Lattice polymers with two competing collapse interactions
We study a generalised model of self-avoiding trails, containing two different types of interaction (nearest-neighbour contacts and multiply visited sites), using computer simulations. This model contains various previously-studied models as special cases.
We find that the strong collapse transition induced by multiply-visited sites is a singular point in the phase diagram and corresponds to a higher order multi-critical point separating a line of weak second-order transitions from a line of first-order transitions.
There are various statistical mechanical models of the polymer collapse phase transition (Gennes 1979, Vanderzande 1998). However, there are two basic elements in each of the models: on the one (Gennes 1979) hand, the configurations of the polymer have some type of so-called excluded volume, that implies molecules are separate in space, and on the other hand the configurations have an attractive force between different parts of the polymer, that drives the transition. On lattices, both self-avoiding walks (SAW), where different sites of walk avoid being on the same site of the lattice, and self-avoiding trails (SAT), which are walks that can share sites though not bonds of the lattice, have been used as the configuration space for collapse models. The attractive force has been modelled both by adding energies for shared sites and also via so-called nearest-neighbour contacts, where sites adjacent on the lattice not joined by a step of the walk are given an energy. When SAW are coupled with nearest neighbour interactions the canonical ISAW model is reproduced (see (Vanderzande 1998) for a review), while when trails are coupled with site interactions the ISAT model (Malakis 1976) is reproduced. Despite what one might expect from the principle of universality the collapse of these models appears to behave differently (Shapir 1984, Owczarek 1995, Owczarek 2007).
A recent study (Bedini 2013) considered self-avoiding trails interacting via nearest-neighbour contacts (INNSAT) as a hybrid of the two models. Evidence from computer simulations showed that the collapse transition in INNSAT is different from the collapse transition in ISAT, which is a strong second order transition, but similar to ISAW which is predicted to be a weak second order transition where the specific heat converges at the transition. In ISAW collapse one needs to consider the third derivative of the free energy to see a divergent quantity and then only weakly divergent. It was also found that the low-temperature phase of the two trail-collapse models differ substantially: the phase associated with multiply visited site interactions is fully dense in the thermodynamic limit (as shown in (Bedini 2013a)). The low-temperature phase associated with nearest-neighbour contacts wasn’t fully dense as is believed to be the case for interacting self-avoiding walks (Owczarek 1993).
The properties of lattice polymers are also related to those of magnetic systems near their critical point (Vanderzande 1998). More precisely, self-avoiding walks configurations appear as the diagrams of the high-temperature expansion of an \(O(n)\) magnetic system when taking the formal limit of zero components (\(n \to 0\)), and their scaling exponents can be obtained from the \(O(n)\) critical point. In this mapping the collapse transition corresponds to a tri-critical point of the magnetic system and one would hope to obtain the critical exponents for the polymer collapse transition from the ones of this tri-critical point.
Various authors (Nienhuis 1982, Nienhuis 1984, Blote 1989, Guo 2006, Nienhuis 2008) have studied critical and tri-critical \(O(n)\) spin systems. For a special choice of the model on the honeycomb lattice, exact results were obtained in (Nienhuis 1982) for two cases: a critical point and a special point governing the low-temperature phase. When \(n \to 0\) these two cases become the dilute and dense polymer phase. The dilute and dense phases were also found along two branches of a square-lattice \(O(n)\) model (Nienhuis 1984, Blote 1989) together with two different branches describing the critical behaviour that occurs when \(O(n)\) and Ising degrees of freedom on the square lattice display a joint critical point.