Velocity dependence of kinetic friction in the Prandtl-Tomlinson model

The Prandtl-Tomlinson model for friction has been used extensively for the interpretation of atomic force microscopy data during the past decade. Up to this point, the kinetic friction $F_k$ has nevertheless not been studied in a range of velocities $v$ that would be sufficiently broad to cover the crossover from the high-velocity logarithmic to the low-velocity linear $F_k\left(v\right)$ dependence. This gap will be closed here through a combination of an asymptotic analysis and direct simulations of the relevant Langevin equation. The simulations span three decades in temperature $T$ and up to six decades in $v$. All numerical data can be fit quite accurately with a $F_k=a\left(T\right)\mathrm{arsinh}[v/v_{\mathrm{c}}\left(T\right)]$ law, where the prefactor $a\left(T\right)$ scales with $T^{2/3}$. Correction terms proportional to odd powers of $\mathrm{arsinh}(v/v_{\mathrm{c}})$, only need to be included at $v\gg v_{\mathrm{c}}$. Reasons are given as to why it is difficult to confirm meticulously the ${\left(\ln v\right)}^{2/3}$ dependence of kinetic friction predicted by recent rate theories, although they can be easily modified to produce the correct prefactor to the $a\left(T\right)\propto T^{2/3}$ law.

Figure 1

Combined substrate and spring potential $V_{\mathrm{tot}}$ for two selected positions of the $k=0.4$ spring as a function of position $x$. For $x_{\mathrm{spring}}=2n\pi$, the two mechanically stable sites are degenerate, which allows one to read off the activation barrier $\Delta E_{\mathrm{b}}=0.6231\left(5\right)$. Near $x_{\mathrm{spring}}=2n\pi +1.132$, the left minimum disappears, which leads to an instability energy of $\Delta E_i=1.87\left(8\right)$, from which the athermal, low-velocity limit kinetic friction force $F_k=\Delta E_i/2\pi =0.298\left(9\right)$ follows.

Figure 2

Kinetic friction $F_k$ as a function of velocity $v$ for $k=0.4$ and $\gamma =0.1$. The low-velocity law $F_k=F_{k,\mathrm{ideal}}+\mathrm{const}{v}^{2/3}$ is accurate up to a velocity of $v=0.1$, where inertial effects are starting to play a role. The value for $F_{k,\mathrm{ideal}}$ is equal to that obtained from the analysis of the instability energy in Fig. 1.

Figure 3

$F_k/\sqrt{v}$ as a function of velocity $v$ for the default model at $T=0.2$. The MD data are fit to two different model functions.

Figure 4

$F_k/\sqrt{v}$ as a function of velocity $v$ for the default model at $T=0.1$. The MD data are fit to two different model functions. The prefactor to the $v^{2/3}$ correction was not a fit parameter but identical to the value identified in Fig. 2.

Figure 5

$F_k\left(v\right)$ as a function of velocity $v$ for the default model at temperatures covering more than two decades in magnitude. Lines are fits according to the same equations used in Fig. 4, where the third-order correction term corresponded to the power law at $T<0.1$, while the third-order arsinh term was used otherwise.

Figure 6

Leading-order prefactor $a\left(T\right)$ as a function of $T$. The line, $0.149T^{2/3}$, is a parameter-free result from a slightly modified rate theory in which the motion of the equilibrium site to which a depinning particle is jumping, was included.

Figure 7

Difference between measured friction and reference friction $\Delta F(T,v)=F_{\mathrm{ref}}(T_{\mathrm{ref}},v_{\mathrm{ref}})-F(T,v)$ as a function of velocity. Damping was varied from overdamped to underdamped. Velocity is divided by the eigenfrequency of the pulling spring, which made it possible to collapse results for different dampings. In one case (open circles), the reference friction was simply the ideal athermal kinetic friction.

Figure 8

$F_k\left(v\right)$ as a function of velocity $v$ for the default model at temperature covering more than two decades in magnitude.

Figure 9

Equilibrium damping ${\gamma }_{\mathrm{eq}}$ as a function of inverse temperature $1/T$. The solid line is a fit with an Eyring equation of the form ${\gamma }_{\mathrm{eq}}=\beta \tilde{\gamma }\exp \left(\beta \Delta F\right)$ with $\tilde{\gamma }=0.85\left(6\right)$ and $\Delta F=0.40\left(1\right)$.

Figure 10

Total potential $V_t(X,x)$ for $k=0.4$ as a function of $x$ at two selected spring positions $X$. A few important points are selected and indicated by small, solid circles: The transition point in the athermal case $x_c$, the absolute minima for $x_m\left(X_c\right)$ and $x_m(X_c-\pi /15)$ as well as the relative minimum $x_{-}$ and the relative maximum $x_{+}$ for $X_c-\pi /15$. Their locations are calculated with the formulas derived in the text.

Figure 11

Solutions for $x^{*}$ at different levels of approximation.