Deciphering the hierarchical structure of phosphate glasses using persistent homology with optimized input radii

The first sharp diffraction peak (FSDP) in the reciprocal-space structure factor S(Q) of glasses has been associated with their medium-range order (MRO) structure, but the real-space origin remains debated. While some progress has been made in the case of silicate and borate glasses, the MRO structure of phosphate glasses has not been studied in detail. Here, we apply persistent homology (PH), a topological data analysis method, to extract the MRO features and deconvolute the FSDP of zinc phosphate glasses. To this end, the oxygen, phosphorus, and zinc atoms in the atomic configurations of the glasses are regarded as vertices weighted by initial atom radii for PH computation before decomposing their contributions to the FSDP. To determine the vertex weights, we vary the oxygen (O) radius systematically and set the radii of zinc (Zn) and phosphorus (P) atoms based on the positions of the first peak in the O-Zn and O-P partial radial distribution functions. These configurations with varying atom radii are used as inputs for PH computation, allowing us to assess the contributions of the different ring structures to the MRO. In turn, this comparison between the computed and measured S(Q) gives rise to an optimized oxygen radius for the best agreement. The optimized vertex weight (oxygen radius) is found to have a physical meaning based on the covalent and ionic bonding characters. Finally, using the optimized atom radii, we are able to decompose the hierarchical structural contributions to the FSDP.