Growing Perfect Decagonal Quasicrystals by Local Rules

Abstract

A local growth algorithm for a decagonal quasicrystal is presented. We show that a perfect Penrose tiling (PPT) layer can be grown on a decapod tiling layer by a three dimensional (3D) local rule growth. Once a PPT layer begins to form on the upper layer, successive 2D PPT layers can be added on top resulting in a perfect decagonal quasicrystalline structure in bulk with a point defect only on the bottom surface layer. Our growth rule shows that an ideal quasicrystal structure can be constructed by a local growth algorithm in 3D, contrary to the necessity of nonlocal information for a 2D PPT growth.

Experimental Observation of Quasicrystal Growth

“…Some ideal growth models, such as those by Onoda et al [8], Jeong [9], and Olami [10], can explain the formation of such ideal quasicrystals. These models incorporate no structural relaxation either during or after the growth; instead, the growth proceeds in a manner such that the quasicrystalline order is always maintained.…”

“…This discrepancy can be resolved in view of different notions of locality in these rules [12]. Jeong [9] has extended Onoda's algorithm for the growth of a perfect 3D decagonal quasicrystal that consists of a periodic stacking of 2D Penrose tilings. Olami [10] has proposed a growth model for a pentagonal quasicrystalline tiling by local rules, generating highly ordered quasicrystals with limited disorder in a tile arrangement, i.e., with limited phason disorder.…”

“…While crystals can grow simply by copying a unit cell via local atomic interactions, the growth of quasicrystals with quasiperiodic long-range order appears to require the atoms adhering to the growth front to obtain nonlocal structural information, which is physically implausible. This problem has attracted much attention ever since the first discovery of a quasicrystal in 1984 [2], and several theoretical growth models, which primarily apply to decagonal and icosahedral quasicrystals, have been proposed [3][4][5][6][7][8][9][10].…”

“…While crystals can grow simply by copying a unit cell via local atomic interactions, the growth of quasicrystals with quasiperiodic long-range order appears to require the atoms adhering to the growth front to obtain nonlocal structural information, which is physically implausible. This problem has attracted much attention ever since the first discovery of a quasicrystal in 1984 [2], and several theoretical growth models, which primarily apply to decagonal and icosahedral quasicrystals, have been proposed [3][4][5][6][7][8][9][10].First, models have been created to illustrate the random aggregation of symmetric decagonal or icosahedral clusters, with the assumption that the clusters join together under some constraint to maintain orientational order [3][4][5][6]. Though these types of models show better translational correlations than one might expect, they are still not sufficient to explain the correlations observed experimentally.…”

Surface scaling properties of decagonal quasicrystals under nonequilibrium vertical growth

“…The heights of both cells are the same and are taken as the unit length in the vertical direction. First, we imagine a decagonal quasicrystal with a flat surface which can be formed by a layer-by-layer growth under near-equilibrium [17]. With this initial flat surface, we consider the vertical growth under the nonequilibrium deposition process.…”