Partition function zeros and phase transitions for a square-well polymer chain

Abstract

The zeros of the canonical partition functions for flexible square-well polymer chains have been approximately computed for chains up to length 256 for a range of square-well diameters. We have previously shown that such chain molecules can undergo a coil-globule and globule-crystal transition as well as a direct coil-crystal transition. Here we show that each of these transitions has a well-defined signature in the complex-plane map of the partition function zeros. The freezing transitions are characterized by nearly circular rings of uniformly spaced roots, indicative of a discontinuous transition. The collapse transition is signaled by the appearance of an elliptical horseshoe segment of roots that pinches down towards the positive real axis and defines a boundary to a root-free region of the complex plane. With increasing chain length, the root density on the circular ring and in the space adjacent to the elliptical boundary increases and the leading roots move towards the positive real axis. For finite-length chains, transition temperatures can be obtained by locating the intersection of the ellipse and/or circle of roots with the positive real axis. A finite-size scaling analysis is used to obtain transition temperatures in the long-chain (thermodynamic) limit. The collapse transition is characterized by crossover and specific-heat exponents of φ≈0.76(2) and α≈0.66(2), respectively, consistent with a second-order phase transition.

Identifying transitions in finite systems by means of partition function zeros and microcanonical inflection-point analysis: A comparison for elastic flexible polymers

“…Ref. [14] considers only the real part of the leading zero, whereas we prefer the absolute value, motivated by the fact that at first-order transitions the zeros lie on a circle whose radius is a unique estimator for the transition temperature. The section of the map for L = 35 shown in Fig.…”

“…[14]. In our case they are strictly confined within a circle with radius 1 (the boundary at 1 corresponds to infinite temperature).…”

“…[14], we here employ a more realistic coarse-grained model for elastic, flexible polymers with continuous, distance-dependent monomermonomer interactions based on van der Waals forces. Recently developed sophisticated simulation methodologies specific to this model [30] enable a very precise estimation of fundamental statistical quantities such as the density of states.…”

“…[9][10][11]), proteins [12,13], and to polymers [14,15]. Most applications of the partition function zero analysis method are considered to be alternative approaches to scaling properties near phase transitions in large systems.…”

Partition function zeros and finite size scaling for polymer adsorption

“…However, with the exception of Ref. 46, these previous polymer studies have been restricted to rather modest chain lengths (typically N ≤ 36) for which the partition function zero "maps" are relatively sparse and accurate finite size scaling is difficult.…”