## 2 Experimental Section  
A simulated brine solution was prepared with a magnesium-lithium ratio of approximately 40:1, based on the composition of brine samples obtained after potassium extraction from the Qinghai salt lake, China. This solution was then subjected to room-temperature evaporation, after the addition of Na\({}_{2}\)SO\({}_{4}\cdot\)10H\({}_{2}\)O, with a composition of Mg\({}^{2+}\) 48 g/L, Na\({}^{+}\) 50.23 g/L, Cl\({}^{-}\) 138.2 g/L, Li\({}^{+}\) 1.236 g/L, K\({}^{+}\) 13.86 g/L, and SO\({}_{4}\)\({}^{2-}\) 77.25 g/L.  
Single-salt solutions of MgSO\({}_{4}\), Li\({}_{2}\)SO\({}_{4}\), LiCl, and MgCl\({}_{2}\), with a concentration of 2.5 mol/L, as well as simulated brine solutions with magnesium-lithium ratios of 160:1,80:1, 40:1, 20:1, 10:1, and 5:1, were prepared and heated under reflux at 80 \({}^{\circ}\)C. The resulting MgSO\({}_{4}\)-LiCl saturated solution was then slowly cooled to around 60 \({}^{\circ}\)C, at which point the solution became supersaturated (i.e., the concentration of MgSO\({}_{4}\) is empirically approximately 1.3 times its standard solubility). The temperature was then further lowered to room temperature (around 20 \({}^{\circ}\)C). During this cooling stage, approximately one-third of the dissolved MgSO\({}_{4}\) crystallized and precipitated, significantly reducing the magnesium-lithium ratio of the brine. This is the most distinctive feature of this separation method.  
Furthermore, because the brine was cooled from a relatively high temperature (60 \({}^{\circ}\)C), the viscosity of the brine was low during the cooling process, which prevented a large amount of Li\({}^{+}\) from being trapped in MgSO\({}_{4}\) crystals and then avoided serious lithium process losses.  
The changes in the temperature and state of the brine solution were recorded. After the solid-liquid separation, the crystals were collected, and the magnesium-lithium separation was completed. FTIR spectroscopy was used to observe changes in the structure of the single-salt and brine solutions before and after cooling, over a range of wave numbers from 3800 cm\({}^{-1}\) to 650 cm\({}^{-1}\). The morphology and composition of the crystals were analyzed using SEM, EDS, and XRD, and the magnesium-lithium separation was confirmed via chemical analysis. The laboratory instruments used in this study included a Fourier transform infrared spectrometer (Thermo Fisher Scientific Nicolet iS50, Waltham, MA, USA), an X-ray diffractometer (Rigaku Corporation Smartlab, Tokyo, Japan), an inductively coupled plasma atomic emission spectrometer (Thermo Fisher Scientific ICAP6300, Waltham, MA, USA), a scanning electron microscope, and an energy dispersive X-ray spectroscopy system (JSM-IT500 HR, JEOL, Tokyo, Japan), among others.  
Different temperature conditions (20-60 \({}^{\circ}\)C) were established for the MgSO\({}_{4}\)-H\({}_{2}\)O, Li\({}_{2}\)SO\({}_{4}\)-H\({}_{2}\)O, LiCl-H\({}_{2}\)O, and MgCl\({}_{2}\)-H\({}_{2}\)O systems. The B3 LYP functional was utilized, and all simulated systems were maintained at a constant pressure and temperature using the Nose-Hoover barostat and Nose-Hoover thermostat, with parameter settings of 1 atm and 298 K, 313 K, and 333 K. The vibrational frequencies and reaction kinetics were calculated with a convergence criterion of 10\({}^{-6}\) eV. The following thresholds were used for geometry optimization: 1 \(\times\) 10\({}^{5}\) Hartree for maximum energy change, 2 \(\times\) 10\({}^{-3}\) Hartree/A for maximum force, and 5 \(\times\) 10\({}^{-3}\) A for maximum displacement. The radial distribution function of anions (SO\({}_{4}\)\({}^{2-}\), Cl\({}^{-}\)) in the solute was analyzed, and the distribution of water molecules around the ions before and after temperature changes was compared to provide a molecular-level explanation for the control of magnesium-lithium separation during cooling. Similar methods have been used in many inorganic salt solution systems [20; 21; 22].