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Analytical Models for Atomic Friction

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Abstract

In this methods article, we describe application of Prandtl–Tomlinson models and their extensions to interpret dry atomic-scale friction. The goal is to provide a practical overview of how to use these models to study frictional phenomena. We begin with the fundamental equations and build on them step-by-step—from the simple quasistatic one-spring, one-mass model for predicting transitions between friction regimes to the two-dimensional and multi-atom models for describing the effect of contact area. The intention is to bridge the gap between theoretical analysis, numerical implementation, and predicted physical phenomena. In the process, we provide an introductory manual with example computer programs for newcomers to the field, and an illustration of the significant potential for this approach to yield new fundamental understanding of atomic-scale friction.

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Abbreviations

a :

Substrate lattice spacing

b :

Tip lattice spacing

C eff :

Effective stiffness (cantilever, tip, and contact)

d :

Superstructure periodicity

f :

Actuation frequency

f 0 :

Attempt frequency

f nt :

Frequency of the tip apex mode (nanocontact)

f PT :

Frequency of the one effective mode of the PT model

F :

Friction force

F c :

Maximum friction at zero temperature

F n :

Normal force

F ts :

Interaction force in the normal direction

k :

System stiffness (cantilever and tip)

k t :

Stiffness of spring connecting neighboring tip atoms

k n :

Normal stiffness

m :

Mass of tip

N :

Number of atoms

p :

Probability of a transition

t :

Time

t v :

Average time for the tip to traverse one lattice spacing

T :

Temperature

U :

Corrugation potential amplitude

U c :

Corrugation potential

v :

Sliding speed of support

v c :

Critical speed

V :

Total potential energy

x :

Displacement of the tip in the sliding direction

x t :

Transition point

x sp :

Displacement of the support

y :

Displacement of the tip perpendicular to applied sliding direction

z :

Displacement of the tip in the normal direction

α:

Parameter that reflects the resonance of normal mode actuation

αa :

Magnitude of amplitude modulation

αc :

Magnitude of centerline modulation

β:

Curvature of the corrugation potential

γ:

Parameter that reflects the resonance of torsional mode actuation

η:

Stick-slip regime transition parameter

κ:

Transition rate

μ:

Viscous friction (damping) coefficient

ξ:

Thermal activation force

τ:

Average time to hop out from a potential well due to thermal activation

ω:

Angular frequency

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Acknowledgments

We are grateful for insightful discussions with Drs. Qunyang Li and Danny Perez. This study was funded by the National Science Foundation through grant CMMI-1068552.

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  1. Purdue University, West Lafayette, IN, 47907, USA

    Yalin Dong & Ajay Vadakkepatt

  2. University of California Merced, Merced, CA, 95343, USA

    Ashlie Martini

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  1. Yalin Dong
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Correspondence to Ashlie Martini.

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Dong, Y., Vadakkepatt, A. & Martini, A. Analytical Models for Atomic Friction. Tribol Lett 44, 367 (2011). https://doi.org/10.1007/s11249-011-9850-2

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