Mean-field treatment of polymer chains trapped between surfaces and penetrable interfaces

We study the exact solutions for the problem of adsorption of polymers in a system containing two interfaces within the mean-field ground-state dominance approximation both for penetrable and nonpenetrable interfaces. We discuss the case of saturation of the polymer double layer for the limiting case of zero bulk concentration. Here, the exact solution is controlled by a single scaling variable which describes the coupling between the interfaces due to the polymer chains. For the case of penetrable interfaces we obtain a nonmonotonous behavior of the amount of adsorbed polymers as a function of the distance between the interfaces. This leads to a high- and a low-energy phase for the double layer with respect to the amount of polymers localized. At the saturation point, the force acting between the interfaces is strictly attractive and monotonously decaying toward zero for increasing interface distance for both types of interfaces. The exact solution for the chemical equilibrium state of the polymer double layer with a semidilute bulk state is governed by two scaling variables and explicit concentration dependence can be removed. The scaling variables describe the ratio between the interface distance and the bulk correlation length and the ratio between the localization length of the interfaces and the bulk correlation length, respectively. Using the exact solution on intervals of constant potentials opens the possibility to treat various localization problems for polymer chains using the appropriate boundary conditions.