Phase transitions of a two-dimensional periodic hydrophilic hydrophobic chain

We study a single self-avoiding hydrophilic hydrophobic polymer chain, through Monte Carlo lattice simulations. The affinity of monomer i for water is characterized by a (scalar) charge lambda(i), and the monomer-water interaction is short-ranged. Assuming incompressibility yields an effective short ranged interaction between monomer pairs (i, j), proportional to (lambda(i) + lambda(j)), In this article, we take lambda(i) = +1 (resp. (lambda(i) = -1)) for hydrophilic (resp, hydrophobic) monomers and consider a chain with (i) an equal number of hydrophilic and -phobic monomers (ii) a periodic distribution of the lambda(i) along the chain, with periodicity 2p. This model may be of interest in various situations (protein folding, polysoaps,...) The simulations are done on the square lattice (d = 2), for various chain lengths N. There is a critical value (p(c)(N) similar to 0.07N) of the periodicity, which distinguishes between different low temperature structures. For p > p(c), the ground state corresponds to a macroscopic phase separation between a dense hydrophobic core and hydrophilic loops. For p < p(c) (but not too small), one gets a microscopic (finite scale) phase separation, and the ground state corresponds to a chain or network of hydrophobic droplets, coated by hydrophilic monomers. These different cases will be explored through a Multiple Markov chain method. The results for the d = 3 case (where p(c)(N) similar to N-1/3) are similar.