Scale- and load-dependent friction in commensurate sphere-on-flat contacts

Contact of a spherical tip with a flat elastic substrate is simulated with a Green's-function method that includes atomic structure at the interface while capturing elastic deformation in a semi-infinite substrate. The tip and substrate have identical crystal structures with nearest-neighbor spacing $d$ and are aligned in registry. Purely repulsive interactions between surface atoms lead to a local shear strength that is the local pressure times a constant local friction coefficient $\alpha$. The total friction between tip and substrate is calculated as a function of contact radius $a$ and sphere radius $R$, with $a$ up to ${10}^{3}d$ and $R$ up to $4\times {10}^{4}d$. Three regimes are identified depending on the ratio of $a$ to the core width of edge dislocations in the center of the contact. This ratio is proportional to $\alpha a^2/Rd$. In small contacts, all atoms move coherently and the total friction coefficient $\mu =\alpha$. When the contact radius exceeds the core width, a dislocation nucleates at the edge of the contact and rapidly advances to the center where it annihilates. The friction coefficient falls as $\mu \sim \alpha {(\alpha a^2/Rd)}^{-2/3}$. An array of dislocations forms in very large contacts and the friction is determined by the Peierls stress for dislocation motion. The Peierls stress rises with pressure, and $\mu$ rises with increasing load.

Figure 1

(a) We consider an idealized spherical tip sliding on a flat elastic substrate with no adhesive interactions. Motion is in the $x$ direction and the substrate surface is normal to the $z$ axis. In the continuum picture the surfaces are smooth on scales less than the sphere radius $R$. (b) Explicit MD simulations include atomic surface structure (Sec. 2a). A bent, commensurate crystalline lattice most closely mimics the continuum picture. In the image, much of the rigid sphere has been cut away to show the substrate atoms that experience a repulsive force (in darker color) and are displaced downwards to form a curved indentation. (c) In the lateral force model (Sec. 2b), shear forces are applied directly to the substrate atoms with a peak shear stress proportional to the local pressure predicted by Hertz theory. There is no normal force on the substrate and it remains flat. Reference [21] used the same geometry but with a constant maximum shear stress to model adhesive contacts.

Figure 2

Variation of the height of an atom as it moves over a rigid substrate at a fixed normal force of $1.0\epsilon /\sigma$. The height is periodic in the atomic spacing $d=2^{1/6}\sigma$ and the black line indicates the minimum-energy path between successive minima. The lateral force is equal to the slope along this path times the normal force and the local static friction coefficient $\alpha$ equals the maximum slope along this path.

Figure 3

Schematic of (a) rigid slip (regime I), (b) slip by dislocation nucleation (regime II), and (c) slip by dislocation unpinning (regime III). Pictures show cross sections along the sliding direction $x$ at different displacements, with the top snapshot corresponding to the maximum friction. Some atoms are labeled with darker colors to show relative motion. In (a), the rigid asperity slides over the substrate and any lateral displacements in the substrate are too small to alter the registry of atoms on opposing surfaces as the relative position of the tip increases by $d$ from the first to third snapshot. In (b), atoms in the top snapshot are pinned but a small displacement leads to the second snapshot where a dislocation loop nucleates near the edges of the contact. The dislocation has Burgers vector $\mathit{b}=d\widehat{x}$ and line tangent $\widehat{\xi }=\widehat{y}$ into the page at the back of the contact (upside down $T$) and $\widehat{\xi }=-\widehat{y}$ at the front of the contact ($T$). The dislocation glides through the contact and self-annihilates, resulting in slip by a Burgers vector. In (c) the contact is so large that many dislocations become arrested in the contact and further sliding of the asperity is required for them to self-annihilate.

Figure 4

Static friction coefficient for nonadhesive surfaces plotted against the ratio of the contact radius $a$ to the dislocation core width $b_{\text{core}}^0$ in the center of the contact [Eq. (6)]. Results are shown for the explicit atomistic model (open triangles, asterisks, and pluses) and lateral force model (closed triangle, circle, and square) as $a$ increases at the indicated sphere radius ($\sigma =2^{-1/6}d$). The lateral force model only depends on $\alpha /R$, and values of $R$ in the legend correspond to the same $\alpha =0.7$ as the explicit model. The three regimes of different sliding mechanism are indicated at the top of the figure. The friction is constant in regime I, drops as a power law in regime II, and rises in regime III. A black dashed line shows the power law of $-2/3$ expected in regime II. Blue dotted lines show the predictions for Regime III obtained by averaging half the Peierls stress [Eq. (19)] over the contact area (Sec. 3b3).

Figure 5

Friction normalized by maximum static friction $\alpha N$ as a function of lateral tip displacement $\mathrm{\Delta }x$ normalized by nearest-neighbor spacing $d$ for the lateral force model with $R=535d$ and different values of $a/b_{\text{core}}^0$: 0.054 (solid line, $a/d=4$), 1 (dash-dotted line, $a/d=16$), 6 (dashed line, $a/d=43$), and 64 (dotted line, $a/d=138$). In regime I (solid line), atoms move in phase and the total lateral force is proportional to the periodic force on each atom [Eq. (2)]. The static friction coefficient $\mu =\alpha$ and the kinetic friction, corresponding to the time-averaged force, is nearly zero. At $a/b_{\text{core}}^0=1$, the static friction is depressed, sharp drops are becoming evident, and the kinetic friction has increased to about a third of the static friction. In regime II (dashed and dotted lines), the tip begins to advance in discrete stick-slip events associated with dislocation motion. The kinetic friction approaches the static friction, while $\mu$ drops with increasing $a/b_{\text{core}}^0$.

Figure 6

Variation in shear stress for points along the (a) $x$ axis (sliding direction) and (b) $y$ axis (transverse direction) in the lateral force model. Diamonds show the shear stress ${\tau }^b$ (black, open) just before dislocation nucleation ($\mathrm{\Delta }x/d=1.35$ in Fig. 5), and just after nucleation and annihilation, ${\tau }^a$ (red, closed). A dashed line shows the maximum local shear stress ${\tau }_{\text{max}}$, which is $\alpha$ times the Hertz prediction for the local pressure. Stresses are normalized by the peak value at the origin $\alpha p_0$ and distances by the Hertz prediction for the contact radius. The solid line shows the Cattaneo-Mindlin prediction for the shear stress. Here $a=43d$, $R=535d$, and $\alpha =0.7$. This corresponds to $F_z=806G{d}^2$, $b_{\text{core}}^0=6.9d$, and $a/b_{\text{core}}^0=6.23$.

Figure 7

Snapshots showing local shear stress relative to the local maximum shear stress using the color map indicated by the bar at the bottom of the figure. Dislocations are located in regions where stress changes sign over the core width. Each has Burgers vector $\vec{b}=d\widehat{x}$ along the sliding direction and the local orientation $\widehat{\xi }$ varies around the loop. Panels (a) and (b) show the lateral force model near the crossover from regime II to regime III ($a/d=126$ and $a/b_{\text{core}}^0=70$). Panel (a) shows the first dislocation that forms, (b) shows a state just before sliding where a second dislocation has formed. After a small additional displacement the second dislocation annihilates, and the system returns to the configuration of (a). Panel (c) shows an example of the anisotropic dislocation loops that form in the adhesive model of Ref. [21] with $a/b_{\text{core}}=126$ and $b_{\text{core}}=d$. Panel (d) shows a case from regime III of the lateral force model ($a/d=510$ and $a/b_{\text{core}}^0=278$). Two dislocation lines with predominantly screw character (nearly horizontal so that $\widehat{\xi }\parallel \widehat{b}$) have nucleated near the top and bottom of the contact and terminate at the contact edge. As dislocation lines move into the center, they develop into closed loops.

Figure 8

Variation in stress across the contact for atoms along the (a) $x$ axis (sliding direction) and (b) $y$ axis (transverse direction) for the explicit model. Diamonds show the shear stress ${\tau }^b$ just before dislocation nucleation, and squares show the stress ${\tau }^a$ just after nucleation and annihilation. Also shown is the local maximum shear stress $\alpha p$ before and after nucleation, where $p^b$ (triangles) and $p^a$ (asterisks) are measured locally from the force on an atom divided by the atomic area $d^2$. Lines give the Hertz prediction for $\alpha p$ (dashed) and CM prediction for $\tau$ (solid). The Hertz prediction for contact radius and $\alpha p_0$ are used to normalize position and stress, respectively. As in Fig. 6, $F_z=806G{d}^2$ and $R=535d$. The static friction is measured to be $F_x=222G{d}^2$, and $\mu =0.28$.

Figure 9

Variation in shear stress (circles) across the contact for atoms along the (a) $x$ axis (sliding direction) and (b) $y$ axis (transverse direction) for the lateral force model in regime III. Stresses are normalized by $\alpha p_0$ and position by the Hertz contact radius. Vertical solid lines indicate dislocation cores and dashed lines indicate the bounds on the local shear stress, $\pm \alpha p$. Horizontal red bars indicate the mean local stress acting on each dislocation and are determined by averaging the stress an equal distance on both sides of the dislocation. A solid gray curve indicates the calculated Peierls stress for the given pressure, which is larger for screw dislocations.

Figure 10

The Peierls stress for edge, screw and mixed character dislocations in the lateral force model as a function of (a) core width and (b) dislocation character. The character is labeled as the angle between the line direction $\widehat{\xi }$ and the Burgers vector $\mathit{b}=d\widehat{x}$.