Entropy of self-avoiding branching polymers: mean field theory and Monte Carlo simulations

TL;DR

This work tackles the entropy and branching statistics of self-avoiding, single-chain polymers with excluded-volume on regular lattices by mapping rooted-directed SATs to a lattice field theory in the $n\to0$ limit and solving via a mean-field saddle point. The authors derive a compact MF entropy functional $s_{\rho}(\{\phi_f\})$ that interpolates between dilute trees and spanning trees, and they show the mean number of branch-nodes is set by branch chemical potentials rather than lattice details. Validation against exact spanning-tree results and new Monte Carlo simulations in $d=2,3,4$ demonstrates good accuracy, improving with dimension. They provide practical tools, including a Monte Carlo scheme for spanning trees and an extension to poor-solvent conditions, predicting universal branching statistics and a coil-globule collapse akin to linear polymers.

Abstract

We study the statistics of branching polymers with excluded-volume interactions, by modeling them as single self-avoiding trees on a generic regular periodic lattice with coordination number $q$. Each lattice site can be occupied at most by one tree node, and the fraction of occupied sites can vary from dilute to dense conditions. By adopting the statistics of rooted-directed trees as a proxy for that of undirected trees without internal loops and by an exact mapping of the model into a field theory, we compute the entropy and the mean number of branch-nodes within a mean field approximation and in the thermodynamic limit. In particular, we find that the mean number of branch-nodes is independent of both the lattice details and the lattice occupation, depending only on the associated chemical potential. Monte Carlo simulations in $d=2,3,4$ provide evidence of the remarkable accuracy of the mean field theory, more accurate for higher dimensions.

Figures (7)

• Figure 1: (a) Illustration of a SAT on the square lattice ($q=4$, $V=4\times 4=16$ total sites) with $N=12$, $N_1=6$, $N_2=3$, $N_3=2$ and $N_4=1$. Notice that $N_1$ and $N_2$ verify Eqs. \ref{['eq:N1']} and \ref{['eq:N2']}. The dashed bond illustrates the role of periodic boundary conditions (p.b.c.). (b, left) The same polymer conformation of panel (a) as a rooted-directed SAT, with the root site marked in orange. (b, right) A different rooted-directed SAT with the root on the same lattice site and the same connectivity structure (through p.b.c.) of the conformation on the left.
• Figure 2: Distribution of equilibrium densities $\langle \phi_f\rangle$ for nodes of functionality $f$ in spanning trees on the triangular lattice ($q=6$). Orange columns correspond to our MF result for an infinite lattice, whereas blue columns refer to the results obtained through the numerical procedure of Pozrikidis Pozrikidis2016 for a periodic lattice with a total number of $256$ sites.
• Figure 3: Asymptotic mean fraction of branch-nodes, $\langle \phi_3\rangle$, as a function of the branch chemical potential $\beta\mu_3$, comparison between MF prediction Eq. \ref{['eq:BranchingProbability']} (dashed line) and MC numerical simulations (symbols, corresponding to the values for the largest $N$'s reported in Fig. \ref{['fig:2Ddata']} in SM) for different node densities $\rho$ (see colorcode/symbol in the legend) and in spatial dimensions $d=2,3,4$. (Insets) Corresponding ratios between MC results and the MF prediction.
• Figure S1: Tree conformations (left column) and their corresponding lattice configurations as undirected trees (central column) and rooted-directed trees (right column) on the $3\times 3$ square lattice.
• Figure S2: Self-avoiding tree conformations on the $3\times3$ square lattice with periodic boundary conditions representing some typical terms of the diagrammatic expansion of Eq. \ref{['eq:ZDiagramaticExpansion']}. Lattice sites are numbered from $0$ to $8$, the spatial position of the tree root is ${\mathbf x}_0$ (bottom left corner, in yellow).