Odd-even effect of melting finite polymer film on square lattice

TL;DR

This paper models how the melting temperature of a finite 2D dimer film on a square lattice depends on the film's width being odd or even, revealing entropy-driven odd-even effects and topological influences.

Contribution

It introduces a mathematical physics model that links melting temperature to external bonds and entropy, highlighting odd-even effects in finite dimer films and their topological implications.

Findings

Melting temperature proportional to external bonding energy and inverse entropy per site.

Odd-length films exhibit different entropy behaviors compared to even-length films.

Topological entropy effects influence the interaction and fusion of finite dimer films.

Abstract

Two dimensional film system bears many exotic thermodynamics behaviors. We proposed a mathematical physics model to explore how the melting temperature of a two dimensional mathematical dimer film depends on the odd-eveness of the finite width of dimer film. A weak external bond between dimers is introduced into the classical dimer model in this dimer film. We derived a general equation of melting temperature and applied it for computing the melting temperature of a dimer film covering a finite square lattice. The melting temperature is proportional to the external bonding energy that we assume it binds neighboring dimers together and proportional to the inverse of entropy per site. Further more, it shows fusing two small rectangular dimer film with odd number of length into one big rectangular film gains more entropy than fusing two small rectangles with even number of length into the same big rectangle. Fusing two small toruses with even number of length into one big torus reduces entropy. Fusing two small toruses with odd number of length increases the entropy. Thus two dimer films with even number of length repel each other, two dimer films with odd length attract each other. The odd-even effect is also reflected on the correlation function of two topologically distinguishable loops in a torus surface. The entropy of finite system dominates odd-even effect. This model has straightforward extension to longer polymers and three dimensional systems.