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Polymers with nearest- and next nearest-neighbor interactions on the Husimi lattice

The exact grand-canonical solution of a generalized interacting self-avoid walk (ISAW) model, placed on a Husimi lattice built with squares, is presented. In this model, beyond the traditional interaction $ω_1=e^{ε_1/k_B T}$ between (nonconsecutive) monomers on nearest-neighbor (NN) sites, an additional energy $ε_2$ is associated to next-NN (NNN) monomers. Three definitions of NNN sites/interactions are considered, where each monomer can have, effectively, at most 2, 4 or 6 NNN monomers on the Husimi lattice. The phase diagrams found in all cases have (qualitatively) the same thermodynamic properties: a non-polymerized (NP) and a polymerized (P) phase separated by a critical and a coexistence surface that meet at a tricritical ($θ$-) line. This $θ$-line is found even when one of the interactions is repulsive, existing for $ω_1$ in the range $[0,\infty)$, i. e., for $ε_1/k_B T$ in the range $[-\infty,\infty)$. Counterintuitively, a $θ$-point exists even for an infinite repulsion between NN monomers ($ω_1=0$), being associated to a coil-"soft globule" transition. In the limit of an infinite repulsive force between NNN monomers, however, the coil-globule transition disappears and only a NP-P continuous transition is observed. This particular case, with $ω_2=0$, is also solved exactly on the square lattice, using a transfer matrix calculation, where a discontinuous NP-P transition is found. For attractive and repulsive forces between NN and NNN monomers, respectively, the model becomes quite similar to the semiflexible-ISAW one, whose crystalline phase is not observed here, as a consequence of the frustration due to competing NN and NNN forces. The mapping of the phase diagrams in canonical ones is discussed and compared with recent results from Monte Carlo simulations.