Predicting frictional aging from bulk relaxation measurements

Abstract

The coefficient of static friction between solids normally increases with the time they have remained in static contact before the measurement. This phenomenon, known as frictional aging, is at the origin of the difference between static and dynamic friction coefficients but has remained difficult to understand. It is usually attributed to a slow expansion of the area of atomic contact as the interface changes under pressure. This is however challenging to quantify as surfaces have roughness at all length scales. In addition, friction is not always proportional to the contact area. Here we show that the normalized stress relaxation of the surface asperities during frictional contact with a hard substrate is the same as that of the bulk material, regardless of the asperities' size or degree of compression. This result enables us to predict the frictional aging of rough interfaces based on the bulk material properties of two typical polymers: polypropylene and polytetrafluoroethylene.

Fig. 1. Stress relaxation of the polymer spheres after bulk and/or surface deformations.

a Schematic of the setup for deformation experiments. A polypropylene or polytetrafluoroethylene sphere is squeezed between a steel plate connected to a rheometer and a glass slide installed on a microscope via a very stiff frame. b Relaxation of the applied force F needed to maintain a constant strain as a function of time for two values of the initial force $F_0$; 0.12 N (blue circles) and 53.05 N (red triangles). Dashed lines indicate logarithmic relaxation, Eq. 1. c Relation between constants C and B as obtained from fitting stress relaxation characteristics to Eq. 1 for different values of $F_0$. d Constant C as a function of $F_0$. Dashed lines in (c) and (d) are linear fits. e Microscopy images of the contact area for four values of $F_0$. These images have been taken $t=60$ s after the squeezing was applied. Colored symbols indicate which data points in (d) correspond to which images in (e).

Fig. 2. Failure of the bulk polymer.

Deformation d of polypropylene spheres (as shown in Fig. 1a) as a function of the initially applied compressive force $F_0$. At small forces ($F_0<10N$), d increases with $F_0$ to the power of 2/3 (blue dashed line), corresponding to Hertz’s theory for elastic deformation. At larger forces, this changes into a linear increase (red dashed line), corresponding to plastic deformation (see Supplementary Fig. 10 for a linear-linear plot).

Fig. 3. The interface aging and the evolution of static friction.

a Schematic of the apparatus for the friction measurements. A rheometer is used to rotate a hollow tube around its symmetry axis. Three polymer spheres are attached to the underside of the tube at equal distances, and these, together with a substrate at the bottom, form the frictional interfaces. b Friction as a function of sliding distance for polypropylene spheres on glass under the combined weight of the tube and spheres (42 mN) after different waiting times t_w. The polymer spheres are not replaced between these measurements. c The ratio of static friction (F_s, the peak friction value) to dynamic friction (F_d, mean friction in steady-state) for multiple aging experiments as in (b): solid triangles are from curves in (b), open right triangles correspond to a subsequent repeat of the experiment with the same spheres, open left triangles are with new polypropylene spheres, $\times$ signs are with a new tube of 87 mN weight, and red diamonds correspond to an experiment on a silicon wafer. In all the above experiments, the imposed sliding velocity is 86 nm/s. + signs and open diamonds are repeats of the last two experiments, respectively, with a sliding velocity of 258 nm/s. The solid line corresponds to equation $F_s\left(t\right)=F_{\tau }+n{F}_{\mathrm{d}}\ln \left(\frac{t_w}{\tau }\right)$ with $n=0.0417$ determined from the stress relaxation experiments (ln is natural logarithm). The free parameter $F_{\tau }$ is the static friction at time $t=\tau$. Here, $\frac{F_s}{F_d}=1+0.0417\ln \left(\frac{t_w}{1\mathrm{s}}\right)$.

Fig. 4. Large changes in surface topography and chemistry do not affect the frictional aging rate.

a Smooth and b rough polypropylene spheres (Supplementary Fig. 4) with root-mean-square (rms) roughness heights of 0.23 and 1.81 μm, respectively, have very different real contact areas under the same normal force of 0.35 N. c These polypropylene (PP) spheres, however, show the same aging characteristics as for the pristine spheres with rms-roughness of around 0.63. The roughness values are obtained after correcting for the macroscopic spherical curvature. The solid line is, as in Fig. 3c, $\frac{F_s}{F_d}=1+0.0417\ln \left(\frac{t_w}{1\mathrm{s}}\right)$. The frictional aging for the polypropylene spheres on a silanized glass substrate (green squares) is also consistent with the slope $n=0.0417$ determined from the bulk stress relaxation experiments, although in this case, there is a clear vertical shift associated to decrease of the friction due to the change in surface chemistry. The dashed line is $\frac{F_s}{F_d}=0.95+0.0417\ln \left(\frac{t_w}{1\mathrm{s}}\right)$. d While silanization of the glass substrate does not change the frictional aging rate, it reduces the absolute dynamic friction by a factor of 3 to 4.