Dense self-interacting lattice trees with specified topologies: From light to dense branching

Abstract

The lattice-cluster theory (LCT) is applied to describe a self-avoiding tree of specified topology on a d-dimensional simple hypercubic lattice. Equivalently, the chain can be viewed as linear and composed of structural units occupying several lattice sites. The self-interacting polymer has a nearest-neighbor contact interaction energy \ensuremath{\varepsilon} and occupies a fraction \ensuremath{\Phi} of the lattice volume. The LCT provides a double series expansion in \ensuremath{\varepsilon} and ${\mathit{d}}^{\mathrm{\ensuremath{-}}1}$ for the specific free energy of the tree. Coefficients in this expansion depend on both the tree architecture and \ensuremath{\Phi}. Our theory is illustrated for some tree topologies by evaluating the specific entropy, energy, and the mean number of contacts per segment. The thermodynamic properties of lightly branched trees are shown to remain identical to those of corresponding linear chains. On the other hand, dense branching (in which the number of branching points grows linearly with tree size) greatly affects the polymer properties.