Pulling a polymer with anisotropic stiffness near a sticky wall

We solve exactly a two-dimensional partially directed walk model of a semi-flexible polymer that has one end tethered to a sticky wall, while a pulling force away from the adsorbing surface acts on the free end of the walk. This model generalizes a number of previously considered adsorption models by incorporating individual horizontal and vertical stiffness effects, in competition with a variable pulling angle. A solution to the corresponding generating function is found by means of the kernel method. While the phases and related phase transitions are similar in nature to those found previously the analysis of the model in terms of its physical variables highlights various novel structures in the shapes of the phase diagrams and related behaviour of the polymer. We review the results of previously considered sub-cases, augmenting these findings to include analysis with respect to the model's physical variables—namely, temperature, pulling force, pulling angle away from the surface, stiffness strength and the ratio of vertical to horizontal stiffness potentials, with our subsequent analysis for the general model focusing on the effect that stiffness has on this pulling angle range. In analysing the model with stiffness we also pay special attention to the case where only vertical stiffness is included. The physical analysis of this case reveals behaviour more closely resembling that of an upward pulling force acting on a polymer than it does of a model where horizontal stiffness acts. The stiffness–temperature phase diagram exhibits re-entrance for low temperatures, previously only seen for three-dimensional or co-polymer models. For the most general model we delineate the shift in the physical behaviour as we change the ratio of vertical to horizontal stiffness between the horizontal-only and the vertical-only stiffness regimes. We find that a number of distinct physical characteristics will only be observed for a model where the vertical stiffness dominates the horizontal stiffness. (paper)