Continuous and first-order polymer adsorption on hierarchical fractal walls

Polymer adsorption on fractally rough walls of varying dimensionality is studied by renormalization group methods on hierarchical lattices. Exact results are obtained for deterministic walls. The adsorption transition is found continuous for low dimension d(w) of the adsorbing wall and the corresponding crossover exponent phi monotonically increases with d(w), eventually overcoming previously conjectured bounds. For d(w) exceeding a threshold value d(w)* phi becomes one and the transition changes to first order. d(w)* > d(saw), the fractal dimension of the polymer in the bulk. An accurate numerical approach to the same problem with random walls gives evidence of the same scenario.