Quaternionic Orientational Order and Defect-Driven Transitions Between Crystalline and Glassy Solids

Abstract This paper develops a unified phenomenological description of solidification by using a quaternionic orientational order parameter to represent local atomic rotations in undercooled liquids. The novelty of the work is that the same topological language is used to describe both crystallization and glass formation. In the crystalline pathway, quantized misorientation defects bind, order, and support long-range orientational and translational coherence. In the glassy pathway, those defects proliferate and remain frustrated, producing a rigid but non-periodic solid. The competition is organized by a non-thermal tuning parameter g = U/J that balances orientational stiffness J against localization U , in close analogy with duality ideas known from Josephson-junction arrays. Within this framework, geometrical frustration explains persistent defect skeletons in topologically close-packed phases and the opposite temperature dependences of thermal conductivity in crystals and glasses are linked to the presence or loss of coherent heat-carrying vibrational modes. The paper is intended as a materials-oriented topological phenomenology that complements, rather than replaces, atomistic and first-principles approaches.