  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.
