Thermally assisted lubricity and negative work tails in sliding friction

Franco Pellegrini, Emanuele Panizon, Giuseppe E. Santoro, and Erio Tosatti

Phys. Rev. B 99, 075428 – Published 19 February 2019

Abstract

We discuss and qualify the connection between two separate phenomena in the physics of nanoscale friction, general in nature and relevant to experiments. The first is thermally assisted lubricity (TAL), i.e., the low-velocity regime where a nanosized dry slider exhibits a viscouslike friction despite corrugations that would otherwise imply hard stick–slip friction. The second is the occurrence of negative dissipated work (NDW) events in sampling the work probability distribution. The abundance, or scarcity due to insufficient sampling, of these NDW events implies experimental fulfillment or violation of the celebrated Jarzynski equality (JE) of nonequilibrium statistical mechanics. We show, both analytically and in simulations of the one-dimensional point slider Prandtl-Tomlinson model, that a general crossover can be individuated as the total frictional work per cycle crosses $k_{B} T$ . Below such crossover, the TAL regime holds, the dissipation is essentially linear, and the numerical validation for the JE is feasible (i.e., does not require an exponentially large sampling size). Above it, the dissipation profile departs from linearity and gains its hard stick–slip features, and the mandatory sampling for the JE becomes exponentially large. In addition, we derive a parameter-free formula expressing the linear velocity coefficient of viscous friction, correcting previous empirically parameterized expressions. With due caution, the connection between friction and work tails can be extended beyond a single degree of freedom to more complex sliders, thus inviting realistic crosscheck experiments. Of importance for experimental nanofriction will be the search for NDW tails in the sliding behavior of trapped cold ions, and alternatively checking for TAL in the sliding pattern of dragged colloid monolayers as well as in forced protein unwinding.

Received 4 October 2018
Revised 8 February 2019

DOI:https://doi.org/10.1103/PhysRevB.99.075428

Physics Subject Headings (PhySH)

Friction Nonequilibrium statistical mechanics

Nonequilibrium systems

Fokker–Planck equation

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Franco Pellegrini¹, Emanuele Panizon¹, Giuseppe E. Santoro^1,2,3, and Erio Tosatti^1,2,3

¹SISSA, Via Bonomea 265, I-34136 Trieste, Italy
²CNR-IOM Democritos National Simulation Center, Via Bonomea 265, I-34136 Trieste, Italy
³International Centre for Theoretical Physics (ICTP), Strada Costiera 11, I-34151 Trieste, Italy

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Issue

Vol. 99, Iss. 7 — 15 February 2019

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Images

Figure 1
A sketch of the evolution of the Prandtl-Tomlinson potential $V (x, t) = U (x) + \frac{k}{2} {(x - v t)}^{2}$ for three different times. At $t = t_{B}$ , the position of the driving force coincides with the maximum of the barrier, forming the symmetric double well (thick black line). For $t = 0 (t = a / v)$ , the minimum of the driving force coincides with a potential minimum giving rise to a single well as shown with a red (blue) dashed line. The renormalized (bare) barrier $V_{B} (2 U_{0})$ and the reduced distance between minima $a_{*}$ are also shown.
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Figure 2
The linear coefficient $η_{eq}$ of frictional dissipation $〈 W 〉 = η_{eq} v$ . Filled black circles indicate results obtained by numerical integration of the Langevin equation (all error bars are smaller than the size of the dots.). In the high temperature $(β V_{B} ≪ 1)$ regime, the dissipation is low, constant, and dominated by the Langevin term: $η_{eq} = γ a$ (dashed line). Parameters for the PT model are $U_{0} = 1, a = π, γ = 1$ and $k = 2.5$ . The solid black line shows the analytical result of Eq. (15a) for $β V_{B} > 1$ with the fitted value of $C = 0.45$ .
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Figure 3
Frictional work between consecutive minima (a distance $a$ apart) obtained with PT simulations. (a) Average dissipated work—normalized to temperature $k_{B} T$ —as a function of sliding speed $v$ . A horizontal line is drawn at $〈W〉 / k_{B} T = 1$ , separating two friction regimes, essentially linear for $〈W〉 / k_{B} T < 1$ and sublinear for $〈W〉 / k_{B} T > 1$ . (b) Same data shown in (a) now presented as a function of the normalized velocity $v e^{β Δ E_{λ_{*}}} \equiv v e^{β V_{B}}$ , i.e., removing the temperature dependence through Eq. (15a). Parameters for the PT model are $U_{0} = 1.0, a = π, γ = 1$ , and $k = 2.5$ . Each point is the average of $10^{5}$ or $10^{6}$ “slip” events (respectively, for $v < 5 \times 10^{- 3}$ and $v \geq 5 \times 10^{- 3})$ .
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Figure 4
Probability density $P (W)$ (solid lines) for two realizations of the Prandtl-Tomlinson model (parameters as in Fig. 3] for temperature $k_{B} T / V_{B} = 0.382$ and velocities $v = 0.8 \times 10^{- 2}$ and $2.5 \times 10^{- 3}$ , respectively, for blue and green. The filled areas represent the total probability for NDW events, where $W < 0$ . The shadow (mirrored) curves represent $P (- W) = P (W) e^{- β W}$ , while data points indicate the numerical results obtained with a sample of $N = 10^{5}$ cycles.
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Figure 5
The value of the estimated free-energy Boltzmann weight $e^{- Δ_{JE} (N)}$ in simulated Prandtl-Tomlinson compared to its true value 1, i.e., to the numerical realization of the Jarzynski equality (shown as a black line). The sample size $N$ is normalized by a term which is exponential in the average dissipation per cycle. Remarkably, all curves, obtained at different temperatures and for different slider velocities, cross at the central point. The curve $x / x + 1$ is shown in blue as a guide to the eye.
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Figure 6
Sliding in the Frenkel-Kontorova model. The linear coefficient $η_{eq}$ of friction $〈 W_{single} 〉 = η_{eq} v$ for the $\frac{r_{0}}{a} = \frac{9}{10}$ kink. The bare parameters are $U_{0} = 1, a = π, k = 2.5, m = 0.1, γ = 1$ , and $N_{p} = 10$ . The effective parameters are calculated as $V_{B}^{*} = 0.1015, m^{*} = 0.08$ , and $k^{*} = 2$ . (All error bars are smaller than the size of the dots.) In the high temperature $(β V_{B} ≪ 1)$ regime, the dissipation is constant and dominated by the Langevin term $η_{eq} = γ m a$ (dashed line). The solid black line shows the analytical result of Eq. (15a) for $β V_{B} > 1$ with the fitted value of $C = 0.25$ . The numerical results are also shown (filled black circles).
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