Mass fractal dimension from 2D microscopy images via an aggregation model with variable compactness

Microscopy-image analysis provides precious information on size and structure of colloidal aggregates and agglomerates. The structure of colloids is often characterized using the mass fractal dimension df$d_f$ , which is different from the two-dimensional (2D) fractal dimension dp$d_p$ that can be computed from microscopy images. In this work, we propose to use a recent morphological aggregation model to find a relationship between 2D image fractal dimension and 3D mass fractal dimension of aggregates and agglomerates. Our case study is represented by scanning transmission electron microscopy images of boehmite colloidal suspensions. The behaviour of the computed df$d_f$ at different acid and base concentration shows a fair agreement with the results of small angle x-ray scattering and with the literature, enabling to use the df$d_f$ versus dp$d_p$ relationship to study the impact of the composition of the colloidal suspension on the density of colloidal aggregates and agglomerates.Boehmite powder is often employed for the manufacturing of γ$\gamma$ -alumina catalyst carriers and absorbents. This process involves boehmite colloidal particles coagulation, the size and shape of the particles affect the porosity of the final solid. As there is a high interest in tuning the porosity via the synthesis parameters, this work is aimed at the use of microscopy analysis to evaluate the mass fractal dimension of boehmite aggregates and agglomerates formed under controlled physical chemical conditions (acid and alkaline content and Brownian motion). Since the mass fractal dimension cannot be directly computed on binary two-dimensional images, a recently developed morphological model has been used to generate three-dimensional clusters characterized by a certain mass fractal dimension df$d_f$ , the projection of the clusters leads to the estimation on the projected fractal dimension dp$d_p$ , being related to the slope on a bi-logarithmic plot perimeter versus area. According to the morphological model, the higher the df$d_f$ , the lower the dp$d_p$ . The relationship between df$d_f$ and dp$d_p$ has been used to estimate the mass fractal dimension from STEM images of boehmite suspensions. The trend of df$d_f$ with respect to the acid and alkaline content in the suspensions agrees with small angle X-ray scattering measurement and with the literature.