Force dependence of energy barriers in atomic friction and single-molecule force spectroscopy: critique of a critical scaling relation
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-(Delta E(f)/kBT), with Delta E(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., this scaling relation is actually incorrect. Rather, the barrier is a double-valued function of force of the form Delta E(f) alpha (kappa/kappa(c) +/- root 1-f/f(0))(3), where f(0) is the maximal force that the system potential can generate, and the characteristic stiffness kappa(c) is not necessarily much larger than.. In particular, for finite kappa, the barrier vanishes at a certain force f(kappa) < f(0), but, in view of the double-valuedness of Delta E(f), the maximal force f(0) can still be reached. We derive the relation between the most probable force at the moment of transition, f(m), and the pulling velocity, v. The usually assumed scaling f(m) alpha (ln v)(2/3) is recovered as the kappa -> 0 limit of our more general result, but becomes increasingly worse as 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, kappa, 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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