Computational Simulations Using Time-Dependent Ginzburg–Landau Theory for Nb–Ti-Like Microstructures

Simulations based on time-dependent Ginzburg–Landau theory are employed to determine the critical current for a model system which represents a Nb–Ti-like pinning landscape at low drawing strain. The system consists of ellipsoids of normal metal, with dimensions $60\xi \times 3\xi \times 3\xi$ , randomly distributed throughout the superconducting bulk with their long axes parallel to the applied current and perpendicular to the field. These preciptates represent the $\boldsymbol {\alpha }$ -Ti elongated precipitates which act as strong pinning centres in Nb–Ti alloys. We present the volume pinning force density as a function of field across the entire range of precipitate volume fractions and find that optimised material in our model system occurs at 32 vol.% ppt., whereas in real materials the optimum occurs at 25 vol.% ppt. The maximum pinning force density in our simulations is slightly higher ( $5.4\times 10^{-3}J_\mathrm{D}B_\mathrm{c2}$ vs. $ 17\,{\rm GN\!\cdot \!m^{-3}} = 4.5\times 10^{-3}J_\mathrm{D}B_\mathrm{c2}$ ) and occurs at a lower reduced field ( $0.2B_\mathrm{c2}$ vs. $ 0.5B_\mathrm{c2}$ ) than in real materials. We conclude that the broad features of Nb–Ti-like systems are captured in our model, but that the details of the precipitate pinning mechanism are not yet included properly.