On the escape transition of a tethered Gaussian chain; exact results in two conjugate ensembles

Abstract

Upon compression between two pistons an end-tethered polymer chain undergoes an abrupt transition from a confined coil state to an inhomogeneous flower-like conformation that is partially escaped from the gap. In the thermodynamic limit the system demonstrates a first-order phase transition. A rigorous analytical theory of this phenomenon for a Gaussian chain is presented in two ensembles: a) the H-ensemble, in which the distance H between pistons plays the role of the control parameter, and b) the conjugate f-ensemble in which the external compression force f is the independent parameter. A loop region for (f(H)) with negative compressibility exists in the H-ensemble, while in the f-ensemble (H(f)) is strictly monotonic. The average lateral forces taken as functions of H (or (H), respectively) have distinctly different behavior in the two ensembles. This result is a clear counterexample of the main principles of statistical mechanics stating that all ensembles are equivalent in the thermodynamic limit. Another theorem states that the thermodynamic potential as a function of volume must be concave everywhere. We demonstrated that the exact free energy in the H-ensemble contradicts this statement. Inapplicability of these fundamental theorems to a macromolecule undergoing the escape transition is clearly related to the fact that phase coexistence in the present system is strictly impossible. This is a direct consequence of the tethering and the absence of global translational degrees of freedom of the polymer chain. © 2006 WILEY-VCH Verlag GmbH & Co. KGaA.