Diffusion in 2D Quasi-Crystals

D Joseph, M Baake, P Kramer, H.-R Trebin
1994 Europhysics letters  
PACS. 61.40 -Amorphous and polymeric materials. PACS. 61.70 -Defects in crystals. P ACS. 66.30F -Self-diffusion in metals, semimetals, and alloys. Abstract. -Self-diffusion induced by phasonic flips is studied in an octagonal model quasi-crystal. To determine the temperature dependence of the diffusion coefficient, we apply a Monte Carlo simulation with specific energy values of local configurations. We compare the results of the ideal quasi-periodic tiling and a related periodic approximant
more » ... odic approximant and comment on possible implications to real quasi-crystals. Recently, Kalugin and Katz [1] proposed a possible mechanism for bulk self-diffusion in quasi-crystals. Their model is based on specific geometric properties of the quasi-crystalline structure. In ordinary crystals, the diffusion process depends on the presence of vacancies in the lattice and is activated by the vacancy production rate and the hopping of atoms in the neighbourhood. Both lead to a Brownian-like motion, and the temperature dependence of the diffusion coefficient thus obeys an Arrhenius law [2] . Due to additional degrees of freedom of quasi-periodic systems-the so-called phasons-one can construct an additional process for bulk diffusion in quasi-crystals. Based on general arguments, Kalugin and Katz conclude that the phason degree of freedom leads to a deviation of the diffusion coefficient from the Arrhenius law. Unfortunately, because of the general nature of their considerations, they were neither able to determine at which temperature these deviations are to be expected nor to estimate their order of magnitude. In this article, we calculate the diffusion coefficient of the eightfold symmetric Ammann-Beenker tiling [3] quantitatively. We start from a mean-field model which takes the additional degrees of freedom into account and we apply a Monte Carlo method to estimate the phason contribution to the bulk diffusion in the quasi-periodic plane. We discuss general properties and similarities with the ansatz of Kalugin and Katz and compare the quasi-periodic tiling with a periodic approximant. The Arnmann-Beenker tiling is chosen because, on the one hand, it provides generic quasi-periodic properties while, on the other hand, it is as simple as possible. Nevertheless, all results stated below can easily be extended to other two-dimensional quasi-periodic tilings like the Penrose tiling [4] or the Ti.ibingen triangle tiling [5] . The Ammann-Beenker tiling consists of squares and 45-degree rhombi which form (modulo operations of the dihedral group d 8 ) 6 different vertex configurations. Combinatorially, another 10 ( + 3 mirror inverted) vertices are possible without gaps or overlaps, which were introduced and
doi:10.1209/0295-5075/27/6/007 fatcat:xcwamgiojvhqlbashyth2wnove