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How drops start sliding over solid surfaces

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Nan Gao
Nan Gao
Nan Gao
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Florian Geyer at Max Planck Institute for Polymer Research
Florian Geyer
Dominik W. Pilat
Dominik W. Pilat
Dominik W. Pilat
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Sanghyuk Wooh at Chung-Ang University
Sanghyuk Wooh
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Abstract and Figures

It has been known for more than 200 years that the maximum static friction force between two solid surfaces is usually greater than the kinetic friction force—the force that is required to maintain the relative motion of the surfaces once the static force has been overcome. But the forces that impede the lateral motion of a drop of liquid on a solid surface are not as well characterized, and there is a lack of understanding about liquid–solid friction in general. Here, we report that the lateral adhesion force between a liquid drop and a solid can also be divided into a static and a kinetic regime. This striking analogy with solid–solid friction is a generic phenomenon that holds for liquids of different polarities and surface tensions on smooth, rough and structured surfaces.
Schematics of friction force measurements a, Textbook configuration for demonstrating solid–solid friction. b, Homemade set-up for measuring liquid–solid friction. A drop of liquid is placed on a solid substrate mounted on a linear stage driven by a step motor. A laser beam incident on the capillary is reflected to a position-sensitive detector (PSD). The contact width between the drop of liquid and the solid surface (orthogonal to the direction of motion) and contact length (parallel to the direction of motion) are simultaneously monitored by cameras (not shown).
Schematics of friction force measurements a, Textbook configuration for demonstrating solid–solid friction. b, Homemade set-up for measuring liquid–solid friction. A drop of liquid is placed on a solid substrate mounted on a linear stage driven by a step motor. A laser beam incident on the capillary is reflected to a position-sensitive detector (PSD). The contact width between the drop of liquid and the solid surface (orthogonal to the direction of motion) and contact length (parallel to the direction of motion) are simultaneously monitored by cameras (not shown).
… 
Lateral adhesion force experiment of a drop of ionic liquid (volume ≈1.5 μl) on a fluorinated silicon wafer The measurement was performed at a constant linear stage velocity of 200 μ m s⁻¹. The surface tension of the ionic liquid 1-butyl-2,3-dimethylimidazolium bis(trifluoromethanesulfonyl)imide is 34.6 mN m⁻¹. a, The measured lateral adhesion force is plotted with blue circles. The error in force measurement is ±1 μN and smaller than the symbol size. The calculated force is plotted with red squares. b, Simultaneously determined contact width and length of the drop during the force measurement shown in a. c, Front and rear contact angles during the force measurement. Supplementary Movie 1 shows the motion of the ionic liquid drop on the fluorinated Si wafer during the lateral adhesion force measurement.
Lateral adhesion force experiment of a drop of ionic liquid (volume ≈1.5 μl) on a fluorinated silicon wafer The measurement was performed at a constant linear stage velocity of 200 μ m s⁻¹. The surface tension of the ionic liquid 1-butyl-2,3-dimethylimidazolium bis(trifluoromethanesulfonyl)imide is 34.6 mN m⁻¹. a, The measured lateral adhesion force is plotted with blue circles. The error in force measurement is ±1 μN and smaller than the symbol size. The calculated force is plotted with red squares. b, Simultaneously determined contact width and length of the drop during the force measurement shown in a. c, Front and rear contact angles during the force measurement. Supplementary Movie 1 shows the motion of the ionic liquid drop on the fluorinated Si wafer during the lateral adhesion force measurement.
… 
Lateral adhesion forces for drops of different liquids on solid surfaces a, Measured (blue circles) and calculated lateral adhesion forces using k = 1 (red squares). The surfaces include silicone nanofilaments (SNFs), silicon wafers (Si), SU-8 square pillar arrays (height, 25 μm; width, 50 μm; centre–centre distance, 100 μm), multilayers of 20-nm titanium dioxide nanoparticles (TiO2 NPs), and crosslinked polydimethylsiloxane (PDMS). All surfaces except for those of polydimethylsiloxane were fluorinated before use. All fabrication details are provided in the Methods section. The adopted surface tensions of water and hexadecane are 72.5 mN m⁻¹ and 27.5 mN m⁻¹, respectively. The front and rear contact angles in the kinetic regimes were 171° ± 1° and 164° ± 1° for water on fluorinated silicone nanofilaments, 83° ± 1° and 59° ± 1° for hexadecane on fluorinated silicon wafers, 169° ± 2° and 122° ± 2° for water on fluorinated SU-8 pillars, 128° ± 1° and 95° ± 1° for water on fluorinated silicon wafers, 164° ± 1° and 128° ± 2° for water on titanium dioxide nanoparticles, and 121° ± 1° and 81° ± 1° for water on crosslinked PDMS, respectively (Supplementary Figs 5–10). Supplementary Movies 2–7 show the motions of drops on the different surfaces during the lateral adhesion force measurements. b, Lateral adhesion force per contact width. Drop volumes between 1.5 and 8.0 μl were chosen to avoid rupturing of drops during motion. c, Ratios of the kinetic friction force FKIN and its threshold force FTHRD of all liquid–solid combinations that are studied. Here we used FKIN and FTHRD prior to normalization with the respective contact widths to point out FKIN/FTHRD ≤ 1. Error bars in b,c indicate variability between different experiments.
Lateral adhesion forces for drops of different liquids on solid surfaces a, Measured (blue circles) and calculated lateral adhesion forces using k = 1 (red squares). The surfaces include silicone nanofilaments (SNFs), silicon wafers (Si), SU-8 square pillar arrays (height, 25 μm; width, 50 μm; centre–centre distance, 100 μm), multilayers of 20-nm titanium dioxide nanoparticles (TiO2 NPs), and crosslinked polydimethylsiloxane (PDMS). All surfaces except for those of polydimethylsiloxane were fluorinated before use. All fabrication details are provided in the Methods section. The adopted surface tensions of water and hexadecane are 72.5 mN m⁻¹ and 27.5 mN m⁻¹, respectively. The front and rear contact angles in the kinetic regimes were 171° ± 1° and 164° ± 1° for water on fluorinated silicone nanofilaments, 83° ± 1° and 59° ± 1° for hexadecane on fluorinated silicon wafers, 169° ± 2° and 122° ± 2° for water on fluorinated SU-8 pillars, 128° ± 1° and 95° ± 1° for water on fluorinated silicon wafers, 164° ± 1° and 128° ± 2° for water on titanium dioxide nanoparticles, and 121° ± 1° and 81° ± 1° for water on crosslinked PDMS, respectively (Supplementary Figs 5–10). Supplementary Movies 2–7 show the motions of drops on the different surfaces during the lateral adhesion force measurements. b, Lateral adhesion force per contact width. Drop volumes between 1.5 and 8.0 μl were chosen to avoid rupturing of drops during motion. c, Ratios of the kinetic friction force FKIN and its threshold force FTHRD of all liquid–solid combinations that are studied. Here we used FKIN and FTHRD prior to normalization with the respective contact widths to point out FKIN/FTHRD ≤ 1. Error bars in b,c indicate variability between different experiments.
… 
Velocity dependence of lateral adhesion forces a,b, Development of the lateral adhesion force of hexadecane drops (≈3 μl) on a fluorinated Si wafer surface at threshold (a) and in the kinetic regime (b). The inset in b shows the front and rear contact angles measured in the kinetic regime. c,d, Velocity dependence of lateral adhesion forces of an ionic liquid (1-butyl-2,3-dimethylimidazolium bis(trifluoromethanesulfonyl) imide) drop (≈2 μl) on a fluorinated Si wafer substrate (c) and water drops (≈7.5 μl) on a fluorinated Si wafer substrate (d). Here velocities below 0.08 mm s⁻¹ were not studied owing to evaporation of water. The error bars correspond to the standard deviation calculated from five to seven independent measurements. The highest velocity that can be probed is given by the threshold force where the liquid drop detaches from the glass capillary (≈81 μN for water, ≈36 μN for ionic liquid and ≈31 μN for hexadecane).
Velocity dependence of lateral adhesion forces a,b, Development of the lateral adhesion force of hexadecane drops (≈3 μl) on a fluorinated Si wafer surface at threshold (a) and in the kinetic regime (b). The inset in b shows the front and rear contact angles measured in the kinetic regime. c,d, Velocity dependence of lateral adhesion forces of an ionic liquid (1-butyl-2,3-dimethylimidazolium bis(trifluoromethanesulfonyl) imide) drop (≈2 μl) on a fluorinated Si wafer substrate (c) and water drops (≈7.5 μl) on a fluorinated Si wafer substrate (d). Here velocities below 0.08 mm s⁻¹ were not studied owing to evaporation of water. The error bars correspond to the standard deviation calculated from five to seven independent measurements. The highest velocity that can be probed is given by the threshold force where the liquid drop detaches from the glass capillary (≈81 μN for water, ≈36 μN for ionic liquid and ≈31 μN for hexadecane).
… 
Lateral adhesion force measurement of a water drop on a goose feather a, Three different lateral adhesion force measurements of a water drop on a goose feather at different areas at a velocity of 6.8 mm s⁻¹ (drop volume of 8.0 μl). The threshold lateral adhesion force varied between 35 ± 1 μN and 67 ± 2 μN. Continuing the kinetic motion of the water drop required a force up to 35 μN. b, An optical image of the feather with resting water drops. c, Scanning electron microscopy images reveal the detailed structure of the feather. The inset is an enlarged area showing the microscopic structure. The scale bar in the inset is 10 μm. d, No velocity dependence of the lateral adhesion force at threshold (black squares) and in the kinetic regime (red circles) could be measured. The error bars correspond to the standard deviations calculated from five to seven measurements each.
Lateral adhesion force measurement of a water drop on a goose feather a, Three different lateral adhesion force measurements of a water drop on a goose feather at different areas at a velocity of 6.8 mm s⁻¹ (drop volume of 8.0 μl). The threshold lateral adhesion force varied between 35 ± 1 μN and 67 ± 2 μN. Continuing the kinetic motion of the water drop required a force up to 35 μN. b, An optical image of the feather with resting water drops. c, Scanning electron microscopy images reveal the detailed structure of the feather. The inset is an enlarged area showing the microscopic structure. The scale bar in the inset is 10 μm. d, No velocity dependence of the lateral adhesion force at threshold (black squares) and in the kinetic regime (red circles) could be measured. The error bars correspond to the standard deviations calculated from five to seven measurements each.
… 
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ARTICLES
PUBLISHED ONLINE: 6 NOVEMBER 2017 | DOI: 10.1038/NPHYS4305
How drops start sliding over solid surfaces
Nan Gao1,2*, Florian Geyer1, Dominik W. Pilat1, Sanghyuk Wooh1†, Doris Vollmer1, Hans-Jürgen Butt1
and Rüdiger Berger1*
It has been known for more than 200 years that the maximum static friction force between two solid surfaces is usually
greater than the kinetic friction force—the force that is required to maintain the relative motion of the surfaces once the
static force has been overcome. But the forces that impede the lateral motion of a drop of liquid on a solid surface are not as
well characterized, and there is a lack of understanding about liquid–solid friction in general. Here, we report that the lateral
adhesion force between a liquid drop and a solid can also be divided into a static and a kinetic regime. This striking analogy
with solid–solid friction is a generic phenomenon that holds for liquids of dierent polarities and surface tensions on smooth,
rough and structured surfaces.
When two solid objects are brought into contact, a
threshold force FTHRD must be overcome in order for
one of the objects to slide1–3. This phenomenon can be
visualized in a typical classroom experiment where a solid block
attached to a spring is pulled over a solid surface (Fig. 1a). The static
force FSis applied to the stationary block and then increased until
it exceeds FTHRD, upon which the block begins to slide. After that, a
lower kinetic force FKIN is required to maintain the block’s motion3.
However, it is not clear whether these forces develop in a comparable
manner when a drop of liquid resting on a solid surface starts to
slide. This gap in our understanding is astonishing, given the fact
that liquid drops are omnipresent in our lives and their motion
is relevant for numerous applications, including microfluidics4,
printing5, condensation6,7, and water collection8,9. Hence insight on
the behaviour of drops that start sliding over solid surfaces is needed.
A sessile drop of liquid is usually in molecular contact with the
supporting solid surface. In contrast, two solid bodies are in direct
contact only at asperities owing to surface roughness10,11. Thus, the
real contact area of a solid–solid contact is much smaller than the
apparent contact area. Consequently, the sliding of drops might be
fundamentally different. However, by simply observing a drop of
water on a pivot window pane, we know that also sessile drops
start sliding when a critical tilt angle is reached—that is, when the
gravitational force acting on the drop overcomes the lateral adhesion
force12. The question may therefore be raised whether a static and
a kinetic regime are also present for sessile drops. The general
question is: How do drops start sliding over solid surfaces and how
do the forces develop while the drops slide?
Owing to higher gravitational forces, larger drops start sliding
at lower tilt angles. Sliding is opposed by capillary forces. They
are associated with a contact angle difference between the rear
and the front of the drop. Indeed, the interactions between solid
surfaces and liquids are described by the liquid–air surface tension
γand the apparent rear and front contact angles of the drop, θRear
and θFront, respectively. Thus, the surface tension, the contact angles
and the drop contact width Ldetermine the lateral adhesion force
FLA by13–17
FLA =kLγ(cosθRear −cosθFront)(1)
The dimensionless factor kaccounts for the precise shape of the
solid–liquid–air three-phase contact line of the drop. Values for k
were calculated to be between 1/2 and π/2 (refs 13,18–20).
Despite the omnipresence of drops, the onset of motion has
never been correlated with the development of lateral adhesion
forces. The lateral adhesion force has been related to external forces
that cause a drop to slide, such as gravitational21,22, centrifugal23,
magnetic24, or capillary forces25–27. The contact angles have also been
experimentally and numerically investigated for the pinned state—
that is, just before and during steady motions28,29. However, once a
drop has started to slide, the lateral adhesion force cannot be tracked
using simple techniques. Astonishingly, it is unclear how the force
develops and how it depends on sliding velocity. We will demon-
strate that, for lateral liquid–solid adhesion, we can distinguish a
static and a kinetic regime, analogous to solid–solid friction.
To measure the lateral adhesion force between a drop of liquid
and a solid substrate, a capillary is positioned in the centre of the
drop. The substrate with the drop is moved sideways against the
capillary at a constant velocity. When the capillary reaches the edge
of the drop, it sticks to the drop. Consequently the motion of the
substrate is accompanied by a deformation of the drop as well as a
deflection of the capillary. Initially, the drop remains pinned to the
substrate (Fig. 1b). Once the capillary exerts a certain critical force,
the drop overcomes the lateral adhesion and is set into translational
motion relative to the substrate—that is, the front and rear side of
the drop start moving. The deflection Dof the capillary is measured
by recording the position of a reflected laser beam with a position-
sensitive detector26. Then, the lateral adhesion force acting on the
drop can be calculated by FLA =κD, where κis the spring constant
of the capillary. Simultaneously, the drop’s shape is monitored by two
cameras, which are synchronized with the force measurement. In
this way the lateral adhesion force can be correlated with the contact
angles. Velocities ranged from 1 µm s−1to 50 mm s−1(see Methods).
As a representative example we start with a drop of an ionic
liquid placed on a fluorinated Si wafer. It forms a contact angle of
approximately 70◦. Moving the wafer laterally increased the force
(blue circles in Fig. 2a) until a maximum force of 50 µN was reached
(after 11 s). This maximum force corresponded to the threshold
force for the drop, upon which it started to slide. Then the force
1Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany. 2Fudan University, 220 Handan Road, Shanghai 200433, China.
†Present address: School of Chemical Engineering & Materials Science, Chung-Ang University, 84 Heukseok-ro, Dongjak-gu, Seoul 06974, Korea.
*e-mail: nann.gao@gmail.com;berger@mpip-mainz.mpg.de
NATURE PHYSICS | VOL 14 | FEBRUARY 2018 | www.nature.com/naturephysics 191
© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.
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The contact angle of a liquid drop moving on a real solid surface depends on the speed and direction of motion of the three-phase contact line. Many experiments have demonstrated that pinning on surface defects, thermal activation and viscous dissipation impact contact line dynamics, but so far efforts have failed to disentangle the role of each of these dissipation channels. Here, we propose a unifying multi-scale approach that provides a single quantitative framework. We use this approach to successfully account for the dynamics measured in a classic dip-coating experiment performed over a unprecedentedly wide range of velocity. We show that the full contact line dynamics up to the liquid film entrainment threshold can be parametrized by the size, amplitude and density of nanometer-scale defects. This leads us to reinterpret the contact angle hysteresis as a dynamical cross-over rather than a depinning transition.
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How Water Advances on Superhydrophobic Surfaces
Article
Full-text available
  • Feb 2016
Superliquid repellency can be achieved by nano- and microstructuring surfaces in such a way that protrusions entrap air underneath the liquid. It is still not known how the three-phase contact line advances on such structured surfaces. In contrast to a smooth surface, where the contact line can advance continuously, on a superliquid-repellent surface, the contact line has to overcome an air gap between protrusions. Here, we apply laser scanning confocal microscopy to get the first microscopic videos of water drops advancing on a superhydrophobic array of micropillars. In contrast to common belief, the liquid surface gradually bends down until it touches the top face of the next micropillars. The apparent advancing contact angle is 180°. On the receding side, pinning to the top faces of the micropillars determines the apparent receding contact angle. Based on these observations, we propose that the apparent receding contact angle should be used for characterizing superliquid-repellent surfaces rather than the apparent advancing contact angle and hysteresis.
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Green-Kubo relation for friction at liquid-solid interfaces
Article
Full-text available
  • Mar 2014
  • PHYS REV E
We have developed a Green-Kubo relation that enables accurate calculations of friction at solid-liquid interfaces directly from equilibrium molecular dynamics (MD) simulations and that provides a pathway to bypass the time-scale limitations of typical nonequilibrium MD simulations. The theory has been validated for a number of different interfaces and it is demonstrated that the liquid-solid slip is an intrinsic property of an interface. Because of the high numerical efficiency of our method, it can be used in the design of interfaces for applications in aqueous environments, such as nano- and microfluidics.
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Energy Dissipation of Moving Drops on Superhydrophobic and Superoleophobic Surfaces
Article
  • Nov 2016
A water drop moving on a superhydrophobic surface or an oil drop moving on a superoleophobic surface dissipates energy by pinning/¬depin¬ning 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 (cosθappr+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 super¬hydro¬phobic array of cylindrical micropillars and a superoleophobic array of stacks of micro¬spheres. For a radius of a protrusion R and a receding materials contact angle Θrwe calculate the energy dissipated per protrusion as WS=πγ R²[A-ln(R/Κ)]f(Θr) . Here, A=0.60 for cylindrical micropillars and 2.9 for stacks of spheres. κ is the capillary length. f(Θr) 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.
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Intermolecular and Surface Forces, 3rd Edition
Book
  • Jun 2011
This reference describes the role of various intermolecular and interparticle forces in determining the properties of simple systems such as gases, liquids and solids, with a special focus on more complex colloidal, polymeric and biological systems. The book provides a thorough foundation in theories and concepts of intermolecular forces, allowing researchers and students to recognize which forces are important in any particular system, as well as how to control these forces. This third edition is expanded into three sections and contains five new chapters over the previous edition.
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Surface and Interfacial Forces
Article
  • Aug 2010
This systematic introduction to the topic includes theoretical concepts to help readers understand and predict surface forces, while also integrating experimental techniques and practical applications with up-to-date examples plus motivating exercises. Starting with intermolecular forces, the authors discuss different surfaces forces, with a major part devoted to surface forces between solid surfaces in liquid media. In addition, they cover surface forces between liquid-vapor interfaces and between liquid-liquid interfaces.
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The Friction and Lubrication of Solids
Article
  • Oct 1951
Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies
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Trilevel-Structured Superhydrophobic Pillar Arrays with Tunable Optical Functions
Article
Water-repelling surfaces inspired by lotus leaves have been developed for their commercial needs in superhydrophobic and self-cleaning coatings on glasses and windows. The extraordinary properties originate from their multiscale structures with waxy materials. To obtain high transparency as well as superhydrophobicity, microhair arrays are designed with large spacing to reduce optical scattering effects caused by microstructures, but with a trilevel hierarchical structure to compensate for the loss of superhydrophobicity. In this study, a soft molding technique on wet pastes consisting of nanoparticles (NPs) is proposed to create a multilevel hierarchical structure of sub-100 nm nanoparticles, which demonstrates excellent water repellency. Additionally, full advantage is taken of the TiO2 NP mesoporous structure for UV protection and for its ability to attach to various kinds of functional (for example, photoresponsive) dyes. Furthermore, the stability of fluorinated surfaces against UV light is enhanced by the passivation of the TiO2 surface with a thin silica coating.
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