Modeling Sliding Contact of Rough Surfaces with Molecularly Thin Lubricants

Abstract

The sliding contact between two rough surfaces in the presence of a molecularly thin lubricant layer is investigated. Under very high shear rates, the lubricant is treated as a semi-solid layer with normal and lateral shear-dependent stiffness components obtained from experimental data. The adhesive force in the presence of lubricant is also adapted from the Sub-boundary lubrication model and improved to account for variation in surface energy with penetration into the lubricant layer. A model is then proposed, based on the Improved sub-boundary lubrication model, which accounts for lubricant contact and adhesion and its validity is discussed. The model is in good agreement with published experimental measurements of friction in the presence of molecularly thin lubricant layers and suggests that a molecularly thin lubricant bearing could be successfully used to reduce solid substrate damage at the interface.

Appendix

The following are the expressions of the solid adhesion, contact, and friction forces of the ISBL model. These are to be added to the lubricant adhesion, contact, and friction forces derived in this study to yield the total forces.

The solid adhesive force comprises the contributions of elastically and elastically plastically contacting asperities. It is a function of the maximum surface energy, as well as the radius and number of asperities in contact:

$$ F_{\text{s,solid}} = 2\pi RN\max \left( {\Updelta \gamma } \right)\left[ {0.98\int\limits_{d}^{d + \omega } {J_{ - 0.29}^{0.298} + } 0.79\int\limits_{d + \omega }^{d + 6\omega } {J_{ - 0.321}^{0.356} + 1.19\int\limits_{d + 6\omega }^{d + 110\omega } {J_{ - 0.332}^{0.093} } } } \right] $$

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The integrands are defined as:

$$ J_{\text{c}}^{\text{b}} = \left( {\frac{\omega }{{\omega_{\text{c}} }}} \right)^{\text{b}} \left( {\frac{\varepsilon }{{\omega_{\text{c}} }}} \right)^{\text{c}} \varphi \left( z \right)dz $$

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The local and critical interference of each asperity is ω and ω _c, respectively, while ε is the equilibrium spacing. A Gaussian distribution of asperity heights φ (z) is assumed.

The solid contact force is:

$$ P_{\text{solid}} = \frac{2}{3}\left( {\pi RKH\omega_{\text{c}} } \right)N\left[ {\int\limits_{d}^{{d + \omega_{\text{c}} }} {I^{1.5} } + 1.03\int\limits_{{d + \omega_{\text{c}} }}^{{d + 6\omega_{\text{c}} }} {I^{1.425} } + 1.4\int\limits_{{d + 6\omega_{\text{c}} }}^{{d + 110\omega_{\text{c}} }} {I^{1.263} + \frac{3}{K}\int\limits_{{d + 110\omega_{\text{c}} }}^{\infty } {I^{1} } } } \right] $$

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The solid contact force includes contributions from elastically, elastically–plastically, and fully plastically deforming asperities. The hardness is H and the hardness coefficient is K = 0.454 + 0.41ν.

Similarly, the solid friction force is:

$$ Q_{\text{solid}} = \frac{2}{3}\left( {\pi RKH\omega_{\text{c}} } \right)N\left[ {0.52\int\limits_{d}^{{d + \omega_{\text{c}} }} {I^{0.982} } + \int\limits_{{d + \omega_{\text{c}} }}^{{d + 6\omega_{\text{c}} }} { - 0.01I^{4.425} } + 0.091I^{3.425} + 0.85I^{1.425} } \right] $$

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The integrands for both the contact and friction forces are:

$$ I^{\text{b}} = \left( {\frac{z - d}{{\omega_{\text{c}} }}} \right)^{\text{b}} \varphi \left( z \right)dz $$

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