# Modeling Bearing and Shear Forces in Molecularly Thin Lubricants

- 318 Downloads
- 13 Citations

## 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

## References

- 1.Vakis, A.I., Polycarpou, A.A.: Head-disk interface nanotribology for Tbit/in
^{2}recording densities: near-contact and contact recording. J. Phys. D Appl. Phys.**43**, 22 (2010)CrossRefGoogle Scholar - 2.Vakis, A.I., Polycarpou, A.A.: Optimization of thermal fly-height control slider geometry for Tbit/in
^{2}recording. Microsyst. Technol.**16**(3), 1021–1034 (2010)CrossRefGoogle Scholar - 3.Vakis, A.I., Lee, S.-C., Polycarpou, A.A.: Dynamic head-disk interface instabilities with friction for light contact (surfing) recording. IEEE Trans. Magn.
**45**(11), 4966–4971 (2009)CrossRefGoogle Scholar - 4.Stanley, H.M., Etsion, I., Bogy, D.B.: Adhesion of contacting rough surfaces in the presence of sub-boundary lubrication. J. Tribol.
**112**(1), 98–104 (1990)CrossRefGoogle Scholar - 5.Fukuzawa, K., Itoh, S., Mitsuya, Y.: Fiber wobbling shear force measurement for nanotribology of confined lubricant molecules. IEEE Trans. Magn.
**39**, 2453–2455 (2003)CrossRefGoogle Scholar - 6.Demirel, A.L., Granick, S.: Relaxations in molecularly thin liquid films. J. Phys. Condens. Matter.
**8**, 9537–9539 (1996)CrossRefGoogle Scholar - 7.Itoh, S., Fukuzawa, K., Hamamoto, Y., Hedong, Z., Mitsuya, Y.: Fiber wobbling method for dynamic viscoelastic measurement of liquid lubricant confined in molecularly narrow gaps. Tribol. Lett.
**30**(3), 177–189 (2008)CrossRefGoogle Scholar - 8.Guo, Q., Chung, P.S., Jhon, M.S., Choi, H.J.: Nano-rheology of single unentangled polymeric lubricant films. Macromol. Theory Simul.
**17**(9), 454–459 (2008)CrossRefGoogle Scholar - 9.Demirel, A.L., Granick, S.: Transition from static to kinetic friction in a model lubricated system. J. Chem. Phys.
**109**(16), 6889–6897 (1998)CrossRefGoogle Scholar - 10.Bonaccurso, E., Butt, H.-J., Craig, V.S.J.: Surface roughness and hydrodynamic boundary slip of a Newtonian fluid in a completely wetting system. Phys. Rev. Lett.
**90**(14), 144501 (2003)CrossRefGoogle Scholar - 11.Zhu, Y., Granick, S.: Limits of the hydrodynamic no-slip boundary condition. Phys. Rev. Lett.
**88**(10), 106102 (2002)CrossRefGoogle Scholar - 12.Zhu, Y., Granick, S.: Rate-dependent slip of Newtonian liquid at smooth surfaces. Phys. Rev. Lett.
**87**(9), 096105 (2001)CrossRefGoogle Scholar - 13.Martini, A., Hsu, H.-Y., Patankar, N.A., Lichter, S.: Slip at high shear rates. Phys. Rev. Lett.
**100**(20), 206001 (2008)CrossRefGoogle Scholar - 14.Priezjev, N.V., Troian, S.M.: Molecular origin and dynamic behavior of slip in sheared polymer films. Phys. Rev. Lett.
**92**(1), 018302 (2004)CrossRefGoogle Scholar - 15.Barrat, J.L., Bocquet, L.: Large slip effect at a nonwetting fluid-solid interface. Phys. Rev. Lett.
**82**(23), 4671–4674 (1999)CrossRefGoogle Scholar - 16.Thompson, P.A., Troian, S.M.: A general boundary condition for liquid flow at solid surfaces. Nature
**389**(6649), 360–362 (1997)CrossRefGoogle Scholar - 17.Hu, H.-W., Granick, S., Schweizer, K.S.: Static and dynamical structure of confined polymer films. J. Non-Cryst. Solids
**172–174**, 721–728 (1994)CrossRefGoogle Scholar - 18.Peachey, J., Van Alsten, J., Granick, S.: Design of an apparatus to measure the shear response of ultrathin liquid films. Rev. Sci. Instrum.
**62**(2), 463–473 (1991)CrossRefGoogle Scholar - 19.Fukuzawa, K., Ando, T., Shibamoto, M., Mitsuya, Y., Zhang, H.: Monolithically fabricated double-ended tuning-fork-based force sensor. J. Appl. Phys.
**99**(9), 094901 (2006)CrossRefGoogle Scholar - 20.Fukuzawa, K., Hayakawa, K., Matsumura, N., Itoh, S., Zhang, H.: Simultaneously measuring lateral and vertical forces with accurate gap control for clarifying lubrication phenomena at nanometer gap. Tribol. Lett.
**37**(3), 497–505 (2009)CrossRefGoogle Scholar - 21.Minn, M., Sinha, S.K., Lee, S.K., Kondo, H.: High-speed tribology of PFPEs with different functional groups and molecular weights coated on DLC. Tribol. Lett.
**24**(1), 67–76 (2006)CrossRefGoogle Scholar - 22.Itoh, S., Takahashi, K., Fukuzawa, K., Amakawa, H., Hedong, Z.: Spreading properties of monolayer lubricant films: effect of bonded molecules. IEEE Trans. Magn.
**45**(11), 5055–5060 (2009)CrossRefGoogle Scholar - 23.Thompson, P.A., Robbins, M.O., Grest, G.S.: Structure and shear response in nanometer-thick films. Isr. J. Chem.
**35**(1), 93–106 (1995)Google Scholar - 24.Dhinojwala, A., Granick, S.: Surface forces in the tapping mode: solvent permeability and hydrodynamic thickness of adsorbed polymer brushes. Macromolecules
**30**(4), 1079–1085 (1997)CrossRefGoogle Scholar - 25.Kogut, L., Etsion, I.: A semi-analytical solution for the sliding inception of a spherical contact. J. Tribol.
**125**(3), 499–506 (2003)CrossRefGoogle Scholar - 26.Yeo, C.-D., Polycarpou, A.A., Kiely, J.D., Hsia, Y.-T.: Nanomechanical properties of sub-10 nm carbon film overcoats using the nanoindentation technique. J. Mater. Res.
**22**(1), 141–151 (2007)CrossRefGoogle Scholar - 27.Lee, S.-C., Polycarpou, A.A.: Microtribodynamics of pseudo-contacting head-disk interfaces intended for 1 Tbit/in
^{2}. IEEE Trans. Magn.**41**(2), 812–818 (2005)CrossRefGoogle Scholar - 28.Demirel, A.L., Granick, S.: Origins of solidification when a simple molecular fluid is confined between two plates. J. Chem. Phys.
**115**(3), 1498–1512 (2001)CrossRefGoogle Scholar - 29.Goldman, A.J., Cox, R.G., Brenner, H.: Slow viscous motion of a sphere parallel to a plane wall. I. Motion through a quiescent fluid. Chem. Eng. Sci.
**22**(4), 637–652 (1967)CrossRefGoogle Scholar - 30.Brinson, H., Brinson, C.: Polymer Engineering Science and Viscoelasticity: An Introduction. Springer, New York (2007)Google Scholar
- 31.Vinogradova, O.I.: Drainage of a thin liquid film confined between hydrophobic surfaces. Langmuir
**11**(6), 2213–2220 (1995)CrossRefGoogle Scholar - 32.Williams, J.: Engineering Tribology. Cambridge University Press, (2005)Google Scholar
- 33.Stahl, J., Jacobson, B.O.: A lubricant model considering wall-slip in EHL line contacts. J. Tribol.
**125**(3), 523–532 (2003)CrossRefGoogle Scholar - 34.Suh, A.Y., Lee, S.-C., Polycarpou, A.A.: Design optimization of ultra-low flying head-disk interfaces using an improved elastic-plastic rough surface model. J. Tribol.
**128**(4), 801–810 (2006)CrossRefGoogle Scholar - 35.Marchon, B., Dai, Q., Nayak, V., Pit, R.: The physics of disk lubricant in the continuum picture. IEEE Trans. Magn.
**41**(2), 616–620 (2005)CrossRefGoogle Scholar - 36.Mate, C.M., Marchon, B.: Shear response of molecularly thin liquid films to an applied air stress. Phys. Rev. Lett.
**85**(18), 3902–3905 (2000)CrossRefGoogle Scholar