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A mechanism is proposed for the nucleus formation at the mutual transformation of austenite and cementite crystals. The mechanism is founded on the interpretation of the considered structures as crystallographic tiling onto non-intersecting rods of triangulated polyhedra. A 15-vertex fragment of this linear substructure of austenite (cementite) can be transformed by diagonal flipping in a rhombus consisting of two adjacent triangular faces into a 15-vertex fragment of cementite (austenite). In the case of the mutual austenite-cementite transformation, the mutual orientation of the initial and final fragments coincides with the Thomson-Howell orientation relationships which are experimentally observed [Thompson & Howell (1988). Scr. Metall. 22, 229-233] in steels. The observed orientation relationship between f.c.c. austenite and cementite is determined by a crystallographic group-subgroup relationship between transformation participants and noncrystallographic symmetry which determines the transformation of triangulated clusters of transformation participants. Sequential fulfillment of diagonal flipping in the 15-vertex fragments of linear substructure (these fragments are equivalent by translation) ensures the austenite-cementite transformation in the whole infinite crystal. The energy barrier for diagonal flipping in the rhombus with iron atoms in its vertices has been calculated using the Morse interatomic potential and is found to be equal to 162 kJ mol -1 at the face-centered cubic-body-centered cubic transformation temperature in iron.

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.

Tessellations by generating clusters are proposed for the high-pressure phases BC8 and R8 of silicon. The structures of both high-pressure phases are represented by the parallel packings of rods. The latter are stacks of distorted icosahedra, joined by a common triangular face and flattened out along the threefold symmetry axis. Along the rod axis there is an alternation of empty and double-centered icosahedra. The empty icosahedra are additionally distorted in the cubic BC8 phase by antiparallel rotation about the rod axis, while the double-centered icosahedra are distorted by that rotation in the rhombohedral R8 phase. A possible mechanism of the reversible BC8 <--> R8 transformation is proposed as the rotation about the rod axis of the common triangular face of the neighboring icosahedra, thus transforming between distorted and undistorted icosahedra. The graphs of the generating clusters for both the BC8 and R8 structures are determined by two subconfigurations of the same construction of the finite projective geometry.