High-pressure phase relations in Zn2SiO4 system: A first-principles study

TL;DR

This study uses first-principles DFT calculations to investigate phase relations in the Zn2SiO4 system, identifying potential high-pressure phases and phase transitions.

Contribution

It provides new insights into high-pressure phase structures of Zn2SiO4, proposing candidate structures for phases III and IV and analyzing phase transition mechanisms.

Findings

Phases III and IV are retrograde phases.

Na2CrO4- and Ag2CrO4-structured phases are likely high-pressure candidates.

Phase II transitions to spinel during optimization.

Abstract

In order to clarify phase relation of Zn2SiO4 system, first-principles DFT calculations of 11 phases were conducted. We confirmed that phases III and IV are retrograde phases. Instead, Na2CrO4- and Ag2CrO4-structured phases are most likely candidate structures for high-pressure phases of III and IV, respectively. A transition of phase II to spinel found during optimization was discussed in relation to similar transition known for nitrides.

Tables

Table 1. Table 1: Equations of states of Zn 2 SiO 4 phases

phase	V0111Volume per formula(Å3)	V(obs)	K0(GPa)	K′	P range
I	86.607(12)	86.92	137.0(11)	1.6(2)	0–11
II	78.894(12)	79.34	149.4(6)	0.63(4)	0–23
III	86.58(8)	87.03	70.7(11)	1.10(8)	0–17
IV	80.33(2)	80.85	119.5(14)	0.6(2)	0–10
V	69.037(2)	69.30	183.1(2)	4.76(2)	0–25
olivine	73.515(4)	74.3222Estimated from Mg-Zn olivine solid solution Syono et al. (1971)	145.9(3)	4.56(3)	0–25
spinel	67.4145(4)	NA	204.21(5)	4.561(4)	0–25
Na2CrO4	72.198(2)	NA	158.9(2)	4.63(2)	0–25
Ag2CrO4	71.577(4)	NA	149.5(3)	5.23(3)	0–25
Ca2RuO4	63.666(2)	NA	213.7(3)	4.52(3)	0–25
III-$\mathrm{𝑃𝑏𝑎𝑚}$	63.714(4)	NA	181.4(5)	5.37(5)	0–25
ilmenite	44.9017(14)	44.73	204.3(3)	4.97(3)	0–25

Table 2. Table 2: Optimized crystal structures of Zn 2 SiO 4 phases at 1 bar and 0 K

willemite				phase II				phase III
$R$$\overline{3}$, $a$ = 8.6197 Å,				$I$$\overline{4}$2$d$, $a$ = 7.0124,				$\mathrm{𝑃𝑛𝑚𝑎}$, $a$ = 10.2804,
$\alpha$ = 107.896∘				$c$ = 6.4193 Å				$b$ = 6.6360, $c$ = 5.0609 Å
site	x	y	z	site	x	y	z	site	x	y	z
Zn1(6f)	0.3986	0.6255	0.2231	Zn(8d)	0.1565	1/4	1/8	Zn(8d)	0.1562	0.9996	0.8248
Zn2(6f)	0.0591	0.2961	0.8897	Si(4b)	0	0	1/2	Si(4c)	0.0923	1/4	0.3231
Si(6f)	0.7339	0.9611	0.5539	O(16e)	0.3060	0.4845	0.1426	O1(4c)	0.1120	1/4	0.6446
O1(6f)	0.8605	0.8543	0.5346					O2(4c)	0.4392	1/4	0.2631
O2(6f)	0.7483	0.0736	0.4325					O3(8d)	0.1632	0.0480	0.2051
O3(6f)	0.8114	0.1017	0.7686
O4(6f)	0.5250	0.8090	0.4743
phase IV				olivine phase				phase V
$\mathrm{𝑃𝑏𝑐𝑎}$, $a$ = 10.8823,				$\mathrm{𝑃𝑛𝑚𝑎}$, $a$ = 10.2520,				$\mathrm{𝐼𝑚𝑚𝑎}$, $a$ = 5.7358,
$b$ = 9.7025, $c$ = 6.0886 Å,				$b$ = 6.0014, $c$ = 4.7805 Å				$b$ = 11.4936, $c$ = 8.3783 Å
site	x	y	z	site	x	y	z	site	x	y	z
Zn1(8c)	0.5618	0.5819	0.3458	Zn1(4a)	0	0	0	Zn1(4a)	0	0	0
Zn2(8c)	0.6606	0.3165	0.0564	Zn2(4c)	0.2797	1/4	0.9846	Zn2(4e)	0	1/4	0.9694
Si(8c)	0.3760	0.1200	0.6366	Si(4c)	0.0958	1/4	0.4268	Zn3(8g)	1/4	0.1259	1/4
O1(8c)	0.2515	0.2162	0.6686	O1(4c)	0.0933	1/4	0.7673	Si(8h)	0	0.1204	0.6172
O2(8c)	0.4008	0.0232	0.8517	O2(4c)	0.4472	1/4	0.2176	O1(4e)	0	1/4	0.2141
O3(8c)	0.4882	0.2302	0.5944	O3(8d)	0.1647	0.0316	0.2791	O2(4e)	0	1/4	0.7161
O4(8c)	0.3642	0.0185	0.4225					O3(8h)	0	0.9887	0.2563
								O4(16j)	0.2625	0.1230	0.9922
spinel phase				Na2CrO4 phase				Ag2CrO4 phase
$\mathrm{𝐹𝑑}$$\overline{3}$$m$, $a$ = 8.1397 Å				$\mathrm{𝐶𝑚𝑐𝑚}$, $a$ = 5.5743,				$\mathrm{𝑃𝑛𝑚𝑎}$, $a$ = 9.3999,
				$b$ = 8.5267, $c$ = 6.0766 Å				$b$ = 6.1644, $c$ = 4.9514 Å
site	x	y	z	site	x	y	z	site	x	y	z
Zn(16d)	5/8	5/8	5/8	Zn1(4b)	0	1/2	0	Zn1(4b)	1/2	0	0
Si(8a)	0	0	0	Zn2(4c)	0	0.1660	1/4	Zn2(4c)	0.1659	1/4	0.9871
O(32e)	0.3681	0.3681	0.3681	Si(4c)	0	0.8513	1/4	Si(4c)	0.8216	1/4	0.0252
				O1(8g)	0.2691	0.4754	1/4	O1(4c)	0.6488	1/4	0.1107
				O2(8f)	0	0.2611	0.5336	O2(4c)	0.8582	1/4	0.7000
								O3(8d)	0.8869	0.0323	0.1670
III-$\mathrm{𝑃𝑏𝑎𝑚}$ phase				Ca2RuO4 phase
$\mathrm{𝑃𝑏𝑎𝑚}$, $a$ = 4.9128 Å				$\mathrm{𝑃𝑏𝑐𝑎}$, $a$ = 4.9236,
$b$ = 9.3540, $c$ = 2.7730 Å				$b$ = 5.0143, $c$ = 10.3174 Å
site	x	y	z	site	x	y	z
Zn(4g)	0.9688	0.1833	0	Zn(8c)	0.9989	0.05376	0.3364
Si(2d)	1/2	0	1/2	Si(4a)	0	0	0
O1(4g)	0.2856	0.0559	0	O1(8c)	0.1972	0.2925	0.0479
O2(4h)	0.6668	0.1715	1/2	O2(8c)	0.8874	0.9532	0.1598

Full text

††preprint: APS

I Introduction

Zn2+ ion prefers tetrahedral sites over octahedral sites Syono et al. (1971). Consequently, zinc silicates exhibit quite different crystalline phases and phase relation compared to those of magnesium silicates at ambient pressure. Partition coefficient of Zn between olivine and a basaltic magma significantly deviates from those of other divalent cations with similar ionic radii, resulting Zn anomaly Matsui et al. (1977). Thus, crystal chemical understanding of zinc silicates might provide better insight of chemical behavior of Zn in nature and in synthetic materials.

Pressure-induced phase transition sequence of Zn2SiO4 determined by previous quench experiments up to 15 GPa is as follow: I$\rightarrow$II$\rightarrow$III$\rightarrow$IV$\rightarrow$V. Phase V then decomposes to ZnSiO3 ilmenite phase plus ZnO (NaCl structure) at around 13 GPa Ito and Matsui (1974). Synthesis of phase VI (structure unknown) was reported at around 12 GPa Doroshev et al. (1983), but the phase has not been confirmed by other studies.

Phase I (mineral name willemite) has a phenacite structure Klaska et al. (1978), whereas phase II adopts a compact and rare structure Marumo and Syono (1971). In both structures, Zn and Si occupy tetrahedral sites. Crystal structures of phases III and IV are reported recently Liu et al. (2013). Phase III adapts a “tetrahedral olivine” structure Baur (1980) in which Zn ions occupying vacant tetrahedral sites of olivine structure, while leaving $\it{M}$1 and $\it{M}$2 sites of olivine structure vacant. Phase IV consists of tetrahedrally coordinated Zn and Si, and it features unique edge-shared Zn2O6 dimers. This study also reveals that these two phases have volumes lower than that of lower pressure phase (i.e., phase II), and the volume of phase III is very close to that of the ambient pressure phase (phase I). Therefore, these phases are apparently not high-pressure phases, and are likely transformed from yet unknown high-pressure phases during the decompression process. This sort of the retrograde phase transformation of high-pressure phase during decompression to a metastable phase are known for other compounds Yusa (2017). Phase V was identified to have modified spinel structure Syono et al. (1971), and its crystal structure was refined recently Kanzaki (2018a). This is the only experimentally known Zn2SiO4 phase in which all Zn ions occupy octahedral sites in the structure.

Recent in-situ high-pressure Raman spectroscopic study of phases III and IV at room temperature revealed that both phases exhibited the pressure-induced phase transitions at 5.5 and 2.5 GPa during compression, respectively Kanzaki (2018b). These transformed phases are abbreviated as III-HP and IV-HP hereafter to avoid confusion. This study certified that phases III and IV are retrograde phases from the supposed III-HP and IV-HP, respectively. However, the crystal structures of phases III-HP and IV-HP are not known yet. In order to understand stabilities of these phases, and to find structure candidates for these structure-unknown phases, first-principles density functional theory (DFT) calculations were conducted in present study. The DFT study of seven phases in this system has been reported before Karazhanov et al. (2009). Their calculations included structures of $\beta$-Ca2SiO4 (mineral name larnite) and olivine which they regarded as model structures for phases III and IV, respectively. However, as shown by Liu et al. Liu et al. (2013), crystal structures of phases III and IV are different from those model structures. Therefore, new thorough calculations including these newly established structures and other possible candidate structures are needed. Nalbandyan and Novikova Nalbandyan and Novikova (2012) reviewed structural chemistry of A2MX4 compounds in terms of packing densities, and their result can be used to estimate high-pressure candidate phases for Zn2SiO4 compounds. In the present study, structural optimizations of 11 selected phases were conducted up to 25 GPa using the first-principles DFT method.

II Calculation method

The first-principles DFT calculations were conducted using the QUANTUM ESPRESSO (ver. 6.2) package Giannozzi et al. (2017). Projector augmented-wave (PAW) method was used with following potentials: Zn.pbesol-dn-kjpaw$\_$psl.0.2.2.UPF, Si.pbesol-n-kjpaw$\_$psl.0.1.UPF, O.pbesol-n-kjpaw$\_$psl.0.1.UPF from PSlibary Corso (2014). Following 21 phases were considered at the very beginning: I, II, III, IV, V, olivine, spinel, III-$\it{Pbam}$, Na2CrO4, Ag2CrO4, Ca2RuO4, thenardite (Na2SO4), $\beta$-Li2SO4, larnite, K2SO4, Na2MnO4, K2MnO4, K2MoO4, Tl2CrO4, NaGdSiO4 and Zn2SiO4-$\it{m}$ (a phase reported in Karazhanov et al. Karazhanov et al. (2009)). However, last 10 phases have enthalpies substantially higher than other phases at 0 and 9 GPa. Therefore, only first 11 phases are considered further. Ilmenite phase of ZnSiO3 was also calculated to compare with experimental compression data.

The kinetic energy cutoff for wave function, the kinetic energy cutoff for charge density, and the scf convergence threshold were set 80, 400 and 10*-14* Ry, respectively. The Brillouin zones were sampled with the Monkhorst-Pack scheme. Used meshes were 4$\times$4$\times$4 for phase II, spinel and ilmenite, 4$\times$2$\times$8 for phases III and III-$\it{Pbam}$ (a high-pressure phase transformed from phase III, see below), 2$\times$4$\times$4 for olivine and Ag2CrO4, 2$\times$2$\times$2 for phase I, 2$\times$2$\times$4 for phase IV, 4$\times$2$\times$2 for phase V, 4$\times$2$\times$4 for Na2CrO4 and 4$\times$4$\times$2 for Ca2RuO4. At each pressure, atomic positions and cell parameters are optimized using BEFG algorithm Giannozzi et al. (2017). Zero-point vibrational energy was not included in the enthalpy calculation.

Initial crystal structural parameters were taken from experimental ones for I, II, III, IV and V Klaska et al. (1978); Marumo and Syono (1971); Liu et al. (2013); Kanzaki (2018a). For olivine, spinel and ilmenite phases, structures of Mg-silicate polymorphs were used Fujino et al. (1981); Sasaki et al. (1982); Horiuchi et al. (1982), and Mg positions are replaced with Zn. For Na2CrO4 and Ag2CrO4 phases, isostructural Na2SO4 polymorphs Rasmussen et al. (1996) were used, and Na and S are replaced with Zn and Si, respectively. For Ca2RuO4 structure, the structural parameters were taken from Karazhanov et al. Karazhanov et al. (2009). The equations of states from the calculated volumes were fitted to the third-order Birch-Murnaghan equations using the non-linear fitting function (nls) of R Team (2018).

III Results and discussion

III.0.1 Structural optimizations

The compression curves of 11 phases up to 25 GPa are shown in Figure 1, and the parameters for the equation of state are listed in Table 1. Because of “transition” during structural optimization, some curves are terminated below 25 GPa. The optimized structures of these phases at 1 bar and 0 K are given in Table 2. Comparison with available experimental cell volumes at ambient pressure for phases I, II, III, IV and V showed that the calculated volumes are slightly smaller than observed ones, mostly within 0.6% (Table 1). We noted that the pbesol-type pseudopotentials which use general gradient approximation reproduce the cell volume well. Karazhanov et al. Karazhanov et al. (2009) conducted the DFT calculations using local density approximation (LDA). Their calculated volumes at ambient pressure for phases I, II and V are a few % smaller than those of experiments and are typical for the LDA calculations Kroll (2003).

There is no experimental compression study for any of Zn2SiO4 phases so far, but the compression data is available for ZnSiO3 ilmenite phase up to 12 GPa Sato et al. (1977). Therefore, the compression curve of ilmenite phase was also calculated to assess reproducibility of our calculation. The reported bulk modulus (216 GPa with fixed K*′=4) can compare well with our calculated one (204.2 GPa with K′=4.97), whereas Karazhanov et al. Karazhanov et al. (2009) reported much smaller bulk modulus (177.89 GPa with K′*= 5.5). Sato et al. Sato et al. (1977) stated that their pressure scale based on LiF lattice parameter is likely overestimated pressure by 5% compared to more popular NaCl scale Decker (1971). If this is the case, their bulk modulus is recalculated as 210 GPa, resulting further agreement with our calculated value. Therefore, similar accuracy in calculated compressibility would be expected for the Zn2SiO4 system too.

The compression curves in Figure 1 demonstrate a general trend that a phase with larger volume is more compressible. Among them, phases I and III revealed anomalous compressional behavior. The compression curve of phase I has a kink at around 14 GPa. Close inspection of the structure reveals that there is no change in the space group (\it{R}$$\bar{3}) up to 25 GPa. Figure 2 shows lattice parameters, tetrahedral volumes and selected inter-tetrahedral angles of phase I with pressure. As shown in Figure 2d, the inter-tetrahedral angles change more effectively above 15 GPa, contributing reduction of void space made by six-membered rings. Therefore, this kink is likely due to a change in dominant compressional mechanism.

Phase III exhibits an unusual low bulk modulus, and is less than a half compared to those of other phases, except phase IV (see Table 1). At ambient pressure, phase III has a volume comparable to that of phase I. However, it surpasses phase II at 14 GPa and almost reaches to that of olivine phase at 18 GPa, where a “transition” happens. Liu et al. Liu et al. (2013) revealed that phase III has tetrahedral olivine structure Baur (1980). “Normal” (Mg,Fe)2SiO4 olivine structure is based on hexagonal closed packing of oxygens, and (Mg,Fe) ions occupy two octahedral sites ($\it{M}$1 and $\it{M}$2), and Si ion occupies a tetrahedral site. Tetrahedral olivine is named by Baur Baur (1980), and it can be derived from normal olivine structure by relocating cations in $\it{M}$1 and $\it{M}$2 sites into one of vacant tetrahedral sites (see Figure 4a for crystal structure of phase III).

Therefore, it is interesting to compare structural changes of phase III with olivine phase. Figure 3 compares the lattice parameters of both structures with pressure. The $\it{a}$- and $\it{c}$-axis of phase III show higher linear compressibilities than those of olivine phase. Figure 3 shows polyhedral volumes of the cation sites of two phases with pressure. It also includes two vacant octahedral sites of phase III corresponding to $\it{M}$1 and $\it{M}$2 sites in olivine structure, and a vacant tetrahedral site in normal olivine, which is occupied by Zn in phase III. Significant reduction in volume for the vacant octahedral sites in phase III is apparent in Figure 3e, while nearly similar changes in the tetrahedral volumes of both occupied and vacant sites in Figure 3d. At 18 GPa, volumes of two vacant octahedral sites in phase III approach to those of $\it{M}$1 and $\it{M}$2 sites in olivine phase, a trend parallel with the volume change (Figure 1). These results suggest that the high compressibilities of the vacant octahedral sites are responsible for the anomalous bulk compressibility of phase III.

Phase III was observed in the experimental product recovered from 8 to 10 GPa Syono et al. (1971). Even though phase III has the high compressibility, the volume of phase III is still higher than that of phase II at those pressures, confirming that phase III is not high-pressure phase. Phase IV showed the compressibility higher than those of other phases, except phase III, and a transition during optimization was found at 11 GPa. Phase IV was observed in experimental products recovered from 9 to 11 GPa Syono et al. (1971); Ito and Matsui (1974). At 10 GPa, the volume of phase IV is nearly same to that of phase II, but much larger compared to that of phase V.

Since we are looking for the candidate structures of phases III-HP and IV-HP, the curves locating between those of phase II and phase V in Figure 1 are of special interest. They are olivine, Na2CrO4 and Ag2CrO4 phases. These phases are also listed as possible high-pressure phases of A2BX4 system (with tetrahedral B ion) Nalbandyan and Novikova (2012). There are other possible candidate phases (such as thenardite) listed in Nalbandyan and Novikova Nalbandyan and Novikova (2012). Those phases generally have large cation sites for A cation, and have much higher enthalpies. Na2CrO4 and Ag2CrO4 structures are not adopted in other M2SiO4 compounds (with M = Mg and the transition metal cations) and are not considered as the candidate high-pressure phases of Zn2SiO4 before.

III.0.2 “Phase transitions” observed during structure optimizations

For phases II, III and IV, we observed pressure-induced “phase transition” during the structural optimization. Since the structural optimization is not exactly following evolution of dynamics of atoms with time, such as molecular dynamics simulations, these transitions should be treated with caution. Nevertheless, it would suggest prospective phase transitions. Therefore, these observed “phase transitions” are briefly described below.

Phase II was stable up to 24 GPa, but at 25 GPa, its volume dropped significantly during the optimization, and the obtained volume was comparable to that of spinel phase. Structure examination reveals that the resultant structure is indeed spinel structure. Spinel phase did not revert to phase II during the optimization even at 1 bar. This transition is discussed separately in the following section.

For phase III, a phase transition was observed at 19 GPa. Close examination of the resultant structure revealed that $\it{b}$-axis becomes half of original cell, and space group changed from $\it{Pnma}$ to $\it{Pcma}$ (= $\it{Pbam}$ in standard setting). Crystal structure of the phase (designated hereafter as phase III-$\it{Pbam}$) is shown in Figure 4 along with that of phase III, and atomic positions are given in Table 2. The structure consists of an octahedral Si site and a penta-coordinated Zn site. Penta-coordinated ZnO5 forms a prism-like polyhedron. For penta-coordinated Zn in the phase, bond distance for sixth nearest oxygen is longer than 2.3 Å, and it was not regarded as octahedral coordination. To our knowledge, no analogue phase with this structure is found in other high-pressure silicates, but we noted some structural similarity with Ca2IrO4 Babel et al. (1966). These polyhedral units are common with Ca2RuO4 structure too. However, SiO6 octahedra are linked by edge-sharing, and form one-dimensional chains in III-$\it{Pbam}$, whereas SiO6 octahedra are linked by sharing corners, and form a layer in Ca2RuO4. These two phases show very similar compressional behaviors (Figure 1). Kanzaki Kanzaki (2018b) observed a phase transition of phase III to III-HP at 5.5 GPa during compression. So natural question is that phase III-$\it{Pbam}$ corresponds to phase III-HP? We are negative to this question. Experimentally observed transition pressure (5.5 GPa) is too low to realize octahedral Si in the structure. Therefore, phase III-$\it{Pbam}$ will not be a stable phase in this system. Phase III-$\it{Pbam}$) did not revert to phase III during the optimization even at 1 bar.

For phase IV, a phase transition was observed at 10 GPa during the optimization. The resultant structure was found to be isostructural to Na2CrO4 structure with $\it{Cmcm}$ space group, and the structure was already included in our DFT calculations. Hereafter, the phase is designated as Na2CrO4 phase. As we will discuss later, we are not sure this phase corresponds to phase IV-HP or not. One of Na2SO4 phases (Na2SO4 III) also takes this structure Rasmussen et al. (1996). Crystal structures of phase IV and Na2CrO4 are shown in Figure 5. It is noted that slight displacements of Zn ions in phase IV would result the structure of Na2CrO4 as shown by arrows in Figure 5a. In this structure, Zn ions occupy a tetrahedral and an octahedral sites, and Si ions occupy single tetrahedral site. Unique feature of this structure is edge-shared ZnO4 and SiO4 tetrahedra. Optimized structural parameters of this phase are given in Table 2. Na2CrO4 phase did not revert to phase IV during the optimization even at 1 bar.

Despite a half of Zn ions are still in the tetrahedral site, Na2CrO4 phase is denser than olivine phase in Zn2SiO4 system as shown in Figure 1, suggesting the phase could be a high-pressure phase of olivine phase in certain A2BX4 compounds. Such transition is actually known for LiFePO4. LiFePO4 adapts olivine structure at ambient pressure. Na2CrO4 phase of LiFePO4 was synthesized at 6.5 GPa and 1173 K Garc$\acute{i}$a-Moreno et al. (2001). It is interesting to note that coordination number of Li is reduced from 6 to 4 by this transition, contrary to the well-known trend of pressure-induced coordination number increase. This structure was also observed as a high-pressure phase of Li2SO4 at 7 GPa Parfitt et al. (2005). However, no isostructural phase is known for silicates so far.

III.0.3 Stability of phases and possible high-pressure candidates

Stability of phases can be evaluated by the enthalpies of 11 phases (at 0 K) as shown in Figure 6. The figure shows the enthalpies relative to that of phase II (always zero). Because phase II transformed to spinel phase at 25 GPa, the relative enthalpies are shown up to 24 GPa. At ambient pressure, phase I (willemite) has the lowest enthalpy, consistent with the experimental observation that phase I is the stable ambient pressure phase. Although phase III is highly compressible, it never becomes lowest enthalpy at any pressures. Situation is similar for phase IV. Thus, present calculations confirmed again that phases III and IV are metastable phases. At ambient pressure, phases III has second lowest enthalpy after phase I, and IV has fourth lowest enthalpy. This would explain why those phases appear as the retrograde phases during decompression.

At 3 GPa, phase II becomes the lowest enthalpy phase. Experimental phase boundary between phases I and II is about 3.5 GPa at 1000 ∘C Syono et al. (1971), and is consistent with present calculation. At 9 GPa, the enthalpy of phase V becomes lowest (see inset of Figure 6). Although lower pressure stability field of phase V is not well constrained experimentally, phase V has been synthesized at pressure as low as 11 GPa Ito and Matsui (1974). This again is roughly consistent with the present calculation. Above 12 GPa, spinel, Ca2RuO4, and III-$\it{Pbam}$ phases have the enthalpies lower than that of phase V. However, phase V is experimentally known to decompose to ZnSiO3 ilmenite plus ZnO at about 13 GPa (e.g., Ito and Matsui (1974)), so spinel phase and other dense phases (III-$\it{Pbam}$ and Ca2RuO4) cannot be appeared as a stable phase in this system.

Although the DFT calculation reproduced the observed phase relation of I, II and V well, there are missing phases between phases II and V: the high-pressure phases of III and IV (i.e., III-HP and IV-HP). As noted before, phases III and IV are “experimentally” observed at a pressure range between 8 to 11 GPa in the recovered samples Syono et al. (1971). In-situ Raman spectroscopic study Kanzaki (2018b) has demonstrated that phases III and IV transformed to the structure-unknown high-pressure phases (III-HP and IV-HP) at 5.5 and 2.5 GPa during compression, respectively. Therefore, there must be two stable phases corresponding those phases between phases II and V. However, no lowest enthalpy phases other than phase V were detected at pressure corresponding to synthesis of phases III and IV (8 to 11 GPa). It should be noted that our calculations are based on the enthalpies at 0 K without considering vibrational entropy contribution. Therefore, we need to consider not only lowest enthalpy phase, but also those phases which have enthalpies next to lowest one, considering the entropy contribution might stabilize those phases. At 7–10 GPa, the enthalpies of selected phases are closely compared (inset of Figure 6). This figure shows the enthalpies relative to phase V, and phase V becomes stable directly from phase II. However, the figure also revealed that the enthalpy of Ag2CrO4 phase is close to the enthalpy of phase V, and Na2CrO4 phase has slightly higher enthalpy, next to Ag2CrO4 phase. Olivine phase has the enthalpy always higher than those of Na2CrO4 and Ag2CrO4 phases. The compression curves (Figure 1) showed same order: olivine $>$ Na2CrO4 $>$ Ag2CrO4 $>$ phase V. Considering these, we suggest following assignments: phase III-HP would have Na2CrO4 structure, whereas phase IV-HP would take Ag2CrO4 structure. Crystal structures of phase IV and Ag2CrO4 phase are shown in Figure 5.

Although this suggestion seems reasonable, there is one inconsistency still remains. Present suggestion implies that phase IV-HP should have a Ag2CrO4 structure. However, as noted before, phase IV transformed to Na2CrO4 structure, not Ag2CrO4 structure, during optimization above 10 GPa. This seems contradict with our interpretation. However, it should be noted that stable phase is ultimately defined to have lowest free energy, not by structural relationship. Similarly, structural similarity between olivine and phase III would suggest that olivine structure as structure candidate for III-HP, however, this is less likely from point of the enthalpies and volumes. In order to confirm present interpretation, the quasi-harmonic approximation (QHA) calculation which accounts the vibrational entropy contribution is necessary. Also, experimental study such as in-situ X-ray diffraction study under pressure will ultimately solve the issues.

III.0.4 Phase II to spinel transition and its relationship with nitrides

The transformation from phase II to spinel structure was observed during the DFT calculation at 24 GPa. Literature survey reveals that structural relation of phase II-like structure and spinel is well known for nitrides Kroll (2003). First-principles study revealed that C3N4 could adapt phase II-like structure with low compressibility Teter and Hemley (1996). They showed that when Zn and Si are replaced with C, and O with N in phase II, the structure slightly relaxes, resulting a cubic phase (designated as willemite II, or “wII” in the literature). Later, Zerr et al. Zerr et al. (1999) and Kroll Kroll (2003) calculated phase wII of Si3N4 as well. Zerr et al. Zerr et al. (1999) experimentally synthesized spinel phase of Si3N4 above 15 GPa using a laser-heated diamond anvil cell high-pressure device. However, phase wII has not been experimentally synthesized to date. Kroll Kroll (2003) calculated relative enthalpies of wII and spinel phases for Si3N4, Ge3N4 and C3N4, and proposed a diffusionless transition mechanism from spinel to phase wII, and also calculated activation energy for the transition. Kroll Kroll (2003) explained this transition mechanism based on the Bain correspondence, which connects the crystal lattice of face-centered cubic (fcc) and body-centered cubic (bcc) by a homogeneous strain. The anion (oxygen or nitrogen) arrangement in phase II (and wII) is nearly bcc Marumo and Syono (1971), whereas that of spinel phase is fcc Sasaki et al. (1982). Therefore, the Bain correspondence relates anion arrangements of two phases, and provides the transition pathway from spinel to phase wII Kroll (2003).

Accordingly, our observed “transition” from phase II to spinel during structural optimization is following this proposed transition route backward. It is interesting to note that this transformation brought Zn from a tetrahedral site to an octahedral site, while Si remains in the tetrahedral site, resulting the normal spinel structure, not inverse spinel structure. For the nitrides, this distinction is not relevant. Spinel phase of Zn2SiO4 has not been experimentally synthesized to date, since at pressures higher than stability field of phase V, it breaks down to ZnSiO3 ilmenite plus ZnO (NaCl structure) Ito and Matsui (1974). The situation is just opposite to Si3N4, in which spinel phase can be synthesized Zerr et al. (1999), but phase wII cannot be synthesized thus far. Kroll Kroll (2003) proposed a metastable route to obtain phase wII starting from spinel phase. Similarly, we propose that spinel phase of Zn2SiO4 can metastably produced by compressing phase II at low temperature. The transition from phase II to spinel will be expected at around 9 GPa based on the enthalpy crossover between two phases as shown in Figure 6. Considering close structural relationship between Si3N4 and Zn2SiO4, crystal chemical insights obtained by present study might be useful to further explore stable and metastable phases of Si3N4, Ge3N4 and C3N4.

IV Conclusions

Phase relation of Zn2SiO4 has been studies using the DFT calculations up to 25 GPa. Total 11 phases were considered. Phase III with tetrahedral olivine structure exhibited extremely high compressibility, which is due to reduction of volumes of vacant octahedral sites. We confirmed that phases III and IV recovered from high-pressure synthetic experiments are retrograde phases. From the calculated enthalpies and volumes, we suggested that the high-pressure phase of III would have a Na2CrO4 structure, whereas that of phase IV would have a Ag2CrO4 structure. Therefore, the pressure-induced phase transition sequence of Zn2SiO4 will be: phase I (willemite)$\rightarrow$phase II$\rightarrow$Na2CrO4 phase$\rightarrow$Ag2CrO4$\rightarrow$phase V(modified spinel structure)$\rightarrow$ZnSiO3-ilmenite plus ZnO (NaCl structure). This sequence is quite different to those of other M2SiO4 systems, and only common structure is modified spinel (that realized in Mg2SiO4 and Co2SiO4 only). This peculiar behavior is likely due to closed 3$\it{d}$ orbital inherent to Zn. A transition from phase II to spinel structure was observed during structural optimization at 25 GPa, and the transition is essentially identical to known transition for nitrides. Metastable route to synthesize spinel phase was suggested.

Acknowledgements.

This study was supported by the regular operational grant from Okayama University.