Partition function, metastability, and kinetics of the escape transition for an ideal chain

Abstract

An end-tethered polymer chain squeezed between two pistons undergoes an abrupt transition from a confined coil state to an inhomogeneous flower-like conformation partially escaped from the gap. We present a rigorous analytical theory for the equilibrium and kinetic aspects of this phenomenon for a Gaussian chain. Applying the analogy with the problem of the adsorption of an ideal chain constrained by one of its ends, we obtain a closed analytical expression for the exact partition function. Various equilibrium thermodynamic characteristics (the fraction of imprisoned segments, the average compression, and lateral forces) are calculated as a function of the piston separation. The force versus separation curve is studied in two complementary statistical ensembles, the constant force and the constant confinement width ones. The differences in these force curves are significant in the transition region for large systems, but disappear for small systems. The effects of metastability are analyzed by introducing the Landau free energy as a function of the chain stretching, which serves as the order parameter. The phase diagram showing the binodal and two spinodal lines is presented. We obtain the barrier heights between the stable and metastable states in the free energy landscape. The mean first passage time, i.e., the lifetime of the metastable coil and flower states, is estimated on the basis of the Fokker-Planck formalism. Equilibrium analytical theory for a Gaussian chain is complemented by numerical calculations for a lattice freely jointed chain model.

Negative compressibility and nonequivalence of two statistical ensembles in the escape transition of a polymer chain

“…A short-range repulsive force exists between the chain segments and the surfaces of the pistons. It was shown 25,30 that the confinement effect of the pistons is equivalent to an effective potential per segment in k B T units u = 1 6 ͑ a / H͒ 2 . The nature of this effective potential is purely entropic ͑u is the loss of entropy per polymer segment in slit͒.…”

“…Then, the two states coexist and the radial density and the free end distribution have a bimodal character. 25 The compressed coil minimum is located at s coil = 0 and its depth is given by ⌽͑s coil ͒ = N −1 ͑ R g / H͒ 2 . The minimum, corresponding to the escaped flower state, is found at s esc = a /3H, and this minimum is ⌽͑s esc ͒ = L / NH.…”

“…This is in conflict with yet another general theorem of statistical mechanics stating the equivalence of various statistical ensembles ͑including the phase coexistence regions in the phase diagram͒ in the appropriate thermodynamic limit. 5 In previous work 25,26 we have mainly concentrated on results for the escape transition in the H-ensemble. In this paper we aim to bring together results for the escape transition for Gaussian chains in both the H-and f-ensembles.…”

“…25. The effects of metastability were analyzed by introducing the Landau free energy as a function a suitable order parameter.…”

“…The resistance force decreases abruptly, indicating a first-order transition. The escape transition was studied rather extensively both by analytical [22][23][24][25][26] and numerical [27][28][29][30][31][32][33][34][35][36] methods.…”