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The conformational properties of a polymer chain interacting with a surface are studied and new features for the cases of a chain free to move in the whole space and for a localized chain fixed with one of its ends at a point are clearly illustrated. The progress achieved is based on exact solutions given in the frame of the continuous model by means of integral equations and Laplace transforms. They permit the study of the behavior of the chain at any distance from the surface as a function of its length, the intensity of the polymer-surface interactions, and the size L of the available space perpendicular to the surface for the nonlocalized chain or the localization position for the localized chain, respectively. The necessary attractions for the gathering of the nonlocalized chains at the surface are larger for larger freedom of the chains which increases with L. From the dependence on the surface-polymer interactions, density profiles are determined for both the depletion and adsorption cases. The behavior of the localized chains is governed by both, the localization constraint and the interactions with the surface.

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The end-to-end distribution function of a linear chain interacting with a penetrable surface with the potential uδ(z) is demonstrated to recover the case of the distribution in the presence of an impenetrable surface with different boundary conditions. The two different boundary conditions of zero probability density and of zero of the gradient of the probability density at the surface correspond to different values of u and the penetrable distribution function can thus be used to describe chains with various degrees of interactions both in the presence of penetrable or impenetrable surfaces. Density profiles of the monomeric units of the chains localized at one or both ends are described, furnishing an insight to the distortion which the interacting surface brings on the shape of a coil. The study includes an extension to the cases of ring and star macromolecules.