A Topological Structure Model and a Nonlinear Formation Model of ZnO Tetrapods

Abstract

Objectives. Development of a generalised mathematical model of hierarchical crystalline nanoforms of zinc oxide tetrapods produced by vapor transport synthesis under the conditions of chaotic particle dynamics. Methods and methodology. Topological analysis of nanoscale polymorphic transformations, nonlinear dynamics of self-assembled structures formation under the conditions of dynamic chaos. An algorithm to construct an attractor of an affi ne system of iterated functions with parameters determined by experimental diagnostic methods: electron microscopy, diffraction studies, and numerical methods of quantum-chemical calculations for core of hierarchical structure. Results. The classifi cation of tetrapods as a hierarchical structures was established; nanotechnological cycle of self-organization of zinc oxide tetrapods chaos - core - percolation system - crystal was determined; the tetrapod structure was mathematically identifi ed as four linked topological spaces of wurtzite crystal growth; the tetrapod formation modelling method was developed by constructing an attractor of iterated function systems; the rule of the relation linking a dot in the nonlinear stochastic dynamic system model attractor with the topological crystallisation space in the formation of the self-assembled hierarchical crystal structure was established. A theoretical approach was developed that allows creating and modifying models of three dimensional space-distributed hierarchical crystal structures characterised by novel physical and topological properties, as compared to a monocrystal, as well as controlling their form and classifying the structures by topological and symmetrical properties. Conclusion. The formation of zinc oxide tetrapods occurs in four linked topological spaces, modelled as continuous maps of a B4 crystal growth space. The transition from the sphalerite core growth to the growth of four wurtzite crystals that constitute the tetrapod hierarchical structure is considered as topological space growth disruption during a polymorphic transformation of nanoscale. The topological approach involving the affi ne iterated function systems may be applied to modelling precrystallisation stages of multipods, as well as dendritic crystal structures and fractal nanocomposites, and has the capacity to predict their morphology, geometrical and structural properties.

Keywords

• topological space
• crystal polymorphism
• hierarchy
• dynamic chaos
• selfassembly
• ifs
• fractal