We consider the adsorption of an isolated, Gaussian, random, and quenched copolymer chain at an interface. We first propose a simple analytical method to obtain the adsorption/depletion transition, by averaging over the disorder the partition function instead of the free energy. The adsorption thresholds obtained by previous authors at a solid/liquid and at a liquid/liquid interface for multicopolymer chains can be rederived using this method. We also compare the adsorption thresholds obtained for bimodal and for Gaussian disorder; they only agree for small disorder. We focus on the specific case of an ideally flat asymmetric liquid/liquid interface, and consider the situation where the chain is composed of monomers of two different chemical species A and B. The replica method is developed for this case. We show that the Hartree approximation, coupled to a replica symmetry assumption, leads to the same adsorption thresholds as obtained from our general method. In order to describe the properties of the adsorbed (or depleted) chain, we develop a new approximation for long chains, within the framework of the replica theory. In most cases, the behavior of a random copolymer chain can be mapped onto that of a homopolymer chain at an asymmetric attractive interface. The values of the effective adsorption energy are different for a random and a periodic copolymer chain. Finally, we consider the case of uncorrelated annealed disorder. The behavior of an annealed chain can be mapped onto that of a homopolymer chain at an asymmetric non attractive interface; hence, an annealed chain cannot adsorb at an asymmetric interface.
PACS. 05.90.+m Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems -68.10.-m Fluid surfaces and fluid-fluid interfaces -82.65.Dp Thermodynamics of surfaces and interfaces List of notations a size of a monomer, D length defined by equation (A.10), D eff length defined by D eff ≡ a/ 6|∆χ eff |, f fraction of charged monomers, i discrete curvilinear coordinate along the chain, s continuous curvilinear coordinate along the chain, u operator defined by u ≡ n α=1 θ(z α ), u(s) real number defined by u(s) ≡ n α=1 θ z α (s) , U energy of adsorption per monomer,potential acting on a monomer located at a position z, z position with respect to the interface (z < 0 in the water phase), δ length defined by δ ≡ a/ ∆χ/6, θ(z) Heaviside function, defined by θ(z < 0) = 0 and θ(z > 0) = 1, ξ(s) parameter describing the chemical nature of the sth monomer: ξ(s) = χ A in the case of a A monomer and ξ(s) = −χ B for a B monomer, a