Methods of Calculation of a Friction Coefficient: Application to Nanotubes

Abstract

In this work we develop theoretical and numerical methods of calculation of a dynamic friction coefficient. The theoretical method is based on an adiabatic approximation which allows us to express the dynamic friction coefficient in terms of the time integral of the autocorrelation function of the force between both sliding objects. The motion of the objects and the autocorrelation function can be numerically calculated by molecular-dynamics simulations. We have successfully applied these methods to the evaluation of the dynamic friction coefficient of the relative motion of two concentric carbon nanotubes. The dynamic friction coefficient is shown to increase with the temperature.PACS numbers: 61.48.+c, 62.20.Qp Thanks to recent developments in nanotechnology, the hope is high to build mechanical devices on the scale of the nanometer. For this purpose, it is important to determine the mechanical properties and, especially, the friction forces in such nanodevices. In this work, we are going to focus on systems of carbon nanotubes. Our particular interest is for a system that Cumings and Zettl [1] observed experimentally with a TEM. They fixed an edge of a multiwalled nanotube to a surface and opened the other edge, they then extracted inner layers from the core for several nanometers and released them. They observed a full retraction of the inner layers and furthermore, they could conclude that the multiwalled nanotubes are self cleaning since amorphous carbon due to the opening of the edge are inside the tube and also do not have any wear, structural change or fatigue after several extraction and retraction processes. This experiment suggest us that the multiwalled nanotubes can be promising systems for future nanometric mechanical parts such as springs, gears or even motors.A short time after the work of Cumings and Zettl, Zheng and Jiang [3] estimated the frequency of the oscillations in this system to be of the order of GHz. This result is also very interesting since in the macroscopic world moving parts with such frequencies does not exist at the present time. But as in the macroscopic world moving parts have friction forces which hinder the motion and dissipate energy. We hence have to know the importance of these forces before conceiving such devices. Our work will thus focus on, firstly methods of calculation of the friction and secondly an application of these theories to the multiwalled nanotubes.The plan of the paper will be as follows, we are first going to introduce the theoretical framework for the description of the mechanics of nanotubes. We will then solve the classical equations of motion and calculate the dynamic friction coefficient by molecular-dynamics simulation and by the autocorrelation-function method developed by Jarzynski, Berry and Robbins [8,9].We have depicted in Fig. 1 two sliding nanotubes. R is the distance between the centers of mass of each nanotube. The Hamiltonian of the system of two nanotubes is given bywhere T 1 and T 2 are respectively the total kinetic energies of th...

Dynamical properties of granular rotors

“…Apart from the theoretical issues of granular matter, and in the context of Brownian motors, rotational motion (being restricted in space) is of relevance to future applications. This is nicely illustrated, e.g., by recent theoretical and simulation work regarding the rotational and dynamical properties of nanosized systems formed by carbon nanotubes [21,22].…”

Friction of low-dimensional nanomaterial systems

“…For oscillators made from long nanotubes, translational energy is mainly dissipated via a wavy deformation in the outer tube as it undergoes radial vibrations. Servantie and Gaspard [32] developed theoretical and numerical methods to calculate the dynamic friction coefficient based on an adiabatic approximation that uses the temporal integral of the autocorrelation function of the force between both sliding objects. They used these methods to evaluate the kinetic-friction coefficient for the relative motion of two concentric carbon nanotubes and found that it increases with temperature.…”

Trans-phonon effects in ultra-fast nanodevices

“…There has been intense interest [3][4][5][6][7][8][9] in investigating energy dissipation mechanisms in CNT-based ultrafast devices via molecular dynamics simulations. These studies have been motivated by the potentials of the nanoscale devices to serve as basic building blocks in next-generation nano-electromechanical systems (NEMS) [10][11][12] .…”