Simple model of kinetic roughening of quasicrystalline surfaces

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 (Δμ${)}^{\mathrm{-}1/3}$ as the driving chemical-potential difference Δμ 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., ${\mathit{V}}_{\mathit{g}}$∝exp{-1/3[Δ${\mathit{u}}_{\mathit{c}}$(T)/Δμ${]}^{4/3}$}. The quantity Δ${\mathrm{\mu }}_{\mathit{c}}$(T), which defines a rounded kinetic roughening transition, is nonuniversal. For ‘‘perfect-tiling models’’ of quasicrystal growth, I find Δ${\mathrm{\mu }}_{\mathit{c}}$(T)∝${\mathit{T}}^{\mathrm{-}3/2}$, which fits recent numerical simulations, while for models which allow bulk phason Debye-Waller disorder, ln(1/Δ${\mathrm{\mu }}_{\mathit{c}}$)∝${\mathit{T}}^{3/2}$. The growing interface is algebraically rough at all temperatures.