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Using exact enumerations of self-avoiding walks (SAWs) we compute the inhomogeneous pressure exerted by a two-dimensional end-grafted polymer on the grafting line which limits a semi-infinite square lattice. The results for SAWs show that the asymptotic decay of the pressure as a function of the distance to the grafting point follows a power-law with an exponent similar to that of gaussian chains and is, in this sense, independent of excluded volume effects.

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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.

We obtain the entropy of flexible linear chains composed of M monomers placed on the square lattice using a transfer matrix approach. An excluded volume interaction is included by considering the chains to be self-avoiding and mutually avoiding, and a fraction rho of the sites is occupied by monomers. We solve the problem exactly on stripes of increasing width m and then extrapolate our results to the two-dimensional limit m--> infinity using finite-size scaling. The extrapolated results for several finite values of M and in the polymer limit M--> infinity for the cases where all lattice sites are occupied (rho=1) and for the partially filled case rho<1 are compared with earlier results. These results are exact for dimers (M=2) and full occupation (rho=1) and derived from series expansions, mean-field-like approximations, and transfer matrix calculations for some other cases. For small values of M, as well as for the polymer limit M--> infinity, rather precise estimates of the entropy are obtained.