Fourth-order Results Research Articles

Lattice cluster theory for the packing of rods on a lattice: Extension to treat anisotropic orientational distributions

The lattice cluster theory for the free energy of a set of mutually avoiding rigid rod polymers is extended to treat anisotropic orientational distributions. The theory permits the systematic evaluation of corrections to the isotropic Flory mean field approximation for arbitrary rod orientational distributions, with the Flory theory being the zeroth order isotropic limit of the full theory. The corrections to the zeroth order mean field entropy are represented as a cluster expansion and may be evaluated as a series expansion in the polymer volume fraction φ. We compute all corrections through order φ3 that survive in the thermodynamic limit for the general anisotropic case, along with new fourth order results, which also extend the isotropic limit theory. The anisotropic rod lattice cluster theory represents an improvement over the DiMarzio theory for the packing entropy of rod polymers. This improvement first emerges at fourth order in φ and arises in the lattice cluster theory from inclusion of correlations between four rods lying along distinct lattice directions, four-rod correlations that are absent in DiMarzio’s theory.

The effect of a pair of vacancy defects on the free energy of harmonic crystals: Cell cluster method

The change in the high-temperature Helmholtz free energy of harmonic crystals due to the interaction of a pair of vacancy defects was calculated by the cell cluster method. Third order results were obtained for the face-centered cubic lattice, and fourth order results were obtained for the two-dimensional triangular lattice.