Simple Law for Third-Body Friction

Fei Deng, Georgios Tsekenis, and Shmuel M. Rubinstein

Phys. Rev. Lett. 122, 135503 – Published 5 April 2019

Abstract

A key difficulty to understanding friction is that many physical mechanisms contribute simultaneously. Here we investigate third-body frictional dynamics in a model experimental system that eliminates first-body interaction, wear, and fracture, and concentrates on the elastic interaction between sliding blocks and third bodies. We simultaneously visualize the particle motion and measure the global shear force. By systematically increasing the number of foreign particles, we find that the frictional dissipation depends only on the size ratio between surface asperities and the loose particles, irrespective of the particle’s size or the surface’s roughness. When the particles are comparable in size to the surface features, friction increases linearly with the number of particles. For particles smaller than the surface features, friction grows sublinearly with the number of particles. Our findings suggest that matching the size of surface features to the size of potential contaminants may be a good strategy for reliable lubrication.

Received 24 August 2018

DOI:https://doi.org/10.1103/PhysRevLett.122.135503

Physics Subject Headings (PhySH)

Friction

Nonlinear DynamicsInterdisciplinary PhysicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Fei Deng^1,†, Georgios Tsekenis^1,†, and Shmuel M. Rubinstein^1,2,*

¹Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
²Kavli Institute for Bionano Science and Technology, Harvard University, Cambridge, Massachusetts 02138, USA

^*Corresponding author. shmuel@seas.harvard.edu
^†These authors contributed equally to this work.

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Vol. 122, Iss. 13 — 5 April 2019

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Images

Figure 1
Experimental setup and raw data. (a) Schematic of experimental setup. (b) The PDMS rings with disordered (left) and ordered (right) surfaces with cosine asperities, as well as two sizes of polyethylene particles. (c) Shear force time series for $Δ = 1.45$ and (d) for $Δ = 0.5$ on disordered surfaces. The initial force times series ( $N = 0$ ) contain oil and no particles. The number of added particles increases downward within each panel.
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Figure 2
Frictional resistance depends only on $Δ$ and $N$ . (a) The average shear force $f_{ave}$ normalized by its single particle ( $N = 1$ ) value as a function of the number of particles, $N$ , for different $Δ = D_{b} / L_{f}$ . The inset shows the same data without normalization. The scaling $f_{ave} \sim N^{β}$ is robust even though the proportionality factor may not be. The error bars represent the variation of particle size according to the manufacturer. (b) $β$ as a function of $Δ$ . ( $β \sim 0.15$ for $Δ = 0.26$ , $β \sim 0.5$ for $Δ = 0.5$ , $β \sim 1.0$ for $Δ = 1.45$ ). (inset) The marker symbols reveal the particle diameters $D_{b}$ and asperity sizes $L_{f}$ we used for the $Δ$ ’s. Open (filled) symbols represent ordered (disordered) surfaces, respectively.
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Figure 3
Friction spikes correlate highly with discontinuities in the motion of third-body particles. (a) Three snapshots showing particles trapped between the two PDMS rings ( $N = 1$ , 10, 50 particles of size $D_{b} = 550 μ m$ on a disordered ring surface with $L_{f} = 500 μ m$ ). (b) Composite image of a single-particle experiment showing the particle’s entire trajectory. The dashed lines indicate where the force discontinuities occur. (c) Radial velocity (top), shear force (middle), and the angular velocity (bottom) of a single particle as a function of time. The dashed lines show where discontinuities of velocity and force occur. (d),(e) The success pairing rate of the cross-correlation of velocity spikes with shear force spikes as indicated by vertical blue lines is 0.8571 for single-particle experiments and 0.8475 for a 10-particle experiment. These success pairing rates are more than 3-sigma away from the Gaussian correlation (red curves) exhibited by synthetic data (stars). (f),(g) Cross-correlation matrix of velocity and shear force spikes for six single particle runs and for one 10-particle run.
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Figure 4
Third-body particles’ velocities statistics. (a),(b) Trajectories of individual particles unfolded in azimuthal angle $θ (t)$ vs time for $Δ = 0.55 < 1$ with $N = 30$ particles (left) and $Δ = 1.1 \geq 1$ with $N = 10$ particles (right). We colored with red and green the trajectory of the particle that traveled the least and most distance out of all the particles that we successfully tracked in the entirety of the experiment. (c),(d) The respective distributions of instantaneous velocities of (a),(b). The thick line is the total distribution of all the particles instantaneous velocities. (e) The total particle instantaneous velocity distributions for $N = 1$ , 30, 100 with $Δ = 0.55$ and $N = 1$ , 10, 50 with $Δ = 1.1$ . (inset) The average particle speed as a function of particle number $N$ . The black downward arrow indicates the imposed rotational speed of the top plate; the blue and red upward arrow indicates the expected fluid speed at the center of a small ( $Δ = 0.55$ ) or large ( $Δ = 1.1$ ) particle for a linear laminar profile from the stationary bottom plate to the top steady moving plate (assuming no roughness present).
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