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An algorithm is described and tested that carries out a non negative matrix factorization (NMF) ignoring any stretching of the signal along the axis of the independent variable. This extended NMF model is called StretchedNMF. Variability in a set of signals due to this stretching is then ignored in the decomposition. This can be used, for example, to study sets of powder diffraction data collected at different temperatures where the materials are undergoing thermal expansion. It gives a more meaningful decomposition in this case where the component signals resemble signals from chemical components in the sample. The StretchedNMF model introduces a new variable, the stretching factor, to describe any expansion of the signal. To solve StretchedNMF, we discretize it and employ Block Coordinate Descent framework algorithms. The initial experimental results indicate that StretchedNMF model outperforms the conventional NMF for sets of data with such an expansion. A further enhancement to StretchedNMF for the case of powder diffraction data from crystalline materials called Sparse-StretchedNMF, which makes use of the sparsity of the powder diffraction signals, allows correct extractions even for very small stretches where StretchedNMF struggles. As well as demonstrating the model performance on simulated PXRD patterns and atomic pair distribution functions (PDFs), it also proved successful when applied to real data taken from an in situ chemical reaction experiment.

Synthesis is a major challenge in the discovery of new inorganic materials. Currently, there is limited theoretical guidance for identifying optimal solid-state synthesis procedures. We introduce two selectivity metrics, primary and secondary competition, to assess the favorability of target/impurity phase formation in solid-state reactions. We used these metrics to analyze 3,520 solid-state reactions in the literature, ranking existing approaches to popular target materials. Additionally, we implemented these metrics in a data-driven synthesis planning workflow and demonstrated its application in the synthesis of barium titanate (BaTiO$_3$). Using an 18-element chemical reaction network with first-principles thermodynamic data from the Materials Project, we identified 82,985 possible BaTiO$_3$ synthesis reactions and selected nine for experimental testing. Characterization of reaction pathways via synchrotron powder X-ray diffraction reveals that our selectivity metrics correlate with observed target/impurity formation. We discovered two efficient reactions using unconventional precursors (BaS/BaCl$_2$ and Na$_2$TiO$_3$) that produce BaTiO$_3$ faster and with fewer impurities than conventional methods, highlighting the importance of considering complex chemistries with additional elements during precursor selection. Our framework provides a foundation for predictive inorganic synthesis, facilitating the optimization of existing recipes and the discovery of new materials, including those not easily attainable with conventional precursors.