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Energy dissipation of moving drops on superhydrophobic and superoleophobic surfaces Hans-Jürgen Butt, Nan Gao, Periklis Papadopoulos, Werner Steffen, Michael Kappl, and Rüdiger Berger Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b03792 • Publication Date (Web): 30 Nov 2016 Downloaded from http://pubs.acs.org on December 4, 2016

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Energy dissipation of moving drops on superhydrophobic and superoleophobic surfaces Hans-Jürgen Butt1*, Nan Gao1, Periklis Papadopoulos2, Werner Steffen1, Michael Kappl1 & Rüdiger Berger1 1

2

Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany

University of Ioannina, Department of Physics, P.O. Box 1186, GR-45110 Ioannina, Greece *[email protected]

Abstract A water drop moving on a superhydrophobic surface or an oil drop moving on a superoleophobic surface dissipates energy by pinning/depinning at nano- and microprotrusions. Here, we calculate the work required to form, extend, and rupture capillary bridges between the protrusions and the drop. The energy dissipated at one protrusion WS is derived from the observable apparent receding contact angle Θrapp and the density of

protrusions n by Ws = γ ( cos Θ rapp + 1) n , where γ is the surface tension of the liquid. To derive

an expression for WS that links the microscopic structure of the surface to apparent contact angles two models are considered: A superhydrophobic array of cylindrical micropillars and a superoleophobic array of stacks of microspheres. For a radius of a protrusion R and a receding materials contact angle Θr we calculate the energy dissipated per protrusion as Ws = πγ R 2  A − ln ( R κ )  f ( Θ r ) . Here, A=0.60 for cylindrical micropillars and 2.9 for stacks

of spheres. κ is the capillary length. f ( Θ r ) is a function which depends on Θr and the specific geometry; f ranges from ≈0.25-0.96. Combining both equations above, we can correlate the macroscopically observed apparent receding contact angle with the microscopic structure of the surface and its material properties.

Introduction A sessile liquid drop moving on a surface dissipates energy.1, 2, 3 Although of great practical relevance it is still not fully understood, how energy is dissipated and, depending on the specific circumstances, which process dominates dissipation. A moving drop dissipates energy by viscous flow, by adsorption and detachment of liquid molecules and pinning and depinning from the surface. For viscous dissipation usually two processes are discriminated: Viscous dissipation near the rim of the apparent contact line and in the bulk. The first tends to dominate for small drops, in particular when they have a low contact angle. The latter ACS Paragon Plus Environment

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becomes important for large drops.4, 5, 6 Small and large refers to the drop height as compared to the capillary length κ = γ ρ g . Here, γ is the surface tension of the liquid, ρ is its density, and g = 9.81 m/s2 is the acceleration of gravity. For water at 25°C κ = 2.71 mm. In addition, pinning of the contact line at specific sites on the surface is another channel of energy dissipation.3, 6, 7, 8, 9, 10, 11 It is observed on chemically heterogeneous surfaces and on structured surfaces. Pinning resists movement of the drop by retarding the contact line until a certain threshold force is overcome. After depinning the capillary bridge will snap back into the drop. It will create capillary waves which undergo viscous dissipation. Here, we address energy dissipation on superliquid repellent surfaces. Superliquid repellent surfaces are characterized by a high apparent receding contact angle of, for example, >140°.12 Superliquid repellency of surfaces is typically realized by nano- and micrometer sized protrusions made of a low surface energy material. These protrusions need to be such that they keep a liquid drop from direct contact with a substrate. As a result, a layer of air is stabilized between the drop and the substrate leading to the so-called Cassie state.13, 14, 15, 16 For water with its high surface tension the protrusions can be simple microposts with vertical walls. For nonpolar liquids, protrusions with overhangs are required.17, 18, 19, 20, 21 On superliquid repellent surfaces moving drops show a low friction and low energy dissipation.1, 22, 23, 24 Such a low friction is particularly interesting for aqueous electrolytes on superhydrophobic surfaces for droplet-based microfluidics.25, 26, 27 Water drops on superhydrophobic surfaces show low viscous dissipation. Viscous dissipation at the rim is reduced because of the high apparent contact angle;28 thus the dissipation per unit line is reduced. In addition, a high contact angle also decreases the length of the rim (for a drop of given volume). Even for large drops bulk viscous dissipation is reduced because drops slip and roll.5, 29 When water drops move down a superhydrophobic surface they first show some slippage; that is the bottom side of the drop shows an apparent slip,30, 31 and then tend to roll rather than slide.4, 32 Energy dissipation is often studied by letting drops slide or roll down an inclined plane. When a sessile liquid drop is placed on a tilted plane and the tilt angle exceeds a certain critical angle the drop will move downhill. To describe this motion we consider the two main energy contributions on superliquid repellent surfaces. Firstly, the rate of decrease of potential gravitational energy. It is given by Pg = ρVgU sin α

,

(1)

where, V is the volume of the drop, U is the sliding/rolling velocity and α is the inclination of the plane with respect to the horizontal. Secondly, energy is dissipated by capillary dissipation which is associated with the adsorption and detachment of liquid molecules and pinning and depinning from the surface. This contribution can be described by:5, 7, 33 Pc = γ wU ( cos Θ app − cos Θaapp ) r

.

(2) 2

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Here, w is the width (extension perpendicular to direction of motion) of the apparent contact area of the sliding drop. Θrapp and Θaapp are the apparent dynamic contact angles at the receding and advancing side, respectively. These angles are measured at a length scale much smaller than the capillary length, to neglect effects of gravitation. On the other hand, the apparent contact angles are measured at a length scale much larger than the range of surfaces forces of few 10 nm and larger than the extension of the nano- and micrometer sized protrusions. Eq. (2) was originally derived by integrating the horizontal component of the surface tension of the liquid around the rim of the drop.34, 35, 36, 37, 38, 39 This integration can, however, also be carried out at a certain height above the substrate, as long as this height is much, much lower than the total height of the drop. Depinning is the main channel of energy dissipation for water drops moving on superhydrophobic surfaces.3, 12, 40, 41 When the drop moves, at its rear capillary bridges are formed between the drop and the top faces of protrusions (Fig. 1A-C). These bridges are stretched and finally rupture. The work required to stretch and rupture the capillary bridge is dissipated and not used to propel the drop forward.3 Here, we calculate the work required to stretch a capillary bridge and we relate it to the energy dissipation of the entire drop. The formalism is developed for water drops on superhydrophobic arrays of vertical cylindrical micropillars and nonpolar liquid drops on pillars composed of spheres partially sintered together. One intrinsic feature of the model is that the contact line moves discontinuously since one pillar is wetted/dewetted after another. Therefore, it is not directly applicable to e.g. striped surfaces or surfaces with cavities 14, 42, 43, where the contact line can slide continuously. The calculations include an analytical equation for the work required to pull a circular, horizontal disk out of a liquid pool. To our knowledge this has not been reported before.

General formalism Outline of the model. As a model we consider a square array of cylindrical micropillars (Fig. 1B). The pillar radius is R and the lattice constant a. We further assume that the drop and its radius of curvature is much larger than a. Each top face of a pillar acts as a pinning center. The pinning sites are distributed with a density n = 1 a 2 in number per unit area. When the rim of a water drop moves over such a pinning site it is deformed. A capillary bridge is formed, stretched and finally ruptures. We denote the extension of the capillary bridge by δ. The force required to stretch the contact line is F(δ). The work required to form a capillary bridge till it breaks is δc

Ws = ∫ Fdδ

(3)

0

Here, δc is the maximal possible extension of the stretched capillary bridge. In the following we call WS the work of adhesion of a single bridge or briefly, work of adhesion. 3 ACS Paragon Plus Environment

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Figure 1: (A) Schematic of a water drop on a superhydrophobic array of micropillars moving to the right. (B) A schematic detail of the rear side of the drop. The schematic figure is based on confocal microscope images, such as the one shown in (C). (C) Confocal microscope image of the rear of a water drop (V = 5 µL) on a superhydrophobic square array of cylindrical micropillars (radius R = 5 µm, pitch a = 40 µm, height 5 µm).44 The pillars were made of SU-8 and were hydrophobized with 1H,1H,2H,2H-perfluorooctyl trichlorosilane. The vertical crosssection is along the diagonal of the array. Water was fluorescently labelled with Alexa488 (0.1 mg/mL). As the pillars have a refractive index of 1.6 and a dry objective was used, they appear shorter. Thus we show artificial pillars in yellow with the correct height. (D) Schematic of the last pillar at the rear side of the drop and the direction in which the bulk drop moves away from the pillar. To model this situation we turned the geometry upside down (E). Before calculating the extension and rupture of a capillary bridge between an infinitely extended water surface and a tilted pillar, we considered the even simpler situation of the horizontal pillar (F). The energy dissipated by a drop of width w moving at a velocity U is

Pd = nWs wU

(4)

In many practical situations there is not only one type of pinning site but a distribution. Then the energy dissipated is 4 ACS Paragon Plus Environment

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Langmuir

Wmax

P = wU

∫

nw (W )WdW

(5)

0

In this case nw(W) is the number of pinning sites per unit area and within a certain energy interval W KW + dW in units m-2J-1. To relate the work of adhesion to experimentally observables we assume that depinning dominates energy dissipation over viscous dissipation and set Eq. (4) equal to Eq. (2), leading to app nWs = γ ( cos Θapp r − cos Θa )

(6)

Therefore, if pinning dominates, the density of micropillars multiplied with the work of stretching and the surface tension is the cause for contact angle hysteresis. In a strict sense Eq. (6) is valid for U→0 where viscous dissipation is zero. A similar equation has already been derived by Joanny and de Gennes for partially wetting drops on heterogeneous surfaces.7 When a partially wetting drop slides down a chemically heterogeneous surface the contact line is typically pinned at lyophilic sites. Only when the contact line exceeds a certain stretching, it is released again and jumps forward. The work done by the drop to stretch the contact line is dissipated. Please note that as long as nWs does not depend on the velocity U, energy dissipation due to pinning does not lead to a steady state velocity. A drop which is only dissipating energy by pinning/depinning would either not move at all or it would accelerate with a constant rate. 28 In contrast to viscous dissipation, which is proportional to U2, energy dissipation due to pinning is only proportional to U, as long as the work required to stretch the contact line at a single site Ws is independent on velocity. For a superhydrophobic array of micropillars, only pinning at the rear needs to be considered; at the front the drop is touching down and the apparent advancing contact angle is 180°.12, 45, 46, 47

= 180° Eq. (6) simplifies to With Θapp a

nWs = γ ( cos Θapp r + 1)

(7)

Thus, from a measurement of the apparent receding contact angle one can determine the work of adhesion for a single protrusion. Vice versa, Eq. (7) allows to calculate the contact angle if Ws is known. In the following we aim at calculating Ws from the microscopic structure of the surface.

Water drop on a superhydrophobic array of micropillars Capillary forces versus extension. To estimate the work of adhesion Ws we need an expression for the force versus extension, F-vs-δ. For a precise calculation one would need to solve the Laplace equation for the specific geometry and obtain the force by integration of the surface tension around the capillary bridge. Here, we derive and discuss an analytical 5 ACS Paragon Plus Environment

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approximation which not only allows us to estimate Ws but also helps us identifying the important dependencies. An exact calculation can only be carried out numerically. Rather than considering the underside of the drop being withdrawn from the top face of a cylinder, we treat the inverted situation: A cylinder with a flat bottom being pulled out of a liquid pool (Fig. 1F, 2). Assuming that the micropillar size is much smaller than the capillary length, the shape of the liquid menisci is dominated by surface tension and the influence of gravity is negligible. Both situations should be dominated by capillarity and lead to a similar force. We discuss the case in detail because, to our knowledge, the work to draw a circular, horizontal disc out of a liquid pool has not been described in the literature. At a certain extension of the liquid meniscus δ the liquid forms an angle φ with respect to the horizontal. The vertical capillary force acting on the cylinder is F = 2π Rγ sin φ

(8)

The extension of the liquid neck formed between the liquid surface and the bottom face of the micropillar for R > 1. For a superhydrophobic array of micropillars the neighbouring micropillar will lead to a flat surface already in the middle between two pillars. Thus, if a/R >> 1 is not fulfilled, δ is overestimated.

Figure 2. Schematic of a cylinder with a flat bottom being drawn out of a liquid pool (first bottom image then top). Turned by 180° the situation is a model for the rear side of a water drop on a superhydrophobic array of micropillars. 6 ACS Paragon Plus Environment

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Eq. (9) was originally derived for a cylinder in a liquid pool at fixed angle φ. We use it to plot Fvs-δ curves by calculating the force with Eq. (8) and extension with Eq. (9) for increasing φ. When plotting F-vs-δ, the curves, turn out to be almost linear (Fig. 3). We can derive a good approximation by setting 1 + cos φ ≈ 1 in Eq. (9), solve for sin φ and insert the resulting expression into Eq. (8): F =

2πγδ 0.8091 − ln ( R κ )

(10)

Thus, the capillary force acts like a spring with spring constant k, which can be described as k=

2πγ 0.8091 − ln ( R κ )

(11)

Please note that the normalized pillar radius R κ is usually much smaller than one and the logarithm is negative. By comparison with the exact solution (Eqs. 8 and 9) we find that approximation Eq. (10) is accurate within 10% for normalized pillar diameters R/κ of 5×10-4 to 0.05. Keeping the dependencies of Eq. (10), which reflect the underlying physics, we tried to maximize accuracy by numerically adjusting the constant. Indeed, replacing the constant 0.8091 by 0.4 even leads to an accuracy of 5%.

Figure 3. Force-versus-extension for capillary bridges formed from a vertical cylinder with a flat bottom face and radius 0.1, 1, and 10 µm. The liquid is water (γ=72.0 mN/m, κ=2.71 mm at 25°C). Thick continuous lines were calculated with Eqs. (8) and (9) and varying φ. The dashed lines were obtained with the linear approximation Eq. (10). Breaking of the capillary bridge. The capillary bridge breaks if one of two conditions will be fulfilled. First, 180 ° − φ falls below the receding contact angle of the material Θr: 180 ° − φ ≤ Θ r

(12)

Here, Θr is the receding materials contact angle as measured for a receding water front on a smooth, homogeneous surface of the same material. As long as condition (12) is fulfilled the 7 ACS Paragon Plus Environment

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liquid bridge is connected to the whole top face of the pillar. When condition (12) is not fulfilled anymore, the contact line slides over the top face of the micropillar, detaches from the front edge, and the capillary bridge collapses. Second, the capillary bridge becomes unstable and breaks, when small changes in length lead to large changes in angle φ. This happens at a certain critical angle φc > 90° (unless condition 12 is fulfilled). This critical angle can be derived by finding the maximal extension possible d sin x ln (1 + cos x ) = − according to Eq. (9). Setting d δ d φ = 0 and using leads to a dx 1 + cos x condition for the critical angle: ln (1 + cos φc ) −

sin 2 φc R = 0.8091 − ln 2 κ cos φc + cos φc

(13)

Unfortunately, it is not an explicit equation for φc. For typical superhydrophobic arrays of micropillars φc varies between 96° and 106° (Fig. 4). For R → 0 the critical angle approaches 90°.

Figure 4. Critical angle versus the normalized radius of the micropillar. By inserting the critical angle into Eq. (9) we obtain the rupture distance:

δ c = R sin φc  0.8091 − ln 

 − ln (1 + cos φc )  κ  R

(14)

With high accuracy this can be approximated with

δ c = R  0.947 − ln 

R κ 

(15)

The approximation is valid for R/κ=5×10-5 – 0.05 within 1% accuracy. Thus, δc increases almost linearly with the radius of the cylinder. Work of adhesion. The work of adhesion is

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Ws =

δc

φc

0

0

∫ Fd δ =

dδ

∫ F dφ dφ

(16)

With Eqs. (8) and (9) this leads to (appendix)  R 1    1 Ws = πγ R 2 sin 2 φc  0.8091 − ln − ln (1 + cos φc )  + cos 2 φc + − cos φc  κ 2   2  

(17)

When plotting the factor in curly brackets with φc from Eq. (13) versus ln ( R κ ) we find an almost linear function. It can be described by R R   Ws ≈ πγ R 2  1.78 − 0.959 ln  ≈ πγ R 2  1.58 − ln  κ κ    

(18)

The first approximation is accurate within 0.5%, the second holds within 1.5% in the range R κ = 5×10-4 – 0.05. The work required to draw a cylinder with a flat bottom face out of a liquid pool scales almost linearly with R2; the dependence on ln ( R κ ) is weak. It is instructive to compare the actual work of adhesion as given by Eq. (18) with the energy of the system before and after removing the cylinder. The work of adhesion is a factor

2.27 − ln ( a κ ) larger than the surface energy required to create fresh liquid surface of π R 2 area. For R κ = 5×10-5 – 0.05 this is a factor 11.5 to 4.6. Reason: When drawing the cylinder out of the liquid much more fresh liquid surface area is created than only π R 2 . The additional surface formed when forming the meniscus is much larger than the top face of the micropillar. The work of adhesion can also be obtained with the linear approximation (10). The work to extend a spring of spring constant k to an extension δc is  0.947 − ln ( R κ )  k W s = δ c2 = πγ R 2  2 0.8091 − ln ( R κ )

2

(19)

Inserting Eqs. (11) and (15) and comparing the results with those calculated with Eq. (17) shows that Eq. (19) underestimates Ws by 3-10% in the range R κ = 5×10-5 – 0.05. Considering the simplicity, it is still a good approximation. Tilted cylinder. In reality the drops’ surface does not detach from the micropillar with a perfectly horizontal surface (Fig. 1 B,C). Rather, the liquid surface is tilted with an angle similar to the apparent receding contact angle (Fig. 1D,E). Therefore, a better approximation for the situation of a rolling drop is that of a tilted cylinder being drawn out of a liquid pool (Fig. 5). The effective additional tilt angle is equal to 180° − Θapp r . As a result, the liquid surface at the rear of the micropillar falls below the receding materials contact already at a smaller extension and condition (12) becomes effective rather than condition (13). Already at a critical angle given by 9 ACS Paragon Plus Environment

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Θ r = 180° − φc − (180° − Θrapp ) = Θ app − φ c ⇒ φ c = Θ app − Θr r r

(20)

the liquid contact line detaches from the rear of the top face of the micropillar. The contact line slides over the micropillar surface and detaches. Inserting condition (20) into Eq. (9):  

δ c = R sin ( Θ app − Θ r ) 0.8091 − ln r ≈ R sin ( Θ

app r

R

κ

 − ln 1 + cos ( Θ app − Θ r )   r 

R  − Θ r )  0.41 − ln  κ  

(21)

The maximal extension of the capillary bridge will in all practical cases depend only on the difference Θapp − Θr . Practically, for superhydrophobic arrays of micropillars Θr is 60-90° and r

− Θr varies between 40-90° (more typically 50-70°). Θrapp = 130-150° so that Θapp r

Figure 5. Schematic of a tilted micropillar drawn out vertically from a water pool. With Eqs. (11), (21) and Ws = kδ c2 2 we get

0.41 − ln ( R κ )  k Ws = δ c2 = πγ R 2 sin 2 ( Θapp − Θr )  r 2 0.8091 − ln ( R κ )

2

(22)

Eq. (22) can be approximated by R  Ws = πγ R 2 sin 2 ( Θ rapp − Θ r ) ⋅  0.60 − ln  κ 

(23)

To estimate the quality of the approximation we inserted condition (20) into Eq. (17) and compared the results. The approximation is better than 10% in the range from a κ = 5×10-4 – 0.05 and for Θ app − Θ r =40-80°. For Θ app − Θ r =60° the approximation is even accurate within r r 1%. Eq. (23) together with Eq. (7) can be used to calculate the apparent receding contact angle. Unfortunately, the resulting equation is not an explicit equation and has to be solved numerically.

Liquid on a superoleophobic surface To repel not only water but also nonpolar liquids the protrusions need to have an overhanging structure. The above model can also be applied to nonpolar liquids on arrays 10 ACS Paragon Plus Environment

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with microscopic tabletops or micro-hoodoos fabricated lithographically.18, 50 The re-entrant structure is formed by the top “plate”. However, for practical applications superoleophobic surfaces are usually not made by complex lithographic processes but more by self-assembly processes. Often, nano-or microscopic spheres are used as building blocks.51, 52, 53, 54 Therefore, as a model for a superoleophobic surfaces we consider a square array of vertical pillars composed of a stack of nano- or microspheres, each with a radius R (Fig. 6).55 When a liquid front recedes the situation is similar to water on superhydrophobic surfaces: Capillary bridges between the drop and the top of protrusions are formed, stretched and finally rupture. In this case, however, the top of the protrusion is not a flat face with defined contact line. Rather it is a sphere and the contact line slides over its surface.

Figure 6. (A) Schematic of the rear side of a liquid drop on a superoleophobic square array of pillars of sintered spheres. The drop is moving to the right. (B) Schematic detail of the last spherical protrusion with the receding liquid. (C) Turned by 180°-Θrapp the situation is a model for the rear side of a drop on a superoleophobic array of stacks of microspheres. The force acting on a sphere being drawn out of a liquid and neglecting gravity is (Fig. 7)56 F = 2π Rγ sin β sin φ = 2π Rγ sin (φ + Θ r ) sin φ

(24)

Here, the angle β describes the position of the contact line on the top of the sphere and

φ = β − Θr . The maximal force is reached for β = (180° + Θ r ) 2 and φ = (180° − Θr ) 2 leading to57 Fc = 2π Rγ cos 2

Θr 2

(25)


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Figure 7. Schematic of a sphere being drawn out of a liquid pool. Turned by 180°-Θrapp the situation is a model for the rear side of a water drop on a superoleophobic array of stacks of microspheres. At the bottom the sphere is in equilibrium without external force and neglecting gravity. The fundamental difference with respect to the cylinder with a pinned contact line (Fig. 2) is that the three phase contact line slides over the surface and the contact radius changes when pulling the sphere out of the liquid. To obtain a F-vs-δ curve we start at φ = 0, calculate β, F, and δ and proceed to increase φ until reaching φ = 90°. Thereby the contact line slides over the particle surface at a constant contact angle. For the normalized length of the capillary bridge we adapt Eq. (9) and replace the radius of the cylinder by R sin β :

δ

R sin β   = sin β sin φ ⋅  0.8091 − ln − ln (1 + cos φ )  + cos Θ r − cos β κ R   (26) R   = sin (φ + Θ r ) sin φ ⋅ 0.8091 − ln − ln 1 + cos φ + sin (φ + Θ r )   + cos Θ r − cos (φ + Θ r ) κ  

Here, the extension of the capillary is slightly smaller than the distance the sphere is moved vertically because the contact line slides over the surface of the particle. In this case δ is the distance the sphere is moved. The last term, cos Θ r − cos (φ + Θr ) , corrects for the sliding of the contact line on the particle surface. The force increases almost linearly with the extension of the capillary bridge. The reason is that both, F and δ, are proportional to sin (φ + Θ r ) sin φ . As examples we plotted force-vs-extension for spheres of different size being drawn out of nhexadecane (Fig. 8). We assumed that the surface of the sphere has a materials receding contact angle of 60°, as for example observed on fluorinated surfaces.


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Figure 8. Force-versus-extension for capillary bridges formed from a sphere with a radius 0.1, 1, and 10 µm in n-hexadecane (γ=27.04 mN/m, κ=1.89 mm at 25°C). Thick continuous lines were calculated with Eqs. (24) and (26) and varying φ. The thin dashed lines were obtained with Eq. (10). Again, a simple approximation can be derived. We start with Eq. (26) and neglect the last part cos Θ r − cos (φ + Θ r ) , which accounts for the sliding of the contact line over the particle

surface. We further set (1 + cos φ ) sin (φ + Θ r ) = 1 in the argument of the logarithm. With these changes we solve Eq. (26) for sin (φ + Θ r ) sin φ

δ

R sin (φ + Θ r ) sin φ =  0.8091 − ln  R κ

−1

(27)

and insert the result into Eq. (24). The result is identical to Eq. (10). Only the interpretation of R is different. Here it is the radius of the spherical top protrusion rather than the radius of the top face of a flat pillar. To obtain the work of adhesion via Ws = kδ c2 2 we calculate the maximal extension from the maximal force (Eq. 25): Fc = 2π Rγ cos 2

Θr 2πγδ 0 Θ  R = ⇒ δ 0 = R cos 2 r  0.8091 − ln  2 0.8091 − ln ( R κ ) 2  κ

(28)

Here, δ0 is the extension at the maximal force. By setting the extension at maximal force equal to the extension of rupture, δ 0 ≈ δ c , we get an approximate expression for the work of adhesion: W s = πγ R 2 cos 4

Θr 2

R  ⋅  0.8091 − ln  κ 

(29)

The work of adhesion has already been calculated with good accuracy by Anachkov et al.58 based on an earlier approximate expression by Pitois & Chateau:59 13 ACS Paragon Plus Environment

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2 2   Ws Θ r   1  Θr Θr    R  Θr  Θr     2 Θr = 1 − sin 7 sin + 8 sin + 3 − 1 + sin + 0.5772           ln   1 + sin  cos  2 2   2  2 2 2    4κ  2  2  πR γ     

Θ  Θ  R +2 cos 4 r ⋅  sin r − ln  2  2  4κ

 Θr  Θr   − 1.5772   1 + sin  cos  2  2   

−1

(30)

When comparing predictions of the simple approximation (29) with calculations based on Eq. (30) the dependencies are largely correct, but the quantitative numbers obtained with Eq. (29) are typically 20% lower than results obtained with the more accurate Eq. (30). Reasons are, first, an underestimation of δ0 and, second, neglecting the work required for the extension between maximal force and rupture of the capillary bridge (δ0 and δc). Better agreement is obtained by adjusting the numerical constant 0.8091 to 2.9 to maximize accuracy Ws = πγ R 2 cos4

Θr 2

R  ⋅  2.9 − ln  κ 

(31)

We tested predictions of Eq. (31) with calculations of Eq. (30) for Θr= 50, 70°, and 90° and R/ κ ranging from of 5×10-5 to 0.05 (Fig. 9). The deviation is below 2% for almost the whole range. Note that both expressions show that Ws is proportional to πγ R 2 .

Figure 9. Ratio of the work of adhesion calculated with the simple approximation Eq. (31) and the much more accurate Eq. (30). Combining Eq. (31) with Eq. (7) allows us to predict the apparent receding contact angle from the microstructure and the material properties:

γ

( cos Θ n

cos Θ

app r

Θr 2

R  ⋅  2.9 − ln  ⇒ κ  Θ  R = nπ R 2 cos 4 r ⋅  2.9 − ln  − 1 2  κ app r

+ 1) = πγ R 2 cos 4


(32)

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At first sight, this result looks very different from a result obtained previously from a force balance 55: cos

Θapp Θ r = π R n cos 2 r 2 2

(33)

However, applying the mathematical identity cos ( x 2 ) = ( cos x + 1) 2 to Eq. (33) leads to Θapp 1 Θ 1 Θ r = cos Θapp + 1) = π R n cos2 r ⇒ ( cos Θapp + 1) = π 2 R 2 n cos4 r ⇒ ( r r 2 2 2 2 2 Θ cos Θapp = 2π 2 R 2 n cos 4 r − 1 r 2 cos

(34)

Thus, except for the weak dependence in ln (κ R ) both equations show the same scaling. The difference is the factor 2π in Eq. (34) rather than 2.9 − ln ( R κ ) in Eq. (32). Even the orders of magnitude of the quantitative results are similar. Both factors completely agree for

R κ = e−2.83 = 0.059 .

Comparison with experiments Superhydrophibic arrays of micropillars. For an array with known density of nano- or micropillars n and measured values for the apparent receding contact angle Θrapp we can calculate the work of adhesion Ws with Eq. (7). Then we can compare the result to the one obtained with the radius of the cylinder and the material properties Θr, γ, and κ (Eq. 23). As one example we take results from Schellenberger et al. 12, who used square arrays of cylindrical micropillars with R=2.5 µm, n=2.5×109 m-2 and R=5 µm, n=6.25×108 m-2. In both cases experiments showed Θrapp = 142°. With Eq. (7) we find a work of adhesion WS=6.1 pJ for R=2.5 µm and 24.4 pJ for R=5 µm. Coming from the microscopic structure and applying Eq. (23) with Θr=85° we find WS=7.5 pJ for R=2.5 µm and 27.4 pJ for R=5 µm. Both values agree within 25%. Bico et al. 14 measured contact angles of water drops on square arrays of micropillars with Θr=100°, a side length of 1 µm, a=4 µm and Θrapp = 155°. With Eq. (7) and setting R=0.5 µm we obtain Ws= 0.11 pJ while the microscopic picture with Eq. (23) leads to Ws= 0.34 pJ. A similar factor of 2-3 difference is found when comparing the results of Öner and McCarthy 60. They measured contact angles on hexagonal arrays of square micropillars with edge lengths 2-32 µm and a lattice constant a of twice that value. When calculating Ws with Eq. (23) the results are a factor of 2-3 higher than values obtained with Eq. (7) and n = a −2 sin −1 60° (Fig. 10). The dependency of WS on the pillar size is correct but the quantitative values predicted are 2 - 3 times too high. One reason is probably the sensitive dependence of Ws in Eq. (7) on the observed apparent receding contact angle. Measuring large contact angles is difficult and apparent receding contact angles measured with a goniometer as in refs. 14, 60 are easily overestimated. Apparent contact angles of Schellenberger were measured by confocal 15 ACS Paragon Plus Environment

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microscopy 12. If the real Θrapp values in 14, 60 were 7° smaller Ws calculated with Eqs. (7) and (23) would agree. Thus, when using Eq. (7) to calculate the work of adhesion care must be taken to measure the corrects apparent receding contact angle.

Figure 10. Values of the work of adhesion per pillar calculated from experimental results of Öner and McCarthy60 with Eq. (7) (black) and Eq. (23) (red) setting the edge length equal to 2R. The three different lines were measured and calculated for different hydrophobic coatings with materials contact angles of Θr = 94, 102, and 110°. Superoleophobic surfaces. A precise and direct comparison with experimental results is difficult because no defined model surface has been tested. We rely on estimations with selfassembled superoleophobic surfaces. As an example we take the soot-templated surfaces.54 With an estimated spacing between protrusion of a≈1 µm and an apparent receding contact angle of Θrapp=152° for n-hexadecane we get the work of adhesion (Eq. 7) Ws = 3.2 × 10−15 J. From the microscopic picture (Eq. 31) we calculate with Θr=64° and R≈80 nm WS = 3.7 × 10 −15 J. The agreement should, however, not be overestimated. The results depend sensitively on the radius R and the pitch of the micropillars a entering via n = a −2 . It does not so much 4 depend on the Θr because for a reasonable range of Θr =40-70° the factor cos ( Θr 2) only

varies between 0.78 and 0.45.

Estimation of the velocity dependence Eqs. (6) and (7) are independent on velocity as long as Ws does not depend on velocity. To get an idea of how fast the capillary bridge relaxes and the equilibrium equations are valid we estimate its relaxation time τ. Therefore, we treat each capillary bridge like a spring with an inverse relaxation time 1 τ = k m cb . Here, k is the spring constant and mcb the moving mass of the capillary bridge. The mass of a capillary bridge can be estimated from the following balance: When withdrawing a cylinder or sphere from a liquid pool (Fig. 2, 5 and 7) the maximal capillary force is equal to the maximal weight of the capillary bridge. With a maximal


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force of the order of F ≈ 2π Rγ we estimate the mass to be 2π Rγ = mcb g ⇒ mcb ≈ 2π Rγ g . Applying Eq. (11) we get 1

τ

k = mcb

=

g R  0.8091 − ln ( R κ ) 

(35)

With good approximation τ ≈ R ln (κ R ) g . The relaxation times are of the order 0.1-10 ms (Fig. 11). They are relatively independent on the specific liquid because the liquid properties only enter via the capillary length of ≈2 mm.

Figure 11. Relaxation time of an aqueous capillary bridge after collapsing from a micropillar of radius R calculated with Eq. (35). The assumption of equilibrium shape of the menisci breaks down if the velocity of the apparent contact line exceeds U=

a

τ

=a

g

(36)

R  R  0.8091 − ln  κ 

Taking two examples from above this is expected to happen at U= 1.1 cm/s (water, R=2.5 µm, a=20 µm) and U= 2.2 cm/s (water, R=5 µm, a=40 µm). Typically, accelerations for water drops of 1-2 mm radius sliding down an inclined superhydrophobic surface at a tilt angle of 2 3, 31 10° are of the order of && A velocity of 2 cm/s is reached after t = U && s = 0.5 m/s . s = 0.04 s or a sliding distance of s = U 2 2&& s = 0.4 mm. We expect that Ws decreases with increasing velocity because the capillary bridge will break before reaching its full extension.

Summary When drops move on superliquid repellent surfaces they dissipate energy at nano- and microscopic protrusions. At the rear side of a moving drop capillary bridges are formed, elongated, and broken. The energy required to stretch such bridges is dissipated. For two 17 ACS Paragon Plus Environment

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model surfaces we calculate the work required to stretch capillary bridges until they break: (1) A water drop moving on a superhydrophobic square array of cylindrical micropillars and (2) a liquid drop moving on a square array of stacks of spheres. In both cases force-versusextension for individual capillary bridges is calculated. Simple approximations which are based on the fundamental physical processes are derived. These expressions are used to calculate the work to stretch and rupture individual capillary bridges. The expressions are verified by comparison with more accurate equations. For the array of cylinders with flat tops the accurate expression is derived by calculating the work required to pull a horizontal vertical disc out of a liquid pool. For the stack of spheres we apply an expression derived earlier.58

Acknowledgments We acknowledge financial support from the ERC grant No. 340391 SuPro and the SFB 1194.

Appendix To carry out the integration in Eq. (16) we need the derivative d dφ

 R   sin φ  0.8091 − ln − ln (1 + cos φ )   κ   

R sin 2 φ   = cos φ  0.8091 − ln − ln (1 + cos φ )  + κ   1 + cos φ

Inserting φc

φ

c  dδ R sin 2 φ    Ws = ∫ F d φ = 2πγ R 2 ∫ sin φ  cos φ  0.8091 − ln − ln (1 + cos φ )  +  dφ dφ κ   1 + cos φ   0 0

φ

c  sin 3 φ  R  = 2πγ R 2 ∫ sin φ cos φ  0.8091 − ln  − sin φ cos φ ln (1 + cos φ ) +  dφ κ 1 + cos φ   0 

We consider the three terms separately. First term: φc

φ

c sin 2 φc 1 2  sin φ cos φ d φ = sin φ = ∫0  2  2 0

Second term: φc

∫ sin φ cos φ ln (1 + cos φ ) dφ 0

φ

=

c 1 3 1 2 2  ln 1 + cos φ − cos φ ln 1 + cos φ − − cos φ + cos φ ( ) ( )  2  2 2 0

=

1 2 1 1 sin φc ln (1 + cos φc ) − cos φc + cos 2 φc +   2 2 2

3rd term: 18 ACS Paragon Plus Environment

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φc

    2 φ 2 φc φc     + + 1 2 tan 1 2 tan 3 sin φ 2  = −2  2 − 1  d = − 2 φ 2 2 ∫0 1 + cos φ    2 φ   2 φc    1 + tan 2     1 + tan 2    0     =2

1 + 2 tan 2

φc

+ tan 4

φc

− 1 − 2 tan 2

2 2 2  2 φc   1 + tan  2 

φc 2 =

2 tan 4

φc

2 2  2 φc   1 + tan  2 

with x 2 tan 2 tan x 2 sin ( 2 x ) = ⇒ sin x = ⇒ sin 2 x = 2 x 1 + tan 2 x 1 + tan 2

x 2 2  2 x  + 1 tan   2  4 tan 2

and sin 3 φ 1 2 1 2 (1 − cos φc ) 1 2 2 φ ∫0 1 + cos φ d φ = 2 sin φc tan 2c = 2 sin φc sin 2 φc = 2 (1 − cos φc )

φc

2

In the last step we used tan  sin 2 φc Ws = 2πγ R 2   2

x 1 − cos x sin x = = . Now we can put everything together: 2 sin x 1 + cos x

R 1 2 1 1 1 2  2  0.8091 − ln  −  sin φc ln (1 + cos φc ) − cos φc + cos φc +  + (1 − cos φc )  κ 2 2 2 2     

R 1 1    = πγ R 2 sin 2 φc  0.8091 − ln  − sin 2 φc ln (1 + cos φc ) + cos φc − cos 2 φc − + 1 − 2 cos φc + cos 2 φc  κ 2 2      R 1    1 = πγ R 2 sin 2 φc  0.8091 − ln − ln (1 + cos φc )  + cos 2 φc + − cos φc  κ 2 2    

to obtain Eq. (17).


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53. Ellinas, K.; Tserepi, A.; Gogolides, E. From superamphiphobic to amphiphilic polymeric surfaces with ordered hierarchical roughness fabricated with colloidal lithography and plasma nanotexturing. Langmuir 2011, 27, 3960-3969. 54. Deng, X.; Mammen, L.; Butt, H.-J.; Vollmer, D. Candle soot as a template for a transparent robust superamphiphobic coating. Science 2012, 335, 67-70. 55. Butt, H. J.; Semprebon, C.; Papadopoulos, P.; Vollmer, D.; Brinkmann, M.; Ciccotti, M. Design principles for superamphiphobic surfaces. Soft Matter 2013, 9, 418-428. 56. Yarnold, G. D. The hysteresis of the angle of contact of mercury. Proc. Phys. Soc. London 1946, 58, 120-125. 57. Scheludko, A.; Toshev, B. V.; Bojadjiev, D. T. Attachment of particles to a liquid surface (Capillary theory of flotation). J. Chem. Soc. Faraday Trans. I 1976, 72, 2815-2828. 58. Anachkov, S. E.; Lesov, I.; Zanini, M.; Kralchevsky, P. A.; Denkov, N. D.; Isa, L. Particle detachment from fluid interfaces: Theory vs. experiments. Soft Matter 2016, 12, 76327643. 59. Pitois, O.; Chateau, X. Small particle at a fluid interface: Effect of contact angle hysteresis on force and work of detachment. Langmuir 2002, 18, 9751-9756. 60. Öner, D.; McCarthy, T. J. Ultrahydrophobic surfaces. Effects of topography length scales on wettability. Langmuir 2000, 16, 7777-7782.


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Figure 1. (A) Schematic of a water drop on a superhydrophobic array of micropillars moving to the right. (B) A schematic detail of the rear side of the drop. The schematic figure is based on confocal microscope images, such as the one shown in (C). (C) Confocal microscope image of the rear of a water drop (V = 5 µL) on a superhydrophobic square array of cylindrical micropillars (radius R = 5 µm, pitch a = 40 µm, height 5 µm).42 The pillars were made of SU-8 and were hydrophobized with 1H,1H,2H,2H-perfluorooctyl trichlorosilane. The vertical cross-section is along the diagonal of the array. Water was fluorescently labelled with Alexa488 (0.1 mg/mL). As the pillars have a refractive index of 1.6 and a dry objective was used, they appear shorter. Thus we show artificial pillars in yellow with the correct height. (D) Schematic of the last pillar at the rear side of the drop and the direction in which the bulk drop moves away from the pillar. To model this situation we turned the geometry upside down (E). Before calculating the extension and rupture of a capillary bridge between an infinitely extended water surface and a tilted pillar, we considered the even simpler situation of the horizontal pillar (F). Figure 1.


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166x247mm (300 x 300 DPI)


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Figure 2. Schematic of a cylinder with a flat bottom being drawn out of a liquid pool (first bottom image then top). Turned by 180° the situation is a model for the rear side of a water drop on a superhydrophobic array of micropillars. Figure 2. 81x87mm (300 x 300 DPI)


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Figure 3. (A) Force-versus-extension for capillary bridges formed from a vertical cylinder with a flat bottom face and radius 0.1, 1, and 10 µm. The liquid is water (γ=72.0 mN/m, κ=2.71 mm at 25°C). Thick continuous lines were calculated with Eqs. (8) and (9) and varying Φ. The dashed lines were obtained with the linear approximation Eq. (10). (B) Force-versus-extension for capillary bridges formed from a sphere with a radius 0.1, 1, and 10 µm in n-hexadecane (γ=27.04 mN/m, κ=1.89 mm at 25°C). Thick continuous lines were calculated with Eqs. (24) and (26) and varying Φ. The thin dashed lines were obtained with Eq. (10). Figure 3. 145x186mm (300 x 300 DPI)


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Figure 4. Critical angle versus the normalized radius of the micropillar. Figure 4. 130x104mm (300 x 300 DPI)


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Figure 5. Schematic of a tilted micropillar drawn out vertically from a water pool. Figure 5. 76x42mm (300 x 300 DPI)


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Figure 6. (A) Schematic of the rear side of a liquid drop on a superoleophobic square array of pillars of sintered spheres. The drop is moving to the right. (B) Schematic detail of the last spherical protrusion with the receding liquid. (C) Turned by 180°-Θr app the situation is a model for the rear side of a drop on a superoleophobic array of stacks of micro¬spheres. Figure 6. 111x107mm (300 x 300 DPI)


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