Supplementary Materials
Large-scale molecular dynamics simulation of
wear in diamond-like carbon at the nanoscale
Zhen-Dong Sha1, Viacheslav Sorkin1, Paulo S. Branicio1, Qing-Xiang Pei1, Yong-Wei Zhang1,* ,
and David J. Srolovitz1, 2**
1
2
Institute of High Performance Computing, A*STAR, 138632, Singapore
Departments of Materials Science & Engineering and Mechanical Engineering & Applied
Mechanics, University of Pennsylvania, Philadelphia, PA 19104, USA
Molecular dynamics (MD) simulations were performed using the large-scale
atomic/molecular massively parallel simulator (LAMMPS) code.1 The essential part of MD
simulations is a reliable force field (empirical interatomic potential function). In the present
work, we employed a Tersoff potential with an extended cutoff and van der Waals interactions to
model the non-hydrogenated diamond-like carbon (DLC).2 This potential together with three
other potentials for DLC, the second generation reactive empirical bond order (REBO), the
standard Tersoff, and the Tersoff/ZBL, were evaluated by comparison with experimental data
and first-principle calculations.2 It was shown that the Tersoff potential with the extended cutoff
most successfully reproduced the first-principles and experimental structural and mechanical
data for DLC.2 Hence, in the present work, we employed this potential to study wear
mechanisms and laws.
* E-mail address: zhangyw@ihpc.a-star.edu.sg
** E-mail address: srol@seas.upenn.edu
1
Although the Tersoff potential describes the formation and rupture of chemical bonds, it
does not consider the dispersive forces. In the present work, we integrate the modified Tersoff
potential with the van der Waals (vdW) potential:3,4
E=
1
∑ ∑[EijTersoff + EijvdW ],
2
i
(1)
j≠i
where
𝐸𝑖𝑗𝑣𝑑𝑊
0
3
c
(𝑟
−
𝑟
)
+
c
(𝑟
−
𝑟𝑘 )2 + c1,k (𝑟𝑖𝑗 − 𝑟𝑘 ) + c0,k
𝑘
2,k 𝑖𝑗
= { 3,k 𝑖𝑗
12
6
4ε [(σ/𝑟𝑖𝑗 ) − (σ/𝑟𝑖𝑗 ) ]
𝑟𝑖𝑗 < 2.45Å
2.45Å ≤ 𝑟𝑖𝑗 < 3.82Å , (2)
3.82Å ≤ 𝑟𝑖𝑗 ≤ 10.0Å
and the cn,k are cubic spline coefficients and for C-C interactions ε = 2.84 meV and σ = 3.4 Å.5,6
It should be noted that this composite potential allows for the formation and rupture of chemical
bonds and non-bonded interactions at the contact interface, making it suitable to study wear. One
should note that the present work is fundamentally different from “wearless” nanoscale contacts
studied in previous MD simulations focussed on friction.7,8
In the present MD simulations, the system consisted of a flat non-hydrogenated DLC
sample and a hemispherical non-hydrogenated DLC (asperity) tip at the density of 2.9 g/cm3.
The tip had a height of 3 nm and radius of 4 nm and contained 12,058 atoms. The tip was
prepared by cutting the hemisphere from a bulk DLC sample (Fig. S1). Note that this tip was
similar to that used in recent wear experiments.9 The tip was first relaxed at 0 K using molecular
statics simulations to minimize its total energy. Subsequently the tip was gradually heated to and
equilibrated at 300 K. A 0.5 nm thick layer near the base of the hemispherical tip was rigidly
held during the simulations in order to maintain the prescribed normal load and sliding rate.
2
Similarly, the contact pad (see Fig. S1) was prepared by cutting a slab from a bulk DLC sample,
statically relaxed at 0K, and gradually heating to and equilibrating at 300 K. The pad contained
1,568,360 atoms and dimensions 45 nm × 45 nm × 5 nm. The bottom 1 nm of the DLC contact
pad was held fixed during the simulations. Periodic boundary conditions were applied in the
directions perpendicular to the contact pad surface.
An indentation simulation was first performed at 300 K in order to determine the normal
load (Tn) vs. penetration depth (h) relation. To do so, the tip was driven towards the contact pad
at 1.0 m/s – slow enough to allow the decay of most transients.10 The normal load was calculated
as the sum of the normal forces on the atoms within the top 0.5nm layer of the tip. The indentation
depth was defined as the difference between the lowest point of the tip and the initial, highest
position of the contact pad surface. The obtained normal load (Tn) vs. penetration depth (h)
relation is shown in Fig. S2.
Simulations of the lateral sliding with an externally applied constant normal load were
performed at 300 K. The external normal load was applied to the top 0.5 nm thick rigid layer of the
tip in order to keep the overall force on this layer at zero. Prior to the lateral sliding, the normal load
was applied for 30 ps to allow for the decay of transients. Then a fixed lateral velocity of 20 m/s was
applied to the top 0.5 nm thick rigid layer of the tip, while its normal velocity and position (in the zdirection) were not constrained to ensure a constant normal load. The sliding velocity is much larger
than the typical velocities in scanning force microscopy (SFM) experiments, but comparable to
the operating conditions in some micromechanical systems (MEMS).8 Previous studies
7
have
also demonstrated that the magnitude of the sliding force and the observed law and mechanisms
depend weakly on sliding velocity. In the sliding simulations, we recorded the instantaneous
3
lateral forces. The lateral forces were obtained by averaging the instantaneous lateral forces over
a sliding distance.
We have also performed MD simulations on the loading and unloading processes during
indentation to investigate the plastic (irreversible) deformation of the tip within the normal load
range. In our work, the final distance between the tip and the substrate is about 0.6 Å and -0.5 Å for
the normal load of 4 nN and 104 nN, respectively. When the distance between the tip and the
substrate is -0.5 Å (i.e., the penetration depth reaches around 0.5 Å), we unload the tip at a very slow
velocity, as shown in Fig. S3. We find that after the unloading, the tip is able to fully recover its
original shape, indicating that there is no plastic (irreversible) deformation in the tip.
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4
Fig. S1. Atomistic models of the hemispherical DLC tip and the flat DLC contact pad.
Temperature is held at 300K. The atoms in the top 0.5 nm thick layer of the tip (brown) are
rigidly held/moved and the bottom 1 nm thick layer of the contact pad (dark blue) are rigidly
held. The remaining atoms (yellow and light blue) displace dynamically according to the MD
equations of motion.
5
Fig. S2. The normal load (Tn) vs. penetration depth (h) relation obtained in a typical indentation
simulation at 300 K.
6
Fig. S3: The tip deformation at the normal loads range. The arrows indicate the motion of the tip. For
clarity, the substrate is represented by the dotted line. h is the penetration depth.
7