Thermodynamic aspects of nanoscale friction

Developing the nonequilibrium thermodynamics of friction is required for systematic design of low-friction surfaces for a broad range of technological applications. Intuitively, the thermodynamic work done by a material sliding along a surface is expected to be partially dissipated as heat and partially transformed into the change of the internal energy of the system. However, general nonequilibrium thermodynamic principles governing this separation are presently unknown. We develop a theoretical framework based on the transition state theory combined with the conventional Prandtl-Tomlinson model, allowing to set explicit expressions for evaluating the heat dissipation and internal energy change produced during the frictional stick-slip motion of a tip of a typical friction force microscope. We use the formalism to quantify the heat dissipation for a range of parameters relevant to materials in practical applications of nanoscale friction.

Figure 1

A sketch of the principle of the one-dimensional Prandtl-Tomlinson model of a FMM experiment. Typical values of the parameters are $E_0\sim 0.1–1$ eV, $V=1$ nm/$\mathrm{s}\text{to}1$ m/s, $C=0.1\text{to}50$ N/m, and $a\approx 0.3$ nm [25].

Figure 2

Time-dependent energy in the Prandtl-Tomlinson model obtained from Eq. (1) for $\gamma =E_0{\left(2\pi \right)}^2/C{a}^2=16.7$, at three different times $t_1$ (I), $t_2$ (II), $t_3$ (III), such that $t_1<t_2<t_3$. The times between (I) and (II) correspond to the system being in the stick state, and at some point between (II) and (III) the slip occurs. The red circle corresponds to the position of the tip, and the green square corresponds to the position of the cantilever at a given instant in time.

Figure 3

The position of the tip (left) and lateral force (right) as a function of time at 0 and 300 K. The position of the cantilever is highlighted by a gray dashed line in the left figure. These trajectories were produced by a kinetic Monte Carlo algorithm consistent with Eqs. (3) and (4).

Figure 4

Average force [Eq. (2b)] at 300 K produced over a variable number of the individually generated stochastic trajectories for velocities 1 nm/s (left) and 100 nm/s (right). Blue continuous line shows expectation value of the force, computed as per Eq. (6). Red dashed line shows (A), (a) Force evaluated over a single trajectory, (B), (b) averaged over 10 independent trajectories of a typical FFM experiment, and (C), (c) averaged over 100 independent trajectories and closely resembling the solution obtain from Eq. (6) (blue continuous line). All trajectories were produced by a kinetic Monte Carlo algorithm consistent with Eqs. (3) and (4). Dotted gray line is the time-average force (friction) of the red dashed trajectories.

Figure 5

Expectation values of (a) position of the tip, (b) lateral force of the substrate, (c) heat exchanged with the surroundings, energy change, work, and (d) entropy production, entropy flow, and entropy change of the tip, at 300 K and $V=10$ nm/s. In the stick stages, (a) the position remains almost constant, (b) the force increases in magnitude, (c) work increases, small amounts of heat is dissipated, and (d) the entropy of the system increases. In the slip stages, (a) the position increases by about one lattice spacing, (b) the force relaxes, (c) work is reduced, heat dissipation peaks, and (d) the entropy of the system decreases to a minimum, and entropy production and entropy flow peak. The total entropy change has been magnified by a factor of 10 for better visualization.

Figure 6

$\overline{F}\text{−}V$ and ${\overline{Q}}_p\text{−}V$ relations. The friction-velocity relation is logarithmic whereas the heat-velocity behavior is linear. $\overline{F}$ and $Q_p$ are calculated for temperatures 200–500 K, substrate corrugation $E_0\approx 0.2–0.8\text{eV}(\gamma$ between 11.2 and 45) and velocities $0.1\mathrm{nm}/\mathrm{s}\le V\le V_{\max }$ [32]. The rest of model parameters are fixed at the default values.

Figure 7

(a) Average internal energy change, (b) average entropy change during stick-slip events, calculated for temperatures 200–500 K, and velocities $0.1\mathrm{nm}/\mathrm{s}\le V\le V_{\max }$ [32]. The rest of model parameters are fixed at the default values.