Bond topology of chain, ribbon and tube silicates. Part II. Geometrical analysis of infinite 1D arrangements of (TO4)n- tetrahedra

Abstract

In Part I of this series, all topologically possible 1-periodic infinite graphs (chain graphs) representing chains of tetrahedra with up to 6-8 vertices (tetrahedra) per repeat unit were generated. This paper examines possible restraints on embedding these chain graphs into Euclidean space such that they are compatible with the metrics of chains of tetrahedra in observed crystal structures. Chain-silicate minerals with T = Si⁴⁺ (plus P⁵⁺, V⁵⁺, As⁵⁺, Al³⁺, Fe³⁺, B³⁺, Be²⁺, Zn²⁺ and Mg²⁺) have a grand nearest-neighbour ⟨T-T⟩ distance of 3.06±0.15 Å and a minimum T...T separation of 3.71 Å between non-nearest-neighbour tetrahedra, and in order for embedded chain graphs (called unit-distance graphs) to be possible atomic arrangements in crystals, they must conform to these metrics, a process termed equalization. It is shown that equalization of all acyclic chain graphs is possible in 2D and 3D, and that equalization of most cyclic chain graphs is possible in 3D but not necessarily in 2D. All unique ways in which non-isomorphic vertices may be moved are designated modes of geometric modification. If a mode (m) is applied to an equalized unit-distance graph such that a new geometrically distinct unit-distance graph is produced without changing the lengths of any edges, the mode is designated as valid (m_v); if a new geometrically distinct unit-distance graph cannot be produced, the mode is invalid (m_i). The parameters m_v and m_i are used to define ranges of rigidity of the unit-distance graphs, and are related to the edge-to-vertex ratio, e/n, of the parent chain graph. The program GraphT-T was developed to embed any chain graph into Euclidean space subject to the metric restraints on T-T and T...T. Embedding a selection of chain graphs with differing e/n ratios shows that the principal reason why many topologically possible chains cannot occur in crystal structures is due to violation of the requirement that T...T > 3.71 Å. Such a restraint becomes increasingly restrictive as e/n increases and indicates why chains with stoichiometry TO_<2.5 do not occur in crystal structures.

Keywords: Euclidean space; [TO4]n− tetrahedra; bond topology; chain graph; chains of tetrahedra; graph embedding.