Quantifying Dispersion of Nanoparticles in Polymer Nanocomposites Through Transmission Electron Microscopy Micrographs

Abstract

The property of nanocomposites is crucially affected by nanoparticle dispersion. Transmission electron microscopy (TEM) is the “golden standard” in nanoparticle dispersion characterization. A TEM Micrograph is a two-dimensional (2D) projection of a three-dimensional (3D) ultra-thin specimen (50–100 nm thick) along the optic axis. Existing dispersion quantification methods assume complete spatial randomness (CSR) or equivalently the homogeneous Poisson process as the distribution of the centroids of nanoparticles under which nanoparticles are randomly distributed. Under the CSR assumption, absolute magnitudes of dispersion quantification metrics are used to compare the dispersion quality across samples. However, as hard nanoparticles do not overlap in 3D, centroids of nanoparticles cannot be completely randomly distributed. In this paper, we propose to use the projection of the exact 3D hardcore process, instead of assuming CSR in 2D, to firstly account for the projection effect of a hardcore process in TEM micrographs. By employing the exact 3D hardcore process, the thickness of the ultra-thin specimen, overlooked in previous research, is identified as an important factor that quantifies how far the assumption of Poisson process in 2D deviates from the projection of a hardcore process. The paper shows that the Poisson process can only be seen as the limit of the hardcore process as the specimen thickness tends to infinity. As a result, blindly using the Poisson process with limited specimen thickness may generate misleading results. Moreover, because the specimen thickness is difficult to be accurately measured, the paper also provides robust analysis of various dispersion metrics to the error of the claimed specimen thickness. It is found that the quadrat skewness and the K-function are relatively more robust to the misspecification of the specimen thickness than other metrics. Furthermore, analysis of detection power against various clustering degrees is also conducted for these two selected robust dispersion metrics. We find that dispersion metrics based on the K-function is relatively more powerful than the quadrat skewness. Finally, an application to real TEM micrographs is used to illustrate the implementation procedures and the effectiveness of the method.