Semi-flexible compact polymers in two dimensional nonhomogeneous confinement

We have studied the compact phase conformations of semi-flexible polymer chains confined in two dimensional nonhomogeneous media, modelled by fractals that belong to the family of modified rectangular (MR) lattices. Members of the MR family are enumerated by an integer $p$ $(2\leq p<\infty)$, and fractal dimension of each member of the family is equal to 2. The polymer flexibility is described by the stiffness parameter $s$, while the polymer conformations are modelled by weighted Hamiltonian walks (HWs). Applying an exact method of recurrence equations we have found the asymptotic behavior of partition function $Z_N$ for closed HWs consisting of $N$ steps. We have established that $Z_N$ scales as $\omega^N \mu^{N^\sigma}$, where the critical exponent $\sigma$ in the stretched exponential term does not depend on $s$, and takes the value 1/2 for each fractal from the family. The constants $\omega$ and $\mu$ depend on both $p$ and $s$, and, in addition, $\mu$ depends on the parity of MR generator. Besides, we have calculated numerically the stiffness dependence of the polymer persistence length and various thermodynamic quantities (such as free and internal energy, specific heat and entropy), for a large set of members of MR family. We have found that semi-flexible compact polymers, on MR lattices, can exist only in the liquid-like (disordered) phase, whereas the crystal (ordered) phase has not appeared. Finally, the behavior of examined system at zero temperature has been discussed.