Bethe approximation for a semiflexible polymer chain

Abstract

We present a Bethe approximation to study lattice models of linear polymers. The approach is variational in nature and based on the cluster variation method. We focus on a model with (i) a nearest-neighbor attractive energy ${\ensuremath{\epsilon}}_{v}$ between a pair of nonbonded monomers, (ii) a bending energy ${\ensuremath{\epsilon}}_{h}$ for each pair of successive chain segments that are not collinear. We determine the phase diagram of the system as a function of the reduced temperature $t=T/{\ensuremath{\epsilon}}_{v}$ and of the parameter $x={\ensuremath{\epsilon}}_{h}/{\ensuremath{\epsilon}}_{v}.$ We find two different qualitative behaviors, on varying t. For small values of x the system undergoes a \ensuremath{\theta} collapse from an extended coil to a compact globule; subsequently, on decreasing further t, there is a first order transition to an anisotropic phase, characterized by global orientational order. For sufficiently large values of x, instead, there is directly a first order transition from the coil to the orientational ordered phase. Our results are in good agreement with previous Monte Carlo simulations and contradict in some aspects mean-field theory. In the limit of Hamiltonian walks, our approximation recovers results of the Flory-Huggins theory for polymer melting.