Packing flexible polymer chains on a lattice

We study the problem of arranging p self-avoiding flexible chains of molecular weight M on a hypercubic lattice of N sites with lattice coordination number z using a field-theoretic approach. The thermodynamic limit of pM becoming infinite, with the packing fraction f=pM/N ranging from zero to 1, is considered. Hence the dimer, the Hamiltonian walk, and the single self-avoiding walk problems are included as particular cases. Mean-field theory is of the Flory type and it becomes exact at z=∞. Systematic corrections in powers of $z^{\mathrm{-}1}$ can be evaluated. We calculate the weight- and volume-fraction-dependent connectivity constant and site entropy to second order in $z^{\mathrm{-}1}$ and compare it with known results in two and three dimensions. The site entropy, which vanishes at f=0, is a convex function of f (for any M) with a maximum at f≳0.6 that moves towards higher f as either M or z increases. We also discuss the most efficient packing of M-mers at a given volume fraction and dimensionality.