Grand-canonical solution of semiflexible self-avoiding trails on the Bethe lattice

Abstract

We consider a model of semiflexible interacting self-avoiding trails (sISATs) on a lattice, where the walks are constrained to visit each lattice edge at most once. Such models have been studied as an alternative to the self-attracting self-avoiding walks (SASAWs) to investigate the collapse transition of polymers, with the attractive interactions being on site as opposed to nearest-neighbor interactions in SASAWs. The grand-canonical version of the sISAT model is solved on a four-coordinated Bethe lattice, and four phases appear: non-polymerized (NP), regular polymerized (P), dense polymerized (DP), and anisotropic nematic (AN), the last one present in the phase diagram only for sufficiently stiff chains. The last two phases are dense, in the sense that all lattice sites are visited once in the AN phase and twice in the DP phase. In general, critical NP-P and DP-P transition surfaces meet with a NP-DP coexistence surface at a line of bicritical points. The region in which the AN phase is stable is limited by a discontinuous critical transition to the P phase, and we study this somewhat unusual transition in some detail. In the limit of rods, where the chains are totally rigid, the P phase is absent and the three coexistence lines (NP-AN, AN-DP, and NP-DP) meet at a triple point, which is the endpoint of the bicritical line.

Solution of semi-flexible self-avoiding trails on a Husimi lattice built with squares

“…As mentioned above, this phase diagram is identical to the ones found in the solution of this model on the Bethe and square lattices, and it is depicted in Fig. 3 (figure 3d in [19]).…”

“…In summary, the AN phase is separated from the NP phase by a discontinuous critical transition at z = 1, and from the DP phase by a similar transition at τ x = 1/z. These rather unusual type of phase transition was discussed in general by Fisher and Berker some time ago [22], and was also found and discussed in some detail in the solution of the present model on the Bethe and square lattices [19]. The NP-DP transition is also discontinuous, but not critical.…”

“…In this limit, both in the solution on the Bethe lattice and on the square lattice the same phase diagram was found [19], with three phases: non-polymerized (NP), anisotropic nematic (AN), and dense polymerized (DP). A similar phase diagram should be expected for the model on the Husimi tree.…”

“…When bends are allowed in the chains, we still have a non-polymerized phase (NP), similarly to rigid rods. The dense polymerized (DP) phase [where ρ z = 1 and (ρ c + ρ x ) = 1] found in the solution of the model on the Bethe lattice [19], gives place to a phase where most of the sites are occupied by two monomers, so that ρ z ≈ 1 and (ρ c + ρ x ) ≈ 1, but which is not dense anymore. Following Ref.…”

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

“…However, recent evidences against this exists coming from numerical [20,21] and field theory [10] works. Controversies exist also on the ISAT collapse transition [22][23][24][25][26][27][28], which have motivated several recent works on this model and generalizations of it [29][30][31][32][33][34][35]. Noteworthy among these works is the field theory by Nahum et al [30] showing that the ISAT collapse transition in 2D is multicritical with infinite order.…”