1
Dynamic Wetting of Nanodroplets on Smooth
and Patterned Graphene-Coated Surface
Shih-Wei Hung, Junichiro Shiomi,*
Department of Mechanical Engineering, The University of Tokyo, 7-3-1 Hongo,
Bunkyo, Tokyo 113-8656, Japan
AUTHOR INFORMATION
Corresponding Author: [email protected]
2
ABSTRACT
Wettability of graphene-coated surface has gained significant attention due to the
practical applications of graphene. In terms of static contact angle, the wettability of
graphene-coated surfaces has been widely investigated by experiments, simulations,
and theories in recent years. However, the studies of dynamic wetting on
graphene-coated surfaces are limited. In the present study, molecular dynamics
simulation is performed to understand the dynamic wetting of water nanodroplet on
graphene-coated surfaces with different underlying substrates, copper, and graphite,
from a nanoscopic point of view. The results show that the spreading on smooth
graphene-coated surfaces can be characterized by the final equilibrium radius of
droplet and inertial time constant. The dynamics of spreading of water nanodroplet on
graphene-coated surface with patterned nanostructures on graphene is also carried out.
Unlike the smooth graphene-coated surfaces, dynamic wetting on the patterned
graphene-coated surfaces depends on the underlying substrate: the inhibition of wetting
by the surface nanostructures is stronger for a hydrophobic underlying substrate than
the hydrophilic substrate.
3
INTRODUCTION
Vast research efforts have targeted on graphene owing to its unique electronic,
optical, chemical and mechanical properties.1–4 The advanced coating technique using
graphene also benefits from its extraordinary properties, e.g. excellent chemical
resistance, impermeability to gases, anti-bacterial properties, and mechanical strength.5
The practical application of graphene coatings requires further understanding of how
the single-atom-thick sheet affects the surface properties of the underlying substrate,
such as wetting. Recently, there have been extensive researches6–19 in quantifying the
wetting behavior of graphene-coated surfaces in terms of static contact angle. An early
work of Rafiee et al.11 found the “wetting transparency” of graphene, i.e. the wetting
behavior of underlying substrate, where van der Waals forces dominate the wetting, is
not significantly altered by the addition of graphene. However, the follow-up
experimental studies demonstrate the graphene is not completely but partially
transparent to wetting.9,10,12,14,15 Computer simulations6,7,12,13,15 and theoretical
calculations6,8,12,15 also support the partial wetting transparency of graphene. Those
researches indicate the monolayer graphene acts like a barrier, reducing the
water-underlying substrate interaction.
The progress in quantifying the static wetting on graphene-coated surface has been
substantial, but the reported study of dynamic wetting on graphene-coated surface is
4
still limited. The dynamic wetting is a multiscale and complex problem as the classical
hydrodynamic description breaks down at the moving contact line, and the nanoscopic
features at the molecular scale needs to be considered.20 Thus, the dynamic wetting
shows different behaviors at different length scales due to the size effect. A recent
study of capillary filling in nanochannels21 demonstrates that the dynamic contact
angle at the nanoscale differs significantly from the static contact angle. While
molecular kinetic theory22 has been formulated to understand the advance of contact
line in terms of kinetics of molecules, the most direct method to investigate the details
of the dynamics is molecular dynamics (MD) simulation, which have been widely used
to study dynamic wetting phenomena, particularly at the nanoscale.23–33 The results of
MD simulations on flat surfaces are generally in agreement with the molecular kinetic
theory.24–26,33 Although there is a limitation in the diameter of the simulated droplet,
typically in the order of tens of nanometers (nanodroplet), MD simulations can trace
the evolution of the droplet spreading, from which scaling of contact radius R can be
directly obtained. When comparing with the macroscopic experiments of initial rapid
wetting on flat surface that are typically studied with millimeter droplets, they exhibit
some similarities in the scaling despite the large difference in the length scale; R
evolves with time, t, as R~t1/2 for completely wetting surfaces23,34,35 in analogy to the
5
scaling of an inertia-dominated coalescence,36,37 and the exponent decreases with
decreasing wettability for partial wetting surfaces.30,31,38,39 The dominant physics that
counteracts capillarity in the initial rapid wetting phase is expected to be mostly inertia
and, for droplets at the microscale, the time histories of R for different droplet sizes
coincide with each other by using the inertial scaling even for partial wetting
surfaces.38 However, the inertial scaling does not capture the dependence on the
wettability. As for the nanodroplets in MD simulations, no scaling for neither diameter
nor wettability has been identified.
On graphene-coated surfaces, Andrews et al.23 studied the nanodroplet spreading
using MD simulation. Unlike the partial wetting-transparency in static wetting, the
dynamic wetting of nanodroplet was found to be independent of the underlying
substrates; the contact radius followed R~t1/2 regardless of the number of graphene
layers or the wetting property of the underlying substrate. The study shed light on the
inertia-dominated dynamic wetting of nanodroplets on graphene-coated surfaces.
However, the scaling that incorporates both dependences on diameter and wettability is
still unknown, and thus we aim to identify such scaling in this work. Furthermore, we
investigate the sensitivity of the dynamic wetting on the surface structure with the
scope of controlling the wetting phenomena through the geometry of the graphene
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layer, following series of works on macroscale droplets where large controllability by
surface structures has been demonstarated.40,41 Here, the surface structure is altered by
forming patterned defects in the graphene, which can be achieved using the ion beam
technique.42,43
SIMULATION DETAILS
MD simulations were performed to study the dynamic wetting of water droplet on a
solid surface down to the nanoscale. Three sizes of water droplet with initial radii, Ri =
6.4, 9.0, and 12.8 nm, corresponding to 25,000, 50,000, and 100,000 water molecules,
were considered to study the size effect. The model surfaces included graphene,
graphite, monolayer graphene coated copper (MGC), bilayer graphene coated copper
(BGC), and copper.
For droplet at the nanoscale, the measured contact angle is affected by the line
tension at the three-phase line.44 To eliminate the line tension effect,45 the
quasi-two-dimensional (2D) system, droplet in cylindrical geometry, was implemented
as shown in Fig. 1. The periodic boundary conditions were applied in all directions.
The system size in the y direction was set to 5.92 nm, leading to an infinitely long
cylindrical droplet. The system size in the x direction was set to 77.00 nm to ensure
that the droplet does not interact with the periodic images. The cylindrical geometry is
7
widely implemented in the studies of dynamic wetting of nanodroplet23,27,31,35 to
achieve sufficiently large droplet sizes in a hydrodynamic regime, which is
computationally expensive in spherical geometry. Besides, the contact problems such
as dynamic wetting are essentially 2D phenomena.35 The previous study31 also
demonstrates that even though the evolution of contact radius is different for droplets
in the spherical and cylindrical geometries, the local processes at the contact line are
independent of the geometry of droplet.
Figure 1. Snapshot of the initial state of dynamic wetting simulation of water droplet
(Ri=12.8nm) on monolayer graphene coated copper (MGC) surface.
The Cu(111) surface was chosen and the graphite surface was represented by six
graphene layers. The separation distance between copper–graphene and
graphene-graphene was 0.326 nm46 and 0.34 nm,44 respectively. The extended simple
8
point charge (SPC/E) model47 was applied for water droplets. The Lennard-Jones (LJ)
potential was implemented for the van der Waals interaction between water and solid
surfaces. The LJ parameters of the interaction between oxygen atom of water molecule
and carbon atom were ε=0.392 kJ/mol and σ=0.319 nm,44 while the LJ parameters
were ε=5.067 kJ/mol and σ=0.272 nm48 for the interaction between oxygen atom of
water molecule and copper atom. The difference between the water contact angles on
rigid and flexible substrates has been shown to be negligible,44 and thus all atoms of
solid surfaces were kept fixed at their equilibrium positions to reduce the
computational cost.
All simulations were performed with the GROMACS 4.6.749 and visualized with the
software PyMOL.50 The simulations were carried out in a canonical ensemble with
integrating time steps of 1.0 fs. The temperature was kept constant at 300 K using the
Nose−Hoover thermostat51,52 with a time constant of 0.2 ps. A cutoff distance of 2.0
nm was used for the nonbonded van der Waals (vdW) interactions. The electrostatic
interaction was treated using particle-mesh Ewald summation method53 with a
real-space cutoff distance of 2.0 nm.
The water droplet was first equilibrated in the system without solid surfaces to
achieve the circular shape. The equilibrated droplet was then placed at a location with
9
a certain distance from the solid surface so that they start to interact. The droplet
spreads on the solid surface spontaneously due to the solid-liquid interaction. To
measure the contact radius of droplet, a square binning of local number of water
molecules along the x and z directions with a bin size of 0.2 nm was performed. The
liquid-gas interface was defined to be the contour of density = 500 kg/m3, which
approximately corresponds to the middle value of bulk liquid and gas densities of
water. The atom positions were recorded every picosecond to evaluate time histories of
the contact radius evolution.
RESULTS AND DISCUSSION
The static contact angle of droplet with Ri=6.4, 9.0, and 12.8 nm on graphene was
calculated to be 89.6±3.5°, 89.9±3.5°, and 89.9±3.4°, respectively. The obtained
contact angle on graphene is consistent with the previous simulation with the same
model; the contact angle on graphene is around 90°.7,17 The static contact angles of
droplet with Ri=12.8 nm are 64.5±3.5°, 78.4±2.5°, and 80.2±2.5° on MGC, BGC, and
graphite, respectively. The copper surface is completely wetting, i.e. contact angle is
zero. The values of static contact angles are in good agreement with the previous MD
study12 using the same force field. The contact angle is increased significantly by the
addition of a monolayer graphene. By adding two layers of graphene, the value of
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contact angle approaches that of graphite, which indicates that the effect from
underlying substrate can be shielded by bilayer graphene.
Figure 2(a) shows how the contact radius of droplets with various initial radii
evolves in time on surfaces with various wettabilities. The late-time spreading shows a
dependence on droplet sizes that larger droplets spread faster than smaller ones. For
surfaces with different wettabilities, the droplets spread faster on more hydrophilic
surfaces. The observation is consistent with the microscale experimental results.38 The
previous studies suggest that the spreading at the nanoscale operates in the
inertial-dominated regime,23,35 thus the inertial scaling is adapted in Figure 2(b). The
characteristic inertia time constant (ρRi3/γ)1/2, where ρ is the liquid density and γ is the
liquid-vapor surface tension, comes from the balance of the inertial pressure inside the
droplet and the capillary pressure associated with the droplet curvature.35 The values
ρ=1000 kg/m3 and γ=72 mN/m are used in this work. The curves of droplets with
different sizes on the same surface collapse onto one curve. However, the curves for
surfaces with different wettabilities scatter, which was also observed in the previous
macroscopic experiments.38 Now, since the final equilibrium radius, Rf, of droplet is
related to the wettability of surface, we scale the radius by Rf instead of Ri, shown in
Fig. 2(c). Note that the final equilibrium radius cannot be obtained for the copper
11
surface, thus the profiles of copper surface are not included in this scaling analysis.
Interestingly, all simulation profiles collapse onto one single curve, which suggests
that the appropriate length scale and time scale to describe the dynamic wetting at the
nanoscale are Rf and (ρRi3/γ)1/2, respectively. We here call this scaling the “modified
inertial scaling”.
12
Figure 2. Time histories of contact radius of droplets with varying sizes on surfaces
with varying wettability (a) dimensional units, (b) inertial scaling, and (c) modified
inertial scaling. The numbers in parentheses in (a) denote the static contact angle of
Ri=12.8 nm droplet.
13
In mind of enhancing the controllability of dynamic wetting, one possible strategy is
to nanostructure the surface by restructuring the coated monolayer graphene. To this
end, patterned holes with various sizes and shapes are formed to understand the effect
of geometries of the surface nanostructures. Note that intentional introduction of such
defects are now possible in experiments using the ion beam technique.42,43 Each
structure was denoted with R(δx, δy) and T(δx, δy), as shown in Fig. 3, where R and T
denote rectangular and triangle and δx and δy are the length of the hole in the x and y
directions integer-multiples of 0.43 nm and 0.37 nm, respectively. For example, R(3, 4)
indicates that a 1.29 nm × 1.48 nm rectangular hole is periodically repeated 20 times in
the x direction and 2 times in the y direction. Two underlying substrate, copper and
graphite, are chosen to study the influence of underlying substrate with different
wettabilities.
14
Figure 3. Top view of the patterned graphene-coated copper surface of (a) rectangular
R(3,4) and (b) triangle T(4,4). δx and δy denote the pattern sizes in the x and y
directions, respectively.
Figure 4 shows the contact radius profiles of droplet with Ri = 12.8 nm spreading on
various patterned graphene-coated surfaces. The results demonstrate the droplets
spread more slowly on the patterned-graphene coated surface than that on the smooth
graphene-coated surface. Comparison between Fig. 4(a) and Fig. 4(b) indicates that the
extent of this wetting inhibition depends on the underlying substrate, where the
inhibition is stronger for the case of graphite underlying substrate (Fig. 4(a)) than the
copper underlying substrate (Fig. 4(b)). The validity of modified inertial scaling is
examined to the patterned graphene-coated surface in Fig. 5. For the graphite
underlying substrate case (Fig. 5(a)), the profiles of different sizes of droplet are also
analyzed. The results show that the profiles do not collapse as well as in the case of
15
smooth graphene-coated surfaces. For the copper underlying substrate case (Fig. 5(b)),
the profiles of patterned graphene-coated surfaces do not collapse and clearly differ
from the profile of smooth graphene-coated surface. It indicates that the effect of the
underlying substrate are enhanced because of the surface nanostructures. The
importance of other effects, for example the line friction force, may be amplified on
the patterned graphene-coated surfaces. Thus, although the modified inertial scaling
works for droplets on the smooth graphene-coated surfaces, it no longer holds on the
patterned graphene-coated surfaces.
16
Figure 4. Time histories of contact radius of droplets on various patterned
graphene-coated surfaces with (a) graphite and (b) copper as underlying substrates.
17
Figure 5. Time histories of contact radius of droplets on various patterned
graphene-coated surfaces using modified inertial scaling with (a) graphite and (b)
copper as underlying substrates.
In order to quantify the effect of pattern size on the dynamic wetting, the spreading
speeds are calculated. The spreading speed is defined as the average contact line speed
over the initial 500 ps (dash line in Fig. 4), i.e., dividing the contact radius at 500 ps by
500 ps. The multiple regression analysis is applied to relate the spreading speed to the
18
size of pattern, δx and δy, as shown in Fig. 6. Note that the results of triangular
patterned cases are not shown in Fig. 6 because the effect of triangular pattern on
dynamic wetting is difficult to separate in the x and y directions, respectively. The
results reveal that the effect of surface nanostructures on dynamic wetting is more
obvious on the graphite underlying substrate case than that on the copper one. The
difference in spreading speed is 18.35% for the graphite underlying substrate case and
it is only 7.77% for the copper one. It implies that the dynamic wetting can be
manipulated more effectively by the design of pattern size for the graphite underlying
substrate than the copper one. Also, it is shown that the spreading speed correlates to
δy, which is perpendicular to the spreading direction, more than to δx, which is parallel
to the spreading direction. The larger values in δy (larger pattern size in the y direction)
induce more resistance for droplet spreading on both underlying substrates. The
behavior can be attributed to the contact line pinning effect of spreading. The pinning
effect has been shown to be important for the spreading on the textured surfaces with
micro-structures.41,54 On nanostructured surfaces, the contact angle hysteresis shows
the dependence on the area density of defects, indicating that the defects pin the
contact line of droplet.55 It was also shown that the macroscopic pinning behavior is
preserved for nanostructures with dimensions down to ~200 nm.56 On the other hand,
19
the contributions of δx to the dynamic wetting are opposite for these two underlying
substrates, i.e. positive coefficient for the graphite underlying substrate case and
negative coefficient for the copper one, as shown in Fig. 6. In order to depict the
phenomena clearly, the top views of spreading of water adjacent to the surface, within
0.5 nm, are displayed in Fig. 7 for these two underlying substrates. For graphite
underlying substrate case, the contact line is pinned on the boundary of hole in both the
x and y directions, as shown from t = 500 ps to 560 ps. The pinning effect in the both
directions can also be observed for copper underlying substrate case from t = 430 ps to
490 ps. When water molecules contact the copper surface (t = 520ps), the spreading is
enhanced because copper is more hydrophilic than graphene. For graphite underlying
substrate case, the larger values in both δx and δy form more resistance for droplet
spreading due to the pinning effect. The slower spreading and higher hydrophobicity
was also observed on the defective graphene surface.8 For copper underlying substrate
case, the larger values in δy slow down the spreading while the larger values in δx
enhance the spreading. Due to the competition between these two effects, the spreading
processes are insensitive to the pattern sizes in the early regime.
20
Figure 6. Spreading speed of droplets as a function of pattern size, δx and δy, on
various patterned graphene-coated surfaces with (a) graphite and (b) copper as
underlying substrates.
21
Figure 7. Top view of spreading of water molecules adjacent to R(7,8) patterned
graphene-coated surface with (a) graphite and (b) copper as underlying substrates.
Coated graphene in green, graphite in gray, and copper in yellow.
22
CONCLUSION
MD simulations are implemented to study the dynamics wetting of water droplet on
graphene-coated surface with different underlying substrates, copper, and graphite. The
influence from underlying substrate on the both static contact angle and dynamic
spreading can be weakened significantly by the addition of one layer of graphene. The
dimensional analysis demonstrates that the dynamic wetting can be characterized by
the inertial time constant and final equilibrium radius for the nanodroplets. For
patterned graphene-coated surfaces, the modified inertial scaling does not hold,
indicating other effects, such as line friction force, become important due to the surface
nanostructures. Unlike the smooth graphene-coated surfaces, the dynamic wetting
shows strong dependence on the underlying substrate. For the graphite underlying
substrate case, the spreading speed decreases and hydrophobicity increases with the
increasing size of holes. The dependence of early-time spreading on pattern size is
weaker for the copper underlying substrate case, because of the competition between
the strong attraction from copper surface and pinning effect. The results indicate that
the manipulation of spreading can be achieved by modifying the nanostructures of
graphene on hydrophobic underlying substrate.
23
Notes
The authors declare no competing financial interests.
AKNOWLEDGEMENTS
This work was financially support by Grant-in-Aid for JSPS Fellows (Grant No.
26-04364). S.-W. H. was financially supported by JSPS Postdoctoral Fellowship for
Overseas Researchers Program (26-04364).
24
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