Hierarchical porosity in real quasicrystals

Abstract

The structure of real icosahedral quasicrystals of the Al± Pd± Mn system can be described in terms of a hierarchical self-similar packing of overlapping atomic clusters. An in¯ation scale factor ¿ 3 preserves long-range order but generates a hierarchy of holes and a fractal structure. Such holes have actually been observed and they might be of basic importance in the growth mechanism and stability of quasicrystals. § 1. IntroductionQuasicrystals are a form of the solid state which possesses a long-range translational order despite a non-crystallographic point group of symmetries (Janot 1994). The debate concerning structure and stability of quasicrystals very often revolves about the relative roles of energy and entropy (Henley 1991) but in both cases the description of the structure is based on tiles representing the short-range order in the material. Perfect quasiperiodic structures can be obtained via several geometrical methods including in¯ation± de¯ation of prototiles, tiling with at least two di erent prototiles plus matching rules and also a physical cut of a higher-dimensional periodic structure. These perfectly quasiperiodic models are irreplaceable when one needs to approach experimentally the structure of real quasicrystals for which they give at least a very useful average description (Cornier-Quiquandon et al. 1991Boudard et al. 1992), but space tiling requires a choice of a small number of prototiles together with the de® nition of rules to ® ll space without overlap nor holes. This is intuitively simple for crystals; it is actually easy to imagine a single building block arising as a low-energy atomic cluster of the given elements and, then, periodicity is obtained by adding identical clusters again and again. The procedure works quite well as long as the building block has proper crystallographic symmetries. The physical conditions required to emulate, say, a Penrose (1974) tiling appear to be much more complex. The energetics must be delicately balanced to allow two distinct clusters to be almost equally stable so that they intermix with a speci® c ratio of densities and according to matching rules. The entropic view point (Henley 1991, Joseph andElser 1997) allows phason defects to stabilize the structure, relaxes the

Special Defects in Quasicrystals

“…The residual broadening of the Bragg peaks may be ascribed to distortion fields around facetted dodecahedral voids as recently observed using contrast imaging and X-ray topography (Fig. 2) [17]; surfaces of the voids would then be coated or tapested by layers of approximants with increasing lattice parameters when going from the void border to the bulk around. Beyond the Bragg peak broadening, diffuse scattering has also been measured around the Bragg peaks, with average intensity increasing as Q 2 c and a 1/q 2 decay from the Bragg peak centre [28±30, 33].…”

“…Models are easily built to demonstrate this [16]. Interestingly enough, phase contrast X-ray imaging techniques have been able to show facetted holes in single grain of the Al±Pd±Mn system [17,18] which, at least, suggests unusual growth defects (Fig. 2).…”

Atomic-level structure and structure–property relationship in metallic glasses

“…In addition to the interpretation in reciprocal space (structure factor), an understanding of the real space fractal packing is also important. First of all, as pointed out by Coddens in an earlier debate on quasicrystals [337][338][339], the fractal in a dense 3-D material, if it ever exists, can only survive up to medium range, and the fractal correlation must break down over long range (otherwise the density of the material would vanish). Second, there should be a basic unit for the fractal packing, which defines the lower limit of the fractal length-scale.…”

The Influence of Growth Rate on Porosity in Al-Pd-Mn Icosahedral Quasicrystals.

“…Additional work by Beeli et al confirmed that pore size and spacing corresponded to diffusion models [4]. Two other hypotheses regarding the origin of porosity were based on quasicrystalline structural arguments [5,6]. The self-similar model proposes that porosity develops from specifically located vacancies that are inherent in the quasicrystal structure [5].…”

“…Two other hypotheses regarding the origin of porosity were based on quasicrystalline structural arguments [5,6]. The self-similar model proposes that porosity develops from specifically located vacancies that are inherent in the quasicrystal structure [5]. A modification of the selfsimilar model, the random covering theory, maintains the inherent and specifically located vacancies but increases the ability for deviation from the perfect arrangement by reducing the strict scaling laws [6].…”