Force–extension relations for adsorbing polymers subject to a force

We consider directed walk models of a polymer that is adsorbing at a surface due to polymer–surface interactions, and is also pulled away from the surface by an elongational force. We obtain force–temperature, force–extension, and density–extension curves for the Dyck path and partially directed walk models for the situation where the polymer is pulled from one end, and for Dyck paths when the polymer is pulled from a central location in the polymer. We obtain force–extension and density–extension curves for these models, and their dependence on the length of the polymer