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: shiomi@photon.t.u-tokyo.ac.jp

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

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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).

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