Partition function zeros and finite size scaling for polymer adsorption

Abstract

The zeros of the canonical partition functions for a flexible polymer chain tethered to an attractive flat surface are computed for chains up to length N = 1536. We use a bond-fluctuation model for the polymer and obtain the density of states for the tethered chain by Wang-Landau sampling. The partition function zeros in the complex eβ-plane are symmetric about the real axis and densest in a boundary region that has the shape of a nearly closed circle, centered at the origin, terminated by two flaring tails. This structure defines a root-free zone about the positive real axis and follows Yang-Lee theory. As the chain length increases, the base of each tail moves toward the real axis, converging on the phase-transition point in the thermodynamic limit. We apply finite-size scaling theory of partition-function zeros and show that the crossover exponent defined through the leading zero is identical to the standard polymer adsorption crossover exponent φ. Scaling analysis of the leading zeros locates the polymer adsorption transition in the thermodynamic (N → ∞) limit at reduced temperature \documentclass[12pt]{minimal}\begin{document}Tc&z.ast\break=1.027(3)\end{document} Tc&z.ast =1.027(3) [\documentclass[12pt]{minimal}\begin{document}\beta -c=1/ Tc&z.ast =0.974(3)\end{document}βc=1/ Tc&z.ast =0.974(3)] with crossover exponent φ = 0.515(25). Critical exponents for the order parameter and specific heat are determined to be \documentclass[12pt]{minimal}\begin{document}\widetilde{\beta }=0.97(5)\end{document}β=0.97(5) and α = 0.03(4), respectively. A universal scaling function for the average number of surface contacts is also constructed.