Tribology Letters

, 44:367 | Cite as

Analytical Models for Atomic Friction

Methods Paper

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.

Keywords

Nanotribology Friction mechanisms AFM Stick-slip Dynamic modeling 

List of Symbols

Variables

a

Substrate lattice spacing

b

Tip lattice spacing

Ceff

Effective stiffness (cantilever, tip, and contact)

d

Superstructure periodicity

f

Actuation frequency

f0

Attempt frequency

fnt

Frequency of the tip apex mode (nanocontact)

fPT

Frequency of the one effective mode of the PT model

F

Friction force

Fc

Maximum friction at zero temperature

Fn

Normal force

Fts

Interaction force in the normal direction

k

System stiffness (cantilever and tip)

kt

Stiffness of spring connecting neighboring tip atoms

kn

Normal stiffness

m

Mass of tip

N

Number of atoms

p

Probability of a transition

t

Time

tv

Average time for the tip to traverse one lattice spacing

T

Temperature

U

Corrugation potential amplitude

Uc

Corrugation potential

v

Sliding speed of support

vc

Critical speed

V

Total potential energy

x

Displacement of the tip in the sliding direction

xt

Transition point

xsp

Displacement of the support

y

Displacement of the tip perpendicular to applied sliding direction

z

Displacement of the tip in the normal direction

Greek Symbols

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

Supplementary material

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Supplemental material 1 (PDF 11 kb)
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Supplemental material 2 (M 4 kb)
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Supplemental material 3 (M 7 kb)
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Supplemental material 4 (M 9 kb)
11249_2011_9850_MOESM5_ESM.m (8 kb)
Supplemental material 5 (M 8 kb)

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Purdue UniversityWest LafayetteUSA
  2. 2.University of California MercedMercedUSA

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