vitrification in these domains. Upon rapid cooling in presence of density gradients caused by presence of confined surfaces, a non-uniform polymer system can therefore be subdivided into the glassy and melt domains. This effect can explain the experimentally observed difference in the role of confining surfaces in the glass formation.

In the present study we address above described effects by developing a lattice model that takes into account the semiflexibility of a chain as well as the interactions with other chains near the adsorbing surface. One of the main advantages of the developed approach is that it goes beyond the Bragg-Williams athermal limit incorporated into the Flory-Huggins model underlying most of the current theories [@DoiEdwards; @dG; @Roe; @Scheutjens-Fleer] of the non-uniform polymer systems. In particular, the mentioned previous theories do not take into account the finite semi-flexibility of the chain that is one of the main issues in the present study of the polymer structure near the adsorbing surface. The developed lattice model allows for a more straightforward incorporation of the semi-flexibility of chains in the many chain problem with excluded volume taken into the consideration, in comparison with the continuous theories based on the worm-like chain model.[@Daoulas; @Schmid; @Ganesan] 