Migration Mechanisms of Al 3+ /Li + Lattice Impurities during Phase Transition from α -Quartz to β -Quartz: An Implication for Puriﬁcation of High-Purity Quartz

: The quality of high-purity quartz (HPQ) that used in strategic industries is normally limited due to lattice impurities. In order to reveal the migration pathway of lattice impurities in quartz particles during phase transition from α -quartz to β -quartz, α -quartz and Al 3+ /Li + -substituted α -quartz (S-α -quartz) was investigated under 846.15 K and 101.325 kPa based on density functional theory. Results showed that β -quartz exhibited more interstitial volume, dominating the migration of lattice impurities. This further indicates that the phase transition process was beneﬁcial for the migration of lattice impurity from a structural point of view. Moreover, Al 3+ and Li + mainly migrated from the intracell to lattice surface along the c axis. In addition, Li + migrated more easily than Al 3+ due to higher mean square displacement values. From a thermodynamic point of view, the lower energy barrier in the case of the Al 3+ and Li + lattice suggested that the presence of lattice impurities promoted phase transition. This study therefore provides an excellent theoretical basis for the removal of lattice impurities of HPQ from an atomic perspective, for the ﬁrst time.


Introduction
As an essentially basic raw material for strategic industries such as semiconductors, new energy, 5G communications, etc., high-purity quartz (HPQ, SiO 2 content ≥ 99.998%) is traditionally purified from quartz particles [1].However, the rapid development in these industries has led to an increasing consumption of HPQ sources, and more alternatives are being sought, such as pegmatite, vein quartz, which, however, have higher impurity contents [2].Generally, the impurities in quartz occur in independent gangue minerals, gas-liquid-solid inclusions, and lattice impurities.Normally, quartz and gangue minerals (such as mica, feldspar, limonite, hematite, etc.) can be easily separated via a flotation process [3][4][5], while the impure elements in quartz lattices are the most difficult to remove [6].Previous studies indicated that there are two kinds of lattice impurities [7][8][9][10], i.e., lattice substitution impurities, such as Si 4+ substituted by Al 3+ , Ga 3+ , Fe 3+ , B 3+ , Ge 4+ , Ti 4+ and P 5+ , in which Al 3+ is the most common ion due to its abundant reserves in Earth's crust and similar ionic radius to Si 4+ (0.54 Å vs. 0.40 Å) [11].Lattice interstitial impurities, such as H + , Li + , Na + , K + , Be + , Rb + and Fe 2+ , can enter into the interstitial lattice positions to balance the charge, with Li + being the most common interstitial impurity to balance Al 3+ charge in igneous and pegmatite quartz [12][13][14].
In order to remove the lattice impurities in quartz particles, high-temperature roasting and acid leaching are usually applied [15][16][17].For instance, Loritsch et al. [16] reported that the alkali metal impurities can diffuse to the quartz surface and thus be removed during Minerals 2023, 13, 1280 2 of 9 the roasting process using HCl gas at 1073.15 K~1873.15K. Lin et al. [17] indicated that the lattice impurities migrated from the interior to the quartz surface during the KCl-doping roasting process at 1173.15 K.It should be noted that quartz experiences evolutions in both its crystal phase and volume during the roasting process, thus affecting the removal efficiency of lattice impurities.The most common phase transition occurs from α-quartz to β-quartz at 846.15 K [18].Many studies focus on the characteristics of α-quartz; for instance, Baur [19] found that there were two long (i.e., 1.613 Å) and two short Si-O bonds (i.e., 1.603 Å) in the tetrahedron α-quartz.Le Page et al. [20] reported that the length of the Si-O bonds of α-quartz did not change at temperatures from 94 K to 298 K, suggesting no structural change at low temperatures.However, the crystal properties of β-quartz, the phase transition process from α-quartz to β-quartz and the migration of lattice impurities during the phase transition process at an atomic level are not yet understood, and a deeper understanding of the removal mechanism of lattice impurities during the preparation of high-purity quartz sand is essential.
As an effective method, density function theory (DFT) has been widely used to reveal the crystal properties of quartz [21][22][23].For instance, Sun et al. [22] employed DFT to investigate the crystal structure of α-quartz and the adsorption of collectors on α-quartz surfaces.Lin et al. [23] examined the migration of interstitial He in α-quartz and βcristobalite based on DFT.However, investigations on the migration pathways of inherent impurities in HPQ have not been reported yet.
Therefore, the phase transition process from α-quartz to β-quartz at 846.15 K and 101.325 kPa was calculated through DFT.Moreover, the substitution of Si 4+ by Al 3+ and Li + was selected as a typical case to reveal the migration mechanisms of the lattice impurities.This study thus provides an excellent theoretical basis for the removal of lattice impurities in quartz particles from an atomic perspective, which is beneficial for further application in HPQ purification in strategic industries.

Methodology
The DFT calculation was performed based on PW91 correlation potential in generalized gradient approximation (GGA) through CASTEP package in Materials Studio [24].The Monkhorst−Pack scheme with 3 × 3 × 4 k-point mesh was considered for α-quartz and β-quartz.Geometry optimization was applied to investigate the bulk properties of both α-quartz and β-quartz.Considering the influence of intermolecular force, both dipole correction and correction potential [25] were recalculated at each calculation step.As shown in Table S1, a kinetic energy cutoff of 360 eV was selected for both α-quartz and β-quartz due to the closer lattice parameters to the experimental values.A 2 × 2 × 2 unit cell was used to display the differences between the α-quartz and β-quartz structures visually.Moreover, the NPT ensemble condition was performed under 846.15K and 101.325 kPa to investigate the phase transition, and data were exported every 20 ps and named transition state n (TSn, n = 1, 2, 3. ..).In order to simulate the migration of lattice impurities, one Si atom in 2 × 2 × 2 α-quartz supercell was substituted by one Al atom, while a Li atom was placed in the nearby gap to balance the charge.The same NPT ensemble condition was applied on the substituted α-quartz and β-quartz (S-α-quartz and S-β-quartz).The data were also exported every 20ps and named S-TSn (n = 1, 2, 3. ..).
The average volume of the silica tetrahedron was calculated via Equation (1): where V t is the average volume of the silica tetrahedron, s and h are the base area and height of the silica tetrahedron.As shown in Figure 1, the average lengths of L O-O and h were calculated via Equations ( 2) and (3).
where L O-O is the average length between O and O atoms, L Si-O is the average Si-O bond length, and A O-O-Si is the average angle of O-O-Si.Since the base was an equilateral triangle, the average base area (s) can be calculated via Equation (4): According to Equations ( 3) and (4), Equation (1) can be converted to Equation (5).
where V i and V B are the interstitial and bulk volumes in α-quartz, respectively.
Minerals 2023, 13, x FOR PEER REVIEW 3 of 9 where Vt is the average volume of the silica tetrahedron, s and h are the base area and height of the silica tetrahedron.As shown in Figure 1, the average lengths of LO-O and h were calculated via Equations ( 2) and (3).

LO-O
where LO-O is the average length between O and O atoms, LSi-O is the average Si-O bond length, and AO-O-Si is the average angle of O-O-Si.Since the base was an equilateral triangle, the average base area (s) can be calculated via Equation ( 4): According to Equations ( 3) and ( 4), Equation ( 1) can be converted to Equation (5).
where Vi and VB are the interstitial and bulk volumes in α-quartz, respectively.Similarly, the average volume of the silica and aluminum oxygen tetrahedron in S-αquartz can be calculated via Equation (7).
where Vt' is the average volume of the silica tetrahedron, LAl-O is the average Al-O bond length, and AO-O-Al is the average angle of O-O-Al.Since the 2 × 2 × 2 supercell of S-α-quartz has 23 silica tetrahedrons and 1 aluminum oxygen tetrahedron (with a formula of Si23Al1Li1O48), the interstitial volume in S-α-quartz can be calculated via Equation (8).
where Vi is the interstitial volume in S-α-quartz.

Optimization of Bulk α-Quartz and β-Quartz
Table 1 shows the lattice parameters of optimized α-quartz and β-quartz based on two repeated experiments.It was clear that the standard deviation was very small, suggesting that calculated parameters herein were reasonable.In addition, the lattice parameters in this work were close to the experimental values reported in Young et al. [26] for α-quartz (i.e., an a value of 4.961 Å vs. 4.910 Å and a c value of 5.472 Å vs. 5.402 Å) and Wright and Lehmann [27] for β-quartz (i.e., an a value of 5.048 Å vs. 5.010 Å and a c value Similarly, the average volume of the silica and aluminum oxygen tetrahedron in S-α-quartz can be calculated via Equation (7).
where V t ' is the average volume of the silica tetrahedron, L Al-O is the average Al-O bond length, and A O-O-Al is the average angle of O-O-Al.Since the 2 × 2 × 2 supercell of S-αquartz has 23 silica tetrahedrons and 1 aluminum oxygen tetrahedron (with a formula of Si 23 Al 1 Li 1 O 48 ), the interstitial volume in S-α-quartz can be calculated via Equation (8).
where V i is the interstitial volume in S-α-quartz.

Optimization of Bulk α-Quartz and β-Quartz
Table 1 shows the lattice parameters of optimized α-quartz and β-quartz based on two repeated experiments.It was clear that the standard deviation was very small, suggesting that calculated parameters herein were reasonable.In addition, the lattice parameters in this work were close to the experimental values reported in Young et al. [26] for α-quartz (i.e., an a value of 4.961 Å vs. 4.910 Å and a c value of 5.472 Å vs. 5.402 Å) and Wright and Lehmann [27] for β-quartz (i.e., an a value of 5.048 Å vs. 5.010 Å and a c value of 5.607 Å vs. 5.470 Å).In addition, the lattice parameters for α-quartz in this work were closer to the experimental value than those in Yang et al. [28] (i.e., a = 5.027 Å and c = 5.513 Å for α-quartz), further indicating that the calculated parameters herein were reasonable.The optimized structures of α-quartz and β-quartz exhibited the same atomic structure (Figure 2), i.e., a Si atom bonded with four surrounding O atoms to form a silica tetrahedron structure, while an O atom bonded with two surrounding Si atoms to connect the adjacent silica tetrahedron structure, giving a possibility of phase transition [18].However, the arrangements of the silica tetrahedron in α-quartz and β-quartz were different.of 5.607 Å vs. 5.470 Å).In addition, the lattice parameters for α-quartz in this work closer to the experimental value than those in Yang et al. [28] (i.e., a = 5.027 Å and c = Å for α-quartz), further indicating that the calculated parameters herein were reason The optimized structures of α-quartz and β-quartz exhibited the same atomic s ture (Figure 2), i.e., a Si atom bonded with four surrounding O atoms to form a silic rahedron structure, while an O atom bonded with two surrounding Si atoms to con the adjacent silica tetrahedron structure, giving a possibility of phase transition [18].H ever, the arrangements of the silica tetrahedron in α-quartz and β-quartz were diffe The O-Si-O angles in the α-quartz silica tetrahedron were 108.63° and 110.13°, while b O-Si-O angles of 109.26° and 117.13° were observed for β-quartz.Moreover, the Si angle between two β-quartz silica tetrahedrons was also greater than that of α-quartz 153.19° vs. 144.47°),leading to trigonal α-quartz and hexagonal β-quartz [29].
In addition, two long (i.e., 1.624 Å) and two short Si-O bonds (i.e., 1.619 Å) wer served in the optimized α-quartz silica tetrahedron [19], while four shorter Si-O bon 1.614 Å was found in the optimized β-quartz silica tetrahedron.Although the volum the silica tetrahedron in α-quartz was larger than that in β-quartz, the bulk volume smaller, i.e., 116.653Å 3 vs.123.761Å 3 , suggesting that β-quartz exhibited a larger int tial volume.In other words, more space is available for the migration of lattice impu in β-quartz.

Phase Transition from α-Quartz to β-Quartz
In order to better understand the phase transition process from α-quartz to β-qu a 2 × 2 × 2 supercell of α-quartz was built to perform the dynamic tasks, and the re are shown in Figures 3 and S1 and Tables S2 and S3.The silica tetrahedrons exhibi In addition, two long (i.e., 1.624 Å) and two short Si-O bonds (i.e., 1.619 Å) were observed in the optimized α-quartz silica tetrahedron [19], while four shorter Si-O bonds of 1.614 Å was found in the optimized β-quartz silica tetrahedron.Although the volume of the silica tetrahedron in α-quartz was larger than that in β-quartz, the bulk volume was smaller, i.e., 116.653Å 3 vs.123.761Å 3 , suggesting that β-quartz exhibited a larger interstitial volume.In other words, more space is available for the migration of lattice impurities in β-quartz.

Phase Transition from α-Quartz to β-Quartz
In order to better understand the phase transition process from α-quartz to β-quartz, a 2 × 2 × 2 supercell of α-quartz was built to perform the dynamic tasks, and the results are shown in Figure 3 and Figure S1 and Tables S2 and S3.The silica tetrahedrons exhibited a nonuniform migration tendency during the phase transition process (Figure 3a).Specifically, the average length of Si-O bonds slightly decreased from 1.622 Å in α-quartz to 1.612 Å in TS1, which was gradually increased, i.e., 1.658 Å in TS4 (Figure 3b).In addition, as shown in Figure 3c, the bulk volume exhibited an increasing tendency, i.e., it slightly increased from 933.324 Å 3 in α-quartz to 937.928 Å 3 and 940.648Å 3 in TS1 and TS2, respectively, while more significant increases to 1007.560Å 3 and 1026.392Å 3 were observed in TS3 and TS4, respectively.In this case, the average Si-O bond length and bulk volume were employed to calculate the interstitial volume, exhibiting a nonlinear increase during the phase transition process from α-quartz to TS4 (Figure 3d), with the minimum and maximum values being in α-quartz (881.048Å 3 ) and TS4 (974.816Å 3 ), respectively, suggesting that the transition process was beneficial for the migration of lattice impurities.However, the interstitial volume decreased to 937.000 Å 3 in β-quartz, which was still higher than that in α-quartz [26,27], further indicating that the whole phase transition provided more space for the migration of lattice impurities.
Minerals 2023, 13, x FOR PEER REVIEW 5 of 9 nonuniform migration tendency during the phase transition process (Figure 3a).Specifically, the average length of Si-O bonds slightly decreased from 1.622 Å in α-quartz to 1.612 Å in TS1, which was gradually increased, i.e., 1.658 Å in TS4 (Figure 3b).In addition, as shown in Figure 3c, the bulk volume exhibited an increasing tendency, i.e., it slightly increased from 933.324 Å 3 in α-quartz to 937.928 Å 3 and 940.648Å 3 in TS1 and TS2, respectively, while more significant increases to 1007.560Å 3 and 1026.392Å 3 were observed in TS3 and TS4, respectively.In this case, the average Si-O bond length and bulk volume were employed to calculate the interstitial volume, exhibiting a nonlinear increase during the phase transition process from α-quartz to TS4 (Figure 3d), with the minimum and maximum values being in α-quartz (881.048Å 3 ) and TS4 (974.816Å 3 ), respectively, suggesting that the transition process was beneficial for the migration of lattice impurities.However, the interstitial volume decreased to 937.000 Å 3 in β-quartz, which was still higher than that in α-quartz [26,27], further indicating that the whole phase transition provided more space for the migration of lattice impurities.In addition, the lattice parameters of ai and bi exhibited the same increasing trend during the phase transition process, thus only Δai and Δci were employed to evaluate the contribution of lattice parameters to ΔV. R 2 values of 0.9979 and 0.9945 suggested a good relationship between lattice parameter changes and ΔV.Overall, the Δai and Δci increased during the phase transition process, thereby increasing the bulk volume.The values for Δai/ΔV and Δci/ΔV of 0.01233 and 0.01851 (Figure S1) suggested that ΔV was mainly In addition, the lattice parameters of a i and b i exhibited the same increasing trend during the phase transition process, thus only ∆a i and ∆c i were employed to evaluate the contribution of lattice parameters to ∆V.R 2 values of 0.9979 and 0.9945 suggested a good relationship between lattice parameter changes and ∆V.Overall, the ∆a i and ∆c i increased during the phase transition process, thereby increasing the bulk volume.The values for ∆a i /∆V and ∆c i /∆V of 0.01233 and 0.01851 (Figure S1) suggested that ∆V was mainly dominated by ∆c i .In this case, the volume mainly increased along the c axis, suggesting that the interstitial lattice impurity may migrate along the c axis.
As shown in Figure 3e, the energy barrier of 0.49 eV indicated that the phase transition from α-quartz to β-quartz was nonspontaneous from a thermodynamic point of view [18,29].The maximum energy barrier value was observed between α-quartz and TS4, indicating that the phase transition from α-quartz to β-quartz needs to overcome an energy barrier of 9.76 eV.
As shown in Figure 4, there was a significant migration of lattice impurities from intracell to lattice surface, which then can be removed by dissolving the quartz surface produce HPQ.In addition, two migration stages were observed during the phase tran tion process.During the first stage (from S-α-quartz to S-TS2), the lengths of the Al-O a Si-O bonds increased 1.009 times (from 1.751 Å to 1.766 Å) and 1.008 times (from 1.624 to 1.637 Å), respectively, as shown in Figure 4b.However, the bulk volume only increas 1.007 times (from 984.354 Å 3 to 991.122 Å 3 , Figure 4c), giving rise to a slight variation the interstitial volume during this stage (0.998 and 1.005 times in S-TS1 and S-TS2).The fore, no migration is expected during this stage.As shown in Figure 4, there was a significant migration of lattice impurities from the intracell to lattice surface, which then can be removed by dissolving the quartz surface to produce HPQ.In addition, two migration stages were observed during the phase transition process.During the first stage (from S-α-quartz to S-TS2), the lengths of the Al-O and Si-O bonds increased 1.009 times (from 1.751 Å to 1.766 Å) and 1.008 times (from 1.624 Å to 1.637 Å), respectively, as shown in Figure 4b.However, the bulk volume only increased 1.007 times (from 984.354 Å 3 to 991.122 Å 3 , Figure 4c), giving rise to a slight variation in the interstitial volume during this stage (0.998 and 1.005 times in S-TS1 and S-TS2).Therefore, no migration is expected during this stage.
An obvious migration of lattice impurities occurred during the second stage from S-TS3 to S-β-quartz, as shown in Figure 4a, where the Al 3+ and Li + were transferred from the intracell to lattice surface, giving mean square displacement (MSD) values of 4.1562 × 10 −8 m 2 •s −1 , 0.3589 × 10 −8 m 2 •s −1 and 0.3519 × 10 −8 m 2 •s −1 for Li + , Al 3+ , and Si 4+ , respectively.Since no bonding effects occurred, Li + easily migrated from intracell to lattice surface during the phase transition process, while Al 3+ and Si 4+ were difficult to separate due to their similar MSD values [29].Similar to the interstitial volume evolution during the phase transition process of α-quartz (Figure 3b), the maximum increase in interstitial volume of 1.038 times (from 931.919 to 967.203 Å 3 ) was observed in S-TS4 (Figure 4d).Although a slight decrease in the interstitial volume of 965.413Å 3 was observed in S-β-quartz, this was, however, still higher than that in S-α-quartz, indicating that the whole phase transition process provided channels for the migration of lattice impurities.Overall, the migration of lattice impurities seems to be dominated by changes in the interstitial volume.In addition, the evolution of interstitial volume during the phase transition process of S-α-quartz was more moderate than that in α-quartz (e.g., 1.038 vs. 1.106 times in TS4 and 1.036 vs. 1.064 times in β-quartz), suggesting that the presence of lattice impurities weakened the increasing potential of interstitial volume [6].
Different to the α-quartz phase transition process, a more complex increasing trend occurred for all lattice parameters of a i , b i , and c i during the S-α-quartz phase transition; thus, ∆a i , ∆b i , and ∆c i were all employed to evaluate the contribution of lattice parameters to ∆V (Figure S2a).R 2 values of 0.9979, 0.9987, and 0.9993 suggested a good relationship between the bulk volume and lattice parameters.More specifically, ∆V was mainly dominated by ∆c i due to the higher value for ∆c i /∆V than that for ∆b i /∆V and ∆a i /∆V (i.e., 0.00375 vs. 0.00340 and 0.00335).It should be noted that the contribution of ∆a i and ∆b i to ∆V in the S-α-quartz phase transition was higher than that in the α-quartz phase transition.In addition, the increasing volume in the S-α-quartz phase transition was mainly along the c axis; thus, Al 3+ and Li + mainly migrated from the intracell to lattice surface along the c axis, as shown in Figure 4a.The changes in Al 3+ and Li + coordination were employed to exhibit the migration pathway of lattice impurities via the c axis (Figure S2b).The results showed that Li + exhibited more changes in coordination than Al 3+ , further indicating that Li + readily migrated from the intracell to lattice surface during the phase transition process.In addition, the change near the lattice surface was significantly greater than that in the intracell (e.g., 3.845 Å from S-TS4 to S-β-quartz vs. 0.842 Å from S-TS3 to S-TS4), which may be due to less interaction between atoms near the lattice surface, suggesting that it was more difficult for the lattice impurities in the intracell to migrate than those near the lattice surface.
As shown in Figure 4e, the phase transition from S-α-quartz to S-β-quartz was thermodynamically unfavorable due to the energy barrier of 0.15 eV.In addition, the energy barrier between S-α-quartz and S-TS4 requires extra energy during the phase transition process.Since the energy barrier in the phase transition from S-α-quartz to S-β-quartz was smaller than that from α-quartz to β-quartz (i.e., 8.51 eV vs. 9.76 eV), the presence of lattice impurities promoted phase transition from a thermodynamic point of view.

Conclusions
Our DFT results indicated that β-quartz exhibited a shorter Si-O bond length but a larger bulk volume than α-quartz; thus, β-quartz possessed a larger interstitial volume, which is beneficial for the migration of lattice impurities.For the phase transition from S-α-quartz to β-quartz, Al 3+ and Li + mainly migrated from the intracell to the lattice surface via the c axis with MSD values of 4.1562 × 10 −8 m 2 •s −1 and 0.3589 × 10 −8 m 2 •s −1 , suggesting that Li + more readily migrated than Al 3+ during the phase transition.Overall, the migration of lattice impurities tended to be dominated by the changes in interstitial volume, and a nonlinear interstitial volume evolution was presented during the phase transition for S-αquartz, e.g., it significantly increased from 931.919 Å 3 (S-α-quartz) to 967.203 Å 3 (S-TS4).From a thermodynamic point of view, the phase transition from S-α-quartz to S-β-quartz was more thermodynamically favorable than that from α-quartz to β-quartz due to the lower energy barrier (8.51 eV vs. 9.76 eV), suggesting that the presence of lattice impurities increased the

Figure 1 .
Figure 1.The structure of the side and bottom view of silica tetrahedron (Red ball is O atom, yellow ball is Si atom).

Figure 1 .
Figure 1.The structure of the side and bottom view of silica tetrahedron (Red ball is O atom, yellow ball is Si atom).

Figure 3 .
Figure 3. (a) Structure of phase transition process from α-quartz to β-quartz, (b) average Si-O bond length, (c) bulk volume, (d) interstitial volume, and (e) energy barrier.Two repeated experiments were performed for error estimates.

Figure 3 .
Figure 3. (a) Structure of phase transition process from α-quartz to β-quartz, (b) average Si-O bond length, (c) bulk volume, (d) interstitial volume, and (e) energy barrier.Two repeated experiments were performed for error estimates.

Figure 4 .
Figure 4. (a) Structure of phase transition process from S-α-quartz to S-β-quartz, (b) average Si-O and Al-O bond lengths, (c) bulk volume, (d) interstitial volume, and (e) energy barrier.Two repeated experiments were performed for error estimates.

Table 1 .
Lattice parameters of bulk α-quartz and β-quartz (Si 3 O 6 ).Two repeated experiments were performed for error estimates.