Fractal-based analysis of the void microstructure of Bi₂Sr₂CaCu₂O_x superconducting filaments and the caused anomalous thermal diffusion

High-T_c multifilamentary composites Bi₂Sr₂CaCu₂O_x/Ag/Ag-alloy (Bi2212) are the only round wires (RWs) to be potential candidates for fabricating high field magnets. However, the complicated microstructure of Bi2212 RWs, and especially the Bi2212 filaments, makes deep understanding of the correlation between macroscopic electrothermal behaviors and the microstructure a challenge. Building on our previous work of fractal-based analysis of the Bi2212/Ag interface, in this paper we focus on the void (gas bubble or pore) structure of individual Bi2212 filaments, taking a fractal-based analysis of the void structure and the resulting anomalous thermal diffusion. For fractal characterization of the void structure of Bi2212 filaments, we developed a fractal-based reconstruction model, and further derived a one-dimensional fractional order differential equation with the exponent α (0 < α ≤ 1), which is related to the porosity (0 ≤ ϕ ≤ 1) and fractal dimension (2 < D < 3) of the void structure of Bi2212 filaments. The calculated results show a local anomalous thermal diffusion (quite different from the diffusion in Euclidean homogeneous space corresponding to the materials without any pores), which depends on the porosity and fractal dimension of porous Bi2212 filaments. The dependence of thermal diffusion on the porosity is stronger than on the fractal dimension of the porous structure. This indicates that reducing voids/bubbles in Bi2212 filaments during the heat treatment process is significant for refining thermal diffusion.