Hierarchical models for the spatial–temporal carbon nanotube height variations

Carbon nanotubes (CNTs) are allotropes of carbon with a cylindrical nanostructure. Due to their low production cost and potentially high demand, the large-scale production of CNTs is urgently needed and will be highly profitable. However, quality control will be a great challenge to a large-scale production due to the delicate nature of the production process. Among the problems involved in the quality control of CNT array production, height variation is one of the primary concerns. The objective of this study is to model the height along both the spatial and temporal dimensions, so that the height variations can be controlled during the production process, thus improving the quality and stability of CNT arrays. Specifically, the height variation of the CNT arrays is decomposed into macro-scale and micro-scale variations. The macro-scale variation is modelled by state-space and regression models, and the micro-scale variation is modelled as a spatial process. The models successfully capture both the macro-trends and the micro-patterns. A practical case study shows the effectiveness of the proposed models in terms of goodness of fit and prediction accuracy, and the distinction of the models is summarised to aid in choosing a model to apply to other spatial–temporal data modelling problems.