Fig. 4: Sequential treatments in stages and passes in DME-FC.

a Scheme depicting stages and passes in DME-FC. DME-FC stages involve treatment of the initial leachate (\({{{{{{\mathscr{l}}}}}}}_{0}\)), dissolution of the the solid product (e.g., \({{{{{{\rm{s}}}}}}}_{1}\)) in water, followed by treatment of the resulting solution to yield a higher-purity solid (e.g., \({{{{{{\rm{s}}}}}}}_{11}\)). DME-FC passes are successive treatments of the liquid stream to recover chemically distinct solid products (\({{{{{{\rm{s}}}}}}}_{1}\), \({{{{{{\rm{s}}}}}}}_{2}\)) from the same aqueous solution. Temperature control can enhance separations in passes (e.g., initial solid fraction \({{{{{{\rm{s}}}}}}}_{1}\) recovered at temperature \({{{{{{\rm{T}}}}}}}_{1}\), followed by treatment at a different temperature (\({{{{{{\rm{T}}}}}}}_{2}\)) to recover a separate solid fraction (\({{{{{{\rm{s}}}}}}}_{2}\))), and can also be used to enhance purification via DME-FC stages. As the chemical character of the solution remains unchanged, albeit with reduced salt concentration, liquid products are suitable for further hydrometallurgical processes upstream or downstream of the separation. ICP-OES metal compositions depicting the purification from original solution (\({{{{{{\mathscr{l}}}}}}}_{0}\)), first solid product (\({{{{{{\rm{s}}}}}}}_{1}\)), and solid product produced through staging (\({{{{{{\rm{s}}}}}}}_{11}\)) with treatments at 25 °C for b Sm-Co magnet leachate and c Nd-Fe-B mixed magnet leachate. Sum of the lanthanide elements (Nd, Pr, Sm, and Dy) is given as Ln. Separation factors for both b and c are listed in Table 2. Source data are provided as a Source Data file.