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Recent experiments reveal that adhesive interactions can play a key role in causing surface instability in soft lubrication. Instances of instability include fluid entrapment in isolated pockets upon a soft sphere’s normal contact with a hard substrate and surface wrinkling of a soft substrate as a hard sphere slides across it. These phenomena underscore a substantial distinction between hard and soft lubrication. They are of paramount importance from a fundamental standpoint, providing an entirely new explanation for the transition mechanism from elasto-hydrodynamic to the mixed lubrication regimes. Here, we introduce a new theory to elucidate these observations. Our theory modifies the Reynolds elasto-hydrodynamic equation by incorporating adhesive interaction across the fluid layer, investigating the interplay between adhesion, fluid flow and elastic instability. Our analysis proposes the addition of a new dimensionless parameter in lubrication theory, that compares the stiffness of the adhesive interaction to that of the substrate. When this parameter exceeds unity, the soft solid surface exhibits instability to small perturba- tions in its shape. In mathematical terms, the Reynolds equation undergoes a transition from a nonlinear diffusion equation to a nonlinear wave equation at this critical point. Post-transition, the diffusivity of the nonlinear diffusion equation turns negative, rendering the problem ill- posed. We investigate the transition using the method of characteristics and present an exact analytic solution. This solution offers insights into the occurrence of a vanishing liquid film thickness at specific locations, resulting in dry contact—initiating transition to mixed lubrication.
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Lubricated soft normal elastic contact of a sphere: a new numerical method and experiment
An important problem in lubrication is the squeezing of a thin liquid film between a rigid sphere and an elastic substrate under normal contact. Numerical solution of this problem typically uses iteration techniques. A difficulty with iteration schemes is that convergence becomes increasingly difficult under increasingly heavy loads. Here we devise a numerical scheme that does not involve iteration. Instead, a linear problem is solved at every time step. The scheme is fully automatic, stable and efficient. We illustrate this technique by solving a relaxation test in which a rigid spherical indenter is brought rapidly into normal contact with a thick elastic substrate lubricated by a liquid film. The sphere is then fixed in position as the pressure relaxes. We also carried out relaxation experiments on a lubricated soft PDMS (polydimethysiloxane) substrate under different conditions. These experiments are in excellent agreement with the numerical solution.
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- Award ID(s):
- 1854572
- PAR ID:
- 10337792
- Date Published:
- Journal Name:
- Soft Matter
- Volume:
- 18
- Issue:
- 6
- ISSN:
- 1744-683X
- Page Range / eLocation ID:
- 1219 to 1227
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1039/D1SM01654G
