Length Bias in the Measurements of Carbon Nanotubes

To measure carbon nanotube lengths, atomic force microscopy and special software are used to identify and measure nanotubes on a square grid. Current practice does not include nanotubes that cross the grid, and, as a result, the sample is length-biased. The selection bias model can be demonstrated through Buffon’s needle problem, extended to general curves that more realistically represent the shape of nanotubes observed on a grid. In this article, the nonparametric maximum likelihood estimator is constructed for the length distribution of the nanotubes, and the consequences of the length bias are examined. Probability plots reveal that the corrected length distribution estimate provides a better fit to the Weibull distribution than the original selection-biased observations, thus reinforcing a previous claim about the underlying distribution of synthesized nanotube lengths.