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Qi2002.pdf
PHYSICAL REVIEW B 66, 085420 共2002兲
Friction anisotropy at Ni„100…Õ„100… interfaces:
Molecular dynamics studies
Yue Qi and Yang-Tse Cheng
GM R&D center, MC: 480-106-224, 30500 Mound Rd, Warren, Michigan 48090-9055
Tahir Çağin and William A. Goddard, III*
Material & Processing Simulation Center, 139-74, California Institute of Technology, Pasadena, California, 91125
共Received 8 March 2002; published 30 August 2002兲
The friction of surfaces moving relative to each other must derive from the atomic interaction at interfaces.
However, recent experiments bring into question the fundamental understanding of this phenomenon. The
analytic theories predict that most perfect clean incommensurate interfaces would produce no static friction,
whereas commensurate aligned surfaces would have very high friction. In contrast recent experiments show
that the static friction coefficient between clean but 45° misoriented Ni共001兲 surfaces is only a factor of 4
smaller than for the aligned surfaces ( ␪ ⬃0°) and clearly does not vanish 共␪ is defined as the rotation angle
between the relative crystallographic orientations of two parallel surfaces兲. To understand this friction anisotropy and the difference between analytic theory and experiment, we carried out a series of nonequilibrium
molecular dynamics simulations at 300 K for sliding of Ni共001兲/Ni共001兲 interfaces under a constant shear
force. Our molecular dynamics calculations on interfaces with the top layer roughed 共and rms roughness of 0.8
Å兲 lead to the static frictional coefficients in good agreement with the corresponding experimental data. On the
other hand, perfect smooth surfaces 共rms roughness of 0 Å兲 lead to a factor of 34 –330 decreasing of static
friction coefficients for misaligned surfaces, a result more consistent with the analytic theories. This shows that
the major source of the discrepancy is that small amounts of roughness dramatically increase the friction on
incommensurate surfaces, so that misaligned directions are comparable to aligned directions.
DOI: 10.1103/PhysRevB.66.085420
PACS number共s兲: 68.35.Af, 71.15.Pd
I. INTRODUCTION
Macroscopic friction follows Amonton’s Law, which
states that the frictional force f needed to slide one object
laterally over another is proportional to the normal load F n ,
f ⫽ ␮ dF n ,
where the constant, ␮ d is the dynamic friction coefficient.1 In
addition, two solid bodies placed together in dry contact exhibit static friction in which no relative motion occurs until a
threshold force is exceeded. The ratio of F c , the force
needed to initiate motion between objects at rest, and the
load F n is defined as the static friction coefficient,
F c⫽ ␮ sF n .
However, the origin of this static friction is not well understood. Analytic theories indicate that static friction vanishes
at most clean, incommensurate crystal interfaces due to the
lack of periodicity, but it is quite large when clean surfaces
are commensurate, when the surfaces deform elastically, and
the interactions between the surfaces are weak.2–5 These analytic models focus on such intrinsic factors as the interactions between constituent atoms, while ignoring such complicating factors as surface roughness, fracture, plastic
deformation, and contaminants.
In a recent study of friction anisotropy at Ni共100兲/Ni共100兲
interfaces Ko and Gellman6 found that the static friction coefficient between two clean Ni共100兲 surfaces has a maximum
of ␮ s ⫽8.6⫾2.5 when aligned and decreases monotonically
to a minimum of ␮ s ⫽2.5⫾1 when the two surfaces are misoriented by 45°. Thus in contradiction with the analytic mod0163-1829/2002/66共8兲/085420共7兲/$20.00
els, they observe a significant static friction at the misoriented interface; however, the reason for this anisotropy was
not clear. This result differs from measurements on mica surfaces, where Hirano et al.7 found that the friction force anisotropy depends strongly on the ‘‘lattice misfit angle.’’
Robbins et al. recently used molecular dynamics 共MD兲
simulations to study the origin of static friction anisotropy,
and proposed that the absorption of a ‘‘third body,’’ such as
small hydrocarbon molecules, can cause the nonvanishing
static friction between two macroscopic objects.8 However,
the model proposed by Robbins et al. cannot explain the experiments at Ni共001兲/Ni共001兲 interfaces, because the experiments were carried out in a highly controlled ultrahigh
vacuum 共UHV兲 environment. These authors were careful to
show that no impurities were present on the surface 关as measured by Auger electron spectroscopy 共AES兲兴 and that the
surfaces were well ordered 共as measured by LEED兲. However, the Ni surfaces were polished mechanically and hence
were not atomically flat.
In order to clarify the issues operating in the Ko and
Gellman6 experiments and to provide a better understanding
of the origin of the friction anisotropy in dry sliding, including the effects of surface roughness, dislocation generation,
and plastic deformation, we performed a series of nonequilibrium molecular dynamics 共NEMD兲 simulations9 for sliding of Ni共001兲 interfaces designed to mimic the experimental
conditions. Section II describes the details of the calculations. We analyze the effect of surface roughness by comparing the differences in static friction coefficients for atomically flat and rough surfaces. These simulation results
and comparison with experimental results are discussed in
Sec. III.
66 085420-1
©2002 The American Physical Society
PHYSICAL REVIEW B 66, 085420 共2002兲
QI, CHENG, ÇAĞIN, AND GODDARD
FIG. 1. Projection along the y direction of the 2D periodic cell
共along the x and y directions兲 for the steady-state nonequilibrium
molecular dynamics simulations. F s is the applied external force on
two moving slabs with 12 layers of atoms, f is the frictional force
during the sliding of two slabs, and 具 F n 典 is the normal load in the z
direction.
II. SIMULATIONS
A. Calculation methods
We used the quantum modified Sutton-Chen 共QMSC兲type many-body force field 共FF兲 to describe the interactions
of Ni atoms. The parameters for this FF 共Ref. 10兲 were determined to match the experimental properties of bulk Ni
共density, cohesive energy, compressibility, elastic constants,
and phonon dispersion兲, including zero-point motion of lattice. This QMSC FF has previously been used to study structural transitions between various phases of Ni, Cu, and other
face-centered-cubic 共fcc兲 metals.11–15
The MD simulations considered finite thickness slabs 共z
direction兲 periodically infinite in the x and y directions.
These ‘‘samples’’ were first prepared separately by equilibrating the upper and lower slabs of Ni for 20 ps at 300 K
共0.001-ps time steps兲 using the Nose-Hoover thermostat with
a relaxation constant of 0.1 ps and fixed volume molecular
dynamics 共ThN MD兲.16,17
The two slabs of Ni were then brought into contact and
equilibrated for another 20 ps using ThN MD.
After equilibrating the sample, external forces were applied to simulate the sliding process. Figure 1 shows the y
projection of the two-dimensional 共2D兲 periodic cell 共x and y
periodic, and z nonperiodic兲 used for the steady-state NEMD
sliding simulations. The z direction is along the 共001兲 axis of
Ni while the x and y directions were based on the orientations of the sliding experiments. All models considered 14
layers of 共001兲 planes in each slab. At each time step, an
external force of F s , was applied along ⫹x direction for the
top N r ⫽12 layers of atoms 共termed a moving slab兲 and along
the ⫺x direction for the bottom N r ⫽12 layers of atoms 共a
moving slab兲. The top layer in the top slab and the bottom
layer in the bottom slab were constrained not to move in the
z direction. This allowed us to simulate the sample under
compression, keeping fixed the length of the sample along
the z direction. The interface zone, consisting of two layers
of atoms on each slab, was fully flexible and allowed to
move freely 共no external forces, no constraints, and no thermal damping兲.
The moving slabs were thermostated to a fixed temperature T⫽300 K 共isokinetic energy兲. The atoms of the interface
zone were subject to frictional heating and allowed to exchange energy with the rest of the slab through lattice vibrations. The averaged normal force per atom, 具 F n 典 , was calculated from the total compressive stress of the system
averaged over the simulation time times the contact area then
divided by the number of atoms. The average of the total
lateral force on the top rigid slab per atom was calculated as
具 f x 典 , which was summed over all atoms of the top rigid slab
and averaged over time and the number of atoms. This force
is equal and opposite to the lateral force on the bottom rigid
slab.
We increased the external force F s until the two slabs
started to slide with respect to each other. The minimum
force needed to initiate motion is defined as F c , and the
static friction coefficient is defined as
␮ s ⫽F c / 具 F n 典 ,
共1兲
where 具 F n 典 is the averaged normal load.
After the two slabs start to move, the average atomic net
forces 具 f x 典 in the upper and lower slabs differ from the applied force. This difference is caused by the frictional force f
at the interface. This frictional force is given by
f ⫽F s ⫺ 具 f x 典 .
共2兲
The average acceleration in the x direction of each atom,
caused by the net force 具 f x 典 is given by a⫽ 具 f x 典 /m, where m
is the atomic mass. From the frictional force, the dynamic
friction coefficient is calculated from Eq. 共3兲,
␮ d⫽ f / 具 F n典 .
共3兲
The unit of force per atom is (kJ/mol)/nm⫽1.6604
⫻10⫺12 N in this paper.
B. Orientations and mismatch angels
We examined the effect of orientation on both the dynamic and static coefficient of friction for three orientations
of the two surfaces shown in Fig. 2:
• ␪ ⫽0° case. In this case both surfaces are aligned. The
direction of sliding is taken as the 具110典 direction for both
slabs, the same as in the experiment. For each slab, the simulation cell size is 7&a⫻7&a⫻14a along the x, y, and z
directions, respectively, while the total number of atoms N in
the system is 5488 per periodic cell. This system is expected
to have a very high coefficient of friction, particularly for the
static friction of the perfectly smooth surface.
• ␪ ⫽45° case, where the 具110典 axis of the lower surface
matches the 具100典 direction of the top surface. In this case,
085420-2
PHYSICAL REVIEW B 66, 085420 共2002兲
FRICTION ANISOTROPY AT Ni共100兲/共100兲 . . .
• The perfect interface, where each surface is atomically flat.
When the surfaces are aligned ( ␪ ⫽0°), the sliding corresponds to slip inside a single crystal. This surface has an rms
roughness of 0 Å.
• The random rough interface, where 25% of the atoms in the
interface zone are randomly deleted from each surface 共thus
for the ␪ ⫽0° case, we eliminated 49 of the 196 atoms in
the top layer of each surface兲. This surface has an rms roughness of 0.8 Å.
This paper considers only random rough interface models
in detail, and we will include different rough surface configurations in the future studies.
III. RESULTS AND DISCUSSION
We studied smooth and random rough surfaces for ␪
⫽0°, 45°, and 30° as summarized in Table I. The applied
force F s , net atomic force 具 f x 典 , friction force f, and normal
force 具 f n 典 were defined as in Sec. II A. The relative displacement of the center of mass of each slab was tracked during
the MD simulations and used to calculate the center of mass
velocity and acceleration along the x direction.
FIG. 2. 共a兲 Schematic diagram showing the lattice mismatch of
Ni共100兲 interfaces. 共b兲 The z projection of the simulation cell. The
dark circles are atoms of upper slab and light circles are atoms of
lower slab. ␪ is the lattice misoriented angle between the upper and
lower slabs. The sliding direction is along the x direction of the
simulation cell.
the direction of sliding is taken as the 具100典 direction of the
upper slab and 具110典 for the lower slab. Since periodic
boundary conditions are applied along the x and y directions,
the sizes for both slabs need to be equal along these two
directions. To minimize the lattice mismatch, we choose
10a⫻10a⫻7a for the lower slab, and 7 冑2a⫻7 冑2a⫻7a
for the upper slab, and then stretch the upper slab by 1%
共tension strain兲 to reach 10a to match the periodicity in the
lower slab. The total number of atoms in this system is 5544
per periodic cell.
• ␪ ⫽30° case 共actually ␪ ⫽29.7°兲. In this case, the sliding
direction is the 具1 1 0典 orientation for the lower slab and the
具 3̄ 11 0 典 orientation for the upper slab. To minimize lattice
mismatch, we choose 4&a⫻4&a⫻7a for the lower slab
and 冑130a/2⫻ 冑130a/2⫻7a for the upper slab, and then
compress the upper slab by 0.7% 共compression strain兲 to
reach the same length as the lower slab. The number of atoms in the simulation cell is 1806 per periodic cell. We also
doubled the cell length in both directions, such that the simulation cell is 7224 atoms, and the cell size did not change our
results.
To obtain the same sliding conditions as used in the experiment, the sliding direction is always along the x direction
and the lattice misorientation angle is kept constant during
each sliding simulation.
C. Surface roughness
We constructed two surface structures, including flat and
rough surfaces, for each orientation.
A. Perfect interfaces „␪ Ä0° and ␪ Ä45°…
Figure 3 shows the relative displacement in the x direction
between upper and lower slabs for the Ni共001兲/Ni共001兲
atomically flat interface under a constant external force F s .
Figure 3共a兲 is for the perfect alignment case ( ␪ ⫽0°), and
Fig. 3共b兲 is for the ␪ ⫽45° misorientation case.
For ␪ ⫽0° we observe oscillatory motion of two slabs for
F s ⭐15, indicating that the two slabs are not sliding with
respect to each other. But for F s ⭓20.17 the slabs do slide.
Therefore 15⬍F c ⭐20.17.
For the ␪ ⫽45° misorientation case there is no such oscillation for F s ⬍0.255, indicating that 0.05⬍F c ⭐0.255.
Clearly, for a perfect surface the friction at the misoriented
interface is only ⬃1% of that for the aligned interface ( ␪
⫽0°).
Figure 4 shows the snapshots of the atomic structure of
the corresponding interfaces whose sliding behaviors are
plotted in Fig. 3. Figure 4共a兲 is for the perfect alignment ( ␪
⫽0°) case with an applied force of F s ⫽20.17 after 5 ps and
Fig. 4共b兲 is for the ␪ ⫽45° misorientation case with F s
⫽0.255 after 10 ps. We note here that the misoriented surface has essentially no damage, whereas the aligned surface
already exhibits some damage. These results for the perfect
surfaces are consistent with the expectations of the analytic
theories.
B. Rough interfaces „␪ Ä0° and ␪ Ä45°…
Figure 5 shows the relative displacement in the x direction
between the upper and the lower slabs for the Ni共001兲/
Ni共001兲 random rough interface. Figure 5共a兲 for perfect
alignment ( ␪ ⫽0°) shows oscillatory motion for F s ⫽10 or
lower, but sliding for F s ⫽12.6 and higher, indicating that
10⬍F c ⭐12.6. For the ␪ ⫽45° misorientation case the range
of the critical force is 0.5⬍F c ⭐2.2. Thus for the rough surface the ratio of the critical force of misoriented versus
085420-3
PHYSICAL REVIEW B 66, 085420 共2002兲
QI, CHENG, ÇAĞIN, AND GODDARD
TABLE I. Summary of results from NEMD simulations of friction at Ni共100兲/Ni共100兲 interfaces. 共The
unit for force is kJ/mol⫽1.66⫻10⫺12 N the unit for velocity is Å/ps⫽100 m/s, and the unit for acceleration
is Å/ps2 ⫽1014 m/s2 .兲
Orientation
␪ ⫽0°; flat
␪ ⫽0°; rough
␪ ⫽30°; flat
␪ ⫽30°; rough
␪ ⫽45°; flat
␪ ⫽45°; rough
具fx典
Friction
f ⫽F s ⫺ 具 f x 典
具fn典
obtained
from stress
5.04
15.13
20.17
0
0.07
3.035
15.13
17.14
0
0.55
2.95
25.2
7.59
17.61
6.36
5.04
10.08
12.6
15.13
0
0
0.62
3.32
5.04
10.08
11.98
11.81
0.52
0.89
1.43
2.71
25.2
9.2
16
3.97
Applied
Fs
Velocity or
acceleration
v ⫽0
v ⫽0
a⫽0.09
2 具 f x 典 /m⫽0.10
a⫽0.246
2 具 f x 典 /m⫽0.258
v ⫽0
v ⫽0
v ⫽0.1
a⫽0.12
2 具 f x 典 /m⫽0.11
a⫽0.35
2 具 f x 典 /m⫽0.31
v ⫽0
0.029
0.057
0.144
0.288
1.44
2.88
4.32
5.76
0
0.006
0.104
0.105
0
0
0.15
0.26
0.029
0.051
0.04
0.183
1.44
2.88
4.17
5.5
0.65
0.77
0.05
0.255
0.509
0
0.130
0.273
0.05
0.125
0.236
1.20
1.19
1.21
2.55
1.31
1.24
1.23
5.09
2.96
2.13
1.22
0.509
2.55
5.09
0
0.06
0.75
0.509
2.49
4.34
1.28
1.24
1.52
2.55
2.7
2.85
v ⫽0
v ⫽0
aligned surfaces is ⬃17.8%, compared to ⬃1.26% for the
perfect flat surfaces. Clearly surface roughness dramatically
increases 共by a factor of 14兲 the ratio of the critical force for
the ␪ ⫽45° misorientation case and that for the ␪ ⫽0° case.
Figure 6 shows snapshots of the atomic structure of the
rough interfaces whose sliding behaviors are plotted in Fig.
5. Figure 6共a兲 is the structure for perfect alignment ( ␪
⫽0°) with F s ⫽12.6 after 6 ps and Fig. 6共b兲 is the snapshot
for ␪ ⫽45° misorientation with an applied force of F s
⫽2.25 after 10 ps. In both cases, the rough interface leads to
disordering. For the perfect alignment case the plastic deformation and disordering generated at the interface propagates
into the bulk of each slab however for the ␪ ⫽45° misoriented interface the plastic deformation was more localized on
the surface layers.
v ⫽0
v ⫽0.082
a⫽0.009
2 具 f x 典 /m⫽0.009
a⫽0.0487
2 具 f x 典 /m⫽0.045
a⫽0.111
2 具 f x 典 /m⫽0.101
v ⫽0
v ⫽0.25 m/s
a⫽0.025
2 具 f x 典 /m⫽0.026
Friction
coefficients
␮ s or ␮ d
␮ s⬵6.84
␮ d⫽2.77
␮ s⫽8.80
␮ d⫽4.35
␮ d⫽4.03
␮ s⫽0.022
␮ d⫽0.015
␮ d⫽0.065
␮ s⫽6.65
␮ d⫽7.15
␮ s⫽0.21
␮ d⫽0.19
␮ d⫽1.01
␮ d⫽1.75
␮ s⫽2.06
␮ d⫽2.85
C. Deformation at commensurate interfaces
In the perfect alignment case ( ␪ ⫽0°), we observed that
the two slabs of materials collapse into one when brought to
a spacing below a critical distance. This ‘‘adhesion avalanche’’ phenomenon was previously predicted for Ni共001兲
using the equivalent crystal method.18
For the perfect alignment case, sliding is equivalent to
shearing of a perfect crystal, leading to very high ‘‘friction.’’
Experiments conducted under similar conditions lead to cold
welding. That is the shearing force was so large that it approached the upper limit of the tribometer. Thus for the case
of perfect alignment ( ␪ ⫽0°), sliding requires generating
dislocations and other defects to accommodate plastic deformation at the interface. This results in a rough interface 关Fig.
085420-4
PHYSICAL REVIEW B 66, 085420 共2002兲
FRICTION ANISOTROPY AT Ni共100兲/共100兲 . . .
FIG. 3. Relative displacement in the x direction between the
upper and lower slabs for Ni共100兲/Ni共100兲 perfect flat interfaces. 共a兲
The ␪ ⫽0° aligned case, F c ⫽20.17; 共b兲 the ␪ ⫽45° misorientation
interface, F c ⫽0.255.
4共a兲兴. In contrast, for the misaligned case ( ␪ ⫽45°) the interface remains atomically flat during sliding with no dislocation generation observed.
We found that for perfect alignment ( ␪ ⫽0°), both flat
and rough surfaces lead to pronounced oscillatory motion
between the slabs when the external force is below F c . For
FIG. 4. Snapshots of the atomic structure of the Ni共100兲/Ni共100兲
interfaces after sliding. 共a兲 perfectly flat with perfect alignment ( ␪
⫽0°) after an applied force of F s ⫽20.17 for 5 ps; 共b兲 perfectly flat
with ␪ ⫽45° misorientation after an applied force of F s ⫽0.255 for
10 ps.
FIG. 5. Relative displacements in the x direction between the
upper and lower slabs for Ni共100兲/Ni共100兲 rough interface 共rms
roughness of 0.8 Å兲. 共a兲 for ␪ ⫽0° aligned case, F c ⫽12.6 and 共b兲
for ␪ ⫽45° misorientation interface, F c ⫽2.25.
the misoriented ␪ ⫽45° case we also observe oscillatory motion but since F c is quite small the oscillations are not large.
As the applied force increases to the critical force F c , we
observe a barrier at the beginning of sliding 共similar to the
first peak in the oscillation兲 for the aligned ␪ ⫽0° case but
not in the misoriented ␪ ⫽45° case. This indicates that sliding is intermittent rather than smooth. This phenomena is
similar to the experimental observations of ‘‘stick-slip’’ behavior in ␪ ⫽0° sliding, but only slip motion in ␪ ⫽45°
sliding.6
FIG. 6. Snapshots of the atomic structure of the Ni共100兲/Ni共100兲
rough interfaces after sliding. 共a兲 perfect alignment ( ␪ ⫽0°) after an
applied force of F s ⫽12.6 for 6 ps and 共b兲 ␪ ⫽45° misorientation
after an applied force of F s ⫽2.25 for 10 ps.
085420-5
PHYSICAL REVIEW B 66, 085420 共2002兲
QI, CHENG, ÇAĞIN, AND GODDARD
The barrier in sliding for the aligned ␪ ⫽0° interface suggests the existence of energy minima or ‘‘lock positions’’
leading to the stick-slip motion. Even for the rough interface
the underlying periodic lattices of the upper and lower slabs
are commensurate, leading to multiple minima. For the misoriented ␪ ⫽45° interface, the lack of commensurateness between the upper and lower slabs 共except that of the supercell兲
works against the presences of such a lock position so that
slip motion is generally observed.
To understand the process of stick slip, we analyzed the
atomic trajectories in detail. For forces less than the critical
force 关F s ⫽15.13 and F s ⫽5.04 in Fig. 3共a兲兴 and before the
displacement reaches the peak 共up to 1.6 ps兲, the whole upper slab of Ni is elastically sheared with respect to the lower
slab 共uniform displacement with time兲. This is followed by a
decrease in displacement 共1.6 –3.2 ps兲 due to the release of
the accumulated elastic shear strain. When the applied force
is larger than the critical force 关F s ⫽20.17 for Fig. 3共a兲兴, only
pure elastic deformation is observed before the first peak
(0⬃1.6 ps), then plastic deformation starts to occur at the
interface 共1.6 –2.0 ps兲, which then leads to a continuous increase in the relative displacement 共2–10 ps兲. This demonstrates that plastic deformation is necessary for sliding,
whereas elastic deformation is responsible for the ‘‘stick’’
motion. Li et al. showed that elastic deformation of the surface layers is the main cause for the stick-slip phenomenon
in Ni-Al alloy.19 They also observed the ‘‘stick-slip’’ phenomena at incommensurate interfaces. In contrast, we observe only ‘‘slip’’ at incommensurate interfaces. However,
our mechanism for the stick-slip behavior is consistent with
their observations.
For the perfectly aligned case, we estimated the critical
shear stress from the critical force for sliding. This leads to a
critical stress between 4.875 GPa 共for F s ⫽15.13兲 and 6.5
GPa 共for F s ⫽20.17兲. The critical stress for sliding is related
to the theoretical shear strength of Ni. Using the experimental tetragonal shear modulus for Ni 共Ref. 20兲 of ␥ ⫽35 GPa
共the calculated value from our force field is 49.6 GPa兲, leads
to a critical shear between ␥/5.4 to ␥/7, which is close to the
theoretical strength predicted by Frenkel’s model.21
The relative displacement of two slabs in perfectly
aligned cases oscillated at a frequency. We found out that the
oscillation frequency does not depend on the magnitude of
the applied force, the number of layers with F s being applied, nor the length of the simulation cell along x or y directions. However, if the total length of slabs along the z
direction is doubled, the frequency doubles. This indicates
that the frequency is related to the time of sound velocity
traveled along the z direction.
D. Comparison with experimental results
The calculated values of the static friction coefficient are
compared with the experimental observations of Ko and
Gellman in Fig. 7. The MD simulation results of both static
friction coefficients and their anisotropy behavior as a function of misorientation angle for rough interfaces agree well
with the experimental. Therefore we believe that surface
FIG. 7. Static friction coefficients as a function of the lattice
misorientation angle between two Ni共100兲 surfaces. The experimental values measured by Gellman and Ko are shown with solid diamonds, where the points with upward arrows represent the lower
limit from the experiment. The solid square symbols indicate simulation results for rough interfaces 共0.8-Å rms兲. These values are in
excellent agreement with experiment. The solid triangles static indicate simulation results for atomically flat interfaces.
roughness and interface disordering are the main causes for
the experimentally observed anisotropic behavior of at the
Ni共001兲/Ni共001兲 interfaces.
For the MD calculations with rough interfaces, the static
frictional coefficients are
␮ s ⫽8.8 for ␪ ⫽0°, ␮ s ⫽6.65 for ␪ ⫽30°,
and ␮ s ⫽2.06 for ␪ ⫽45°.
The corresponding experimental data were
␮ s ⫽8.6⫾2.5 for ␪ ⫽0°, ␮ s ⫽5.5⫾2 for ␪ ⫽30°,
and ␮ s ⫽2.5⫾1.0 for ␪ ⫽45.
Thus the simulation results for random rough interface agree
very well with the experimental data.
In contrast, MD simulations with atomically flat interfaces
lead to static frictional coefficients of
␮ s ⬃6.8 for ␪ ⫽0°, ␮ s ⬃0.02 for ␪ ⫽30°,
and ␮ s ⬃0.21 for ␪ ⫽45°.
For the aligned case ␮ is comparable to the values for the
rough surface obtained with MD and with the value from
experiment. However, the static friction coefficients on perfect misoriented interfaces 共for both ␪ ⫽30° and ␪ ⫽45°兲 are
much lower: a factor of 10 for ␪ ⫽45° and a factor of 330 for
␪ ⫽30°. This shows a strong dependence on the misorientation angle ␪. This anisotropic behavior agrees well with the
analytic theories,2–5 which conclude that there is no static
friction on most clean incommensurate interfaces. Because
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FRICTION ANISOTROPY AT Ni共100兲/共100兲 . . .
we used a supercell with periodic boundary condition in our
simulation, there is a long-range ordering of the interface
which could account for the small but nonzero static friction
for the 30° and 45° cases, and for the dramatic differences
between these two cases.
The MD simulations show that roughness in only the top
layer or two of the surface is necessary to account for the
experimental observation that the static friction coefficients
varies by less than a factor of 4 for various orientations. The
similar values for perfect and rough aligned surfaces arise
because even just 5 ps of MD is sufficient to roughen the
surface 关see Fig. 4共a兲兴. We observed that only ⬃1 Å of
roughness is sufficient to lead to the observed anisotropy in
roughness. Thus to achieve a large ratio in friction anisotropy
共say a factor of 10–100兲 would probably require a roughness
of less than 1 Å, which would be very difficult to achieve
experimentally. Both theory and experimental agree that
there is a minimum static friction coefficient occurs for ␪
⫽45°.
IV. SUMMARY
We describe an approach for using molecular dynamics
simulations to elucidate the phenomena resulting in friction
*Author to whom correspondence should be addressed. Email address: [email protected]
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of Ni共001兲/Ni共001兲 interfaces.
ACKNOWLEDGMENTS
We want to thank Professor Andy Gellman for alerting us
to his work prior to publication. This work was carried out at
both GM and the Materials and Processing Simulation Center 共MSC兲 at Caltech. The facilities of the MSC are supported by grants from DOE-ASCI-ASAP, NSF
共CHE9985574兲, ARO-MURI, Chevron, 3M, Seiko Epson,
GM, Avery Dennison, Beckman Institute, Asahi Chemical,
and Nippon Steel.
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