Topology and geometry of chain, ribbon and tube silicates: generation and analysis of infinite one-dimensional arrangements of (TO4)n- tetrahedra

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dc.contributor.authorDay, Maxwell C.
dc.contributor.examiningcommitteeFayek, Mostafa (Earth Sciences)en_US
dc.contributor.examiningcommitteeHerbert, David (Chemistry)en_US
dc.contributor.examiningcommitteeKrivovichev, Sergey V. (Crystallography, St. Petersburg State University)en_US
dc.contributor.supervisorHawthorne, Frank C.
dc.contributor.supervisorSokolova, Elena
dc.date.accessioned2022-10-04T18:32:42Z
dc.date.available2022-10-04T18:32:42Z
dc.date.copyright2022-10-04
dc.date.issued2022-10-04
dc.date.submitted2022-10-04T15:57:59Zen_US
dc.degree.disciplineEarth Sciencesen_US
dc.degree.levelDoctor of Philosophy (Ph.D.)en_US
dc.description.abstractA structure hierarchy for chain, ribbon and tube silicate minerals is proposed. Chains of (SiO4)4- tetrahedra are described topologically as chain graphs, where tetrahedra are represented as vertices and the linkages between them as edges. Chain arrangements are organized into classes and sub-classes based the number and connectivity of tetrahedra or vertices in the repeat unit (unit cell) of each chain arrangement. Chain graphs that correspond to the most abundant minerals have O:T = 3:1 - 2.75:1 and chains with O:T < 2.5:1 – 2:1 are not observed in minerals despite being topologically possible. A graph-theoretical method is proposed for converting each infinite chain graph to a finite wrapped graph such that it can be described using an adjacency matrix. A method (MatLab script) for generating all possible non-isomorphic chain graphs (up to a boundary condition, ∑r = 8) is proposed. Approximately 1500 non-isomorphic chain graphs are generated, ~50 of which are observed in chain silicate minerals. A software program (GraphT-T) is given for embedding chain graphs in Euclidean space to gauge their geometrical compatibility with the observed metrics of crystal structures. The average distance between linked T-cations (T-T distances) and the minimum distance between unlinked T-cations (T…T separations) were determined for all chain-silicate minerals to be 3.06 ± 0.15 Å and 3.71 Å, respectively. These values are used to constrain the geometry of chain graphs once embedded. If the resultant chain graphs have geometries that satisfy the T-T and T…T constraints, they are compatible with the metrics of crystal structures and may occur; if they do not, they are incompatible and are unlikely to occur. As the average vertex connectivity (1-4) increases, the e/n (edge/vertex) ratio increases and all chains with e/n < 1.5 are compatible and many chains with e/n > 1.5 are incompatible. This compatibility change at e/n = 1.5 coincides with decrease in chain flexibility and many chains with e/n ≥ 1.5 require distortion to satisfy the T-T and T…T constraints. As e/n = 1.5 corresponds to an O:T = 2.5, the above observations explain why chains with O:T = 2.5 – 2.0 are not observed in minerals.en_US
dc.description.noteFebruary 2023en_US
dc.identifier.urihttp://hdl.handle.net/1993/36942
dc.language.isoengen_US
dc.rightsopen accessen_US
dc.subjectSilicate mineralsen_US
dc.subject(SiO4)4- tetrahedraen_US
dc.subjectChains of tetrahedraen_US
dc.subjectStructure hierarchyen_US
dc.subjectInfinite graphsen_US
dc.subjectTopologyen_US
dc.subjectGraph theoryen_US
dc.subjectCrystal structuresen_US
dc.subjectMineral stabilityen_US
dc.subjectGraph embeddingsen_US
dc.titleTopology and geometry of chain, ribbon and tube silicates: generation and analysis of infinite one-dimensional arrangements of (TO4)n- tetrahedraen_US
dc.typedoctoral thesisen_US
local.subject.manitobanoen_US
oaire.awardTitleUniversity of Manitoba Graduate Fellowshipen_US
oaire.awardURIhttps://umanitoba.ca/graduate-studies/funding-awards-and-financial-aid/university-manitoba-graduate-fellowship-umgfen_US
project.funder.identifierhttps://doi.org/10.13039/100010318en_US
project.funder.nameUniversity of Manitobaen_US
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