Comment pubs.acs.org/Langmuir
Misconceptions in Wetting Phenomena ABSTRACT: In a recent paper (’t Mannetje, D.; Banpurkar, A.; Koppelman, H.; Duits, M. H. G.; van den Ende, D.; Mugele, F. Electrically Tunable Wetting Defects Characterized by a Simple Capillary Force Sensor. Langmuir 2013, 29, 9944−9949), there are a few misconceptions regarding the interpretations of theories emanating from Shanahan and de Gennes in describing centrifugal adhesion balance (CAB) experiments, making their results seemingly contradictory to the theory. These are clarified here. We show that their results, if interpreted correctly, do not contradict the theories mentioned above. angular velocity, and α is the tilt angle of the CAB chamber (Figure 1). Equation 2 could not be validated using the tilt stage method because as the stage is tilted both normal and lateral forces are changed at the same time and, as CAB experiments demonstrate, the normal force significantly affects the lateral retention force in a seemingly nonintuitive way with a minimum at zero normal force and typically higher f∥ values for pendant drops than for sessile drops. Nonetheless, eq 2 enables a reasonable fit to CAB measurements. ’t Mannetje et al. proceed with an experiment in which they use a small glass capillary tube of known spring constant to pull on a water drop back and forth along a surface and measure the force from the deflection of the capillary. With this they argued that “simultaneous variation of the tangential and normal components of the gravitational force” is “clearly avoid(ed)” and move on to draw conclusions about experiments involving different drop sizes that seemingly do not agree with eq 2. However, because they change the drop size in the gravitational field they also change the normal force (i.e., weight) experienced by the drop. Therefore, this is not a system of constant normal force although it is on a horizontal surface. But this is not the only reason that their system cannot be used to test eq 2. ’t Mannetje et al. coated their surface with a layer of silicon oil and slid a water drop on the oiled surface. Because the surface is oil-coated, most of the deformation of the solid surface under the oil cannot be expressed, and hence eq 2 is not expected to be valid for this case. Additionally, the process of the solid surface deformation requires a time during which the drop rests on the surface without moving (the exact time required for the system to reach a plateau in the force can be short6 or long4 depending on the system). ’t Mannetje et al., however, applied a back-andforth motion by never allowing any resting of the drop on the surface. This again suppresses the physical phenomena for which eq 2 is valid. Emphasizing this point, we note that their ref 19 (Pilat et al.15) shows that even the speed with which one applies the back-and-forth motion influences the force required to slide the drop (Figure 6 in ref 15). ’t Mannetje et al. also plot the force that they obtain versus the drop diameter and show that within their (not small) scatter they can obtain a linear fit according to eq 1 (their Figure 2c). However, they do not check the actual variation of
’t Mannetje et al.1 consider what they call friction force (Ff) and what we will call here “lateral force” ( f∥) required to slide a drop on a surface. They start with the theoretical relation of the force required to slide a drop on a perfectly flat surface to the receding and advancing contact angles, θR and θA, respectively. This (for a perfectly flat surface) is as written in eq 1 below (which is also numbered eq 1 in their paper) f|| = cγW(cos θR − cos θA)
(1)
where γ is the surface tension of the drop, W is the width of the drop perpendicular to the direction of motion (W = 2r for a drop that makes a circular three-phase contact line of radius r), and c is a constant that ’t Mannetje et al. eventually takes as 1 (the correct coefficient for this equation based on the Dussan theory2). However, according to Shanahan and de Gennes,3 solid surfaces with drops on them are not perfectly flat: there is a deformation of the solid surface at the three-phase contact line that is inversely proportional to the modulus, GS, that is associated with the outermost layer of the solid surface (subscript S stands for surface) and is proportional to the Laplace pressure difference between the inside and outside of the drop, ΔP = 2γ/R, where R is the radius of the curvature of the drop−vapor surface (r = R sin θ for spherical drops). The Shanahan−de Gennes deformation gives rise to a different intermolecular interaction at the three-phase contact line.4−12 To account for this, eq 1 needs to be multiplied by a factor of 2γ/(GSR) to describe properly systems in which the surface deformation contributes to the retention force.4−6 This, for c = 1 and W = 2r, results in the equation f|| =
2γ 2 sin θ(cos θR − cos θA) GS
(2)
where θ is the as-placed contact angle. Equation 2 shows no dependence of the force on the drop size and was experimentally validated using the centrifugal adhesion balance (CAB).4−6,9 The CAB (produced by Wet Scientific) uses both centrifugal and gravitational forces as shown in Figure 1 to enable any combination of normal (f⊥) and lateral ( f∥) forces through eqs 3 and 4 13,14
f = m(ω 2L cos α − g sin α)
(3)
f⊥ = m(ω 2L sin α + g cos α)
(4)
where m is the drop’s mass, g is the gravitational acceleration, L is the distance of the drop from the axis of rotation, ω is the © 2013 American Chemical Society
Received: September 17, 2013 Published: November 20, 2013 15474
dx.doi.org/10.1021/la403578q | Langmuir 2013, 29, 15474−15475
Langmuir
Comment
Figure 1. (a) Schematic of the basic CAB setup that combines centrifugal and gravitational forces. (b) Vectorial representation of the forces acting on a drop in the CAB chamber (cf. eqs 3 and 4).
θR and θA with drop size but instead assume that they are constant and that only the drop size varies. Finally, I do not mean to imply that eq 1 is wrong. On the contrary, there are many situations in which it is the best way to describe a system, and I believe ’t Mannetje et al.’s system is an example. At the same time, eq 2 is valid for describing other systems.4−6,8,9 A third and common situation is when a system contains elements from the underlying assumptions of eqs 1 and 2, in which case both equations govern the system partially. Care should be given to relate the right equation to the right system. In summary, we make the following points: (1) To verify eq 1 or 2, there is a need to maintain a constant normal force. For this, it is not sufficient to retain a horizontal surface but rather a centrifugal adhesion balance (CAB) should be used. (2) For a proper experiment, one needs to consider the time effect; namely, how does the retention force change with the time the drop rests motionless on the surface. (3) For a proper experiment, one also needs to consider the advancing and receding angles of every force measurement because they may vary with the drop size and other conditions. Avoiding this may compromise the interpretation of eq 1 or 2. (4) An oil coating on a surface makes the surface more liquid, thereby obscuring effects associated with solid surfaces such as those expressed in eq 2.
solid micro-deformation on contact angle equilibrium. J. Phys. D: Appl. Phys. 1987, 20, 945. (4) Tadmor, R.; Bahadur, P.; Leh, A.; N'guessan, H. E.; Jaini, R.; Dang, L. Measurement of Lateral Adhesion Forces at the Interface between a Liquid Drop and a Substrate. Phys. Rev. Lett. 2009, 103, 266101-1−266101-4. (5) Tadmor, R. Approaches in Wetting Phenomena. Soft Matter 2011, 7, 1577−1580. (6) N’guessan, H. E.; Leh, A.; Cox, P.; Bahadur, P.; Tadmor, R.; Patra, P.; Vajtai, R.; Ajayan, P. M.; Wasnik, P. Water Tribology on Graphene. Nat. Commun. 2012, 3, 1242. (7) Xu, W.; Leeladhar, R.; Kang, Y. T.; Choi, C.-H. Evaporation Kinetics of Sessile Water Droplets on Micropillared Superhydrophobic Surfaces. Langmuir 2013, 29, 6032−6041. (8) Bormashenko, E.; Bormashenko, Y.; Gendelman, O. On the Nature of the Friction between Nonstick Droplets and Solid Substrates. Langmuir 2010, 26, 12479−12482. (9) Leh, A.; N’guessan, H. E.; Fan, J.; Bahadur, P.; Tadmor, R.; Zhao, Y. On the Role of the Three-Phase Contact Line in Surface Deformation. Langmuir 2012, 28, 5795−5801. (10) Belman, N.; Jin, K.; Golan, Y.; Israelachvili, J. N.; Pesika, N. S. Origin of the Contact Angle Hysteresis of Water on Chemisorbed and Physisorbed Self-Assembled Monolayers. Langmuir 2012, 28, 14609− 14617. (11) Ling, W. Y. L.; Ng, T. W.; Neild, A.; Zheng, Q. Sliding Variability of Droplets on a Hydrophobic Incline Due to Surface Entrained Air Bubbles. J. Colloid Interface Sci. 2011, 354, 832−842. (12) In nanostructured materials, this can also induce nanomechanical fracture and yield.9 (13) Guilizzoni, M. Drop Shape Visualization and Contact Angle Measurement on Curved Surfaces. J. Colloid Interface Sci. 2011, 364, 230−236. (14) Santini, M.; Guilizzoni, M.; Fest-Santini, S. X-ray Computed Microtomography for Drop Shape Analysis and Contact Angle Measurement. J. Colloid Interface Sci. 2013, 409, 204−210. (15) Pilat, D. W.; Papadopoulos, P.; Schaffel, D.; Vollmer, D.; Berger, R.; Butt, H.-J. Dynamic Measurement of the Force Required to Move a Liquid Drop on a Solid Surface. Langmuir 2012, 28, 16812−16820.
Rafael Tadmor
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Dan F. Smith Department of Chemical Engineering, Lamar University, Beaumont, Texas 77710, United States
AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS This study was made possible by financial support from NSF Grant CBET-0960229. I thank M.E.R. Shanahan, P.J. Cox, J.V.I. Timonen, and J. Gossage for going over the manuscript and providing important comments.
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REFERENCES
(1) ’t Mannetje, D.; Banpurkar, A.; Koppelman, H.; Duits, M. H. G.; van den Ende, D.; Mugele, F. Electrically Tunable Wetting Defects Characterized by a Simple Capillary Force Sensor. Langmuir 2013, 29, 9944−9949. (2) Dussan, E. B. On the Ability of Drops or Bubbles to Stick to Non-Horizontal Surfaces of Solids 0.2. Small Drops or Bubbles Having Contact Angles of Arbitrary Size. J. Fluid Mech. 1985, 151, 1−20. (3) Shanahan, M. E. R.; de Gennes, P. G. The Ridge Produced by a Liquid near the Triple Line Solid Liquid Fluid. C. R. Acad. Sci. Ser. II 1986, 302, 517−521. See also: Shanahan, M. E. R. The influence of 15475
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