2-Point correlation function of nanostructured materials via the grey-tone correlation function of electron tomograms: A three-dimensional structural analysis of ordered mesoporous silica

Electron tomography is a unique technique for imaging the microstructure of materials with a nanometer resolution. The signal-to-noise ratio of electron tomograms is, however, often too low for a reliable segmentation-based image analysis. We derive a general relation between the grey-tone correlation function of the tomograms and the 2-point correlation function of the morphology, which enables us to analyse quantitatively the grey-tone correlation function with a morphological model of the material. The methodology is applied to SBA-15 ordered mesoporous silica. The three-dimensional grey-tone correlation function obtained from electron tomography is analysed in terms of a hexagonal array of Gaussian independent pores. The model enables us to relate the morphology obtained from the 2-point correlation function to macroscopic characterization data of the material, notably small-angle X-ray scattering and nitrogen adsorption.