Article pubs.acs.org/Langmuir
Contact Angle and Local Wetting at Contact Line Ri Li*,† and Yanguang Shan‡ †
School of Engineering, University of British Columbia, 1137 Alumni Avenue, Kelowna, BC, Canada V1V 1V7 School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China
‡
ABSTRACT: This theoretical study was motivated by recent experiments and theoretical work that had suggested the dependence of the static contact angle on the local wetting at the triple-phase contact line. We revisit this topic because the static contact angle as a local wetting parameter is still not widely understood and clearly known. To further clarify the relationship of the static contact angle with wetting, two approaches are applied to derive a general equation for the static contact angle of a droplet on a composite surface composed of heterogeneous components. A global approach based on the free surface energy of a thermodynamic system containing the droplet and solid surface shows the static contact angle as a function of local surface chemistry and local wetting state at the contact line. A local approach, in which only local forces acting on the contact line are considered, results in the same equation. The fact that the local approach agrees with the global approach further demonstrates the static contact angle as a local wetting parameter. Additionally, the study also suggests that the wetting described by the Wenzel and Cassie equations is also the local wetting of the contact line rather than the global wetting of the droplet.
I. INTRODUCTION Since the well-regarded Young equation was established1 for the wetting of smooth surfaces in the early 1800s, the static contact angle has long been used as an important parameter to quantify the wettability of the solid surface by the liquid. In the 1930s, the Wenzel equation2 was proposed for the wetting of rough surfaces, and the Cassie equation3,4 was developed in the 1940s for the wetting of heterogeneous surfaces. All of these early theories are considered to form the fundamental basis for the research of contact angle and wetting. During the past few years, there has been a renewed interest in the fundamental research of contact angle, mainly driven by the intensive research on superhydrophobic surfaces. Inspired by biological surfaces that exhibit significant water-repellent phenomena,5,6 hydrophobicity can be augmented by texturing hydrophobic surfaces with micro or even submicro structures to achieve superhydrophobicity.7−9 The research on superhydrophobicity has received tremendous attention, due to both the fast increase of related engineering applications10−12 and rapid advancement in micro/nano fabrication technology.13,14 The static contact angle of water droplets on these textured surfaces has been commonly used to represent superhydrophobicity. The understanding of the relationship between the contact angle and droplet−surface interactions15,16 is very often based on the Wenzel and Cassie equations.2,3 The two theories were developed on the basis of uniform assumptions, which include uniform distributions of surface features (surface chemistry and texture) and uniform liquid−solid contact under the bulk droplet. However, in many cases, heterogeneous © 2012 American Chemical Society
conditions exist. For example, the droplet−surface contact area consists of spots with different chemistry, roughness, or wetting state, and contact angles measured under these heterogeneous conditions are referred to as heterogeneous contact angles. A number of experimental studies on heterogeneous contact angles have been reported and stimulated intensive discussion on the fundamentals of heterogeneous wetting. For the contact angle on heterogeneous surfaces, two recent experiments should be mentioned here. Extrand17 experimentally investigated two-component smooth surfaces, where a single circular heterogeneous spot was completely contained within the droplet footprint. It was found that the contact angles were equal to the angles exhibited by the homogeneous periphery, showing no influence from the heterogeneous spot. Gao and McCarthy18 showed that the existence of a spot with different chemistry or roughness within the droplet footprint did not affect contact angle behaviors. Both papers concluded that the wetting in the vicinity of the contact line, not the entire droplet base, determines the contact angle. Heterogeneous wetting may also occur on rough surfaces with uniform textures, and the relationship between the contact angle and wetting state has been studied.19−23 A few studies have shown that the contact angle on these surfaces strongly depends on the way the droplet has been deposited24,25 or the pressure inside the sessile droplet.20,26 This indicates the existence of metastable wetting states, which has been Received: September 11, 2012 Revised: October 11, 2012 Published: October 16, 2012 15624
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experimentally observed.26 Moulinet and Bartolo26 also found that the penetration of the liquid meniscus within surface microstructures under the bulk droplet was not uniform. As the droplet size decreases due to evaporation, penetration at the center of the droplet footprint was observed to increase more significantly than the circumferential region. However, the contact angle remained almost unchanged, showing significant dependence on the local wetting at the contact line. Convincingly, the experimental studies have shown that static contact angles under heterogeneous conditions are solely determined by the local wetting at the contact line. Several studies17,18 claimed that this conclusion disagrees with the Wenzel and Cassie equations, and questioned the validity of the early theories. It was argued that the equations support incorrect concepts:18 (1) Contact area is important for determining the contact angle. (2) Interfacial free energies dictate the contact angle. The questioning initiated intensive discussion,27−30 with focus on the role of local wetting and understanding of the Wenzel and Cassie equations. McHale’s theoretical study27 suggested that the Wenzel and Cassie equations are valid for heterogeneous contact angles if the equations take local wetting values at the contact line rather than the global wetting of the droplet. Marmur and Bittoun30 argued that the contact angle can be predicted by the Wenzel or Cassie equation if the droplet is sufficiently large compared with the size of heterogeneity. Showing theoretical agreement with the previously published experiments, these recent theoretical efforts have demonstrated the static contact angle as a local wetting parameter. However, confusion and misunderstanding still exist in the research community regarding the relationship between contact angle and wetting. It is still very common that the static contact angle is used as a global wetting parameter to interpret and characterize the overall wetting of a droplet on a surface. In view of this, we think further theoretical work is necessary. The present work does not seek to make fundamentally new conclusions other than the recent theoretical work (e.g., ref 27), but it aims to clearly demonstrate the physical meaning of the static contact angle by applying two theoretical approaches based on different fundamental principles.
Figure 1. (a) Schematic representation of a static droplet on a composite surface composed of concentrically distributed surface components. (b) Top view of the surface with the dashed line being the contact line.
is the ratio of actual solid−liquid contact area to the area’s projection. Hence, each surface component in Figure 1 is specified by θe,i, f i, and ri. In Figure 1, the sessile droplet has a cap area A* and a projected footprint area B*, and these two areas have been normalized by the surface area of the droplet in free spherical shape. The droplet footprint B* covers components #1 through #n, and the contact line of the sessile droplet is currently located in the #n component at static state. Therefore, the present static contact angle is denoted by θs,n. The area of each component accounts for a fraction Fi of B*, and F1 + F2 + ... + Fn = 1. Here the fraction Fn refers to the covered portion of the #n component. For the heterogeneous wetting shown in Figure 1, we are interested in the relationship between the static contact angle and the wettings of all the components covered by the droplet. We hope the relationship can be described by a general equation, which will be derived in the following section using two approaches: global and local. For the global approach, the entire droplet and surface will be considered to investigate the relationship between the contact angle and the global wetting. For the local approach, only the contact line will be investigated for the relationship between the contact angle and local forces acting on the contact line. II.1. Global Approach Based on Thermodynamics. In the first approach, we define a closed thermodynamic system consisting of the droplet and solid surface and treat surface tensions as energy per unit area. To analyze the free surface energy of the system, we define a reference state where the droplet is a free sphere and has no contact with the surface. From the reference state to the present state (see Figure 1), the change of bulk liquid−vapor area is A* − 1, and the liquid− vapor area existing between microstructures is B*∑Fi(1 − f i) − 1. For each component, the generation of solid−liquid area (also the decrease of solid−vapor area) is B*Fi f iri with interfacial surface tensions of σSV,i and σSL,i. Hence, the total change of free surface energy against the reference state can be expressed by
II. THEORETICAL ANALYSIS Theoretical analysis will be carried out for the static contact angle of a liquid droplet on a solid surface composed of heterogeneous components. For simplicity, as shown in Figure 1, the heterogeneous components are laid out concentrically, and the components are numbered starting from the center. A droplet is put at the center of the concentric configuration such that the area covered by the droplet base is axisymmetric. To be general, all the surface components have different surface chemistries and roughnesses. The surface chemistry is represented by the equilibrium contact angle θe, which, for flat surfaces, satisfies the Young equation given by cos θe, i = (σSV, i − σSL , i)/σ
(1)
where σSV, σSL, and σ are surface tensions for solid−vapor, solid−liquid, and liquid−vapor interfaces, respectively. The wetting state of liquid on roughness structures can be characterized by two parameters: wetting fraction f and wetting roughness r. The wetting fraction is the fraction of projected area that has solid−liquid contact, while the wetting roughness 15625
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ΔG = (A* − 1)σ + B*
is in the #n component at static state, for the whole system shown in Figure 1 to be at local lowest-energy level, the surface energy and contact angle must satisfy
∑ Fi(1 − fi )σ 1→n
+ B*
∑ Ffi i ri(σSL ,i − σSV,i) 1→n
dΔG*/dθ = 0
(2)
Dividing through by σ and substituting eq 1, we reorganize eq 2 as ΔG* = A* − 1 − B*
∑ Fiχi 1→n
Inserting eq 3 into eq 5 gives dA * − dθ
(3)
∑ 1 → (n − 1)
d(B*Fi ) d(B*Fn) χi − χn = 0 dθ dθ
(6)
With the contact line in the #n component, an infinitesimal change of θ does not change the wetted area of other components, which, for 1 ≤ i ≤ n − 1, is
where ΔG* is the nondimensional surface energy normalized by the surface energy of the free droplet in spherical shape and χ is χi = fi ri cos θe, i + fi − 1
(5)
d(B*Fi ) =0 dθ
(4)
Here χi is a wetting parameter specific for each surface component. It is a function of surface chemistry (θe,i) and wetting state ( f i, ri) but is independent of the size and location of the surface component. Clearly, the free surface area expressed by eq 3 is a function of wetting condition, and the wetting condition can be characterized by the base diameter of the droplet maintaining the shape of a spherical cap on the surface. To understand how the free surface energy changes with the wetting condition, we consider the base diameter increasing from zero at the center of the #1 component. The increase of the base diameter can also be alternatively described by the contact angle decreasing from 180°, as these two geometric parameters are interrelated for spherical caps with a constant volume. In Figure 2, −ΔG* is qualitatively plotted against the contact angle descending from 180°. When the contact line moves
(7)
And any change of B* will only affect the covered portion of the #n component, which is d(B* − ∑1 → n − 1 B*Fi ) d(B*Fn) dB * = = dθ dθ dθ
(8)
Hence, eq 6 reduces to dA * dB * − χ =0 dθ dθ n
(9)
For a spherical cap, the following relation exists dA * = cos θ dB *
(10)
Combining eqs 9, 10, and 4, we obtain cos θs, n = fn rn cos θe, n + fn − 1
(11)
We have changed θ to θs,n, as there must only be one contact angle that satisfies the local lowest-energy level within the #n component. Equation 11 indicates that, if the contact line can maintain a static state in a surface component, the static contact angle is determined solely by the surface chemistry and wetting state of that component, χ, and the angle has nothing to do with other components covered within the droplet base. Now we can explain the other two trends depicted in Figure 2. The free surface energy could decrease or increase monotonically, which suggests nonexistence of static contact angle for components 2 and 4. This is simply because the static contact angle cos−1(χi), which is determined by θe,i, f i, and ri, does not fall into the range of the geometric contact angle (+θi, − θi), which is determined by the location and size of the surface component (−Ri, +Ri) as well as the droplet volume. It is easy to show that, for the #2 component, cos−1(χ2) < −θ2, while, for the #4 component, cos−1(χ4) > +θ4. The above analysis can be extended for an ideal wetting scenario, in which all the surface components within the droplet base are uniformly “mixed”, and this mixed pattern is applied over the entire surface. Hence, the local wetting at the contact line is the same as the global wetting under the bulk droplet. This is equivalent to considering a surface composed of uniformly distributed small patches of different materials or different textures. Thus, no matter how B* varies, area fractions Fi remain constant. Two composite parameters can be introduced to characterize this “well-mixed” wetting, which are
Figure 2. Qualitative depiction of the free surface energy versus decreasing contact angle for the wetting shown in Figure 1.
through a surface component located between −Ri and +Ri, the contact angle changes within a range bounded by +θi and −θi. Due to different values of θe, f, and r for adjoining components, a singular change of the energy appears when the contact line moves from one surface component to another. Within each component, the energy varies with the contact angle in three possible trends. The first trend can be seen for components 1, 3, and n, where local minima of energy exist. The second trend is depicted for the #2 component where the energy decreases monotonically, whereas the third trend is that the energy increases monotonically for the #4 component. Our analysis will start with the first trend, and the second and third trends will be discussed later on. For the first trend shown in Figure 2, since a local minimum exists, the static contact angle can be derived. If the contact line
(fr cos θe)C = 15626
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∑ fi Fi
(12)
For this ideal wetting scenario, eq 6 can be rewritten as dA * dB * − dθ dθ
∑ Fiχi
=0
(13)
Combining eqs 10 and 13 gives cos θs =
∑ Fiχi
(14)
which is the Cassie equation. Here θs denotes a static contact angle, and there must be only one such angle specific for any surface with uniformly mixed heterogeneity. If we plot the free surface energy of a droplet on this type of surface as a function of contact angle, there should be only one lowest-energy level, as opposed to the multiple minima shown in Figure 2 for heterogeneous surfaces. It should be noted that the above discussion of “well-mixed” wetting redemonstrates Marmur and Bittoun’s conclusion,30 because the size of an individual heterogeneous component in the well-mixed surface pattern is extremely smaller than the droplet size. If we combine eqs 14, 4, and 12, the Cassie equation can be rewritten as cos θs = (fr cos θe)C + fC − 1
Figure 3. (a) Contact line with infinitesimal displacement Δx within the surface component #n as shown in Figure 1 (L, liquid; V, vapor; S, solid). The dashed line at the L−S interfaces indicates partial wetting of microstructures. (b) Surface tension forces acting on a differential element Δl of the triple-phase contact line.
by each force is equal to the force (e.g., σSLΔl) multiplied by the effective distance (e.g., Δxf nrn). We will use the positive and negative signs to show the force and displacement are in the same and opposite directions, respectively. Hence,
(15)
WLV,1 = +Δx( −cos θ )σ Δl
Clearly, eq 15 has the same form as eq 11. This suggests that the Cassie equation is also based on the local wetting at the contact line. A similar discussion can also be made for the Wenzel equation. In Figure 1, if the #n component is rough and all the microstructures are completely wetted (f n = 1), eq 11 becomes
WLV,2 = −Δx(1 − fn )σ Δl
cos θs, n = rn cos θe, n
WSV = −Δxfn rnσSV Δl WSL = +Δxfn rnσSLΔl
(17)
This force−displacement analysis can also be interpreted from the perspective of the change of surface energy. For example, WSL can be considered as surface energy by reorganizing its expression as WSL= σSL × Δxf nrnΔl . Here σSL is the solid− liquid interfacial energy per unit area, while Δxf nrnΔl is the increase of solid−liquid area due to the displacement of the contact line. If the four forces are in balance, the net amount of work done for the infinitesimal displacement of the contact line must be zero, which is
(16)
Hence, if there is complete wetting at the contact line, the contact angle follows the Wenzel equation regardless of the overall wetting under the bulk droplet. This shows that the fundamental basis for the Wenzel equation is also the local wetting at the contact line. II.2. Local Approach Based on Forces. The analysis above using a global approach resulted in a local conclusion related to the local wetting at the contact line. This motivated us to redo the analysis using a local approach, which focuses on the contact line rather than the droplet−surface system. A similar local approach has been used in previous studies on wetting (e.g., refs 31 and 32) for investigating the work done by the displacement of the contact line. In the second approach, we consider only the contact line as shown in Figure 3 and treat surface tensions as force per unit length. To be consistent with the first approach, the wetting values (θe,n, f n, rn) in the vicinity of the contact line in Figure 3 are the same as those of the #n component in Figure 1. We assume that the contact line is subject to forces associated with only the three types of surface tensions (σ, σSV, σSL). For an infinitesimal displacement Δx of the contact line as shown in Figure 3a, both the change of contact angle and the change of contact line length are negligible. Figure 3b shows the forces acting on a differential element of the contact line, and its length is Δl. There are a total of four forces, which, in addition to the three well-understood forces, also include another force σΔl in the same direction of σSLΔl toward the center of the droplet base. This force exists if surface microstructures are not completely wetted. Due to the displacement of the contact line, the work (e.g., WSL) done
lim
Δx → 0
∑W =0 Δx
(18)
Substituting eq 17 into eq 18 and applying eq 1 gives eq 11. Therefore, the local force balance approach results in the same equation as the free surface energy approach. The fact that the local approach is applicable further demonstrates the static contact angle as a local wetting parameter. It should be noted that for the second approach only the surface tension forces were considered. If other contact line pinning forces exist, the force balance derived from the second approach would not agree with eq 11, and the contact angle must satisfy the force balance rather than the thermodynamic equilibrium, thereby forming nonequilibrium contact angles.
III. CONCLUSIONS The two approaches based on the global free surface energy and local force balance at the contact line resulted in one same equation, which describes the relationship of the static contact angle to wetting. The relationship is that the static contact angle is determined only by the local wetting at the contact line. Hence, as a local wetting parameter, a measured static contact 15627
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(20) Lafuma, A.; Quéré, D. Superhydrophobic states. Nat. Mater. 2003, 2, 457−460. (21) Ishino, C.; Okumura, K.; Quéré, D. Wetting transitions on rough surfaces. Europhys. Lett. 2004, 68, 419−425. (22) Wang, S.; Jiang, L. Definition of superhydrophobic states. Adv. Mater. 2007, 19, 3423−3424. (23) Li, R.; Alizadeh, A.; Shang, W. Adhesion of liquid droplets to rough surfaces. Phys. Rev. E 2010, 82, 041608. (24) He, B.; Patankar, N. A.; Lee, J. Multiple equilibrium droplet shapes and design criterion for rough hydrophobic surfaces. Langmuir 2003, 19, 4999−5003. (25) Kwon, H.-M.; Paxson, A. T.; Varanasi, K. K.; Patankar, N. A. Rapid deceleration-driven wetting transition during pendant drop deposition on superhydrophobic surfaces. Phys. Rev. Lett. 2011, 106, 036102. (26) Moulinet, S.; Bartolo, D. Life and death of a fakir droplet: Impalement transitions on superhydrophobic surfaces. Eur. Phys. J. E 2007, 24, 251−260. (27) McHale, G. Cassie and Wenzel: Were they really so wrong? Langmuir 2007, 23, 8200−8205. (28) Nosonovsky, M. On the range of applicability of the Wenzel and Cassie equations. Langmuir 2007, 23, 9919−9920. (29) Panchagnula, M. V.; Vedantam, S. Comment on how Wenzel and Cassie were wrong by Gao and McCarthy. Langmuir 2007, 23, 13242−13242. (30) Marmur, A.; Bittoun, E. When Wenzel and Cassie are right: Reconciling local and global considerations. Langmuir 2009, 25, 1277−1281. (31) Extrand, C. W. A thermodynamic model for wetting free energies from contact angles. Langmuir 2003, 19, 646−649. (32) Extrand, C. W. Work of Wetting Associated with the Spreading of Sessile Drops. In Contact Angle, Wettability and Adhesion; Mittal, K. L., Ed.; VSP/Brill: Leiden, The Netherlands, 2009; Vol. 6, pp 81−93.
angle cannot be used to characterize the global wetting of liquid droplets on heterogeneous surfaces. For a droplet on a heterogeneous surface, each heterogeneous component has a specific static contact angle, which is a function of surface chemistry and wetting state. However, this static contact angle may or may not be achievable, depending on the size of that component, its relative location from the droplet, and the volume of the droplet. Discussion was also made on the Cassie and Wenzel equations by applying the derived general equation to some ideal surface conditions. It suggests that both equations fundamentally also refer to local wetting rather than global wetting.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
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