Grand-canonical and canonical solution of self-avoiding walks with up to three monomers per site on the Bethe lattice

Abstract

We solve a model of polymers represented by self-avoiding walks on a lattice which may visit the same site up to three times in the grand-canonical formalism on the Bethe lattice. This may be a model for the collapse transition of polymers where only interactions between monomers at the same site are considered. The phase diagram of the model is very rich, displaying coexistence and critical surfaces, critical, critical endpoint and tricritical lines, as well as a multicritical point.From the grand-canonical results, we present an argument to obtain the properties of the model in the canonical ensemble, and compare our results with simulations in the literature. We do actually find extended and collapsed phases, but the transition between them, composed by a line of critical endpoints and a line of tricritical points, separated by the multicritical point, is always continuous.This result is at variance with the simulations for the model, which suggest that part of the line should be a discontinuous transition. Finally, we discuss the connection of the present model with the standard model for the collapse of polymers (self-avoiding self-attracting walks), where the transition between the extended and collapsed phases is a tricritical point.

Collapse transition in polymer models with multiple monomers per site and multiple bonds per edge

“…For β 1,Θ −0.45, notwithstanding, an inverse situation seems to arise, with β 2,Θ becoming larger for the RF model. Though with our data we can only infer this, one remarks that such scenario have in- deed been found in the mean-field solutions of these models on the Bethe lattice [43]. For instance, for a Bethe lattice with coordination q = 6, the tricritical line in the limit β 2 → −∞ for RA (RF) model is located at β 1 ≈ 0.53 (β 1 = 0), whilst the line of critical-end-points in the limit of β 1 → −∞ is at β 2 ≈ 0.86 (β 2 ≈ 1.13) [43].…”

“…However, the rising of a stable ordered (crystalline) phase in the MMS models, without the addition of any local chain stiffness on them, seems quite unexpected. In fact, no evidence of the existence of a third phase in the canonical phase diagrams was found here or elsewhere [36,42,43].…”

“…Moreover, it is not possible to analyze the effect of dimensionality on the behavior of the models, because these lattices have infinite dimension. Anyhow, the results in [42,43] strongly suggest that the collapse transitions in MMS models: i) are not dependent on their details; and ii) are always continuous.…”

“…Following the notation in [36,43], we define the parameters β 1 ≡ ε 1 /k B T and β 2 ≡ ε 2 /k B T , where k B is the Boltzmann's constant and T is the temperature. So, the canonical partition function of a system with walks of N steps is Z N = S e −E(S)/kBT = S e M2(S)β1+M3(S)β2 .…”

“…In such solutions, coil and globule phases are always separated by lines of continuous transitions, being a tricritical line in the region of β 2 < 0 (with β 1 > 0) and a line of critical-end-points (CEP) for β 1 < 0 (with β 2 > 0), both meeting at a multicritical point [43]. As an aside, note that the region β 1 > 0 and β 2 < 0 is somewhat related to the field theory by des Cloizeaux and Duplantier [44,45] considering attractive (repulsive) two-(three-) body interactions.…”