Statistical mechanics of adsorption and collapse transitions in branched polymer models

The numerical and theoretical study of lattice trees provides a natural model for calculating the properties of branched polymers in dilute solution. The metric exponent, which is related to the size of a tree, and the branch exponent, which measures the internal structure of a tree, are estimated by fitting Monte Carlo data to empirical finite size formulae in dimensions from 2 to 7. The effects of corrections to scaling on the estimated values of exponents are also examined.

Polymer molecules in dilute solution will undergo a collapse transition if the quality of its solvent deteriorates beyond a certain critical point, called the &thetas;-point. Polymers also exhibit an adsorption transition when the temperature is changed. A natural model for these phase transitions in branched polymers is self-interacting lattice trees interacting with a penetrable (or an impenetrable) wall. We derive qualitative phase diagrams of this model using rigorous mathematical arguments. Moreover, the adsorption process is investigated numerically using umbrella sampling methods. We report results about the thermodynamic and metric properties of the trees, and estimate the location of the adsorption transition and cross-over exponent.

Having discussed lattice trees (homopolymers), in which all vertices behave in the same way, we are also interested in copolymers which consist of more than one type of monomer. A natural lattice model of a branched copolymer is a coloured lattice tree, where each colour corresponds to a different type of monomer. The location of the adsorption critical points in quenched and annealed models of lattice branched copolymers are related and compared to the adsorption of branched homopolymers. Finally, we show that a certain quenched model of adsorbing branched copolymers is self-averaging.