Modelling the adsorption of a polymer subject to an elongational force by directed walk models

We discuss several different directed walk models of a homopolymer adsorbing at a surface when the polymer is subject to an elongational force which hinders the adsorption. These models have the advantage that, within the generating function approach, they can often be solved exactly and that in the stretching regime they give a fair representation of the configurational properties of the polymer. By using combinatorial arguments we analyse how the critical temperature for adsorption depends on the magnitude of the applied force and show that the order of the adsorption transition changes from continuous to first order when a force is applied. We discuss which are the minimal ingredients to add into the model in order to obtain a reentrant phase diagram similar to the one observed in models of DNA mechanical denaturation. Another advantage in using directed walk models is that they can be generalized to study, to some extent, the adsorption transition of random copolymers. Indeed, despite the fact that the quenched model cannot be solved exactly we show that a partial knowledge of the full random problem can be obtained by resorting to an approximation in which the quenched average is approximated by a partial annealing procedure, the so-called Morita approximation.