Thermodynamic behavior of a polymer with interacting bonds on a square lattice

Semianalytical solutions of Ising-like and Potts-like magnetic polymers on the Bethe lattice

We study magnetic polymers, defined as self-avoiding walks where each monomer i carries a "spin" s_{i} and interacts with its first neighbor monomers, let us say j, via a coupling constant J(s_{i},s_{j}). Ising-like [s_{i}=±1, with J(s_{i},s_{j})=ɛs_{i}s_{j}] and Potts-like [s_{i}=1,...,q, with J(s_{i},s_{j})=ɛ(s_{i})δ(s_{i},s_{j})] models are investigated. Some particular cases of these systems have recently been studied in the continuum and on regular lattices and are related to interesting applications. Here, we solve these models on Bethe lattices of branching number σ, focusing on the ferromagnetic case in zero external magnetic field. In most cases, the phase diagrams present a nonpolymerized (NP) and two polymerized phases: a paramagnetic (PP) and a ferromagnetic (FP) one. However, quite different thermodynamic properties are found depending on q in the Potts-like polymers and on whether one uses the Ising or Potts coupling in the two-state systems. For instance, for the standard Potts model [where ɛ(1)=⋯=ɛ(q)] with q=2, beyond the θ-point (where the critical and discontinuous NP-PP transition lines meet), a second tricritical point connecting a critical and a discontinuous transition line between the PP-FP phases is found in the system. A triple point where the NP-PP, NP-FP, and PP-FP first-order transition lines meet is also present in the phase diagram. For q≥3 the PP-FP transition is always discontinuous, and the scenario with the triple and θ points appears for q≤6. Interestingly, for q≥7, as well as for the Ising-like model the θ point becomes metastable and the critical NP-PP transition line ends at a critical end point (CEP), where it meets the NP-FP and PP-FP coexistence lines. Importantly, these results indicate that when q≤6 the spin ordering transition is preceded by the polymer collapse transition, whereas for q≥7 and in the Ising case these transitions happen together at the CEPs. Some interesting nonstandard Potts models are also studied, such as the lattice version of the model for epigenetic marks in the chromatin introduced by Michieletto et al. [Phys. Rev. X 6, 041047 (2016)2160-330810.1103/PhysRevX.6.041047]. In addition, the solution of the dilute Ising and dilute Potts models on the Bethe lattice are also presented here, since they are important to understand the PP-FP transitions.

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

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.

Nature of the collapse transition in interacting self-avoiding trails

We study the interacting self-avoiding trail (ISAT) model on a Bethe lattice of general coordination q and on a Husimi lattice built with squares and coordination q=4. The exact grand-canonical solutions of the model are obtained, considering that up to K monomers can be placed on a site and associating a weight ω_{i} with an i-fold visited site. Very rich phase diagrams are found with nonpolymerized, regular polymerized, and dense polymerized phases separated by lines (or surfaces) of continuous and discontinuous transitions. For a Bethe lattice with q=4 and K=2, the collapse transition is identified with a bicritical point and the collapsed phase is associated with the dense polymerized (solidlike) phase instead of the regular polymerized (liquidlike) phase. A similar result is found for the Husimi lattice, which may explain the difference between the collapse transition for ISATs and for interacting self-avoiding walks on the square lattice. For q=6 and K=3 (studied on the Bethe lattice only), a more complex phase diagram is found, with two critical planes and two coexistence surfaces, separated by two tricritical and two critical end-point lines meeting at a multicritical point. The mapping of the phase diagrams in the canonical ensemble is discussed and compared with simulational results for regular lattices.

Isotropic-nematic phase diagram for interacting rigid rods on two-dimensional lattices

The phase behavior of interacting rigid rods of length k (k-mers) on two-dimensional square and triangular lattices has been studied by theoretical calculations in the framework of the lattice-gas model. The process was analyzed by comparing the dependence on coverage of the free energy per site of an isotropic submonolayer of interacting k-mers f(iso)(θ) with that corresponding to a fully aligned (nematic) system f(nem)(θ). The existence of an intersection point between the curves f(iso)(θ) and f(nem)(θ), which is indicative of the occurrence of an isotropic-nematic phase transition in the adlayer, allowed us to obtain the complete (temperature, coverage, k-mer size) phase diagram of the system.

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

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 end point, 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 end points 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.

Solution of a model of self-avoiding walks with multiple monomers per site on the Husimi lattice

We solve a model of self-avoiding walks which allows for a site to be visited up to two times by the walk on the Husimi lattice. This model is inspired in the Domb-Joyce model and was proposed to describe the collapse transition of polymers with one-site interactions only. We consider the version in which immediate self-reversals of the walk are forbidden. The phase diagram we obtain for the grand-canonical version of the model is similar to the one found in the solution of the Bethe lattice, with two distinct polymerized phases: a tricritical point and a critical endpoint.

Solution of a model of self-avoiding walks with multiple monomers per site on the Bethe lattice

We solve a model of self-avoiding walks with up to two monomers per site on the Bethe lattice. This model, inspired in the Domb-Joyce model, was recently proposed to describe the collapse transition observed in interacting polymers [J. Krawczyk, Phys. Rev. Lett. 96, 240603 (2006)]. When immediate self-reversals are allowed (reversion-allowed model), the solution displays a phase diagram with a polymerized phase and a nonpolymerized phase, separated by a phase transition which is of first order for a nonvanishing statistical weight of doubly occupied sites. If the configurations are restricted forbidding immediate self-reversals (reversion-forbidden model), a richer phase diagram with two distinct polymerized phases is found, displaying a tricritical point and a critical end point.