Friction is often associated with the generation of vibrations in the solids that are sliding over each other. Mathematically, these vibrations are described by the phonons in the two solids, the vibrational eigenmodes. Here, we are confronted with a conceptual difficulty, because the frictional interaction between the solids is strongly localized in the contacting asperities, while the phonons are collective vibrations of the solids as a whole and there is nothing local about phonons. How can these two aspects, the local excitation of the vibrations and the delocalized nature of the phonons, be merged into a single, consistent view?
A research team at the Advanced Research Center for Nanolithography in Amsterdam demonstrated in a recent publication that the answer is simple and elegant. Each time that the solids slip with respect to each other, a simultaneous package of phonons generated – a large number of phonons, each with a different wavelength. By adding up their effects, such a combination of phonons can locally result in large displacements and large velocities, while everywhere else in the solid the phonons interfere destructively and lead to practically zero ‘action’. This phonon wave-packet picture is confirmed by both molecular dynamics simulations and lattice dynamics calculations. The initial situation changes rapidly with time, because the phonons also have different frequencies. As a consequence, the large initial vibrational displacement at the location of the slipping asperity is reduced quickly through the destructive interference of the phonons that run out of sync with each other as a function of time. The resulting motion at that location is close to critically damped. This damped character emerges even if the solids exhibit absolutely zero intrinsic damping, i.e. when they are completely harmonic and each of their phonons has an infinite lifetime. It is this interference-based, near-critical damping that makes the effective dissipation during the slip phase of the motion sufficiently rapid to validate simple stick-slip models, such as those classical descriptions due to Prandtl and Tomlinson.
The video below shows the set up of the model. The movie shows the damped motion, obtained by the numerical integration of the equations of motion of the bcc slab of harmonically interacting atoms. The periodically repeated supercell is viewed along one of the <110>-directions, with the central surface atom colored in red. The red atom has been displaced over a fixed distance along the x-direction and all atoms in the system have relaxed their positions in response, in order to minimize the total energy. The relaxed configuration serves as the starting point of the movie. After the central atom is released from its displaced starting position, the movie demonstrates that the red atom effectively comes to rest within a small number of vibrational periods, even though absolutely no explicit damping is present in the calculation. For the sake of better visibility, all displacements in this movie have been exaggerated by a factor 20 with respect to their actual values.
Further information: Renfeng Hu, Sergey Yu. Krylov, Joost W. M. Frenken, On the Origin of Frictional Energy Dissipation, https://doi.org/10.1007/s11249-019-1247-7, Open Access.
 Renfeng Hu, Sergey Yu. Krylov, Joost W. M. Frenken, On the Origin of Frictional Energy Dissipation, https://doi.org/10.1007/s11249-019-1247-7