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J Phys Condens Matter. 2015 Apr 1;27(12):125004. doi: 10.1088/0953-8984/27/12/125004. Epub 2015 Feb 19.

Force dependence of energy barriers in atomic friction and single-molecule force spectroscopy: critique of a critical scaling relation.

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1
Department of Physics and Physical Oceanography, Memorial University of Newfoundland, St. John's, NL, A1B 3X7, Canada.

Abstract

Friction force microscopy and single-molecule force spectroscopy are experimental methods to explore multistable energy landscapes by means of a controlled reduction of the energy barriers between adjacent potential minima. This affects the system's interstate transition rates proportional to e(-ΔE(f)/kBT), with ΔE(f) being the barrier height, k(B)T the thermal energy, and f the elastic force applied. It is often assumed that, at large forces, the barrier height scales as (f(c) - f)(3/2), where f(c) is the critical force, at which the barrier vanishes. We show that, for the elastic forces produced by a pulling device of finite stiffness κ, this scaling relation is actually incorrect. Rather, the barrier is a double-valued function of force of the form E(f) ∝ (κ/κ(c) ±√1 − f/f(0))(3), where f(0) is the maximal force that the system potential can generate, and the characteristic stiffness κ(c) is not necessarily much larger than κ. In particular, for finite κ, the barrier vanishes at a certain force f(κ) < f(0), but, in view of the double-valuedness of ΔE(f), the maximal force f0 can still be reached. We derive the relation between the most probable force at the moment of transition, fm, and the pulling velocity, v. The usually assumed scaling f(m) ∝ (ln v)(2/3) is recovered as the κ → 0 limit of our more general result, but becomes increasingly worse as κ grows. We introduce a new data analysis method that allows one to quantitatively characterize the system potential and evaluate the stiffness of the pulling device, κ, which is usually not known beforehand. We demonstrate the feasibility of our method by analyzing the results of a numerical experiment based on the standard Prandtl-Tomlinson model of nanoscale friction.

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