1. Theory of Nanoscale Friction Theory of Nanoscale Friction Mykhaylo Evstigneev CAP Congress University of Ottawa June 14, 2016

2. Macroscopic vs. Nanoscale Friction Results from the interaction of hundreds of asperities on two rough surfaces Steady sliding after overcoming an initial static friction threshold Kinetic friction is approximately independent of the pulling velocity F N ~ N, v ~ m/s Interaction of a single asperity – the tip of an AFM – with an atomically flat surface Stick-slip motion of the AFM tip Friction force increases logarithmically with velocity F N ~ nN, v ~ μm/s

3. Stick-slip motion at T = 0 Potential energy: U(x, X) = u(x) + κ(x – X) 2 / 2, u(x + a) = u(x) Tip base position: X(t) = vt Friction force: f (X) = κ [X – x min (X)] Approximation of surface potential near inflection point x 0 : Barrier height: Critical tip position:

4. Stick-slip motion at T = 0 : arbitrary κ Friction force: Force-dependent barrier height: M.E. and P. Reimann, J. Phys.: Condens. Matt. 27 125004 (2015)

5. Stick-slip motion at T = 0: small κ Potential energy: U(x, X) = u(x) + κ(x – X) 2 / 2 ≈ u(x) – f (X) x + const Friction force: f (X) ≈ κ X Force-dependent barrier height: Y. Sang, M. Dubé, and M. Grant, PRL 87 174301 (2001)

6. Rate theory at small κ Barrier height: ΔE( f ) = ΔE 0 (1 – f / f 0 ) 3/2 Escape rate: ω( f ) = ω 0 exp[–ΔE(f ) /k B T] Elastic force: f (t) = κvt No-jump probability: In the force domain: Most probable slip force: E. Riedo et al., PRL 91, 084502 (2003) Y. Sang, M. Dubé, and M. Grant, PRL 87 174301 (2001); O.K. Dudko, A.E. Filippov, J. Klafter, and M. Urbakh, Chem. Phys. Lett. 352, 499 (2002);

7. Rate equation modeling Probability to find the tip in the n th lattice site at time t : p n (t) Force in the n th stick phase Transition rates Rate equation In the long-time limit In the force domain: Normalization

8. Regimes of stick-slip motion Logarithmic regime Linear response regime Box-like p(f ) Bell-shaped p(f ) M.E. and P. Reimann, Phys. Rev. B 87, 205441 (2013) Simulation parameters:

9. Solution of the rate equation Solution Ansatz: Approximations: –Large : –Small : where M.E. and P. Reimann, Phys. Rev. B 87, 205441 (2013)

10. Analytics vs. Numerics M.E. and P. Reimann, Phys. Rev. B 87, 205441 (2013) κ = 0.1 N/m (black), 0.5 N/m (red), 3 N/m (green), 5 N/m (blue)

11. Slip statistics Rate equation for the no-jump probability Transition rate follows immediately: Experimentally: Better to work with Validity test of the rate equation: g v (f) should be independent of pulling velocity M.E. and P. Reimann, PRE 68, 045103 (2003) π ( f )

12. Experimental results Experimental details: –HOPG –Pressure = 2 x 10 -10 mbar –Transitions in between hexagon centers Collapse of the experimental g -functions onto a single master curve for v > 90 nm/s No collapse for slower pulling speeds indicates contact aging Aging time: t aging ~ (0.25 nm)/(90 nm/s) ~ 3 ms M.E., A. Schirmeisen, L. Jansen, H. Fuchs, and P. Reimann PRL 97, 240601 (2006)

13. Contact aging The model: Rates: Additional assumptions: Fit values: M.E., A. Schirmeisen, L. Jansen, H. Fuchs, and P. Reimann, J. Phys.: Condens. Matt. 20 354001 (2008)

14. Nonmonotonic T- and v-dependent friction The model: Rates: Slow pulling or high temperature state B Fast pulling or low temperature state A Intermediate pulling speed or intermediate temperature anomalous friction behavior:

15. Nonmonotonic T- and v-dependent friction The model: Rates: Symbols: experiment (a): HOPG Jansen et al, PRL 104, 256101 (2010) (b): NaCl Barel et al, PRB 84, 115417 (2011) Lines: simulations M.E. and P.Reimann, PRX 3, 041020 (2013)

16. Conclusions Interstitial jumps of the AFM cantilever tip are thermally activated events, resulting in a logarithmic force-velocity relation Analytical results for the force probability distribution are obtained within the rate theory The data analysis method is introduced, which allows one to check the validity of the rate equation and deduce the transition rates Experimental application of this method to the real system indicated that the rate equation holds only at high pulling speeds Discrepancy at slow pulling is explained as resulting from contact aging A simple contact aging model explains the statistics of the slip events and a non-monotonic temperature and velocity dependence of friction