Monte Carlo Simulation of the Θ-Point in Lattice Trees

Abstract

Branched Polymers in solution are known to undergo a collapse transition driven by the quality of the solvent at the Θ-point. The collapse of the polymer is in a characteristic length, usually taken to be the root mean square radius of gyration, R, of the polymer. In the “good solvent” regime, one expects that R ~ M ν , where M is the molecular mass of the polymer. ν is a critical exponent, commonly called the metric exponent (and it describes the scaling of R with M). In three dimensions, it is believed that \(\nu = \tfrac{1}{2}\), and the branched polymer is said to be “expanded”. Beyond the collapse transition, it is believed that \(v = \frac{1}{2} \) , so that the polymer scales like a solid object (d is the spatial dimension). Branched polymers can be modeled as trees in the cubic lattice, with a short-ranged interaction between vertices which are nearest neighbour in the lattice. Trees can be efficiently sampled by a Metropolis Monte Carlo algorithm. We collect data on the Θ-transition by finding the peak in the specific heat of trees using a Robbins-Monro scheme. In addition, Monte Carlo simulations on trees over a wide range of the short-ranged force using umbrella sampling is described. The data strongly support the notion that the collapse transition is a continuous (second order) transition with a divergent specific heat. We also report values of computed critical exponents.