Mechanisms and Conditions of Ti and Cr Incorporation in Mantle Phlogopite: The Results of Atomistic Simulation

doi:10.21203/rs.3.rs-1990265/v1

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Mechanisms and Conditions of Ti and Cr Incorporation in Mantle Phlogopite: The Results of Atomistic Simulation

https://doi.org/10.21203/rs.3.rs-1990265/v1

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published 10 Feb, 2023

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Modeling of eight mechanisms for the incorporation of Ti⁴⁺ and Cr³⁺ impurity components into phlogopite was carried out by a semi-empirical method using the GULP (General Utility Lattice Program) software. The calculation of thermodynamic mixing properties in the range of 1–7 GPa and 373–1573 K and the analysis of the structure geometry for the simulated solid solutions provided the following energy-preferred schemes of isomorphic substitution: ^VI(Mg²⁺) + 2^IV(Si⁴⁺) = ^VI(Ti⁴⁺) + 2^IV(Al³⁺) and ^VI(Mg²⁺) + 2^IV(Al³⁺) = ^VI(□)+ 2^IV(Ti⁴⁺), ^VI(Mg²⁺) + ^IV(Si⁴⁺) = ^VI(Cr³⁺) + ^IV(Al³⁺), and 3^VI(Mg²⁺) = ^VI(Al³⁺) + ^VI(Cr³⁺) + ^VI(□). It is shown the scheme 2^VI(Mg²⁺) = ^VI(Ti⁴⁺) + ^VI(□) illustrating entrance of Ti with the formation of a vacancy is realized in the case of microconcentrations of Ti only. Accumulation of high Ti contents associates with the formation of a vacancy in the octahedral site. This provides incorporation of Ti via the schemes ^VI(Mg²⁺) + 2^IV(Al³⁺) = ^VI(□)+ 2^IV(Ti⁴⁺) and (Mg,Fe²⁺) + 2OH^- = Ti⁴⁺ + 2O^2- only. It is shown that incorporation of high Cr concentrations (>5.5 wt % Cr₂O₃) is accompanied by an increase in the number of vacancies in the octahedral site with an increase in the proportion of the dioctahedral component K(Al,Cr,□)AlSi₃O₁₀(OH)₂.

Ti-bearing phlogopite

Cr-bearing phlogopite

isomorphic substitution

atomistic simulation

Earth’s mantle

Phlogopite is a trioctahedral mica with a structure composed of three-layer packages of two tetrahedral layers [(AlSi₃O₁₀)⁵⁻] interconnected by cations (Fe,Mg)²⁺ in the octahedral sites. The three-layer packages are bound by singly charged cations (K⁺, Na⁺) with a coordination number of 12.

Phlogopite is a common mineral in alkaline basalt, ultrapotassic rocks, kimberlite, lamproite, lamprophyre, granulite, carbonatite (Brod et al. 2001), and rocks of ophiolite complexes (Peng et alю 1995). In addition, phlogopite is mica of metasomatized mantle peridotite. At the same time, phlogopite inclusions in diamonds are rarely considered as an indicator of assemblages, since such finds are extremely rare being often interpreted as epigenetic (Meyer 1987; Harris 1992). Nevertheless, classification of diamonds into eclogitic and ultramafic (dunite–harzburgite, wehrlite, and lherzolite) associations should account for minor elements rather than major components in the composition of minerals only. As was shown in (Erlank et al. 1987; Sobolev et al. 1997; 1998; 2009) the content of such elements as Ti and Cr in phlogopite syngenetic with diamond may indicate that the mineral belongs to the eclogitic or peridotitic assemblage.

The incorporation of Ti⁴⁺ and Cr³⁺in phlogopite and some other trioctahedral micas proceeds via various substitution mechanisms involving both tetrahedral and octahedral sites (Table 1). The most common substitution mechanisms (1–4) (Dymek 1983; Henry et al. 2005) imply the presence of Al in the tetrahedral site of phlogopite, which is characteristic of igneous rocks including high-pressure ones (Arima, Edgar, 1981). The schemes (5), (6), and (7), which are a combination of schemes (1–4) and two major mechanisms that provide incorporation of Al into the octahedral site, ^VI(R²⁺) + ^IV(Si⁴⁺) = ^VI(Al³⁺) + ^IV(Al³⁺) and 3^VI(R²⁺) = 2^VI(Al³⁺) + ^VI(□), are discussed as well and considered as basic ones for Mg–Fe micas from high-temperature metapelites (Thu et al. 2016). The scheme (8) of subordinate deprotonation substitution (Ti-oxy substitution) is considered to be one of the main mechanisms for the incorporation of Ti into phlogopite at high temperatures of the granulite facies (Dymek et al. 1983; Henry et al. 2005), which has recently received special attention (e.g. (Ventruti et al. 2020) and references therein).

Table 1

Mechanisms of Ti⁴⁺ and Cr³⁺ incorporation into phlogopite and corresponding end-members
	Substitution mechanism	End-member	References
Incorporation of Ti⁴⁺ into phlogopite
1	^VI(Mg²⁺) + 2^IV(Si⁴⁺) = ^VI(Ti⁴⁺) + 2^IV(Al³⁺)	K(Mg₂Ti)(Al₃Si)O₁₀(OH)₂	(Czamanske and Wones 1973; Robert 1976a,b; Guidotti et al. 1977; Tracy and Robinson 1978)
2	2^VI(Mg²⁺) = ^VI(Ti⁴⁺) + ^VI(□)	K(Mg,Ti,□)AlSi₃O₁₀(OH)₂	(Dahl 1969; Forbes and Flower 1974; Dymek and Albee 1977)
3	^VI(Mg²⁺) + 2^IV(Al³⁺) = ^VI(□) + 2^IV(Ti⁴⁺)	K₂(Mg₅,□)Ti₂Si₆O₂₀(OH)₄	(Zhang et al. 1993)
4	^IV(Si⁴⁺) = ^IV(Ti⁴⁺)	KMg₃AlTi₃O₁₀(OH)₂	(Kunitz 1936)
5	2^VI(Al³⁺) + ^VI(Mg²⁺) = 2^VI(Ti⁴⁺) + ^VI(□)	K₂□Mg₃Ti₂Si₄Al₄O₂₀(OH)₄	(Ikeda 1990)
6	^VI(Al³⁺) + ^IV(Si⁴⁺) = ^VI(Ti⁴⁺) + ^IV(Al³⁺)	KMg₂TiAl₃SiO₁₀(OH)₂	(Dymek 1983 and references therein)
7	2^VI(Al³⁺) = ^VI(Ti⁴⁺) + ^VI(Mg²⁺)	K(Mg₅,Ti)Al₄Si₄O₂₀(OH)₄
8	^VI(Mg²⁺) + 2(OH)⁻ = ^VI(Ti⁴⁺) + 2(O²⁻) + H₂	KMg₂TiSi₃AlO₁₂
Incorporation of Cr³⁺ into phlogopite
9	^VI(Mg²⁺) + ^IV(Si⁴⁺) = ^VI(Cr³⁺) + ^IV(Al³⁺)	K(Mg₂,Cr)Al₂Si₂O₁₀(OH)₂	(Ferenc et al. 2016)
10	3^VI(Mg²⁺) = 2^VI(Cr³⁺) + ^VI(□)	K(Cr₂,□)AlSi₃O₁₀(OH)₂	(Ferenc et al. 2016; Guiliani et al. 2016)
11	3^VI(Mg²⁺) = ^VI(Al³⁺) + ^VI(Cr³⁺) + ^VI(□)	K(Al,Cr,□)AlSi₃O₁₀(OH)₂
12	^IV(Al³⁺) = ^IV(Cr³⁺)	KMg₃CrSi₃O₁₀(OH)₂

In this paper, we discuss the schemes (1–4) only to illustrate incorporation of Ti⁴⁺ into phlogopite (Table 1), since we focus on high-pressure Ti-bearing mantle phlogopite, which is not characterized by the high concentrations of Al in the octahedral site (Arima and Edgar 1981).

The mechanisms of Cr³⁺ incorporation into phlogopite are poorly discussed in literature. Basically, only the schemes (9) and (10) are considered (e.g., (Ferenc et al. 2016) for high-chromium triochahedral micas (up to 6.54 wt % Cr₂O₃) from Muranska Zdychava listvenite). Cr-bearing phlogopites are reported as inclusions in diamonds as well (up to 3.5 wt % Cr₂O₃) (Sobolev et al. 2009) and in other rocks of different geodynamic environments, in which the concentration of Cr₂O₃ impurity reaches 2.3 wt % Cr₂O₃ (Delaney 1980; Shee 1985; Lorand and Cottin 1987; Dawson 2002; Bosi et al. 2012; Guiliani et al. 2016). Thus, here we study the schemes (10) and (11) and simulate two new end-members, using the same approach to Cr³⁺ ion incorporation as for Ti end-members. Note that the scheme (11) suggests accumulation of Al³⁺ in the octahedral site in association with Cr³⁺ entrance. These schemes are acceptable from the point of the local charge balance and correlate with the Ti incorporation schemes; however, additional factors, such as the preference energy for the octahedral coordination (McClure 1957, may impose extra restrictions on such mechanisms.

It is important that the choice of one or another scheme dominating for phlogopite is related to the coexisting mineral assemblage that controls the composition of the studied phase. Due to the wide variations in natural and experimental mineral associations, and, consequently, in the composition of phlogopite in terms of the content of both major and minor components, the data on preferable mechanisms of Ti and Cr incorporation into phlogopite remain ambiguous.

The inconsistency of experimental information, which makes possible to give preference to one or another mechanism for Ti and Cr incorporation into phlogopite, has led to necessity of theoretical models for determination of the most preferable defect formation scheme. The aims of this study are (1) to identify the most realistic schemes for the incorporation of impurity components into phlogopite and (2) to estimate the maximum concentrations of Ti and Cr in the mineral by atomistic computer simulation under various thermodynamic conditions of the Earth's upper mantle.

The mechanisms of minor element incorporation into the phlogopite were simulated using the GULP (General Utility Lattice Program) software (Gale and Rohl 2003), which is based on the method of minimizing the structural energy of a crystal using semi-empirical interatomic potentials.

A set of interatomic interaction potentials (Pedone et al. 2006) was used as a starting point to develop a force-field model for phlogopite, which has proven itself in molecular dynamics modeling of various silicates and silicate glasses with a fixed bond iconicity of f = 0.6 for all interatomic interactions. In addition, the Morse-type O–H potential (Saul et al. 1985) combined with the Buckingham-type potential (Steele et al. 2000) was applied for oxygen atoms in hydroxyl groups. The parameters of interatomic interaction potentials are given in Table 2. Crystallographic data of Redhammer and Roth (2002) was used for the model structure of phlogopite.

Table 2

Parameters of pair potentials of interatomic interaction used in this study
Type Morse	D_ij, eV	a_ij, Å⁻²	r₀, Å
K^0.6-O^− 1.2	0.011612	2.062605	3.305308
Mg^1.2-O^− 1.2	0.038908	2.281000	2.586153
Ti^2.4-O^− 1.2	0.024235	2.254703	2.708943
Cr^1.8-O^− 1.2	0.399561	1.785079	2.340810
Al^1.8-O^− 1.2	0.361581	1.900442	2.164818
Si^2.4-O^− 1.2	0.340554	2.006700	2.100000
O^− 1.2-O^− 1.2	0.042395	1.379316	3.618701
H^0.6-O^OH−1.2	7.052500	2.198600	0.948500
Type Buckingham	A, eV	p, Å	C, eV Å⁶
H^0.6-O^− 1.2	312.0000	0.250000	0.000000

Modeling of structural defects in phlogopite was carried out in two different ways, which allowed us to control the reproducibility and correctness of the results. The first approach was to use the standard procedure for calculation of point defects by the Mott–Littleton two region strategy (Mott and Littleton 1938) implemented in the GULP software. Such approach suggests that a point defect in the crystal structure D and the immediate region around the defect defined by a sphere of radius r, participate in the procedure of minimizing the energy of interatomic interaction with the symmetry restrictions completely removed. A screening layer defined by the radius r₂ is introduced between the defect area and the periodic crystal. For a correct calculation of a point defect, it is usually sufficient to use the r₁ value of ~ 6–7 Å (150–300 atoms) and the thickness of the screening layer (r₂–r₁) of 10 Å (1500–2000 atoms). However, the need to calculate the associates of point defects given in Table 1 required an increase in the dimension of these parameters to 10 and 20 Å, respectively. As is shown by our test simulations, the use of spheres of a larger radius turned out to be inexpedient, since in this case the calculation time only increased significantly, with an extremely slight change in the simulation accuracy.

The second way to study of the defect energy in the crystal structure was to simulate the associate in the central regions of supercells of different dimensions of the model structure of phlogopite with the non-translational symmetry removed (the calculation was carried out within the space group P1) directly. Two types of supercells were used: 4×2×2 (704 atoms) and 6×3×3 (2376 atoms) (Fig. 1).

The calculations of defect complexes upon simulation using supercells were carried out for several topologically different configurations of defects separated from each other over the space of the supercell by a certain distance. The choice of topological configurations was based on the selection of atoms with such coordinates that the distance between them was minimal (from 3.2 to 5.5 Å).

The results of the simulation of the crystal structure of phlogopite in comparison with the experimental values from (Redhammer and Roth 2002) are shown in supplementary information (Table S1). As is evident from the table, the unit-cell volume is reproduced quite accurately, and the deviations of the cell parameters are characterized by an error of ≤ 2.7%. This allowed us to use the selected set of potentials for a comparative analysis of the energies of defect formation in the crystal structure according to the isomorphic schemes described above.

Simulation of Ti- and Cr-phlogopite end-members

The hypothetical Ti and Cr end-members isostructural to phlogopite were simulated using the model of interatomic potentials in order to estimate thermodynamic properties of solid solution mixing and isomorphic capacity of phlogopite for Ti⁴⁺ and Cr³⁺.

The energies of the structures and those of the formation of point defects were calculated without a pressure and temperature applied to the structure for the considered end-members of solid solutions. The results of simulations via different methods for each considered isomorphic scheme are reported in Table 3. The energy of defect calculated for each mechanism in the selected supercells is an averaged value in the calculation for nonequivalent atomic configurations. The associate energy was normalized to 1 point defect.

Table 3

Energies of structures and formation of point defects of impurity incorporation into the phlogopite structure calculated according to the considered mechanisms by two different methods
Defect formation mechanism	Calculation/ Method	Defect energy, eV	Number of configurations	Defect energy, eV	Number of configurations
(Mg²⁺)^VI + 2(Si⁴⁺)^IV = (Ti⁴⁺)^VI + 2(Al³⁺)^IV		KMg₃AlSi₃O₁₀(OH)₂		K(Mg₂Ti)(Al₃Si)O₁₀(OH)₂
	Energy, eV	-285.3		-275,35
	Supercell 4×2×2	3.364	9	-3.211	9
	Supercell 6×3×3	3.406	9	-3.253	9
	Two-region strategy	3.417	18	-3.261	18
2(Mg²⁺)^VI = (Ti⁴⁺)^VI + (□)^VI		KMg₃AlSi₃O₁₀(OH)₂		K(MgTi□)AlSi₃O₁₀(OH)₂
	Energy, eV	-285.3		-308,16
	Supercell 4×2×2	-11.220	13	11.822	12
	Supercell 6×3×3	-11.210	7	11.833	8
	Two-region strategy	-11.056	20	11.665	20
(Mg²⁺)^VI + 2(Al³⁺)^IV = (□)^VI + 2(Ti⁴⁺)^IV		KMg₃AlSi₃O₁₀(OH)₂		K₂(Mg₅,□)Ti₂Si₆O₂₀(OH)₄
	Energy, eV	-285.3		-596.3
	Supercell 4×2×2	-4.343	5	4.165	4
	Supercell 6×3×3	-4.309	3	4.483	4
	Two-region strategy	-3.987	8	4.517	8
(Si⁴⁺)^IV = (Ti⁴⁺)^IV		KMg₃AlSi₃O₁₀(OH)₂		KMg₃AlTi₃O₁₀(OH)₂
	Energy, eV	-285.3		-263,97
	Supercell 4×2×2	7.269	7	-7.037	7
	Supercell 6×3×3	7.219	7	-6.985	7
	Two-region strategy	7.233	14	-7.006	14
(Mg²⁺)^VI + (Si⁴⁺)^IV = (Cr³⁺)^VI + (Al³⁺)^IV		KMg₃AlSi₃O₁₀(OH)₂		K(Mg₂,Cr)Al₂Si₂O₁₀(OH)₂
	Energy, eV	-285.3		-282.27
	Supercell 4×2×2	1.632	11	-1.423	10
	Supercell 6×3×3	1.574	5	-1.481	6
	Two-region strategy	1.593	16	-1.444	16
3(Mg²⁺)^VI = 2(Cr³⁺)^VI + (□)^VI		KMg₃AlSi₃O₁₀(OH)₂		K(Cr₂,□)AlSi₃O₁₀(OH)₂
	Energy, eV	-285.3		-302.80
	Supercell 4×2×2	-5.684	10	6.018	10
	Supercell 6×3×3	-5.742	5	5.981	5
	Two-region strategy	-5.723	15	5.963	15
3(Mg²⁺)^VI = (Al³⁺)^VI + (Cr³⁺)^VI + (□)^VI		KMg₃AlSi₃O₁₀(OH)₂		K(Al,Cr,□)AlSi₃O₁₀(OH)₂
	Energy, eV	-285.3		-303.48
	Supercell 4×2×2	-6.009	10	6.094	10
	Supercell 6×3×3	-6.059	5	6.075	5
	Two-region strategy	-6.023	15	6.127	15
(Al³⁺)^IV = (Cr³⁺)^IV		KMg₃AlSi₃O₁₀(OH)₂		KMg₃CrSi₃O₁₀(OH)₂
	Energy, eV	-285.3		-282.83
	Supercell 4×2×2	2.443	10	-2.388	10
	Supercell 6×3×3	2.429	5	-2.403	5
	Two-region strategy	2.479	15	-2.354	15

As is evident, the data obtained in the 4×2×2 supercell correlate with the results of simulation in the 6×3×3 cell and calculations via the Mott–Littleton model. The difference between the defect energies for all simulated end-members obtained by different methods does not exceed 0.4 eV. Such consistence allows us to consider the results obtained as quite reliable. Note that the defect energy is negative and minimal for the vacancy schemes of the isomorphic incorporation of minor elements into phlogopite. Thus, the most energetically preferable schemes are the vacancy ones upon the formation of one point defect when an impurity enters the phlogopite structure.

Mixing properties of Ti- and Cr-phlogopite end-members

The selected set of potentials reproduces correctly both the properties of the end-members of solid solutions and the structures of phlogopites with one point defect. The results obtained allowed us to apply a consistent set of potentials for simulation of a series of solid solutions in order to identify the most energetically preferable isomorphic scheme under the condition of various P-T parameters, as well as to estimate the isomorphic capacity of phlogopite by the content of impurity Cr³⁺ and Ti⁴⁺.

The mixing entropy ∆S_mix was calculated in the pressure range of 1–7 GPa and temperatures of 373–1573 K for each series of solid solutions via the formula:

∆S_mix = S_conf +∆S_vib,

where S_conf is the configuration and ∆S_vib is the vibrational contributions calculated by the formulae:

S _conf = kN[x lnx + (1-x)ln(1-x)],

where k is the Boltzmann's constant, N is the Avogadro's number.

∆S_vib = S_ss – xS_str^(Phl) – (1-x)S_str^(Imp−Phl),

where S_ss and S_str are the vibrational entropies of the given solid solution and pure components: Phl, phlogopite; Imp-Phl impurity (Ti or Cr) end-member.

The simulations showed that S_conf often provides the main contribution to the mixing entropy. The solid solutions formed via the mechanisms 2^VI(Mg²⁺) = ^VI(Ti⁴⁺) + ^VI(□) and ^VI(Mg²⁺) + ^IV(Si⁴⁺) = ^VI(Cr³⁺) + ^IV(Al³⁺), for which, based on the calculations, the vibrational contribution exceeds the configuration one, are exceptions.

The results of calculations of the mixing enthalpy ∆H_mix for the considered isomorphic substitution mechanisms are shown in Figs. 2. The calculation of the enthalpy of mixing ∆H_mix was carried out using the equation:

∆H_mix = x₁x₂(x₁Q₁ + x₂Q₂),

where x₁ and x₂ are the mole fractions of solid solution components; Q₁ and Q₂ are the interaction parameters defined as follows:

Q ₁ = E_defect^{(Imp in Phl)} + E_str^(Phl) – E_str^(Imp−Phl)

Q ₂ = E_defect^{(Phl in Imp−Phl)} + E_str^(Imp−Phl) – E_str^(Phl)

where E_defect is the isolated defect energy; E_str is the structure energy of corresponding solid solution end-member per formula. Calculations within the limits of infinite dilution were carried out in the approximation of a linear dependence of the interaction parameter Q on the composition of isomorphic mixtures (see supplementary information Table S2).

The results obtained allowed us to evaluate the change in the Gibbs free energy (∆G_mix= ∆H_mix – T∆S_mix) depending on the compositions of the considered solid solutions in a given P-T range. The coexisting compositions of solid solutions were obtained at a minimum of ∆G_mix by searching for the minimum of the polynomial function for each series, and the solvus curves were plotted for some of them (Fig. 3).

Let us consider the conditions for the decomposition of each of the considered solid solutions.

Phlogopite–Ti-phlogopite solid solution

The highest isomorphic capacity in the series of the studied solid solutions of Ti-bearing phlogopite was gained via the mechanism ^VI(Mg²⁺) + 2^IV(Si⁴⁺) = ^VI(Ti⁴⁺) + 2^IV(Al³⁺) (Fig. 3a). The solvus of such solid solution is asymmetric (see supplementary information Table S3) and its maximum is shifted towards Ti-phlogopite, which is consistent with the polarity rule of isomorphic substitutions (an ion with a high charge is more preferably incorporated into mineral than an ion with a lower charge occupying the same crystallographic site upon heterovalent isomorphism). The vacancy mechanism ^VI(Mg²⁺) + 2^IV(Al³⁺) = ^VI(□) + 2^IV(Ti⁴⁺) suggests limited miscibility (Fig. 3b), and the maximum isomorphic capacity of phlogopite detected at 7 GPa is almost identical to the capacity of its Ti end member (22 mol % TiPhl). The isomorphic capacity in the KMg₃AlSi₃O₁₀(OH)₂–KMg₃AlTi₃O₁₀(OH)₂ solid solution is significantly lower (Fig. 3c). The entry of Ti⁴⁺ ions into phlogopite is extremely limited (microconcentrations) up to a temperature of 1373 K. The solvus curve is characterized by a slight asymmetry being shifted towards the Ti end-member, which is consistent with the polarity rule for isomorphism. The G-x sections demonstrate complete immiscibility over the entire range of the studied P-T parameters for a solid solution simulated via the vacancy mechanism 2^VI(Mg²⁺) = ^VI(Ti⁴⁺) + ^VI(□); therefore, this mechanism may provide minor concentrations of impurities only.

Phlogopite –Cr-phlogopite solid solution

Analysis of the G-x sections for a series of solid solutions with KMg₃CrSi₃O₁₀(OH)₂ and K(Al,Cr,□)AlSi₃O₁₀(OH)₂ end-members showed the complete miscibility even at room conditions. At the same time, it should be noted that a substitution like ^IV(Al³⁺) = ^IV(Cr³⁺) is most likely hypothetical and obviously unfavorable from the point of the crystal field theory (Cr³⁺ ions are preferentially incorporated into the octahedral rather than tetrahedral sites) (McClure, 1957). The crystal field stabilization energy (CFSE) is − 1.2∆_o in the octahedral site and − 0.8∆_t in the tetrahedral site for Cr³⁺ ions; since ∆_t = 4/9∆_o, the CFSE in the tetrahedron is 0.35∆_o. Hence, the energy of octahedral site preference in the crystal field for the Cr³⁺ is ∆E_oct = 0.85. Thus, energy of the octahedral site preference by Cr ions provides an additional, and rather significant, energy effect. Nevertheless, the values of the mixing enthalpy and entropy obtained for simulated solid solutions of phlogopite with K(Mg₂,Cr)Al₂Si₂O₁₀(OH)₂ and K(Cr₂,□)AlSi₃O₁₀(OH)₂ end-members allowed us to define the limited miscibility when Cr³⁺ ions enter the octahedron. Isomorphic substitution via the scheme ^VI(Mg²⁺) + ^IV(Si⁴⁺) = ^VI(Cr³⁺) + ^IV(Al³⁺) suggests limited miscibility over the entire tested temperature range (Fig. 3d). An increase in pressure leads to an expansion of the stability fields of both solid solutions, and this effect increases with temperature. The critical temperature is estimated as 1075 ± 17 K for the vacancy mechanism (Fig. 3e) in the pressure range of 3–5 GPa, (see supplementary information Table S3); however, at a pressure of 1 GPa, the critical temperature is much higher, 1370K. Note that the solvus curve is relatively symmetrical and its maximum is close to the composition of 50 mol % K(Cr₂,□)AlSi₃O₁₀(OH)₂ (x(Cr) = 0.46–0.50 in the selected pressure range).

The depression isomorphism rule characteristic of the simplest isovalent substitutions most often does not work for the studied solid solutions, which are characterized by limited solubility. This shows that the isomorphic capacity at various pressures in each case is controlled by a whole complex of parameters, such as the compressibility of various polyhedra, volume of vacancy regions, and their mutual arrangement.

Change of the structure geometry as a function of impurity concentration

Analysis of the local structure of the simulated solid solutions was carried out with account for changes in both individual and average interatomic distances and volumes of coordination polyhedra within the framework of the phenomenological theory (Urusov, 1992).

The geometry of isolated impurity polyhedra of titanium [TiO₄], [TiO₆] and chromium [CrO₄], [CrO₆] in phlogopite for each of the considered mechanisms for the incorporation of impurities is shown in supplementary information (Fig. S4).

Figure S5 (see supplementary information) shows the change in the structure geometry depending on the composition of the solid solution upon incorporation of the considered impurities via the competing substitution mechanisms. Deviations from the additivity of structural parameters depending on the composition of the considered solid solutions are reported in supplementary information (see supplementary information Table S6).

Note that the dependences of structural parameters and cell volume on the composition of solid solution are close to linear upon Ti incorporation into phlogopite via to the non-vacancy substitution mechanisms ^IV(Si⁴⁺) = ^IV(Ti⁴⁺) and ^VI(Mg²⁺) + 2^IV(Si⁴⁺) = ^VI(Ti⁴⁺) + 2^IV(Al³⁺) (see supplementary information Fig. S5a). There are slight deviations of these mechanisms from additivity, which do not exceed 0.09 Å and 3.678 Å³ for bond length and cell volume, respectively (see supplementary information Table S6). At the same time, change in the structural parameters for vacancy mechanisms has a pronounced non-linear character. The mechanism ^VI(Mg²⁺) + 2^IV(Al³⁺) = ^VI(□) + 2^IV(Ti⁴⁺) is characterized by a negative deviation of a, c, and cell volume V, and a positive deviation of b from additivity. Such regularities are rather rare; however, according to (Urusov 1977), they may be one of the reasons for the observed increase in isomorphic capacity with increasing pressure.

The Vegard’s and Retger’s additivity rules are systematically violated upon the substitution like 2^VI(Mg²⁺) = ^VI(Ti⁴⁺) + ^VI(□) (see supplementary information Table S6). There is a regular increase in the unit-cell volume V and c parameter up to 40 mol % of Ti component. A further increase in the Ti component ceases to have a noticeable effect on the value of these parameters. This pattern may explain the detected immiscibility in the K(Mg,Ti,□)AlSi₃O₁₀(OH)₂ – KMg₃AlSi₃O₁₀(OH)₂ series described in the previous section.

We also note that vacancy mechanisms are characterized by a decrease in the structure density (see supplementary information Fig. S5). At the same time, the opposite effect is observed for non-vacancy mechanisms. According to the mechanism ^VI(Mg²⁺) + 2^IV(Si⁴⁺) = ^VI(Ti⁴⁺) + 2^IV(Al³⁺), the structure density increases with increasing concentration of the Ti component of the solid solution, which is naturally explained by an increase in the unit-cell parameters. However, the density in the KMg₃AlTi₃O₁₀(OH)₂–KMg₃AlSi₃O₁₀(OH)₂ series is constant, despite increase in the unit-cell parameters, which is compensated by the difference in interatomic distances: Ti-O bond length is larger than Si-O bond length.

Cr incorporation into phlogopite shows no obvious deviations from the additivity rules, which do not exceed 0.11 Å and 28.13 Å³ for bond length and unit-cell volume, respectively (see supplementary information Table S6). A linear change in the structure density of Cr phlogopites is recorded for all mechanisms, with the greatest increase for the vacancy mechanisms (see supplementary information Fig. S5b).

Let us consider the patterns of the local structure of a solid solution in terms of changes in interatomic distances, volumes of coordination polyhedra, compliances of cation sites (Dollase 1980), and degree of their relaxation λ (Vegard 1928; Urusov 1992) depending on the impurity content. The values of λ for most polyhedra are close to the bond length alternation model, in which it is equal to 1 (Table 4, supplementary information Fig. S7a,b,e,f, Fig. S8a,b,e,f), which demonstrates the constancy of the lengths of individual bonds in the structures of analyzed solid solutions, regardless of the composition of the solid solution. This is explained by the key influence of the size of the common structural unit of an isomorphic mixture, which follows the isomorphism assistance rule (Urusov 1977).

Table 4

Estimated relaxation parameters λ^R, bond lengths, and polyhedron volumes λ^V
	(Si⁴⁺)^IV = (Ti⁴⁺)^IV	(Mg²⁺)^VI + 2(Si⁴⁺)^IV = (Ti⁴⁺)^VI + 2(Al³⁺)^IV	(Mg²⁺)^VI + 2(Al³⁺)^IV = (□)^VI + 2(Ti⁴⁺)^IV	2(Mg²⁺)^VI = (Ti⁴⁺)^VI + (□)^VI
λ^R (Mg-O)		1.00	1.00	0.89
λ^R (Ti-O)	1.00	1.00	0.88	0.95
λ^R (Si-O)	1.00	0.89
λ^R (Al-O)		0.95	0.70
λ^V (Mg-O)		0.92		0.64
λ^V (Ti-O)	1.00	1.00	0.75	0.94
λ^V (Si-O)	1.00	1.00
λ^V (Al-O)		1.00	0.78
	(Al³⁺)^IV = (Cr³⁺)^IV	(Mg²⁺)^VI + (Si⁴⁺)^IV = (Cr³⁺)^VI + (Al³⁺)^IV	3(Mg²⁺)^VI = (Al³⁺)^VI + (Cr³⁺)^VI + (□)^VI	3(Mg²⁺)^VI = 2(Cr³⁺)^VI + (□)^VI
λ^R (Mg-O)		0.90	0.88	0.95
λ^R (Cr-O)	0.97	0.97	0.90	0.96
λ^R (Si-O)		0.99
λ^R (Al-O)	0.993	1.00	0.96
λ^V (Mg-O)		0.81	0.77	0.76
λ^V (Cr-O)	1.00	0.94	0.82	0.80
λ^V (Si-O)		1.00
λ^V (Al-O)	0.99	1.00

The greatest deviations from the bond length alternation model are typical (see supplementary information Fig. S7c,g, Fig. S8c,g), when an impurity enters with the formation of a vacancy, and there is a non-linearity in the change in the considered parameters from the content of impurity as well (see supplementary information Fig. S7h, Fig. S8h). For the K(Mg,Ti,□)AlSi₃O₁₀(OH)₂ – KMg₃AlSi₃O₁₀(OH)₂ solid solution, Significant deviations from additivity are fixed for the K(Mg,Ti,□)AlSi₃O₁₀(OH)₂–KMg₃AlSi₃O₁₀(OH)₂ solid solution (see supplementary information Fig. S7d, Fig. S8d), which may be another criterion for the described immiscibility.

Comparison of incorporation of Cr and Ti in phlogopite

Comparison of the patterns of Cr³⁺ and Ti⁴⁺ incorporation into phlogopite by the competing substitution mechanisms shows that the formation of Cr-bearing solid solutions requires less energy consumption. Although the radii of impurity ions in the octahedral site have very close values (rTi⁴⁺ = 0.75Å, rCr³⁺ = 0.76Å), incorporation of Ti⁴⁺ ions distort the structure of the solid solution stronger than Cr³⁺ ions compared to pure phlogopite (see supplementary information Fig. S5). This is due to the greater difference in charges and electronegativity of the Ti⁴⁺–Mg²⁺ pair compared to the Cr³⁺–Mg²⁺ pair. In addition, such distortion results in a stronger change in the structure energy in relation to the energy of pure phlogopite in the case of Ti⁴⁺ incorporation (Table 3).

Comparison of isomorphic substitution schemes

Analysis of thermodynamic mixing properties, as well as structure geometry of the considered solid solutions, allowed us to identify the most energetically preferable schemes. Thus, the most likely scheme of Ti incorporation into phlogopite is ^VI(Mg²⁺) + 2^IV(Si⁴⁺) = ^VI(Ti⁴⁺) + 2^IV(Al³⁺), which stimulates complete miscibility at relatively low temperatures in the entire pressure range of phlogopite stability. The simplest isovalent substitution (Si⁴⁺)^IV = (Ti⁴⁺)^IV is energetically inferior to that described above, despite the smaller deviations of structural parameters from additivity, and exhibits highly limited miscibility over the entire considered P–T range. The vacancy mechanism 2^VI(Mg²⁺) = ^VI(Ti⁴⁺) + ^VI(□) is most preferable in the case of very low Ti concentrations only, which is explained by the minimum energies of defect formation. Moreover, such substitution is unlikely in significant scale, as indicated by the facts that the additivity rules, as well as the Vegard’s and Retger’s rules, do not work, the interaction parameters have extremely high values, and, as a result, the immiscibility is detected. The formation of a vacancy in the octahedral site upon Ti incorporation into the tetrahedron most likely proceeds via the scheme ^VI(Mg²⁺) + 2^IV(Al³⁺) = ^VI(□) + 2^IV(Ti⁴⁺). Such substitution is energetically more favorable in comparison with the scheme 2^VI(Mg²⁺) = ^VI(Ti⁴⁺) + ^VI(□) and allows the accumulation of higher Ti concentration (Table 3).

According to the results of the simulations, the schemes 3^VI(Mg²⁺) = ^VI(Al³⁺) + ^VI(Cr³⁺) + ^VI(□) and ^VI(Mg²⁺) + ^IV(Si⁴⁺) = ^VI(Cr³⁺) + ^IV(Al³⁺) turn out to be the most energetically preferable. The incorporation of chromium into phlogopite is less preferable via the substitution 3^VI(Mg²⁺) = 2^VI(Cr³⁺) + ^VI(□). Despite the absence of anomalies with change in the structure geometry, following the additivity rules, and detection of complete miscibility, this mechanism is the least favorable energetically (Table S2). As was mentioned above, the scheme ^IV(Al³⁺) = ^IV(Cr³⁺) is obviously unfavorable in accordance with the crystal field theory, despite the extremely low values of the mixing enthalpy (Fig. 2).

Some reasoning for the schemes of isomorphic substitution under the natural conditions

The quantitative estimates of the isomorphic capacity of phlogopite in terms of the content of Ti and Cr impurities in comparison with their calculated concentrations in the studied end-members of solid solutions, as well as the highest Ti and Cr concentrations recorded in phlogopites from inclusions in natural diamonds, other natural mineral assemblages, and those synthesized in different experimental systems are reported in Table 5.

Table 5

Concentration of impurities in phlogopites and Ti and Cr end-members
	System	Reference	TiO₂, wt %	System	Reference	Cr₂O₃, wt %
Experiment	Ti-KNCMASH (4 GPa, 1250°C)	(Konzett 1997)	3.7	Chromite + ilmenite with H₂O-CO₂-K₂CO₃ fluids (5 GPa, 1200°C)	(Safonov et al. 2019: Butvina et al. 2019)	2.7
Inclusions in diamonds	Eclogitic association	(Sobolev et al. 2009)	12	Ultramafic association	(Sobolev et al. 2009)	3.23
Natural phlogopites	Mongolian basaltoid rocks	(Koval et al. 1988)	12.5	Phlogopite harzburgite xenolith (Wesselton kimberlite)	(Griffin et al. 1999)	2.26
Natural phlogopites	Mongolian basaltoid rocks	(Koval et al. 1988)	12.5	Muranska Zdychava listvenites	(Ferenc et al. 2016)	6.54
	Scheme	Composition		Scheme	Composition
Atomistic modeling	^IV(Si⁴⁺) = ^IV(Ti⁴⁺)	KMg₃AlTi_0.15Si_2.85O₁₀(OH)₂	2.29	^IV(Al³⁺) = ^IV(Cr³⁺)	-	-
	^IV(Si⁴⁺) = ^IV(Ti⁴⁺)	KMg₃AlTi₃O₁₀(OH)₂	44.7	^IV(Al³⁺) = ^IV(Cr³⁺)	KMg₃CrSi₃O₁₀(OH)₂	17.18
	^VI(Mg²⁺) + 2^IV(Si⁴⁺) = ^VI(Ti⁴⁺) + 2^IV(Al³⁺)	-	-	^VI(Mg²⁺) + ^IV(Si⁴⁺) = ^VI(Cr³⁺) + ^IV(Al³⁺)	K(Mg_2.7,Cr_0.3)Al_2.3Si_1.7O₁₀(OH)₂	5.48
	^VI(Mg²⁺) + 2^IV(Si⁴⁺) = ^VI(Ti⁴⁺) + 2^IV(Al³⁺)	K(Mg₂Ti)(Al₃Si)O₁₀(OH)₂	15.1	^VI(Mg²⁺) + ^IV(Si⁴⁺) = ^VI(Cr³⁺) + ^IV(Al³⁺)	K(Mg₂,Cr)Al₂Si₂O₁₀(OH)₂	17.12
	^VI(Mg²⁺) + 2^IV(Al³⁺) = ^VI(□) + 2^IV(Ti⁴⁺)	K(Mg_2.9,□)Ti_0.3Al_0.7Si₃O₁₀(OH)₂	3.54	3^VI(Mg²⁺) = ^VI(Al³⁺) + ^VI(Cr³⁺) + ^VI(□)	-	-
	^VI(Mg²⁺) + 2^IV(Al³⁺) = ^VI(□) + 2^IV(Ti⁴⁺)	K(Mg_2.5,□)TiSi₃O₁₀(OH)₂	15.58	3^VI(Mg²⁺) = ^VI(Al³⁺) + ^VI(Cr³⁺) + ^VI(□)	K(Al,Cr,□)AlSi₃O₁₀(OH)₂	17.95
	2^VI(Mg²⁺) = ^VI(Ti⁴⁺) + ^VI(□)	-	-	3^VI(Mg²⁺) = 2^VI(Cr³⁺) + ^VI(□)	-	-
	2^VI(Mg²⁺) = ^VI(Ti⁴⁺) + ^VI(□)	K(Mg,Ti,□)AlSi₃O₁₀(OH)₂	15.95	3^VI(Mg²⁺) = 2^VI(Cr³⁺) + ^VI(□)	K(Cr₂,□)AlSi₃O₁₀(OH)₂	33.9

As is evident, the results of atomistic simulation do not contradict the patterns of impurity accumulation detected in natural samples and in run products. The maximum Ti concentrations calculated for the most energetically preferable stoichiometric mechanism ^VI(Mg²⁺) + 2^IV(Si⁴⁺) = ^VI(Ti⁴⁺) + 2^IV(Al³⁺) exceed those in natural assemblages (Table 5). At the same time, the formation of a vacancy in the octahedral site is often reported in the literature. In the absence of X-ray diffraction data, information on the content of Fe³⁺ and OH, as well as unconfirmed Ti-oxy substitution (Mg,Fe²⁺) + 2OH^- = Ti⁴⁺ + 2O^2- (Ventruti et al. 2020; Cruciani and Zanazzi 1994; Schingaro et al. 2005; Mesto et al. 2006), the 2^VI(Mg²⁺) = ^VI(Ti⁴⁺) + ^VI(□) substitution is discussed in many studies (e.g., (Arima and Edgar 1981; Tronnes et al. 1985; Abrecht and Hewitt 1988; Thu et al. 2016)); based on the results of our study, we suggest only minor Ti incorporation via this scheme. In addition, the TiO₂ content of > 7.81 wt % (> 50 mol % K(Mg,Ti,□)AlSi₃O₁₀(OH)₂) should result in the formation of transitional di-/trioctahedral mica varieties, which were not recorded in the literature (Koval et al. 1988; Sobolev et al. 2006).

At the same time, Ti incorporation via the scheme ^VI(Mg²⁺) + 2^IV(Al³⁺) = ^VI(□) + 2^IV(Ti⁴⁺) may result in accumulation of 15.58 wt % TiO₂ in the composition of the K₂(Mg₅,□)Ti₂Si₆O₂₀(OH)₄ component (Table 5) without significant structural changes and with retention of three-layer packages of divalent cations. It is important that such Al substitution for Ti leads to the complete absence of Al in the tetrahedron, which is unlikely under the natural conditions. Thus, comparing the substitutions 2^VI(Mg²⁺) = ^VI(Ti⁴⁺) + ^VI(□) и ^VI(Mg²⁺) + 2^IV(Al³⁺) = ^VI(□) + 2^IV(Ti⁴⁺), we may suggest that the accumulation of Ti will be more energetically favorable in the tetrahedron, while the high Al concentrations may be gained in the octahedral site via one of the schemes: ^VI(R²⁺) + ^IV(Si⁴⁺) = ^VI(Al³⁺) + ^IV(Al³⁺) or 3^VI(R²⁺) = 2^VI(Al³⁺) + ^VI(□). Such multistage substitution can compete with Ti-oxy substitution, and the choice of a certain mechanism will be determined by the conditions of mineral crystallization.

The data of atomistic modeling for Cr incorporation of into phlogopite are consistent with the literature data as well. An exception is the high Cr₂O₃ content (up to 6.54 wt %) reported for trioctahedral Ni-bearing micas from listvenite (Ferenc et al. 2016). The limited miscibility detected for the substitution mechanism ^VI(Mg²⁺) + ^IV(Si⁴⁺) = ^VI(Cr³⁺) + ^IV(Al³⁺) provides Cr₂O₃ accumulation up to 5.5 wt % (Table 5). Ferenc et al. (2016) point to the predominant role of the schemes ^VI(R²⁺) + ^IV(Si⁴⁺) = ^VI(Cr³⁺) + ^IV(Al³⁺) и 3^VI(R²⁺) = 2^VI(Cr³⁺) + ^VI(□), where R²⁺ are divalent cations. At the same time, the described micas contain the high Al₂O₃ concentrations (up to 23 wt %), which cannot be realized, if Cr is incorporated via the scheme 3^VI(R²⁺) = 2^VI(Cr³⁺) + ^VI(□); according to this scheme, at a content of ~ 6.5 wt % Cr₂O₃, the concentration of Al₂O₃ will not exceed 12.1 wt %. Note that only the mechanisms ^VI(R²⁺) + ^IV(Si⁴⁺) = ^VI(Cr³⁺) + ^IV(Al³⁺) и 3^VI(R²⁺) = ^VI(Al³⁺) + ^VI(Cr³⁺) + ^VI(□) provide the high concentrations of Al in mica. The energetically more preferable vacancy mechanism 3^VI(Mg²⁺) = ^VI(Al³⁺) + ^VI(Cr³⁺) + ^VI(□) stimulating the complete miscibility in a wide P–T range seems to be more likely for accumulation of high Cr₂O₃ concentrations (up to 18 wt %) in the composition of dioctahedral component K(Al,Cr,□)AlSi₃O₁₀(OH)₂. Thus, an increase in the Cr content in trioctahedral micas will always stimulate the formation of transitional varieties between di- and trioctahedral micas.

The thermodynamic calculations and structure geometry analysis performed for the end-members of solid solutions and the change in the structure geometry of phlogopite depending on the concentration of impurities allowed us to suggest the most energetically favorable mechanisms for the isomorphic incorporation of Ti⁴⁺ and Cr³⁺ impurities. Ti mainly accumulates via the following schemes: ^VI(Mg²⁺) + 2^IV(Si⁴⁺) = ^VI(Ti⁴⁺) + 2^IV(Al³⁺) и ^VI(Mg²⁺) + 2^IV(Al³⁺) = ^VI(□) + 2^IV(Ti⁴⁺). Accumulation of Cr requires less energy consumption via the substitutions like ^VI(Mg²⁺) + ^IV(Si⁴⁺) = ^VI(Cr³⁺) + ^IV(Al³⁺) и 3^VI(Mg²⁺) = ^VI(Al³⁺) + ^VI(Cr³⁺) + ^VI(□). It is shown the formation of a vacancy via the scheme 2^VI(Mg²⁺) = ^VI(Ti⁴⁺) + ^VI(□) may provide only low Ti concentrations in phlogopite. A comparative analysis of the results obtained with the available literature allowed us to argue that the high Ti contents accompanied by the formation of a vacancy in the octahedral site may be achieved only via the schemes ^VI(Mg²⁺) + 2^IV(Al³⁺) = ^VI(□) + 2^IV(Ti⁴⁺) и (Mg,Fe²⁺) + 2OH⁻ = Ti⁴⁺ + 2O²⁻. It is shown that the incorporation of high concentrations of Cr (> 5.5 wt % Cr₂O₃) will be accompanied by an increase in the number of vacancies in the octahedral site and content of the K(Al,Cr,□)AlSi₃O₁₀(OH)₂ component. Such high-Cr micas will be transitional between di- and trioctahedral varieties.

The results of this work do not allow us to assess the predominant role of a certain mechanism for the incorporation of an impurity into phlogopite unambiguously. However, it became possible to reveal the competing relations of vacancy and stoichiometric mechanisms; therefore, these results may provide important criteria for evaluating of the conditions of Ti and Cr incorporation into mica. Experimental synthesis and subsequent X-ray diffraction analysis of high-Ti and high-Cr phlogopites in the systems with the different Al content may provide more complete understanding of the mechanisms of considered isomorphic substitutions.

Acknowledgments

This study was conducted as part of the scientific plan of the Laboratory of Deep Geospheres, Geological Faculty, Moscow State University, State Task of the Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences, and was supported by the Russian Science Foundation, project no. 21-17-00147 (review and atomistic simulation), and by the Russian Foundation for Basic Research, project no. 21-55-50011 (discussion).

Author contribution

All authors have read and agreed to the published version of the manuscript. AVB and NNE: Conceptualization. AVB: Funding acquisition. AAB and NNE: Computer simulation and Methodology. AAB: Writing original draft. NNE and AVB: General supervision.

Conflict of interest. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Mechanisms and Conditions of Ti and Cr Incorporation in Mantle Phlogopite: The Results of Atomistic Simulation

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Abstract

Figures

Introduction

Simulation Methodology

Results And Discussion

Simulation of Ti- and Cr-phlogopite end-members

Mixing properties of Ti- and Cr-phlogopite end-members

Phlogopite–Ti-phlogopite solid solution

Phlogopite –Cr-phlogopite solid solution

Change of the structure geometry as a function of impurity concentration

Comparison of incorporation of Cr and Ti in phlogopite

Comparison of isomorphic substitution schemes

Some reasoning for the schemes of isomorphic substitution under the natural conditions

Conclusions

Declarations

References

Additional Declarations

Supplementary Files

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