Tribology Letters

, 44:367 | Cite as

Analytical Models for Atomic Friction

Methods Paper


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.


Nanotribology Friction mechanisms AFM Stick-slip Dynamic modeling 

List of Symbols



Substrate lattice spacing


Tip lattice spacing


Effective stiffness (cantilever, tip, and contact)


Superstructure periodicity


Actuation frequency


Attempt frequency


Frequency of the tip apex mode (nanocontact)


Frequency of the one effective mode of the PT model


Friction force


Maximum friction at zero temperature


Normal force


Interaction force in the normal direction


System stiffness (cantilever and tip)


Stiffness of spring connecting neighboring tip atoms


Normal stiffness


Mass of tip


Number of atoms


Probability of a transition




Average time for the tip to traverse one lattice spacing




Corrugation potential amplitude


Corrugation potential


Sliding speed of support


Critical speed


Total potential energy


Displacement of the tip in the sliding direction


Transition point


Displacement of the support


Displacement of the tip perpendicular to applied sliding direction


Displacement of the tip in the normal direction

Greek Symbols


Parameter that reflects the resonance of normal mode actuation


Magnitude of amplitude modulation


Magnitude of centerline modulation


Curvature of the corrugation potential


Parameter that reflects the resonance of torsional mode actuation


Stick-slip regime transition parameter


Transition rate


Viscous friction (damping) coefficient


Thermal activation force


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


Angular frequency



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.

Supplementary material

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Supplemental material 1 (PDF 11 kb)
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Supplemental material 5 (M 8 kb)


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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

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

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