# Записки Российского минералогического общества, 2021, T. 150, № 5, стр. 79-91

##### Mechanisms of Phase Transitions between Al_{2}SiO_{5} Polymorphs

* A. I. Samtsevich*^{ 1, *}, * A. R. Oganov*^{ 1, **}

^{1} Skolkovo Institute of Science and Technology, Skolkovo Innovation Center

121205 Moscow, Bolshoy Boulevard, 30, 1, Russia

^{*} E-mail: A.Samtsevich@skoltech.ru^{**} E-mail: A.Oganov@skoltech.ru

Поступила в редакцию 26.05.2021

После доработки 7.07.2021

Принята к публикации 18.08.2021

Аннотация

In nature Al_{2}SiO_{5} exists as three polymorphs: kyanite, andalusite and sillimanite, often coexisting
in the same rock. Here, we have studied in detail the mechanisms of structural transitions
between all three phases of Al_{2}SiO_{5} – kyanite-andalusite, andalusite-sillimanite and kyanite-sillimanite at the pressures
of 0 and 10 GPa. The phase transition pathways with the lowest energy barriers are
found by constructing a number of geometrically likeliest pathways and optimizing
them using the variable-cell nudged elastic band method (VCNEB). We have analyzed
the structural changes along the obtained lowest-energy pathways. These results have
provided insights into the nature of structural relationships between Al_{2}SiO_{5} polymorphs, their coexistence with each other and their transformation pathways.
In particular, we confirm that phase transition barriers are very high, which allows
these phases to coexist during geological timescales – thus serving as geothermobarometers.

_{2}SiO

_{5}polymorphs, phase transitions mechanisms, density functional theory

## INTRODUCTION

Minerals andalusite, kyanite and sillimanite are polymorphic modifications of Al_{2}SiO_{5} (see Schmidt et al., 1997; Harben, 2002). Important for Earth sciences, they also
have wide practical usage as ceramic and refractory materials (McMichael, 1990) and
they are used to produce lightweight aluminum-silicon alloys for making metallic fiber,
which in its turn is used in supersonic aircraft and spaceships, etc. (Skoog, Moore,
1988; Aryal et al., 2008; Belogurova, Grishin, 2012; Zhang et al., 2013). A closely
related material, mullite, the main component of porcelain, has recently become a
promising candidate for structural and functional ceramics due to its low thermal
expansion, low thermal conductivity, and excellent creep resistance along with high-temperature
strength and stability under severe chemical environments (Schneider, Komarneni, 2005;
Schneider et al., 2008).

All three structures have some common features: all Si atoms are tetrahedrally coordinated
and half of all Al atoms are in the octahedral coordination and form chains of edge-sharing
AlO_{6} octahedra (see Burnham, 1963; Ohuchi et al., 2006). The other half of Al atoms are
in the tetrahedral coordination in sillimanite, 5-fold coordination in andalusite,
and in the octahedral coordination in kyanite. Kyanite crystallizes in the triclinic
system with space group $P\bar {1}$ (Yang et al., 1997b), while sillimanite and andalusite have orthorhombic structures
with space groups *Pnnm* and *Pbnm*, respectively (Yang et al., 1997a, 1997b). All three polymorphs of Al_{2}SiO_{5} are found commonly in metamorphic rocks and are geologically important markers since
they provide information about pressure and temperature of their formation and the
type of metamorphism (Atherton, Brotherton, 1974; Klein, Hurlbut, 1995; Whitney, 2002).

Andalusite is a low-pressure and low-temperature phase, while kyanite is formed at high pressures and low temperatures, and sillimanite is formed at medium or low pressures and high temperatures (Klein, Hurlbut, 1995; Kerrick, 2018). The entropies and Gibbs free energies of the three minerals are very similar (Klein, Hurlbut, 1995; Oganov et al., 2001).

In nature, often two or three polymorphs of Al_{2}SiO_{5} are found coexisting in the same rock. There are numerous examples with each of two-polymorph
assemblages, i.e. andalusite + kyanite, kyanite + sillimanite and andalusite + sillimanite
(Evans, Berti, 1986; Lux et al., 1986; Pattison et al., 1991), and in many cases all
three Al_{2}SiO_{5} polymorphs coexist in a rock (Hietanen, 1956; Carey et al., 1992; Grover et al.,
1992; Garcia-Casco, Torres-Roldan, 1996; Hiroi et al., 1998; Whitney, 2002; Sepahi
et al., 2004; Gibson et al., 2004; Allaz et al., 2005; Sayab, 2006; Likhanov et al.,
2009; Ali, 2010; Kim, Ree, 2010; Palin et al., 2012; Whitney, Samuelson, 2019; Baharfar
et al., 2019; Gervais, 2019). Such coexistence indicates conditions of formation at
a rock, corresponding to two- or three-phase equilibrium. Coexistence at normal conditions
is due to high barriers of transitions, leading to metastable persistence of phases.

Numerous theoretical and experimental studies explored structural stability of different
phases, the phase diagram, electronic and optical properties of Al_{2}SiO_{5} phases. For example, Oganov and colleagues studied stable (Oganov, Brodholt, 2000)
and metastable (Oganov et al., 2001) pressure-induced transitions. Zhu et al. predicted
phase transition mechanisms between stable Al_{2}SiO_{5} polymorphs using the evolutionary metadynamics method (Zhu et al., 2011), which is
capable of giving crude atomistic mechanism, but not the kinetics of the transition.

Generally, transition state theory (TST) (Eyring, 1935) is used to estimate the reaction rate constants. A simplification to TST – the harmonic approximation – is most commonly used and transforms the kinetics estimation problem into another – seeking first-order saddle points on the complex multidimensional free energy surface (FES) and properties of this saddle point characterize the transition. One of the most popular techniques to study reaction paths is nudged elastic band (NEB) method (Jonsson et al., 1998; Henkelman et al., 2000; Henkelman, Jónsson, 2000), but the results of this method are highly dependent on the initial path. There are extensions of the NEB method allowing the study of solid-solid phase transitions, such as the solid-state NEB (SSNEB) (Caspersen, Carter, 2005), and in particular the generalized solid-state NEB (G-SSNEB) (Sheppard et al., 2012) and variable-cell NEB (VCNEB) (Qian et al., 2013).

Still, the search for the lowest-energy saddle point is not an easy task, especially for periodic systems – there are a huge (strictly speaking, infinite) number of possible initial paths (various atom-to-atom mappings and lattice-to-lattice mappings). The problem of mapping of crystal structures is essential for the initial path(s) generation. There are very few approaches that may help to do so, such as the algorithm developed by Stevanović and co-workers (Stevanović et al., 2018; Therrien et al., 2020) and the one developed by Munro et al. (2018), which are purely geometrical (dealing with mapping cell parameters and atomic positions, interatomic distances, angles, coordination polyhedra). However, the computational complexity of such methods grows exponentially with the number of atoms. Once a good initial approximation to the transition path is constructed, it can be refined by such methods as VCNEB or its analogs mentioned above.

Here we present a detailed study of the atomistic mechanisms of phase transitions between andalusite, sillimanite and kyanite.

## METHODOLOGY

Initial structural models of the Al_{2}SiO_{5} polymorphs were taken from experimental studies of Ralph et al. (Ralph et al., 1984)
for andalusite and from Yang et al. (Yang et al., 1997a, 1997b) for kyanite and sillimanite.
Then, these structures were relaxed. Structure relaxations and total energy calculations
were performed using density functional theory (DFT) (Hohenberg, Kohn, 1964; Kohn,
Sham, 1965) within the generalized gradient approximation (Perdew–Burke–Ernzerhof
functional) (Perdew et al., 1996), and the projector augmented wave method (Blöchl,
1994; Kresse, Joubert, 1999) as implemented in the VASP (Kresse, Hafner, 1993, 1994;
Kresse, Furthmüller, 1996) package. The plane-wave energy cutoff of 600 eV was used,
ensuring excellent convergence of total energies, forces and stresses. Crystal structures
were relaxed until the maximum net force on atoms became less than 0.01 eV/Å. The
Monkhorst–Pack scheme (Monkhorst, Pack, 1976) was used to sample the Brillouin zone,
using 4 × 3 × 3 meshes for all three Al_{2}SiO_{5} phases.

Paths of phase transitions were optimized using the variable-cell nudged-elastic-band
(VCNEB) method (Qian et al., 2013) as implemented in the USPEX code (Oganov, Glass,
2006; Oganov et al., 2010, 2011). As we mentioned above, the VCNEB method requires
an initial path to be selected. Here, the initial paths of the transition between
different phases of Al_{2}SiO_{5} were constructed using the algorithm of Stevanović et al. (Stevanović et al., 2018;
Therrien et al., 2020). This algorithm searches for the mapping between the two structures,
such that it minimizes the Euclidean distance between the positions of each atom in
the initial and final structures. The algorithm consists of two steps. At the first
stage, the most convenient representations of the initial and final structures are
sought; it allows the optimal representation of unit cells of these structures (Stevanović
et al., 2018; Therrien et al., 2020). At this stage, two unit cells are transformed
to have the same number of atoms taking into account the total symmetry of nonequivalent
supercells whose number is determined within the Hart–Forcade theory (Hart, Forcade,
2008). For each of the two structures, we seek for such unit cell settings (among
all possible choices of the unit cell) that are closest to each other. At the second
stage, atoms of the structure are placed back into two generated supercells and the
algorithm finds such correspondence, or mapping, between the two structures that the
total distance traveled by all the atoms from the initial to the final structure be
minimal (Stevanović et al., 2018; Therrien et al., 2020). It is important to note
that the mapping algorithm is not commutative; thus, for each pair of structures,
the algorithm was used to generate two sets of pathways – forth and back (structure
A → structure B and structure B → structure A). This geometric approach is crude and
there is absolutely no guarantee that the generated path is indeed the optimal one.
To increase our confidence that the optimal path is indeed found, for each of the
transitions andalusite-kyanite, kyanite-sillimanite and andalusite-sillimanite, we
took top ten paths produced by Stevanović algorithm, refined them using VCNEB method
and for each transition, took the one with the lowest barrier.

The VCNEB calculations were done using the USPEX code (Oganov, Glass, 2006; Qian et
al., 2013) employing forces and stresses calculated using the VASP code (Kresse, Hafner,
1993, 1994; Kresse, Furthmüller, 1996). The VCNEB calculation began with an initial
transition path consisting of 20 intermediate structures (“images”). All three polymorphs
have 32 atoms in the unit cell, and in all trajectories we considered all intermediate
structures also had 32 atoms/cell. For accurate determination of transition states
and intermediate minima corresponding to metastable transition states, we used the
climbing image–descending image technique (Henkelman et al., 2000). The spring constants
for the VCNEB method were varied from 3 to 6 eV/Å^{2}. The halting criterion for the calculation was set as RMS (Root Mean Square forces)
on images that are less than 0.003 eV/Å. Crystal structures were visualized using
VESTA software (Momma, Izumi, 2011).

## COORDINATION NUMBER ANALYSIS

To understand better these reconstructive transitions, we focused on the changes in the bond network. For the determination of interatomic contacts, we used two approaches.

One approach was based on the Voronoi–Dirichlet partitioning of crystal structures. This is an attractive, unbiased and automatic, way to determine the (integer) coordination number of atom in any crystal structure.

https://www.tandfonline.com/doi/abs/10.1080/08893110412331323170?casa_token=jBJw-5hZxjpIAAAAA:5R-LJpsQ4T3WIDl0At7w2FqrJ0U6pl8ejwSQQnu21TVdRCdR4mofSbZ9Iqn-jx_GhKVTUtaPkFxTJ4pw

Another approach is to use Hoppe’s effective coordination number (ECoN) approach (Hoppe, 1979; Hoppe et al., 1989). This approach gives non-integer coordination numbers which change continuously as structure is continuously distorted, and its advantage is its continuous nature, where greater weight is given to stronger interactions. Here, each cation-anion interaction comes with a weight, in general non-integer. The ECoN approach is based on the exponential decrease of the importance of interatomic contacts with distance.

For a given atom, the ECoN is defined as ${\text{ECoN}} = \sum\nolimits_i {{{\omega }_{i}}} $, where the sum runs over all the atoms in the environment of the chosen central atom.
The weight of the *i*-th atom (${{\omega }_{i}}$) is defined as:

##### (1)

${{\omega }_{i}} = \exp \left[ {1 - {{{\left( {\frac{{{{d}_{i}}}}{{{{d}_{{{\text{av}}}}}}}} \right)}}^{6}}} \right],$*I*and ${{d}_{{{\text{av}}}}}$ the weighted average distance, defined as:

##### (2)

${{d}_{{{\text{av}}}}}\frac{{\sum\limits_i {{{d}_{i}}\exp \left[ {1 - {{{\left( {\frac{{{{d}_{i}}}}{{{{d}_{{\min }}}}}} \right)}}^{6}}} \right]} }}{{\sum\limits_i {\exp \left[ {1 - {{{\left( {\frac{{{{d}_{i}}}}{{{{d}_{{\min }}}}}} \right)}}^{6}}} \right]} }},$

These two approaches allow us to understand how many interatomic contacts are broken and which new ones are formed during the transition.

## RESULTS AND DISCUSSION

Crystal structures of the Al_{2}SiO_{5} phases are shown in Fig. 1*a–c*. Gray, orange and green polyhedra shown in Fig. 1*e–f* correspond to 6-,5- and 4-coordinate Al atoms. Blue polyhedra in Fig. 1*g* corresponds to silicon atoms, which are 4-fold coordinated in all of these structures.

First, we relaxed structures of all three Al_{2}SiO_{5} phases at the pressures of 0 and 10 GPa. The lowest-enthalpy phase is andalusite
at 0 GPa and kyanite at 10 GPa, which is in agreement with experiments and previous
calculations (Oganov, Brodholt, 2000). As one can see in Fig. 2, all three phases are energetically close to each other. At 0 GPa, sillimanite is
10 meV/atom higher in energy than andalusite, and kyanite is 12 meV/atom above sillimanite.
At 10 GPa, sillimanite is 60 meV/atom higher in enthalpy than kyanite and andalusite
is 8 meV/atom above sillimanite.

First, let us consider the transition from andalusite to sillimanite at pressures of 0 and 10 GPa. Both structures have relatively similar orthorhombic cell parameters listed in Table 1. Given the similarity of unit cells and even of crystal structures, one might think that it is trivial to construct the optimal phase transition pathway and that it should be within the orthorhombic cell. We did this both at 0 GPa and 10 GPa, at each pressure constructing the initial pathways by using smooth variation of orthorhombic cell parameters and choosing such mapping of atomic coordinates that minimizes the total distance traveled by all the atoms. Then, we optimized these paths by the VCNEB method. The resulting pathways are presented in Figs. 3 and 4. As one can see, the energy barrier equals 0.389 eV/atom at 0 GPa and 0.353 eV/atom at 10 GPa. It is instructive that these are not the lowest-barrier paths. Considering non-trivial cell mappings, we found lower-barrier paths.

##### Table 1.

VASP, 0GPa | Experiment, 0 GPa | VASP, 10 GPa | |
---|---|---|---|

Andalusite | $a = 5.610$ Å $b = 7.868$ Å $c = 7.973$ Å |
$a = 5.557$ Å $b = 7.798$ Å $c = 7.903$ Å |
$a = 5.534$ Å $b = 7.647$ Å $c = 7.821$ Å |

Sillimanite | $a = 5.816$ Å $b = 7.568$ Å $c = 7.772$ Å |
$a = 5.777$ Å $b = 7.488$ Å $c = 7.681$ Å |
$a = 5.754$ Å $b = 7.415$ Å $c = 7.559$ Å |

Thus, to describe phase transitions we generated six sets of initial pathways for
the optimized structures at each pressure (kyanite → sillimanite, sillimanite → kyanite,
kyanite → andalusite, andalusite → kyanite, andalusite → sillimanite and sillimanite
→ andalusite). The lowest-barrier transition profiles for each transformation at 0
and 10 GPa are shown in Figs. 5 and 6, respectively. For all transitions the symmetry of intermediate states is *P1*. These are all reconstructive phase transitions; barriers for each transition at
both pressures are very high and the schemes of transitions with barrier values are
presented in Fig. 7. Such high barriers imply that these transitions are kinetically feasible only at
high temperatures (the lowest energy barrier is 0.255 eV/atom).

Supporting Materials present the full set of optimized barriers for each kind of transition (10 pathways for A → B and 10 pathways for B → A).

Looking at the changes of coordination numbers along transition pathways, we found
that SiO_{4} tetrahedra are preserved during all transitions and there are no changes in the coordination
number of silicon. Indeed, Si–O bonds are the strongest here and, naturally, they
are perturbed the least. The change of coordination of Al atoms is much more complex
and informative – it provides more insights into the transition nature. Figures 8 and 9 show the change of the average coordination number of Al atoms (obtained using Voronoi–Dirichlet
partitioning) during all transitions at 0 GPa and 10 GPa, respectively. Figures 10 and 11 present the change of average ECoN of Al atoms during all transitions at 0 GPa and
10 GPa, respectively. Note that the averaging involves all Al atoms (including those
which are 6-coordinate in three polymorphs – during transitions, their coordination
numbers also change). Transition states have the lowest average coordination number
(CN) of Al atoms along the pathway, which is easy to understand – a large number of
all Al–O bonds have been broken, while the new ones have not yet been formed.

We find that the lowest-barrier transition mechanisms are different at 0 and 10 GPa. For andalusite-sillimanite transition at 0 GPa, the changes of coordination are more complex than at 10 GPa. In general, at high pressure the decrease of the average coordination number in the activated state is smaller.

Two remarks must be made. First, the discussed mechanisms of transitions are the best
among those tested, i.e., have the lowest activation barrier and other mechanisms
with a lower transition barrier are not excluded since global optimization of phase
transition paths was not performed and robust methods for doing so do not exist yet.
However, the procedure we use, combining a geometric selection of the “easiest” paths
and detailed exploration of these to find the lowest-energy path, should give results
close or identical to a full global search. Se-cond, the presented phase transition
mechanisms are based on the mean-field approximation, where all unit cells undergo
the same changes simultaneously. In reality, first-order phase transitions proceed
via nucleation and growth, making the mean-field approximation a rough, but crystallographically
and intuitively attractive, model. A more realistic study of nucleation and growth
phenomena requires much larger systems (including thousands of atoms) and advanced
sampling methods such as transition path sampling (Bolhuis et al., 2002). Transition
path sampling method (also implemented in the USPEX code) requires a good mean-field
starting model (and our work shows how to obtain it), but this method expensive to
be done at *ab initio* level, and requires a very accurate force field – machine learning force fields are
promising in this regard.

## CONCLUSIONS

We show that even if structures are geometrically similar and have similar unit cell
parameters (as andalusite and sillimanite), the construction of the transition path
is non-trivial and one should not rely on intuitively “obvious” mappings. Moreover,
we showed that the mechanism of the same transition changes with pressure. Using state-of-the-art
methodologies, we have studied the atomistic mechanisms of phase transitions between
the Al_{2}SiO_{5} polymorphs (andalusite, sillimanite and kyanite) at pressures of 0 and 10 GPa. First,
we generated a number of closest mappings between each pair of structures. Then, the
variable-cell nudged-elastic-band (VCNEB) method was used for optimizing these paths,
allowing us to choose the lowest-barrier path, for which we analyzed the evolution
of coordination numbers along the transition path. Our work shows that due to significant
changes of coordination numbers (breaking of many bonds), all transitions among Al_{2}SiO_{5} polymorphs have very high barriers, which explains the coexistence of these polymorphs
for many millions of years in nature – and that coexistence allows Al_{2}SiO_{5} polymorphs to be widely used for determining *P–T*-conditions of rock formation.

Список литературы

*Ali A.*The tectono-metamorphic evolution of the Balcooma Metamorphic Group, north-eastern Australia: A multidisciplinary approach.*J. Metamorph. Geol.***2010**. Vol. 28. P. 397–422.*Allaz J., Maeder X., Vannay J.C., Steck A.*Formation of aluminosilicate-bearing quartz veins in the Simano nappe (Central Alps): Structural, thermobarometric and oxygen isotope constraints.*Schweize-rische Mineral. und Petrogr. Mitteilungen*.**2005**. Vol. 85. P. 191–214.*Aryal S., Rulis P., Ching W.Y.*Density functional calculations of the electronic structure and optical properties of aluminosilicate polymorphs (Al_{2}SiO_{5}).*Amer. Miner*.**2008**. Vol. 93. P. 114–123.*Atherton M.P., Brotherton M.S.*Metamorphyic index minerals in the Eastern Dalradian.*Scottish J. Geol.***1974**. Vol. 9. P. 321–324.*Baharfar A.-A., Whitney D.L., Pang K.-N., Chung S.-L., Iizuka Y.*Petrology, geothermobarometry, and*P–T*path of spinel-bearing symplectite migmatites from the Simin area, Hamedan, Sanandaj-Sirjan Zone, Iran.*Turkish J. Earth Sci*.**2019**. Vol. 28. P. 275–298.*Belogurova O.A., Grishin N.N.*Carbidized heat insulation materials from kyanite ore. R*efract. Ind. Ceram*.**2012**. Vol. 53. P. 26–30.*Blöchl P.E.*Projector augmented-wave method.*Phys. Rev. B.***1994**. Vol. 50. P. 17953–17979.*Bolhuis P.G., Chandler D., Dellago C., Geissler P.L.*Transition path sampling: throwing ropes over rough mountain passes, in the dark.*Annu. Rev. Phys. Chem.***2002**. Vol. 53. P. 291–318.*Burnham C.W.*Refinement of the crystal structure of sillimanite.*Zeit.**Krist*.**1963**. Vol. 118. P. 127–148.*Carey J.W., Rice J.M., Grover T.W.*Petrology of aluminous schist in the Boehls Butte region of northern Idaho; geologic history and aluminum-silicate phase relations.*Amer. J. Sci.***1992**. Vol. 292. P. 455–473.*Caspersen K.J., Carter E.A.*Finding transition states for crystalline solid-solid phase transformations.*Proc. Natl. Acad. Sci.***2005**. Vol. 102. P. 6738–6743.*Evans B.W., Bert J.W.*Revised metamorphic history for the Chiwaukum Schist, North Cascades, Washington.*Geology*.**1986**. Vol. 14. P. 695.*Eyring H.*The activated complex in chemical reactions.*J. Chem. Phys.***1935**. Vol. 3, P. 63–71.*Garcia-Casco A., Torres-Roldan R.L.*Disequilibrium induced by fast decompression in St−Bt−Grt−Ky−Sil−And metapelites from the Betic Belt (Southern Spain).*J. Petrol*.**1996**. Vol. 37. P. 1207–1239.*Gervais F.*Three modes of isograd formation in the northern Monashee Complex of the Canadian Cordillera.*Geol. Soc. London, Spec. Publ*.**2019**. Vol. 478. P. 373–388.*Gibson G.M., Peljo M., Chamberlain T.*Evidence and timing of crustal extension versus shortening in the early tectonothermal evolution of a Proterozoic continental rift sequence at Broken Hill, Australia.*Tectonics*.**2004**. Vol. 23. TC5012. P. 1–20.*Grover T.W., Rice J.M., Carey J.W.*Petrology of aluminous schist in the Boehls Butte region of northern Idaho; phase equilibria and*P–T*evolution.*Amer. J. Sci*.**1992**. Vol. 292. P. 474–507.*Harben P.W.*The industrial minerals handbook: a guide to markets, specifications and prices. Industrial Minerals Information, Ltd.,**2002**.*Hart G.L.W., Forcade R.W.*Algorithm for generating derivative structures.*Phys. Rev. B – Condens. Matter Mater. Phys*.**2008**. Vol. 77. P. 1–12.*Henkelman G., Jónsson H.*Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points.*J. Chem. Phys*.**2000**. Vol. 113. P. 9978–9985.*Henkelman G., Uberuaga B.P., Jónsson H.*A climbing image nudged elastic band method for finding saddle points and minimum energy paths.*J. Chem. Phys*.**2000**. Vol. 113. P. 9901–9904.*Hietanen A.*Kyanite, andalusite and sillimanite in the schist in Boels Butte quadrangle, Idaho.*Amer. Miner.***1956**. Vol. 41. P. 1–27.*Hiroi Y., Kishi S., Nohara T., Sato K., Goto J.*Cretaceous high-temperature rapid loading and unloading in the Abukuma metamorphic terrane, Japan.*J. Metamorph. Geol*.**1998**. Vol. 16. P. 67–81.*Hohenberg P., Kohn W.*Inhomogeneous electron gas.*Phys. Rev.***1964**Vol. 136. P. B864–B871.*Hoppe R.*Effective coordination numbers (ECoN) and mean fictive ionic radii (MEFIR).*Zeitschrift für Krist. –**Cryst. Mater.***1979**. Vol. 150. P. 23–52.*Hoppe R., Voigt S., Glaum H., Kissel J., Müller H.P., Bernet K.*A new route to charge distributions in ionic solids. J.*Less-Common Met.***1989**. Vol. 156. P. 105–122.*Jonsson H., Mills G., Jacobsen K.W.*Nudged elastic band method for finding minimum energy paths of transitions. In:*Classical and Quantum Dynamics in Condensed Phase Simulations.*World Scientific,**1998**. p. 385–404.*Kerrick D.M.*The Al_{2}SiO_{5}polymorphs. Walter de Gruyter GmbH & Co KG,**2018**.*Kim H.S., Ree J.H.*P-T modeling of kyanite and sillimanite paramorphs growth after andalusite in late Paleozoic Pyeongan Supergroup, South Korea: Implication for metamorphism during the Mesozoic tectonic evolution.*Lithos*.**2010**. Vol. 118. P. 269–286.*Klein**C.,**Hurlbut**C.*Manual of mineralogy (by James D. Dana).**1995**.*Kohn W., Sham L.J.*, Self-consistent equations including exchange and correlation effects.*Phys. Rev.***1965**. Vol. 140. P. A1133–A1138.*Kresse G., Furthmüller J.*Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set.*Phys. Rev. B.***1996**. Vol. 54. P. 11169–11186.*Kresse G., Hafner J.*Ab initio molecular-dynamics simulation of the liquid-metalamorphous- semiconductor transition in germanium.*Phys. Rev. B.***1994**. Vol. 49. P. 14251–14269.*Kresse G., Hafner J.*Ab initio molecular dynamics for liquid metals.*Phys. Rev. B.***1993**. Vol. 47. P. 558–561.*Kresse G., Joubert D.*From ultrasoft pseudopotentials to the projector augmented-wave method.*Phys. Rev. B.***1999**. Vol. 59. P. 1758–1775.*Likhanov I.I., Reverdatto V.V., Kozlov P.S., Popov N.V.*Kyanite–sillimanite metamorphism of the Precambrian complexes, Transangarian region of the Yenisei Ridge.*Russ. Geol. Geophys*.**2009**. Vol. 50. P. 1034–1051.*Lux D.R., DeYoreo J.J., Guldotti C.V., Decker E.R.*Role of plutonism in low-pressure metamorphic belt formation.*Nature*.**1986**. Vol. 323. P. 794–797.*McMichael**B.*Aluminosilicate minerals. Refractories steel the Show.*Ind. Miner.***1990**. P. 27–43.*Momma K., Izumi F.*VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data.*J*.*Appl. Crystallogr.***2011**. Vol. 44. P. 1272–1276.*Monkhorst H.J., Pack J.D.*Special points for Brillouin-zone integrations.*Phys. Rev. B.***1976**. Vol. 13. P. 5188–5192.*Munro J.M., Akamatsu H., Padmanabhan H., Liu V.S., Shi Y., Chen L.Q., Vanleeuwen B.K., Dabo I., Gopalan V.*Discovering minimum energy pathways via distortion symmetry groups.*Phys. Rev. B.***2018**.*Oganov A.R., Brodholt J.P.*High-pressure phases in the Al_{2}SiO_{5}system and the problem of aluminous phase in the Earth’s lower mantle: ab initio calculations.*Phys. Chem. Miner.***2000**. Vol. 27. P. 430–439.*Oganov A.R., Glass C.W.*Crystal structure prediction using ab initio evolutionary techniques: Principles and applications.*J. Chem. Phys.***2006**. Vol. 124. P. 1–15.*Oganov A.R., Lyakhov A.O., Valle M.*How evolutionary crystal structure prediction works – and why.*Acc. Chem. Res.***2011**. Vol. 44. P. 227–237.*Oganov A.R., Ma Y., Lyakhov A.O., Valle M., Gatti C.*Evolutionary crystal structure prediction as a method for the discovery of minerals and materials.*Rev. Miner. Geochem*.**2010**. Vol. 71. P. 271–298.*Oganov A.R., Price G.D., Brodholt J.P.*Theoretical investigation of metastable Al_{2}SiO_{5}polymorphs.*Acta Crystallogr. Sect. A Found. Crystallogr.***2001**. Vol. 57. P. 548–557.*Ohuchi F.S., Ghose S., Engelhard M.H., Baer D.R.*Chemical bonding and electronic structures of the Al_{2}SiO_{5}polymorphs, andalusite, sillimanite, and kyanite: X-ray photoelectron-and electron energy loss spectroscopy studies.*Amer. Miner*.**2006**. Vol. 91. P. 740–746.*Palin R.M., Searle M.P., Waters D.J., Horstwood M.S.A., Parrish R.R.*Combined thermobarometry and geochronology of peraluminous metapelites from the Karakoram metamorphic complex, North Pakistan; New insight into the tectonothermal evolution of the Baltoro and Hunza Valley regions.*J. Metamorph. Geol.***2012**. Vol. 30. P. 793–820.*Pattison**D.R.M.,**Tracy**R.J.,**Kerrick**D.M.*Contact metamorphism. In:*Contact Metamorphism*. Ed. D.M. Kerrick. Rev. Miner. Ceochem. P. 105–206.*Perdew J.P., Burke K., Ernzerhof M.*Generalized gradient approximation made simple.*Phys. Rev. Lett.***1996**. Vol. 77. P. 3865–3868.*Qian G., Dong X., Zhou X., Tian Y., Oganov A.R., Wang H.-T.*Variable cell nudged elastic band method for studying solid–solid structural phase transitions.*Comput. Phys. Commun*.**2013**. Vol. 184. P. 2111–2118.*Ralph R.L., Finger L.W., Hazen R.M., Ghose S.*Compressibility and crystal structure of andalusite at high pressure.*Amer. Miner*.**1984**. Vol. 69. P. 513–519.*Sayab M.*Decompression through clockwise P-T path: implications for early N-S shortening oroge-nesis in the Mesoproterozoic Mt Isa Inlier (NE Australia).*J. Metamorph. Geol*.**2006**. Vol. 24. P. 89–105.*Schmidt M.W., Poli S., Comodi P., Zanazzi P.F.*High-pressure behavior of kyanite: Decomposition of kyanite into stishovite and corundum.*Amer. Miner*.**1997**. Vol. 82. P. 460–466.*Schneider H., Komarneni S.*Basic properties of mullite, in:*Mullite.*Wiley Online Library,**2005**. P. 141–155.*Schneider H., Schreuer J., Hildmann B.*Structure and properties of mullite – a review.*J. Eur. Ceram. Soc.***2008**. Vol. 28. P. 329–344.*Sepahi A.A., Whitney D.L., Baharifar A.A.*Petrogenesis of andalusite-kyanite-sillimanite veins and host rocks, Sanandaj-Sirjan metamorphic belt, Hamadan, Iran.*J. Metamorph. Geol.***2004**. Vol. 22. P. 119–134.*Sheppard D., Xiao P., Chemelewski W., Johnson D.D., Henkelman G.*A generalized solid-state nudged elastic band method.*J. Chem. Phys*.**2012**. Vol. 136. P. 074103.*Skoog A.J., Moore R.E.*Refractory of the past for the future: mullite and its use as a bonding phase.*Amer. Ceram. Soc. Bull.***1988**. Vol. 67. P. 1180–1185.*Stevanović V., Trottier R., Musgrave C., Therrien F., Holder A., Graf P.*Predicting kinetics of polymorphic transformations from structure mapping and coordination analysis.*Phys. Rev. Mater.***2018**. Vol. 2. P. 033802.*Therrien F., Graf P., Stevanović V.*Matching crystal structures atom-to-atom.*J. Chem. Phys.***2020**. Vol. 152.*Whitney D.L.*Coexisting andalusite, kyanite, and sillimanite: Sequential formation of three Al_{2}SiO_{5}polymorphs during progressive metamorphism near the triple point, Sivrihisar, Turkey.*Amer. Miner*.**2002**. Vol. 87. P. 405–416.*Whitney D.L., Samuelson W.J.*Crystallization sequences of coexisting andalusite, kyanite, and sillimanite, and a report on a new locality: Lesjaverk, Norway.*Eur. J. Miner*.**2019**. Vol. 31. P. 731–737.*Yang H., Downs R.T., Finger L.W., Hazen R.M., Prewitt C.T.*Compressibility and crystal structure of kyanite, Al_{2}SiO_{5}, at high pressure.*Amer. Miner*.**1997a**. Vol. 82. P. 467–474.*Yang H., Hazen R.M., Finger L.W., Prewitt C.T., Downs R.T.*Compressibility and crystal structure of sillimanite, Al_{2}SiO_{5}, at high pressure.*Phys. Chem. Miner*.**1997b**. Vol. 25. P. 39–47.*Zhang W., Meng Q., Dai W.*Research on application of kyanite in plastic refractory.*Chinese J. Geochem.***2013**. Vol. 32. P. 326–330.*Zhu Q., Oganov A.R., Lyakhov A.O.*Evolutionary metadynamics: a novel method to predict crystal structures.*Cryst. Eng. Comm*.**2011**. P. 1–7.

Дополнительные материалы отсутствуют.

Инструменты

Записки Российского минералогического общества