Selected topics in polymer physics

This thesis explores simple generalization of the self-avoiding walk problem with the aim of approaching a better description of such complicated systems. We will verify that peculiar interaction mechanisms can influence the nature of the transition and induce a collapse towards different compact (and even non compact) phases. These possibilities imply a richer scenario and a wider degree of non universality than those contemplated in the usual theta point physics. In chapter 1 we study the collapse transition of a linear homopolymer model, which presents two competing interactions: an attractive potential between pair of monomers and an energy penalty which discourages a single monomer from having more than a certain number of contacts. The first interaction favors the collapse of the chain, the second enhances the ramification into thin branch-like structures. The model is directly inspired by a lattice model recently introduced to describe protein behavior and, in particular, to take into account that different amino acids have different sizes, (and, therefore, a different capability of admitting surrounding molecules). We study the phase diagram and the critical behavior of the system through exact enumerations and Monte-Carlo simulations. In addition we present an analytical argument which determines qualitatively the phase boundaries. Our findings suggest the existence of a complex phase diagram, with a swollen, a compact and a branched phase. In chapter 2 we consider Hamiltonian walks on a lattice, i.e. self-avoiding walks that visit all the sites. As globular protein in their native state form compact structures, Hamiltonian walks have recently become one of the model of choice for protein folding studies. We formulate the polymer problem in terms of a particular O(n) model, in the limit n-+ 0. We then use a suitable high-temperature expansion to estimate the total number of Hamiltonian walks on a lattice. This approach enables us to derive a mean-field result and to generate corrections. to it in power of 1/ d in a systematic way ( d is the dimensionality of the space). We calculate the coefficients of the series up to third order, extending of one order previous results. The spin representation can be extended to Hamiltonian walks with a bending energy which favors straight segments of the chain. This is the so called Flory model for polymer melting. In this case, our mean field results are in agreement with approaches based on a field-theoretic representation of the partition function. In chapter 3 we present a novel Bethe approximation to study lattice models of linear polymers. The approach is variational in nature and based on the cluster variation method. We apply it to study the phase diagram of a semifiexible chain model which includes both an attractive potential and a bending rigidity. This study should be relevant to the case of stiff polymer, as for example DNA. Our findings support the existence of an open coil at high temperature, a collapse globule at intermediate temperature and low stiffness, and a stretched phase at low temperature and large stiffness. The transition from the coil to the globule is a second order theta collapse, whereas the transition toward the stretched phase is a first order transition. We find evidences for a multicritical point, where the two transition lines meet. As a consequence, for sufficiently stiff polymers the globular phase disappears and the system undergoes a direct first order transition from the random coil to the orientational ordered state. These results contradict in several aspects mean-field theory and are in good agreement with previous Monte Carlo simulations of the model. In the limit of Hamiltonian walks, moreover, our approximation recovers results of the Flory-Huggins theory for polymer melting.