On the Escape Transition of a Tethered Gaussian Chain; Exact Results in Two Conjugate Ensembles

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.