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Nanoscale Friction Theory and Experiment:

Explore the theory and experimental results of nanoscale friction at the University of Ottawa CAP Congress. Analyze stick-slip motion, force-velocity relationships, and transition rates in AFM interactions.

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Nanoscale Friction Theory and Experiment:

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  1. 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 • FN ~ 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 • FN ~ 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 – xmin (X)] • Approximation of surface potential near inflection point x0: • Barrier height: • Critical tip position:

  4. Stick-slip motion at T = 0 : arbitraryκ • 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: smallκ • Potential energy: U(x, X) = u(x) + κ(x – X)2/ 2 ≈ u(x) – f (X)x + const • Friction force: f (X)≈κX • 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 ) = ΔE0(1 – f / f0)3/2 Escape rate: ω( f ) = ω0exp[–ΔE(f ) /kBT] 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 thenth lattice site at time t: pn(t) • Force in the nth 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 ) Simulation parameters: M.E. and P. Reimann,Phys. Rev. B 87, 205441 (2013)

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

  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: gv(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: taging ~ (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

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