Self-assembly mechanism for limit-periodic structure

Limit-periodic (LP) structures, which are the union of an infinite set of periodic lattices with ever increasing lattice constants, present a challenge for self-assembly protocols. We consider the possibility of forming a LP phase in a slow quench of a collection of colloidal particles designed to mimic the Taylor-Socolar monotile system.\footnote{J.\ E.\ S.\ Socolar and J.\ M.\ Taylor, {\it J.\ Comb.\ Theory A} {\bf 118}: 2207 (2011).} A toy model with discrete tile orientations and mismatch energies yields the LP state through an infinite sequence of phase transitions.\footnote{C.\ Marcoux, T.\ W.\ Byington, Z.\ Qian, P.\ Charbonneau, and J.\ E.\ S.\ Socolar, {\it Phys. Rev. E} {\bf 90}, 012136 (2014).} Here we present the results of Monte Carlo simulations of slow quenches of identical hard disks with embedded magnetic dipoles, allowing for continuous rotations of the close-packed disks. Surprisingly, an extremely slow quench still results in the spontaneous emergence of the LP state even when the system has a periodic ground state. The series of phase transitions preempts the formation of the periodic phase, leading to low energy states separated from the ground state by insurmountable free energy barriers.