Adhesive contact mechanics: From deterministic to stochastic description

Abstract

The theoretical models in adhesive contact mechanics have been successfully applied in fields such as colloidal assembly, drug delivery as well as virus-cell interaction. This report briefly reviews the classical models in describing particle-substrate adhesion. Johnson, Kendall and Roberts (JKR) first obtained the critical tensile load leading to ballistic particle pull off from a substrate. Derjaguin, Muller and Toporov (DMT) proposed an alternate theory with a different prediction on the critical load. Maugis was able to show the JKR-DMT transition using a Dugdale cohesive description of surface interaction. More extensions of adhesive contact models have involved material anisotropy, elasticity gradation, viscoelasticity, and large deformation in nonlinear elastomers. All of these existing results are inherently deterministic, sharing the viewpoint that a critical level of pulling force, commonly referred to as the pull-off force, is needed for particles to detach from substrates. Below the critical pull-off force, particle detachment never occurs. This deterministic understanding from classical theories of adhesive contact mechanics fails in describing nanoparticle-surface interactions, such as those in drug delivery and cellular uptake systems. As the characteristic size of particles falls into the nanometer range, the energy confining the state of adhesion equilibrium is expected to be comparable to the scale of thermal energy, and thus susceptible to random excitations from the environment. It is therefore imperative to develop a statistical framework in examining the thermally-assisted and stochastic failure of the adhesive contact between a single nanoparticle and a compliant substrate. The problem is described by Kramers theory as a thermally activated diffusion process along an energy landscape. The time-varying survival probability of nanoparticle-substrate adhesion naturally leads to an expected detachment time that, despite being influenced by factors such as particle size, substrate compliance and applied load, can typically be on the order of seconds, highlighting the relevance and implications of the present study in problems such as drug delivery and self (or forced) separation in cell-nanoparticle interaction. More importantly, the detachment of nanometer-sized particles from compliant substrates is shown to occur statistically, in contrast to any existing models that are deterministic. The adhesive state of multiple nanoparticles on a compliant substrate exhibits strong spatiotemporal coupling, as demonstrated by a two-nanoparticle system. Nanoparticle pairs on a compliant substrate show a form of communication through the elastic interaction. The adhesive state of one nanoparticle can be effectively influenced by the behavior of neighboring nanoparticle through the overlapping fields generated by individual adhesion sites. Moreover, the spatiotemporal coupling between nanoparticle pairs is more pronounced as the substrate stiffness is reduced. This principle on statistical and mutual interaction of nanoparticle pairs is important in understanding the targeting and docking processes of nano-sized particles on compliant substrates, and provides a promising strategy to control the detachment of a target nanoparticle by adding and manipulating another nanoparticle nearby.