Exactly solvable model with stable and metastable states for a polymer chain near an adsorbing surface.
We report on the conformational properties and transitions of an ideal polymer chain near a solid surface. The chain is tethered with one of its ends at distance z(0) from an adsorbing surface. The surface is characterized by an adsorption parameter c. The exact expression for the partition function is available. We obtained the distribution of complex zeros of this function. The comparison with the Yang-Lee theory allows the characterization of the phase transitions. A first-order conformational transition from a coil to a (adsorbed) flower conformation occurs at c(*)=6z(0)/N. The flower is composed of a strongly stretched stem and a pancake that collects the remaining adsorbed segments. The degree of stretching of the coil or of the stem serves as an order parameter which parametrizes the analytical expressions of the Landau free energy. The phase diagram with one binodal and two spinodal lines is presented. The height of the barriers between metastable and stable states is obtained and the lifetime of metastable states is estimated. A two-state ansatz is used to develop scaling arguments to account for the effects of excluded volume.