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Nonlinear dynamical aspects of atomic scale friction

W. G. Conley, A. Raman, C. M. Krousgrill
Purdue Univerrsity, US

Keywords: atomic scale friction, stick-slip dynamics, friciton force microscopy

Abstract:
This work presents a detailed computational and analytical investigation of Tomlinson's model for atomic scale dry friction [1]. The model (Figure 1) describes a prototypical mechanism of energy dissipation from an atom dragged across a periodic atomic lattice and is often used to model the dynamics of friction force microscope tips [2] and for the atomistic sliding of adsorbate layers on surfaces [3]. While a number of articles in the literature utilize this model [1-13], there is little work in the literature that utilizes the computational and theoretical tools of modern nonlinear dynamical systems for the analysis of this model. This research demonstrates that the use of computational nonlinear dynamics techniques provides a deep insight into the mechanisms of chaotic stick-slip phenomena, the speed dependence of frictional forces and thermodynamics of atomic scale friction. The nonlinear response of the oscillator model is investigated for slow, moderate and high speeds and for different magnitudes of surface potential. The equations of motion are highly nonlinear and time dependent. AUTO [14]- a sophisticated continuation tool for dynamical systems is used to compute the periodic response of the oscillator as a function of translation speed. At very low speeds, the oscillator undergoes stable, periodic "stick-slip" like motions (Figure 2). However at slow to moderate speeds, the response demonstrates superharmonic and harmonic resonances along with a pair of period doubling bifurcations that destabilize the periodic stick-slip response in a certain range of speeds (Figure 3). The origin of the period doubling bifurcations is investigated using perturbation methods and is shown to be related directly to the onset of parametric instabilities in Tomlinson's model. Regions in parameter space (surface potential magnitude and translation speed) are computed where the periodic solutions are destabilized through period doubling bifurcations. The oscillator demonstrates very complex dynamics in such speed ranges including period doubling cascades to chaotic stick-slip dynamics with large vibration amplitudes (Figure 4). The dynamic response of the model is also computed numerically through MATLAB for varying speeds and surface potential magnitudes. The root mean square friction force on the oscillator is computed for varying speeds (Figure 5). It can be clearly seen that the speed ranges at which friction forces are large correspond directly to the regions of harmonic and superharmonic resonances, and parametric instabilities computed earlier. At sufficiently high speeds, the oscillator returns to low amplitude periodic motions resulting in a corresponding decrease in friction force. This explains the transition to the traditional sliding friction regime observed in macroscale experiments. References [1] J. S. Helman and W. Baltensperger, Simple Model for Dry Friction, Phys. Rev. B, 49, 3831 (1994). [2] Y.S. Leng and S. Jiang, Slow Dynamics in Atomic-Force Microscopy, Phys. Rev. B, 63 193406 (2001). [3] G. He and M. Robins, Simulations of Static Friction due to Absorbed Molecules. Phys. Rev. B, 64, 25413 (2001). [4] H. Matsukawa and H. Fukuyama, Theoretical Study of Friction: One Dimensional Clean Surfaces, Phys. Rev. B, 49 17286 (1994). [5] J. Adams, L. Hector, and D. Siegel, Adhesion, Lubrication, and Wear on the Atomic Scale, Surface and Interface Analysis, 31 619 (2001). [6] H. Suda, Origin of Friction Derived from Rupture Mechanics, Langmuir, 17 6045 (2001). [7] L. Wenning, Friction Laws for Elastic Nanoscale Contacts, Europhys. Letters, 54 693 (2001) [8] Y. Sang and M. Dube, Thermal Effects on Atomic Friction, Phys. Rev. Lett., 87, 174301 (2001) [9] B. Li and P. Clapp, MD simulations of Stick-Slip, J. App. Phys., 90, 3090 (2001). [10] W. Zhong and D. Tomanek, First Principles Theory of Atomic Scale Friction, Phys. Rev. Lett., 64, 3054 (1990). [11] E. Gnecco and R. Benewitz, Friction Experiments on the Nanometer Scale, Journal of Physics C, 13, R619 (2001). [12] G. V. Dedkov, Experimental and Theoretical Aspects of the Modern Nanotribolgy, Physica Satus Solidi A, 179, 3 (2001). [13] G. V. Dedkov, Friction on the Nanoscale: New Physical Mechanisms, Materials Letters, 38, 360 (1999). [14] E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede, and X. Wang, AUTO 97: Continuation and Bifurcation software for Ordinary Differential Equations (with HomCont), http://www.cs.concordia.ca/auto, 1998.

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