# Modeling Bearing and Shear Forces in Molecularly Thin Lubricants

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## Abstract

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.

### Keywords

Nanotribology Magnetic data storage Sub-boundary lubrication Non-Newtonian hydrodynamic effects Viscoelasticity Rheology### List of Symbols

*A*Oscillation amplitude

*a*,*b*Viscosity-gap model coefficients

*a*′,*b*′Viscosity-shear rate model coefficients

*C*Molecular coverage of surface area

*c*Damping coefficient (for fluid ‘contact’)

*d*_{0}Liquid gap

*f*Driving frequency of oscillation

*f**Dimensionless slip factor

*G*′Storage modulus

*G*″Loss modulus

*G**Complex modulus

*H*Heaviside function

*h*Film thickness

*h*_{B}Average molecular height of bonded layer

*h*_{0}Solid-solid gap (minimum film thickness)

*L*_{s}Slip length

*p*Pressure

*P*_{fluid}Normal (bearing) hydrodynamic force

*P*_{solid}Normal solid force

*P*_{trans}Normal transitional force

*Q*_{fluid}Shear hydrodynamic force

*Q*_{solid}Solid friction force

*Q*_{trans}Shear transitional force

*R*Sphere (probe or asperity) radius

*S*Spherical cap area

*t*Total lubricant layer thickness

*U*Shearing velocity

*u*Fluid velocity along

*x*-direction*u**Dimensionless form of

*u**u*_{s}Slip velocity

- \( u_{\text{s}}^{*} \)
Dimensionless form of

*u*_{s}*u*_{w}Wall velocity (as boundary condition)

*v*Fluid velocity in

*y*-direction*v**Dimensionless form of

*v*- \( \dot{\gamma } \)
‘Apparent’ shear rate

- \( \dot{\gamma }_{\text{true}} \)
‘True’ shear rate (accounting for slip)

- ζ
Normalized

*y*-coordinate- η
‘Apparent’ viscosity

- η
_{true} ‘True’ viscosity (decoupled from slip)

- η
_{w} Wall friction (liquid–solid coupling)

- η
_{0} Maximum limiting viscosity

- η
_{∞} Bulk viscosity

- η′
Viscosity (

*η*=*η*′)- η″
Elasticity

- η*
Complex viscosity

- κ
Minimum liquid gap

- λ
Normalized

*z*-coordinate- ξ
Normalized

*x*-coordinate- σ
Root-mean-square roughness of substrate

- τ
Shear stress on sphere surface

*Ω*Geometric factor

*Ω*_{f-s}Geom. f. for fully submerged sphere

*Ω*_{p-s}Geom. f. for partially submerged sphere

- ω
Solid interference

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