Linear Aging Behavior at Short Timescales in Nanoscale Contacts

Kaiwen Tian, Zhuohan Li, Yun Liu, Nitya N. Gosvami, David L. Goldsby, Izabela Szlufarska, and Robert W. Carpick

Phys. Rev. Lett. 124, 026801 – Published 16 January 2020

Abstract

Nanoscale silica-silica contacts were recently found to exhibit logarithmic aging for times ranging from 0.1 to 100 s, consistent with the macroscopic rate and state friction laws and several other aging processes. Nanoscale aging in this system is attributed to progressive formation of interfacial siloxane bonds between surface silanol groups. However, understanding or even data for contact behavior for aging times $< 0.1 s$ , before the onset of logarithmic aging, is limited. Using a combination of atomic force microscopy experiments and kinetic Monte Carlo simulations, we find that aging is nearly linear with aging time at short timescales between $\sim 5$ and 90 ms. We demonstrate that aging at these timescales requires the existence of a particular range of reaction energy barriers for interfacial bonding. Specifically, linear aging behavior consistent with experiments requires a narrow peak close to the upper bound of this range of barriers. These new insights into the reaction kinetics of interfacial bonding in nanoscale aging advance the development of physically based rate and state friction laws for nanoscale contacts.

Received 30 May 2019

DOI:https://doi.org/10.1103/PhysRevLett.124.026801

Physics Subject Headings (PhySH)

Aging Chemical bonding Friction Surface & interfacial phenomena Tribology

Atomic force microscopy Monte Carlo methods

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Kaiwen Tian ^1,†, Zhuohan Li^2,†, Yun Liu^3,‡, Nitya N. Gosvami^4,¶, David L. Goldsby⁵, Izabela Szlufarska², and Robert W. Carpick ^4,*

¹Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
²Department of Materials Science and Engineering, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
³Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, USA
⁴Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
⁵Department of Earth and Environmental Science, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

^*Corresponding author. carpick@seas.upenn.edu
^†K. T. and Z. L. contributed equally to this work.
^‡Present address: Apple Inc., Cupertino, California 95014, USA.
^¶Present address: Department of Materials Science and Engineering, IIT Delhi, Hauz Khas, New Delhi, Delhi 110016, India.

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Vol. 124, Iss. 2 — 17 January 2020

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Images

Figure 1
Friction drop vs effective contact time. Relative $humidity = 80 %$ , applied normal $load = 284 nN$ , temperature $T = 297 K$ , loading point velocity $1.53 μ m / s$ , and adhesion undetermined. Upper left inset: data points from an effective contact time of ca. 5 ms to 0.11 s, with a linear timescale. The friction drop increases linearly with effective contact time from ca. 5 to 90 ms. The best-fit line is the red solid line in the inset. Results for contact times of 1, 10, and 100 s fall on a best-fit line, which is the purple solid line in the main plot, extended by a purple dashed line. Data points for a contact time of 0.1 s are clearly above this extrapolated line. The lower limit of the logarithmic trend is between 0.1 and 1 s. The lower right inset shows one lateral force vs lateral displacement curve for a 0.07 s hold time, and the friction drop. Hold times are varied randomly to exclude systematic errors.
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Figure 2
KMC simulations of number of bonds and energy barrier as a function of contact time. [(a) and (b)] Time evolution of the normalized number of interfacial bonds plotted on logarithmic (full range) and linear (early times) timescales. The number of bonds is normalized by the total number of available reaction sites before aging occurs. [(c) and (d)] Energy barrier for sites that form bonds at a given point in time as a function of time for Gaussian and uniform barrier distributions. The Gaussian distribution has a mean value of 0.8 eV with a standard deviation of 0.2 eV; the uniform distribution has a lower bound of 0.2 eV with a range of 1.2 eV. Dashed lines in (d) indicate the range of effective barriers within 5–90 ms.
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Figure 3
Artificial energy barrier distribution and KMC simulation results with different parameter sets. (a) Schematic plot of artificial barrier distribution. The barriers are decomposed into three parts. Low ( $E_{b, low}$ ) and high ( $E_{b, high}$ ) barriers are described by a $δ$ distribution, and middle ( $E_{b, mid}$ to $E_{b, mid} + E_{b, range}$ ) barriers are described by a uniform distribution. In our simulation, we set the percentage of reaction sites with low, middle, and high barriers to be 4%, 30%, and 66%, respectively. We use $E_{b, low} = 0.4$ , and $E_{b, high} = 0.95 eV$ . $E_{b, mid}$ and $E_{b, range}$ are two free parameters in the model. (b) Time evolution of interfacial number of bonds normalized by $N_{0, total}$ within 5–90 ms, with different values of $E_{b, mid}$ and $E_{b, range}$ . In (b), we show every 10th data points of the simulations for clarity.
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Figure 4
Qualitative and quantitative evaluation of aging curves with various sets of parameters. In all plots, different values of $E_{b, mid}$ and $E_{b, range}$ are considered. (a) Linearity of aging curves are quantified by Pearson correlation coefficient (higher coefficient means higher linearity). The three markers in this plot correspond to the three parameter sets used in Fig. 3 as denoted by the symbol used. (b) Number of bonds normalized by $N_{0, total}$ at 5 ms. (c) Increase in the normalized number of bonds during 5–90 ms. (d) Possible parameter space of middle barrier distribution shown in brown, and the impossible region shown in white. The possible region is selected when all the three criteria (both qualitative and quantitative) are met, i.e., Pearson correlation $coefficient > 0.98$ , normalized number of bonds at $5 ms < 6 %$ , and increase in the normalized number of $bonds < 10 %$ . Simulation results here are averaged over 10 simulations with different random seeds.
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