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We introduce a framework for adsorption of a single polymer in which the topology of the polymer is quenched before adsorption, in contrast to more standard adsorption models having annealed topology. Our "topology" refers either to the precise branching structure of a branched polymer (in any dimension), or else to the knot type of a ring polymer in three dimensions. The quenched topology is chosen uniformly at random from all lattice polymers of a given size in one of four classes (lattice animals, trees, combs, or rings), and we then consider adsorption of the subclass of configurations that have the quenched topology. When the polymer-surface attraction increases without bound, the quenched topological structure keeps a macroscopic fraction of monomers off the surface, in contrast with annealed models that asymptotically have 100% of monomers in the surface. We prove properties of the limiting free energy and the critical point in each model, although important open questions remain. We pay special attention to the class of comb polymers, which admit some rigorous answers to questions that otherwise remain open. Since the class of all combs was not previously examined rigorously in full generality, we also prove the existence of its growth constant and its limiting free energy for annealed adsorption.