Simple model of kinetic roughening of quasicrystalline surfaces.

Abstract

A simple relaxational model of the dynamics of the surface of a growing quasicrystal is studied. As in a crystal, growth proceeds through the nucleation of steps on the surface. Unlike the crystal, the heights ${\mathit{h}}_{\mathit{s}}$ of these steps diverge like (\ensuremath{\Delta}\ensuremath{\mu}${)}^{\mathrm{\ensuremath{-}}1/3}$ as the driving chemical-potential difference \ensuremath{\Delta}\ensuremath{\mu} between quasicrystal and fluid goes to zero. The exponent 1/3 is universal for all quasicrystals with structures derived from quadratic irrationals. This large step size leads to unusually low growth velocities ${\mathit{V}}_{\mathit{g}}$; i.e., ${\mathit{V}}_{\mathit{g}}$\ensuremath{\propto}exp{-1/3[\ensuremath{\Delta}${\mathit{u}}_{\mathit{c}}$(T)/\ensuremath{\Delta}\ensuremath{\mu}${]}^{4/3}$}. The quantity \ensuremath{\Delta}${\mathrm{\ensuremath{\mu}}}_{\mathit{c}}$(T), which defines a rounded kinetic roughening transition, is nonuniversal. For ``perfect-tiling models'' of quasicrystal growth, I find \ensuremath{\Delta}${\mathrm{\ensuremath{\mu}}}_{\mathit{c}}$(T)\ensuremath{\propto}${\mathit{T}}^{\mathrm{\ensuremath{-}}3/2}$, which fits recent numerical simulations, while for models which allow bulk phason Debye-Waller disorder, ln(1/\ensuremath{\Delta}${\mathrm{\ensuremath{\mu}}}_{\mathit{c}}$)\ensuremath{\propto}${\mathit{T}}^{3/2}$. The growing interface is algebraically rough at all temperatures.