Записки Российского минералогического общества, 2021, T. 150, № 5, стр. 79-91
Mechanisms of Phase Transitions between Al2SiO5 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 Al2SiO5 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 Al2SiO5 – 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 Al2SiO5 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.
INTRODUCTION
Minerals andalusite, kyanite and sillimanite are polymorphic modifications of Al2SiO5 (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 AlO6 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 Al2SiO5 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 Al2SiO5 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 Al2SiO5 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 Al2SiO5 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 Al2SiO5 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 Al2SiO5 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 Al2SiO5 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 Al2SiO5 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/Å2. 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)
${{\omega }_{i}} = \exp \left[ {1 - {{{\left( {\frac{{{{d}_{i}}}}{{{{d}_{{{\text{av}}}}}}}} \right)}}^{6}}} \right],$(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 Al2SiO5 phases are shown in Fig. 1a–c. Gray, orange and green polyhedra shown in Fig. 1e–f correspond to 6-,5- and 4-coordinate Al atoms. Blue polyhedra in Fig. 1g corresponds to silicon atoms, which are 4-fold coordinated in all of these structures.
First, we relaxed structures of all three Al2SiO5 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 SiO4 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 Al2SiO5 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 Al2SiO5 polymorphs have very high barriers, which explains the coexistence of these polymorphs for many millions of years in nature – and that coexistence allows Al2SiO5 polymorphs to be widely used for determining P–T-conditions of rock formation.
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