Velocity-weakening and -strengthening friction at single and multiasperity contacts with calcite single crystals

Abstract

SignificanceThe empirical nature of rate-and-state friction (RSF) equations remains a drawback to their application to predict earthquakes. From nanoscale friction measurements on smooth and rough calcite crystals, a set of parameters is analyzed to elucidate microscopic processes dictating RSF. We infer the influence of roughness on the velocity dependence of friction in dry environment and that atomic attrition leads to stick-slip instabilities at slow velocities. In fault dynamics, stick-slip is associated with seismic slips. The aqueous environment eliminates atomic attrition and stick-slip and dissolves calcite under pressure. This yields remarkable lubrication, even more so in rough contacts, and suggests an alternative pathway for seismic slips. This work has implications for understanding mechanisms dictating fault strength and seismicity.

Keywords: atomic force microscopy; calcite; contact aging; friction; rate-and-state friction law.

Fig. 1.

Representative AFM topographical images (500 nm × 500 nm) and corresponding cross-sections of calcite surfaces #1, #2, #3, and #4. AFM images were taken in air in tapping mode with a sharp tip. The asperity size significantly increased from #1 to #4, as indicated by the profiles (note the different y axes). Representative 6-µm × 6-µm and 1-µm × 1-µm images of calcite surfaces are shown in SI Appendix, Figs. S1 and S2.

Fig. 2.

(A) Pull-off force on smooth and roughened calcite single crystals measured with a blunt Si tip (190-nm radius) as a function of normal load. The rms roughness of each surface is given. The error bar gives the standard variation of the pull-off force at each load. (B) Average contact stress as a function of normal load estimated via the DMT model on smooth surfaces and on single asperities with radii of 23, 24.6, and 29 nm. The green dashed line assumes that the load is carried out by four asperities of 23 nm in radius. Note that the asperities are assumed to be smooth, and hence, this is a simplified geometry. (C) Conceptual picture of the lubricated contact between the tip and calcite, illustrating the balance between the normal stress, fluid pressure, and disjoining pressure П (D). This balance is assumed to be maintained during sliding with concurrent dissolution of calcite.

Fig. 3.

Friction force between a blunt Si tip and calcite with rms roughness values of (A) 0.1 nm (#1; smooth), (B) 5 nm (#2), (C) 10 nm (#3), and (D) 15 nm (#4) as a function of sliding velocity and normal load in dry N₂ to avoid water contamination. The marker legend used in this paper is as follows: empty circles for 10 nN (black), diamonds for 20 nN (green), triangles for 50 nN (dark green), squares for 75 nN (red), and crosses for 100 nN (red). The error bars give the variation in friction over eight friction loops at each speed. The lines illustrate the logarithmic change of friction during velocity weakening (regime D-I; dashed lines) and velocity strengthening (regime D-II; full lines). The plateau between these $V^{*}$ and $V^{**}$ is significant. Note that the x axis is on a logarithmic scale and that the y axis is on a linear scale. The lines indicate logarithmic trends and are to guide the eye. The quantitative agreement between replica experiments is outstanding except for surface #4, which sometimes showed a less pronounced velocity-weakening regime, likely depending on the local topography. The fits of the models to the experimental results are in SI Appendix, Fig. S6, and the fitting parameters are in SI Appendix, Table S1.

Fig. 4.

Fitting parameters for the friction of smooth and rough calcite single crystals in dry and aqueous environments. (A) Rate slopes ${\alpha }_D$ and ${\beta }_D$ in the velocity-weakening [D-I, ${\beta }_D\text{ln}(V)$] and velocity-strengthening [D-II, ${\alpha }_D\text{ln}(V)$] regimes, respectively, in dry nitrogen. Each box contains the results at all loads except for surface #2, which does not include 10 nN. (B) Transitions velocities $V^{*}$ and $V^{**}$ of each surface. Rate parameters (C) ${\alpha }_D$ and (D) ${\beta }_D$ as a function of load. (E) Rate parameters ${\alpha }_{PS}$ and ${\alpha }_W$ for regimes W-I and W-II, respectively, and correlation length $\lambda =kT/{\alpha }_{W}L$, all in an aqueous environment. (F) Transition velocity $V_{PS}$, (G) ${\alpha }_W$, and (H) ${\alpha }_{PS}$ as a function of normal load.

Fig. 5.

(A) Pull-off force vs. loading time for a smooth calcite surface in dry nitrogen. A sharp AFM tip was used to measure the pull-off force under the normal loads of 10 nN (black) and 75 nN (red). (B) Lateral force experienced by a blunt Si tip sliding at a velocity of 0.5 µm/s on smooth calcite after being held for a contact time between 0 and 60 s at 100 nN in dry nitrogen. (C) Representative stick–slip in friction loops on smooth calcite at sliding velocities of 1 µm/s (red) and 400 µm/s (pink) in dry nitrogen under a normal load of 100 nN. (D) AFM images (2 × 2 µm) of a smooth calcite surface after eight contact mode images (normal load of 75 nN and sliding velocity of 1 µm/s) were taken in the area limited by the white square (800 × 800 nm). The 2- × 2-µm images were taken in contact mode as well but at a small load of 5 nN to avoid wear of the surface.

Fig. 6.

Friction measurements between (A) smooth or (B–D) rough calcite surfaces and a blunt Si tip as a function of velocity from 0.1 to 600 μm/s. Three different regimes are distinguished. At small loads (10 nN), friction increases with the logarithm of velocity. At higher loads, regime W-I extends to V < $V_{PS}$, where $F\sim V$ and the pressure solution is most prominent, and regime W-II (V > $V_{PS}$), where $F\sim \ln V$. Black dashed lines represent logarithmic fits to the friction force in regime W-II, and full lines represent the linear fit in regime W-I. Note the different scales of the y axes of the plots. The fits of the models to the experimental results are in SI Appendix, Fig. S11, and the fitting parameters are in SI Appendix, Table S3.

Fig. 7.

(A) Friction coefficient in the dry environment and in the two regimes of the aqueous environment at low loads (labeled as water) and high loads (labeled as PS). Each box includes results at velocities of 0.1, 1, 10, 100, and 600 µm/s. (B) Friction vs. load for calcite surface #4 at various velocities (0.1, 1, 10, 100, and 600 µm/s) illustrating how the slope of friction vs. load ($d{F}_L/dL$) in the aqueous environment was obtained in two regimes at low and high loads (labeled as water and PS in A, respectively). The friction vs. load curves for other conditions are shown in SI Appendix, Figs. S5 and S10.