Critical properties of a branched polymer growth model

Abstract

We study the branched polymer growth model (BPGM) introduced by Lucena et al. [Phys. Rev. Lett. 72, 230 (1994)] in two dimensions. First the BPGM was simulated in very large lattices with concentrations of impurities q=0 and q=0.2. The scaling of the mass in chemical space gives accurate estimates of the critical branching probabilities b(c) and of the chemical dimensions Dc at criticality, improving previous results. Estimates of the fractal dimension D(F) at criticality are consistent with a universal value along the critical line. Our results for q=0 suggest small deviations of Dc and D(F) from the percolation values. We also simulated the BPGM in finite lattices of lengths between L=32 and L=512 for the same concentrations q. Using finite-size scaling techniques, we confirm the previous estimates of D(F) and the universality along the critical line, and obtain the correlation exponent nu=1.43+/-0.06. It proves that the BPGM is not in the same universality class of percolation in two dimensions. Finally, we simulate random walks on the critical polymers grown in very large lattices with q=0 and q=0.2, and obtain the random walk dimension Dw and the spectral dimension Ds. Dw is larger and Ds is smaller than the corresponding values in critical percolation clusters, due to the lower connectivity of the polymers. The scaling relation Ds=2D(F)/Dw is not satisfied, as observed in other tree-like structures.

Modeling highly branched structures: Description of the solution structures of dendrimers, polyglycerol, and glycogen

“…By studying the molecular weight distribution, Buzza found a power‐law scaling of the polymer's radius of gyration with its molecular weight 55. Botelho and Reis studied the critical properties of the Branched Polymer Growth Model on a two‐dimensional lattice,56, 57 found fractal structures and concluded that the model is not in the same universality class as percolation. We discuss below distinctly non‐fractal behavior for several polymers.…”

Mechanistic Information from Analysis of Molecular Weight Distributions of Starch

“…A full description of the microstructure of a starch sample requires a function of very high dimensionality which includes not only the overall MWD, but the MWDs of central chains, branches off the central chains, branches off branches, etc., as well as the branching density at each level. There are also means of describing the architecture in terms of more compact quantities such as topological and fractal-dimension characterization of branched systems, − although they have never been applied to starch. Unfortunately, a full description is not only beyond the reach of current experimental techniques, but also such a plethora of information would be very hard to reduce to mechanistic knowledge.…”

The branched polymer growth model revisited

“…This generalized model (which became known as the branched polymer growth model -BPGM) was found to exhibit an interesting phase transition (due to competition between hindrances and branching) separating infinite from finite growth regimes [7]. In the following years, several authors have studied the BPGM [8][9][10][11][12][13][14][15]. The topological * Electronic address: ubiraci@ffclrp.usp.br and dynamical aspects of the model were investigated [8].…”

“…On the Bethe lattice, this proposal is based on analytical results [11]. A further analysis using finite-size scaling techniques led to the conclusion that the BPGM is not in the same universality class of percolation in two dimensions [15].…”