Symmetric and Asymmetric Capillary Bridges between a Rough

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Symmetric and Asymmetric Capillary Bridges between a Rough Surface and a Parallel Surface Yongxin Wang, Stephen Michielsen, and Hoon Joo Lee* College of Textiles, North Carolina State University, 2401 Research Drive, Raleigh, North Carolina 27695-8301, United States S Supporting Information *

ABSTRACT: Although the formation of a capillary bridge between two parallel surfaces has been extensively studied, the majority of research has described only symmetric capillary bridges between two smooth surfaces. In this work, an instrument was built to form a capillary bridge by squeezing a liquid drop on one surface with another surface. An analytical solution that describes the shape of symmetric capillary bridges joining two smooth surfaces has been extended to bridges that are asymmetric about the midplane and to rough surfaces. The solution, given by elliptical integrals of the first and second kind, is consistent with a constant Laplace pressure over the entire surface and has been verified for water, Kaydol, and dodecane drops forming symmetric and asymmetric bridges between parallel smooth surfaces. This solution has been applied to asymmetric capillary bridges between a smooth surface and a rough fabric surface as well as symmetric bridges between two rough surfaces. These solutions have been experimentally verified, and good agreement has been found between predicted and experimental profiles for small drops where the effect of gravity is negligible. Finally, a protocol for determining the profile from the volume and height of the capillary bridge has been developed and experimentally verified.

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INTRODUCTION The study of capillary bridges goes to the first half of the 1800s. Young1 and Laplace2 first introduced the concepts of surface tension and Laplace pressure, and Delaunay3 provided the first solution to the capillary bridge profile. The Laplace pressure refers to the pressure difference across the interface between the inside (liquid phase) and the outside (vapor phase), which arises from the surface tension of the liquid−vapor interface. When a liquid drop is in contact with two or more surfaces, the liquid forms a bridge between the surfaces. This bridge is called a capillary bridge. This occurs between grains of sand in an oil reservoir, between a flat surface and a cloth when wiping up a spill, between a silicon wafer and a polishing cloth, and between the skin and clothing, as well as many other applications. The profile contributes to the interfacial area between the liquid and the surface and to the forces exerted on the surfaces. This leads to clothing sticking to the skin in one case, or to polishing debris moving from the silicon wafer to the polishing cloth in another case. Forces exerted by capillary bridges have been studied extensively, often with the goal of obtaining the capillary force of adhesion between the liquid and the surface. The capillary force of adhesion depends on the contact perimeter of the bridge with the surface and the Laplace pressure; thus, the profile of the bridge is required to obtain the capillary force of adhesion. Fisher,4 Mason and Clark,5 and Gillespie et al.6 proposed numerical methods to solve for the capillary force of adhesion between two identical spheres as well as the relationship between the capillary force and the Laplace pressure.7 Clark et al.8 and Rabinovich et al.9 derived the © 2013 American Chemical Society

capillary force, Laplace pressure, and volume mathematical expressions for the capillary bridge between a sphere and a flat surface, based on the assumption that the profile of the bridge is part of a circle in one dimension. Similarly, De Souza et al.10 presented a numerical analysis of the capillary forces of a capillary bridge between two chemically different parallel surfaces as the separation distance changed, and revealed that the capillary force and the rupture separation distance decreased as the asymmetry in contact angles (CAs) increased. McCarthy et al.11 employed the same model as Orr et al.12 to study the effect of CA hysteresis on the measurement of capillary forces. Restagno et al.13 applied a constant velocity disturbance (1 μm/s) to the length of a capillary bridge to study the contact angle hysteresis from its formation to its rupture. The disturbance affected the capillary force, the capillary profile, and the liquid−solid contact area. Kusumaatmaja and Lipowsky14 computed the equilibrium distance of the capillary bridge as a function of liquid volume, contact angle, and radius of the liquid−solid contact area. They also derived the effective spring constant as the capillary bridge was perturbed from equilibrium and verified this behavior with Surface Evolver simulations. Vagharchakian et al.15 explored the profile of a capillary bridge between a sphere and a flat surface by calculating the surface curvature, and concluded that the velocity, viscosity, and volume of the drop can affect the shape of the bridge as the two Received: April 15, 2013 Revised: August 4, 2013 Published: August 5, 2013 11028

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also shown that a rough surface formed by using a fabric for one or both of the surfaces can be treated in the same manner where one uses the apparent advancing or receding contact angle for the liquid on the fabric. Finally, we show that the profile can be obtained directly from the drop volume, the distance between the two surfaces on either end of the capillary bridge, and the contact angles between the drop and the surfaces.

surfaces move toward or away from each other. Van Honschoten et al.16 analyzed the profiles of the capillary bridge between a spherical probe tip of an atomic force microscope (AFM) and a flat surface. They extended this to two plane surfaces by increasing the radius of the spherical tip to infinity. The curvature of the capillary bridge could only be obtained from the segments of self-intersecting periodic loops. They used Surface Evolver to generate the final shape of their bridges. Hotta et al.17 analyzed the meridian curve and discussed using a circular approximation for the profile of a capillary bridge. Fortes18 studied the capillary bridges between two identical parallel plates and divided them into two configurations, r bridge and θ bridge, depending on whether the edge of the liquid touches the circumference of the plate. An r bridge is one where the radius of the plate limits the spreading, while a θ bridge is one where the contact angle limits the spreading. Orr et al.12 also described the profile of a capillary bridge between a sphere and a flat surface, and predicted the curvature of a capillary bridge with the advancing or receding CAs, the mean surface curvature, and the surface tension. Butt19 studied the influence of roughness with an asperity distribution function on the capillary bridges, while Chen et al.20 predicted that capillary bridges are only stable when their lengths are smaller than their circumferences. In a Gillette and Dyson study,21 the stability of a capillary bridge between circular coaxial parallel plates was investigated. Their theory predicted the minimum volume needed to form a capillary bridge between two plates, as well as the maximum possible volume for a stable bridge between two plates at a given separation distance. Slobozhanin et al.22 investigated the instability of an axisymmetric capillary bridge between two coaxial disks for a given radii of disks, the separation distance, and the liquid volume. Mazzone et al.23 and Adams et al.24 studied the effect of gravity on the profile of a capillary bridge between two spheres, and concluded that gravity had a significant effect on the characteristics of the capillary bridge when the bond number was large compared with the dimensionless curvature 2HR, where H is the mean surface curvature and R is the characteristic length defined as the radius of a spherical drop of the same volume. A Bond number larger than 1 indicates that the effect of gravity is larger than the surface tension effect. De Boer et al.25 and Alexandrou et al.26 studied the break-up of capillary bridges and the work required to rupture a bridge between a sphere and a flat surface. Cohen et al. studied the antiadhesive properties of a surface using capillary profiles.27 The antiadhesive force refers to the ability of a surface to rid itself of liquids and keep itself clean. Vagharchakian et al.,15 inspired by the Johnson, Kendall, and Roberts (JKR) theory,28 developed a quantitative test to characterize extremely weak adhesive surfaces through the formation and breakage of capillary bridges. McFarlane and Tabor29 investigated the adhesion force of a liquid drop with a solid flat surface by forming a capillary bridge between a spherical bead and a flat plate. They also discussed the effects of surface roughness, surface tension, viscosity, and thickness of the liquid film. Thus, the study of capillary bridge profiles continues to be used to address many important technological problems. In this research, an analytical solution for the profile of a drop forming a capillary bridge between two flat surfaces is derived on Delaunay’s results3 and extended from a capillary bridge that is symmetric about the midplane to capillary bridges that are asymmetric about the midplane. The predicted profiles are then compared to experimentally determined profiles. It is

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BACKGROUND The profile of a capillary bridge between two surfaces can be obtained from the combination of Laplace pressure and mean surface curvature expressions.30−33 Figure 1 shows a schematic

Figure 1. Schematic diagram for a liquid bridge between identical parallel surfaces and definition of symbols.

diagram of a concave capillary bridge between two identical parallel surfaces, and the XOZ coordinate system. The capillary bridge is assumed to be in thermodynamic equilibrium, and gravitational forces are assumed to be insignificant. Although several recent papers provide approximate solutions to this problem, it was first solved analytically in 1841 by Delaunay.3 In Figure 1, the bridge is symmetric about both the X- and Z-axes, since the two surfaces are identical. This means that the profile of the bridge can be determined using only one quadrant. Points A and B are the points of intersection of the liquid− vapor interface with the X-axis and with the three-phase contact line at the top surface, respectively. The advancing or receding CA between the capillary bridge and the flat surface at point B is defined as φ. When the surfaces are moving toward each other, the liquid is being compressed, and the CA is the advancing CA; when the surfaces are moving away from each other, the liquid is being stretched, and CA is the receding CA. The principal radii of curvature defining the liquid−vapor interface at each (X, Z) point are defined in the following way. First, a tangent (green dash-dot) line and the corresponding normal (blue dash-dot) line are drawn at a point on the interface. See, for example, point B in Figure 1. Two tangent circles with their centers lying on the normal line can be constructed with their curvatures matching the curvatures of the liquid surface at point B. The green dashed tangent circle is in the xz plane, with radius R2. Likewise, the blue solid circle lies in the plane normal to the plane of the paper and tangent to the liquid surface at point B with radius R1 and tilt angle ε between R1 and the positive Z direction. The liquid−vapor interfacial energy is donated as γLV. For other points along the BA curve, the angle ε varies, since it is the angle between the normal and the Z-axis. At position B, ε = π − φ. Since the capillary bridge is in thermodynamic equilibrium and since the effect of gravity is negligible, the Laplace pressure is the same at every point on the liquid−vapor surface. From 11029

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the definitions of Laplace pressure, ΔP = γLV(1/R1 + 1/R2), and mean surface curvature, H = 0.5(1/R1 + 1/R2). From the geometry, we obtain11,15,16 2H =

dz dx

⎡ x ⎢1 + ⎣

1/2 dz 2 ⎤ dx ⎥ ⎦

( )

+

once, with a solution of reagent alcohol and deionized water once where the weight ratio was 1:1, with deionized water twice; cure the fabric with oven at 120 °C for 6 min;35 and (3) then the same procedure used to treat the FR fabric was repeated for nylon fabric. This fabric is referred to as (TCMS-FS) treated fabric. Capillary Bridge Instrument. A custom built instrument was used to form, compress, and extend capillary bridges between two surfaces. A picture of the equipment and schematic diagram is shown in Figure 2; details can be found in the Supporting Information.

d2z dx 2

⎡ ⎢⎣1 +

3/2 dz 2 ⎤ dx ⎥ ⎦

( )

(1)

To simplify the equation, we define u = −sin ε, which gives dz −sin ε u = −tan ε = = >0 dx cos ε − 1 − u2

(2)

where ε ∈ [π/2, π − φ] and so cos ε = −(1 − u ) . For a biconcave bridge as shown in Figure 1, 1/R1 = −u/x > 0, while 1/R2 = −du/dx < 0. Therefore, 2H = −u/x − du/dx and 2 1/2

u 2 − 4Hc (3) 2H where c is a constant. Then, H and c are determined by the boundary conditions: x=

−u ±

Figure 2. Schematic diagram of the capillary bridge instrument. In our experiments, we used several combinations of surfaces: glass−glass, PTFE−PTFE, glass−FS treated glass, glass−FS treated fabric, PTFE film−FS treated fabric, PTFE film−(TCMS-FS) treated fabric, and FS treated FR−FS treated nylon fabrics. For each combination of liquid, top surface and bottom surface, at least three compression and extension cycles were performed. Photographs of the capillary bridges were recorded during the compression phase or during the extension phase to ensure that the CA was either the advancing CA (compression) or receding CA (extension).

⎛π ⎞ At point A: x = x1, u = −sin⎜ ⎟ = −1 ⎝2⎠

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At point B: x = x 2 , u = −sin(π − φ) = −sin φ

(4)

EXPERIMENTAL SECTION

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Materials. Three liquids were used to form capillary bridges: deionized water (surface tension γ = 72.8 mN/m, density ρ = 0.997 g· cm−3, viscosity η = 0.997 mPa·s), Kaydol (surface tension γ = 31 mN/ m, density ρ = 0.81−0.89 g·cm−3, viscosity η = 160 mPa·s, CBM group, Inc.), and dodecane (surface tension γ = 25 mN/m, density ρ = 0.75 g·cm−3, viscosity η = 1.344 mPa·s, Sigma-Aldrich). In addition, (heptadecafluoro-1,1,2,2-tetrahydrodecyl)trimethoxysilane (FS, Gelest Inc.), isopropyl alcohol (VWR International LLC), reagent alcohol (VWR International LLC), toluene (Sigma-Aldrich), poly(acrylic acid) (PAA), and ammonium hydroxide (Sigma-Aldrich) were used without further purification to treat glass slides as described below. Capillary bridges were formed using poly(tetrafluoroethylene) (PTFE) films (SmallParts LLC), glass slides (Corning, Inc.), and glass slides treated by the condensation of FS.34 In addition to the smooth surfaces, fire retardant (FR) fabric (80% FR rayon and 20% para-polyaramid), FS treated FR fabric, nylon 6, and FS treated nylon 6 fabrics were used as rough surfaces. For each test, two new surfaces were used to avoid complications that could arise due to cleaning. Treatments onto Glass and Fabrics. The FS treated glass slides were prepared by first immersing them in isopropyl alcohol to prewet them. A solution consisting of 2 parts by weight of FS, 100 parts of isopropyl alcohol, and 1 part of ammonium hydroxide was prepared just before use. This solution was poured onto the glass slides in a weight ratio of 5.15:1. After 2 h immersion, the glass slides were removed from the solution and washed with isopropyl alcohol and allowed to dry in air. FR fabric was treated in a similar way. It was dipped into isopropyl alcohol to prewet it, and excess isopropyl alcohol was removed by squeezing it between two rollers. (This process is called “padding”.) The wet fabric immersed into a freshly prepared FS, isopropyl alcohol, and ammonium hydroxide solution after which the fabric was padded to remove the excess liquid. The fabric was dried in an oven at 120 °C for 2 min and then washed with isopropyl alcohol twice and dried in an oven at 100 °C for another 2 min. The nylon fabric was treated as follows: (1) the fabric was immersed in a 4 g/L aqueous solution of PAA for 4 h; (2) the fabric was treated with a solution of methyltrichlorosilane and toluene for 2 h with a weight ratio of 1:20; the fabric was washed with toluene twice, with reagent alcohol

RESULTS AND DISCUSSION In this section, the analytical solution for the profile of a capillary bridge between two parallel surfaces is given under the assumption that gravity and other external forces can be ignored. The experimentally obtained capillary profiles are presented for different combinations of surfaces, including glass−glass, PTFE−PTFE, glass−FS treated glass, glass−FS treated fabric, PTFE−FS treated fabric, PTFE film−(TCMSFS) treated fabric, and FS treated FR−FS treated nylon fabrics. The profiles of capillary bridges are symmetric when identical surfaces are used, while the shapes are asymmetric when different surfaces are used. In addition, the effects of liquids with different surface tensions and different volumes are explored both for symmetric and asymmetric bridges. Analytical Solution for Capillary Profiles. Recent studies4−6,36 of the profiles of a capillary bridge, primarily using numerical methods, do not seem to be aware of an analytical solution provided by Delaunay3 in 1841. Following his approach, we have rederived the analytical solution, which is shown below followed by experimental verification and extended it to bridges that are asymmetric about the midplane. According to eq 3 for any point on the liquid−air interface shown in Figure 1, x is sin 2 ε − 4Hc (5) 2H when ε ∈ [(π/2), π − φ]. For a concave bridge, the “−” sign is chosen, while, for a convex bridge, the “+” sign is chosen. Equation 5 can be solved for the integration constant c using the boundary conditions, where, at point A, c = cA = x1 − Hx12 and, at point B, c = cB = sin φ·x2 − Hx22. Since c, cA, and cB are all integration constants, they must be equal for the same capillary bridge, cA = cB = c. Then, H becomes x=

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x1 − x 2 sin φ x12 − x 2 2

At point B1: x = x1, u = −sin(π − φ1) = −sin φ1 , c1 = sin φ1·x1 − Hx12

(6)

Furthermore, dz/du = dz/dx × dx/du. Therefore, z is given by z=

1 2H

⎡ sin ζ ⎢ −1 ± ⎢⎣ π /2

∫

ε

1 {cos ζ ± = 2H

At point B2 : x = x 2 , u = −sin(π − φ2) = −sin φ2 , c 2 = sin φ2 ·x 2 − Hx 2 2

⎤ sin ζ ⎥ dζ sin 2 ζ − 4Hc ⎥⎦

(9)

Once again, the two c values must be the same, so c1 = c2 and H are given by sin φ1·x1 − sin φ2 ·x 2 H= x12 − x 2 2 (10)

ε

−4Hc [E(sin ζ , k) − F(sin ζ , k)]} π /2

(7)

where k = (1/4Hc)1/2 and ζ is the integration variable; F and E are the elliptical integrals of the first and second kinds. For a concave bridge, the “+” sign is chosen, while, for a convex bridge, the “−” sign is chosen. Then, the final expression for the profile of a capillary bridge is simply the locus of points (x, z) where x and z are given by eqs 5 and 7, respectively. This solution is valid when gravity is insignificant and can be ignored. In the following experiments, the Bond number is