Stochastic thermodynamics of nanoscale friction

P. C. Torche, P. Nicolini, T. Polcar, and O. Hovorka

Phys. Rev. E 103, 052104 – Published 3 May 2021

Abstract

Developing the thermodynamics of nanoscale friction is needed in a wide range of tribological applications, where the key objective is to optimally control the energy dissipation. Here we show that modern stochastic thermodynamics allows us to interpret the measurements obtained by friction force microscopy, which is the standard tool for investigating the frictional properties of materials, in terms of basic thermodynamics concepts such as fluctuating work and entropy. We show that this allows the identification of the heat produced during the friction process as an unambiguous measure of thermodynamic irreversibility. We have applied this procedure to quantify the heat produced during the frictional sliding in a broad velocity range, and we observe velocity-dependent scaling behavior, which is useful for interpreting the experimental outcomes.

Received 4 February 2021
Accepted 12 April 2021

DOI:https://doi.org/10.1103/PhysRevE.103.052104

Physics Subject Headings (PhySH)

Entropy production Fluctuation theorems Friction Irreversible processes Nonequilibrium & irreversible thermodynamics Nonequilibrium statistical mechanics Stochastic processes Tribology Work theorems

Atomic force microscopy Markovian processes Master equation Noncontact atomic force microscopy Stochastic processes & statistics

Statistical Physics & ThermodynamicsCondensed Matter, Materials & Applied PhysicsGeneral Physics

Authors & Affiliations

P. C. Torche^1,*, P. Nicolini ², T. Polcar¹, and O. Hovorka¹

¹Engineering and Physical Sciences, University of Southampton, Southampton SO17 1BJ, United Kingdom
²Department of Control Engineering, Faculty of Electrical Engineering, Czech Technical University in Prague, Technicka 2, Prague 6, 16627, Czech Republic

^*pc.torche@soton.ac.uk

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Issue

Vol. 103, Iss. 5 — May 2021

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Images

Figure 1
(a) A sketch of the one-dimensional Prandtl-Tomlinson model of FFM experiments with $C = 2$ N/m, $V t = 1.2$ nm, $a = 0.25$ nm, and $E_{0} = 0.5 eV$ , in the range of typical values for FFM experiments [36]. Part (b) shows the total energy landscape (continuous line), the substrate-tip interaction term (dashed line), and the elastic term (dotted line). The instantaneous position of the tip in this scenario is $x \approx 0.4$ nm, and energy $u \approx 4 eV$ .
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Figure 2
Kinetic Monte Carlo simulation based on Eqs. (1, 2, 3, 4) of thermodynamics of a FFM tip subject to forward (solid line) and reversed driving (dotted line). Shown are the stochastic processes for (a) tip position, (b) friction force, and cumulative changes of (c) internal energy, (d) work, (e) heat exchanged with the environment, and (f) heat produced during the sliding process. The forward process starts at time $t = 0$ with the cantilever at position $x = 0$ and the tip in thermal equilibrium, and it follows the driving protocol $λ = V t$ until the finite time $t = t_{f}$ . The reverse process starts in thermal equilibrium at time $t_{f}$ , with the cantilever at position $x = V t_{f}$ , and it follows the protocol $λ = V t_{f} - V t$ from $t = 0$ until $t = t_{f}$ . All cumulative changes $Δ$ are calculated by summing up all the time contributions, subject to resetting the counter at $t_{f}$ . Simulation parameters fixed as in Fig. 1, with $V = 10$ nm/s, $T = 300 K$ , and $ω_{0} = 20 kHz$ .
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Figure 3
A plot analogous to Fig. 2 containing 500 individual trajectories. The lines are the mean values obtained over the 500 trajectories, and the shaded areas imply two standard deviations around the mean.
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Figure 4
(a),(b) Examples of probability distributions of cumulative fluctuating work for the forward ( $P$ ) and reverse process ( $P_{R}$ ) obtained as histograms over 1000 different trajectories for velocities $V = 1$ and 2 nm/s, $C = 9$ N/m, $ω_{0} = 100 kHz$ , and the rest of the parameters fixed as in Fig. 2. The estimates using the Crooks ( $Δ F_{c}$ ) and Jarzynski relations ( $Δ F_{J}$ ) are shown for comparison. (c),(d) The free-energy difference estimated from the Crooks and Jarzynski relations as a function of the mean fluctuating work for a variable number $N$ of repeated fluctuating trajectories corresponding to $C$ in the range 5–9 N/m, $E_{0}$ between 0.3 and 0.7 eV, $V$ between 0.5 and 2 nm/s, $T$ in the range 300–500 K, fixed $a = 0.25$ nm, and $ω_{0} = 100 kHz$ . The dashed line in all the plots corresponds to exact reference free energy $Δ F = 0$ .
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Figure 5
A plot of an estimate of the cumulative entropy production $T Δ_{i} s$ , or equivalently the heat $Δ q_{i}$ according to Eq. (16), as a function of the cumulative fluctuating work. The exact value (black, solid line) is found using the stochastic thermodynamics developed in Sec. 3. The estimations are based on Eq. (19) using the Crooks (red, dashed line) and Jarzynski relations (blue, dotted line) to obtain the free energy $Δ F$ . Simulation data equivalent to Fig. 4.
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Figure 6
The mean friction force $\bar{f}$ vs the tip sliding velocity $V$ . Each symbol corresponds to a different fluctuating trajectory starting in equilibrium. (a) A comparison of the accuracy of the least-squares fits of the force-velocity relations ${\bar{f}}_{a} = α_{1} + α_{2} | ln {(V / V_{0})}^{2 / 3} |$ derived in [31], the implicit formula ${(f^{★} - {\bar{f}}_{b})}^{3 / 2} / b = ln (V_{0} / V) - ln (1 - {\bar{f}}_{b} / f^{★}) / 2$ obtained in [32], the series expansion ${\bar{f}}_{c} = \sum_{n (odd) = 1}^{3} α_{n} arcsinh {(v / v_{c})}^{n}$ from [35], and ${\bar{f}}_{d}$ calculated from Eq. (20). The different marker types represent the thermal drift and thermal and nonthermal stick-slip friction regimes in the order of increasing velocity. (b) Validation of Eq. (20) for a broad range of parameters corresponding to data in Fig. 4 but with an extended velocity range. The fit parameter variations are discussed in the text.
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