Apparent Contact Angle Calculated from a Water Repellent Model with

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Apparent Contact Angle Calculated from a Water Repellent Model with Pinning Effect Shojiro Suzuki, and Kazuyuki Ueno Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b03832 • Publication Date (Web): 06 Dec 2016 Downloaded from http://pubs.acs.org on December 10, 2016

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Apparent contact angle calculated from a water repellent model with pinning effect Shojiro Suzuki and Kazuyuki Ueno* Graduate School of Engineering, Iwate University, 4-3-5 Ueda, Morioka 020-8551, Japan KEYWORDS Contact angle, Hydrophobicity, Pinning effect, Periodic structure, Cassie–Baxter equation

ABSTRACT

A set of new theoretical equations for apparent contact angles is proposed. The equations are derived from an equilibrium of interfacial tensions of a three-phase contact line pinned at the edges of a fine structure. These equations are validated by comparison with contact angle measurement results for 2 µL water droplets on poly(methyl methacrylate) microstructured samples with square pillars or holes. The equilibrium contact angles predicted by the new equations reasonably agree with the experimental results. In contrast, the values predicted by the Cassie–Baxter equation or the Wenzel equation do not qualitatively agree with the experimental results in pillar pattern cases because the Cassie–Baxter equation and the Wenzel equation do not account for the pinning effect.

ACS Paragon Plus Environment

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INTRODUCTION Controlling surface wettability introduces the possibility of decreasing floating-water resistance and preventing adhesion of a water droplet. In addition, appropriate wettability control can ensure visibility or antifouling characteristics. Thus, such control is critical in many industrial applications and is a research topic of fundamental interest. The most common approach to increasing the apparent contact angle is a combination of surface structure modification and surface chemical treatment.1-3 Examples of hydrophobic rough surfaces are found not only in artificial materials but also in nature. Lotus leaves form a superhydrophobic surface attributable to hierarchical micro- and nanostructures comprising papillae with a dense coating of agglomerated wax tubes and epicuticular wax crystals on top.4,5 Thus, raindrops bead up and wash dust from lotus leaves on rainy days; this unique property is called the “self-cleaning effect.”3 Surfaces of rose petals also have hierarchical micro- and nanostructures, both of which are larger than the corresponding structures of lotus leaves. Water droplets are expected to enter into large-scale grooves of rose petals but not into smaller ones. Thus, a high contact angle can coexist with strong adhesion between water and the surface of the rose petal.5-7 Figure 1 shows an equilibrium of interfacial tensions at a three-phase contact line of a droplet on a flat surface. The equilibrium contact angle of the droplet on a flat surface, θ0, is given by Young equation: cos θ 0 =

γ SV − γ SL γ LV ,

(1)


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where

γ SV , γ SL , and γ LV are interfacial tensions per unit length of the solid–vapor, solid–liquid,

and liquid–vapor interfaces, respectively. The equilibrium angle can be calculated from the Tadmor equation8 for advancing and receding angles.

Figure 1. Equilibrium of interfacial tensions of a droplet on a flat surface. The symbol θ0 represents the equilibrium contact angle on the flat surface. The equilibrium in the horizontal direction is called “Young equation.”

In the Cassie–Baxter theory,9 a solid surface is assumed to comprise two components of materials A and B in a microscopically mixed state. In this theory, the contact angle of a droplet on the microscopically mosaic surface is expressed as cos θ' = f A cosθ A + f B cosθ B ,

(2)

where θA and θB are the contact angles of droplets on homogeneous surfaces comprising components A and B, respectively, and θ′ is the apparent contact angle of the heterogeneously mixed surface. The parameters fA and fB are surface-area fractions of components A and B in the mixed solid surface, respectively. When θA is rewritten as θ0 and component B is air (θB = 180°), Equation 3 reduces to cos θ' = f A cosθ 0 + f A − 1

(3)


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where fA + fB = 1.9 This equation is widely used for expressing the contact angle of a droplet on a rough surface when the droplet only contacts the tops of pillars or when the top surface of a hole pattern, and air is trapped inside grooves. Figure 2a shows a rough solid surface microstructured with periodic square pillars of height h. For this surface, the surface area fraction fA is given by fA =

a2 (a + b )2

(4)

where a is the width of the square pillar and b is the distance between pillars.10-12 In contrast, a rough surface microstructured with a periodic structure with square holes of depth h is shown in Figure 2b. In this case, fA is given by fA =1−

b2 (a + b )2

(5)

where a is the distance between holes and b is the width of a square hole.

Figure 2. Schematics of the microstructure geometry on the solid surface, where h, a, and b are the height, the width of the upper part, and the width of the lower part, respectively: (a) pillar pattern and (b) hole pattern.

According to the Cassie–Baxter equation, the contact angle is a monotonically decreasing function of the contact area. However, the Cassie–Baxter equation is often inconsistent with


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experimental results. Local contact angle behaviors (i.e., advancing, receding, and hysteresis) are determined by force balance at the three-phase contact line. This relation can be translated to a problem of interfacial energy and work. The interfacial force theory and the interfacial energy theory for local contact angles are equivalent to each other.8 Nevertheless, we have not yet obtained a clear translation between the interfacial force theory13 and the interfacial energy theory9, 14 for apparent contact angles on rough surfaces. In a recent study, the contact angle was presumed to be dominated by the pinning effect, which stops the three-phase contact line at the edges of pillars or holes and increases the contact angle.1, 15-18 Figure 3 shows top-view optical micrographs of a 2 µL water droplet on a hole-patterned surface of a Si wafer. The cross section of the aligned holes is square. The pitch of the holes a + b is 150.0 µm, and the depth of the holes h is 60.0 µm. These images show that the bottom of the water droplet is not circular and that a three-phase contact line stops along the edges of the holes. Liu et al. also proposed that the apparent contact angle is dominated not by the contact area but by the three-phase contact line, and constructed a new model that predicts the contact angle more accurately than the Cassie–Baxter model.19 Moreover, to explore wetting behaviors from an experimental perspective, Forsberg et al. derived a new equation that considers the pinning effect at the tops of pillars.20


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Figure 3. Optical micrographs of a water droplet on a Si wafer with a periodic structure with square holes; pitch a + b is 150.0 µm, and height h is 60 µm.

The contact angle of a water droplet on a pillar-patterned surface is experimentally known to be influenced by the height of the pillars h; however, no equation has been proposed that can be used to evaluate the influence of h on the contact angle when the water repellent effect is substantial. Therefore, our objectives in this study are to derive a new equation that considers the influence of the height of pillars h on the contact angle by an equilibrium of interfacial tensions at a pinned three-phase contact line and to validate the new equation by comparison with experimental results. In addition, we derive a new equation for hole-patterned surfaces.

DERIVATION OF NEW WATER REPELLENT EQUATIONS Pinning Effect


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Figure 4 shows how interfacial tensions affect a droplet fixed by the pinning effect at the edge of a pillar or a hole. The roundness at the corner of the pillar is negligible. The equilibrium of interfacial tensions in the horizontal direction is given by − γ LV cos θ1 = γ SL

(6)

where θ1 denotes the contact angle of the droplet at the pinning edge. The following relation is obtained from Equation 6: cos θ1 = −

γ SL γ LV

(7)

The relation cos θ0 − cos θ1 = γ SV / γ LV > 0 is obtained from Equations 1 and 7. Because cos θ is a monotonously decreasing function, θ1 is always greater than θ0.

Figure 4. Equilibrium of interfacial tensions dominated by the pinning effect at the edge of a pillar or a hole. The angle θ1 represents the contact angle at the pinning edge.


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Vapor Liquid θ' h ∆h

a b Solid

(a) Droplet completely penetrating the grooves. The liquid goes around the bottom of the pillar, and one of the four lateral faces of the pillar is not covered by the liquid. Vapor Liquid θ' h a b Solid

(b) Droplet trapping air under it. The spread of the droplet stops at the edge of the top surface of the pillars because of the pinning effect.

Figure 5. Equilibrium of interfacial tensions of a droplet on a rough surface with a periodic structure with square pillars. The angle θ' represents the apparent contact angle on the rough surface.

Pillar Pattern Figure 5 shows an equilibrium of interfacial tensions at a pinned three-phase contact line of a droplet on a pillar-patterned surface. In the model of Figure 5a, the droplet fills grooves. The following two assumptions are made about wetting of pillar-patterned surfaces: the roundness at


8

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the corners of the pillars is negligible, and the liquid goes around the bottom of the pillar, and one of the four lateral faces of the pillar is not covered by the liquid. The equilibrium of interfacial tensions in the horizontal direction is expressed as − (a + b)γ LV cos θ' + (a + b)γ SV = (a + b)γ SL + aγ SL + 2(h − ∆h)γ SL

(8)

where ∆h is a height of the liquid that goes around the bottom of the pillar (Figure 5a). When ∆h is much smaller than h, Equation 8 reduces to (a + b)γ LV cos θ' = (a + b )γ SV − {(a + b ) + a + 2h}γ SL

(9)

By substitutingEequations 1 and 7 into Equation 9, we obtain

cos θ' = cos θ 0 +

a + 2h cos θ1 a+b

(10)

Figure 5a shows a penetrating droplet, whereas Figure 5b shows a droplet that only contacts the tops of the pillars, where an air layer is trapped between the pillars. In this case, the equilibrium of interfacial tensions in the horizontal direction is expressed as − (a + b)γ LV cos θ' = bγ LV + aγ SL

(11)

By combining Equations 7 and 11, we obtain

cos θ' =

a b cos θ1 − a+b a+b

(12)


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Equation 10 expresses the change in contact angle with the height of the pillar h when a droplet fills up grooves, whereas Equation 12 indicates that the contact angle is unrelated to the height of pillars h when the droplet only contacts the tops of the pillars.

Hole Pattern Figure 6 shows the equilibrium of interfacial tensions at a pinned three-phase contact line of a droplet on a hole-patterned surface. The following two assumptions are made about wetting of hole-patterned surfaces: the roundness at the corner of the hole is negligible, and the spread of the droplet stops at the edge of the holes because of the pinning effect. In this case, the contact angle is not affected by whether the droplet penetrates grooves. The equilibrium of interfacial tensions in a horizontal direction is expressed as − (a + b)γ LV cos θ' + aγ SV = (a + b )γ SL

(13)

By substituting Equations 1 and 7 into Equation 13, we obtain

cos θ' =

a b cos θ0 + cos θ1 a+b a+b

(14)

Equation 14 expresses that the apparent contact angle of a droplet on a hole-patterned surface is unrelated to the depth of holes h.


10

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Vapor Liquid θ' a b

h

Solid

Figure 6. Equilibrium of interfacial tensions of a droplet on a rough surface with a periodic structure with square holes. The spread of the droplet stops at the edge of the holes because of the pinning effect.

EXPERIMENTAL SECTION Contact angles of droplets were measured using poly(methyl methacrylate) (PMMA) samples with 33 different pillar patterns and 26 different hole patterns. The dimensions of test pieces were measured by scanning probe microscopy (SPM; Nanonavi IIs/NanoCute, Hitachi HighTech Science Co., Japan). The pitches of the pillar patterns and hole patterns (a + b) were within the ranges 0.50–1.0 and 0.60–5.0 µm, respectively, and the maximum value of height h was 0.61 µm. Prior to observation of a water droplet, the test piece was washed with water and dried for 20 min at 20 °C in a thermostatically controlled chamber. Immediately afterwards, 2 µL of water was dropped onto the rough part of the PMMA sample, and the contact angle of the droplet was measured using an optical microscope (VH-E500, Keyence Co., Japan). The amount of the water was set to 2 µL using a microsyringe. The contact angle was measured with the aid of image-


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analysis software (Image J, open source). Figure 7 shows images of water droplets on a flat surface and a pillar-patterned surface of PMMA samples. It is not able to confirm whether the water droplet contacts the bottom of groove or not. In the next section, experimental results are compared with theoretical values calculated using Equations 3, 10, 12, and 14.

Figure 7. Micrographs of 2 µL of water droplets on PMMA samples: (a) flat surface and (b) pillar-patterned surface (a = 0.73 µm, b = 0.92 µm, h = 0.64 µm).

RESULTS AND DISCUSSION The surface tension of water γ LV is 72.8 mN/m,21-23 and the surface tension of PMMA γ SV is 40.6 mN/m.24 The Young angle θ0 of PMMA is 74.7°. Substituting these values into Equation 1, we obtain the interfacial tension γ SL = 21.4 mN/m. Consequently, a pinning angle θ1 = 107.1° is obtained from Equation 7.


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Figure 8a shows the relation between the apparent contact angles and groove fraction b / (a + b) for the droplets on pillar-patterned surfaces. The predictions of the Cassie–Baxter equation (Equation 3) do not agree with most of the experimental results, and this incongruity becomes remarkable for large b / (a + b) values. Theoretical values calculated using Equation 10 are consistent with experimental results in cases where the contact angle is smaller than θ1. Moreover, when the groove fraction b / (a + b) is approximately 0.1, some of the experimental results corresponding to contact angles greater than θ1 agree with the predictions based on Equation 12. These results suggest that penetration of water into grooves is difficult (or occasional) when the groove fraction b / (a + b) is small. Equation 10 well reproduces changes in contact angles due to not only the groove fraction b / (a + b) but also due to the height ratio of pillars h / (a + b). Figure 8b shows the relation between the apparent contact angles and (a + 2h) / (a + b). Equation 10 is a monotonically increasing function of (a + 2h) / (a + b), and it reasonably agrees with the experimental results corresponding to contact angles smaller than θ1. The apparent contact angles corresponding to some of the experimental results are larger than θ1 when (a + 2h) / (a + b) is approximately 1.0. Apparent contact angle by Wenzel equation25 is given by

cos θ' =

(a + b) 2 + 4ah cos θ 0 ( a + b) 2

(15)

Wenzel equation predicts θ’ < θ0 when θ0 = 74.7°. No observation, however, was found in the range of θ’ < θ0 in our experiments. Figure 8 suggests that the apparent contact angle of droplets penetrating the grooves is given not by Wenzel equation but by Equation 10 in square pillar cases.


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Figure 8. Comparisons between the theoretical predictions and experimentally measured contact angles of 2 µL of water droplets on pillar-patterned surfaces: (a) pillar-width dependency and (b) pillar-height dependency.


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Figure 9 shows the contact angles of droplets on hole-patterned surfaces. Both the theoretical values calculated using Equations 3 and 14 reasonably agree with the experimental results. The mean square error between the theoretical and experimental contact-angle results is 10.0° in the case of the Cassie–Baxter model (Equation 3), whereas it is 7.2° in the case of the pinning model (Equation 14). Thus, the pinning model developed in the present work is better than the Cassie– Baxter model. The error of the Cassie–Baxter equation becomes particularly large when b / (a + b) approaches 1.0.

Figure 9. Comparisons between the theoretical predictions and experimentally measured contact angles of 2 µL of water droplets on hole-patterned surfaces.


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The apparent contact angles on a pillar-patterned superhydrophobic surface were also predicted using the pillar-pattern model Equations 10 and 12. Figure 10 shows the experimental apparent contact angles reported by Yoshimitsu et al.25 The predictions based on model Equations 10 and 12 in the case of γ SV = 16.6 mN/m for C13H13F17O3Si

26

agree with the

experimental results. This agreement suggests that the appropriate pillar width and height for a superhydrophobic surface are given by the following equations:

a 1 + cos150° < a+b 1 + cos θ1

(16)

h 1  a  > cos θ1   cos150° − cos θ 0 − a + b 2 cos θ1  a+b 

(17)

and

where cos θ1 is always negative, and cos θ0 is also negative in many superhydrophobic cases. The condition for superhydrophobicity without groove–bottom contact is obtained if cos150° is replaced with − 1 (= cos180°) in Equation 17.


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: Experiments by Yoshimitsu et al. (2002) : Contact angle by the pillar-pattern model 10 : Contact angle by the pillar-pattern model 12 : Equilibrium pinning angle θ1 = 129.4° by equation 7 : Equilibrium contact angle θ0 = 114° by Young equation 1 : Contact angle by Wenzel equation 15 when a/(a+b) = 1/3

Contact angle θ´(°)

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With transition groove– bottom contact

Superhydrophobicity without groove–bottom contact

(a+2h) / (a + b)

Figure 10. Contact angles of water droplets on a pillar-patterned superhydrophobic surface: comparison between the theoretical predictions of the present study and experimental results reported by Yoshimitsu et al.25 ( γ SV = 16.6 mN/m, γ LV = 72.8 mN/m, θ0 = 114°, a = 50 µm, b = 100 µm).

CONCLUSION In the present paper, we proposed new theoretical equations for the apparent contact angle of droplets on rough surfaces. These equations were derived from equilibria of interfacial tensions at the pinned three-phase contact line. One of the equations includes the height effect of the pillars. Theoretical values for the apparent contact angles calculated using the Cassie–Baxter equation, the Wenzel equation and the new equations in consideration of pinning effect were


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compared with experimental results. We observed numerous cases in which the Cassie–Baxter equation and the Wenzel equation did not agree with the experimental results. In contrast, the predictions of the new equations in consideration of pinning effect agreed reasonably well with the experimental results. Thus, we expect the new equations derived in this study to be helpful in elucidating the wetting mechanism and designing more accurate water-repellent surfaces.

ACKNOWLEDGMENT The authors would like to thank Mr. Kazuhito Shibata for his survey of previous studies and Professor Masaki Yamaguchi for his guidance and suggestions.

AUTHOR INFORMATION

Corresponding Author *E-mail: [email protected]

Author Contributions The manuscript was written through equal contributions of all authors. All authors have given approval to the final version of the manuscript.

Notes The authors declare no competing financial interests.


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Langmuir

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TOC Graphic


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Apparent Contact Angle Calculated from a Water Repellent Model with

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