The escape transition of a polymer: a unique case of non-equivalence between statistical ensembles.

A flexible polymer chain under good solvent conditions, end-grafted on a flat repulsive substrate surface and compressed by a piston of circular cross-section with radius L may undergo the so-called "escape transition" when the height of the piston D above the substrate and the chain length N are in a suitable range. In this transition, the chain conformation changes from a quasi-two-dimensional self-avoiding walk of "blobs" of diameter D to an inhomogeneous "flower" state, consisting of a "stem" (stretched string of blobs extending from the grafting site to the piston border) and a "crown" outside of the confining piston. The theory of this transition is developed using a Landau free-energy approach, based on a suitably defined (global) order parameter and taking also effects due to the finite chain length N into account. The parameters of the theory are determined in terms of known properties of limiting cases (unconfined mushroom, chain confined between infinite parallel walls). Due to the non-existence of a local order parameter density, the transition has very unconventional properties (negative compressibility in equilibrium, non-equivalence between statistical ensembles in the thermodynamic limit, etc.). The reasons for this very unusual behavior are discussed in detail. Using Molecular Dynamics (MD) simulation for a simple bead-spring model, with N in the range 50<or=N<or=300, a comprehensive study of both static and dynamic properties of the polymer chain was performed. Even though for the considered rather short chains the escape transition is still strongly rounded, the order parameter distribution does reveal the emerging transition clearly. Time autocorrelation functions of the order parameter and first passage times and their distribution indicate clearly the strong slowing down associated with the chain escape. The theory developed here is in good agreement with all these simulation results.