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

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.


Introduction
Friction is mechanical kinetic-energy loss or the transformation of sliding motion into heat and other excitations [1][2][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][5][6]. Berman and Israelachvili postulated that in the cobblestone model, upon each molecular collision during sliding, only part of the kinetic energy is 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 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][11][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][14][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.

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 v RD 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 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.
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 the damping coefficient, which is calculated using Eq. (2), where c (0.6 < c < 2.0) is a dimensionless parameter, and   c 2 k m is the critical resonance frequency of the system [23].
The driving support moves with a constant velocity of DR v = 1  / m s; thus, at every instant, the elongation of the spring is x t is the displacement of the oscillator at the moment t. During an entire stick-slip period, the system energy can be divided into four forms: the surface potential (P s ), elastic potential of the spring (P e ), kinetic energy of the oscillator (K), and dissipation energy (D). The surface potential is modeled in the sinusoidal form for simplicity. The spring connecting the oscillator and the driving support is harmonic, following Hooke's law. Thus, the elastic potential and surface potential are calculated using the following formulas: where the system potential energy consists of two parts: the elastic potential energy (P e ) and surface potential energy (P s ).
The total system energy total ( ) E , including the dissipation energy, is then calculated as follows: Both the kinetic energy K(t) and dissipation energy D(t) are related to the velocity of the oscillator, ( ) x t  .
Here, the dissipation energy is calculated by summing the work done by the viscous damping forces, i.e., by performing a time integration of the damping force, as shown in Eq. (8). The damping coefficient is a comprehensive, system-dependent quantity that is affected by several factors. Phononic damping, electronic damping, and viscous damping can all contribute to the damping coefficient [3]. However, in the present paper, the damping force is calculated as   D f m x  , which is a widely accepted approximation [23][24][25]. Distinguishing the exact channel of the frictionalenergy dissipation remains a challenge and is required for a more accurate description of the damping coefficient. This issue is worth studying further in future works.
The source of the energy, which is the work done by the external driving support (W), is calculated as follows: The driving support moves uniformly. Thus, the lateral force F should be always equal to the spring force, as indicated by Eq. (10): 3 Results and discussions

Energy transitions and reversibility
The energy transitions during a complete stick-slip period are shown in Fig. 2. Most of the energy dissipation occurs during the slipping. Therefore, in this paper, we focus particularly on the slip stage. To clearly illustrate the energy transitions, a complete stick-slip period can be divided into four stages, the first of which is the accumulation of potential energy (hereinafter called stage I). In this stage, the oscillator is pulled by the driving support, and potential energy accumulates gradually. When the energy accumulates to a certain extent, a slip takes place, which comprises two stages (II and III). Stage II is the major energy dissipation, i.e., the transformation of part of the potential energy into non-conservative energy.
The sudden decreases in P e and P s and the increase in K and D occur synchronously. Stage III is the transformation of the residual energy. The potential energy and kinetic energy convert into each other during the oscillation. After the slip stage, the potential energy is not completely dissipated, and the reversible part is released to the external system. As shown in the inset of Fig. 2, at the end of stage III, 0.08 eV of P e and 0.05 eV of P s , i.e., RP e and RP s , respectively, remain. RP e and RP s constitute the reversible potential energy (RP): Stage IV involves the release of reversible energy and the onset of the next stick-slip. Here, the potential energy of the system decreases, but there is no change in the dissipation or kinetic energy, indicating that the residual energy is gradually transferred to the external system. Figure 3(a) demonstrates the variation of the total system energy total ( ) E and external work (W) during sliding, as indicated by Eqs. (6) and (9), respectively. The overlapping of the external work and total energy curves shows that all the external work of the driving support is used only to counteract the energy dissipation and maintain sliding. As presented in Fig. 3(a), the total energy accumulates during the stick stage, corresponding to the positive work performed by the driving support. Upon a slip, the total energy decreases, and negative work is performed by the driving support, indicating that not all the energy that accumulates during the stick dissipates permanently. Rather, the external system can retrieve part of the energy. The peak value 1 W is the total energy that accumulates in the stick stage, and 2 W represents the reversible energy (or work) transferred back to the external system. Thus,  1 2 W W represents the energy dissipated during one stick-slip period. As shown in Fig. 2, the reversible energy 2 W comes from the potential energy RP e and RP s at the beginning of stage IV. Figure 3(b) shows the variation of the system potential energy P during the stick-slip motion. Part of the potential energy is converted into dissipation energy (DP), and the remainder is reversible energy (RP).
To measure the degree of energy dissipation or energy reversibility, we present two definitions for the EDR: EDR W and EDR P , which are described in Eqs. (12) and (13), respectively. Equation 12, derived from Fig. 3(a), is used to calculate the fraction of dissipated external work (W), i.e., EDR W . Equation 13, deduced from Fig. 3(b), is used to calculate the fraction of reversible potential energy during the slip stage, i.e., EDR P . These two definitions for the EDR are equivalent under the assumption that most of the Fig. 3 Variations of system energy in stick-slip motion. (a) Variation of external work and total system energy during sliding. The peak value W 1 is the total energy that accumulates in the stick stage, and W 2 is the reversible energy (or work) transferred back to the external system. Thus, (W 1 − W 2 ) is the energy dissipated during one stick-slip period. (b) Variation of the system potential energy P during the stick-slip motion. Part of the potential energy is converted into dissipation energy (DP), and the remainder is reversible energy (RP) that is transferred to the external system. The whole stick-slip period is denoted by T. The simulation is performed under the conditions indicated in Fig. 2. energy dissipation occurs at the slip stage because of the system instability, which applies to most tribological systems. However, the two definitions yield different EDR results under smooth sliding or similar conditions with a very small energy dissipation, where the dissipation due to the slip instability is negligibly small compared with the contribution of the viscous dissipation, which is proportional to the driving speed. In this paper, we investigate the energy dissipation caused by the stick-slip instability, which is better described by EDR P , i.e., the proportion of the irreversible potential energy during slipping to the total potential energy that accumulates in the stick stage. Therefore, unless otherwise specified, in the following calculations and discussions, EDR P is applied as the definition of the EDR.
What causes the energy reversibility in our system? To answer this question, the variation of the lateral force and the potential energy during the stick-slip are depicted in Fig. 4. In a complete period (denoted as "T"), both the potential energy and lateral force vary in a periodic manner. The potential energy is slowly released after its suddenly decrease at the slip stage as the lateral force changes from positive to negative when the spring stiffness is 3 N/m, as shown in Fig. 4(a). However, when the spring stiffness is 1 N/m, the stage of the potential release is absent, and the potential increases gradually after the rapid decrease in the slip stage. In this case, no negative lateral force appears, as shown in Fig. 4(b). We believe that a negative lateral force is necessary for part of the energy to be reversible, because a negative lateral force indicates that part of the internal energy of the system can be transferred back to the external driver. The emergence of the negative lateral force is essentially caused by the over-slipping of the oscillator when the oscillator slips over the driving support, i.e., According to Eq. (9), this results in negative work, thereby leading to energy reversibility.

Controlling factors of EDR
According to Lyapunov's criterion for stability [26,27], the dimensionless parameter  shown in Eq. (14) is appropriate to describe the system stability:   Fig. 5.
Notably,  has a crucial impact on the slip style (e.g., single or multiple slips) [6,30], which affects the energy dissipation but is not the focus of this paper.
To eliminate the influence of the slip style, calculations are conducted with a spring stiffness ranging from 0.1 to 100 N/m. This stiffness range ensures a single slip and is wide enough to cover both the stick-slip and smooth-sliding regimes. The surface potential corrugation U and spring stiffness k are equivalent for determining the EDR. This equivalency is also clearly indicated by Eq. (14). The parameter  is used in our discussions, although the simulations presented in this section are based on a varying stiffness with a constant U value, unless otherwise specified, for simplicity. The aforementioned result demonstrates that higher spring stiffness can contribute to a lower EDR; i.e., a stiffer spring improves the energy reversibility. This dependency is verified by our previous molecular-dynamics simulation, where the interlayer friction of few-layer graphene is found to be dependent on the number of layers. When the number of layers is decreased below three, the lateral stiffness of the system is sufficiently large, and the stick-slip phenomenon disappears, causing the friction to vanish [31].
To understand the effects of the spring stiffness, we first recall that the EDR indicates the ratio between the dissipative energy and the total energy that accumulates during sticking (Eq. (13)). Specifically, the energy transition can be represented by two important points: the slip starting point s x and the slip ending point e x of the oscillator. The spring stiffness determines how much energy is invested in the system during the stick stage, as proposed by Krylov and Frenken in a recent review paper [32]. Generally, less potential is stored in the case of a stiffer spring. Moreover, the stiffness k can influence the EDR by affecting the slipping distance. In order to explain this clearly, three oscillator trajectories are shown in Fig. 6, with k = 3, 6, and 9 N/m, respectively. In these three cases, the s x positions of the oscillator are relatively close to each other (labeled as 1, 2, 3), and the slipping distance (from 1−1', 2−2', and 3−3', respectively) decreases as the spring stiffness increases. The slip ending point moves continuously from the peak point to the valley point of the surface potential, as shown in the inset of Fig. 6, which explains the continuous variation of the EDR from 100% to 0. Consequently, the e x positions are given by the coordinates of 2.8, 2.3, and 1.9 Å (denoted as 1, 2', and 3', respectively). These points correspond to surface potential energies of 0.011, 0.112, and 0.208 eV, as indicated by the Fig. 6 inset and Table 1. According to the definition shown in Fig. 3, this potential energy is exactly equal to the RP s and is released from point 1' (or 2', 3') to the re-equilibrium point (E). When the stiffness varies from 3 to 9 N/m, the RP s varies by a factor greater than 20 and plays an important role in determining the amount of reversible energy. Thus, the RP s greatly influences the EDR values listed in Table 1. Consequently, the friction dissipation depends not only on the amount of energy invested in the system but also, more importantly, on the amount of reversible energy. The stiffness can greatly influence the slipping distance and thus the amount of RP s , which plays an important role in determining the amount of reversible energy.

Influence of damping coefficient
According to the Langevin equation, damping causes energy dissipation. Thus, it is straightforward to discuss the influence of the damping coefficient. We used a damping constant c ranging from 0.6 to 2.0 to ensure a single slip during the sliding, while covering all the under-damping, critical-damping, and overdamping states. As shown in Fig. 7, the EDR increases slightly as the damping coefficient increases. The influence of the damping coefficient is less than 10%. Nonetheless, it is inappropriate to conclude that the damping coefficient has a small effect on the EDR, although it is the case under this specific condition ( DR v = 1 μm/s, T = 0 K). Nevertheless, the damping coefficient profoundly impacts how the energy dissipates. In Fig. 2, we divide the whole stick-slip period into four stages with  regard to the energy transitions. Correspondingly, from the viewpoint of dynamics, these can be regarded as the stick, primary-slip, oscillation, and re-equilibrium stages, as shown in Fig. 8. Considering  = 8.8 (thus, EDR value is 100%) as an example, different damping coefficients lead to completely different results with regard to both the dynamics and energy dissipation, as shown in Fig. 8. In the case of low damping (c = 0.6), a strong and long-lived oscillation stage is observed at the end of the primary slip, which accounts for ~22% of the total dissipation energy ( Fig. 8(a)). However, in the case of high damping (c = 2.0), there is almost no oscillation, and over 99% of the energy dissipation occurs in the primary slip stage (Fig. 8(b)). Apart from this, the damping coefficient greatly influences the speed of energy dissipation. As shown in Fig. 8, the slip stage lasts 3 and 11 μs with high and low damping coefficients, respectively. Hence, compared with its less obvious effect on the EDR, the damping coefficient has a remarkable influence on the system dynamics.
A system with a high damping coefficient can quickly achieve its stable state.
Energy dissipation is a complex process involving various pathways. The stick-slip phenomenon is an important cause of the energy dissipation in atomicscale friction. The sudden slip and oscillation result in lattice vibrations and phononic dissipation, and phonons are the quantum representation of the lattice vibration energy [33]. The collisions among phonons introduce inharmonic acoustic modes, which can be rapidly converted to heat. The lattice vibrations can also be transformed into acoustic waves or luminous flashes rather than coupling with the heat generation. Moreover, recent studies indicate that energy can be dissipated in the stick stage as well, which is attributed to thermoelastic damping [34][35][36]. Various factors, including the humidity, viscosity, and temperature, may contribute crucially to the damping coefficient.

Influence of temperature
The thermal effect on atomic-scale friction have been reported for years [10,29,32,37,38]. One of the explanations is that random fluctuation provides extra energy input and helps the oscillator surpass the energy barrier, thus reducing the mean friction and even achieving a superlow friction [32]. Here, we provide a detailed analysis of the thermal effect on atomic-scale stick-slip motion with regard to the energy dissipation. The Langevin equation for describing the PT model comprises both deterministic dynamics and stochastic processes. Thermal fluctuation is a common stochastic process that is frequently used in dynamic simulations. In this paper, the thermal fluctuation is represented by a random force  ( ) t that follows the fluctuationdissipation relation, as shown in Eq. (15). In this expression, the mean value of the autocorrelation function of the thermal force is used to measure the randomness of the thermal effect. B k is the Boltzmann constant, and  indicates the Dirac delta function. Equation (15) indicates that  ( ) t is a random term whose the average amplitude is proportional to T : To investigate the thermal effect, the friction and system potentials at three temperatures-0, 100, and 300 K-are compared in Fig. 9. It is immediately apparent that a higher temperature yields more intense fluctuations. The average lateral force ( av F ) decreases rapidly with the temperature. As shown in Figs. 9(a), 9(c), and 9(e), av F is 0.1485, 0.0521, and nearly 0 nN. With regard to the energy, a thermally induced fluctuation can greatly increase the probability of the oscillator jumping across the energy barrier. This reduces the potential-energy accumulation during the stick stage, which decreases from 0.29 at 0 K to 0.18 eV and 0.12 eV at 100 and 300 K, as indicated by the lower profiles of the curves shown in Figs. 9(b), 9(d), and 9(f), respectively. This is similar to the previously proposed thermolubricity mechanism. The proportion of the reversible energy (RP) in the slip stage also increases with the temperature, as estimated by the potential energy curves. With the combined effects of decreasing the energy accumulation and increasing the energy reversibility, the EDR exhibits a remarkable decrease with the temperature (95.8%, 57.3%, and nearly 0 at 0, 100, and 300 K, respectively). In an extreme case, superlow friction can be achieved at 300 K, as shown in Figs. 9(e) and 9(f). This superlubricity regime should be distinguished from the smoothsliding regime as described in Section 3.2, where the system is stable with a very low  value. In this case, the thermal fluctuation is sufficiently strong to enable the "forward jump" and "backward jump" [32] across the energy barrier. In this superlubricity regime, the stick-slip instability remains, and the major reason for the extremely small energy dissipation is the reversibility of the mechanical energy that accumulates in the stick stage during the slipping.
To understand the synergetic effects of  and the thermal effect on the friction energy dissipation,  −T phase diagrams of the EDR are drawn for driving velocities of DR v = 1 μm/s ( Fig. 10(a)) and 0.01 μm/s ( Fig. 10(b)). In both graphs, two white curves indicate the boundaries of the regions for EDR = 100% and 0. For a fixed temperature, the variation of the EDR as  decreases is similar to the curve shown in Fig. 4. This phase diagram suggests two possible pathways to achieve superlubricity, as indicated by the arrows in Fig. 10: by increasing the system stability (one example is the structural lubricity with a very low potential-energy corrugation) and by increasing the temperature to reach a state of thermolubricity. However, both mechanisms are actually unified with regard to the energy dissipation and can be regarded as energy-reversible superlubricity, as the EDR is mainly determined by the proportion of the reversible energy after slipping. It should be mentioned that the sliding velocity also has important effects on the friction dissipation. This was systematically reviewed by Krylov and Frenken [32] and is not discussed in the present paper.

Analytical formulation of EDR
In the previous sections, the EDR is calculated according to the energy-transition process, which is difficult to measure in real experimental systems. To validate our theoretical prediction, an analytical formulation of the EDR is derived directly from the lateral-force curve shown in Fig. 11(a). Here, we use the definition of EDR W shown in Eq. (12) for the comparison with the experimental force curves. For simplicity, the lateral force is approximated by a perfect sawtooth curve. In Fig. 3(a), the area enclosed by the positive lateral force 1 W indicates positive work done by the external system during a stick-slip period. Meanwhile, the area enclosed by the negative lateral force 2 W indicates the energy retrieval from the internal system.
Considering the geometry of the lateral-force curve, Eq. (12) can be rewritten as Eq. (16), where S is the slip length of the oscillator (   e s S x x ): In Eq. (16),  and U are intrinsic system parameters, while av F and S are two variables that depend on various parameters, such as the damping coefficient, temperature, and particularly the  value. Figure 11( (16) demonstrates the consistency and relationship between the lateral force and energy dissipation in atomic-scale friction. More importantly, the proposed method is applicable for the prediction of atomicscale friction in real AFM or friction force microscopy (FFM) experiments [39]. The expression of the EDR can be further simplified (Eq. (17)) by defining a dimensionless parameter   S L , where L is the maximum elongation of the spring, and  can be viewed as the degree of recovery of the spring elongation during slipping.  1  indicates the partial or complete recovery of the spring deformation.
   1 2 indicates the over-recovery of the spring deformation or over-slipping of the oscillator, which causes energy reversibility. Although  is dependent on various system factors such as  , μ, T, and DR v , this expression provides an intuitive understanding of the energy dissipation according to the system sliding behaviors (slip distance and maximum elongation of spring).

Conclusion
In this study, atomic-scale friction is examined with regard to energy, and the energy dissipation and transition during sliding are examined systematically. First, a reversible energy exchange between internal and external systems is observed under atomic friction, and the concept of the EDR is proposed to measure the degree of energy reversibility. The spring stiffness and surface potential, including the comprehensive parameter  , are the main factors influencing the EDR. We observe a continuous variation of the EDR from 100% to 0 with  covering the stick-slip, intermediate, and smooth-sliding regimes. For a sufficiently small  , smooth-sliding superlubricity is achieved, and no energy is dissipated. Furthermore, both the damping coefficient and temperature are found to impact the amount and dynamics of energy dissipation. In particular, the thermal effect tends to decrease the potential energy accumulation during sticking and increase the proportion of reversible energy during slipping, yielding a state of thermolubricity. On the basis of  −T phase diagrams of the EDR, we propose that the smooth-sliding superlubricity and thermolubricity are actually unified with regard to the energy dissipation and transition. Finally, an analytical formulation of the EDR is derived. This not only deepens our understanding of the energy dissipation during atomic-scale friction but also is applicable for judging energy-dissipation properties for real AFM/FFM experiments and provides criteria for saving energy in the design and fabrication of micro-frictional systems.