Analytic understanding and control of dynamical friction

Recent model simulations discovered unexpected nonmonotonic features in the wear-free dry phononic friction as a function of the sliding speed. Here we demonstrate that a rather straightforward application of linear-response theory, appropriate in a regime of weak slider-substrate interaction, predicts frictional one-phonon singularities which imply a nontrivial dependence of the dynamical friction force on the slider speed and/or coupling to the substrate. The explicit formula which we derive reproduces very accurately the classical atomistic simulations when available. By modifying the slider-substrate interaction the analytical understanding obtained provides a practical means to tailor and control the speed dependence of friction with substantial freedom.

Figure 1

(a) A sketch of the one-dimensional model [17] considered in this paper. (b) Graphical solutions of the energy-conservation condition $v_{\mathrm{SL}}Q=\omega \left(Q\right)$ which determines the inelastically excited phonons for three different slider speeds $v_{\mathrm{SL}}$ expressed in units of the sound velocity $v_{\mathrm{s}}$. (c) The resonant magnitude evaluated as the term $R\left(Q\right)=\gamma /\left(2\pi \right)Q^2/\left\{{[Q{v}_{\mathrm{SL}}-\omega \left(Q\right)]}^2+{(\gamma /2)}^2\right\}$ of Eq. (1). (d),(e) Friction as a function of $v_{\mathrm{SL}}$ at several distances $d$ between the slider and the chain. $V_{\mathrm{ext}}$ is defined by a Lennard-Jones potential with $\sigma =a/2$ and $V_0=5\times {10}^{-4}K{a}^2$. The harmonic chain particle motion has a damping coefficient $\gamma =0.1{(m/K)}^{1/2}$. Parameters $a,m$, and $K$ are the spacing, mass, and spring constant of the harmonic chain, as indicated in panel (a).

Figure 2

Comparison of the slider-speed dependence of the friction force $F$ obtained from the two analytical expressions (8) (dotted, no dissipation included) and (1) (dashed, dissipation included), with that obtained for the same conditions by numerical simulations carried out for a chain of 500 atoms (solid) [17]. All calculations have spacing $a$, nearest-neighbor spring constant $K$, damping rate $\gamma =0.1{(m/K)}^{1/2}$, LJ slider chain interaction with $\sigma =0.5a$, and distance $d=0.475a$. Peaks for $v_{\mathrm{SL}}\le 0.22v_{\mathrm{s}}$ correspond to speeds listed in Table 1 and evolve from true divergences without dissipation to sharp peaks with dissipation. The agreement between the analytical and the simulation results is genuinely striking.

Figure 3

Load dependence of the dynamic friction force computed according to Eq. (1) for a large range of distances at two sliding speeds. Comparison with a linear increase (Amonton-like) is also shown. The effective load $L$ is estimated as the average vertical force acting on the slider in one sliding period, Eq. (14). The model parameters are the same as for Fig. 1. The connecting lines are guides to the eye. In the inset the scale is logarithmic on both axes to better illustrate the quadratic scaling of friction for large loads, at all speeds. At small positive loads, friction becomes very nonmonotonic and even exhibits a deep minimum at the larger considered speed. The reentrance of both friction curves at negative loads exhibits the standard attractive, large-distance regime observed in AFM experiments.

Figure 4

The $|V_{\mathrm{ext}}\left(Q\right)|$ “form factors” for varying LJ parameter $\sigma$ and distance $d$ in the slider-chain interaction potential, assuming an interaction depth $V_0=5\times {10}^{-4}K{a}^2$. (a) Distance is a fixed ratio $d=0.92\sigma$ to the LJ parameter. (b) Detail of the region where $V_{\mathrm{ext}}\left(Q\right)$ vanishes, for fixed $\sigma =0.5a$, and for three similar values of $d$. The bold arrow marks the $Q$ value $2.8204/a$ corresponding to the $v_{\mathrm{SL}}=0.7v_{\mathrm{s}}$ line crossing the dispersion in Fig. 1.

Figure 5

Friction-velocity profiles obtained for the same $\sigma$ values as in Fig. 4. Distance is always $d=0.92\sigma$. Other parameters are slider interaction depth $V_0=5\times {10}^{-4}K{a}^2$ and damping rate $\gamma =0.1{(m/K)}^{1/2}$.

Figure 6

Friction profile for simulated composite tips consisting in the sum of $N_{\mathrm{tip}}$ equal LJ contributions displaced at intervals of $l=0.7a$. All other parameters are the same as used previously: $\sigma =0.5a,d=0.465a,V_0=5\times {10}^{-4}K{a}^2$, and $\gamma =0.1{(m/K)}^{1/2}$.

Figure 7

Comparison of the Fourier transforms $|V_{\mathrm{ext}}{\left(Q\right)|}^2$ of the slider-chain interaction, for simulated sliders consisting in the sum of $N_{\mathrm{tip}}$ LJ contributions placed regularly at an interval $l=0.7a$. The other parameters for $V_{\mathrm{ext}}$ are the standard ones used in this work: $\sigma =0.5a,d=0.465a,V_0=5\times {10}^{-4}K{a}^2$.