Adsorption of polymer chains at penetrable interfaces

We investigate the problem of adsorption (localization) of polymer chains in the system of two penetrable interfaces within the mean-field approximation. The saturation of the polymer system in the limit case of zero bulk concentration is studied. We find the exact solution of this mean-field polymer adsorption problem that opens the possibility to treat various localization problems for polymer chains in such environments using appropriate boundary conditions. The exact solution is controlled by a single scaling variable that describes the coupling between the interfaces due to the polymer chains. We obtain a nonmonotonic behavior of the amount of adsorbed polymers as a function of the distance between the interfaces. This leads to a high-energy and a low-energy phase for the double layer with respect to the amount of polymers localized. At the saturation point, we find the total energy of the system and determine the force acting between the interfaces to be strictly attractive and to monotonically decay to zero when the interface distance increases.