Tribology Letters

, Volume 41, Issue 3, pp 573–586 | Cite as

Modeling Bearing and Shear Forces in Molecularly Thin Lubricants

  • Antonis I. Vakis
  • Melih Eriten
  • Andreas A. Polycarpou
Original Paper


Under the effects of high shear rate and confinement between solid surfaces, the behavior of a thin lubricant film deviates from that of the bulk, resulting in significant increases of lubricant viscosity and interfacial slip. A semi-empirical model accounting for the breakdown of continuum theory at the nanoscale is proposed—based on film morphology and chemistry from available experimental and molecular dynamics simulation data—to describe lubricant behavior under shear. Viscosity stiffening and interfacial slip models are introduced into the formulations of the normal (bearing) and shear forces acting on a sphere that moves within a thin lubricant film parallel to a rigid plane. The experimentally measured ‘apparent’ viscosity confounding the effects of both stiffening and slip is used to predict the hydrodynamic forces acting on a fully or partially submerged sphere for the purposes of describing lubricant contact in magnetic storage. The proposed sphere-on-flat model forms the basis of a future, dynamic contact with friction model that will account for lubricant contact in the context of molecularly thin lubricated rough surface contact.


Nanotribology Magnetic data storage Sub-boundary lubrication Non-Newtonian hydrodynamic effects Viscoelasticity Rheology 

List of Symbols


Oscillation amplitude

a, b

Viscosity-gap model coefficients

a′, b

Viscosity-shear rate model coefficients


Molecular coverage of surface area


Damping coefficient (for fluid ‘contact’)


Liquid gap


Driving frequency of oscillation


Dimensionless slip factor


Storage modulus


Loss modulus


Complex modulus


Heaviside function


Film thickness


Average molecular height of bonded layer


Solid-solid gap (minimum film thickness)


Slip length




Normal (bearing) hydrodynamic force


Normal solid force


Normal transitional force


Shear hydrodynamic force


Solid friction force


Shear transitional force


Sphere (probe or asperity) radius


Spherical cap area


Total lubricant layer thickness


Shearing velocity


Fluid velocity along x-direction


Dimensionless form of u


Slip velocity

\( u_{\text{s}}^{*} \)

Dimensionless form of u s


Wall velocity (as boundary condition)


Fluid velocity in y-direction


Dimensionless form of v

\( \dot{\gamma } \)

‘Apparent’ shear rate

\( \dot{\gamma }_{\text{true}} \)

‘True’ shear rate (accounting for slip)


Normalized y-coordinate


‘Apparent’ viscosity


‘True’ viscosity (decoupled from slip)


Wall friction (liquid–solid coupling)


Maximum limiting viscosity


Bulk viscosity


Viscosity (η = η′)




Complex viscosity


Minimum liquid gap


Normalized z-coordinate


Normalized x-coordinate


Root-mean-square roughness of substrate


Shear stress on sphere surface


Geometric factor


Geom. f. for fully submerged sphere


Geom. f. for partially submerged sphere


Solid interference


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Antonis I. Vakis
    • 1
  • Melih Eriten
    • 1
  • Andreas A. Polycarpou
    • 1
  1. 1.Department of Mechanical Science and EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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