Asymptotic theory of the elastohydrodynamic adhesion and gliding motion of a solid particle over soft and sticky substrates at low Reynolds numbers

Abstract

This analysis makes use of asymptotic analyses and numerical methods to address, in the limit of small Reynolds and ionic Péclet numbers and small clearances, the canonical problem of the forces exerted on a small solid spherical particle undergoing slow translation and rotation in an incompressible fluid moving parallel to an elastic substrate, subject to electric double-layer and van der Waals intermolecular forces, as a representative example of particle gliding and the idealized swimming dynamics of more complex bodies near soft and sticky surfaces in a physiological solvent. The competition of the hydrodynamic, intermolecular and surface-deformation effects induces a lift force, and drag-force and drift-force perturbations, which do not scale linearly with the velocities, and produce a non-additivity of the intermolecular effects by reducing the intensity of the repulsive forces and by increasing the intensity of the attractive forces. The lift force enhances a reversible elastohydrodynamic adhesion regime in both ionized and deionized solvents, in which lateral motion and lift-off from the substrate can occur. An irreversible elastohydrodynamic adhesion regime, produced by elastic instabilities in the form of surface bifurcations in the substrate, is found to exist for both positive and negative lift forces and is enhanced by small gliding velocities and large substrate compliances, for which critical thresholds are calculated for both ionized and deionized solvents. Elastohydrodynamic corrections are derived for the critical coagulation concentration of electrolyte predicted by the Derjaguin–Landau–Verwey–Overbeek (DLVO) standard theory of colloid stabilization. The corrected DLVO critical concentration is unable to describe the adhesion process when the substrate is sufficiently compliant or when the solvent is deionized. These effects may have consequences on the lateral motility and adhesion of small particles and swimming micro-organisms to soft and sticky substrates, in which the reversible or irreversible character of the adhesion process may be influenced not only by the solvent ionic strength, as described by the DLVO theory, but also by the motion kinematics and the substrate mechanical properties.