Analytic model for the development of bamboo microstructures in thin film strips undergoing normal grain growth

Abstract

The kinetics of the transformation of polygranular thin film strips to bamboo structures through two-dimensional (2D) normal grain growth are studied. A differential model for the evolution of the polygranular cluster length distribution is developed. It is observed, and demonstrated using a dimensional analysis, that the rate of bamboo-segment nucleation per unit time and unit of untransformed length is proportional to $\ensuremath{\mu}{/w}^{3},$ and is negligible in the growth-dominated steady state. It is also demonstrated that the cluster shrinkage velocity reaches a constant steady-state value proportional to $\ensuremath{\mu}/w$ (assuming constant and uniform \ensuremath{\mu}). This is shown to lead to a time-invariant, steady-state exponential cluster length distribution with an average cluster length proportional to the strip width, and a cluster length fraction decaying exponentially with $\ensuremath{\tau}=\ensuremath{\mu}{t/w}^{2}.$ The analytic model is validated through comparison with data generated using a 2D computer simulation of grain growth. The distribution of grain lengths in the resulting final bamboo grain structure is well fit by a log-normal distribution, with a median grain length scaling with the linewidth, and a linewidth-independent normalized deviation in the grain length.