A model for the transformation of an Al-Cu-Fe icosahedral quasicrystal into a crystal with a B2-type phase is proposed. The model is based on two assumptions: (1) the main building block for the quasicrystal structure is a hierarchical dodecahedron composed of two icosahedral clusters, coinciding with two different sections of the {3, 3, 5} polytope; (2) the transformation of the quasicrystal into a B2-type crystal phase can be described as the transition between 3D sections of two polytopes, namely {3, 3, 5} and {3, 4, 3}. In the framework of the proposed model, two experimental facts gain plausible explanations: the transformation of the Al-Cu-Fe quasicrystal into the BCC phase specifically and the orientational relationships observed between this BCC phase and the initial icosahedral quasicrystal.

Thermodynamic properties of molten Al-Mn, Al-Cu and Al-Fe-Cu alloys in a wide temperature range of 1123-1878 K and the whole range of concentrations have been studied using the integral effusion method and Knudsen mass spectrometry. Thermodynamic functions of melts were described by the associated solution model. The possibility of icosahedral quasicrystal (i-QC) precipitation from liquid Al-Mn and Al-Cu-Fe alloys was found to be a consequence of the existence in liquid associates (clusters). A geometric model is suggested for the structure of associates in liquid.

A model structure of the aperiodic cubic phase (a cubic quasicrystal) has been constructed as a periodical packing of hierarchical octahedral clusters which were composed of truncated tetrahedra (Friauf-Laves polyhedra) and chains of Frank-Kasper polyhedra with 14 vertices. The construction of the hierarchical model for the cubic aperiodic phase became possible due to the discovery of a new space subdivision with equal edges and with vertices belonging to two orbits of the space group Fm3m. The subdivision is characterized by unique values and unique relations between the coordinates of the starting points of two orbits. Calculated x-ray diffraction patterns for the proposed hierarchical model are in qualitative agreement with published experimental x-ray patterns for aperiodical phases observed in melt-quenched Mg-Al and Fe-Nb-B-Si alloys.