A visual-probabilistic framework for analyzing nanoparticle size distribution characteristics in polymer nanocomposites containing two reinforcements

The dispersion quality of rigid nano-inclusions significantly impacts polymer nanocomposite properties, necessitating a rigorous quantitative post-characterization of particle size distribution (PSD). This study combines probabilistic analysis with deep learning-based particle recognition for precise PSD measurement and dispersion characterization. A ternary nanocomposite of polyoxymethylene (POM), carbon black (CB), and nanoprecipitated calcium carbonate (NCC) was analyzed. Z-score filtering removed outliers, enabling accurate PSD histograms. Goodness-of-fit tests—Chi-Square (CS), Kolmogorov-Smirnov (KS), and Anderson-Darling (AD)—identified the Gamma distribution as the optimal PSD model. Gamma probability distribution analysis revealed that 3 wt% NCC reduced skewness and kurtosis by 18.23 % and 33.13 %, respectively, compared to POM/CB, thereby indicating improved PSD uniformity. Proportional size distribution (PRSD) analysis compared percentiles $P_{10,A}$ and $P_{90,A}$ with a benchmark, alongside undersized ($P_{\mathrm{UPP}}$) and oversized ($P_{\mathrm{OPP}}$) particle proportions. Introducing 1.5 wt% NCC reduced $P_{10,A}$ and $P_{90,A}$ by 34.21 % and 60.17 %, increased $P_{\mathrm{UPP}}$ by 42.17 %, and decreased $P_{\mathrm{OPP}}$ by 90.54 %, indicating finer, more consistent PSD.