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

Abstract

Summary: 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.

What is the order of the two-dimensional polymer escape transition?

“…At the transition point, the average order parameter S =< s > jumps from s imp eq to s esc eq . The Landau function allows one to analyze metastable states and to define the two lines where either one of the metastable minima vanishes [16]. Theoretical predictions for the escape transition of a Gaussian chain are summarized in Table I and…”

“…A phenomenon that was called escape transition occurs upon progressive squeezing an end-grafted polymer chain between two pistons and has attracted great interest [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. At weak deformation the chain is compressed uniformly into a relatively thick pan-cake conformation.…”

“…A rigorous analytical theory for the equilibrium and kinetic aspects of the escape transition for a Gaussian chain was constructed in [13,14]. Metastability effects, negative compressibility and the nonequivalence of the escape transition in two conjugate ensembles were analyzed for the same model in [15,16].…”

Coil-bridge transition in a single polymer chain as an unconventional phase transition: Theory and simulation

“…Another example of a polymer system with non-conventional phase transition is an end-grafted chain compressed between two pistons and undergoing the so-called escape transition. 24,37…”

Ensemble inequivalence in single-molecule experiments

“…In the isometric ensemble the position of the chain ends is employed as a control parameter, fixing also the end-to-end distance, while in the isotensional case the control parameter is represented by the force applied on one loose end. The conditions under which the equivalence between these two ensembles can be obtained have been investigated in many works, both from a theoretical point of view [21,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40] and by means of computer simulations [41,42,43,44,45,46,47,48,49,50]. Some of the authors concluded that ensembles are equivalent in the infinitely long chain limit for a Gaussian chain, [30,47] as well as for a generic chain [21] and toy lattice chain models [21,35,36].…”