

2

― (sin 𝜃)

(31)

In the case of hydrophobic outer surface or hydrophilic inner surface, the item 1 ― (sin 𝜃)2 is negative, whereas for hydrophilic outer surface or hydrophobic inner surface, it is positive. Equation (31) demonstrates that, when the tube diameter is extremely large or the droplet size is extremely small (i.e., R  r), the apparent contact angle is close to its intrinsic value. In this case, the influence of curvature is greatly suppressed, therefore the measurement is similar to that on flat surface. Equation (31) also indicates that, when the surface imparts a contact angle close to 0 or 180, the apparent contact angle approximates the intrinsic one. Although the intrinsic contact angle is expressed in terms of tube radius, droplet radius, and apparent contact angle in Equation (31), it is a property of the surface itself, only depending on the interaction between surface and liquid at molecular or atomic level. Equation (31) is inapplicable for extremely small tubes that are of small contact angles, as shown in Figure S14 (Supporting Information). For instance, in the case of a hydrophilic tube with a contact angle of 30, it requires an inner diameter of 10 mm


24

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to obtain a valid evaluation. For outer surface, the dimeter limit would be relatively larger than that of the inner surface. In the case of extremely small outer diameters, the Carroll’s method would be applicable.34 The exact limitation and boundary condition are currently under investigation in our group. Using Equation (31), the intrinsic contact angle is determined to be 80 and 156 on bare titanium tubes (Figure 1e and 1f) and superhydrophobic tubes (Figure 4d and 4f), respectively, consistent with the corresponding flat surfaces. Therefore, Equation (31) could eliminate the curvature effect by considering the tube and droplet geometry.

CONCLUSIONS Uniform and superhydrophobic coatings of TiO2 nanotube arrays are prepared at the outer surface of Ti tubes. This is achieved under a tube-in-tube coaxial anodization scheme. Well-exposed nanotubes are obtained by using an alternating voltage in the form of square wave. A general equation is derived to calculate the intrinsic contact angle at either inner or outer surfaces of tubular substrates of different wetting properties. The equation correlates the intrinsic contact angle with apparent contact angle by considering the tube radius and droplet size. It is verified by systematically altering the tube radius and droplet size. These findings set a theoretical basis for determining intrinsic contact angle at tubular substrates of either opaque or transparent feature, regardless of the geometric configuration.

ASSOCIATED CONTENT


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Supporting Information. The following files are available free of charge. Detailed method for contact angle measurement, square-wave anodization parameters, FESEM images for titanium foils and tubes, typical contact angle data, plot of diameter limitation (PDF) AUTHOR INFORMATION Notes The authors declare no competing financial interests. ACKNOWLEDGMENT L. S. acknowledges the financial support from the National Natural Science Foundation of China (No. 51501024, 51871037), the Fundamental Research Funds for the Central Universities (No. 2018CDQYCL0027), the Chongqing Entrepreneurship and Innovation Program for the Returned Overseas Chinese Scholars (No. cx2017035), and Open Fund of National Engineering Laboratory of Highway Maintenance Technology, Changsha University of Science & Technology, (No. kfj170107).


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General Way To Compute the Intrinsic Contact ... - ACS Publications

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