Contact and friction of nanoasperities: Effects of adsorbed monolayers

Molecular dynamics simulations are used to study contact between a rigid, nonadhesive, and spherical tip with radius of order 30 nm and a flat elastic substrate covered with a fluid monolayer of adsorbed chain molecules. Previous studies of bare surfaces showed that the atomic scale deviations from a sphere that are present on any tip constructed from discrete atoms lead to significant deviations from continuum theory and dramatic variability in friction forces. Introducing an adsorbed monolayer leads to larger deviations from continuum theory but decreases the variations between tips with different atomic structure. Although the film is fluid, it remains in the contact and behaves qualitatively like a thin elastic coating except for certain tips at high loads. Measures of the contact area based on the moments or outer limits of the pressure distribution and on counting contacting atoms are compared. The number of tip atoms making contact during a time interval $\Delta t$ grows as a power of $\Delta t$ when the film is present and as the logarithm of $\Delta t$ for bare surfaces. Friction is measured by displacing the tip at a constant velocity or pulling the tip with a spring. Both static and kinetic friction rise linearly with load at small loads. Transitions in the state of the film lead to nonlinear behavior at large loads. The friction is less clearly correlated with contact area than load.

Figure 1

(Color online) A tip (top) in contact with a substrate (bottom) covered with an adsorbed fluid layer (middle).

Figure 2

Number density $\rho$ of adsorbate atoms vs height $h$ above the top layer of substrate atoms before the tip is brought into contact. Only points with a nonzero value are plotted. The line is a guide for the eye. The height $1.65\sigma$ of the density minimum (arrow) after the first peak is taken as the top of the fluid layer.

Figure 3

(Color online) Normal pressure distribution in the contact zone for different tip geometries: (a) dense; (b) amorphous; (c) incommensurate; (d) commensurate out of registry; (e) commensurate in registry; and (f) stepped. The pressure on each substrate atom is shown by a dot in the panels and the average over all atoms near a given radius is shown by circles. Only the average (triangles) is shown for the pressure on the tip atoms. Solid lines indicate the Hertz prediction, which is the same in all cases. The normal load is relatively high $N/\left(R^2{E}^{\ast }\right)=2.01\times {10}^{-3}$.

Figure 4

Number of monomers per unit area $n\left(r\right)$ as a function of radial distance $r$ from the central axis $\left(r=0\right)$ of the tip for different tip geometries: (a) dense; (b) amorphous; (c) incommensurate; (d) commensurate out of registry; (e) commensurate in registry; and (f) stepped. The normal load $N/\left(R^2{E}^{\ast }\right)=2.01\times {10}^{-3}$ is the same as for Fig. 3 and $n\left(r\right)$ is normalized by the value before contact $n_0=0.794{\sigma }^{-2}$.

Figure 5

(Color online) Atoms of an amorphous tip that contact over a time interval of $50\tau$ $(+$, in red) or $2500\tau$ (○, in blue) for a substrate (a) with or (b) without an adsorbed film at $N=158ϵ/\sigma$. Large dashed circles show the radius $a_o$ determined from the pressure distribution, which is insensitive to the measurement time interval in steady state.

Figure 6

The number of atoms $N_c$ on an amorphous tip that contact the substrate or adsorbed layer over some part of an interval containing $\Delta t/dt$ MD time steps. For both wet (a) and bare (b) contacts, the amorphous tip is used. Other tips produce similar results. For the wet contact, $\text{log}{N}_c\propto \text{log}\Delta t$ at large $\Delta t$, which means that $N_c$ has a power-law dependence on $\Delta t$. The slope of the dashed line in (a) is 0.26. For the bare contact, $N_c$ increases linearly with $\text{log}\Delta t$ at large $\Delta t$. The slope of the dashed line in (b) is 13.3. The inset in (a) shows the autocorrelation function for an atom to remain in contact.

Figure 7

Variation in contact radius $a$ with normal load $N$ for different tip geometries: dense $(\times )$, amorphous (○), incommensurate (◻), commensurate out of registry (△), commensurate in registry (◇), and stepped (▽). Results from Hertz theory (solid lines) and for a bare amorphous tip $(+)$ at the same temperature $\left(T=0.175ϵ/k_{\text{B}}\right)$ are shown for comparison. Three methods are used to measure the contact radius: (a) $a_o$ from the edge of $p\left(r\right)$; (b) $a_s$ from the second moment of $p\left(r\right)$; and (c) $a_c$ from the mean number of contacting atoms at each MD time step.

Figure 8

(Color online) The normal displacement of the tip $\delta$ (filled symbols) and of the substrate ${\delta }_s$ (open symbols) vs load for the indicated tip geometries. The solid line is the Hertz prediction with $E^{\ast }=63ϵ/{\sigma }^3$, $R=100\sigma$, and a correction for finite substrate size [17]. The dotted line shows the uncorrected Hertz prediction and the dashed line is the corrected result with a constant offset.

Figure 9

(Color online) Variation in friction force with time for (a) out-of-registry commensurate and (b) amorphous tips driven with constant tip velocity (solid lines) and through a compliant spring (dashed lines). Since the velocity is $0.01\sigma /\tau$, a time interval of $150\tau$ corresponds to a displacement by the period $d=1.5\sigma$ of the substrate along the direction of motion. The normal load is $N=1267.2ϵ/\sigma$ and the spring compliance is $k_s=40ϵ/{\sigma }^2$.

Figure 10

(Color online) Snapshots showing the location of all film atoms (dots) and those tip atoms that contact the film (open red circles) or substrate (filled blue circles) over an interval of $0.5\tau$ before (left) and after (right) the tip is pulled over the substrate. The tip geometries are out-of-registry commensurate [(a) and (b)]; incommensurate [(c) and (d)]; amorphous [(e) and (f)]; and stepped [(g) and (h)]. The vertical height of the panels is $50\sigma$ and the horizontal dimension is the same for the left panels and $85\sigma$ for the right panels. Large circles show the contact radii determined in Fig. 7, with the largest corresponding to $a_o$ and the smallest to $a_c$. The only place where tip atoms directly contact the substrate is within the inner circle in panel (h).

Figure 11

Static $\left(F_s\right)$ and kinetic $\left(F_k\right)$ friction vs load $\left(N\right)$ for the indicated tip geometries: amorphous (○), incommensurate (◻), out-of-registry commensurate (△), and stepped (▽). The static friction in (a) and the kinetic friction in (b) are measured by dragging the tip along the $x$ direction with a spring at a constant velocity $0.01\sigma /\tau$. The kinetic friction in (c) is measured by displacing the tip uniformly along the $x$ direction at a velocity $0.002\sigma /\tau$. Straight lines are linear fits to the friction. Except in (c), they go through the origin.

Figure 12

Friction on a bare amorphous tip as a function of dimensionless load at $v=0.01\sigma /\tau$ and $k_{\text{B}}T/ϵ={10}^{-4}$. The static friction (open symbols) and kinetic friction (filled symbols) are shown for constant tip velocity (squares) and springs with stiffness $40ϵ/{\sigma }^2$ (downward triangles), $20ϵ/{\sigma }^2$ (upward triangles), and $10ϵ/{\sigma }^2$ (circles).

Figure 13

Static friction from Fig. 11a vs contact radius squared $a_c^2$ for the indicated tip geometries: amorphous (○), incommensurate (◻), out-of-registry commensurate (△), and stepped (▽).