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 $\lambda_{i}$ along the chain, with periodicity $2p$. The simulations are done for various chain lengths $N$, 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, $p \sim O(N)$ and $p\sim O(1)$ to illustrate the physics of the various phase transitions. A tentative variational approach is also presented.