Polytypism of Compounds with the General Formula Cs{Al₂[TP₆O₂₀]} (T = B, Al): OD (Order-Disorder) Description, Topological Features, and DFT-Calculations

Abstract

The crystal structures of compounds with the general formula Cs{^[6]Al₂[^[4]TP₆O₂₀]} (where T = Al, B) display order-disorder (OD) character and can be described using the same OD groupoid family. Their structures are built up by two kinds of nonpolar layers, with the layer symmetries Pc(n)2 (L_2n+1-type) and Pc(a)m (L_2n-type) (category IV). Layers of both types (L_2n and L_2n+1) alternate along the b direction and have common translation vectors a and c (a ~ 10.0 Å, c ~ 12.0 Å). All ordered polytypes as well as disordered structures can be obtained using the following partial symmetry operators that may be active in the L_2n type layer: the 2₁ screw axis parallel to c [– – 2₁] or inversion centers and the 2₁ screw axis parallel to a [2₁ – –]. Different sequences of operators active in the L_2n type layer ([– – 2₁] screw axes or inversion centers and [2₁ – –] screw axes) define the formation of multilayered structures with the increased b parameter, which are considered as non-MDO polytypes. The microporous heteropolyhedral MT-frameworks are suitable for the migration of small cations such as Li⁺, Na⁺ Ag⁺. Compounds with the general formula Rb{^[6]M³⁺[^[4]T³⁺P₆O₂₀]} (M = Al, Ga; T = Al, Ga) are based on heteropolyhedral MT-frameworks with the same stoichiometry as in Cs{^[6]Al₂[^[4]TP₆O₂₀]} (where T = Al, B). It was found that all the frameworks have common natural tilings, which indicate the close relationships of the two families of compounds. The conclusions are supported by the DFT calculation data.

1. Introduction

Borophosphates (as well as borophosphate ceramics and glasses) attract interest because of their wide technological applications as materials with optical [1,2,3,4,5], electrochemical [6,7,8,9], magnetic [10,11,12], and catalytic [13,14,15] properties. Moreover, crystalline borophosphates and metal borophosphates with microporous structures are considered as zeolite-like materials [16,17,18,19,20]. Borophosphates are characterized by a wide diversity of tetrahedral and mixed triangular-tetrahedral anionic motifs [21,22,23,24], owing to the different possible coordination environments of boron. At present, more than 300 representatives of this class are known, which are characterized by anionic motifs with different dimensionalities (from isolated groups to 3D frameworks).

Compounds with the general formula Cs{^[6]Al₂[^[4]TP₆O₂₀]} (where T = B [25], Al [26]) are based on microporous heteropolyhedral frameworks formed by tetrahedral borophosphate or aluminophosphate [TP₆O₂₀]-layers linked by isolated AlO₆ octahedra. The large framework cavities are filled by Cs⁺ cations. As was previously shown, both Cs{Al₂[BP₆O₂₀]} and Cs{Al₂[AlP₆O₂₀]} are of modular character [27] and can be considered as polytypes belonging to the same OD family [20,25]. However, the corresponding groupoid family has not been reported so far.

In this paper we provide a complete OD-theoretical analysis of the compounds with the general formula Cs{^[6]Al₂[TP₆O₂₀]} (where T = B [25], Al [26]) and derive symmetry and atom coordinates for the hypothetical MDO2 polytype. The energies of the observed and hypothetical structures of the family are calculated using the density functional theory (DFT). Possible ion-migration paths inside the microporous frameworks of the family are estimated for different alkaline ions using the topological analysis.

2. Methods

The symmetrical relations between the compounds have been analyzed using the OD theoretical approach [27,28,29,30] for the OD families containing more than one (M > 1) kind of layers [31]. The OD layers have been chosen in accordance with the equivalent region (ER) requirements [32]. As a reference structure for the further analysis, the MDO1 polytype observed in Cs{Al₂[AlP₆O₂₀]} [26] was used. This compound was reported in the non-standard setting of the space group C2cb [a = 10.0048(7) Å, b = 13.3008(10) Å, c = 12.1698(7) Å], which was transformed into the standard setting Aea2 using the [00–1/010/100] matrix (the resulting unit cell parameters are: a = 12.1698(7) Å b = 13.3008(10) Å, c = 10.0048(7) Å). The unit-cell parameters and space groups of the crystal structures of Cs{Al₂[BP₆O₂₀]} polytypes have been transformed accordingly in order to preserve the orientation and stacking direction of the OD-layers.

Topological analysis of the frameworks was performed by means of natural tilings (the smallest polyhedral cationic clusters that form a framework) of the 3D cation nets [33]. The complexity parameters of the frameworks in different polytypes were calculated as Shannon information amounts per atom (I_G) and per reduced unit cell (I_G,total) [34,35]. To analyze the migration paths of alkaline cations in the structures, the Voronoi method [36], which has proven itself in the study of cationic conductors of various types [37,38], was used. Topological and complexity parameters for the whole structures as well as ion migration paths have been calculated using the ToposPro software [39].

DFT calculations on the existing MDO-, non-MDO-4O, as well as hypothetical MDO2 type polytypes (T = Al, B) were performed using the PBE exchange-correlation functional [40] of the GGA-type utilizing the projector augmented wave method (PAW) as implemented in the Vienna ab initio simulation package (VASP) [41,42]. The energy cut-off was set at 500 eV with a 10 × 8 × 8 (MDO1, MDO2), and 6 × 4 × 4 (non-MDO-4O) Monkhorst−Pack [43] k-point mesh used for Brillouin zone sampling. The convergence towards the k-point mesh was checked. Full optimization of the unit cell parameters and atomic coordinates was performed for all the structures except the MDO1 polytype of Cs{Al₂[BP₆O₂₀]}, for which the original cell parameters were retained and atomic coordinates optimized (as the compound was found to have the lowest energy, cell parameter optimization was deemed unnecessary). For the optimization, the structures were converted to the space group P1.

3. Results

3.1. OD (Order-Disorder) Relationships

The crystal structures of Cs{^[6]Al₂[^[4]TP₆O₂₀]} (where T = B [25], Al [26]) belong to the same OD family of category IV [31] with two types of nonpolar OD layers and can be described by an OD groupoid [27]. The layers are as following:

• Nonpolar L_2n+1 type with the layer symmetry pcn2 [or Pc(n)2 in terms of the OD notation, where braces indicate the direction of missing periodicity [44]] was reported previously [20] and is represented by the tetrahedral [^[4]TP₆O₂₀]-layer (Figure 1);
• Nonpolar L_2n type consists of aluminum and oxygen atoms on the borders of a thin slab with the layer symmetry pcam [Pc(a)m or P2₁/c (2/a) 2₁/m].

Layers of both types (L_2n and L_2n+1) alternate along the b direction and have common translation vectors a and c (a ~ 10.0 Å, c ~ 12.0 Å), with b₀, the distance between the two nearest equivalent layers, corresponding to one half of the b parameter of the compound studied by Lesage et al. [26]. Because the symmetry of the L_2n type layers is higher than that of the L_2n+1 type layers, polytypic relations are possible. All ordered polytypes as well as disordered structures can be obtained using the following symmetry operators that may be active in the L_2n type layer: the 2₁ screw axis parallel to c [– – 2₁] or inversion centers and the 2₁ screw axis parallel to a [2₁ – –] (Figure 2) [20]. The symmetry relation common to all polytypes of this family are described by the OD groupoid family symbol:

$$\begin{array}{ccc}Pc(n)2& & P{2}_1/c(2/a)2_1/m\\ & [r,0]& \end{array},$$

(1)

where r = 0; the first line contains the layer-group symbols of the two constituting layers, while the second line indicates positional relations between the adjacent layers [46].

In accordance with the NFZ relation [27,28], there is only one kind of the (L_2n, L_2n+1, L_2n+2) triples and two kinds of the (L_2n-1, L_2n, L_2n+1) triples. Consequently, the smallest possible number of different triples in a structure is two and only two MDO polytypes are possible:

$$\begin{array}{cccccc}\mathrm{OD}-\mathrm{layer}& \mathrm{Layer}\mathrm{group}& \mathrm{Subgroup}\mathrm{of}\mathsf{\lambda }\text{-}\mathsf{\tau }\text{-}\mathrm{operations}& N& F& Z\\ {\mathrm{A}}^1=L_{2n}& P{2}_1/c2/a{2}_1/m& Pc2m& 4\searrow & & \nearrow 1\\ & \mathrm{Symmetry}\mathrm{of}\mathrm{a}\mathrm{layer}\mathrm{pair}\to & Pc11& & 2& \\ {\mathrm{A}}^2=L_{2n+1}& Pcn2& Pc11& 2\nearrow & & \searrow 2\end{array},$$

(2)

The first MDO structure (MDO1 polytype) (Figure 3, left) can be obtained when the [– – 2₁] operator is active in L_2n type layer. Through the action of this operator the asymmetric unit at x, y, z (I) is converted into the asymmetric unit at –x, ½–y, ½+z (II); the latter unit is converted by the [– – 2] operator in the L_2n+1 layer into the asymmetric unit at x, ½+y, ½+z (III). I and III are related by the translation vector t = b₀ + c/2, which is the generating operation, giving rise by the continuation to an A-centered structure with the basis vectors a, b = 2b₀, c and the space group Aea2. The MDO1 polytype corresponds to the structure of Cs{Al₂[AlP₆O₂₀]} with the following unit cell parameters: a = 12.1698(7) Å b = 13.3008(10) Å, c = 10.0048(7) Å [26].

The second MDO structure (MDO2 polytype) (Figure 3, right) can be obtained when the inversion centers and [2₁ – –] operators are both active in the L_2n type layer. Through the action of the operator [2₁ – –] the asymmetric unit at x, y, z (I) is converted into the asymmetric unit ½+x, –y, ½–z (II); the latter unit is converted by the [– n –] operator in the L_2n+1 layer into the asymmetric unit x, ½+y, –z (III); (I) and (III) are related by a b glide normal to c, with translational component b₀, which is the generating operation: its continuation also generates an orthorhombic structure with the basis vectors a, b = 2b₀, c (the same for the MDO1 polytype) and the space group Pcnb (or Pbcn in the standard setting). The MDO2 polytype has not yet been observed for the compound with the general formula Cs{^[6]Al₂[^[4]TP₆O₂₀]}. The calculated atomic coordinates for the MDO2 polytype are given in Table S1 (Supplement Materials).

Different sequences of operators active in the L_2n type layer ([– – 2₁] screw axes or inversion centers and [2₁ – –] screw axes) define the formation of structures with the increased b parameter, which are considered as non-MDO polytypes (because of the presence of more than one kind of (L_2n–1, L_2n, L_2n+1) triples) [27]. The compound Cs{Al₂[BP₆O₂₀]} [25] contains four L_2n and L_2n+1 types layers, where each L_4n type layer has active [2₁ – –] screw axes, while in the L_4n+2 type the inversion centers and [– – 2₁] screw axes are active (Figure 4). The AlO₆ octahedra in the L_2n+2 and L_2n+4 type layers are tilted slightly differently, which can be explained by the “desymmetrization” effect of OD structures [27,47,48], when the ideal symmetry suffers slight (in some cases severe) distortions and the symmetry of OD layers in the polytype is lower than the idealized one. The orthorhombic structure of Cs{Al₂[BP₆O₂₀]}–4O is characterized by the basis vectors a, b = 4b₀, c (where a = 11.815(2) Å, b = 26.630(4) Å, c = 10.042(2) Å [25]) and the space group Pcab (nonstandard setting of the space group Pbca).

3.2. Topological Features

Compounds with the general formula Cs{^[6]Al₂[^[4]TP₆O₂₀]} (where T = B [25], Al [26]) are characterized by the heteropolyhedral MT-frameworks [20,49,50,51] of MO₆-octahedra and TO₄-tetrahedra related to classic zeolites and zeolite-type materials where all oxygen ligands are bridged between two cations only [52]. In accordance with the theory of mixed anionic radicals [53,54,55], the general crystal chemical formula of the framework (taking into account the degree of sharing of oxygen ligands) can be written as [20]:

$${\left\{M_m\left[{\left(T_1\right)}_{n_1}{\left(T_2\right)}_{n_2}{\mathrm{O}}_{3m+2(n_1+n_2)}\right]\right\}}^{m\left(V_M-6\right)+n_1\left(V_{T_1}-4\right)+n_2\left(V_{T_2}-4\right)},$$

(3)

where where m and n_i, V_M and $V_{T_i}$ are the valences of the M and T_i cations, respectively. If M = M³⁺, T₁ = T³⁺, T₂ = P⁵⁺, m = z, n₁ = y, n₂ = z, the Formula (3) can be rewritten as:

$${\left\{M_x\left[T_y{\mathrm{P}}_z{\mathrm{O}}_{3x+2(y+z)}\right]\right\}}^{-3x-y+z}.$$

(4)

Taking into account the observed ratio between the x, y, and z coefficients, the stoichiometry of the heteropolyhedral MT-framework is:

$${\left\{M_2\left[T{\mathrm{P}}_6{\mathrm{O}}_{20}\right]\right\}}^{1-}.$$

(5)

Topological features of the MDO1 and non-MDO 4O polytypes have been described previously [20]. The cationic 3D net corresponding to the heteropolyhedral MT-framework of MDO2 polytype consists of four natural tiles (Figure 5): [4.6²]₂[3.5.6²]₂[4⁴.5².7²][3².4².6⁶.7²]. The (6T1M)-[4.6²] and (6T2M)-[3.5.6²] tiles are topologically equal to those observed in the MDO1 and non-MDO 4O polytypes; the (10T4M)-[4⁴.5².7²] tile is equal to that in the non-MDO 4O polytype. The (16T6M)-[3².4².6⁶.7²] tile is unique and is present in the MDO2 polytype only (

Table 1. The natural tiles in the MT-frameworks of the polytypes of compounds with the general formula Cs{^[6]Al₂[^[4]TP₆O₂₀]} (where T = B, Al).

Polytype	Natural Tiles
MDO1	[4.62]2	[3.5.62]2	[64.74]	[3.42.5.72]2	[4.72]2
MDO2	[4.62]2	[3.5.62]2	[44.52.72]	[32.42.66.72]
non-MDO 4O	[4.62]4	[3.5.62]4	[44.52.72]	[3.42.5.72]2	[4.72]2	[3.4.65.73]2

Note. The point symbol of the 3D net has the form A^a. B^b… indicating that there are a angles with shortest cycles that are A-cycles, b angles with shortest cycles that are B-cycles, etc., with A < B,<⋯ and a + b + ⋯ = n(n–1)/2 [33]. The topologically equivalent tiles are colored in the same color.

).

The complexity parameters of the heteropolyhedral MT-framework of MDO2 polytype are: v = 116 atoms; I_G = 3.892 bits/atom; I_G,total = 451.526 bits/unit cell. The complexity parameters increase in the row MDO1 → MDO2 → non-MDO 4O.

3.3. Ion Migration Path

Migration maps of Na⁺ cation were constructed for the MDO1, MDO2, and non-MDO 4O polytypes (

Table 2. The natural tiles in the MT-frameworks of the polytypes of compounds with the general formula Cs{^[6]Al₂[^[4]TP₆O₂₀]} (where T = B, Al).

Polytype	Natural Tiles
	Li+	Na+	Ag+	K+	Rb+	Cs+
MDO1	3D	2D	2D	–	–	–
MDO2	3D	1D	1D	–	–	–
non-MDO 4O	3D	2D	2D	–	–	–

Note: The following significance criteria for elementary channels (R_chan) and voids (R_sd) have been used for the construction of migration maps: Li⁺ (R_chan = 2.02 Å; R_sd = 1.38 Å); Na⁺ (R_chan = 2.16 Å; R_sd = 1.54 Å); Ag⁺ (R_chan = 2.20 Å; R_sd = 1.58 Å), K⁺ (R_chan = 2.30 Å; R_sd = 1.70 Å); Rb⁺ (R_chan = 2.38 Å; R_sd = 1.78 Å); Cs⁺ (R_chan = 2.47 Å; R_sd = 1.88 Å).

). Despite the presence of large pores filled by large Cs⁺ ions, the size of the effective windows between them is not enough for the migration of large alkaline cations. However, all the types of the microporous heteropolyhedral MT-framework are suitable for the migration of smaller ions such as Li⁺, Na⁺ Ag⁺. The types of migration maps depend on the topological type of the MT-framework (Figure 6), in particular, for Na⁺ ions, the maps are represented by 2D layers parallel to (100) for the MDO1 and non-MDO 4O polytypes, while for the MDO2 polytype it is represented by the system of parallel 1D channels directed along [010] (Figure 6). In the case of Li⁺ ions, the migration 3D maps are similar for all the types of the frameworks.

3.4. DFT Calculations

In order to gain more insight into the stability of various polytypes, energy-wise, we have performed DFT calculations on the existing as well as hypothetical compounds with the general formula Cs{Al₂[TP₆O₂₀]} (T = Al, B) with the structures belonging to MDO1, MDO2, and non-MDO 4O type polytypes, for T = Al; B. The comparative data and optimized unit cell parameters are given in

Table 3. Comparative data for the frameworks of different polytypes.

Parameter	MDO1 Polytype		MDO2 Polytype		Non-MDO 4O Polytype
	T = B	T = Al	T = B	T = Al	T = B	T = Al
Unit cell parameters (Å), a, b, c	n.d.	12.170, 13.301, 10.005	n.d.	n.d.	11.815, 26.630, 10.042	n.d.
Volume (Å3)	n.d.	1619.46	n.d.	n.d.	3159.55	n.d.
Optimized unit cell parameters (Å), a, b, c	12.0296, 13.2109, 9.9017	12.1698,* 13.3008, * 10.0048 *	11.7893, 13.4876, 10.1609	11.9479, 13.6593, 10.3157	11.8248, 26.7192, 10.0423	12.2217, 26.9351, 10.1760
Optimized volume (Å3)	1573.60	1619.46	1615.68	1683.52	3172.86	3349.86
Z	4		4		8
Energy per formula unit (eV)	–219.1885	–218.2701	–219.2479	–217.7659	–219.2780	–218.2109
FD [(M + T)/1000 Å3]		19.76	19.81	19.01	22.69	21.49
v (atoms), framework, all	58, 60		116, 120		232, 240
IG (bits/atom), framework, all	3.892, 3.974		3.892, 3.974		4.858, 4.907
IG, total (bits/unit cell), framework, all	225.763, 238.413		451.526, 476.827		1127.052, 1177.654

n.d.—no data, because of the absence of structural information; original unit cell parameter.

(for MDO1, T = Al, original unit cell metrics were retained).

As seen from the comparison between the original and optimized cells of Cs{Al₂[BP₆O₂₀]} of the non-MDO 4O type, they are in a very good agreement, with the difference in volume of ca. 13 Å, i.e., ca. 0.4% (see Table 3). The optimized coordinates in all structures showed only minimal shifts from their original positions, mostly associated with a very small rotation of tetrahedra. It is important to note that, despite unconstrained optimization, all the structures, observed as well as hypothetical, retained their original cell symmetries.

As seen from Table 3, for the T = Al series, the structure with the lowest energy was the MDO1-type polytype. However, the non-MDO 4O-type structure was only ca. 0.06 eV higher in energy, which corresponds to ca. 6.2 kJ/mol. This difference is not large, yet is arguably outside the margin of error for the computational method used, which is commonly estimated as 1–2 kJ/mol. The important thing here is that both experimentally observed types of structures (albeit not both of them for T = Al), showed comparable energies. Moreover, our calculations indicate that, under the right conditions, it might be possible to obtain the non-MDO 4O polytype for aluminum. Regarding the MDO2-type structure, the optimization gave us a stable minimum structure with the energy of ca. 0.5 eV (ca. 49 kJ/mol) higher than MDO1. This means that, potentially, such a structure might exist, however, the energy difference to the lowest energy structure is significant, and thus it might be difficult to stabilize such a polytype.

For the T = B series, once again the lowest energy corresponds to the experimentally observed structure, this time it is the non-MDO 4O polytype (see Table 3). In this case, however, its energy is only ca. 0.03 eV (ca. 3 kJ/mol) lower than that of the hypothetical MDO2-type structure. The difference is on the border of the perceived accuracy of the computational method, thus the MDO2 polytype appears to be a good candidate for the experimental discovery. The MDO1-type structure in this case looks like the least favorable, energy-wise, with the difference between its energy and minimal structure being ca. 0.09 eV (ca. 8.6 kJ/mol). This is clearly outside the margin of error; however, the difference is small enough to be compensated by various effects in real crystals. It must also be noted regarding all our calculations, that by their very nature they simulate ideal periodic crystals in their ground state at 0 K. In addition, in our computations we cannot account for potential kinetic hindrance of certain paths of compound formation.

4. Discussion

The heteropolyhedral MT-frameworks with similar stoichiometry (3) have been found in compounds with the general formula Rb{^[6]M³⁺₂[^[4]T³⁺P₆O₂₀]}, where M = Al, Ga; T = Al, Ga [26,56]. The unit cell parameters are similar to those for MDO1 and MDO2 polytypes of Cs{Al₂[TP₆O₂₀]} (T = Al, B): a = 9.876–10.002 Å; b = 12.885–13.082 Å; c = 11.919–12.071 Å; space group C222₁. Their crystal structures contain mixed tetrahedral [TP₆O₂₀]-chains, which are linked by the MO₆-octahedra (Figure 7). The tetrahedral chain is formed by the condensation of FBU, an open-branched heptamer with the descriptor 7□:[3□]2□|2□|□|□ similar to that for the tetrahedral [TP₆O₂₀]-layers in Cs{Al₂[TP₆O₂₀]} (T = Al, B). The negative charge of the framework is balanced by Rb⁺ ions.

Despite of the absence of the tetrahedral layers, the MT-framework can also be considered as the result of alternation along b of two types of nonpolar OD layers parallel to (010):

• The first one corresponds to a layer with the symmetry P2(2)2₁ consisting of tetrahedral chains. The tetrahedral layer in Cs{Al₂[TP₆O₂₀]} and tetrahedral pseudolayer in Rb{M₂[TP₆O₂₀] are formed by the same FBU and demonstrate the symmetrical relationship (Figure 8) indicating the possible OD-character as was previously shown for compounds with tetrameric [57] and pentameric [20] borophosphate FBUs, as well as for the silicate layers [58,59];
• The second one consists of an octahedral layer with the symmetry P2₁(2)2₁ similar to that observed in Cs{Al₂[TP₆O₂₀]} (T = Al, B) (the layer group P2₁22₁ is a subgroup of the layer group Pcam). To date, there are no other polytypes of this type of framework, however they may be found later.

Topological features of the MT-framework are reflected in the sequence of its natural tiles: [4.6²]₂[4.7²]₂[3.5.6²]₂[3.4².5.7²]₂[6⁴.7⁴]. It should be noted that three tilings ([4.6²], [4.7²]), and [3.5.6²]) are topologically equivalent to those in the Cs{Al₂[TP₆O₂₀]} (T = Al, B) compounds, which indicate the relation of the two types of the {^[6]M³⁺₂[^[4]T³⁺P₆O₂₀]}-frameworks.

5. Conclusions

The polytypism of compounds with the general formula Cs{Al₂[TP₆O₂₀]} (T = Al, B) has been described using the OD theory approach. The crystal structure of the hypothetical MDO2 polytype has been proposed and optimized using DFT calculations. It was shown that the heteropolyhedral MT-frameworks of all the polytypes contain similar natural tilings. The compounds with the general formula Rb{^[6]M³⁺₂[^[4]T³⁺P₆O₂₀]} (M = Al, Ga; T = Al, Ga) have the heteropolyedral MT-frameworks with the same stoichiometry. It was found that all the frameworks had common natural tilings, which indicates the relationship of both families of compounds. Our computational data agree well with those which are experimentally available and, we believe, provide a reasonable basis for an internally consistent picture which supports crystallographic considerations concerning the formation of the polytypes of compounds with the general formula Cs{Al₂[TP₆O₂₀]} (T = Al, B). Thus, it is seems possible to synthesize the MDO2 polytype as well as the “missing” members, such as MDO1 polytype of Cs{Al₂[BP₆O₂₀]} and non-MDO 4O polytype of Cs{Al₂[AlP₆O₂₀]} using hydrothermal techniques.