Structural and phase transitions of one and two polymer mushrooms

A polymer mushroom here refers to a group of $n$ homopolymer chains end-grafted at the same point on a flat, impenetrable and homogeneous substrate. Using lattice self-consistent field (LSCF) calculations with the Kronecker $\delta $-function interactions (instead of the commonly used nearest-neighbor interactions), we have studied the structures of one and two polymer mushrooms in an explicit solvent as a function of the polymer volume fraction, the solvent quality characterized by the Flory-Huggins \textit{$\chi $} parameter, and the distance between the two mushrooms. We have constructed phase diagrams of these systems showing the coil-globule transition (CGT) of one mushroom and how it is coupled with the fused-separated transition (FST) of two mushrooms. Since LSCF results are exact only in the limit of $n\to \infty $, we also use the newly proposed fast lattice Monte Carlo (FLMC) simulations$^{1}$ with the same Hamiltonian as in LSCF theory to examine how this limit is approached with increasing $n$. Direct comparisons between LSCF and FLMC results without any parameter-fitting quantify the fluctuation/correlation effects neglected in LSCF theory. We also find a second-order symmetric-asymmetric transition (SAT) for one-mushroom system in the globule state, and examine its coupling with CGT and FST. [1]~\textit{Q. Wang}, \textbf{Soft Matter, 5}, 4564 (2009); \textbf{6}, 6206 (2010).