Origins of Wetting - Langmuir (ACS Publications)

Invited Feature Article pubs.acs.org/Langmuir

Origins of Wetting Charles W. Extrand* CPC, Inc., 1001 Westgate Drive, St. Paul, Minnesota 55114, United States ABSTRACT: This feature article provides an overview of wetting phenomena. Much of the analysis done on wetting in the last 100 years assumes that the phenomena are determined by molecular interactions within the interfacial area between the liquid and solid. However, there is now ample evidence that wetting is controlled by interactions in the vicinity of the contact line where the liquid and solid meet. Recent experiments and modeling that demonstrate this are reviewed.

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the first careful observations were reported by Hauksbee in 1706.4 He noted that liquids rise higher in smaller-diameter tubes and that vacuum has no influence. A decade later, Jurin5 showed that the final rise height is inversely proportional to the tube diameter. In 1751, Segner established surface tension as an intrinsic property of a liquid that causes its surface to behave as if it were an elastic membrane.10 At the beginning of the 19th century, Young observed that the angle of contact between a given liquid and solid is independent of geometry, which ultimately led him to the conclusion that “all the phenomena of capillary action may be accurately explained and mathematically demonstrated from the general law of equable tension of the surface of the fluid, together with the consideration of the angle of contact appropriate to every combination of a fluid with a solid.”7 Young avoided mathematical symbols and equations in his writings. Instead, he gave verbal descriptions of various wetting phenomena as well as his now famous Young equation, which describes the balance between forces at the contact line or triple point where gas, liquid, and solid meet.7 In contrast, Laplace, active at the same time as Young, embraced symbols and equations. He performed complex mathematical analyses of a variety of wetting phenomena, including capillary rise in a tube.8 Laplace showed that the final height (h) to which liquid rises can be quantified in terms of the tube diameter (D), the surface tension of the liquid (γ) and its density (ρ), and the contact angle between the advancing liquid and solid wall of the tube (θa)

INTRODUCTION Explorations of the origins of wetting can be traced back hundreds of years.1−8 Much of the early work attempted to explain the seemingly spontaneous rise of liquids in smalldiameter “capillary” tubes, depicted in Figure 1. Consequently, still to this day, wetting phenomena between liquids and solids are often referred to as capillary action or capillarity.9 Some of

h=

4γ cos θa ρgD

(1)

where g is the acceleration due to gravity. The analyses of wetting by Young and Laplace were based on forces and pressures. Figure 1. Depiction of liquid rising inside a small-diameter tube, oriented vertically, with both ends open. (a) The tube contacts the liquid. (b) Liquid rises vertically inside the tube. (c) The liquid stops at a final height of h. (d) A close-up view of the meniscus. © XXXX American Chemical Society

Received: May 20, 2016 Revised: July 12, 2016

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DOI: 10.1021/acs.langmuir.6b01935 Langmuir XXXX, XXX, XXX−XXX

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Invited Feature Article

In their time, the concepts of energy, including heat and work, were ill-defined and poorly understood. The foundations of modern thermodynamics would be laid by Carnot, Joule, Clapeyron, Thomson, Clausius, and Helmholtz in the first half of the mid-19th century.11 These new energy concepts were first applied to wetting phenomena by Dupré and Gibbs during the latter portion of the 19th century.12,13 In the 20th century, problems previously solved in terms of forces, including capillary rise,14 were often recast in terms of energies.15−21 Many of these energy models assume that wetting is controlled by interfacial interactions within the wetted area between the liquid and solid. Area-based models have been criticized for not reflecting the underlying physical processes or producing accurate estimates.22−30 These models generally assume that surfaces show a single equilibrium contact angle, whereas real surfaces almost always show a range of contact angles that lie between a maximum advancing angle (θa) and a minimum receding angle (θr). In this article, recent experiments that demonstrate the origins of wetting are reviewed. Next, it is shown how wetting phenomena can be modeled by assuming that interactions in the vicinity of the contact line control wetting. Finally, the contact line itself is discussed.

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A FEW SIMPLE WETTING EXPERIMENTS

In this section, two experimental studies are summarized. The first involves the spontaneous rise of water in capillary tubes, and the second, liquid drops spreading on flat surfaces. Capillary Rise.28 Consider the capillary tubes shown in Figure 2. They are hydrophilic borosilicate glass from Pyrex and have an internal diameter of D = 1.2 mm. In Figure 2a, a capillary is lowered into contact with water, just touching its surface. Water wets the bottom of the tube and establishes an advancing contact angle (θa) of ∼0°, which creates a spherically shaped meniscus inside the tube. The concave curvature of the meniscus creates an upward Laplace pressure. Water initially rises rapidly, then slows and finally stops where the hydrostatic pressure of the water in the tube equals the Laplace pressure of the air−liquid interface. The final height relative to the surrounding water is 24 mm, which agrees well with the height of h = 24.5 mm predicted by eq 1. In Figure 2b, the same experiment is repeated with one variation. Rather than just touching the water, the capillary tube is plunged below its surface. This has no influence on the relative rise of the water. The water still climbs 24 mm above its surrounding surface. Thus, the change in liquid−solid interfacial area does not influence capillary rise. If an increase in interfacial energies were to control this phenomenon, then the rise height would have been expected to increase with immersion depth. It does not. In Figure 2c,d, the experiment was repeated with an additional variationthe bottom portion (about 10 mm in length) of the capillary tube was dipped in a polystyrenetoluene solution and allowed to dry. This rendered the lower portion of the tube hydrophobic with an advancing contact angle of θa ≈ 90°. If the coated end of the capillary is lowered a short distance below the surface, then water does not rise (Figure 2c). However, if the coated capillary is lowered farther until water reaches the uncoated hydrophilic portion of the tube, then the water rises to the same height as the uncoated tube (Figure 2d). Again, the area of the wetted solid does not influence the rise height. Moreover, applying the hydrophobic

Figure 2. (a) A 1.2 mm glass capillary tube touching the surface of water. (b) The same capillary plunged below the surface. (c) The polystyrene-coated portion of a 1.2 mm capillary contacting the surface of water. No rise occurs. (d) The coated portion of the tube is immersed such that water contacts the uncoated portionwater rises. Arrows show the positions of the menisci. Adapted with permission from ref 28. Copyright 2012 American Chemical Society.

polystyrene coating to only the inside or outside produced the same capillary rise as the uncoated tubes. From these experiments, it can be concluded that the liquid− solid interfacial area does not control capillary rise. This leaves two possibilities: wetting and rise are controlled by interactions either in the vicinity of the contact line or within the air−liquid interface of the meniscus. One of the well-known properties of liquids is the tendency to minimize their air−liquid interfacial area. If the air−liquid interface were to control capillary rise, then the meniscus in tubes would always be flat (or nearly flat except at the periphery where the liquid contacts the wall of the capillary). However, this is generally not true. Unless θa = 90°, the air−liquid interface exhibits curvature. If gravity is not a factor, then the interface is spherically shaped. The air−liquid interface minimizes its area within the constraints imposed at the contact line. Or mathematically, forces at the contact line dictate the boundary conditions for the minimization of the air−liquid interface. If gravity is significant, then the interface must accommodate this force too.16,31−33 Therefore, through process of elimination, it can be concluded that capillary rise is controlled by neither the liquid−solid nor the air−liquid interface. Rather, capillary forces generated in the vicinity of the contact line control the rise of liquids in small-diameter tubes. The surface tension and density of the liquid play supporting roles. This behavior is not B


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unique to capillary rise. A wide variety of wetting phenomena are controlled by interactions in the vicinity of the contact line, including the spreading of sessile drops described in the next section. Spreading of Sessile Drops.25,34 Consider the case of a sessile drop depicted in Figure 3. When a small liquid drop is

Figure 4. Depiction of wetting experiments involving a single heterogeneous island. (a) A water drop is deposited on a clean silica surface, θa = 7 ± 3°. (b) A small circular heterogeneous island is created on the wafer by depositing a drop of polystyrene (PS) solution and allowing it to dry. (c) If a small drop of water is deposited on the PS island such that the drop is contained within the periphery of the island, then θa = 95 ± 2°. (d) If water is added to push the contact line onto the surrounding silica, then the contact angle abruptly decreases to the contact angle observed on the homogeneous silica, θa = 6 ± 2°.

Figure 3. Small sessile drop on a solid surface. (a) As liquid is added, the contact line advances. Each time motion ceases, the drop exhibits an advancing contact angle (θa). (b) Alternatively, if liquid is removed from the drop, then the contact angle decreases to a receding value (θr), and then the contact line retreats.

angles occurs. Receding contact angles and hysteresis also are indifferent to heterogeneous islands contained within their wetted perimeter. These experiments were repeated with other liquids and solids.25 Again, the liquids did not sense the heterogeneous islands beneath them. Thus, apparent contact angles are determined by interactions at the contact line. These findings for smooth, chemically heterogeneous surfaces are analogous to those presented by Bartell and Shepard23 and Gao and McCarthy26 for chemically homogeneous surfaces with rough islands. Modeling of Wetting Phenomena. A wide variety of wetting phenomena have been successfully modeled by assuming that wetting is controlled by interactions in the vicinity of the contact line: capillary rise in homogeneous tubes37 and tubes with chemical gradients,38 inception of flow in horizontal tubes with smooth surfaces39,40 or rough surfaces,41 sessile drops spreading on rough surfaces,13,42 chemically heterogeneous surfaces,25 curved surfaces,43 deformable surfaces (i.e., elasto-capillary phenomena),44,45 directionally biased wetting,41 synthetic superhydrophobic surfaces with rectilinear features,24,46 naturally occurring superhydrophobic surfaces with curvilinear features,47,48 optimization of superhydrophobicity,49 structured superwetting surfaces,50 capillary bridges,32,51 retention of liquid drops on an inclined plane52,53 or retention on a rotating disk,54 flotation of cylinders,32 disks55 or spheres,56 wetting of porous media composed of fibers57 or spherical particles,58 and Wilhelmy tensiometry.59 In this section, three examples are given that show how wetting phenomena can be modeled by assuming interactions at the contact line control wetting: capillary rise in a vertical tube, directionally biased inception of flow in a tube, and the repellency of a naturally occurring structured surfacethe lotus leaf. Capillary Rise in a Vertical Tube. The ultimate height (h) that liquid rises in a capillary (eq 1) can be derived by considering the forces and energies that arise from molecular interactions in the vicinity of the contact line.37 The energy balance for slow capillary rise consists of three terms: the work done in lifting the liquid (w), viscous dissipation (K) in the bulk of the flowing liquid, and a potential energy term that accounts for energy stored in the liquid,

deposited on a smooth solid surface, it spreads until the cohesive forces of the liquid equal the adhesive forces between the liquid and solid. If additional liquid is added to the drop, then the contact line advances. Each time motion ceases, the drop exhibits an advancing contact angle (θa). Alternatively, if liquid is removed from the drop, then the contact line remains pinned until the contact angle decreases to a receding value (θr), and then the contact line retreats. This difference between θa and θr is referred to as contact angle hysteresis (Δθ), Δθ = θa − θr

(2)

Hysteresis can arise from molecular interactions between the liquid and solid or from surface anomalies, such as roughness or heterogeneities.35 In the first studies of heterogeneous surfaces, Wenzel36 and Cassie18,19 concluded that wetting and the resulting apparent angles are controlled by the interfacial contact area between the liquid and solid. Early experiments on the wetting of heterogeneous materials often involved complex domains, which were circuitous, poorly defined, or too small to be easily observed. More recently, simpler tests have been performed using flat surfaces with a single macroscopic heterogeneity.25,34 Consider the experiment depicted in Figure 4. A silicon wafer with a thin native oxide layer is cleaned by oxygen plasma. A drop of water is deposited on its surface. The drop nearly wets the clean silica surface, yielding an advancing contact angle of θa = 7 ± 3°. Next, a small circular heterogeneous island is created on the wafer by depositing a drop of polystyrene (PS) solution and allowing it to dry (Figure 4b). If a small drop of water is deposited on the PS island such that the drop is contained within the periphery of the island, then θa = 95 ± 2° (Figure 4c). However, if water is added to the drop, causing the contact line to advance onto the surrounding silica, then the contact angle changes abruptly, dropping to θa = 6 ± 2° (Figure 4d). Even though the underlying contact area contained a mixture of domains, the contact angle is equal to that exhibited by the homogeneous silica. Thus, the water does not sense the underlying heterogeneity, and no area averaging of the contact

w − K = −ΔU C

(3) DOI: 10.1021/acs.langmuir.6b01935 Langmuir XXXX, XXX, XXX−XXX

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straw with a flexible elbow.41 If a slug of water is introduced into such a straw while oriented horizontally, the slug will sit motionless. If the straw is tilted slowly, the slug will remain stationary until the gravitational force exerted on the slug exceeds the surface forces and then the slug will begin to move.39,40 For the smooth portions of the straw, the impeding surface force is equal in both directions. That is to say, the straw must be rotated to the same extent, left or right, to trigger motion. When the liquid slug encounters the ridges of the articulated elbow, shown in Figure 5, resistance to movement increases.

Assume that the tube diameter (D) is small and homogeneous (Figure 1). As the bottom of the tube touches the liquid, a meniscus forms. The vertical component of the surface tension (γ) acting around the contact line (πD) creates an upwarddirected capillary force ( fc),

fc = πDγ cos θa

(4)

The work done as surface tension lifts liquid upward through a tube (w) can be calculated by integrating the capillary force (fc) from z = 0 to h, w=

∫0

h

fc dz

(5)

Combining eqs 4 and 5 and integrating gives w = πDhγ cos θa

(6)

The energy lost due to viscous dissipation (K) for laminar flow through a tube of constant cross section generally can be estimated from the liquid volume (V) and pressure change (Δp),60 K=

∫0

Δp

Vd(Δp)

For capillary rise, eq 7 becomes K=

∫0

h

(7) 14

Figure 5. Articulated elbow of a transparent plastic drinking straw with a blue stripe. Its diameter is ∼5 mm.

π 2 π D zρgdz = ρgD2h2 4 8

Because the ridges are asymmetric, the resistance to flow is no longer the same in both directions. For a slug of a given volume, if tipped vertically in one direction, the slug will remain anchored in the articulated portion of the straw, but if tipped vertically in the other direction, the slug will move. This directionally biased wetting can be modeled by considering forces arising from the contact line. Figure 6

(8) 61

According to elementary thermodynamics, the change in the potential energy (−ΔU) can be calculated by integrating the hydrostatic force (f i) as liquid rises from z = 0 to h, −ΔU =

∫0

h

fi dz

(9)

The hydrostatic force is the product of the local hydrostatic pressure (Δp) and the cross-sectional area of the tube (Ac), which in turn can be framed in terms of tube diameter, liquid density, and height, π fi = ΔpAc = ρgD2z (10) 4 Combining eqs 9 and 10 and then integrating produces an expression for the change in potential energy due to capillary rise,14 −ΔU =

∫0

h

π π ρgD2z dz = ρgD2h2 4 8

(11)

The various terms for work, viscous dissipation, and potential energy from eqs 6, 8, and 11 can be plugged into the energy balance (eq 3) and rearranged to obtain the well-known Laplace expression for capillary rise (eq 1). This energy approach is simpler and less circuitous than the route of Brown,14 which assumes that wetting is driven by changes in solid surface energy and requires the use of the Young equation.7 Alternatively, capillary rise can be modeled by balancing the Laplace and hydrostatic pressures8 or for homogeneous tubes of constant cross section by equating the capillary force to the weight of the liquid column.9 The pros and cons of these various approaches are discussed elsewhere.28 Directionally Biased Inception of Flow in a Horizontal Tube. A liquid slug inside a tube with asymmetric “sawtooth” features moves more easily in one lateral direction than the other. This point can be demonstrated using a plastic drinking

Figure 6. (a) Depiction of a horizontal tube with asymmetric features. The retention forces (f 2 and f 3) that resist incipient motion differ in the two directions along the axis of the tube. (b) Enlarged side view of asymmetric sawtooth features with rise angles of ω2 and ω3.

depicts a horizontal tube with asymmetric features. The retention force that resists incipient motion differs in the two directions along the axis of the tube. The difference between those forces (f 2 and f 3) is determined by the tube diameter (D), the liquid surface tension (γ), the intrinsic advancing and receding contact angles (θa and θr), and the rise angles that define the shape of the asymmetric features (ω2 and ω3),39,40 D


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f3 − f2 = πγD[cos(θr − ω3) − cos(θa + ω3) − cos(θr − ω2) + cos(θa + ω2)]

(12)

By employing a few trigonometric functions and a bit of algebra, eq 12 can be recast as a ratio, f3 f2

=

( 1 ) 1 sin(ω2 + 2 Δθ) sin ω3 + 2 Δθ

(13)

where Δθ is the intrinsic contact angle hysteresis of the featureless tube surface (eq 2). According to eq 13, to maximize directional bias, one should minimize hysteresis and maximize the asymmetry of the ratchet features. In principle, the force ratio can be quite large. For example, if ω2 = 10°, ω3 = 90°, and Δθ = 10°, then f 3/f 2 ≈ 4. This general approach has been used to model the directionally biased wetting of sessile drops on films composed of tilted poly(p-xylylene) nanorods.62 Repellency of Structured Surfaces. Structured surfaces that exhibit extreme repellency, or superhydrophobicity, usually combine highly geometric topography with intrinsic hydrophobicity. Synthetic versions are often designed to mimic nature.63,64 Two examples are depicted in Figure 7. These

Figure 8. Scanning electron micrograph of the top surface of the lotus (Nelumbo nucifera) leaf. The white scale bar is 20 μm long. Image kindly provided by Prof. W. Barthlott.

papillae takes the form of prolate hemispheroids. The protuberances are covered with a secondary structure of wax crystalloids that project orthogonally outward, creating myriad sharp edges. Figure 9 shows a model of the lotus leaf. The model surface consists of rigid hemispheres centered within a hexagonal array (Figure 9a). The base diameter of the hemispherical

Figure 7. Small sessile drops suspended on superhydrophobic surfaces covered with (a) rectilinear or (b) curvilinear features. The apparent advancing contact angle (θapa) is effectively 180°. The apparent receding contact angle (θapr) determines adhesion.

surfaces are covered with rectilinear or curvilinear features. Small liquid drops have been deposited on them. In both cases, the drops are suspended on the features, often referred to as the Cassie state.18 The apparent advancing contact angle (θapa) is effectively 180°.24,30,48,65−69 Thus, repellency is controlled by interactions at the receding contact line.24,27,30,48 At the extreme limit, apparent receding contact angles (θapr) can approach 180° and adhesion vanishes (θapa − θapr → 0°).27 What is suspending drops atop these structured surfaces? To prevent intrusion, capillary forces acting around the features must be directed upward and of greater magnitude than any downward forces from gravity, inertia, Laplace curvature, and so forth.24 Also, the features must be sufficiently tall that any liquid protruding between them does not contact the underlying solid.46 Otherwise, downward forces will drive liquid into the structure, completely wetting it, creating the so-called Wenzel state.36 The repellency of structured surfaces with rectilinear or curvilinear features has been modeled by considering the interactions in the vicinity of the contact line.24,46−49,58 An example is given for the lotus leaf.47,48 The surface of the lotus leaf is covered with protuberances that have structural hierarchy (Figure 8). The primary structure of these protuberances or

Figure 9. (a) Plan view of a liquid suspended on the model lotus leaf. The larger black circles represent the bases of the protuberances, and the smaller blue circles show the contact lines of the suspended liquid. The perimeter contact line at the outer edge of the drop is depicted as straight blue lines connected to blue semicircles. (b) Side view under the drop or near the advancing edge. (c) Side view at the receding edge. The curvature of the air−liquid interface is exaggerated to emphasize the liquid orientation and local inherent contact angle. The density of the secondary features is intentionally sparse to allow for a clear depiction of the liquid orientation and the various parameters. E


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protuberances is 2R. The unit cell dimension is 2y. The hemispheres have a secondary roughness that radiates orthogonally (Figure 9b,c). In the model, each secondary feature effectively forms a continuous edge that extends around the entire circumference of the hemispheres, much like latitude lines on a globe. 2R extends to the tips of these secondary features. From Figure 8, 2R and 2y for the lotus leaf were estimated to be 11.0 ± 1.4 and 18.6 ± 3.3 μm.58 Figure 9 also depicts the extent of intrusion of water into the interstitial spaces between the hemispheres, as described by the angle ϕ. Although the orientation of the air−water interface at the contact line varies locally with ϕ, it is assumed that globally the air−water interface can be approximated as flat and horizontal. The model assumes that the secondary features pin the contact line13,70,71 and prevent intrusion into the hierarchical structure. The sharp edges also reorient the air− water interface in the vicinity of the contact line,13,24,41,46,62,70,72−75 increasing its apparent contact angle by 90°.47 Any increase in downward pressure drives water deeper between the rough hemispheres, causing the contact line to jump from one secondary feature to the next. At each step, the water is pinned at the waxy edges that exhibit an intrinsic advancing contact angle of θa = 105°.76 In the absence of trapped or compressed gas, the ability of the lotus leaf to resist the intrusion of static liquid can be cast as a competition between the capillary forces acting at the contact line25,26 and the gravitational force acting on the mass of the liquid, where the critical hydrostatic pressure for complete intrusion into the primary structure (Δpc) can be estimated as Δpc =

2Rγ (4 3 /π )y 2 /[cos(θa /2) + sin(θa /2)]2 − R2

Figure 10. Depiction of an inclined plane used to measure the sliding angle (α) at which drops begin to move.

force acting on the mass of the drop.78 The working equation for the sliding angle takes the following form,52−54,79 48 γa sin α = 3 (cos θapr − cos θapa) π ρgV (17) where V is the volume of the liquid drop and a is the radius of its contact patch (Figure 7). (Values of a can be estimated by using tabulated data from Bashforth and Adams.16,48) In the second approach, it is assumed that the retention force arises from capillary bridges on the back half of the drop.80−85 For the drop to roll forward, capillary bridges along the receding contact line must rupture. Balancing capillary rupture at the contact line against the body force acting on the bulk of the drop yields an alternative equation for estimating the sliding angles,

sin α =

(15)

and θapr = λp(θr + ϕ − 270°) + 180°

(18)

For water drops (V = 3 to 30 μL) on the lotus leaf, eqs 14−18 estimate a critical intrusion pressure of Δpc = 12−15 kPa, apparent contact angles of θapa = 180° and θapr = 158−166°, and sliding angles of α = 1−11°. These predictions generally agree with experimental observations.30,48,64,86−88 Contact Line. Throughout this article, the phrase “in the vicinity of the contact line” has been used repeatedly. In this section, the meaning of “contact line” at the macroscopic and molecular level is discussed. Early investigators understood that wetting is controlled by interactions that occur over short distances. They performed measurements to understand not only the seemingly spontaneous rise of liquids in tubes but also the nature of cohesive and adhesive forces.9 Early on, it was assumed that these attractive forces occurred in a “stratum” or thin layer of liquid adjacent to the wetted solid. In the early 1700s, Hauksbee used tubes of similar composition and inner diameter but varied wall thickness to gain insight into the attractive forces that caused the spontaneous rise.89 Liquid rose to the same height in tubes with thin and thick walls. Thus, he concluded that attraction occurred over “insensible” lengths, that is to say, distances that were unresolvable with the unaided eye or magnified light. In the 19th century, Quincke estimated that these forces operate over ranges of less than 50 nm.90 Today we know that the forces that control all sorts of atomic and molecular interactions (e.g., van der Waal forces) operate over even shorter distances, often less than 10 nm. In light of the experiments with capillary tubes and sessile drops described earlier, some of the area-based theories of the 20th century have been retooled.29,91−93 These revised models assume that the wetting is not controlled by the entire wetted area but by an arbitrarily defined strip underneath the liquid,

(14)

Figure 9 shows the air−water interface at the leading and retracting edges of the sessile drop. At the leading edge, the water advances by draping itself across the next protuberance (Figure 9b). The apparent advancing contact angle (θapa) depends not only on the extent of contact but also on the reorientation of the air−water interface. At the receding edge, retreating water is expected to pinch onto the sides of the protuberances and rupture (Figure 9c). Assuming that these apparent contact angles manifest themselves as simple averages along the contact line, expressions have been derived that include the extent of intrusion as well as the influence of curvature and sharp edges. These expressions for the apparent advancing and receding contact angles (θapa and θapr) on the lotus leaf are θapa = λp(θa + ϕ − 90°) + 180°

π 2γaR sin ϕ ρgVy

(16)

where λp is the linear fraction of the perimeter contact line that resides on the protuberances and ϕ is the intrusion angle. (Both λp and ϕ can be estimated analytically.48,77) The angle of inclination (α) at which drops begin to move or “slide” can be measured experimentally using a hinged plane, where α = 0° is horizontal and α = 90° is vertical (Figure 10). Sliding angles for the lotus leaf were estimated from first principles using two different approaches. In the first approach, apparent contact angles are used. Here the global capillary force acting around the perimeter contact line is equated to the body F


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contiguous with the contact line, as denoted in Figure 11. These improvised local area models suffer from many of the

ΔG =

⎡ (1 − cos θ )2 (2 + cos θ ) ⎤ 1 a a ⎥ RT ln⎢ 3 4 ⎣ ⎦

(19)

where R is the ideal gas constant and T is the absolute temperature. For all values of θa, ΔG is negative, as expected from a spontaneous process. The liquid is coerced into spreading. Interactions in the vicinity of the contact line pull the liquid forward and downward onto the solid surface. Without these interactions, liquids do not spread. Note that this model does not depend explicitly on the volume or surface tension/surface energy of the liquid. Figure 12 shows the wetting free energy (ΔG) versus advancing contact angle (θa) at room temperature. For large

Figure 11. Contact line as a three-dimensional zone composed of liquid and solid molecules. The blue circles represent the liquid molecules, and the black circles, the solid molecules.

same pitfalls as the earlier global area models. They assume the existence of a single equilibrium contact angle. They do not reflect the underlying physical processes. They often are built on surface energy. Because energy is a scalar quantity, edges or directional asymmetry cannot be easily incorporated. This is more readily accomplished by starting with forces that are vector quantities. It is noteworthy that J. Willard Gibbs, the father of Gibbs free energy, explained interactions of liquids with structured surfaces in terms of geometry and pinning.13 From the macroscopic perspective, it is reasonable to think of the zone that controls wetting as a one-dimensional (1D) line. However, at the nanoscopic scale, the picture is quite differentthe contact line is a three-dimensional (3D) space where the molecules of the liquid, solid, and gas phases interact. The interactions that control wetting occur in the space immediately adjacent to the intersection of the liquid and solid (Figure 11). At the advancing edge, liquid molecules jump into contact with solid molecules to create an adhesive bond. At the receding edge, liquid must overcome the same forces to molecularly debond from the solid. This paradigm allows one to explain the origins of wetting in the same way that one explains other surface phenomena. For example, the tip of an atomic force microscope (AFM) jumps into contact with a liquid or solid because of molecular interactions operating over short distances.94 This mindset arguably has several advantages over the existing area-based approaches. It provides a rational explanation for intrinsic contact angles being geometry-independent; for example, water gives the same advancing contact angle for poly(tetrafluoroethylene) sheets, spheres, and tubes, even though the relative proportion of interfacial areas for the three geometries will almost always be quite different. Additionally, if the contact line is expanded into a contact zone, then theoretical artifacts, such as stress singularities, could be avoided by spatially averaging the adhesive forces operating there. If one assumes that wetting is indeed controlled by molecular interactions adjacent to the intersection of the liquid and solid, then wetting can be modeled as adsorption. By starting with the Gibbs adsorption equation, the change in the free energy of a smooth, solid surface (ΔG) that occurs when a liquid spreads and establishes an advancing contact angle (θa) can be estimated as95

Figure 12. Wetting free energy (ΔG) versus advancing contact angle (θa) at room temperature (eq 19).

angles, θa > 150°, the wetting free energies are effectively zero. This finding agrees with practice because very large contact angles are difficult to achieve with water and organic liquids by chemical means alone; i.e., angles >120−130° usually require solid surfaces to have geometric structure. (A notable exception would be liquid mercury, which may exhibit contact angles of 150° or more on some polymer surfaces.) Progressively lower contact angles give negative ΔG values of greater magnitude. ΔG values are small for moderate contact angles but increase exponentially as θa → 0°. For θa < 5°, |ΔG| > 10 kJ/mol. Wetting free energies calculated using eq 19 and contact angles of various liquid−solid pairs agree reasonably well with bond strengths measured by other experimental techniques, such as spectroscopy.95 This approach avoids the uncertainty of contact line width and ties wetting to principles that are broadly employed in surface science, physical chemistry, and mainstream science as a whole.

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CLOSING REMARKS The main themes of this article can be summarized as follows. Simple wetting experiments with capillary tubes and heterogeneous surfaces demonstrate that wetting is controlled by interactions in the vicinity of the contact line. A variety of wetting phenomena can be successfully modeled by considering the forces and energies near the contact line. The forces that control wetting occur in the nanoscopic space just beyond the contact line. Even though wetting has been studied for hundreds of years, there are still exciting opportunities. The phenomenon will remain central to many important industrial processes, such adhesion, coatings, fluid handling, lubrication, microfluidics, G


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Open at Both Ends is the Same in Vacuo as in the Open Air. Philos. Trans 1706, 25 (1), 2223−2224. (5) Jurin, J. An Account of Some Experiments Shown before the Royal Society; With an Enquiry into the Cause of the Ascent and Suspension of Water in Capillary Tubes. Philos. Trans. 1717−1719, 30 (1), 739−747. (6) Newton, I. Opticks, 4th ed.; William Innys: London, 1730. (7) Young, T. An Essay on the Cohesion of Fluids. Philos. Trans. R. Soc. London 1805, 95 (1), 65−87. (8) Laplace, P. S. Mécanique Celeste; Courier: Paris, 1805; Vol. t. 4, Supplément au Xe Livre. (9) Maxwell, J. C. Capillary Action. In Encyclopaedia Britannica, 9th ed.; Baynes, T. S., Ed.; Henry G. Allen & Co.: New York, 1888; Vol. 5, pp 56−71. (10) von Segner, J. A. Comment. Soc. Reg. Götting 1751, i, 301. (11) Müller, I. A History of Thermodynamics, The Doctrine of Energy and Entropy; Springer-Verlag: Berlin, 2007. (12) Dupré, A. Théorie Mécanique del la Chaleur; Gauthier-Villars: Paris, 1869. (13) Gibbs, J. W. On the Equilibrium of Heterogeneous Substances. In The Collected Works of J. Willard Gibbs; Yale University Press: New Haven, CT, 1961; Vol. 1, pp 326−327. (14) Brown, R. C. Note on the Energy Associated with Capillary Rise. Proc. Phys. Soc. 1941, 53 (3), 233−234. (15) Adam, N. K. The Physics and Chemistry of Surfaces, 3rd ed.; Oxford University Press: London, 1941. (16) Bashforth, F.; Adams, J. C. An Attempt to Test the Theories of Capillary Action by Comparing the Theoretical and Measured Forms of Drops of Fluid; University Press: Cambridge, England, 1883. (17) Thorpe, W. H.; Crisp, D. J. Studies on Plastron Respiration. I. The Biology of Aphelocheirus [Hemiptera, aphelocheiridae (naucoridae)] and the Mechanism of Plastron Retention. J. Exp. Biol. 1947, 24, 227−269. (18) Cassie, A. B. D.; Baxter, S. Wettability of Porous Surfaces. Trans. Faraday Soc. 1944, 40 (21), 546−551. (19) Cassie, A. B. D. Contact Angles. Discuss. Faraday Soc. 1948, 3 (1), 11−16. (20) Good, R. J. A Thermodynamic Derivation of Wenzel’s Modification of Young’s Equation for Contact Angles; Together with a Theory of Hysteresis. J. Am. Chem. Soc. 1952, 74 (20), 5041− 5042. (21) Johnson, R. E., Jr.; Dettre, R. H. Contact Angle Hysteresis. I. Study of an Idealized Rough Surface. In Contact Angle, Wettability, and Adhesion; Gould, R. F., Ed.; American Chemical Society: Washington, DC, 1964; Vol. 43, pp 112−135. (22) Pease, D. C. The Significance of the Contact Angle in Relation to the Solid Surface. J. Phys. Chem. 1945, 49 (2), 107−110. (23) Bartell, F. E.; Shepard, J. W. Surface Roughness as Related to Hysteresis of Contact Angles. II. The Systems Paraffin−3 Molar Calcium Chloride Solution−Air and Paraffin−Glycerol−Air. J. Phys. Chem. 1953, 57 (4), 455−458. (24) Extrand, C. W. Model for Contact Angles and Hysteresis on Rough and Ultraphobic Surfaces. Langmuir 2002, 18 (21), 7991− 7999. (25) Extrand, C. W. Contact Angles and Hysteresis on Surfaces with Chemically Heterogeneous Islands. Langmuir 2003, 19 (9), 3793− 3796. (26) Gao, L.; McCarthy, T. J. How Wenzel and Cassie Were Wrong. Langmuir 2007, 23 (26), 3762−3765. (27) Gao, L.; McCarthy, T. J. Wetting 101°. Langmuir 2009, 25 (4), 14105−14115. (28) Extrand, C. W.; Moon, S. I. Which Controls Wetting? Contact Line versus Interfacial Area: Simple Experiments on Capillary Rise. Langmuir 2012, 28 (44), 15629−15633. (29) Erbil, H. Y. The Debate on the Dependence of Apparent Contact Angles on Drop Contact Area or Three-phase Contact Line: A Review. Surf. Sci. Rep. 2014, 69 (4), 325−365.

fabrication of microelectronics, printing, and repellency, to name a few. Looking ahead, the field could benefit from unified comprehensive analyses, which include an evaluation of forces, energies, and kinetics.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +1-651-9991859. Notes

The author declares no competing financial interest. Biography

Chuck “C. W.” Extrand is Director of Research at Colder Products Company (CPC) in St. Paul, Minnesota. Over his 20+ year industrial career, he has developed a broad range of materials-based technologies for industrial markets, such as semiconductor and life sciences. He holds 14 U.S. patents and has published more than 100 papers in technical journals, books, and conference proceedings on topics that include wetting, adhesion, contamination, fracture, friction, permeation, polymer processing, and spin coating. Prior to starting his industrial career, Chuck worked in Japan at the National Institute for Materials and Chemical Research and in Paris at the Collège de France and the Ecole Normale Supérieure. He received a Ph.D. in polymer engineering from The University of Akron and a B.S. in chemical engineering from the University of Minnesota.

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ACKNOWLEDGMENTS I thank M. Acevedo, T. Adamson, L. Castillo, D. Downs, J. Doyon, R. Komma, K. Long, D. Meyer, S. I. Moon, K. Sekeroglu, G. Wilhelm, and J. Wittmayer for their support and their suggestions on the technical content and text. The image of the surface of the lotus leaf was kindly provided by Prof. W. Barthlott.

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REFERENCES

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Origins of Wetting - Langmuir (ACS Publications)

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