Collapse transitions of a 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 hydro-philic and -phobic monomers (ii) a periodic distribution of the Xi along the chain, with periodicity 2p. The simulations are done for various chain lengths NI in d = 2 (square lattice) and d = 3 (cubic lattice). There is a critical value p(c)(d, N) 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. We restrict our study to two extreme cases, p similar to O(N) and p similar to O(1) to illustrate the physics of the various phase transitions. A tentative variational approach is also presented.