Structure and properties of CaF₂–B₂O₃ glasses

Abstract

FTIR spectroscopy has been employed to investigate the structure of CaF₂–B₂O₃ glasses. It is proposed that CaF₂ partially modifies the borate network forming ${\text{Ca}}_{ 1 / 2}^{ 2+ } [{\text{BO}}_{ 3 / 2} {\text{F]}}^{ - }$units. The rest of CaF₂ is assumed to build an amorphous network formed of CaF₄ tetrahedra. Analysis of density and molar volume revealed that the volume of CaF₄ tetrahedron in the studied glasses is slightly greater than that in the crystalline form. Data of density, molar volume, and electric conductivity have been correlated with the glass structure. As far as the authors know, CaF₂–B₂O₃ glasses are investigated for the first time.

Introduction

Since publishing of the pioneer work of Gressler and Shelby on PbF₂–B₂O₃ [1] and PbO–PbF₂–B₂O₃ glasses [2], the structure and properties of borate glasses containing F⁻ ions have been the subject of growing interest. These types of glass show a wide variety of physical properties and are characterized with their anionic conduction. They are classified among the well-known fast ionic conductors. However, there is a main controversy about the role of F⁻ ions.

X-ray photoelectron spectroscopy (XPS) of F (1s) in (70−x)PbO·xPbF₂·30B₂O₃ glasses [3] showed two peaks. A peak is assigned to F⁻ ions in Pb–F–Pb clusters (free F⁻) and the other is related to non-bridging fluorine atoms in the borate network (B–F⁻…Pb²⁺). Gopalakrishnan et al. [4] indicated that in xPbF₂·(1−x)B₂O₃ glasses fluorine is incorporated predominantly as B–O–F at high B₂O₃ content, whereas it forms Pb–O–F bonds at high PbF₂ content.

Chowdari and Rong [5] indicated that upon increasing the fluorine content in fluorinated lithium borate glasses, the number of four-coordinated boron atoms increases and the asymmetric BO₃ groups with non-bridging oxygen atoms decreases. These results are assumed as an evidence that fluorine participates in the boron–oxygen network. X-ray photoelectron spectra of F (1s) show two peaks. One of them is related to fluoride ions associated with lithium ions forming Li–F bond and the other is assigned to non-bridging fluorine atoms (B–F…Li⁺). The latter may contribute to the formation of BO₃F and BO₂F₂ units with non-bridging bonds. Formation of these structural units was confirmed by NMR investigations on NaF–Na₂O–B₂O₃ glasses [6, 7] with an estimation of the highest probability for the formation of BO₃F units. Suresh and Chandramouli [8] discussed structural variations in (30−x) Na₂O·xNaF·50B₂O₃·20Bi₂O₃ glasses, with increasing the NaF content, on the basis of the formation of BO₃F and BO₂F₂ units. Both structural units are also considered to be formed in Li₂O–SrO–SrF₂–B₂O₃ glasses [9].

Alkali fluoroborate glasses with alkali fluoride content up to 45 mol% LiF or 55 mol% NaF or KF showed composition-properties trends similar to that observed for alkali borate glasses. To explain the differences in behavior between the alkali fluoroborate glasses and the corresponding alkali borate glasses Shelby and Baker [10] proposed a structural model based on the replacement of BO₄ tetrahedra by BO₃F tetrahedra. The model is based on NMR investigations in sodium fluoroborate glasses [11, 12].

Formation of some different types of fluoroborate units like BO₂F (or BOF₂) is assumed in B₂O₃–xLi₂O–yLiX (X = F, Cl, Br, and I) [13]. The B–F in these units is a terminal one and the bond can be described as a non-bridging bond.

Changes in the electric conductivity with fluorine content in MeF₂–Na₂B₄O₇ glasses (Me = Mg, Ca, Sr, and Ba) have been attributed to formation of a number of structural units. Sokolov et al. [14] assumed the structure of those glasses is built up of nonpolar BO_3/2 units and various polar groupings such as Na⁺[BO_4/2]⁻, Na⁺[F⁻BO_3/2]⁻, Me_1/2[BO_4/2]⁻, Me_1/2[F⁻BO_3/2]⁻, [MeF_4/2], and [MeF_6/3]. Among these units, Na⁺[BO_4/2]⁻ and Na⁺[F⁻BO_3/2]⁻ are the most probably formed species [15, 16]. The negative charge on the regular [BO_4/2] tetrahedron has a uniform distribution, whereas it is localized at F⁻ ion in the distorted [F⁻BO_3/2] tetrahedron.

Another type of structural units is proposed by Sokolov et al. [17]. They proposed formation of Na⁺[F⁻BO_4/2]⁻ oxyfluoride structural units in NaF–Na₂O–B₂O₃ and NaF–B₂O₃ glasses. Although these units are electrically unbalanced, it has been indicated that fluorine ions are not consumed only in these units but they can also be bound to Na⁺ ions in F⁻…Na⁺[F⁻BO_4/2] structural units.

Hager and El-Hofy [18] used IR spectroscopy to study the structure of (70−x)B₂O₃·30BaF₂·xLiX glasses, where X = F, Cl and Br. They concluded that these glasses mainly contain BO₃, BO₂F triangles and BO₄, and BOF₃ tetrahedra. Formation of BO₂F triangles is, however, ruled out in NaF–Na₂O–B₂O₃ glasses [7, 19].

Multi nuclei ⁷Li, ¹¹B, ¹⁹F, and ²⁰⁷Pb single- and double-resonance NMR experiments indicated that units such as BO_4/2 ⁻, BO_3/2F⁻ are expected to form in 50B₂O₃–(50−x)PbO–xLiF glasses (10 ≤ x ≤ 40 mol%) [20]. Furthermore, formation of BO_2/2F units is also possible. The ¹⁹F–⁷Li dipole–dipole coupling of 50B₂O₃–10PbO–40LiF glass is weaker than that of crystalline LiF. This result stands against considering formation of large LiF-like domain in these glasses. At the same time, it agrees well with the assumption that LiF contributes in the formation of the above structural units. The authors also assumed a random distribution of F⁻ ions with respect to Pb²⁺ ions without preferential formation of PbF₂-like domains.

Ayta et al. [21] investigated Li₂O–B₂O₃–Al₂O₃ glasses containing CaF₂. The compositions they investigated are [(50Li₂O–45B₂O₃–5Al₂O₃) (mol%) + xCaF₂ (wt%)], with 0 ≤ x ≤ 50. Thermoluminescence glow curves presented an evidence for CaF₂-crystal formation in these glasses.

The above literature survey reveals that a little work has been carried out on alkaline earth oxyfluoro borate glasses. In addition, as far as the authors know, no work has yet been done on binary alkaline earth fluoroborate glasses. In this article, the structure of CaF₂–B₂O₃ glasses would be explored by means of infrared spectroscopy. Density and electric conduction parameters would be correlated with the glass structure.

Experimental

Glasses having the formula xCaF₂–(100−x)B₂O₃ (30 ≤ x ≤ 45 mol%) were prepared from reagent grade chemicals. Boric acid H₃BO₃ and calcium fluoride CaF₂ were used as sources for the corresponding oxides. The glasses were prepared under normal atmospheric conditions by melting the mixture of raw materials in porcelain crucibles in an electric furnace for about 30 min at a temperature ranged between 800 and 900 °C depending on the composition. To get a homogeneous glass the melt was swirled from time to time. After the melt becomes homogeneous, it was poured on steel plates and then pressed by a ceramic plate to obtain glass disks at room temperature. The disks were used for the measurement of the electrical conductivity. Samples prepared by pouring the melt onto steel plates without pressing were used for the density measurement. All the obtained samples were visually homogeneous and transparent. The glass samples were kept in desiccators until required.

The amorphous nature of the samples was confirmed by X-ray diffraction investigations. A Bruker D8 Advance powder XRD was used. It is fitted with a Vantech Super Speed position sensitive detector and a Cu K _α X-ray tube with a Gobel Mirror. Measurements were made over the range 4°–130° in 2-theta. X-ray diffraction patterns of all the studied glasses show only broad humps typical of amorphous materials.

The density (D) of glasses was determined at room temperature using the Archimedes method with xylene as an immersion fluid. At least, three samples of each glass were used to determine the density. Density values are precise to ±0.02 g/cm³.

For measuring the dc resistivity, polished disks with a thickness of about 1 mm were coated with silver paste to serve as electrodes. The resistance was measured using a type TM14 insulation tester (Levell Electronics Ltd, UK) with a range of 10³–10¹³ Ω. Three samples of each glass were used to measure the resistance. The experimental error in determining the activation energy for conduction is estimated to be less than 0.025 eV, whereas the relative error in the conductivity is expected to be ±5%.

The infrared spectra of the glasses were recorded at room temperature using the KBr disk technique. A Mattson 5000 FTIR spectrometer was used to obtain the spectra in the wavenumber range between 400 and 2000 cm⁻¹ with a resolution of 2 cm⁻¹. At least two spectra for each sample were recorded. Infrared spectra were corrected for the dark current noises and background using the two-point baseline correction. After correction, the IR spectra were analyzed using the deconvolution method. More details about this method are reported elsewhere [22, 23].

Results

Figure 1 shows normalized IR spectra of the investigated xCaF₂·(100−x)B₂O₃ glasses (30 ≤ x ≤ 45 mol%). There are two main strong absorption bands in the regions 800–1200 and 1200–1800 cm⁻¹. In addition, a small absorption band appears around 700 cm⁻¹ in all the spectra. There is mostly no change of band centers with composition of glass.

Fig. 1

Infrared spectra of the investigated xCaF₂–(100−x)B₂O₃ glasses. Values of x are given at the plots

Figure 2 shows the dependence of density (D) and molar volume (V _m) on the CaF₂ content in CaF₂–B₂O₃ glasses. There is a linear increase in the density, whereas a linear decrease is observed in V _m. The increase in D reveals that the developed structural units, by replacing CaF₂ for B₂O₃, are denser than the BO₃ units. On the other hand, the decrease in V _m indicates that the resultant volume of the formed structural units is smaller than that of the start glass (30CaF₂·70B₂O₃).

Fig. 2

Dependence of the density D and molar volume V _m of the studied glasses on the CaF₂ content. Open symbols represent the experimental values and the closed ones are calculated. Eq. 7 and the experimental D data were used to get experimental V _m values. Estimated error limit in the latter is about 0.9%. Eqs. 6 and 8 are used to get the calculated D and V _m values. Lines are fitting plots of the data

All investigated glasses show a linear dependence of the logarithm of conductivity (log σ) on the reciprocal of absolute temperature (1/T), Fig. 3. This a feature of ionic conduction process that can be described by the Arrhenius relation

$$ \sigma = \sigma_{ 0} \exp ( - E /kT) $$

(1)

In this relation σ ₀ is a constant that depends on the glass composition and its thermal history, E is the activation energy for the conduction process, k is the Boltzmann’s constant and T is the absolute temperature.

Fig. 3

Logarithm of the electrical conductivity (log σ) of the investigated xCaF₂–(100−x)B₂O₃ glasses on the reciprocal of absolute temperature (1/T). Lines are fitting plots of the data

Discussion

Infrared spectra

The IR spectra shown in Fig. 1 can be described based on the assignment of those of borate glasses modified with oxides. The absorption of borate glasses in the region ~850–1100 cm⁻¹ is related to B–O bond stretching vibration of BO₄ tetrahedra and the absorption peaks in the region ~1150–1600 cm⁻¹ are attributed to stretches of B–O in BO₃ units [24–28]. Formation of BO₄ tetrahedra in oxide glasses results from modifying the borate matrix with the modifier oxide. In this study, where there is no modifier oxide, B₂O₃ is modified only with CaF₂. The presented literature survey on the structure of fluorborate glasses reveals that, among various structural units that may form in such glasses, [BO_3/2F]⁻ distorted tetrahedra are suggested to be the single [10–12] or the most probably formed [6, 7, 15, 20] oxyfluoro borate species. In this study, four-coordinated boron atoms are assumed to be in the form of BO₃F (B₄ units) and the absorption in the region ~800–1200 cm⁻¹ might be correlated with their vibrations.

IR spectra of various glasses [27–29] could be quantitatively analyzed by deconvoluting the absorption peaks into their individual component bands. Figure 4 shows, as an example, the deconvolution of IR spectrum of the glass 45CaF₂·65B₂O₃.

Fig. 4

Deconvolution of the infrared spectrum of the glass 45CaF₂·55B₂O₃

Figure 5 shows a slight increase in the fraction N ₄ of four-coordinated boron atoms with increasing the CaF₂ content. The fraction N ₄ of four-coordinated boron atoms can be calculated from the area under the component bands since the area is related to the concentration of the borate groups originating it. N ₄ is defined as the ratio of [concentration of B₄ units/concentration of (BO₃ + B₄) units]. The increase in N ₄ indicates that BO₃ units convert successively into B₄ units. The values obtained of N ₄ (Fig. 5) agree quite well with those found for CaO–B₂O₃ glasses by analyzing the IR spectra [30] and pulsed neutron diffraction and molecular dynamics studies [31]. N ₄ values are nearly the same at similar concentration of CaF₂ and CaO. Because of the comparable masses and nearly equal radii of fluorine and oxygen, drastic changes are not expected to be observed in the mid-IR spectra when introducing fluorine at the expanse of oxygen [32].

Fig. 5

Fraction of four-coordinated boron N ₄ as a function of the CaF₂ content in xCaF₂–(100−x)B₂O₃ glasses. Error limit for N ₄ values is estimated as ±5%. Line is a fitting plot of the data

N ₄ values obtained from the equation of fitting line in Fig. 5

$$ N_{4} = 0. 0 0 3 5x + 0. 2 8 6 $$

(2)

can be employed to obtain the concentration of various structural units in the studied glasses. The quantity (B₂O₃)₄ (mol%) of B₂O₃ converted into B₄ units, is given as

$$ ({\text{B}}_{ 2} {\text{O}}_{ 3} )_{ 4} = N_{ 4} ({\text{B}}_{ 2} {\text{O}}_{ 3} ) $$

(3)

where (B₂O₃) is the concentration of B₂O₃ in the glass (mol%). The quantity (B₂O₃)₄ equals to the quantity, CaF₂(m), of CaF₂ that modifies B₂O₃ to B₄ units at a rate of two units per CaF₂ molecule. The rest of B₂O₃ that remains as BO₃ units (B₂O₃)₃, can be given by

$$ ({\text{B}}_{ 2} {\text{O}}_{3} )_{3} = {\text{ B}}_{ 2} {\text{O}}_{3} - N_{4} ({\text{B}}_{2} {\text{O}}_{3} ). $$

(4)

In addition, the quantity CaF₂(f), of CaF₂ that enters the structure as network former is given as

$$ {\text{CaF}}_{2} ({\text{f}}) = {\text{CaF}}_{2} - N_{4} ({\text{B}}_{2} {\text{O}}_{3} ). $$

(5)

In crystalline CaF₂, Ca is tetrahedrally coordinated with F⁻ ions [33]. It is, therefore assumed that CaF₂(f) may be present in the structure in the form of CaF₄ tetrahedra. Figure 6 shows X-ray diffraction patterns of crystalline CaF₂ and CaF₂–B₂O₃ glasses having 30, 40, and 45 mol% CaF₂. There is coincidence of the 2θ values of the sharp peaks in the spectrum of crystalline CaF₂ with the centers of the broad humps in the diffraction patterns of the glasses. This suggests that CaF₂ is present in its amorphous form in the studied glasses. In Fig. 7 is shown the dependence on CaF₂ content of (B₂O₃)₄, (B₂O₃)₃, and CaF₂(f), as obtained from Eqs. 3–5, respectively. There is a linear decrease in both (B₂O₃)₃ and (B₂O₃)₄ while CaF₂(f) increases linearly overall the CaF₂ content. The decrease in both of (B₂O₃)₃ and (B₂O₃)₄ can mainly be attributed to the low rate of N ₄ increase (Fig. 5) and partially to the decrease in the content of B₂O₃ when increasing that of CaF₂. The change in the content of (B₂O₃)₄ and CaF₂(f) indicates that CaF₂ tends preferentially to form its own matrix rather than modifying the borate network.

Fig. 6

XRD patterns of crystalline CaF₂ (plot a) and CaF₂–B₂O₃ glasses having 30, 40, and 45 mol% CaF₂ (plots b, c, and d, respectively)

Fig. 7

Dependence on the CaF₂ content of B₃ (triangular boron (B₂O₃)₃), B₄ (tetrahedral boron (B₂O₃)₄) and CaF₂(f) in the studied glasses. Concentration of CaF₂(m) equals to that of B₄. Lines are fitting plots of the data

Density and molar volume

The information obtained from the IR spectra can be used to calculate D and V _m. As previously indicated [34], the density of a glass can be expressed as

$$ D = \frac{{\sum\nolimits_{u} {n_{u} M_{u} } }}{{\sum\nolimits_{u} {n_{u} V_{u} } }} $$

(6)

where n _u is the number of the structural unit u per mole of glass. M _u and V _u are, respectively, the mass and volume of the structural unit.

As revealed from the IR spectra, the investigated glasses contain different structural units. These are symmetric BO₃ units, B₄ units and the network former units of CaF₂. On the basis of Eq. 6 the density can then be expressed as

$$ D = (n_{4} M_{4} + n_{3} M_{3} + n_{\text{f}} M_{\text{f}} ) /(n_{4} V_{4} + n_{3} V_{3} + n_{\text{f}} V_{\text{f}} ). $$

(7)

Here n ₄, n ₃ and n _f represent, respectively, the number (per mole of glass) of B₄, BO₃, and the network former units of CaF₂ (CaF₄ tetrahedra). M _f and V _f are, respectively, the mass and volume of the former units, CaF₂(f), of CaF₂. The masses are taken as M ₄ = (B + 1.5O + F + 0.5Ca), M ₃ = (B + 1.5O) and M _f = (Ca + 2F). To calculate n ₄, n ₃ and n _f Eq. 1 would be used, where x is the CaF₂ content in the glass expressed in mol%. N ₄ values obtained from Eq. 1 can be used in Eqs. 3–5 to get n ₃, n ₄, and n _f as

$$ n_{4} = 2N_{4} ({\text{B}}_{ 2} {\text{O}}_{ 3} )N_{A} /100, $$

(3a)

$$ n_{3} = \, 2 [ {\text{B}}_{ 2} {\text{O}}_{ 3} - N_{4} ( {\text{B}}_{ 2} {\text{O}}_{ 3} )]N_{A} /100 $$

(4a)

and

$$ n_{\text{f}} = [{\text{CaF}}_{2} - N_{4} ({\text{B}}_{ 2} {\text{O}}_{ 3} )]N_{A} /100 $$

(5a)

where N _A is Avogadro’s number. The factor 2 appears in Eqs. 3a and 4a because each CaF₂ converts two BO₃ units into two B₄ units and each B₂O₃ molecule produces two BO₃ units.

The density of vitreous B₂O₃ is 1.838 g/cm³ [35] and thus a value of V ₃ = 3.14 × 10⁻²³ cm³ could be estimated. This value of V ₃ has been found mostly the same in various borate glasses, such as PbO–B₂O₃ [36], Al₂O₃–PbO–B₂O₃ [29] and Na₂O–Al₂O₃–B₂O₃ [37]. In the light of these findings, it is assumed that V ₃ has the same value in CaF₂–B₂O₃ glasses.

To get the volumes V ₄ and V _f (of B₄ and the network former units of CaF₂, respectively) it is adequate to start by analyzing the molar volume. The latter is given as

$$ V_{\text{m}} = M /D $$

(8)

where M is the relative molecular mass of glass. In terms of the concentration and volume of structural units it can be expressed as

$$ V_{\text{m}} = (n_{3} V_{3} + n_{4} V_{4} + n_{\text{f}} V_{\text{f}} )N_{A} .$$

(9)

Taking V ₃ = 3.14 × 10⁻²³ cm³, Eq. 9 can be solved simultaneously for two V _m values close to each other to get V ₄ and V _f. In the seek of getting reliable values, the equation of fitting line for the experimental data of V _m in Fig. 2 would be used. This equation gives the molar volume as a function of the concentration x of CaF₂ (mol%) as

$$ V_{\text{m}} = - 0.063x + 28.719. $$

(10)

Figure 8 shows the dependence of the volume of B₄ and CaF₂(f) on the CaF₂ content. It is shown that V ₄ is mostly constant whereas V _f decreases linearly with increasing the content of CaF₂. The decrease in V _f reaches about 7% at 44 mol% CaF₂. The average values of V ₄ and V _f are 3.10 × 10⁻²³ and 4.32 × 10⁻²³ cm³, respectively. These values, together with that of V ₃, can be used, respectively, in Eqs. 7 and 9 to calculate both D and V _m. Reasonable agreement between calculated and experimental data could be obtained. However, a better agreement (Fig. 2) could be attained by means of V ₃ = 3.12 × 10⁻²³ , V ₄ = 3.1 × 10⁻²³ and V _f = 4.25 × 10⁻²³ cm³. It is worthy to indicate that the density of CaF₂ is 3.18 g/cm³ [38] that gives V _f = 4.077 × 10⁻²³ cm³. This reveals that the mean value of V _f in the amorphous form is about 1.04 times that in the crystalline form. This result is consistent with the conclusions of Ayta et al. [21]. Thermoluminescence glow curves presented an evidence for CaF₂-crystal formation in [(50Li₂O–45B₂O₃–5Al₂O₃) (mol%) + xCaF₂ (wt%)] glasses, with 0 ≤ x ≤50. The decrease in V _f with increasing CaF₂ content (Fig. 8) can be attributed to competition of the former CaF₂(f) units to occupy space among other units in the glass matrix. The decrease in V _f with increasing the CaF₂ content (Fig. 8) reveals that CaF₄ units tend to reach their volume in the pure crystalline form.

Fig. 8

The volume V ₄ (of ${\text{Ca}}_{ 1 / 2}^{ 2+ } [{\text{BO}}_{ 3 / 2} {\text{F]}}^{ - }$unit) and V _f (of CaF₂(f) unit) as a function of the CaF₂ content in the investigated xCaF₂–(100−x)B₂O₃ glasses. Lines are fitting plots of the data

The difference observed between the volume of CaF₄ units in the investigated glasses and in crystalline CaF₂ is consistent with various studies. Watanabe et al. [39] investigated CaO–CaF₂–SiO₂ glasses (5 ≤ CaF₂ ≤ 25 mol%) using ¹⁹F NMR. They reported that Ca–F clusters are formed and the cluster size, as well as the Ca–F distance, increase with increasing the CaF₂ content. Brauer et al. [40] used ¹⁹F and ²⁹Si MAS NMR to investigate SiO₂–P₂O₅–Na₂O–CaO–CaF₂ glasses (0 ≤ CaF₂ ≤ 32.7 mol%). It is deduced that fluorine does not form Si–F bonds. Instead, fluorine forms mixed sodium–calcium species. However, by increasing the CaF₂ content, fluorine preferentially complexes Ca²⁺ rather than Na⁺ ions. They attributed the higher chemical shift of F–Ca(n), compared with F–Ca(4) or F–Ca(3) in crystalline species, to the slightly longer Ca–F distance in the amorphous form [41]. Recently, an investigation on the density of SiO₂–P₂O₅–Na₂O–CaO–CaF₂ glasses (0 ≤ CaF₂ ≤ 25.54 mol%) revealed that CaF₂ enters the structure as CaF₄ tetrahedra [42]. The mean volume found for CaF₄ unit (4.142 × 10⁻²³ cm³) is close of that in the present work.

The density of the studied glasses can be calculated by using the predicted volumes in Eq. 7. Figure 2 shows that the calculated density and molar volume agree well with the experimental data, which supports the structural view of the studied glasses.

Electric conduction

Figure 9 shows that there is a small change in both E and logσ ₅₂₃ (log conductivity at 523 K) when changing the CaF₂ content. Between 30 and 45 mol% CaF₂ there is an increase of about one order in logσ ₅₂₃. These features are greatly different from those of PbF₂–B₂O₃ glasses [1], where a fast increase in logσ ₄₇₃ (about six orders) takes place between 35 and 60 mol% PbF₂. The marked change in the conductivity of PbF₂–B₂O₃ glasses is attributed to the ease of transport of F⁻ ions as charge carriers [1]. The small increase in logσ ₅₂₃ and the activation energy (Fig. 9) can be taken as a basis to rule out the role of F⁻ ions as the major type of charge carriers in the studied CaF₂–B₂O₃ glasses. It is therefore to assume that the electric conduction in these glasses is mainly due to transport of Ca²⁺ ions rather than F⁻ ions.

Fig. 9

Dependence of both the logarithm of conductivity at 523 K and activation energy for electric conduction on the CaF₂ content in the studied glasses. Lines are guide for the eyes

In Fig. 7 is shown that the content of CaF₂(f) (the former CaF₂) increases whereas that of B₄ (the modifier CaF₂) slightly decreases with increasing the concentration of CaF₂. It can be indicated that the concentration of these components expressed as number/cm³ changes in the same manner. The increase in conductivity with increasing CaF₂ content (Fig. 9) leads likely to consider Ca²⁺ ions in CaF₂(f) being the main charge carriers in the studied CaF₂–B₂O₃ glasses. Because of the small increase in conductivity it can be deduced that the mobility has a pronounced effect. The conductivity is generally given by [43] as

$$ \sigma = nq\mu $$

(11)

where n is the concentration of mobile ions expressed as (number/cm³), q is the ionic charge, and μ is the mobility of the charge carrier. It must be noted that the concentration n of mobile ions is just a fraction of ionic species that is assigned as charge carriers. This fraction is usually unknown and therefore it is not possible to determine the mobility of charge carriers from Eq. 11. To get an idea about the change of mobility with composition of glass it can be assumed that all Ca²⁺ ions in CaF₂(f) are mobile ions. Thus, Eq. 11 can be used to calculate the mean mobility ($\overline{\mu }$) of Ca²⁺ ions. Figure 10 shows that, in spite of the marked increase (>four times) in the total concentration n _Ca of Ca²⁺ ions in CaF₂(f), the logarithm of $\overline{\mu }$ decreases with increasing the CaF₂ content. The decrease in $\overline{\mu }$ can be correlated with the decrease in the molar volume (Fig. 2) and thus the free volume of glass when the CaF₂ content increases (Fig. 11). The decrease in free volume can also be the reason that E does not change when changing the CaF₂ content (Fig. 9).

Fig. 10

Concentration n _Ca of Ca²⁺ ions in CaF₂(f) and the logarithm of mean mobility of Ca²⁺ ion (log $\overline{\mu }$) in dependence of the CaF₂ content in the investigated xCaF₂–(100−x)B₂O₃ glasses. Lines are fitting plots of the data

Fig. 11

Change of the free volume of xCaF₂–(100−x)B₂O₃ glasses with composition. Lines are fitting plots of the data

In addition to the effect of decreasing free volume, another factor may contribute to the low conductivity and its limited increase between 30 and 45 mol% CaF₂. This is the nature of binding of Ca²⁺ ions in the studied glasses. These ions are bound to F⁻ ions either in [F⁻BO_3/2], where the negative charge is localized at F⁻ ion in the distorted [F⁻BO_3/2] tetrahedron [15, 16], or in CaF₄ tetrahedra. A reduction in the mobility of Na⁺ ions in xNa₂O–y(NaF)₂–0.64B₂O₃ and y(NaF)₂–(1−y)B₂O₃ glasses were reported upon increasing the concentration of fluoride ions. The effect is attributed to a stronger interaction between Na⁺ ions and F⁻ ions, compared to that with O²⁻ ions [32]. Similar results were obtained by Jain et al. [44] upon replacing 1% of total oxygen concentration by fluorine in 0.35(Li, Na)₂O·0.65B₂O₃ glasses. In this study, an increase in the CaF₂ content means an increase F/O ratio and this leads to a decrease in the mobility of Ca²⁺ ions.

Conclusion

IR absorption in the region ~850–1200 cm⁻¹ is assumed to be related to stretching vibrations of ${\text{Ca}}_{1 /2}^{2 + } [{\text{BO}}_{3 /2} {\text{F]}}^{ - }$ units. A part of CaF₂ is incorporated in the borate network to form these units. The rest of CaF₂ forms its own matrix, presumably in the form of CaF₄ tetrahedra. Calculated density agrees well with the experimental density. Calculated density has been obtained by considering that the studied CaF₂–B₂O₃ glasses may contain BO₃, ${\text{Ca}}_{ 1 / 2}^{ 2+ } [{\text{BO}}_{ 3 / 2} {\text{F]}}^{ - }$and CaF₂(f) units. Calculated volume of the latter has mostly the same value as in the crystalline CaF₂. Ca²⁺ ions are the main charge carriers in the studied glasses. A limited increase in conductivity is related to the decrease in the mean mobility, in spite of the increase in the total concentration of Ca²⁺ ions. The decrease in the mean mobility is attributed to a decrease in the free volume with increasing the CaF₂ content.