Surface adsorption and the collapse transition of a linear polymer chain: some exact results on fractal lattices

Abstract

We study the surface adsorption and collapse transition of a flexible self-attracting self-avoiding polymer chain on truncated 4- and 5-simplex lattices using real-space renormalization-group techniques. We find phase diagrams that exhibit many different universality domains of critical behavior. In the desorbed ordinary bulk regime, the polymer undergoes a collapse transition from a swollen to a compact-globule phase on a 4-simplex lattice, but not on a 5-simplex lattice, where it always remains in the swollen state. In the adsorbed region, on a 4-simplex lattice, the polymer remains in the swollen state with a critical behavior characterized by that of a 3-simplex lattice, whereas, on a 5-simplex lattice, it has both swollen and compact-globule regions separated by a tricritical (θ) line. The phase diagram of a 4-simplex lattice has a pentacritical point which separates a region, where the point at which the adsorbed (swollen) polymer coexists with both the desorbed polymer and desorbed globule is a tetracritical point, from one in which it appears as the intersection of three lines of continuous transition. On a 5-simplex lattice, the adsorbed phase θ line bends in the neighborhood of the special adsorption tricritical line and does not appear to meet it, even for a very large value of the monomer-monomer attraction.