Thermal Expansion of Alkaline-Earth Borates

Abstract

The thermal expansion of four alkaline-earth borates, namely Ca₃B₂O₆ (0D), CaB₂O₄ (1D), Sr₃B₁₄O₂₄ (2D) and CaB₄O₇ (3D), has been studied by in situ high-temperature powder X-ray diffraction (HTXRD). Strong anisotropy of thermal expansion is observed for the structures of Ca₃B₂O₆ (0D) and CaB₂O₄ (1D) built up from BO₃ triangles only; these borates exhibit maximal expansion perpendicular to the BO₃ plane, i.e., along the direction of weaker bonding in the crystal structure. Layered Sr₃B₁₄O₂₄ (2D) and framework CaB₄O₇ (3D) built up from various B–O groups expand less anisotropically. The thermal properties of the studied compounds compared to the other alkaline-earth borates are summarized depending on the selected structural characteristics like anion dimensionality, residual charge per one polyhedron (BO₃ BO₄), cationic size and charge, and structural complexity. For the first time, these dependencies are established as an average for both types of polyhedra (triangle and tetrahedron) occurring in the same structure at the same time. The most common trends identified from these studies are as follows: (i) melting temperature decreases with the dimensionality of the borate polyanion, and more precisely, as the residual charge per one polyhedron (triangle or tetrahedron) decreases; (ii) volumetric expansion decreases while the degree of anisotropy increases weakly when the residual charge decreases; (iii) both trends (i) and (ii) are most steady within borates built by triangles only, while borates built by both triangles and tetrahedra show more scattered values.

1. Introduction

Over the past few decades, newly discovered borates have been the focus of research due to their diversity and well-known applications; the crystal structures of borates have been described and systematized [1,2,3,4,5,6,7,8,9]. The crystal chemistry of borates is diverse because boron atom orbitals can be hybridized to either an sp² or sp³ configuration, forming BO₃ or BO₄-polyhedra in the same crystal structure. Recently, it has been shown [6] that in the case of sp-hybridization, boron can also occur in the double coordination of oxygen, although this configuration is quite rare. Typically, BO₃ and BO₄ polyhedra are connected through vertices to form more complicated borate anions. Additionally, the structural diversity of borate crystal chemistry is supplemented by the edge-sharing BO₄ tetrahedral units formed in high-pressure/high-temperature borates [7,10]. Recently, edge-sharing borate groups have also been obtained in borates at normal pressure [11,12]. At present, borate systems are considered to be the most promising optical materials in the UV range of nonlinear optical (NLO) materials or host phases for luminophores. Their wide practical application is due to the unique properties of borates, in particular their wide transparency range, outstanding NLO, luminescence, piezoelectricity, relatively high resistance against laser-induced damage, etc. [8,13,14]. In recent years, a more modern application of these materials with near-zero or negative thermal expansion was suggested, that they might be used as a basis of phosphors without thermal quenching or with anti-thermal quenching of luminescence [15,16,17,18,19].

Due to the increasing application of borates in optics, the thermal behavior of these materials is currently being intensively investigated. For two dozen years, the high-temperature crystal chemistry of borates has been investigated by our team [20,21,22,23]. It was shown that borates expand sharply anisotropically up to negative areas [20,24]. In the last decade, overall interest in thermal expansion studies has increased sharply [25,26,27,28,29,30].

Herein, we summarize data on the thermal expansion of alkaline-earth borates. Some phases of the same stoichiometry like 3MO:1B₂O₃, 2:1, 1:1, 1:2, 1:3, and 1:4 are common to most systems MO–B₂O₃ (M = Mg, Ca, Sr, Ba). Since the atomic radii of Ca and Sr are close to each other (R_cryst = 1.26 (Ca) and 1.4 Å (Sr) for CN = 8 [31]), isotypical phases are encountered. Meanwhile, the radii of magnesium (R_cryst = 0.86 Å for CN 6) and barium (R_cryst = 1.61 Å, CN 9) differ substantially.

Among Ca borates, the following compounds have been structurally characterized: Ca₃B₂O₆ (CaO:B₂O₃ = 3:1) [32,33,34,35], γ-Ca₂B₂O₅ (2:1) [36,37], α-Ca₂B₂O₅ (2:1), CaB₂O₄ (1:1) [38,39,40,41,42,43,44], Ca₂B₆O₁₁ (2:3) [45], CaB₄O₇ (1:2) [46,47,48], and CaB₆O₁₀ (1:3) [49]. An interesting sequence of reversible-phase transitions was first described for Sr₂B₂O₅ from thermal analysis data [50]. The crystal structures of Sr₂B₂O₅ polymorphs resulting from the γ′↔β↔α’↔α reversible first-order phase transitions were studied by high-temperature single-crystal X-ray diffraction, high-temperature X-ray powder diffraction, differential scanning calorimetry (DSC), and impedance spectroscopy [51]. A similar sequence of reversible first-order phase transitions (γ ↔ β ↔ α) was found for Ca₂B₂O₅ [52]: the γ-Ca₂B₂O₅ low-temperature polymorph exists up to 520 °C, the β-Ca₂B₂O₅ intermediate modification exists in the range of 520–580 °C, and the high-temperature modification α-Ca₂B₂O₅ exists above 580 °C. All Ca₂B₂O₅ polymorphs are structurally similar to those of Sr₂B₂O₅.

By doping calcium borates with various lanthanides (Ln), prospective luminescent properties were obtained. The thermoluminescent properties of CaB₄O₇ doped with a number of lanthanide ions (Ce³⁺, Sm³⁺, Eu³⁺, Dy³⁺ and Yb³⁺) were studied in [53,54,55]. Ca₃(BO₃)₂:Eu³⁺ [56] and Ca₂B₂O₅:REE (REE = Eu³⁺, Tb³⁺, Dy³⁺) [57] exhibited high photoluminescence intensity.

The thermal expansion of Mg- [58], Sr- [51,59], Ba- and (Sr,Ba)- [60] borates has been investigated, while that of Ca-borates has not been practically studied until now. The exception is the thermal expansion of γ- and α-Ca₂B₂O₅ modifications: the polymorphic transitions of Ca₂B₂O₅ and the crystal structure of the high-temperature polymorph were determined from powder HTXRD data at 600 °C [52,54]. Therefore, one of the goals of this work was to study the thermal expansion of Ca-borates and the unique triple-layered Sr-borate found recently [61].

In the present paper, we intend to clarify how borate’s structure affects its thermal expansion using the example of alkaline-earth borates. For this purpose, our data on thermal expansion of four alkaline-earth borates, namely Ca₃B₂O₆ (0D), CaB₂O₄ (1D), Sr₃B₁₄O₂₄ (2D) and CaB₄O₇ (3D), are supplemented by the literature data and summarized. The main values of the thermal expansion tensor and the orientation of relatively crystallographic axes are reported. The parameters of the thermal expansion of Ca-borates are compared with those of other alkaline-earth metal (Mg, Sr, Ba) borates to reveal the common trends in the thermal expansion anisotropy of alkaline-earth borates. The volume of thermal expansion—depending on the B₂O₃ content, cation radius, and structural complexity—is considered.

2. Materials and Methods

2.1. Synthesis

Polycrystalline samples of borates Ca₃B₂O₆, CaB₂O₄, CaB₄O₇ were obtained by solid-state reactions in air from a mixture of high-purity H₃BO₃ and pre-dried CaCO₃ (600 °C, 3 h). The reagent mixtures were ground in an agate mortar and pressed into tablets. Synthesis was performed in air in platinum crucibles at 500–1400 °C. The thermal treatment time varied from 3 to 22 h. Synthesis of Ca₃B₂O₆, CaB₂O₄ and CaB₄O₇ was carried out under repeated thermal treatments with a sequential increase in temperature to 1450, 1200, 950 °C, respectively.

Sr₃B₁₄O₂₄ powder samples were synthesized by glass crystallization. A stoichiometric mixture of SrCO₃ and high-purity H₃BO₃ that had been preliminarily calcinated at 600 °C was kept at 1250 °C for 40 min. The resulting melt was then poured onto a metal substrate. A cooled glassy sample (unground) was placed into a furnace at 780 °C for 30 min for crystallization.

2.2. Methods

2.2.1. X-ray Diffraction

The phase composition of polycrystalline samples was studied at room temperature in air in the range of 5–60° 2θ using a Rigaku MiniFlex II CuKα radiation, 30 kV/10 mA, Bragg–Brentano geometry, X-ray diffractometer.

2.2.2. Powder High-Temperature X-ray Diffraction

The thermal expansion of Ca₃B₂O₆, CaB₂O₄ and CaB₄O₇ was studied in air via high-temperature X-ray powder diffraction (HTXRD) data collected using a Rigaku Ultima IV powder X-ray diffractometer CuKα/CoKα radiation, 40 kV/30 mA, Bragg–Brentano geometry, PSD D-Tex Ultra, equipped with a high-temperature camera. The samples were prepared from a heptane suspension on a Pt–Rh plate. Standard temperature steps of 20–40 °C were adopted.

The procedure for tensor calculations using the ThetaToTensor and RietveldToTensor 2.0 software [62] was as follows: (i) unit-cell parameters at every temperature step were refined by the least-squares or Rietveld method; (ii) the temperature dependencies of the unit-cell parameters were approximated by linear and quadratic polynomial functions; (iii) using the approximation coefficients, the eigenvalues of the thermal expansion tensor and the orientation of its principal axes with respect to the crystallographic axes’ directions were determined; (iv) the three-dimensional surface and the main sections of the figure of the thermal expansion tensor were drawn. Each radius vector of this figure represents the value of the thermal expansion coefficient (TEC) α in this direction. For comparison to the tensor axes’ orientation, crystal structures were visualized using VESTA 3 [63].

2.2.3. Structural Complexity Calculation

Structural complexity calculations provide theoretical estimation of the configurational contributions to the total entropies of crystalline substances [64]. Structural complexity parameters were calculated using the ToposPro 5.5.2.2 program package [65]. The formal calculations are presented in [66,67].

3. Results

3.1. Thermal Expansion of Ca- and Sr-Borates vs. Dimensionality of Borate Anion

Here, we present data on thermal expansion of four Ca-and Sr-borates, in which the dimensionality of borate anion increases from isolated (0D) groups via chain (1D) and layered (2D) structures up to the framework (3D). The studied borates crystallize with various symmetrical Ca₃B₂O₆ (0D) borates with isolated BO₃ triangles in a trigonal manner; CaB₂O₄ (1D) borates with chains of triangles are orthorhombic, and layered Sr₃B₁₄O₂₄ (2D) and framework CaB₄O₇ (3D) borates are monoclinic. The temperature dependencies of the unit-cell parameters and the volume of these borates (Figure 1) were approximated by linear or quadratic polynomial functions (Table S1, see in Supplementary Materials). The calculated TECs (particularly thermal expansion tensor eigenvalues (α₁₁, α₂₂, α₃₃) and their orientation with respect to the crystallographic axes) and the TECs along a, b, and c (α_a, α_b, and α_c) are given in Table S2. Due to the symmetry in the case of trigonal crystals, the expansion in the ab plane is isotropic α₁₁ = α_a = α₂₂ = α_b (Table S2), whereas α₃₃ = α_c differs from α₁₁. When the symmetry is reduced to monoclinic, the orientation of the α₁₁ and α₃₃ tensor axes is determined by the μ_c,3 and μ_a,1 angles in the monoclinic ac plane (Table S2), where μ_c,3 = (c^∧α₃₃) is the angle between the c axis and α₃₃ tensor axis and μ_a,1 = (a^∧α₁₁) is the angle between a and α₁₁ (Table S2). The principal axes of the thermal expansion tensor are oriented relative to the crystallographic axes using TTT and RTT software [62]. With an increase in temperature, the values of α_V rise for each of the four borates, and the degree of anisotropy decreases for all borates except Ca₃B₂O₆. Both tendencies are usually due to an increase in the thermal motion of atoms. Our estimation of the degree of anisotropy of thermal expansion, its interpretation, and the possible structural factors of Ca- and Sr-borates and other alkaline-earth borates are summarized below (pp. 3.2–3.3).

3.2. Thermal Expansion of Alkaline-Earth Borates

Selected characteristics of thermal expansion (such as eigenvalues of the tensor, a₁₁, a₂₂, a₃₃; volume TECs (a_V = α₁₁ + α₂₂ + α₃₃); and anisotropy parameters) for 20 alkaline-earth borates are summarized in Table 1. Thermal expansion of some of these borates has been examined by HTXRPD before [6,51,52,58,59,60,68], supplemented by data from the present work. Here, except for the formula of the compound, its symmetry (system and space group) and the reference of the source, the volume expansion coefficient equal to the sum a₁₁ + a₂₂ + a₃₃, and the degree of anisotropy thermal expansion are given. The thermal expansion coefficients are represented in images of 3D and 2D sections.

Due to the anisometry and symmetry of the crystals, it is obvious that thermal expansion is different in various directions in the crystal, with the exception of cubic crystals, which expand isotropically. To estimate the maximum degree of anisotropy in a plane in a structure, we should choose the plane of maximal anisotropy. The degree of anisotropy in a plane can be described quantitatively as ∆_plane = ∑ (α_max − α_min)/(α_max + α_min) = ∑ (α_max − α_min)/2α_av [69,70]. Here, we introduce a new volume criterion for the anisotropy of thermal expansion:

∆_V = ∑ |α_ii − α_jj|/α_av = (|α₁₁ − α₂₂| + |α₁₁ − α₃₃| + |α₂₂ − α₃₃|)/3α_V, where α_V = α₁₁ + α₂₂ + α₃₃.

In the case of isotropic thermal expansion in a plane and in a volume, both ∆_max and ∆_V are equal to zero. As the difference in the eigenvalues of tensor, α_ii − α_jj, increases, the degrees of anisotropy, ∆_plane and ∆_V, rise as well.

Table 1. Main characteristics of the thermal expansion of alkaline-earth borates.

Chemical Formula	System, Space Group	nΔ:m□ Ratio *	T, °C	α × 106 °C−1				∆plane	∆V	Refs.
				α11	α22	α33	αV
Isolated BO Groups (0D)
Isolated BO3 (−3) **
Mg3B2O6	Orth., Pnmn	2Δ	25	13	9	8	30	0.24	0.33	[58]
			1100	20	13	13	46	0.21	0.30
Ca3B2O6	Trigon., R-3c	2Δ	25	8	8	27	43	0.54	0.88	***
			900	9	9	38	57	0.62	1.04
Sr3B2O6	Trigon., R-3c	2Δ	25	5	5	34	45	0.74	1.32	[59]
			900	5	5	39	48	0.77	1.39
Ba3Sr3B4O12	Tetragon., I4/mcm	4Δ	100	12	=α11	18	43	0.20	0.29	[60]
			800	17	17	29	63	0.26	0.38
Average							〈47〉4	0.45	0.74
Isolated mixed BO3 and B2O5 pyrogroups (−2.5)
Ba5B4O11	Orth.,	12Δ	100	10	15	13	38	0.20	0.26	[60]
	P212121		800	15	13	27	55	0.35	0.51
Ba2Sr3B4O11	Monocl.,	5Δ	100	3	6	34	43	0.84	1.44	[60]
	C2/c		800	5	6	51	62	0.82	1.48
Average							〈49.5〉2	0.55	0.92
Isolated B2O5 pyrogroups (−2)
γ-Ca2B2O5	Monocl., P21/c	2Δ	25	19	7	−1	25	1.11	1.6	[52]
			500	27	7	3	37	0.80	1.3
α-Ca2B2O5	Monocl.,	2Δ	600	31	7	−5	33	1.38	2.18	[52]
	P21/c		900	33	7	2	42	0.89	1.48
γ-Sr2B2O5	Monocl., P21/c	2Δ	25	20	7	1	28	0.90	1.36	[51]
α-Sr2B2O5	Monocl., P21/c	2Δ	828	32	3	4	39	0.83	1.49
Average							〈34〉4	0.96	1.5
Isolated cyclic (triborate) groups from BO3 (−1)
α-BaB2O4	Hex., R-3c	3Δ	20–700	6	6	28	40	0.65	1.1	[6]
β-BaB2O4	Trigon., R3c	6Δ	20–700	3	3	45	51	0.88	1.65	[6]
Average							〈46〉2	0.77	1.37
1D Borates (−1)
CaB2O4	Orth., Pnca	1Δ	25	22	6	1	28	0.91	1.45	**
			900	33	6	6	45	0.69	1.2
SrB2O4	Orth., Pbcn	1Δ	25	4	4	32	39	0.78	1.4	[59]
			900	4	4	35	43	0.79	1.44
Average							〈39〉2	0.79	1.37
Layered 2D Borates (−0.43)
Sr3B14O24	Monocl.,	8Δ:6□	30	2	11	14	27	0.75	0.8	**
	P21/c		800	11	10	13	33	0.12	0.2
Average							〈30〉	0.44	0.57
3D Borates
−0.5
α-CaB4O7	Monocl., P21/n	4Δ:4□	25	8	8	3	19	0.45	0.53	**
			900	13	9	6	28	0.37	0.5
SrB4O7	Orth., Pmn21	4□	25–900	7	9	8	24	0.13	0.17	[59]
BaB4O7	Monocl., P21/c	4Δ:4□	20–700	23	−12	5	16	3.18	4.38	[6]
Average							〈21〉3	1.24	1.69
−0.25
SrB8O13	Monocl., P21/c	12Δ:4□	25	21	9.6	3.9	34	0.69	1.00	[59]
			740	20	9.6	7.3	37	0.47	0.69
LT-BaB8O13	Orth., P22121	12Δ:4□	100–400	6.9	11	−0.5	17	1.1	1.32	[68]
			500	9.4	11.5	−1.7	19.1	1.35	1.38
Average							〈27〉2	0.9	1.09

*—The number of triangles and tetrahedra in the nΔ:m□ ratio is taken from FBB; **—residual charge per polyhedron (BO₃/BO₄); ***—present work.

Table 1. Main characteristics of the thermal expansion of alkaline-earth borates.

Chemical Formula	System, Space Group	nΔ:m□ Ratio *	T, °C	α × 106 °C−1				∆plane	∆V	Refs.
				α11	α22	α33	αV
Isolated BO Groups (0D)
Isolated BO3 (−3) **
Mg3B2O6	Orth., Pnmn	2Δ	25	13	9	8	30	0.24	0.33	[58]
			1100	20	13	13	46	0.21	0.30
Ca3B2O6	Trigon., R-3c	2Δ	25	8	8	27	43	0.54	0.88	***
			900	9	9	38	57	0.62	1.04
Sr3B2O6	Trigon., R-3c	2Δ	25	5	5	34	45	0.74	1.32	[59]
			900	5	5	39	48	0.77	1.39
Ba3Sr3B4O12	Tetragon., I4/mcm	4Δ	100	12	=α11	18	43	0.20	0.29	[60]
			800	17	17	29	63	0.26	0.38
Average							〈47〉4	0.45	0.74
Isolated mixed BO3 and B2O5 pyrogroups (−2.5)
Ba5B4O11	Orth.,	12Δ	100	10	15	13	38	0.20	0.26	[60]
	P212121		800	15	13	27	55	0.35	0.51
Ba2Sr3B4O11	Monocl.,	5Δ	100	3	6	34	43	0.84	1.44	[60]
	C2/c		800	5	6	51	62	0.82	1.48
Average							〈49.5〉2	0.55	0.92
Isolated B2O5 pyrogroups (−2)
γ-Ca2B2O5	Monocl., P21/c	2Δ	25	19	7	−1	25	1.11	1.6	[52]
			500	27	7	3	37	0.80	1.3
α-Ca2B2O5	Monocl.,	2Δ	600	31	7	−5	33	1.38	2.18	[52]
	P21/c		900	33	7	2	42	0.89	1.48
γ-Sr2B2O5	Monocl., P21/c	2Δ	25	20	7	1	28	0.90	1.36	[51]
α-Sr2B2O5	Monocl., P21/c	2Δ	828	32	3	4	39	0.83	1.49
Average							〈34〉4	0.96	1.5
Isolated cyclic (triborate) groups from BO3 (−1)
α-BaB2O4	Hex., R-3c	3Δ	20–700	6	6	28	40	0.65	1.1	[6]
β-BaB2O4	Trigon., R3c	6Δ	20–700	3	3	45	51	0.88	1.65	[6]
Average							〈46〉2	0.77	1.37
1D Borates (−1)
CaB2O4	Orth., Pnca	1Δ	25	22	6	1	28	0.91	1.45	**
			900	33	6	6	45	0.69	1.2
SrB2O4	Orth., Pbcn	1Δ	25	4	4	32	39	0.78	1.4	[59]
			900	4	4	35	43	0.79	1.44
Average							〈39〉2	0.79	1.37
Layered 2D Borates (−0.43)
Sr3B14O24	Monocl.,	8Δ:6□	30	2	11	14	27	0.75	0.8	**
	P21/c		800	11	10	13	33	0.12	0.2
Average							〈30〉	0.44	0.57
3D Borates
−0.5
α-CaB4O7	Monocl., P21/n	4Δ:4□	25	8	8	3	19	0.45	0.53	**
			900	13	9	6	28	0.37	0.5
SrB4O7	Orth., Pmn21	4□	25–900	7	9	8	24	0.13	0.17	[59]
BaB4O7	Monocl., P21/c	4Δ:4□	20–700	23	−12	5	16	3.18	4.38	[6]
Average							〈21〉3	1.24	1.69
−0.25
SrB8O13	Monocl., P21/c	12Δ:4□	25	21	9.6	3.9	34	0.69	1.00	[59]
			740	20	9.6	7.3	37	0.47	0.69
LT-BaB8O13	Orth., P22121	12Δ:4□	100–400	6.9	11	−0.5	17	1.1	1.32	[68]
			500	9.4	11.5	−1.7	19.1	1.35	1.38
Average							〈27〉2	0.9	1.09

*—The number of triangles and tetrahedra in the nΔ:m□ ratio is taken from FBB; **—residual charge per polyhedron (BO₃/BO₄); ***—present work.

3.2.1. Anisotropy of Thermal Expansion vs. Dimensionality of Borate Anions

Anion dimensionality (0D, 1D, 2D, 3D): The borates in the MO–B₂O₃ systems in which M is an alkaline-earth metal were assigned to the following groups according to the dimensionality of the B–O–polyanion (Table 1): zero-dimensional (0D), chain or one-dimensional (1D), layered or two-dimensional (2D), and framework or three-dimensional (3D). Zero-dimensional borates can additionally be divided into subgroups as the polymerization degree of BO₃ and BO₄ polyhedra increases in B–O polyanions. The dimensionality of BO-polyanions or the degree of polymerization depends highly on chemical composition [20]; it increases as the boron oxide content increases. The reason for this trend is evident: the higher the degree of polymerization, the lower the residual charge per the boron–oxygen coordination polyhedron Z [69,70].

Degree of polymerization/residual charge per an averaged boron–oxygen polyhedron Z: Although OD borates (Table 1) occur in MO–B₂O₃ systems over a wide range (0–50 mol.% B₂O₃), as the content of B₂O₃ increases, the degree of polymerization increases gradually [69,71] from an isolated single [BO₃] triangle (a simple group with the greatest $Z$ residual charge equal to −3) through to an isolated mixed anion (BO₃ + B₂O₅) (a triangle and a pyrogroup of two triangles condensed by an oxygen atom in a ratio of 2:1 ($Z$ is −2.5)), to a group (B₂O₅) of two condensed triangles (−2), and finally to a B₃O₆ cyclic (triborate) group of three triangles (−1). It is noteworthy that these isolated anion groups are built up from boron atoms in triangular coordination only. Chains of 1D borates (1:1 stoichiometry, 50 mol.% B₂O₃) are also composed of boron triangles only. With a further increase in anion polymerization (or a decrease in Z residual charge per polyhedron), BO₄ tetrahedra appear in borate anions, forming 3D borates with 1:2 stoichiometry (66.7 mol.% B₂O₃), such as MgB₄O₇ ([72] (# 34397-ICSD [73]), CaB₄O₇ [46,47,48], SrB₄O₇ [74], and BaB₄O₇ [75]. These crystal structures are built up by both triangles and tetrahedra, except the SrB₄O₇ structure, which is composed of tetrahedra only. Among anhydrous borates of alkaline-earth metals, there exist only Sr₃B₁₄O₂₄ layered borates with 3:7 stoichiometry (70 mol.% B₂O₃) [61], located within a compositional range of 3D borates that is not normal but not unusual. A similar situation was observed in alkali borate systems, where 2D borates are located within the “3D borates range” of composition; examples include α-Na₂B₄O₇ (# 2040-ICSD [73]), Rb₃B₇O₁₂ (# 76344-ICSD), Cs₃B₇O₁₂ (# 98582-ICSD), Cs₃B₁₃O₂₁ (# 95728-ICSD), etc. It should be noted that the listed 2D borates can be considered a pseudo-framework; their layer thickness is usually large, the anion consists of several B–O groups, and all oxygen atoms are bridging. This somewhat contradicts the general principle of dimensional reduction [70], according to which the increasing content of the ’ionic’ MO component in the systems is associated with a gradual decrease in the dimensionality of anions. This phenomenon may be due to the so-called “boron anomaly” found for predominantly anhydrous alkali and alkaline-earth borate crystals and glasses by J. Krogh-Moe in the 1960s, as described by Wright [76,77]. It was stated that the number of tetrahedrally coordinated boron atoms increases as the non-boron metal content increases up to MO:2B₂O₃, the greatest amount of tetrahedra (2□:2∆ = 1) being observed in the MO:2B₂O₃ oxide ratio. Then, the number of tetrahedra decreases as the MO content increases further. As seen in Table 1, within the compositional ranges of 0D and 1D borates (B₂O₃ content ≤ 50 mol.%), borate anions are built up from triangles only, and tetrahedra appear in 2D and 3D borates (B₂O₃ content > 50 mol.%).

Anisotropy of thermal expansion: Borates often exhibit sharply anisotropic thermal expansion [21,24]. This anisotropy is primarily determined by the distribution of the sharply anisotropic thermal oscillations of atoms. In the BO₃ triangles, dimers or cyclic triborate groups (comprising triangles only), the maximal exes of ellipsoids of thermal displacements of atoms and the maximal thermal expansion of structure occur perpendicular to the plane of a triangle or a ring, whereas the minimal expansion is parallel to the BO₃ plane [24].

Strong bonds inside boron–oxygen groups and the ability of these groups to rotate with respect to each other around the common oxygen atoms ensure the plastic thermal behavior of borate crystals. As a result, most borates demonstrate greatly anisotropic thermal expansion up to negative values along certain directions. Here, steady high expansion of anisotropy, estimated as ∆_plane and ∆_V, was observed for M₂B₂O₅ (M = Ca, Sr, Ba) (0D) and MB₂O₄ (M = Ca, Sr, Ba) (0D and 1D) based on BO₃ triangles only. Their structures exhibit maximal expansion strongly perpendicular to the BO₃ planes, i.e., along the direction of weak bonds in the crystal structure. Figure 2 shows that for the alkaline-earth borates built by BO₃ triangles, anisotropy increases slightly but steadily as the residual charge per the anionic polyhedron [BO₃]³⁻ decreases down to Z = −1. A further decrease in the residual charge per anionic polyhedron ([BO₃]³⁻ and [BO₄]⁵⁻) results in more scattered ∆_plane and ∆_V values (Figure 2).

3.2.2. Thermal Expansion as a Function of Cationic and Anionic Properties

Thermal expansion is a multiparametric property, and the influence of these parameters can change with temperature. Therefore, it is important to identify their contribution to the expansion. The main characteristics of thermal expansion—magnitude and anisotropy—depend on the size, content, charge of cations, structure and charge of the polyanion, their ability to oscillate in a solid, symmetry, etc. The thermal vibrations of M atoms and those of borate groups can play an important role in thermal expansion.

Cationic properties: It has previously been shown for alkali metals borates [24] that the average volume of thermal expansion (α_V) increases with an increase in the cationic radius [31]. A similar trend is shown in Figure 3. Borates with relatively small cations exhibit the minimum thermal expansion; the mean values of α_V change for alkaline-earth borates, from α_V = 30 for Mg to 36 × 10⁻⁶ C⁻¹ for Ba. An increase in α_V is observed with an increase in the cationic radius. This trend is caused by the increase in both the average M–O bonds and the coordination number (CN) in MO_n polyhedrons from M = Mg (n = 6) to Ba (n = 8–9). Earlier, Hazen et al. [78,79] showed that bond lengths with a higher degree of ionicity, such as M-O, are lengthened more than covalent bonds. Thus, the magnitude of volume expansion will mainly depend on the properties of cations.

Anionic properties: The dependence of α_V on B₂O₃ content is shown in Figure 3. With an increase in the B₂O₃ content in MO–B₂O₃ systems (M = Mg, Ca, Sr, Ba), the number of triangles and/or tetrahedra in the structure increases, which leads to their condensation and a decrease in the number of oxygen bonds with metals, thereby further weakening the weakest bonds and lowering the strength-related properties of the compounds. There is a tendency for volumetric expansion to decrease as a result of an increase in the degree of polymerization. The average volume expansion gradually decreases with a decrease in the residual charge by one polyhedron (Table 1, Figure 3).

With a decrease in the strength of the structure, its melting point and hardness, compressibility, solubility, etc., decrease. The correlations between the melting point, the volume thermal expansion, and the residual charge of borates of alkaline-earth metals are shown in Figure 3c,d. Analysis of the data for borates of alkaline-earth metals with triangular and tetrahedral radicals showed that there is a clear tendency for the melting temperature to decrease as the residual charge of the radical decreases. With equal values of Z, the melting point is practically independent of the type of the cation and the structure and degree of polymerization of radicals (either isolated cyclic groups of three triangles and chains of triangles or frameworks containing various borate groups). The volumetric thermal expansion also decreases slightly with a decrease in the residual charge per polyhedron. If we examine the effect of the residual charge on the volumetric expansion of borates built only from triangles only, we can see that the volumetric expansion decreases very slightly, but it is almost constant (Figure 3).

Structural complexity: An effective framework for assessing complexity is Shannon’s information theory [80], as modified by S.V. Krivovichev for crystal structures [64,66,67]. According to the complexity classification of crystal structures, most compounds of MO–B₂O₃ systems are either simple (20–100 bits/cell) or intermediate (100–500 bits/cell). The structure of Ba₅(BO₃)₂(B₂O₅) is very complex (1417.65 bits/cell) (this point is not represented on the graphs due to the large difference in values). This sharp increase in the structural complexity of this barium borate is associated with content of two types of anionic group (isolated BO₃ triangles and B₂O₅ pyroborate groups). In general, with an increase in the B₂O₃ content in MO–B₂O₃ systems (M = Mg, Ca, Sr, Ba), the complexity of the borates also increases (Figure 4).

The correlation between the melting point and structural complexity of borates is shown in Figure 4. The observed decrease in the melting temperature of alkaline-earth metal borates with increasing structural complexity shows that both properties are a function of the polymerization of boron–oxygen radicals or residual charge per polyhedron. The reason for this is evident: chemical compounds with greater complexity melt or decompose at lower temperatures.

3.3. Structural Interpretation of Thermal Expansion of Borates in Earth-Alkaline Metals

Thermal vibrations of B–O groups: As has been shown before, when a group of atoms have bonds as strong as in BO₃, the entire group may produce oscillations called rigid-body motion. The thermal vibrations of a group of atoms were first described as rigid-body motion by Cruickshank [81], and much later, these studies were generalized by Downs [82]. The thermal vibrations of any atom in a group are dictated by the common motion of the group. In the case of a BO₃ triangle, oxygen and boron atoms mainly oscillate perpendicular to the strong and significantly covalent B–O bonds [24,83]. There are three B–O bonds in the triangle plane; thus, B and three O atoms vibrate maximally and approximately perpendicular to the plane of BO₃ triangles. Consequently, the triangle as a whole has to vibrate in the same direction, i.e., nearly perpendicular to the triangle plane. If a structure is based on isolated triangles in a preferred nearly parallel orientation, highly anisotropic thermal expansion is usually observed. In the case of a BO₄ tetrahedron, B and O atoms also vibrate practically perpendicular to the strong B–O bonds. However, the B–O bonds of the BO₄ tetrahedron are isometrically distributed in 3D space. Thus, the BO₄ tetrahedron itself oscillates randomly, and its thermal motion does not significantly affect the anisotropy of the thermal expansion of the structure. In most structures containing boron atoms in a triangular coordination, BO₃ triangles are arranged in a preferred orientation almost parallel to each other [83] (see Figure 5 as an example). However, sometimes, this trend of self-assembly is not realized; illustrative examples are borates that are structurally similar to the anti-zeolite [60] and godefroyite families [84]. In this case, the thermal expansion is close to isotropic. This is clearly visible, for instance, in the Sr₂CaBi(REEO)₃(BO₃)₄ borate of the godefroyite family [84], where α_a = 12 and α_c = 13 × 10⁻⁶ C⁻¹; meanwhile, in the Ba₃Sr₃B₄O₁₂ anti-zeolite, the thermal expansion is complicated by the splitting of the oxygen positions [60].

Borate rigid groups in addition to the simple polyhedra include diborate (2B), cyclic triborate (3B), tetraborate (4B), pentaborate (5B) groups, etc. [21,83]. These more complex units retain their configuration and can oscillate or rotate as a whole. Among alkaline-earth borates, there are 0D borates based on isolated triangles (M₃B₂O₆), 0D borates based on pyroborate groups of two or three corner-sharing triangles (M₂B₂O₅ and MB₂O₄), 1D borates built up from the chains of triangles (MB₂O₄), and 2D and 3D borates based on various rigid groups. An approach based on the thermal vibrations of rigid groups [21,22,83] is applied below to explain the dramatic anisotropy of many borates.

Thermal expansion as a function of cationic properties

Borates based on isolated groups of BO₃ triangles

M₃B₂O₆ (M = Mg, Ca, Sr, (Sr,Ba)). (0D). Four phases occur in borates of this stoichiometry which contain isolated BO₃ triangles. The size of the cation and its CN increase from Mg (C.N. 6, R = 0.86 Å) via Ca (C.N. 8, R_cr = 1.26 Å) and Sr (C.N. 8, R_cr = 1.4 Å) to (Sr, Ba) (C.N. 6–11, R_cr = 1.49–1.54 Å). The crystal structures of Mg-, Ca-, and Sr-borates of this stoichiometry are similar. Moreover, Ca- and Sr-borates are isotypical, while Ba₃Sr₃[BO₃]₄ is related to the “anti-zeolite” family in which isolated BO₃ triangles are orientationally disordered over four and eight orientations as a result of the splitting of O sites.

Mg₃B₂O₆ [85] (ICSD-31385 [73]), 1B:1Δ:Δ (hereafter, notations introduced by Burns et al. [86] are used). In the orthorhombic Mg₃B₂O₆, the planes of the isolated BO₃ triangles are not parallel to each other. As a result, it expands less than the other members of the group, and its anisotropy is insignificant (Table 1). This is clearly caused by short and strong Mg–O bonds in MgO₆ polyhedra [58].

Ca₃B₂O₆ (ICSD-1894 [73]) and Sr₃B₂O₆ (ICSD-93395 [73]), 1B:1Δ:Δ. In these isotypical trigonal structures, Ca₃B₂O₆, and Sr₃B₂O₆, isolated BO₃ triangles are arranged perpendicular to the c axis (Figure 5). For this reason, high-expansion anisotropy is observed; both structures expand strongly perpendicular to the planes of the BO₃ triangles (Table 1). The values of TECs are quite close (α_V = 53 and 57 × 10⁻⁶ C⁻¹ for Ca and Sr, respectively) due to the structural similarity.

Ba₃Sr₃(BO₃)₄, 4B:4Δ:Δ,Δ,Δ,Δ [60] is a member of the “anti-zeolite” borate family and is structurally close to the Ba₃B₂O₆ borate. Oxygen atoms bonded to B atoms are split; therefore, the BO₃ triangles are orientationally disordered over four and eight orientations, and triangles are oriented in a non-parallel manner. In this case, it is expected that anisotropy will be weak. As expected, the thermal expansion of Ba₃Sr₃B₄O₁₂ is less anisotropic.

0D Borates with pyroborate groups

M₅B₄O₁₁ (M = (Sr,Ba), Ba), 3B:3Δ:Δ,ΔΔ is based on isolated BO₃ triangles and B₂O₅ pyroborate groups in a ratio of 1:2. Ba₅B₄O₁₁ (# ICSD-250321) is orthorhombic CN Ba 8–10, R = 1.56–1.66 Å. The monoclinic Ba₂Sr₃B₄O₁₁ structure contains B₂O₅ pyroborate groups orientationally disordered over two sites. Thermal expansion in this group is strongly anisotropic for monoclinic Ba₂Sr₃B₄O₁₁ (Table 1) and less anisotropic for orthorhombic Ba₅B₄O₁₁ [60].

M₂B₂O₅ (M = Ca, Sr), 2B:2Δ:ΔΔ. γ-Ca₂B₂O₅ (ICSD-66516 [73]) and α-Ca₂B₂O₅ [52]: as mentioned above, the structures of both modifications contain isolated B₂O₅ pyroborate groups composed of two corner-sharing BO₃ triangles (Figure 3). Both monoclinic structures expand highly anisotropically up to compression for the γ-phase (Table 1). Thermal displacement of atoms occurs in a direction almost perpendicular to the plane of both triangles. Strong anisotropy of expansion is also caused by shear deformations in the monoclinic plane, which are sharply anisotropic in nature [24]. A decrease in monoclinic angle is accompanied by the most intensive expansion along the [101] direction and minimal thermal expansion including contraction along [101].

The volume expansion values of γ-, β- and α-Sr₂B₂O₅ are close to each other and Ca₂B₂O₅, except that of α′-Sr₂B₂O₅. The α_V value of α′-Sr₂B₂O₅ (181 × 10⁻⁶ C⁻¹) is unusually large. The thermal expansion of the Sr₂B₂O₅ modifications is strongly anisotropic, even with a negative thermal expansion in some directions (Table 1). The anisotropy of all of them is similar: the maximal (α₁₁) tensor axis is directed practically perpendicular to planes of BO₃ pyroborate groups [51].

0D Borates with cyclic (triborate) groups of B₃O₆

BaB₂O₄ (0D), 3B:3Δ:<3Δ> exists in two polymorphic modifications (α and β), and the transition temperature is 925 °C [75,87]. Both modifications are based on the same isolated groups of B₃O₆. Both modifications crystallize in the trigonal system, and the space groups are R3c for β-BaB₂O₄ (a = 12.53, c = 12.72 Å; 69319-ICSD) and R-3c for α-BaB₂O₄ (a = 7.235, c = 39.192 Å; 14376-ICSD). The anisotropy of expansion is dictated by the orientation of rigid 3B groups in the structure, where the expansion in the plane is minimal, and the expansion perpendicular to the plane is maximal. The comparison between both modifications shows that the nonlinear-optical β-polymorph expands more anisotropically [6] than the centrosymmetric phase, α-BaB₂O₄ (Table 1) [24], although their volumetric expansion is comparable (α_V = 40 and 51 × 10⁻⁶ C⁻¹ for α- and β-BaB₂O₄, respectively).

1D Borates based on chains of BO₃ triangles

MB₂O₄ (M = Ca, Sr, Ba), 1B:Δ:Δ (1D). CaB₂O₄, Pnca: Ca CN 8 (ICSD-62430). The structure contains the chains of BO₃ triangles along the c axis. Planes of the triangles are approximately parallel to the (001) plane (Figure 5b). The structure expands intensively along the a axis, in a direction perpendicular to the plane of borate triangles. (Table 1). Minimal expansion occurs along the boron–oxygen chains parallel to the c axis since the strongest bonds occur in this direction.

The structures of CaB₂O₄ and SrB₂O₄, Pbcn (203226-ICSD) borates are similar. These borates contain chains of BO₃ triangles; therefore, the nature of their thermal expansion is similar. Thus, an expected anisotropy of thermal expansion is observed: the structure also expands maximally in the direction perpendicular to the plane of [BO₃] triangles (α_a and α_b << α_c (Table 1) [59].

2D and 3D Borates

Borates of 2D and 3D dimensionality are built up from different groups connected via vertices—single-triborate (3B), multi-triborate (double-tetraborate (4B), and pentaboratea (5B) groups).

Sr₃B₁₄O₂₄ (2D), 14B 8Δ6□:[B₁₄O₃₀]:<2Δ□><2Δ□>=<2Δ2□><2Δ□>□ΔΔ□ (ICSD-136651 [73]). The thick complex layer of the structure (Figure 5c) is composed of two [B₄O₁₀] units and a [B₄O₉] double ring connected via two triangles [BO₃] forming a polyanion [B₁₄O₃₀] [61]. The anisotropic expansion is typical of layered structures, with maximal expansion perpendicular to the ab layer plane. Half of the triangles (four of eight) are arranged almost parallel to the plane of the layer.

MB₄O₇ (M = Ca, Sr, Ba) (3D)

α-CaB₄O₇, 8B: 4Δ4□:<Δ2□>=<Δ2□>□<2Δ□>, monocl., P2₁/n, CN Ca 7, 8 (ICSD-200081 [73]). The crystal structure (Figure 5d) is characterized by a boron–oxygen polyanion consisting of four crystallographically independent BO₃ triangles and four BO₄ tetrahedra [47]. The eight triangles and tetrahedra form a repeated [B₈O₁₄]⁴⁻ unit, which consists of three B–O-groups linked via common vertices—a single tetrahedron, single 3B ring composed of two triangles and a tetrahedron, and a 4B double ring in which two 3B rings from two tetrahedra and a triangle are condensed via two tetrahedra. The Ca atoms are in seven- and eight-vertex polyhedra of oxygen atoms. The maximal direction of thermal expansion in the structure (α₁₁ = 8 × 10⁻⁶ °C⁻¹) coincides with an acute angle bisector of the ac parallelogram, and the minimum expansion occurs in the other diagonal direction (Figure 5d, left). The structure expands along the b axis less intensely: α₂₂ = α_b = 8 × 10⁻⁶ °C⁻¹. Such sharp anisotropic behavior could be explained by the shear deformation of the monoclinic plane [69,70]. The reason for such anisotropy of expansion is the arrangement of the 3B and 4B groups. The planes of these groups are located along the b axis and are almost perpendicular to the bisector of the acute angle.

SrB₄O₇, 4B:4□: [φ] <3□>|□| (3D) (ICSD-27404) has almost isotropic thermal expansion. This structure is composed of BO₄ tetrahedra only. Moreover, it contains the units built by three tetrahedra sharing one common oxygen atom, which is presumably responsible for the isotropic thermal expansion [59].

BaB₄O₇, 8B: 4Δ4□:<Δ2□><2Δ□>–<Δ2□>, is monoclinic, P2₁/c [75]. The framework consists of <Δ2□> 3B and <2Δ□>–<Δ2□> 5B groups composed of two single 3B rings; in the cavity of the framework, there are two nonequivalent barium atoms. The thermal expansion of the BaB₄O₇ borate is the largest among anisotropic alkaline-earth borates; the expansion is maximal in the ac monoclinic plane, whereas the contraction occurs along the b axis. The anisotropy of deformations of the ac monoclinic plane is caused by the considerable change in the angle β and the related shear deformations as well as the arrangement of the 5B groups. As discussed before [6,21,83], in the case of structures with 5B groups, the internal oxygen and boron atoms of both single rings vibrate perpendicular to the planes of these rings, while the group as a whole oscillates relative to the axis of the 5B group—the line drawn parallel to the plane of both rings. The maximum expansion occurs perpendicular to the axis of the group, and the minimal expansion is along the axis of group. A similar form of thermal expansion is typical of hydrous and anhydrous alkali pentaborates of 0D, 1D, 2D, and 3D dimensionality [21,24].

It is reasonable to conclude that the 5B groups are arranged in parallel, and in this α₃₃ direction, the structure expands weakly in comparison to the α₁₁ direction in the ac plane [21]. The thermal contraction of the structure along the b axis can be associated with the displacement of the Ba atoms so that the B–O framework is adapted to the displacement of cations due to the rotation of the B–O groupings according to the hinge mechanism.

MB₈O₁₃ (M = Sr, Ba) (3D)

Sr₂B₁₆O₂₆, 16B: 12Δ4□:2(<2Δ□><2Δ□>–<2Δ□>) [88]. There are two independent Sr atoms, 16 B atoms, and 26 O atoms in asymmetrical unit. Each of two penetrating frameworks consists of two [B₃O₅]⁵⁻ 3B rings (two triangles and a tetrahedron) and two [B₅O₈]⁵⁻ 5B groups (two triborate rings sharing a common tetrahedron). The direction of the maximal thermal expansion is close to the c axis and close to the axis of one of the 5B groups. There are a few reasons for this. First of all, planes of both 3B rings are practically perpendicular to the c axis; another explanation could be the shear character of deformations due to the change in the monoclinic β angle not fixed by symmetry, as noted in [59].

Low-temperature BaB₈O₁₃ polymorph, 16B: 12Δ4□:<2Δ□>–<2Δ□><2Δ□><2Δ□><2Δ□>ΔΔ [68]. The crystal structure is based on the heteropolyhedral framework comprised of vertex-sharing [B₅O₁₀] 5Bgroups, three [B₃O₇] 3Bgroups and [B₂O₅] 2Bgroup. In the fundamental building block of the heteropolyhedral borate framework, the [B₂O₅]-diborate and 5Bgroups are connected exclusively by triborate groups via common oxygens. The thermal expansion of the BaB₈O₁₃ borate is strongly anisotropic with anisotropy increasing with temperature (Table 1). The crystal structure shrinks along [001] over the studied temperature range (30–680 °C).

4. Conclusions

The thermal expansion of four alkaline-earth borates, Ca₃B₂O₆, CaB₂O₄, Sr₃B₁₄O₂₄ and CaB₄O₇, was investigated by in situ powder HTXRD. A comparative analysis of their thermal expansion with the data on the other alkaline-earth borates showed that most of them expand sharply anisotropically.

As can be expected, the strongest anisotropy of thermal expansion was observed for the 0D and 1D structures based on BO₃ triangles only from the 0–50 mol.% B₂O₃ range of composition. These borates expand most intensively perpendicular to the BO₃ plane, i.e., along the direction of weaker bonding in the crystal structure. Such a form of expansion is caused by the minimal thermal atomic vibration within the plane of the triangle and the maximal vibration in the perpendicular direction. The borates built by complex rigid groups expand dramatically anisotropically when the complex units contain more triangles than tetrahedra or their quantities are comparable. In contrast, SrB₄O₇, composed of tetrahedra only, demonstrates almost isotropic thermal expansion due to the random distribution of the rigid B–O bonds in the crystal structure [6,24,59]; moreover, there is weak anisotropy in layered Sr₃B₁₄O₂₄ and in a few borates in which BO₃ planes are depicted not in parallel and the amount of BO₃ is not significant.

Our analysis showed that ∆_plane can serve as an optimal quantitative estimation of the degree of anisotropy of thermal expansion. Both ∆_plane and ∆_V gave similar results. Thus, in order to estimate the degree of anisotropy, it is generally enough to estimate the anisotropy of the most anisotropic plane, ∆_plane.

The most common trends in the thermal expansion of alkaline-earth borates emerged from the correlations of thermal properties like volume thermal expansion, degree of anisotropy, dimensionality of B–O anion, residual charge per one polyhedron (BO₃/BO₄), cationic size, and structural complexity. First, we can conclude that sharp anisotropy is caused by the strong bonding within the triangle plane; melting temperatures and volume expansion decrease as the dimensionality of the borate polyanion (and, more precisely, the residual charge) decreases. The structural complexity increases and the melting point values decrease with the increase in B₂O₃ content, i.e., with the increase in the degree of polymerization. The most steady trends were found for borates based on BO₃ triangles only.