Energy dissipation of atomic-scale friction based on one-dimensional Prandtl-Tomlinson model

Friction 3(2): 170–182 (2015) DOI 10.1007/s40544-015-0086-2 ISSN 2223-7690 CN 10-1237/TH RESEARCH ARTICLE Energy dissipation of atomic-scale friction based on onedimensional Prandtl–Tomlinson model Zi-Jian WANG1, Tian-Bao MA1,*, Yuan-Zhong HU1, Liang XU2, Hui WANG1 1 State Key Laboratory of Tribology, Tsinghua University, Beijing 100084, China 2 Beijing Institute of Nanoenergy and Nanosystems, Chinese Academy of Sciences, Beijing 100083, China Received: 27 March 2015 / Revised: 16 May 2015 / Accepted: 01 June 2015 © The author(s) 2015. This article is published with open access at Springerlink.com Abstract: The energy transition and dissipation of atomic-scale friction are investigated using the one-dimensional Prandtl–Tomlinson model. A systematic study of the factors influencing the energy dissipation is conducted, indicating that the energy that accumulated during the stick stage does not always dissipate completely during stick-slip motion. We adopt the energy-dissipation ratio (EDR) to describe the relationship between the energy dissipated permanently in the system and the conservative reversible energy that can be reintroduced to the driving system after the slip process. The EDR can change continuously from 100% to 0, covering the stick-slip, intermediate, and smooth-sliding regimes, depending on various factors such as the stiffness, potential-energy corrugation, damping coefficient, sliding velocity, and the temperature of the system. Among these, the parameter η, which depends on both the surface potential and the lateral stiffness, is proven in this paper to have the most significant impact on the EDR. According to η–T phase diagrams of the EDR, the smooth-sliding superlubricity and thermolubricity are found to be unified with regard to the energy dissipation and transition. An analytical formulation for the EDR that can be used to quantitatively predict the amount of energy dissipation is derived from a lateral-force curve. Keywords: atomic-scale friction; energy reversibility; energy dissipation ratio; superlubricity 1 Introduction Friction is mechanical kinetic-energy loss or the transformation of sliding motion into heat and other excitations [1–3]. Although atomic-scale frictional behavior and its influencing factors have been studied extensively, the process and amount of energy transfer and dissipation during stick-slip friction—which are essential for an in-depth understanding of atomic-scale friction—are rarely quantified. The elastic energy that accumulates during the stick stage can be dissipated irreversibly by heat generation during the slip stage [4–6]. Berman and Israelachvili postulated that in the cobblestone model, upon each molecular collision during sliding, only part of the kinetic energy is *Corresponding author: Tian-Bao MA. E-mail: dissipated, and the rest is reflected back to the system [7]. There have been experimental studies wherein a transition from a highly dissipative stick-slip motion to continuous sliding was observed with a gradual decrease in the friction [8], indirectly supporting this hypothesis, suggesting that there exist some intermediate states rather than an abrupt transition between the stick-slip and frictionless sliding. In theoretical studies, Rozman et al. attempted to divide the frictional force into the potential and dissipative components, where the friction can be viewed as a reversible, adiabatic process with a vanishing dissipative contribution in a quasi-static state [9]. However, the quantitative estimation of the amount of energy dissipation remains a challenge. The fraction of energy that dissipates during sliding is dependent on both the intrinsic system properties and influencing factors such as the sliding velocity and the temperature, which is the Friction 3(2): 170–182 (2015) most intricate parameter and the most important to determine [7]. In this paper, the energy transition and dissipation under stick-slip friction are quantitatively examined using the Prandtl–Tomlinson (PT) model [10–12]. Factors influencing the energy dissipation, such as the stiffness, potential-energy corrugation, damping coefficient, sliding velocity, and temperature of the system, are systematically investigated. We find that the stiffness not only affects the energy entering the system but also, more importantly, affects the energy that can flow back to the driven system after a slip. Two mechanisms of superlubricity [13–15]—smooth sliding and thermolubricity—are discussed in the context of the energy dissipation. The formulation of the energy-dissipation ratio (EDR) is derived analytically to characterize the ratio between the dissipative energy and the total energy that accumulates during sticking. 2 Simulation methodology All the simulations are based on the one-dimensional (1D) Prandtl–Tomlinson model with the assumption of an oscillator having a mass of m (10–12 kg) sliding over a sinusoidal potential with amplitude U (0.01 eV < U < 1 eV) and periodicity a (3 Å). The oscillator is connected to a driving support with a constant speed vRD by a harmonic spring with stiffness k (1 N/m < k < 100 N/m), as shown in Fig. 1. The PT model is a classical model for describing phenomena ranging from macro rigid mechanics to atomic-scale friction. There have been several important advances in the extension of the PT model. The Frenkel–Kontorova (FK) model [16, 17] employs a 1D chain of atoms connected by springs, instead of a single oscillator. The Frenkel− Kontorova−Tomlinson (FKT) model [18] considers the size of contact. In the composite-oscillator model [19], to better describe the thermal lattice vibration, a macroscopic oscillator having a low frequency is coupled with micro oscillators having a high frequency. Nevertheless, presently, the 1D PT model is widely employed to examine the friction on both the macroscopic and microscopic scales. This reduced-order, atomic-scale friction model is particularly suitable to describe the atomic force microscopy (AFM) tip-sample interaction [20], which simplifies the single-asperity 171 Fig. 1 Schematic of the 1D PT model. The upper surface is represented as an oscillator, and the lower surface is represented as a potential field with corrugation U and period a. friction into one point-mass (oscillator) pulled along the periodic lattice (potential energy profile) by an elastic cantilever (spring). The dynamics of the system are solved using the Langevin equation [21, 22], which is shown in Eq. (1), with the fourth-order Runge−Kutta algorithm and a time step of t = 100 ns to achieve a high precision. mx  m x   P( x , t )   (t ) x (1) Here, x is the coordinate of the oscillator along the sliding direction, P is the potential energy in the system (including both the elastic and surface potential), and  (t ) refers to the stochastic thermal-activation force. A system temperature of 0 K is used throughout this paper, except for Section 3.4.  denotes t (...truncated)