Advances in Contact Angle, Wettability and Adhesion

By K.L. Mittal

The topic of wettabilty is extremely important from both fundamental and applied aspects. The applications of wettability range from self-cleaning windows to micro- and nanofluidics.

This book represents the cumulative wisdom of a contingent of world-class (researchers engaged in the domain of wettability. In the last few years there has been tremendous interest in the "Lotus Leaf Effect" and in understanding its mechanism and how to replicate this effect for myriad applications. The topics of superhydrophobicity, omniphobicity and superhydrophilicity are of much contemporary interest and these are covered in depth in this book.

• Language: English
• Publisher: Wiley
• Release date: Aug 16, 2013
• ISBN: 9781118795613

Book preview

PART 1

FUNDAMENTAL ASPECTS

Chapter 1

Correlation between Contact Line Pinning and Contact Angle Hysteresis on Heterogeneous Surfaces: A Review and Discussion

Mohammad Amin Sarshar, Wei Xu, and Chang-Hwan Choi*

Department of Mechanical Engineering, Stevens Institute of Technology, Hoboken, New Jersey, USA

Abstract

Micro- or nano-textured hydrophobic surfaces have attracted considerable interest due to their highly water-repellent property, and are called superhydrophobic. Although such superhydrophobic surfaces typically exhibit high contact angles for water droplets, their adhesion and frictional properties such as contact angle hysteresis are significantly affected by the dynamics of contact line pinning at the droplet boundary. However, a clear correlation between the contact line pinning and the contact angle hysteresis has not been revealed yet. In this paper, we review the literature reporting on their correlation, both for chemically and physically patterned heterogeneous surfaces, including our recent discovery on superhydrophobic surfaces. Then, we propose and discuss an appropriate new physical parameter that shows close and consistent correlation between the dynamics of contact line pinning and the contact angle hysteresis.

Keywords: Contact angle hysteresis, contact line pinning, heterogeneous surfaces, superhydrophobic

1.1 Introduction

When hydrophobic surfaces are roughened or patterned in proper length scale and morphology, air can be entrapped between the surface structures (typically called as Cassie state) and the surfaces show highly non-wetting and slippery, so-called superhydrophobic property [1] that would be of great significance in many applications such as in self-cleaning [2], hydrodynamic friction reduction [3], anti-icing [4, 5], anti-corrosion [6], thermal/ energy system [7], biotechnology [8], and micro- and nano-devices [9]. Known as lotus effect [10], such superhydrophobic surfaces generally result in high contact angle and low contact angle hysteresis for water droplets so that water droplets can easily roll off from the surfaces. However, also known as petal effect [11], if water droplets wet the surfaces either partially or uniformly with no air void retained (typically called as Wenzel state), the surfaces exhibit high contact angle hysteresis despite high apparent contact angle. In such a case, water droplets get strongly pinned on the surfaces and do not roll off even when the surfaces are tiled even greater than 90°. Such sticky surfaces for droplets are also of great importance in many applications such as in spraying/coating [12], ink-jet printing [13], liquid transportation/analysis [14], and microfluidics [15]. Recently it has also been shown that superhydrophobic surfaces, even in Cassie state, can cause more significant contact line pinning and hence behave stickier than non-patterned planar hydrophobic surfaces, depending on the geometry and dimensions of surface patterns [16]. Such reports suggest that the pinning phenomena of droplets are affected in a complicate way by many surface parameters including physical morphology, chemical heterogeneity, and interfacial wetting states [17–21]. To date, a few different approaches have been applied to explain the direct correlation between the behaviors of contact line pinning and the adhesion or frictional properties such as contact angle hysteresis for moving droplets. One of them is based on the effective contact area between the droplet and the solid surface [22–27], while the other one is based on the effective contact length [16, 28–32]. In this paper, we review the literature and discuss which physical parameters would be more relevant to correlate the dynamics of contact line pinning and the adhesion properties of heterogeneous surfaces such as contact angle hysteresis of superhydrophobic surfaces. Based on these, we also propose a non-dimensional surface parameter that can be universally applied to determine their correlation. Despite being simple, the new physical parameter revealed in this paper should serve as a quick and efficient criterion for the design and engineering of heterogeneous or superhydrophobic surfaces with tailored adhesion properties.

1.2 Contact Line Pinning on Chemically Heterogeneous Flat Surfaces

The physical insights into the dynamics of three-phase contact line on heterogeneous or superhydrophobic surfaces can first be obtained from the previous studies on the chemically heterogeneous surfaces with a finite number of flat spots of higher hydrophobicity, or defects. For example, Joanny and de Gennes [33] analytically studied the contact angle hysteresis of solid surfaces with a single chemical defect at lower wettability state. As shown in Figure 1.1, the chemical defect with greater hydrophobicity causes pinning and distortion of contact line as the droplet crosses the defect and moves forward. In this case, the localized pinning force (F) exerted on the defect due to the deformation of the contact line can be estimated by Hooke’s law (F = kx) considering the spring of stiffness (k) defined as:

Figure 1.1 Progress of three-phase contact line as it passes over a single defect. The contact line moves to upward direction with time (t) as a sequence of t1 → t2 → t3. While the dotted lines represent the contact lines before pinning (at t = t1) and after depinning (at t = t3), respectively, the solid line (at t = t2) shows the pinning and deformation of the contact line at the defect.

(1.1) equation

where γ is the surface tension of liquid, θ0 is the quasi-equilibrium contact angle for an ideal surface with no defect, d is the diameter of a circular defect, and L is a cut-off length scale which can be either the droplet diameter or the capillary length. As a result, the pinning force exerted on the defect by the deformed contact line would be F = kxm where xm is the maximum amplitude of the distortion of the contact line. This force vanishes to zero away from the defect. In this way, if the distortion of the contact line (or the overall shape of the contact line) is precisely known, the amount of the pinning force can be estimated precisely. Under the conditions of small equilibrium contact angle on the substrate with the small distortion of the contact line [33, 34], this approach gives reasonably good estimation of the pinning force. Equation 1.1 also implies what variables would be important in the dynamics of contact line movement. This approach can also be applied to the case of a small number density of the defects where the defects are not densely populated so that they do not act collectively to cause the deformation of the interface. In the case the number density of defects increases and the defects are populated close to each other, the defects would behave collectively to deform the contact line. In such a case, the deformation of the contact line at each defect is comparatively smaller so that it results in a faceted droplet shape and the approach explained above is not appropriate to apply.

When the contact line recedes or advances over the multiple defects with a small number density, the pinning force per unit length of the contact line can be reduced to [34]:

(1.2) equation

(1.3) equation

where WR and WA represent the dissipation energies per unit area (or pinning force per unit length) due to the deformation of contact line at a single defect in the receding (θR: receding contact angle) and the advancing (θA: advancing contact angle) motions of the droplet, respectively, and n is the total number of defects engaged in the contact line movement. By combining Equations 1.2 and 1.3, it leads to obtain an equation for contact angle hysteresis (cosθR − cosθA), such as:

(1.4) equation

Physically, WR and WA have the same meaning as the localized pinning force (F) described in Equation 1.1. Thus, if the localized pinning force (F) is known for each defect engaged in the contact line movement, the effect of key surface parameters determining the contact angle hysteresis can be understood, such as the diameter of the defect (d) shown in Equation 1.1.

Experimentally, Cubaud and Fermigier [35] also studied the pinning force of a contact line on chemically heterogeneous surfaces. In the case of a small number density of defects, they proposed that one of the useful parameters, which would be more relevant to define the pinning force than probing the mechanical deformation of the contact line (x, Figure 1.1), would be the angle which the two tangents to the contact lines at each side of the defect make (called depinning angle, Figure 1.1). In order to correlate with the depinning angle, they also introduced a new physical parameter fs for the defects, defined as:

(1.5) equation

where Δs is the difference in spreading coefficients between the substrate and the defect and h is the thickness (height) of the droplet. They regulated the value of fs by varying the diameter of the defect (from 100 μm to 1800 μm) and examined how the depinning angle would change for a single defect. Based on their experimental observation, they concluded that there should be a critical value for fs so if fs is less than the critical value there is only little change in the depinning angle, resulting in weak pinning. However, if fs is greater than the critical value, the depinning angle significantly decreases, resulting in strong pinning. In the case of the higher density of defects, an increase of fs also results in stronger pinning of the contact line, which consequently transitions the droplet shape from a circular drop to a faceted drop. For example, the droplet shape would be transitioned to a square shape with a square array of defects. They also reported that the roundness of the faceted shape, defined as P²r/4πψ where Pr is the perimeter of the droplet and ψ is the wetted area, tends to increase linearly with respect to fs. Cubaud and coworkers [36] also investigated the dynamics of contact line movement over the chemically patterned surfaces by fixing the defect diameter at 400 μm while varying the distance between them from 600 μm to 4000 μm. They observed that when the distance between two defects was less than twice the defect diameter, the deformation of the contact line by a single defect was not pronounced and the defects acted collectively as a cluster to deform the contact line globally.

Although Cubaud and coworkers [35, 36] showed how the number density of defects would affect the contact line morphology and the pinning force, it was not clearly discussed how the contact angle hysteresis would also be affected. In contrast, Di Meglio [37] directly measured the forces required for advancing and receding non-wetting liquids such as hexadecane and heptane on chemically heterogeneous surfaces by connecting a force measurement sensor to the surface samples while they were dipped into or pulled out of the liquids at a constant velocity. The surfaces of samples consisted of planar defects with two distinct diameters of 500 μm and 1500 μm at varying number densities with random distribution. In their experiment, hysteresis was defined as the difference between the advancing and receding forces. They found that the amount of hysteresis increased with the defect density in a non-linear manner.

1.3 Contact Line Pinning on Hydrophobic Structured Surfaces

Now we change our focus from chemically heterogeneous surfaces to physically patterned hydrophobic surfaces and the correlation between the dynamics of contact line pinning and the contact angle hysteresis on superhydrophobic surfaces. The fundamentals and overviews of the superhydrophobic wetting properties can be found in many review papers and references therein [1, 38–42]. In general, the concept used in the derivation of Equation 1.4 can still be applied to estimate how much force is necessary to make a droplet move on superhydrophobic surfaces [24, 43, 44] or roll off in inclination [21, 26, 31, 43]. In order to find out the key physical parameters associated with the depinning force and the contact angle hysteresis more specifically, the influence of surface morphology of superhydrophobic patterns on the droplet pinning has also been studied from the perspective of both contact area [22–24, 27, 43] and contact length [16, 28–30, 32, 45, 46].

McHale and coworkers [22, 23] have commented on the pinning phenomena and contact angle hysteresis as a consequence of the effective contact area between liquid and solid. They measured the contact angle hysteresis on systematically designed superhydrophobic surfaces (Figure 1.2), where the contact perimeter of the liquid-solid interface was varied but the effective contact area of the liquid-solid interface was held constant. Then, they observed that the contact angle hysteresis was invariant despite the differences in contact perimeters.

Figure 1.2 The superhydrophobic surfaces with different patterns tested in the work of McHale et al [22]. The fractions of the solid area (black squares) of the three tested surfaces are the same, while the perimeters are different.

Reyssat and Quéré [24] also examined the contact angle hysteresis over superhydrophobic surfaces with square arrays of pillars and reported that the hysteresis increased in a non-linear way with the pillar density. Considering the liquid-air meniscus as a spring similar to what was discussed in Equation 1.1, they presented an analytical model to relate the contact angle hysteresis with geometric parameters of the surface, especially in the context of solid area fraction. Assuming that the number density of the pillar structures is small (i.e., the pillar diameter is much less than the distance between them), that the long liquid/air meniscus tail over pillar structures follows cosh form as the droplet is displaced, and that the liquid/solid contact area is fixed as the liquid/air meniscus is deformed (Figure 1.3), the following equations for the spring stiffness (k) and the contact angle hysteresis (Δ cos θ = cos θR − cos θA) were derived:

Figure 1.3 Contact line deformation in a receding motion of a droplet over a superhydrophobic surface patterned with circular pillar structures.

(1.6) equation

(1.7) equation

where a is an empirical correction factor, b is the radius of the pillar, p is the pitch of the pillar array, and ϕ is the solid area fraction. Their experimental results showed good agreement with the analytical model (Equation 1.7), especially when the number density of the pillar structures was relatively small (i.e., the solid area fraction ϕ is small).

For a rolling droplet on inclined superhydrophobic surfaces, Lv et al. [43] proposed a theoretical model based on the total interfacial energy and derived an equation such as:

(1.8) equation

where is ρ the density of liquid, g is gravitational acceleration, V is the volume of the droplet, α is the roll-off (or sliding) angle, and R is the radius of the wetted area. On the other hand, by considering the force balance between the weight of the droplet and the pinning force associated with contact angle hysteresis, the following equation can also be derived [43]:

(1.9) equation

where it is assumed that a half of the droplet experiences an advancing motion and the other half experiences a receding motion as the droplet rolls (slides) down [24, 43]. Comparing Equations 1.8 and 1.9, the contact angle hysteresis can be related to the geometric parameters of the surface as follows:

(1.10) equation

Equation 1.10 suggests that the contact angle hysteresis is rather determined by the square root of the solid fraction (i.e., length scale such as a contact line) than the solid fraction (area) itself. If Equation 1.10 is plotted together with Equation 1.7 for comparison, they show a similar relationship between the contact angle hysteresis and the solid fraction. It further suggests that the length scale (e.g., contact line) is a more relevant parameter to determine the contact angle hysteresis than the contact area (e.g., solid fraction) itself.

The importance of the dynamics of contact line to the droplet pinning and the contact angle hysteresis was also discussed by McCarthy and coworkers [16, 28, 29]. They pointed out that what happens at the contact line during the advancing and receding motions of the droplet would be the key in determining the contact angle hysteresis. It is because only the events that occur at the contact line can contribute to the pinning phenomenon and the contact angle hysteresis (Figure 1.4). This suggests that the pinning force and the resultant contact angle hysteresis on superhydrophobic surfaces should significantly be dependent on the dynamics of contact line movement (e.g., deformation and shape) when the droplet advances or recedes. Dorrer and Rühe [46] numerically studied the deformation of contact line of a water droplet moving on square arrays of square post structures and reported a significant distortion of the contact line at the droplet boundary. They found the local contact angles to be different from apparent ones due to the local pinning effect at the surface structures. Especially, in an advancing motion, the contact line moves to the edges of the post structures and gets pinned at the edges until the local contact angle approaches 180°. Dorrer and Rühe [45] also experimentally observed that the advancing contact angle was not affected by the change in geometric parameters of the surface structures. In contrast, it was shown that the receding contact angle was strongly dependent on the geometric parameters of the surface structures. Mognetti and Yeomans [27] also numerically studied the morphology of contact line when the droplet moves on superhydrophobic surfaces with square arrays of post structures. They found that as the solid fraction decreased, the shape of the contact line in the receding motion would be of cosh form and the contact angle hysteresis would follow the same trend as shown in Equation 1.10.

Figure 1.4 The schematic of a droplet moving on a solid substrate. Only the liquid molecules near the contact line (unfilled circles) move during this process. No movement happens to the liquid molecules in the inner region of the interface between liquid and solid surface (filled circles) [28].

Recently, Xu and Choi [32] also proposed that the ratio of the actual contact line to the apparent contact line was a simple and efficient parameter to describe the pinning force and the contact angle hysteresis, instead of the solid fraction (area). They examined an evaporating droplet on superhydrophobic surfaces with square arrays of micropillar structures whose diameters were fixed at 5 μm while the inter-pillar distances were varied from 5 to 50 μm. By observing the contact interface between the droplet and the superhydrophobic surface directly using reflection interference contrast microscopy, they found that the actual contact line on a superhydrophobic surface comprised of both two-phase (liquid-air) and three-phase (liquid-solid-air) interfaces was significantly different from an apparent three-phase contact line as shown in Figure 1.5. Such multi-modal contact line state was dynamically altered when the droplet receded during evaporation (Figure 1.6), and the onset of the contact line depinning occurred when the three-phase contact line reached the maximum (i.e., covering the whole periphery of the circular pillar surface). Then, the depinning force (Fd) [47] defined as

Figure 1.5 The multi-modal droplet boundary on a pillar-patterned superhydrophobic surface, observed from the backside of a transparent superhydrophobic substrate using reflection interference contrast microscopy. The interference fringe, formed by the light reflected from the water-air interface and the solid substrate, indicates the location of the actual droplet boundary. This actual droplet boundary is different from the apparent boundary, and is constituted by the three-phase (liquid-solid-air) as well as the two-phase (liquid-air) interfaces [32].

Figure 1.6 The reflection interference contrast microscopy images show the evolution of the actual boundary of an evaporating droplet on a micropillar patterned superhydrophobic surface. The three-phase (liquid-solid-air) contact line and the two-phase (liquid-air) interface on the droplet boundary are shown by the solid and dashed lines, respectively. The depinning from pillars on the surface occurs when the three-phase contact line reaches the maximum (i.e., covering the whole periphery of the circular pillar top surface) [32].

(1.11) equation

was found to display a linear correlation with the normalized maximal three-phase contact line at the droplet boundary (i.e., the ratio of maximal actual three-phase contact line to apparent droplet boundary), called δ (Figure 1.7). The result shows that when δ is greater than unity (the value on a planar surface with no pattern), a higher depinning force is required even on superhydrophobic surfaces than that on a planar hydrophobic surface. Therefore, such superhydrophobic surfaces with δ > 1 behave as stickier surfaces for droplets than a planar hydrophobic surface. On the contrary, if δ is less than unity, a lower depinning force is required with the superhydrophobic surfaces, and hence a superhydrophobic surface with δ < 1 is slippery compared to a planar hydrophobic surface. The new non-dimensional parameter, δ, defined as the ratio of maximal actual three-phase contact line to apparent droplet boundary, is directly proportional to the square root of solid fraction, when the three-phase contact line covers the whole perimeter of surface structures as the droplet advances or recedes. Thus, this result also agrees with the general trend shown in Equation 1.10.

Figure 1.7 The linear dependence of the depinning force on the normalized maximal three-phase contact line (δ) at the droplet boundary. The superhydrophobic surfaces can behave as sticky or slippery surfaces depending on the surface parameter (δ). The result from Öner and McCarthy [16] was also analyzed with the new parameter (δ). In the experiments of Xu and Choi [32], the liquid-solid contact area fractions on the tested surfaces were systematically varied from 1.00, 0.20, 0.09, 0.03, to 0.01, denoted as Φ1.00, Φ0.20, Φ0.09, Φ0.03, and Φ0.01, respectively.

Xu and Choi [32] also analyzed the previous works reported by McHale and coworkers [22, 23] by using the parameter δ, and found that the normalized maximal three-phase contact lines (δ) of their tested surfaces (Figure 1.2) had the same value. Thus, according to the observation made by Xu and Choi [32], such surface morphologies with a constant δ value should result in the same depinning force and hence the same contact angle hysteresis, which was indeed observed by McHale and coworkers [22, 23]. Therefore, the experimental results of McHale and coworkers [22, 23] turn out to agree with the model proposed by Xu and Choi [32]. Xu and Choi [32] also compared their results with those of McCarthy and coworkers [16, 28, 29]. The three-phase contact line referred in the works of McCarthy and coworkers [16, 28, 29] represents a static and apparent one, instead of the dynamically altered actual three-phase contact line measured in the work of Xu and Choi [32]. Xu and Choi [32] pointed out that the static or apparent surface parameters would not be appropriate to understand and explain the dynamic behaviours of the contact line pinning and contact angle hysteresis on superhydrophobic surfaces. As commented by Xu and Choi [32], direct correlation between the surface morphology and contact angle hysteresis was not found in the work of McCarthy and coworkers [16, 28, 29] using the apparent three-phase contact line as a geometric parameter. Thus, Xu and Choi [32] reanalyzed the previous works of McCarthy and coworkers [16, 28, 29] by using their parameter δ, and found that their experimental results also agreed well with the model proposed by Xu and Choi [32] as shown in Figure 1.7.

Based on the observation made by Xu and Choi [32], we also analyzed the data reported by Reyssat and Quéré [24] as well as Dorrer and Rühe [45] with respect to the parameter δ, which is shown in Figure 1.8. Overall, the depinning force shows a linear correlation with the non-dimensional geometric parameter δ, agreeing with the model proposed by Xu and Choi [32]. As pointed out by them [32] and also in Equation 1.10, the different linearity (slope) is attributed to the different material chemistry (hydrophobicity, i.e., cos θ0) of the surface structures prepared in the experiments. It should also be noted that in the work of Xu and Choi [32] δ was calculated based on the fact that the contact line depinning from their pillar structures occurred when the three-phase contact line reached the maximum, i.e., covered the whole periphery of the circular pillar top surface. Thus, the normalized maximal three-phase contact line (δ) for pillar-type structures is simply determined by the pitch between solid pillars (l) and the perimeter of each pillar (πd) and is reduced to δ = πd/l. In plotting Figure 1.8, the same condition (δ = πd/l) was used in the estimation of δ for the data reported by Reyssat and Quéré [24] and Dorrer and Rühe [45]. However, the maximum three-phase contact line configurable on superhydrophobic surface structures would be dependent on the morphology (e.g., patterns, dimensions, and geometries) of surface structures. Thus, such details should be more carefully examined in the future to verify the unified linear correlation model between the new surface parameter and the depinning force, and hence the contact angle hysteresis as well.

Figure 1.8 Comparison of the depinning forces of superhydrophobic surfaces reported by Reyssat and Quéré [24] as well as Dorrer and and Rühe [45] with respect to the non-dimensional geometric parameter δ proposed by Xu and Choi [32].

1.4 Summary and Conclusion

To date, the correlation between surface parameters and contact angle hysteresis has been studied using many different approaches. Recently, due to the advancement of micro- and nanofabrication technologies, such studies have been extended to micro- or nanopatterned hydrophobic surfaces (typically called superhydrophobic surfaces) because of their highly non-wetting properties and their great potentials in many scientific and engineering applications. However, a unified model to correlate the surface parameters of superhydrophobic surfaces and the contact angle hysteresis has not been revealed yet. In this paper, we reviewed the literature which has reported such correlation both theoretically and experimentally. We first reviewed the droplet pinning and contact angle hysteresis on chemically heterogeneous surfaces with hydrophobic defects. Such studies showed that the deformation of contact line and its state (e.g., morphology) are critical in determining the pinning force and the contact angle hysteresis. Similar analysis can be applied to the physically heterogeneous superhydrophobic surfaces where a significant pinning occurs periodically on the array of hydrophobic structures. Most studies have used the solid wet-area fraction as the key surface parameter to interpret the pinning force/energy and the contact angle hysteresis. In contrast, a few recent studies proposed that the three-phase contact line should be a more relevant and direct parameter to correlate with the pinning force and contact angle hysteresis. Based on the new surface parameter proposed in recent studies, we revisited and analyzed the previous data. It shows that the depinning force of a receding droplet has overall linear correlation with a non-dimensional geometric parameter, defined as the ratio of the maximum length of the three-phase contact line configured at the droplet boundary to the circular perimeter of the apparent droplet boundary. It suggests that the morphology and pinning state of the contact line at the droplet boundary are more critical parameters than the effective contact area to determine the pinning force and the contact angle hysteresis on superhydrophobic surfaces. The morphology and pinning state of the contact line are significantly dependent on the morphology and surface chemistry of the patterned structures on superhydrophobic surfaces. Thus, more systematic and extensive studies are necessary to verify the linear model proposed by recent studies in order to use it as a universal and unified model to predict the pinning force and contact angle hysteresis for a variety of heterogeneous and superhydrophobic surfaces.

References

1 A. Carré and K. L. Mittal (Eds.) Superhydrophobic Surfaces, VSP/Brill, Leiden (2009).

2 R. Fürstner, W. Barthlott, C. Neinhuis, and P. Walzel, Wetting and self-cleaning properties of artificial superhydrophobic surfaces, Langmuir 21, 956–961 (2005).

3 C.-H. Choi and C.-J. Kim, Large slip of aqueous liquid flow over a nanoengineered superhydrophobic surface, Phys. Rev. Lett. 96, 066001 (2006).

4 M. A. Sarshar, C. Swarctz, S. Hunter, J, Simpson, C.-H. Choi, Effects of contact angle hysteresis on ice adhesion and growth on superhydrophobic surfaces under dynamic flow conditions, Colloid. Polym. Sci. 291, 427–435 (2013).

5 K.V. Kripa, D. Tao, J.D. Smith, H. Ming, and B. Nitin, Frost formation and ice adhesion on superhydrophobic surfaces, Appl. Phys. Lett. 97, 234102 (2010).

6 P.M. Barkhudarov, P.B. Shah, E.B. Watkins, D.A. Doshi, C.J. Brinker, and J. Majewski, Corrosion inhibition using superhydrophobic films, Corrosion Sci. 50, 897–902 (2008).

7 J.B. Boreyko, and C.-H. Chen, Self-propelled dropwise condensate on superhydrophobic surfaces, Phys. Rev. Lett. 103, 184501 (2009).

8 C.-H. Choi, and C.-J. Kim, Droplet evaporation of pure water and protein solution on nanostructured superhydrophobic surfaces of varying heights, Langmuir 25, 7561–7567 (2009).

9 W. Xu, R. Leeladhar, Y.-T. Tsai, E.-H. Yang, and C.-H. Choi, Evaporative self-assembly of nanowires on superhydrophobic surfaces of nanotip latching structures, Appl. Phys. Lett. 98, 073101 (2011).

10 A. Marmur, The lotus effect: Superhydrophobicity and metastability, Langmuir 20, 3517–3519 (2004).

11 L. Feng, Y. Zhang, J. Xi, Y. Zhu, N. Wang, F. Xia, and L. Jiang, Petal effect: A superhydrophobic state with high adhesive force, Langmuir 24, 4114–4119 (2008).

12 I.B. Rietveld, K. Kobayashi, H. Yamada, and K. Matsushige, Electrospray deposition, model, and experiment: Toward general control of film morphology, J. Phys. Chem. B 110, 23351–23364 (2006).

13 P. Calvert, Inkjet printing for materials and devices, Chem. Mater. 13, 3299–3305 (2001).

14 X. Hong, X. Gao, and L. Jiang, Application of superhydrophobic surface with high adhesive force in no lost transport of superparamagnetic microdroplet, J. Am. Chem. Soc. 129, 1478–1479 (2007).

15 W.K. Cho and I.S. Choi, Fabrication of hairy polymeric films inspired by geckos: Wetting and high adhesion properties, Adv. Funct. Mater. 18, 1089–1096 (2008).

16 D. Öner and T.J. McCarthy, Ultrahydrophobic surfaces. Effects of topography length scales on wettability., Langmuir 16, 7777–7782 (2000).

17 P.S.H. Forsberg, C. Priest, M. Brinkmann, R. Sedev, and J. Ralston, Contact line pinning on microstructured surfaces for liquids in the Wenzel state, Langmuir 26, 860–865 (2009).

18 C. Priest, R. Sedev, and J. Ralston, Asymmetric wetting hysteresis on chemical defects, Phys. Rev. Lett. 99, 026103 (2007).

19 M. Liu, Y. Zheng, J. Zhai, and L. Jiang, Bioinspired super-antiwetting interfaces with special liquid-solid adhesion, Acc. Chem. Res. 43, 368–377 (2010).

20 K.-H. Chu, R. Xiao, and E.N. Wang, Uni-directional liquid spreading on asymmetric nanostructured surfaces, Nature Mater. 9, 413–417 (2010).

21 N.A. Malvadkar, M.J. Hancock, K. Sekeroglu, W.J. Dressick, and M.C. Demirel, An engineered anisotropic nanofilm with unidirectional wetting properties, Nature Mater. 9, 1023–1028 (2010).

22 G. McHale, N.J. Shirtcliffe, and M.I. Newton, Contact-angle hysteresis on super-hydrophobic surfaces, Langmuir 20, 10146–10149 (2004).

23 N.J. Shirtcliffe, S. Aqil, C. Evans, G. McHale, M.I. Newton, C.C. Perry, and P. Roach, The use of high aspect ratio photoresist (SU-8) for super-hydrophobic pattern prototyping, J. Micromech. Microeng. 14, 1384–1389 (2004).

24 M. Reyssat and D. Quéré, Contact angle hysteresis generated by strong dilute defects, J. Phys. Chem. B 113, 3906–3909 (2009).

25 C. Priest, T.W.J. Albrecht, R. Sedev, and J. Ralston, Asymmetric wetting hysteresis on hydrophobic microstructured surfaces, Langmuir 25, 5655–5660 (2009).

26 K.Y. Yeh, L.J. Chen, and J.Y. Chang, Contact angle hysteresis on regular pillar-like hydrophobic surfaces, Langmuir 24, 245–251 (2008).

27 B.M. Mognetti and J.M. Yeomans, Modeling receding contact lines on superhydrophobic surfaces, Langmuir 26, 18162–18168 (2010).

28 L. Gao and T.J. McCarthy, Contact angle hysteresis explained, Langmuir 22, 6234–6237 (2006).

29 K.A. Wier and T.J. McCarthy, Condensation on ultrahydrophobic surfaces and its effect on droplet mobility: Ultrahydrophobic surfaces are not always water repellant, Langmuir 22, 2433–2436 (2006).

30 C.W. Extrand, Model for contact angles and hysteresis on rough and ultraphobic surfaces, Langmuir 18, 7991–7999 (2002).

31 W. Choi, A. Tuteja, J.M. Mabry, R.E. Cohen, and G.H. McKinley, A modified Cassie-Baxter relationship to explain contact angle hysteresis and anisotropy on non-wetting textured surfaces, J. Colloid Interface Sci. 339, 208–216 (2009).

32 W. Xu and C.-H. Choi, From sticky to slippery droplets: Dynamics of contact line depinning on superhydrophobic surfaces, Phys. Rev. Lett. 109, 024504 (2012).

33 J.F. Joanny and P.G. de Gennes, A model for contact angle hysteresis, J. Chem. Phys 81, 552–562 (1984).

34 L. Leger and J.F. Joanny, Liquid spreading, Rep. Prog. Phys. 55, 431 (1992).

35 T. Cubaud and M. Fermigier, Faceted drops on heterogeneous surfaces, Europhys. Lett. 55, 239–245 (2001).

36 T. Cubaud, M. Fermigier, and P. Jenffer, Spreading of large drops on patterned surfaces, Oil Gas Sci. Technol. 56, 23–31 (2001).

37 J.-M. Di Meglio, Contact angle hysteresis and interacting surface defects, Europhys. Lett. 17, 607 (1992).

38 M. Ma and R.M. Hill, Superhydrophobic surfaces, Curr. Opin. Colloid Interface Sci. 11, 193 (2006).

39 M. Nosonovsky and B. Bhushan, Hierarchical roughness optimization for biomimetic superhydrophobic surfaces, Ultramicroscopy 107, 969–979 (2007).

40 P. Roach, N.J. Shirtcliffe, and M.I. Newton, Progress in superhydrophobic surface development, Soft Matter 4, 224–240 (2008).

41 B. Bhushan, Y.C. Jung, and K. Koch, Micro-, nano- and hierarchical structures for superhydrophobicity, self-cleaning and low adhesion, Phil. Trans. R. Soc. A 367, 1631–1672 (2009).

42 A. Lafuma and D. Quéré, Superhydrophobic states, Nature Mater. 2, 457–460 (2003).

43 C. Lv, C. Yang, P. Hao, F. He, and Q. Zheng, Sliding of water droplets on micro-structured hydrophobic surfaces, Langmuir 26, 8704–8708 (2010).

44 L. Shi, P. Shen, D. Zhang, Q. Lin, and Q. Jiang, Wetting and evaporation behaviors of water-ethanol sessile drops on PTFE surfaces, Surf. Interface Anal. 41, 951–955 (2009).

45 C. Dorrer and J. Rühe, Advancing and receding motion of droplets on ultrahydrophobic post surfaces, Langmuir 22, 7652–7657 (2006).

46 C. Dorrer and J. Rühe, Contact line shape on ultrahydrophobic post surfaces, Langmuir 23, 3179–3183 (2007).

47 K. Sefiane, Effect of nonionic surfactant on wetting behavior of an evaporating drop under a reduced pressure environment, J. Colloid Interface Sci. 272, 411–419 (2004).

*Corresponding author: [e-mail]

Chapter 2

Computational and Experimental Study of Contact Angle Hysteresis in Multiphase Systems

Vahid Mortazavi, Vahid Hejazi, Roshan M D’Souza, and Michael Nosonovsky*

College of Engineering & Applied Science, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin, USA

Abstract

Contact angle (CA) hysteresis exists due to the surface roughness and chemical heterogeneity. It is caused by a variety of factors, including adhesion hysteresis in the solid-water contact area (2D effect) and by pinning of the solid-water-air triple line (1D effect). In this study we show that CA hysteresis is present also in complex systems, such as an organic liquid (oil) in contact with a solid immersed in water. We study experimentally CA hysteresis in solid-water-air (droplet), solid-air-water (bubble), solid-water-oil and solid-water-air-oil systems. Then, we use the Cellular Potts Model (CPM) to discuss dependency of CA hysteresis on the surface structure and other parameters. This analysis allows decoupling of the 1D (pinning of the triple line) and 2D effects (adhesion hysteresis in the contact area) and provides new insights into the nature of the CA hysteresis.

Keywords: Contact angle hysteresis, hydrophobicity, oleophobicity, surface roughness, Cellular Potts model, multiphase system

2.1 Introduction

Contact angle (CA) is the main parameter in the study of wetting, which characterizes wetting of a solid surface by a liquid. It was introduced by Young [1] along with the concept of the interfacial tensions of the solid, liquid and vapor interfaces. Young related the CA to these interfacial tensions, using simple considerations of the equilibrium at the triple line, i.e., the line where the solid, liquid and vapor phases come in contact

(2.1) equation

where γSL, γSV, and γLV are the interfacial energies of the solid-liquid, solid-vapor and liquid-vapor interfaces, respectively (Figure 2.1). These interfacial energies can be thought of either as energies per unit area needed to create an interface or as generalized forces per unit length acting along the interfaces at the triple line in equilibrium. Note that unlike conventional mechanical forces, the tension forces are not applied to the triple line (which is a geometrical line and not a material object with mass) but rather constitute derivatives of the interfacial energies by the distance for which the triple line advances. These forces reflect the tendency of the system to reduce its energy (and increase entropy). The high solid-vapor energy of an interface causes wetting of the solid surface and spreading of liquid on it (low θ) while low energy produces non-wetting interfaces (high θ).

Figure 2.1 Three-phase interface of solid, water and vapor [28].

The molecules sitting at the free surface of materials have less binding with adjacent molecules than the molecules in the bulk, so they have potential to make new bindings. Materials with higher potential have higher wetting ability. The concept of interfacial free energy was introduced by J. W. Gibbs in the 1870s. Although the surface tension (measured in Nm−1) and interfacial energy (measured in Jm−2) are often assumed to be identical, they are not exactly the same. The surface tension or, more exactly, the surface stress is the reversible work per unit area needed to elastically stretch a pre-existing surface. The surface stress tensor is defined as

(2.2) equation

where εij is the elastic strain tensor and δij is the Kronecker delta. For a symmetric surface, the diagonal components of the surface stress can be calculated as

(2.3) equation

For liquids, the interfacial free energy does not change when the surface is stretched, however, for solids ∂γ/∂ε is not zero because the surface atomic structure of a solid is modified in elastic deformation [2].

Since the concept of the CA was introduced, it was realized that this single parameter cannot completely characterize wetting. Furthermore, there is no one single value of the CA, but it can have a range of values θrec ≤ θ ≤ θadv, where θrec and θadv denote the receding and advancing contact angles, respectively. The contact angle can be measured also on a tilted surface (Figure 2.2), although it