Nanostructures with coherent boundaries and the local approach

Abstract

In our recent works [1,2], we analyzed the structural diversity of nanoparticles and the fragmentariness of their structure and formulated the statement that the structural inhomogeneity is a fundamental property of the nanostate. This statement was experimentally confirmed for a number of materials.In particular, reasoning from the results of neutron diffraction and X-ray diffraction investigations of ZrO 2 nanoparticles, Burkhanov et al. [3] proposed a twophase model allowing for the difference between the central and peripheral regions of a particle and their pseudomorphic conjugation. Palosz et al. [4] also used a two-phase model to interpret their diffraction data for SiC, GaN, and diamond nanoparticles. The concept of an "apparent lattice parameter" clearly indicates strong deviations of the position of Bragg peaks (especially at small angles) from the predicted crystallographic positions. The inference was made that a unique lattice parameter for microcrystallites cannot be determined by standard powder diffractometry. The behavior of nanocrystalline powders under pressure cannot be satisfactorily explained in terms of a unique compressibility coefficient. This also counts in favor of the twophase structure of nanoparticles.Earlier [5][6][7], structurally inhomogeneous zirconia nanoparticles that consist of interpenetrating fragments with different symmetries and are characterized by the orientational relationships "incompatible" from the standpoint of classical crystallography were experimentally found for the first time. Regularly oriented fragments with different symmetries, for which the requirements of classical crystallography are not satisfied (the interface need not be a plane, so that the orientational relationships do not necessarily correspond to the Miller indices and the fragments themselves need not be crystals), were referred to as centaurs [5]. The boundaries of fragments are coherent, and, therefore, these particles can be defined as nanostructures with coherent boundaries.Alok Singh and Tsai [8] revealed that the cubic and icosahedral phases in metal alloys can intergrow in a regular oriented manner. It was found that the threefold axes of the cubic phase coincide in direction with the twofold axes of the icosahedral phase and the cubic twofold axes are almost parallel to the icosahedral fivefold axes. The interface is not a plane, and the orientational relationships do not correspond to rational ratios of the corresponding Miller indices and are inconsistent with the assumption that this cubic phase is a quasicrystalline approximant of the icosahedral phase.The structural inhomogeneity of nanoparticles has become evident owing to widespread use of high-resolution electron microscopy [7,9]. Of special interest are materials prepared by ultrafast solidification, because extreme conditions often lead to the formation of vitreous or unstable crystalline structures and quasicrystals. Specifically, it has been demonstrated that fragments of cubic crystals can intergrow to form a hierarchic st...

Icosahedral diamond

“…These structures can exist only in the nanoworld. Let us demonstrate how an icosahedral diamond-like nanoparticle (icosahedral diamond) can be constructed if it is treated as a nanostructure that has coherent boundaries and is composed of (insignificantly distorted) fragments of diamond and lonsdaleite (hexagonal diamond).In our earlier works [12][13][14], we developed the local approach. Within this approach, nanoparticles with coherent boundaries in the general case are assembled from a limited set of building blocks (geometrical structural complexes) determined by the fundamental (specifically, projective) manifolds and the principles of assembling are governed by the topological properties of a fiber space.…”

“…Within this approach, nanoparticles with coherent boundaries in the general case are assembled from a limited set of building blocks (geometrical structural complexes) determined by the fundamental (specifically, projective) manifolds and the principles of assembling are governed by the topological properties of a fiber space. The geometrical structural complexes of tetrahedrally coordinated structures were derived and it was shown that fragments of crystalline and quasicrystalline structures can be joined together into a unified nanostructure with coherent boundaries [12]. In [15], it was demonstrated that the local approach allows one to explain different types of icosahedral packings [16,17] when they are considered mappings (onto the three-dimensional Euclidean space) of specific substructures of high-symmetry ndimensional structures ( n ≥ 3) of a fiber space specified by the Hopf bundle for the eight-dimensional root lattice E 8 .…”

“…In our earlier works [12][13][14], we developed the local approach. Within this approach, nanoparticles with coherent boundaries in the general case are assembled from a limited set of building blocks (geometrical structural complexes) determined by the fundamental (specifically, projective) manifolds and the principles of assembling are governed by the topological properties of a fiber space.…”

Steiner systems ( schemes) and special features of the nanostructures symmetry description

“…The joining of the geometrical structural complex to an elementary-similar cluster (embedded in the same algebraic construction) provides a way of assembling nanostructures with coherent boundaries. 17,18 …”

“…For the given algebra and the root lattice E 8 the ring of invariants have a basis of homogeneous polytopes of degrees 2, 8,12,14,18,20,24,30, and the orbit of its isomorphism group is its spherical seven-scheme. Therefore, it is appropriate to consider v-schemes for v = 4, 6, 8, 12, 14 and the values of μ for known polytopes, which, in their turn, correspond to coordination spheres of the E 8 lattice and its sublattices.…”