Generating functions for stoichiometry and structure of single- and double-layer sheet-silicates

Abstract

Two-dimensional nets may be used to generate the stoichiometry and structure of single-layer and double-layer sheet-silicate minerals. Many sheet-silicate minerals are based on the 3-connected plane nets 63, 4.82, (4.6.8)2(6.82)1and (52.8)1(5.82)1, and some more complicated nets, e.g. (5.6.7)4(5.72)1(62.7)1, (4.122)2(42.12)1, (52.8)1(5.62)1(5.6.8)2(62.8)1,have one or two representative structures. Many complicated sheet-silicate minerals are based on sheets of 2-, 3- and 4-connected tetrahedra that may be developed from 3- and 4-connected plane nets by a series of oikodoméic operations on 3- or 4-connected nets that change the topologyof the parent net. There are three classes of oikodoméic operations: (1) insertion of 2- and 3-connected vertices into 3- and 4-connected plane nets; (2,3) replication of single-layer sheets by topological mirror or two-fold-rotation operators, and condensation of the resulting twosingle-layer sheets to form double-layer sheets. The topological aspects of these sheet structures may be described by functions that express stoichiometry in terms of tetrahedron connectivities (formula-generating functions) and functions that associate these formula-generating functionswith specific two-dimensional nets. Using these functions, we may generate formulae and structural arrangements of single-layer and double-layer silicate structures with specific local and long-range topological features.

A structure hierarchy for silicate minerals: chain, ribbon, and tube silicates

“…Hawthorne et al (2019) dealt with the large number of silicate minerals by dividing them into four categories and considering them separately according to the dimensional polymerisation of their tetrahedra: (1) cluster silicates that do not show infinite polymerisation; (2) chain, ribbon and tube silicates that are infinitely polymerised in one dimension; (3) sheet silicates that are infinitely polymerised in two dimensions; and (4) framework silicates that are infinitely polymerised in three dimensions. Hawthorne (2015a) and Hawthorne et al (2019) introduced the first comprehensive structure hierarchy for sheet-silicate minerals. Hawthorne (2015a) represented sheet structures as n-connected plane nets (2 < n ≤ 4), and showed that combining such nets with topological building operations allows one to generate sheet-silicate structures.…”

“…Hawthorne (2015a) and Hawthorne et al (2019) introduced the first comprehensive structure hierarchy for sheet-silicate minerals. Hawthorne (2015a) represented sheet structures as n-connected plane nets (2 < n ≤ 4), and showed that combining such nets with topological building operations allows one to generate sheet-silicate structures. Hawthorne (2015a) also developed formula-and structuregenerating functions to show how the chemical composition and structure of sheet silicates can be algebraically generated from such plane nets and associated building operations.…”

A structure hierarchy for silicate minerals: sheet silicates

“…Single-layer sheets: 3-and 4-connected nets with inserted 2-and 3-connected vertices Many single-sheet silicate minerals contain 4-connected tetrahedra in addition to tetrahedra of lower connectivity. Hawthorne (2015a) showed how suitable planar nets may be derived for these minerals by insertion of lower-connectivity vertices into planar 3-and 4-connected nets. It is not straightforward to insert vertices into a 3-connected net to produce a net of 3-and 4-connected vertices that forms a suitable basis for a sheet of tetrahedra, although it is feasible.…”

“…Wells, 1962, 1977; Smith, 1977, 1978, 1988; Hawthorne and Smith, 1986 a , b , 1988; Krivovichev, 2008, 2009). Hawthorne (2015 a ) described how nets may be used to theoretically derive possible atomic arrangements of the silicate components of minerals. With regard to the present work on sheet-silicate minerals, the salient issues are dealt with in the following sections.…”

“…Here, we examine the structure hierarchy of sheet-silicate minerals. Hawthorne (2015 a ) discussed the structures of sheet silicates in terms of n -connected plane nets (2 < n ≤ 4), showed how such nets can be combined with various oikodoméic operations (topological building operations) to generate sheet-silicate ( sensu lato ) structures, and went on to develop formula-generating and structure-generating functions for such nets and their associated oikodoméic operations. Here, we examine observed sheet-silicate structures, see how their chemical compositions and structures may be generated from n -connected plane nets and associated building operations, and arrange them into a hierarchy based on increasing degree of connectivity of their silicate structural-unit.…”