Development and assessment of atomistic models for predicting static friction coefficients

The friction coefficient relates friction forces to normal loads and plays a key role in fundamental and applied areas of science and technology. Despite its importance, the relationship between the friction coefficient and the properties of the materials forming a sliding contact is poorly understood. We illustrate how simple relationships regarding the changes in energy that occur during slip can be used to develop a quantitative model relating the friction coefficient to atomic-level features of the contact. The slip event is considered as an activated process and the load dependence of the slip energy barrier is approximated with a Taylor series expansion of the corresponding energies with respect to load. The resulting expression for the load-dependent slip energy barrier is incorporated in the Prandtl-Tomlinson (PT) model and a shear-based model to obtain expressions for friction coefficient. The results indicate that the shear-based model reproduces the static friction coefficients ${\mu }_s$ obtained from first-principles molecular dynamics simulations more accurately than the PT model. The ability of the model to provide atomistic explanations for differences in ${\mu }_s$ amongst different contacts is also illustrated. As a whole, the model is able to account for fundamental atomic-level features of ${\mu }_s$, explain the differences in ${\mu }_s$ for different materials based on their properties, and might be also used in guiding the development of contacts with desired values of ${\mu }_s$.

Figure 1

Structures observed during an FPMD simulation of MgO undergoing slip along the [110](110) slip direction. The leftmost image illustrates the equilibrium structure of MgO. The central image shows MgO subjected to a shear deformation that has caused the top of the system to move a distance $r$ along the slip direction. The rightmost image shows the TS for this slip event. The symbol $h_3$ designates the thickness of the system. Red and pink spheres indicate oxygen and Mg atoms, respectively.

Figure 2

Structures of the systems considered in this work. In all cases, the dashed line designates the slip plane and slip occurs by the atoms above the slip plane moving rightward relative to those beneath the slip plane. (a) BN and graphite. In BN, the sheets contain alternating B (pink) and N (blue) atoms. In graphite, the system consists entirely of carbon. (b) Graphane, FG, and HFG. In all cases, black spheres indicate carbon atoms. In graphane, the white spheres indicate hydrogen atoms, while they indicate fluorine atoms in FG. In HFG, the white spheres below each layer of carbon atoms represent hydrogen atoms, while those above each layer of carbon atoms correspond to fluorine atoms. (c) Si and Ge. The brown spheres indicate Si or Ge atoms as appropriate. (d) ${\mathrm{MoS}}_2$ and ${\mathrm{WS}}_2$. The yellow spheres indicate sulfur atoms and the pink spheres indicate Mo or W as appropriate.

Figure 3

(a) Comparison of values of $\mathrm{\Delta }{E}_{\mathrm{slip}}$ obtained with Eq. (7), $\mathrm{\Delta }{E}_{\mathrm{slip}}^{\mathrm{approx}}$, and GSS-NEB calculations, $\mathrm{\Delta }{E}_{\mathrm{slip}}^{\text{GSS-NEB}}$. (b) Comparison of the deformation energies $\mathrm{\Delta }E$ of MgO as a function of shear distance $r$ obtained with FPMD simulations, the shear model [Eq. (13)], and the PT model with a sinusoidal potential [Eq. (9)], at loads of 0 and 0.9 nN.

Figure 4

(a) Applied force $F$ versus shear distance $r$ for the MgO [110](110) slip system obtained with FPMD simulations, the PT model, and the shear model at loads of 0 and 0.9 nN. (b) Friction forces $F_s$ obtained with the shear and PT models compared against corresponding values obtained through FPMD simulations. Horizontal error bars represent standard deviation in quantities obtained through five independent FPMD simulations, and vertical error bars indicate standard deviations on the slopes obtained through linear least-squares fits to the $F_s$ values predicted by the shear and PT models over the same values of $L$ used in the FPMD simulations.

Figure 5

Friction coefficients ${\mu }_s$ obtained with the shear and PT models compared against corresponding values obtained through FPMD simulations. Horizontal error bars represent standard deviation in quantities obtained through five independent FPMD simulations, and vertical error bars indicate standard deviations on the slopes obtained through linear least-squares fits to the $F_s$ values predicted by the shear and PT models over the same values of $L$ used in the FPMD simulations.