New Insights into Silicon Puri ﬁ cation by Alloying − Leaching Re ﬁ ning: A Comparative Study of Mg − Si, Ca − Si, and Ca − Mg − Si Systems

: In the present work, a comparative study of the silicon alloying − leaching puri ﬁ cation process was carried out on the recently developed Mg − Si system, the Ca − Si system, and the novel ternary Ca − Mg − Si system. Insights were provided into the integrated process from aspects of thermodynamic assessment, microstructural analysis, experimental observation, computational simulation, and analytical modeling. The main silicide precipitates of the three Mg − Si, Ca − Si, and Ca − Mg − Si alloys studied were determined as Mg 2 Si, CaSi 2 , and ternary Ca 7 Mg 7.5 ± δ Si 14 , respectively. Other metallic impurities were found to form complex silicides embedded inside the main precipitate, where P also segregated and precipitated according to the interaction with the alloying elements. All of the impurities were further carried away with the removal of the main precipitates through the subsequent leaching process. It was found that the ternary Ca − Mg − Si alloy exhibits a cleaner leaching process due to the unique crystal structure of Ca 7 Mg 7.5 ± δ Si 14 . A novel cracking − shrinking principle-based kinetics model was developed to further describe the impurity removal process. The segregation behavior of P was also modeled through a thermodynamic approach and Ca was found to have stronger P a ﬃ nity compared to Mg. It was ﬁ nally concluded that the novel Ca − Mg − Si ternary alloy system exhibited better performance overall as compared to the other two binary alloys.


■ INTRODUCTION
The contribution of solar energy to the world's total energy supply has grown significantly in the past two decades. With the rapid photovoltaic market growth and installation of Si-based solar modules worldwide, it has become increasingly critical to develop an environmentally friendly approach for solar-grade silicon (SoG-Si) production. 1 Owing to the low energy consumption and low carbon footprint features, the emerging metallurgical route of SoG-Si production has received increasing attention. 1−3 As one of the recent successfully commercialized processes, the Elkem Solar process, which is operated by REC Solar Norway, is known to purify crude metallurgical-grade Si (MG-Si) through the combination of a series of metallurgical methods of slag refining, acid leaching, and direction solidification. The goal of acid leaching among these methods is to separate nonmetallic impurity P, which is one of the essential impurities in MG-Si needed to meet the strict SoG-Si impurity requirement, and also to remove the majority of metallic impurities at the same time.
The principle of acid leaching is mainly based on the digestion of segregated impurities from silicon grains during Si solidification. Impurities with lower segregation coefficients will precipitate more in the last formed solid phases (precipitates) in solidification due to their strong tendency to stay in the liquid phase. By exposing the solidified silicon to the acid solution, impurities located at grain boundaries of precipitates between the primary silicon grains can be carried away, and pure Si can be obtained. Considerable research has been performed to study the impurity extraction from MG-Si by acid leaching. 4−12 It has been found that most of the metallic impurities can be effectively extracted by adjusting a variety of processing parameters like leaching temperature, time, acid combination, and particle size, but the effective P extraction by direct leaching of MG-Si still remains a challenge. Even though it has been theoretically and experimentally confirmed that P tends to segregate along Si grain boundaries, 13 its segregation coefficient (k p = 0.35) is still relatively high compared to metallic impurities, which is usually less than 10 −3 . To further improve the extraction efficiency of P and other impurities, considerable attention has been paid to redistribute impurities by adding alloying elements as impurity getters such as calcium (Ca), 14−18 magnesium (Mg), 19−23 titanium (Ti), 17 aluminum (Al), 24−29 copper (Cu), 30−33 tin (Sn), 34,35 iron (Fe), 36−38 and nickel (Ni). 39 The widely known solvent refining usually involves alloying by Al, Sn, Cu, and Fe with a large amount of metal addition (around or more than 50 wt %). The low-temperature operation and high impurity removal degree features make the solvent-refining process promising. However, the industrialization process is hampered by the concerns of the potentially high production cost and low Si productivity caused by the need for large metal addition. From this perspective, small alloying metal addition has a better costeffective feature, but this requires a distinctly strong impuritygathering ability of the alloying elements.
Alkaline-earth metals are known for their strong affinity to P. Thus, metals in this group are of increasing interest for P removal from MG-Si. Anders 14 doped Ca into MG-Si and found that pure Si can be obtained after acid leaching. Shimpo et al. 15 measured the interaction coefficient of Ca and P in the Ca−Si−P system and observed the formation of Ca 3 P 2 along the Si grain boundary. It was also reported that efficient P removal could be achieved by altering the Ca alloying concentration, cooling rate, and leaching conditions. 16−18 Mg is a relatively new alloying impurity getter proposed by the NTNU team in recent years, 19−23 and more knowledge about Mg is needed, in particular for comparison with the Ca−Si alloying−leaching system. In addition, according to the Zintl−Klemm concept, 40 the mixing of Ca and Mg in Si may show a pronounced variation of the electronic structure compared to the binary silicide as a result of unbalanced valence electron transfer. It is believed that the segregation behavior of the impurities and also the leaching process of the Ca−Mg−Si ternary alloying system may differ for both Ca−Si and Mg−Si binary alloying systems. Hence, it is essential to study Si purification by Ca and Mg alloying together.
In this work, a comparative study of Ca−Si, Mg−Si, and Ca− Mg−Si systems is carried out experimentally and theoretically to provide essential insights into MG-Si purification for the whole integrated process including aspects of thermodynamic assessment, alloy microstructure and impurity distribution, acid leaching experiments, kinetics modeling, and purification efficiency evaluation.
■ METHODOLOGY Experimental Procedure. The Si sources used in this work are commercial MG-Si (99.5% purity, P: 15 ppmw) and mixed with commercial impurity-free Si produced from the fluidized bed reactor process (FBR-Si) to minimize the influence of other minor impurities in the present study and attain a lower P impurity concentration at several ppmw levels. The MG-Si and FBR-Si mixing ratio was fixed as 40% MG-Si and 60% FBR-Si for all of the alloys. A high-purity Mg rod (99.8% purity, Alfa Aesar) and granular Ca (99% purity, Sigma-Aldrich) were employed as the alloying metal sources, and the initial charging composition of the alloys is 5.2 wt % Mg, 4.2 wt % Ca, and 2.2 wt % Ca−2.6 wt % Mg. Considering the higher volatility of Mg at high temperatures, it was doped into Si after melting, while Ca granulates were directly charged with Si before heating. Thus, about 400 g of Si and Ca granules were first put in a high-density isotropic graphite crucible (inner diameter: 70 mm, height: 150 mm). The crucible was further heated in an induction furnace under a high-purity Ar flow (99.9999% purity) with the chamber pressure maintained at around 1030−1050 mbar. After the samples were fully melted at around 1500°C, metallic Mg pieces were doped into the molten alloy to form the target Ca−Mg−Si and Mg−Si alloys. The molten alloys were then homogenized through induction stirring for 10 min and cooled down to room temperature. The cooling rate was kept at 20−30°C/min as faster cooling may suppress impurity segregation, and slower cooling may lead to higher evaporation loss of the volatile alkaline-earth alloying metals. The final composition of obtained bulk alloys is listed in Table 1.
The obtained bulk alloys were crushed by a ring mill (Retsch Vibratory Disc Mill RS200, tungsten carbide sets) into fine particles within the specific particle size range from 0.2 to 1 mm before leaching. Acid leaching was performed by mixing 20 g of particles with 120 mL of reagent-grade 10% HCl (VMR International) in a beaker on a magnetic stirring hot plate, the stirring speed was fixed at 200 rpm, and the leaching temperature was fixed at 60°C throughout the whole leaching period. Samples were taken at specific leaching times (5, 10, 20, 60, and 180 min). After 3 h leaching, samples were further cleaned by deionized water and ethanol.
The microstructure of the obtained alloys was studied by an electron probe micro-analyzer (EPMA, JXA-8500F) and the precipitate composition was measured by wavelength-dispersive spectroscopy (WDS). The impurity contents were measured by high-resolution inductively coupled plasma mass spectrometry (ICP-MS, Agilent 8800). The particle size distribution (PSD) was measured by a Horiba LA-960 instrument with the laser diffraction method. The refractive index value of pure Si is adopted for the analyzed Si-rich alloy samples as an approximation since the real value remains unknown but should reasonably be close to the value of pure Si. The particle size measurement range of the instrument is 10 nm to 5 mm. It is also worth noting that the particle size measured by laser diffraction is usually larger than the sieving particle size, as it is the diameter of a sphere with the particle's equivalent volume and the sieving particle size is the minimum diameter passing through the sieve aperture.
Computational Methods. The structural and electronic properties of the main precipitate compounds studied, Mg 2 Si, CaSi 2 , and Ca 7 Mg 7.5±δ Si 14 , are investigated based on density functional theory (DFT) calculations. The crystal structures were fully optimized with the DFT simulation employing the Vienna ab initio simulation package (VASP). 41,42 The projectoraugmented wave (PAW) method 43,44 for the pseudopotential, and generalized gradient approximation (GGA) with the Perdew−Burke−Ernzerhof (PBE) exchange−correlation functional are adopted. 45 The van der Waals (vdW)-corrected ■ RESULTS AND DISCUSSION Assessment of the Si−Ca−Mg−P System. Alkaline-earth metals are known for their strong potential to form stable phosphides as M 3 P 2 , where M denotes a metal. 49 Hints of the strong P affinity of Ca and Mg can also be obtained from the mixing enthalpy calculation via the Miedema model, 50 which  To further reassess the phase equilibria of the Ca−Mg−Si−P system, a quaternary phase diagram was constructed using the commercial software Factsage using a reassessed database, as shown in Figure 1b. In the Si-rich corner, P is predicted to exist in the form of Ca 3 P 2 and Mg 3 P 2 , mainly depending on the alloying concentration. In the Mg−Si alloy system, it is seen that only one intermetallic exists in the binary Mg−Si system, where Mg and Si form a eutectic Mg 2 Si phase at 57.31 atom % Si. The eutectic temperature is 930°C, and the Mg 2 Si phase melts at temperatures above 1085°C. The Ca−Si system becomes much more complicated when five Ca-based silicides exist, which also indicates the strong interaction between Ca and Si with each other in the melt and during solidification. In the Si-rich portion, CaSi 2 is the only available binary compound in the alloying− leaching approach that is known to form at 1023°C with a eutectic point at 69.4 atom % Si. Two ternary compounds (τ 1 and τ 2 ) exist in the Ca−Mg−Si system, according to the study by Grobner et al. 51 The exact stoichiometry remains controversial for the τ 2 phase as several stoichiometries (Ca 7 Mg 6 Si 14,

51,52
Ca 7 Mg 7.5±δ Si 14, 53 and Ca 7 Mg 7.25 Si 14 54 ) have been used. Nesper et al. 53 obtained the stoichiometry of Ca 7 Mg 7.6 Si 14 from singlecrystal diffraction measurements, but their simulation yields a composition of Ca 7 Mg 7.25 Si 14 . Thus, the correct stoichiometry is adopted as Ca 7 Mg 7.5±δ Si 14 , where δ is a measure of insecurity from the X-ray refinement. In our work, the WDS-EPMA measurement also suggests a stoichiometry of Ca 7 Mg 7.6 Si 14 , but both the compositions Ca 7 Mg 6 Si 14 and Ca 7 Mg 7.25 Si 14 lead to a stable crystal structure from DFT calculations. Thus, the stoichiometry of the τ 2 phase presented in our work is still recommended as Ca 7 Mg 7.5±δ Si 14 .
Microstructure and Impurity Redistribution. Microstructural analysis is crucial for understanding the impurity distribution during the solidification of a silicon alloy. EPMA elemental mapping was conducted for the microstructure of the studied Mg−Si, Ca−Si, and Ca−Mg−Si alloys; the results are presented in Figure 2.
The microstructure of the utilized raw MG-Si has been studied in the authors' previous work, 21−23 where only fine and dispersed Fe-bearing phases were observed in the silicon grains. However, in the alloyed MG-Si, the main precipitates were found to be strongly dependent on the alloying metals and with a thickness of around 100−200 μm after furnace cooling. The detailed compositions of the observed phases are given in Table  2. In the Ca−Si sample, the dominant precipitate phase is observed to be CaSi 2 , with a small amount of other diversely distributed phases, such as Al 6 CaFe 4 Si 8 and Fe 3 Si 7 (known as high-temperature FeSi 2 , also called FeSi 2.4 or α-leboite), 55 as well as surrounded by some minor quantity of CaAl 2 Si 2 . Ti impurity was found to only form the ternary TiFeSi 2 phase, and no binary TiSi 2 phase was observed due to the larger amount of Fe impurity. In the Mg−Si sample, Mg 2 Si is found as the main precipitate, and with a typical eutectic morphology, while other Fe-bearing phases like Al 6 CaFe 4 Si 8 , Fe 3 Si 7 , and TiFeSi 2 were found embedded inside.
In the Ca−Mg−Si sample, the Ca 7 Mg 7.5±δ Si 14 ternary phase was observed as the main precipitate, and a small amount of the Mg 2 Si phase was also found to coexist in the sample due to a slight Mg excess. It is also worth noting that the Al impurity found is more like to stay as a solid solution in the Ca 7 Mg 7.5±δ Si 14 phase (∼1 atom %) rather than forming specific precipitates like the CaAl 2 Si 2 and Al 6 CaFe 4 Si 8 phases in Ca−Si and Mg−Si alloys. It is also seen that Fe 3 Si 7 and FeTiSi 2 were found to be the common phases among the Mg−Si, Ca−Si, and Ca−Mg−Si alloys, which may imply the partial substitution effect of transition metals during the solidification of Si.
The evolution of impurity redistribution was further determined by equilibrium computation by Factsage using a reassessed database shown in Figure 3. Since the temperature effect caused by the recalescence of the small amount of precipitates can be ignored, the impurity solidification sequence can be subsequently examined as well. It is seen that the primary Si is always the first nucleate phase, followed by a series of eutectic reactions that take place in the temperature range between 800 and 1000°C, while the remaining liquid phase percentage is around 10%. The calculated main precipitates (CaSi 2 , Ca 7 Mg 7.5±δ Si 14 , Mg 2 Si) in each sample also show good consistency with the EPMA results, and P suggested as the form of Ca 3 P 2 and Mg 3 P 2 , which is dependent on the doping metal concentration. However, since their amounts are too little to detect, this cannot be confirmed by the EPMA results in this work. The Fe-bearing phases are calculated as the first Since no FeSi 2 phase is detected in our work, it may indicate that the decomposition kinetics is slow. Al-bearing phases are suggested to form at the end of solidification. The CaAl 2 Si 2 phase observed in the Ca−Si alloy sample is not presented in Figure 3a, which may be owing to the following reaction 56 The above reactions may also explain why the observed CaAl 2 Si 2 , Al 6 CaFe 4 Si 8 , and the Fe 3 Si 7 phases in the Ca−Si alloy sample are often located close to each other. Nevertheless, Al 6 CaFe 4 Si 8 precipitation was not found in the Ca−Mg−Si sample; instead, Al was found as a solid solution in the Ca 7 Mg 7.5±δ Si 14 phase. Thus, the calculated precipitation of the quaternary phase in Figure 3b is marked as a dashed line.
Leaching Phenomena and Mechanism. In the initial leaching stage of all of the studied alloys, heterogeneous exothermic reactions took place rapidly along with the formation of a large number of gas bubbles. Different leaching phenomena were observed in the studied alloys, as shown in Figure 4. In the leaching of the Ca−Si sample, a yellow-green color byproduct named siloxene was formed, which is foamy and floating over the surface. Consequently, foaming bubbles were observed on top of the solution as shown in Figure 4a. In the leaching of the Mg−Si sample, intensive sparks were observed above the solution surface. Obvious minor residue (fine solid particles) can be seen in Figure 4b. Unlike the leaching of Ca−Si and Mg−Si samples, neither viscous siloxane nor intensive sparks were observed in the leaching of the Ca−Mg−Si sample, shown in Figure 4c, which makes the leaching of the Ca−Mg−Si system a cleaner process.
Leaching Mechanism. To further understand the leaching mechanism, the crystal structure and electronic characteristics of the main precipitates were investigated. The perspective views on crystal structures, Brillouin zone shape, and the band structure of CaSi 2 , Ca 7 Mg 7.5±δ Si 14 , and Mg 2 Si along [001] are presented in Figure 5a−f, respectively. The differences between the three compounds are also listed in Table 3.
The formation of the foamy siloxene and massive amounts of hydrogen is the main feature of CaSi 2 leaching. The total leaching reaction corresponds to where the parameter x indicates the hydrolysis degree during the leaching process and usually equals to 1. 57 As the crystal structure of CaSi 2 presented in Figure 5a shows, the Ca 2+ cations formed a planar layer and were sandwiched between the   Subsequently, the siloxene forms with an intense visible luminescence, which is owing to its unique band structure. 58 The formed siloxene is water-soluble but leads to a more viscous solution and increases the difficulty of the post-cleaning procedure.
The leaching of Mg 2 Si in a HCl aqueous solution is known to yield silane gas, which further combusts on exposure to air at the surface. The main reaction of Mg−Si alloy leaching is briefly written as As a half-silicide, Mg 2 Si has a face-centered anti-CaF 2 -type structure, as shown in Figure 5e, where the Mg−Si bond shows a mixed ionic and covalent nature. 59 The silane formation is thus proposed to be related to the violent breakage of the Mg−Si bonds of the isolated Si 4− by an acidic hydrolysis reaction. The reaction kinetics has been found to be dependent on the acid concentration in our previous work. 23 Moreover, the yield of silane gas is reported to increase with increasing temperature and acid concentration in a dilute HCl aqueous solution. 60 The leaching mechanism of Ca 7 Mg 7.5±δ Si 14 has barely been studied so far. To the best of the authors' knowledge, only Nesper et al. 53 reported that it reacts with mineral acids with the formation of silane gas and amorphous Si. According to the experimental observations in our work, a massive gas bubble formed but with much fewer sparks compared to the leaching of the Mg−Si alloy system, and there is also no siloxene formation. Apparently, this unusual leaching behavior of Ca 7 Mg 7.5±δ Si 14 is attributed to its unique crystal structure. 53 As shown in Figure  5c, Ca 7 Mg 7.5±δ Si 14 has a highly anisotropic hexagonal crystal structure. Two types of Si atoms can be found in the unit cell: the Si 12 planar ring type with a Ca atom in between and the isolated Si type that is coordinated by Mg atoms. The band structure obtained by DFT calculations also suggests that it is a metallic compound. Since hydrogen evolution in an acidic solution is a common phenomenon for a number of metallic silicides, 61 it is also reasonable that the generated massive gas is hydrogen. To further characterize the electronic bonding structures of Ca 7 Mg 7.5±δ Si 14, we present the atomic bonding styles via the spatial charge density in Figure 6, which is based on the jellium electron gas model. The bonding behavior of Ca 7 Mg 7.5±δ Si 14 is also strongly anisotropic with the mixing of covalent, ionic bonding features. It is also seen that in the Si 12 planar ring, Si has a smaller charge than the isolated Si surrounded by Mg. Thus, the stoichiometry of Ca 7 Mg 7.5±δ Si 14 can also be written as (Ca 7 Mg 7.5±δ )[Si 12 ](Si 2 ).
According to the above discussion, the leaching reaction of Ca 7 Mg 7.5±δ Si 14 is, therefore, suggested to be Cracking Effect and Si Recovery. As shown in Figure 7a−c, the particle size of all studied samples decreases considerably after leaching, especially for the leaching of Ca−Si and Ca−Mg− Si samples, such that after shrinkage the size is almost half of the original size. It is apparent that strong cracking effects occurred during leaching and seem to be closely be related to the Ca concentration. By directly comparing the particle size distribution after leaching, as shown in Figure 7d, it is found that the cracking effect follows the trend Mg−Si sample < Ca− Mg−Si sample < Ca−Si sample. As a result of the cracking, the particles are more open, and accordingly, more surface is exposed to the acid solution so that it significantly accelerates the reaction progression.
Another parameter strongly related to the cracking effect is Si recovery, which is also crucial for the purification process evaluation. Its value was measured by comparing the sample weight before and after leaching, shown in Figure 8. A distinct trend can be seen where the filtration method shows the highest Si recovery systematically, followed by nonfiltration (direct pouring of the solution after leaching). The nonfiltration process   with additional sieving obtains the least recovery, which further removes insoluble fine particles that are concentrated on the impurities. For the leached samples, the Si recovery rate exhibits the sequence Mg−Si sample > Ca−Mg−Si sample > Ca−Si sample, especially when the fine particles with a diameter smaller than 0.2 mm are sieved out. This is attributed to the increase in particle cracking with increasing Ca content, as shown in Figure  7. Consequently, the generated ultrafine particles were easily lost due to the buoyancy and hydrodynamic thrust in the postcleaning and washing process, and most of them were sieved out after drying. Thus, the lowest recovery of Si was observed in the    Figure 9. It can be distinctly seen that the Ca−Si leaching displays the fastest kinetics, where most of the impurities were removed in around 10 min. Meanwhile, for the leaching of Ca−Mg−Si and Mg−Si alloys, even though for both of them the P removal shows similar efficiency, in general, impurity removal kinetics in the Ca−Mg− Si alloys is faster than that in the Mg−Si alloys, especially for the removal kinetics of Al, Fe, and Ti impurities. In fact, leaching kinetics is mainly affected by the properties of the main precipitated silicide. As shown in Figure 2, although the size of the main precipitates is similar, significant morphology differences can be observed in the Mg−Si sample. The complex twophase eutectic pattern may impede the acid attack and suppress the cracking effect during leaching.
As one of the most important factors that affect the kinetics, the cracking of particles has been widely reported in the literature for several Si-rich alloys, such as the Fe−Si, 38,62 Ca− Si, 63 and Cu−Si 30 systems. Margarido et al. 38,64 proposed a cracking−shrinking model to describe the kinetics of Fe−Si leaching. It was also found that this model works for the leaching of the slag-treated Ca−Si alloy system as well. In the aqua regia leaching of Cu−Si briquettes by Huang et al., 30 a modified shrinking mechanism controlled by both interfacial transfer and product layer diffusion was observed after cracking. However, in the present work, neither the mentioned models nor other commonly used and reviewed models 65 can be adopted to fit our leaching results. Thus, a novel cracking−shrinking model was developed to describe the leaching behavior of Mg−Si and Ca− Mg−Si alloys.
Modeling Assumptions. To simplify the problem, the purification process is assumed to take place in two separate stages, a leaching−cracking stage and a leaching−shrinking stage, as shown in Figure 10a. The total impurity conversion during the leaching−cracking stage is summarized as X c . The time period of the leaching−cracking stage is set as t c , which mainly depends on the alloy composition. More details are described below.
• Leaching−cracking stage: at t = 0, acid leaching starts with a total impurity conversion X = 0, precipitates start to be attacked by chemicals, while the microcracks keep propagating; it is worth noting that when the cracking finishes at t = t c , the leaching−shrinking stage starts with the conversion X = X c .
• Leaching−shrinking stage: from t ≥ t c , to the end of leaching, the core shrinking stage follows the kinetic model. The total impurity conversion at this stage varies as X c < X ≤ 1, while the impurity conversion in the shrinking stage is specifically denoted as α, which can be seen in Figure 10b, and is described later in Modeling Approach. Figure 10c illustrates the heterogeneous reaction that occurs during the leaching−shrinking stage. A large number of gas bubbles form at the reaction surface, and particularly in the early shrinking stage. It is assumed that a product layer is formed due to the massive outspreading bubbles and also some solid byproducts. Therefore, the diffusion of acidic species is significantly limited. Meanwhile, the growing bubble also forms a bubble−liquid film, which also inhibits the diffusion process. Consequently, the concentration gradient is formed, as can be seen from Figure 10d.
Modeling Approach. In the shrinking stage, impurity conversion is defined as where X is the measured impurity removal degree and α is the fraction conversion during shrinking. Considering a layer of the leachable phase around the Si grain particle (Figure 10b), the geometric relationship of the impurity removal degree can be written as where R and r indicate the initial radius and unreacted radius of particles in the shrinking stage, respectively, and l indicates the thickness of the reacted eutectic phase in radial dimensions, which is a reacted shell. Rearranging eq 7, we obtain As described before, the massive erupted gas bubble actively suppresses the diffusion and reveals time-dependent characteristics. Several fluid−solid kinetic models were tested, and it was found that the product layer assumption of the Kroger−Ziegler model 65−67 fits our case. In this case, the growth rate of the product layer is inversely proportional to its thickness and time, which can be expressed as where k is a constant that incorporates physical and chemical parameters. However, it is known that the Kroger−Ziegler model is directly modified from the original Jander model, 68,69 where both are based on the assumption of a constant reaction surface. Nevertheless, for a shrinking particle in our case, the surface curvature should be considered since the reaction area is not constant anymore and keeps decreasing. 69−71 Thus, in the current model, the approach of the Ginstling−Brounshtein model 69−71 is introduced to describe the shrinking of spherical particles under spherical coordinates Integrating eq 11 by considering the boundary conditions gives It yields Combining eqs 9 and 13, we then obtain the following shrinking shell model equation Finally, after introducing the cracking term and let K k R S Since the conversion rate for different impurities is different, the value of X C for each impurity should be different. For instance, the conversion of Ca and Mg was observed to be the most rapid, but the conversion of Fe and Ti was much slower. Herein, a term ξ i ∈ [0, 1] is introduced to evaluate the impurity i removal extent in connection with the Ca and Mg silicide conversions. Therefore, ξ Mg+Ca = 1 is defined as the benchmark. Subsequently, the total conversion of impurities in the cracking stage can be expressed as X i,c = ξ i X c Mg+Ca . Hence, a more general expression of the derived kinetics model is rewritten as It can be seen that when t = t c , we have X i = X i,C = ξ i X c Mg+Ca ; subsequently, the term X X The sign paradox between 0 < t < 1 and 1 < t is also avoided. Furthermore, the effect integration constant is also ruled out through definite integration in eq 12. Nevertheless, the cost of this modification is that the total rate constant is split into two parts: the cracking stage rate constant K C and the shrinking stage rate constant K S . As a result, the two rate constants should be considered together to compare the overall kinetics of different impurities or of different materials.
Model Evaluation. Based on the developed model above, a series of calculations were carried out to evaluate it. It is also worth noting that special modification should be applied to P prior to the calculations because of its relatively high segregation coefficient, which makes a part of P remain in Si grains and theoretically is not involved in the conversion process. Thus, the modified P conversion is expressed as 17) where N bulk is the measured P concentration in bulk material and N t is the measured P concentration at leaching time t. η is the percentage of P that segregated outside Si; ηN bulk also indicates the effective concentration of P that takes part in the leaching process.
In the calculation, η is set as 0.85 approximately for both alloys based on the P segregation model in the Si alloy. 23 Since the error range is quite narrow for η, the impact on the calculation accuracy can be ignored. As shown in Figure 11 Table 4, where K C is calculated by It can be seen from Table 4 that t c in Mg−Si is longer than that in the Ca−Mg−Si alloy accompanied by a minor value of X C , which reflects a more intense cracking effect for the leaching of the Ca−Mg−Si alloy. Impurity Removal. Overall Impurity Removal Degree. The impurity removal degree is calculated as where C initial and C final represent the impurity concentrations of leaching particles before and after leaching, respectively. Ca and Mg can be nearly completely removed from all of the alloys by leaching as shown in Figure 12, while for the removal of other metallic impurities like Al, Fe, and Ti, the following order is determined: Ca−Si sample > Ca−Mg−Si sample > Mg−Si sample. The P removal degree of the investigated systems is similar and close to 80% under the applied leaching conditions. Thus, it is considered that the P removal of the three studied alloys is at an equivalent level since the differences are insignificant to distinguish, and it is necessary to consider a certain level of uncertainty when dealing with trace amounts of impurity. However, it is worth noting that this equivalent P removal degree is reached by different Ca and Mg alloying amounts. By comparing the required alloying concentration for the equivalent P removal, as presented in Figure 13, it can be seen that more Mg addition is required than Ca based on both molar and weight percentage. Consequently, with a fixed total alloying amount, the P removal efficiency from high to low should follow the trend Ca addition > Ca−Mg addition > Mg addition. Further theoretical comparison of the efficiency of alloying elements is described in the following section through thermodynamic calculations. P Segregation Coefficient. Impurity elements are subject to solid/liquid segregation during the Si alloy solidification when the liquid alloy is solidified between the liquidus and eutectic temperatures. Knowledge of how the alloying metals affect the impurity segregation ratio is necessary to have a better understanding of the impurity removal by the alloying−leaching approach. In the Ca−Mg−Si alloying system, the segregation coefficient of P between solidified Si grains and the remaining liquid can be expressed as where X P in Si(s) and X P in Ca−Mg−Si(l) are the P concentrations in solid primary Si and the remaining Ca−Mg−Si alloy melt, respectively, during the solidification. While the system is under an equilibrium state, the chemical potential of P in the coexisting solid and liquid phases is the same. Assuming that the dissolution energy difference of P in liquid Si and the liquid Si-rich alloy is insignificant due to the ultralow P concentration, and considering the negligible amount of metallic impurities in the solid Si, one obtains X Neglecting the self-interaction term of P due to its sufficiently low content, and also ln γ Si(l) ≈ 0, eq 22 can be derived by combining eqs 19 It can be seen that the P segregation coefficient in its dilute solutions in Si is mainly influenced by the alloyed Mg and Ca concentrations and their interaction coefficient value with P. The averaged value of ε Mg in Si P during the solidification of the Mg−Si system was estimated to be −10.8 in our previous work, 23 and the value of ε Ca in Si P at 1450°C was determined to be −14.6(±1.7) according to Shimpo et al. 15 Thus, if we assume that the averaged value of ε Ca in Si P during the whole solidification range is equivalent to −14.6, reasonably, it can be expected that Ca leads to more reduction of the P segregation coefficient than that of Mg with the same amount. Figure 14 is plotted to investigate further the effect of Ca and Mg alloying on the P segregation behavior, and the isocurve of the P segregation coefficient with varying alloying concentrations is also shown as the solid lines. The P segregation coefficient is seen to decrease distinctly with increasing concentrations of Ca and Mg components. Meanwhile, it also follows the trend Ca−Si sample < Ca−Mg−Si sample < Mg−Si sample. However, since the leaching results suggest that the studied alloys show almost equivalent P removal, several reasons may be attributed to this inconsistency. First, the stronger cracking effect of the Ca−Si system may promote the P removal. Second, the averaged value of ε Ca in Si P during the whole solidification range might be more negative than the measured value at 1450°C. To investigate the effect of the interaction coefficient value, a tentative empirical value of ε Ca in Si P = −20.0 was also employed to calculate the isocurves as presented in Figure 13 by the dashed lines. It can be seen that a more negative interaction coefficient value fits better with the experimental results. Nevertheless, no matter which value is appropriate, it can be safely concluded that Ca exhibits stronger P affinity than Mg based on the above thermodynamic analysis.

■ CONCLUSIONS
In the present work, a comparative study of MG-Si leaching purification with Mg, Ca, and Mg−Ca additions to MG-Si was carried out. The following conclusions were drawn: • The doping metal significantly affects the alloy microstructure, leaching behavior, and Si purification efficiency. • In the Ca−Si alloy, the main precipitate was CaSi 2 and it led to the formation of a viscous byproduct siloxene during acid leaching along with a strong cracking effect. This causes the fastest leaching kinetics and the highest impurity removal efficiency. However, this cracking effect lowered the Si recovery compared to other samples. • The Mg−Si alloy system achieved the highest Si recovery with the highest impurity removal efficiency, especially P removal. Nevertheless, the main precipitate Mg 2 Si led to the slowest leaching kinetics. • In the Ca−Mg−Si alloy system, the main phase was determined as Ca 7 Mg 7.5±δ Si 14 with a unique crystal structure that contains both a planar Si 12 ring and isolated Si. As a result, the leaching of the Ca−Mg−Si alloy was found to be a cleaner process that avoids the disadvantages of both Ca−Si and Mg−Si systems. In addition, high Si recovery remained with fast leaching kinetics and high impurity removal, especially P removal. • A cracking−shrinking principle-based kinetics model was developed for the leaching kinetics of Mg−Si and Ca− Mg−Si alloys. Thermodynamic analysis further revealed that the P removal ability is related to the doping metal concentration and its interaction coefficient with P, while Ca shows a stronger affinity to P than that of Mg. • The novel Ca−Mg−Si ternary alloy system exhibited more sustainable performance as compared to the other two binary alloys.

Notes
The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS
This work was performed at NTNU within the Research Center for Sustainable Solar Cell Technology (FME SuSolTech, project number 257639), co-sponsored by the Norwegian Research Council and industry partners.