1SUPPLEMENTARY INFORMATIONEvidence of Multi-step Nucleation Leading to Various Crystallization Pathways from an Fe-O-Al MeltG. C. Wang1,2,3*, Q. Wang1,2*, S. L. Li1,2, X. G. Ai1,2 and C. G. Fan31Key Laboratory of Chemical Metallurgy Engineering, Liaoning Province, University of Science and Technology Liaoning, Anshan, Liaoning, 114051, China2School of Materials and Metallurgy, University of Science and Technology Liaoning, Anshan, 114051, China3Jiangxi University of Science and Technology, Ganzhou, 341000, China*Corresponding author: Tel.: +86-412-5929573Email address: wang_guocheng163.com; wangqi8822sina.com. Simulation Methods to Calculate Thermodynamic Properties Using Density Functional TheoryAll the simulations for calculating thermodynamic properties in the present work are performed using density functional theory (DFT) with the molecular orbital theory computational program Dmol3, a module of the commercial software Materials Studio (MS) 6.0. The initial structures are established using the Visualizer module of MS 6.0. The geometry optimization is performed using the BFGS (the abbreviation corresponds to the first letters of the names of the following researchers: Broyden, Fletcher, Goldfarb, and Shanno) method41 based on a quasi-Newton algorithm. The hybrid density functional BLYP (the abbreviation corresponds to the first letters of the names of the following researchers: Becke, Lee, Yang and Parr) method42, 43 using the generalized gradient approximation (GGA) is used as the exchange- correlation potential function. The thermodynamic properties of various structures are calculated using the atomic harmonic vibrational frequency. The vibrational frequencies are computed by diagonalizing the mass-weighted second-derivative matrix, F42, which is given by(S1) jijiijqqE mmF21where qi and qj represent two Cartesian coordinates of atoms i and j, respectively, and mi and mj are the 2masses of the atoms. The square roots of the eigenvalues of Fij are the vibrational frequencies. The second derivatives are computed as the finite differences of the first derivatives. The gradients are computed at displaced geometries, and the second derivatives are computed numerically using two-point (or central) differencing. Displacements are taken to obtain the second derivatives(S2) 22i ii ijiqqqEqqqEwhere the two terms represent the analytic derivatives at the equilibrium geometry and at a geometry with coordinate qj displaced by distance . Therefore, the heat capacity (CP) is computed using the vibrational frequencies43:(S3) RhhCCCrotp4/kTexp1/kTexp/kThRCi2 ii2 i vibtranswhere , and are the heat capacities of translation, rotation and vibration, respectively; R is transCrotCvibCthe ideal gas constant (8.314 Jmol-1K-1); k is the Boltzmann constant; h is Plancks constant; T is the absolute temperature; and i is the vibrational frequency. The enthalpy (H) is given by45(S4) RTkThkThhkRhkRRTTHTHTHTHiiii i ivib4/exp1/exp2transrotwhere , and are the enthalpies of translation, rotation and vibration, respectively.vibHrotHtransHThe entropy (S)45 is given byvibrottransSSSSRhckThcIhcIhcIRpRwRTRCBA 23888ln23482. 2lnln23ln253222 (S5)ii iiiikThRkThkThkThR)/exp(1ln)/exp(1)/exp(/where , and are the entropies of translation, rotation and vibration, respectively; w is the transSrotSvibSmolecular mass; p is the pressure; is the symmetry number; c is the molar concentration of the molecules; and IA(B,C) is the moment of inertia. The vibrational free energy (GV) is calculated using the expression45 GV = H TS (S6). Details of the Dmol3 Simulation of the Alumina ClustersThe initial structures of the alumina clusters, (Al2O3)n with n = 1 - 10, 15 and 30, are established using Visualizer. The precision of the BFGS algorithm is set as follows: energy 2.010-5 Ha, tension 0.004 3Ha/, and shift 0.005 . The self-consistent field (SCF) method is used with the precision of the total energy and charge density set at 110-5 Ha, and the thermal smearing effect is used at a precision of 0.005 Ha. The cut-off radius of the DNP basis set of the d orbital is 3.5 . Electrons outside the atomic nucleus are handled using the effective core potentials (ECP) method46, 47. Details of the Dmol3 Simulation of the -Al2O3 CrystalThe initial structure of the -Al2O3 crystal is adopted from the MS 6.0 structural database. The precision of the BFGS algorithm is set as follows: energy 2.010-5 Ha, tension 0.004 Ha/, and shift 0.005 . The Brillouin zone integral is calculated using the Monkhorst-Pack48, 49 method with a grid size of 332 k- points. The precision of the total energy and charge density is set as 110-5 Ha, and the thermal smearing effect is used at a precision of 0.055 Ha. The cut-off radius of the DNP basis set of the d orbital is