Size-dependent transitions in grafted polymer brushes.

The partition function of a grafted polymer brush was calculated as a sum over all possible configurations, each of them being an ensemble of n1, n2, ...n(i), ... loops having 2, 4, ..., 2i, ... segments. A system of equations for the most likely configuration has been obtained, and it was concluded that no solution exists for an infinite chain, with nonvanishing interactions between segments and surface. This implies that an infinite chain is either collapsed on the surface (for attractive interactions) or is forming a stretched brush (for repulsive interactions). However, for finite chains, a solution could be found. When the attractive segment-surface interaction becomes sufficiently strong, the brush collapses on the surface (the "loops" to "trains" transition). When the interaction is repulsive and sufficiently strong, the brush becomes stretched, with most segments belonging to open loops which do not return to the surface (the "loops" to "tails" transition). The critical values of the segment-surface interaction, for which the above transitions occur, depend on the length of the polymer chain as well as on other physical properties of the brush, such as the segment-segment and segment-solvent interactions and the grafting density.