Noncontact friction: Role of phonon damping and its nonuniversality

While obtaining theoretical predictions for dissipation during sliding motion is a difficult task, one regime that allows for analytical results is the so-called noncontact regime, where a probe is weakly interacting with the surface over which it moves. Studying this regime for a model crystal, we extend previously obtained analytical results and confirm them quantitatively via particle based computer simulations. Accessing the subtle regime of weak coupling in simulations is possible via use of Green-Kubo relations. The analysis allows us to extract and compare the two paradigmatic mechanisms that have been found to lead to dissipation: phonon radiation, prevailing even in a purely elastic solid, and phonon damping, e.g., caused by viscous motion of crystal atoms. While phonon radiation is dominant at large probe-surface distances, phonon damping dominates at small distances. Phonon radiation is furthermore a pairwise additive phenomenon so that the dissipation due to interaction with different parts (areas) of the surface adds up. This additive scaling results from a general one-to-one mapping between the mean probe-surface force and the friction due to phonon radiation, irrespective of the nature of the underlying pairwise interaction. In contrast, phonon damping is strongly nonadditive, and no such general relation exists. We show that for certain cases, the dissipation can even decrease with increasing surface area the probe interacts with. The above properties, which are rooted in the spatial correlations of surface fluctuations, are expected to have important consequences when interpreting experimental measurements, as well as scaling with system size.

Figure 1

Sketch of the investigated noncontact friction setup, with the blue particle being the (point) probe at ${\mathit{r}}^p$. The square block represents the semi-infinite solid in the continuum description. The red particles are the crystal atoms in the discrete counterpart, where the dashed lines are the bonds among the nearest and the next nearest neighbors. The next nearest neighbor bonds render the shear modulus finite. The distance vector $\mathit{d}$ denotes the distance between the probe and the lattice position of a surface atom with $h$ being the height of the probe, and $\mathit{u}$ the fluctuating displacement vector field. These two vectors thus form a fluctuating distance vector, from which the instantaneous pairwise force $\mathit{f}$ between the probe and a surface atom is evaluated.

Figure 2

The dimensionless prefactors of the viscous contribution, ${\zeta }_{xx}^{\mathrm{vis}}$ and ${\zeta }_{zz}^{\mathrm{vis}}$, as a function of power $n$ at $b=\sqrt{3}$. Solid lines represent fits, using the forms given in the figure.

Figure 3

The friction tensor as a function of height $h$ at different material properties. In (a) and (b), the viscosity $\gamma$ is varied at fixed $c_{\mathrm{T}}^{\prime }=\sqrt{10}[\mathrm{s}.\mathrm{u}.]$. In (c) and (d), the real part of the speed of sound is varied at fixed $\gamma =8[\mathrm{s}.\mathrm{u}.]$. The data points are the corresponding simulation results, in which we set $a=1, M=1, \alpha =1$ and $n_{\mathrm{A}}=1$. The height $h$ is given in the unit of lattice spacing $a$, and the system size is $(X,Y,Z)=(99a,99a,101a)$. The black lines in (a) and (c) denote the analytic prediction for the master curve, respectively. The colored solid lines in (b) and (d) are the analytic predictions for the color matched material properties, and the gray dashed lines draw the inverse power laws in $h$. The inset in (b) shows a closer look at smaller $h$ with a linear scale on both axes, where the viscous contribution dominates. In Eq. (21), the crossover from the viscous to elastic contributions es expected at a height of $h^{*}=1.56\gamma /c_{\mathrm{T}}^{\prime }$. The error bars, representing statistical errors, are estimated from the results found when using one fifth of the data.

Figure 4

The friction tensor as a function of material properties at fixed heights $h$ in the unit of lattice spacing $a$. In the first column, we change the viscosity at fixed real part of the speed of sound $c_{\mathrm{T}}^{\prime }=\sqrt{10}[\mathrm{s}.\mathrm{u}.]$, and in the second column, we change the real part of the speed of sound $c_{\mathrm{T}}^{\prime }$ at fixed viscosity $\gamma =8[\mathrm{s}.\mathrm{u}.]$. The data points denote the simulation results, and the black lines are the analytic predictions for each case. In (g) and (h) the gray dashed lines are the indicators for the power laws in $c_{\mathrm{T}}^{\prime }$, and the arrows mark the expected crossover from the viscous to the elastic contribution, given by $c_{\mathrm{T}}^{\prime *}=1.58\gamma /h$. The statistical error bars, representing one standard deviation, are calculated by partitioning the time series of data into 5 pieces.

Figure 5

The interaction range dependence of the friction tensor, the parallel (a) and perpendicular (b) components measured in the simulation units. The interaction is $\tilde{V}(x,y,h)=\alpha exp-\frac{x^2+y^2}{l^2}{(x^2+y^2+h^2)}^{-3/2}.$ The probe height is fixed at $h=30$ in the unit of lattice spacing $a$. The parallel ${\mathrm{\Gamma }}_{xx}$ and perpendicular ${\mathrm{\Gamma }}_{zz}$ components are normalized by the viscosity $\gamma$ and by the elastic contribution ${\mathrm{\Gamma }}_{zz}(\gamma =0)$, respectively. The black dash lines indicate various power laws. The statistical error bars, representing one standard deviation, are estimated by partitioning the time series of data into five pieces.

Figure 6

The real (a) and imaginary (b) part of the dispersion relations for the longitudinal and transverse modes of the simple cubic crystal with $M=a=1, {\kappa }_1={\kappa }_2=10$, and ${\gamma }_1={\gamma }_2=1$. Note that there exist two transverse modes, and they are degenerate.

Figure 7

The probe-surface force covariance function of the (a) $xx$ and (d) $zz$ component, as a function of time, for parameters $h,c_T^{\prime },\gamma$ as indicated. Panels (b) and (e) show the respective running value of the Green-Kubo integral, which ideally should converge to a plateau. Panel (c) shows the double running integral of the $xx$ component, where a regime of linear growth is clearly visible (extending to the vertical dashed line, marking the upper integration bound). The plateau in (b), represented by the red line, is determined from the slope of the double running integral. For the $zz$ component, the double integral method proved unsuccessful, and so here the first-dip method was used; panel (d) indicates the first-dip time, where the covariance function becomes zero for the first time. The red line in (e) indicates the corresponding estimate of the $zz$ component of the friction tensor. The black dashed lines in (b) and (e) denote the predictions of our theory.

Figure 8

The branch points, the corresponding branch cuts, and the pole on the complex plane of $k_{\left|\right|}$, when $c_{\mathrm{L}}\left(\omega \right)=\sqrt{3}{c}_{\mathrm{T}}\left(\omega \right)$. The branch points and the pole are aligned with the slop defined by $\psi /2$. The contour path is indicated by the arrows, and the radius of the arc is infinitely large. The small angle $\epsilon$ is introduced to ensure the convergence of the integration.

Figure 9

The branch points and the pole on the complex plane of $k_{\left|\right|}$ after truncating $\psi \left(\omega \right)$ at the leading order of $\omega$, assuming $c_{\mathrm{L}}\left(\omega \right)=\sqrt{3}{c}_{\mathrm{T}}\left(\omega \right)$.

Figure 10

The analytic expression of ${\mathrm{\Gamma }}_{xx}$ and ${\mathrm{\Gamma }}_{zz}$ compared against the numerical results by directly integrating Eq. (D2) with Eq. (D4).