DOI 10.1140/epje/i2008-10442-0 Regular Article Eur. Phys. J. E 29, 9–25 (2009) T HE EUROPEAN P HYSICAL JOURNAL E The escape transition of a polymer: A unique case of non-equivalence between statistical ensembles D.I. Dimitrov 1,3 , L.I. Klushin 2 , A. Skvortsov 4 , A. Milchev 1,5, a , and K. Binder 1 1 Institut f¨ ur Physik, Johannes Gutenberg Universit¨at Mainz, Staudinger Weg 7, 55099 Mainz, Germany 2 American University of Beirut, Department of Physics, Beirut, Lebanon 3 University of Food Technologies, 4002 Plovdiv, Bulgaria 4 Chemical-Pharmaceutical Academy, Prof. Popova 14, 197022 St. Petersburg, Russia 5 Instute for Chemical Physics, Bulgarian Academy of Sciences, Sofia, Bulgaria Received 8 December 2008 Published online: 3 April 2009 – c  EDP Sciences / Societ`a Italiana di Fisica / Springer-Verlag 2009 Abstract. 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 param- eters 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 behav- ior are discussed in detail. Using Molecular Dynamics (MD) simulation for a simple bead-spring model, with N in the range 50 ≤ N ≤ 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. PACS. 05.70.Fh Phase transitions: general studies – 07.05.Tp Computer modeling and simulation – 64.60.My Metastable phases 1 Introduction With the development of novel experimental techniques in recent years the physics of single polymer chains is gain- ing increasing interest both in view of possible applica- tions and from the standpoint of basic science. Atomic force microscope techniques make experiments with single grafted macromolecules possible, including manipulations like stretching and compressing isolated polymer mush- rooms at substrates [1–5]. Studying the relation between such external forces acting on a polymer and its structure has become particularly popular in the context of under- standing the relation between structure and function of biopolymers. a e-mail: milchev@ipchp.ipc.bas.bg An idealized but generic model for the interaction between an end-grafted polymer (a “polymer mush- room” [6]) and the tip of an atomic force microscope replaces the latter by a flat cylindrical piston of radius L [7–21]. This model has gained particular interest since it was found that for L and the number N of effective monomers of the flexible polymer in the suitable range the polymer undergoes a very unusual conformational transi- tion, the so-called “escape transition” [7,8]. At weak com- pression the chain is deformed uniformly into a relatively thick “pancake” [22]. The resistance force due to the com- pression of the chain increases monotonously as the sepa- ration between the substrate and the piston decreases, un- til at a critical separation the chain conformation changes abruptly: one part of the chain forms a stem stretching