Monomer distributions in Dyck Path models of grafted polymers

Polymer coatings on surfaces or on suspended or colloid particles have applications in the steric stabilisation of colloid dispersions, or in drug delivery systems (such as nanoparticle-polymer systems, or drug eluding stents). In these applications a polymer is grafted onto a surface or a hard wall and it coils away from the surface to form a polymer layer. In a drug delivery system the drug absorbs in this layer until it is released at a target site. If the polymer is hydrophylic and in an aqueous solution then it forms a thicker and less dense coating, and if it is hydrophobic it forms a thinner and denser layer. These physical properties affect the adsorption of molecules into the layer. In this talk this is modelled using a directed Dyck path model of a grafted linear polymer. The density profile of monomers along the path is determined in the scaling limit for the path in good or in poor quality solvents. In either case the density profile is determined by calculating the exact probability distribution of monomers in the scaling limit as a function of their position along the Dyck path. This shows that, in a good solvent, the entropic repulsion between the Dyck path and a hard wall creates a low density regime next to the hard wall wherein particles can adsorbed and be stabilised until they are released. This will be, for example, the case in nanoparticle-polymer system for targeted drug delivery.