Adhesion in a liquid medium is fundamentally important for understanding a myriad of physiological and technological issues such as nanoparticle or bacteria uptake by cells, attachment of viruses on bacterial surfaces, adhesion of a bacteria on a preformed biofilm, biofouling of ships and marine vehicles, and many more. In this paper, we provide a theory to analyze the under-water adhesion of a rigid spherical particle on a soft, charged surface, which is represented as a layer of grafted polyelectrolyte layer (PEL). Our model is based on calculating and minimizing the free energy, appropriately modified to account for the PEL electric double layer (EDL) induced electrostatic energies. The central result of our paper is that the presence of surface charge typically enhances the adhesion, indicated by a larger negative value of the equilibrium free energy and larger value of the equilibrium depth of indentation. Such a behavior can be explained by noting that the lowering of EDL electrostatic energy due to adhesion better balances the increase in elastic energy caused by the adhesion-induced deformation. We anticipate that our theory will provide the hitherto unknown basis of quantifying the effect of surface charge in under-liquid adhesion, which is central to the vast number of phenomena involving charged bio-systems, like cells, bacteria, and viruses.

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