Statistical mechanics of the packing of rods on a lattice: Cluster expansion for systematic corrections to mean field

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Abstract
We introduce a lattice spin field theory which formally provides an exact description of the statistics of nonoverlapping, nonintersecting rods on regular lattices. The theory is applicable to arbitrary rod length distributions and arbitrary volume fractions. The mean field approximation reproduces Flory–Huggins theory for rods on a strict lattice, but our theory permits the rigorous and systematic evaluation of corrections to the mean field entropies. We present those corrections as a cluster expansion for the entropy density which involves a double expansion in powers of the volume fraction ψ and the inverse of the lattice coordination number z−1. Our results are compared for specific cases with those of DiMarzio’s theory which is more accurate than mean field. For dimers which are equally distributed among the possible lattice directions of a hypercubic lattice, we find very good agreement between our theory and DiMarzio’s with ours giving a slightly larger entropy density when corrections to mean field to order ψ4 are included. Through order ψ3, our entropy for longer rods is identical to that of DiMarzio’s expanded similarly. Our dimer results, however, demonstrate that the ψ4 term for rods contains contributions which we attribute to locally ordered configurations and which are absent in DiMarzio’s theory. Our results are compared with the exact entropy density for dimers fully covering an infinite square planar lattice; the inclusion of corrections to mean field up to order z−2 narrows the error from 33.8% for mean field to 7.85%. Our theory, however, also describes the dimer–monomer problem which has not been very amenable to analytic theories.

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