3.1. Friction Behavior
The instantaneous normal load
L and instantaneous friction force
f are calculated by summing all the normal and lateral interface forces of the substrate atoms acting on the tip atoms during the simulation. To eliminate the high-frequency vibration induced by the stick-slip phenomenon, the
L–d and
f–d curves are smoothed using adjacent-averaging method. The typical instantaneous
L and
f signals are displayed in
Figure 2 as a foundation of the sliding distance
d =
v·t with
v = 100 m/s. For the nanopatterned surface, the
L–d and the
f–d curves appear to be dominated by peaks and troughs with periodic distributions, as shown in
Figure 2a,b. In contrast, the normal load and friction force fluctuate with relative small amplitude and high frequency for the smooth surface, as shown in
Figure 2c,d.
The periodic oscillations of the normal load and friction force for the nanopatterned surface were analyzed using fast Fourier transformations (FFTs), and a sample of the amplitude spectra is shown in
Figure 3. The spectra of the normal load and friction force exhibit well-defined peaks at 1/2.77 nm
−1, indicating a coincident periodic oscillation of the normal load and friction force with a period of 2.77 nm. The oscillation periods are slightly larger than the nanopattern period of 2.17 nm. In addition, the oscillation amplitude of the normal load increases with an increase in the average normal load, as well as the friction force, as shown in the inset in
Figure 3. Moreover, the high frequency signals overlap with the periodic friction force signals, and the oscillation amplitude becomes nonuniform with a decrease in the normal load (
Figure 2b), indicating inconspicuous periodic oscillations of the friction force.
The periodic oscillations of the normal load are easy to understand: the presence of rectangular grooves and protrusion arrays on the surface facilitates the periodic normal load when the separation between the tip and the substrate surface is held constant. The periodic oscillations of the friction force thus appear to be the product of the normal load oscillations. The synchronous oscillations of the normal load and friction force imply a linear friction law, as will be discussed later.
Figure 4 shows the typical
L–d and the
f–d curves for the three nanopatterned surfaces with
lp = 2.17 nm and λ = 25%, 50% and 75%. The three nanopatterned surfaces exhibit similar periodic oscillations between the normal load and friction force with same oscillation periods of 1/2.77 nm
−1 but with different oscillation amplitudes. In this figure, the average normal load of the three nanopatterned surfaces is carefully selected to be nearly equal. Notably, the nanopatterned surface with λ = 25% has a much larger average friction force than the other two nanopatterned surface; however, the friction force does not always decrease with in an increase in the area ratio.
Figure 5 shows the instantaneous normal load and the friction force related to the period width
lp of the nanopatterned surfaces with λ = 50%. The increase in the oscillation period and oscillation amplitude with an increase in
lp is arrestive. The oscillation period increases from 1/2.77 nm
−1 for the nanopatterned surface with
lp = 2.17 nm to 1/4.25 nm
−1 for the surface with
lp = 4.34 nm to 1/6.25 nm
−1 for the surface with
lp = 6.52 nm, indicating the correlation between the nanopattern period and oscillation period. The large nanopattern period results indicate that the tip can barely feel the substrate when it moves into the groove, and it is rapidly captured by the substrate when it moves above the protrusions. In fact, the valleys of the normal load and friction force approximate to zero, as shown in
Figure 5.
Moreover, the velocity effect on the friction behavior was also investigated. In the velocity range of 5–100 m/s, the locations of the peaks and troughs with a periodic distribution overlap for the normal load, as well as the friction force, but with different amplitudes, as shown in
Figure 6. The same periodic oscillations of the normal load and friction force for the nanopatterned surface are observed for nonadhesive contact. Hence, the area ratio, the sliding velocity and the interface adhesion do not influence the oscillation period, but they affect the oscillation amplitude of the normal load and friction force, indicating the different average normal loads and friction forces.
At the macroscale, the micropattern-induced periodic oscillations of the normal load and friction force have been generally reported [
27,
28] and used to understand human tactile perception and to design tactile robotic sensing devices. Similar oscillations induced by different surface topologies have also been observed by atomic force microscopy (AFM) when a very low velocity (0.01–10 mm/s) and relatively sharp tip are used [
20]. In our MD simulations, the high velocity and blunt tip relative to the pattern size are used because of computing power limitations. Our results indicate that these surface pattern-induced friction oscillations can also be found at the nanoscale, even with a very high sliding velocity and a blunt tip.
3.2. Friction Law
To discuss the friction law, the real contact area Areal is defined as the aggregate area of all atoms within contact if the atoms is within the range of 0.19475 nm interaction of any atom of the tip.
Figure 7a shows the average friction force as a function of the real contact area for the nanopatterned surface with
lp = 2.17 nm and λ = 50%. Notably, the friction force f is linear with the real contact area
Areal both in nonadhesive contact and adhesive contact, i.e.,
where
Af0 is a small offset. In nonadhesive contact,
Af0 approximates to zero, i.e.,
f = τ
Areal. In adhesive contact, the disappearance of
Af0 requires the normal load to become negative. The linear
f–
Areal relationship is also valid for the flat contact surface, as shown in
Figure 8a. Our simulation data reveal that the friction force
f depends linearly on the real contact area Areal for both the nanopatterned surface and the flat contact surface in either adhesive contact or in nonadhesive contact at the nanoscale. This linear real contact area dependence of the friction force has been found for H-terminated diamond surfaces [
2] and atomically rough Fe surfaces [
29] at the nanoscale.
Figure 7b shows the average real contact area as a function of the normal load for the nanopatterned surface with
lp = 2.17 nm and λ = 50%. The linear load dependence of the real contact area is found for the nanopatterned surface in both adhesive and nonadhesive contact, i.e.,
where
a is the slope and
AL0 is the real contact area at zero normal load. In nonadhesive contact, the
AL0 is very small, but it becomes very large in adhesive contact. In macroscale Bowden and Tabor roughness theory, the real contact area
∑Aasp is proportional to the normal load
L, i.e.,
∑Aaspµ
L, while the nominal contact area is constant. The linear dependence of
f–
Areal and
Areal–
L indicates that the roughness theories capture the nature of the friction at the nanoscale.
After inspecting the linear dependence of
f–
Areal and
Areal–
L, one can immediately conclude that the friction force varies linearly with the normal load by substituting Equation (2) into Equation (1), i.e.,
where μ* is equal to
τa, and
L0* is equal to (
Af0 −
AL0)/
a. Equation (3) shows the formal Amontons’ friction law, i.e.,
f = μ(
L −
L0), where
μ is the friction coefficient and
L0 is an offset. However, this result cannot show that Amontons’ friction law is valid at the nanoscale for the nanopatterned surface because μ* is not independent of
L0*, as explained by Eder et al. [
29] As shown in
Figure 7c, the nanopatterned surface shows a high but finite friction force at
L = 0 even at negative loads in adhesive contact, while a nearly no friction force at
L = 0 is observed in nonadhesive contact. The friction force in adhesive contact is much greater than that in adhesive contact for the nanopatterned surface. Therefore, the observed linear dependence of
f–L is a result of the synergistic effect between the linear dependence of
f–Areal in Equation (1) and the load dependence of the real contact area in Equation (2). For the nanopatterned surface, the friction law is only the formal Amontons’ friction law, while the significant linear dependence of
f–Areal and
Areal–L captures the general features of nanoscale friction in both adhesive and nonadhesive single-asperity contact.
Regarding the flat contact surface, the linear dependence of
f–Areal is also found for the nonadhesive contact, but the sublinear dependence of
Areal–L is observed for the flat contact surface in adhesive contact, as shown in
Figure 8b. We thus can conclude that the friction force
f varies linearly with normal load
L in nonadhesive contact, but the friction force
f depends sublinearly on the normal load
L in adhesive contact, as shown in
Figure 8c. To quantitatively validate the continuum model in adhesive contact, we fit the simulated data to the Maugis-Dugdale model, a classical adhesive contact model, using an interpolation formula developed by Carpick, Ogletree, and Salmeron (COS) [
30]. The fitted COS transition parameter is equal to 0.71, which corresponds to a Tabor parameter of 1.04. The fitted results bring the friction behavior into correspondence with the continuum model for the flat surface in adhesive contact. Our results confirm that the continuum model breaks down in nonadhesive single-asperity contact but qualitatively holds in adhesive single-asperity contact with a flat or atomically rough contact surface at the nanoscale [
2,
10,
12,
13,
14]; these results agree well with the friction behavior of the H-terminated diamond surface reported by Szlufarska et al. [
2].
The present results demonstrate that an increase in the adhesion force can lead to a transition from linear to nonlinear
f–L dependence for a flat contact surface. In contrast, the linear
f–L dependence is always observed for the nanopatterned surface in both adhesive or nonadhesive contact. Hence, two approaches, decreasing the adhesion force and introducing a nanopattern, can induce a transition from nonlinear to linear
f–L dependence. Because the introduction of a nanopattern on the contact surface can change the adhesion force, one could consider that the linear
f–L dependence for the nanopatterned surface in adhesive contact may be the result of the changing adhesion force. To evaluate this hypothesis, we evaluated the effect of the nanopattern on the adhesion force between the contact surfaces. The adhesion force is directly obtained from the pull-off force by simulating the trace-retrace process of the tip. The resulting adhesion forces are listed in
Table 1. An adhesion reduction is observed only for the nanopatterned surface with λ = 25%. In contrast, a significant increase in adhesion is found for the nanopatterned surfaces with λ = 50% and 75%. As this article shows later, the
f–L relationship for the nanopatterned surface is independent of the area ratio. Hence, the linear
f–L dependence for the nanopatterned surface is attributed to the nature of the nanopattern rather than the resulting adhesion change. The incremental and decremental adhesion force for the nanopatterned surface is caused by two competitive mechanisms, the well-known contact splitting phenomenon and the decreasing real contact area, as reported previously [
31].
The linear dependence of
f–Areal and
Areal–L leads us to conclude that the friction law of the nanopatterned surface in single-asperity contact correlates with the macroscale Bowden and Tabor roughness theory. Roughness theories predict that the normal load depends linearly on the real contact area of all contacting asperities, i.e.,
f = τ
∑Aasp, rather than the nominal contact area. To investigate the roughness of the nanopatterned surface in single-asperity contact, the maps of the real contact area are calculated and shown in
Figure 9. The real contact area is comprised of numerous discrete atoms. The contact atoms become increasingly sparse away from the contact center. For the nanopatterned surface, the real contact surface is divided into several discrete atom islands by the nanopattern, as indicated in
Figure 9c,d. Thus, the real contact area is far less than the nominal contact area, which is defined as the convex hull that encloses all contact atoms, both for the flat surface and nanopatterned surface. This result highlights the atom-scale rough nature of dry contact. The configuration of real contact atoms can be modulated by the nanopattern and results in an ordered corrugation of surface potential energy for the nanopatterned surface. This additional ordered corrugation of surface potential energy enhances the friction of the nanopatterned surface, even though the nanopatterned surface has fewer contact atoms than the flat surface, as shown in
Figure 9.
In adhesive contact, the model of the classical continuum adhesive contact can accurately predict the nanoscale friction behavior of the flat contact surface, as previously mentioned, but this model becomes invalid for the nanopatterned surface. Introducing the nanopattern on the contact surfaces affects the friction in at least two ways. On the one hand, the nanopattern can modulate the adhesive force between the contact surfaces, thus influencing the friction force. This effect on friction is relatively complex because the adhesive force may be enhanced or may be weakened depending the geometrical parameters of the nanopattern. On the other hand, the nanopattern forms an additional geometrical interlock, which enhances the friction force. This interlock effect can be described by Euler’s interlocking asperity model, devised in the 18th century, which predicts a linear
f–L dependence in which the friction coefficient is equal to the local slope. Therefore, the friction force of the nanopatterned surface in adhesive contact can be phenomenologically expressed as:
where
fflat describes the friction force of the flat contact surface,
fadh represents the additional contribution of the adhesion change originating from the nanopattern, and
fgeo is the contribution of the geometrical interlock. If a nanopattern does not exist, then the friction force
fflat is a sublinear function of the normal load in adhesive contact. In addition,
fgeo has a linear dependence on the normal load. Thus, the relationship depends on the relative intensity of
fflat,
fadh and
fgeo. If the geometrical interlock effect dominates the friction behavior or if the nanopattern strongly suppresses the adhesion reduction, then the nanopatterned surface will show a linear load dependence on the friction force; otherwise, the sublinear
f–L relationship will hold.
3.3. Geometry Effect on the Friction
The geometry effect of the nanopatterned surface on the friction law was further investigated.
Figure 10a shows the
f–L dependence as a function of the area ratio λ with a constant nanopattern period
lp of 50%. The linear dependences of
f–L are maintained when the area ratio of the nanopatterned surface changes within the range of 25% to 75%. Both the fitted slope μ* and offset
L0* monotonically decrease with an increase in the area ratio, and thus the three
f–L curves only intersect at one point, as indicated in
Figure 9a. Hence, for a large normal load, the friction force decreases with an increase in the area ratio, but the opposite occurs for a small normal load. This result reveals that although the linear load dependence of the friction force is independent of the geometry parameters, the fitted slope μ* and offset
L0* vary significantly with the geometry parameters of the nanopatterned surface. Moreover, the nanopatterned surfaces with
lp = 50% exhibit greater friction forces than the flat surface at any load.
Figure 10b shows the
f–L dependence as a function of the nanopattern period
lp with a constant area ratio λ of 50%. The linear load dependence of the friction force is valid for all cases. Both the fitted slope μ* and offset
L0* monotonically decrease with an increase in the nanopattern period. However, the decrease in the fitted slope μ* and offset
L0* is very small when the nanopattern period increases from 4.34 nm to 6.52 nm. Interestingly, the desired friction reduction can be obtained when the nanopattern period
lp is larger than 4.34 nm, but significant friction enlargement is observed for the nanopattern with a small
lp of 2.17 nm.
Our simulation data indicates that the linear load dependence of the friction force always holds in the lp range of 2.17 nm to 6.52 nm and λ range of 25% to 75%. The nanopattern can not only enlarge the friction force but can also significantly reduce the friction force. The enlargement and reduction of the friction force critically depends on the nanopattern period rather than the area ratio. The large nanopattern period facilitates the friction reduction.