THE STATISTICAL-MECHANICS OF A MELT OF POLYMER RINGS

Topological restrictions are introduced into a melt of flexible polyme r rings by specifying the linking number between each pair of rings. A ttention is focused on the configurational properties of a single ring in the melt, where the winding number between each pair of rings is z ero: The configurations of all the other chains are averaged out in th e partition sum. The effect of the topological constraint on the remai ning chain is expressed as a product of two configurational weighting factors which involve unusual geometrical properties of the configurat ion. One is a phase factor depending on the torsion of the configurati on and the other is a Boltzmann-like factor with the self-inductance o f the configuration in the role of the interaction energy. The term ba sed on the torsion has the remarkable property of transforming random walk-like configurations into those of stiff rods, while the inductanc e term promotes a transition to a completely collapsed state. Our prel iminary results suggest that the actual configuration of the loop is a non-trivial balance between these opposing tendencies with the size R of the loop scaling with the length as R(2) similar to L(1-1/(3 pi)) similar to L(0.89).