Introduction

In recent years, bismuth-containing layered ferroelectrics, first described by Aurivillius [1–3], have been the subject of researchers’ ever-increasing attention. The general formula of such compounds can be represented as Am−1Bi2BmO3m+3, where m is typically from 1 to 5 [4]. The A atoms most frequently have a valence of I, II, or III (or a combination of these) and a coordination number CNA > 6. The B site is usually occupied by transition metals in octahedral coordination (CNB = 6). The structure of these stoichiometric compounds is made up of slabs and blocks. Each slab consists of n layers of anion octahedra. For n → ∞, its structure approaches the cubic perovskite structure. The blocks can be thought of as structural units of cubic BiF3 [3]. Characteristically, the bismuth-containing layered ferroelectrics have low dielectric permittivity, high Curie temperatures, low temperature coefficients of their resonance frequency, highly anisotropic electromechanical coupling coefficients, and low aging rates [5–7].

The goals of this work include calorimetric determination of the temperature dependence of the heat capacity \( C_{\text{p}}^{\text{o}} \) = f(T) of Bi4Ti3O12 from 7 to 346 K, detection of the possible phase transitions, and calculation of the standard thermodynamic functions \( C_{\text{p}}^{\text{o}} \)(T), H o(T) − H o(0), S o(T) − S o(0), and G o(T) − H o(0) in the range from T → 0 to 346 K. We also present results of X-ray diffraction studies depending on the temperature in order to obtain information on thermal deformation of this material.

Experimental

Sample

Tetrabismuth trititanium oxide was prepared by the solid-state reaction between titanium oxide and bismuth(III) nitrate pentahydrate (reaction 1). The synthesis was performed in a porcelain crucible, into which the reaction mixture with the atomic ratio 4Bi: 3Ti was loaded. The mixture was calcined at 1173 K for 50 h, undergoing regrinding every 10 h.

$$ 4{\text{Bi}}\left( {{\text{NO}}_{3} } \right)_{3} \cdot5{\text{H}}_{2} {\text{O}} + 3{\text{TiO}}_{2} \to {\text{ Bi}}_{4} {\text{Ti}}_{3} {\text{O}}_{12} + 12{\text{NO}}_{2} + 3{\text{O}}_{2} + 20{\text{H}}_{2} {\text{O}} $$
(1)

Apparatus and measurement procedure

For structural investigations, an X-ray diffraction pattern of a Bi4Ti3O12 sample was recorded on a Shimadzu X-ray diffractometer XRD-6000 (Cu Kα radiation, geometry θ–2θ) in the 2θ range from 10° to 120° with scan increment of 0.02°. Rietveld analysis and structure refinement [8] were carried out using RIETAN-97 software [9]. The X-ray data and estimated impurity content (0.5–1 mass%) in the substance led us to conclude that the tetrabismuth trititanium oxide sample studied was an individual crystalline compound. The X-ray diffraction studies depending on the temperature were carried out on a Shimadzu X-ray diffractometer XRD-6000 using Attachment TTK-450 (Anton Paar).

Thermal analysis was carried out with a Setaram LABSYS DSC 1600 differential scanning calorimeter in an argon atmosphere at a heating rate of 0.0833 K s−1.

To measure the heat capacity \( C_{\text{p}}^{\text{o}} \) of the tested substance in the range from 6 to 337 K, a BKT-3.0 automatic precision adiabatic vacuum calorimeter with discrete heating was used. The calorimeter design and the operation procedure were described earlier [10]. The calorimeter was tested by measuring the heat capacity of high-purity copper and reference samples of synthetic corundum and K-2 benzoic acid. The analysis of the results showed that measurement error of the heat capacity of the substance at helium temperatures was within ±2 %, then it decreased to ±0.5 % as the temperature was rising to 40 K, and was equal to ±0.2 % at T > 40 K. Temperatures of phase transitions can be determined with the error of ±0.02 K.

Results and discussion

Crystal structure

The structure of Bi4Ti3O12 was refined assuming space group Fmmm. The initial model included the atomic coordinates in the structure of tetrabismuth trititanium oxide, which was solved by Aurivillius [11]. The details of the X-ray diffraction experiments and structure refinement data are listed in Table 1. Figure 1 represents the measured, simulated, and difference X-ray diffraction patterns for Bi4Ti3O12 (T = 173 K), as well as a pattern of lines corresponding to reflection maxima. There is a good agreement between the measured and simulated patterns. Table 2 lists the coordinates of the atoms and their isotropic thermal parameters.

Table 1 Details of the X-ray diffraction experiment and the results of the structure refinement for Bi4Ti3O12
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Fig. 1
figure 1

Observed (1), simulated (2), and four difference X-ray diffraction patterns and three Bragg reflections for Bi4Ti3O12

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Table 2 Coordinates and isotropic thermal parameters of atoms in the structure of Bi4Ti3O12
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The crystal structure of Aurivillius-phase layered perovskite Bi4Ti3O12 is shown in Fig. 2a. Bi4Ti3O12 has a general formula of [Bi2O2]2+[Bm−1MmO3m+1]2−, where B is the 12-fold-coordinated cation with low valence in the perovskite sublattice, M denotes the octahedral site occupied by the ions with high valence, and m is the number of perovskite layers between the [Bi2O2]2+ blocks. The perovskite slabs of Bi4Ti3O12 are composed of TiO6 octahedrons and 12-fold-coordinated Bi3+ (atoms Bi1) and are three layers in the thickness as shown in Fig. 2a. The [Bi2O2]2+ blocks are constructed of square pyramids (atoms Bi2 are located in the central vertices).

Fig. 2
figure 2

The crystal structure (a) and fragments of the structure of Bi4Ti3O12 (b—slab, c—block, orange arrows—atomic displacements)

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Table 3 lists interatomic distances in the crystal structure of Bi4Ti3O12. The bond lengths Ti–O in octahedra vary in the range from 1.66 to 2.27 Å. It should be noted that the octahedra with a central atoms Ti2 are the most distorted.

Table 3 Bond lengths in the structure of Bi4Ti3O12
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Heat capacity

The \( C_{\text{p}}^{\text{o}} \) measurements were taken between 7 and 346 K. The mass of the sample loaded in the calorimetric ampoules of the BKT-3.0 device was 1.2657 g. Hundred and sixty-eight experimental \( C_{\text{p}}^{\text{o}} \) values were obtained in three series of experiments (Table 1S). The heat capacity of the sample varied from 20 to 50 % of the total heat capacity of calorimetric ampoule + substance over the range between 7 and 346 K. The experimental points of \( C_{\text{p}}^{\text{o}} \) in the temperature intervals between T = (10 and 30) and (21 and 346) K were fitted by means of Eqs. (2, 3) of the \( C_{\text{p}}^{\text{o}} \) versus temperature have been obtained. The corresponding coefficients (A, B, C, etc.) are given in Table 4

Table 4 Coefficients in the fitting polynomials for Bi4Ti3O12
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$$ \begin{aligned} C_{\text{p}}^{\text{o}} & = \, A_{1} + B_{1} \cdot \left( \frac{T}{30} \right) + C_{1} \cdot \left( \frac{T}{30} \right)^{2} + D_{1} \cdot \left( \frac{T}{30} \right)^{3} + E_{1} \cdot \left( \frac{T}{30} \right)^{4} + F_{1} \cdot \left( \frac{T}{30} \right)^{5} \\ & \quad + G_{1} \cdot \left( \frac{T}{30} \right)^{6} + \, H1 \cdot \left( \frac{T}{30} \right)^{7} + \, I_{1} \cdot \left( \frac{T}{30} \right)^{8} + \, J_{1} \cdot \left( \frac{T}{30} \right)^{9} + \, K_{1} \cdot \left( \frac{T}{30} \right)^{10} \\ \end{aligned} $$
(2)
$$ \begin{aligned} \ln C_{\text{p}}^{\text{o}} & = \, A_{2} + \, B_{2} \cdot \ln \left( \frac{T}{30} \right) + \, C_{2} \cdot \ln^{2} \left( \frac{T}{30} \right) + \, D_{2} \cdot \ln^{3} \left( \frac{T}{30} \right) + \, E_{2} \cdot \ln^{4} \left( \frac{T}{30} \right) + \, F_{2} \cdot \ln^{5} \left( \frac{T}{30} \right) \\ & \quad + \, G_{2} \cdot \ln^{6} \left( \frac{T}{30} \right) + \, H_{2} \cdot \ln^{7} \left( \frac{T}{30} \right) + \, I_{2} \cdot \ln^{8} \left( \frac{T}{30} \right) + \, J_{2} \cdot \ln^{9} \left( \frac{T}{30} \right) + \, K_{2} \cdot \ln^{10} \left( \frac{T}{30} \right) + \, L_{2} \cdot \ln^{11} \left( \frac{T}{30} \right) \\ & \quad + \, M_{2} \cdot \ln^{12} \left( \frac{T}{30} \right) + N_{2} \cdot \ln^{13} \left( \frac{T}{30} \right) \\ \end{aligned} $$
(3)

Their root mean square deviation from the averaging \( C_{\text{p}}^{\text{o}} \) = f (T) curve was ±0.15 % in the range T = (7 to 55) K, ±0.075 % from T = (55–80) K and ±0.050 % between T = (80 and 346) K.

The experimental values of the molar heat capacity of Bi4Ti3O12 over the range from 7 to 346 K and the averaging \( C_{\text{p}}^{\text{o}} \) = f (T) plot are shown in Figure 1S. The heat capacity \( C_{\text{p}}^{\text{o}} \) of this substance in interval from 7 to 346 K gradually increases with rising temperature and does not show any peculiarities.

From the experimental \( C_{\text{p}}^{\text{o}} \) values in the range 25–50 K, the value of the fractal dimension D of the tetrabismuth trititanium oxide was evaluated. According to the fractal theory of the heat capacity [12], D is the most important parameter that specifies the character of heterodynamics of the substance structure. For solids of a chain structure, the relation \( C_{\text{p}}^{\text{o}} \) versus T at lower temperatures is proportional to T 1, of a layer structure to T 2, and of steric one to T 3 [13]. In the fractal theory of the heat capacity, an exponent on T is the heat capacity function that is denoted by D and is called the fractal dimension. This follows specifically from Eq. (4) [12]:

$$ C_{\text{v}} = \, 3D\left( {D + 1} \right)kN\gamma \left( {D + 1} \right)\xi \left( {D + 1} \right)(T/\theta_{\hbox{max} } )^{\text{D}} , $$
(4)

where N is the number of atoms in a formula unit, k is the Boltzmann constant, γ(D + 1) is the γ-function, ξ(D + 1) is the Riemann ξ-function, and θ max is the characteristic temperature. As follows from inferences [12], D can be evaluated from the experimental data on the temperature-dependent heat capacities from a slope of the corresponding rectilinear sections of the plot ln C v versus ln T. Without a substantial uncertainty, it may be assumed that at T < 50 K \( C_{\text{p}}^{\text{o}} \) = C v. From the ln C v versus ln T plot and Eq. (4), it was found that in the range 25–50 K, D = 1.6, θ max = 237.0 K for tetrabismuth trititanium oxide. With these values of D and θ max, Eq. (4) reproduces the experimental \( C_{\text{p}}^{\text{o}} \) values in the temperature range mentioned with an uncertainty of ±0.36 %. The D-value points to the chain-layered structure of Bi4Ti3O12 [14–17].

Standard thermodynamic functions

To calculate the standard thermodynamic functions (Table 5) of the tetrabismuth trititanium oxide, its \( C_{\text{p}}^{\text{o}} \) values were extrapolated from the temperature of the measurement beginning at approximately 7 to 0 K by graphic method [18]. The calculations of H o(T) − H o(0) and S o(T) − S o(0) were made by the numerical integration of \( C_{\text{p}}^{\text{o}} \) = f (T) and \( C_{\text{p}}^{\text{o}} \) = f (ln T) curves, respectively, and the Gibbs function G o(T) − H o(0) was estimated from the enthalpies and entropies at the corresponding temperatures [19]. It was suggested that the error of the function values was ±1 % at T < 40 K, ±0.5 % between 40 and 80 K and ±0.2 % in the range from 80 to 346 K.

Table 5 Thermodynamic functions of crystalline Bi4Ti3O12; M = 1171.515 g mol−1, p o = 0.1 MPa
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The absolute entropies of tetrabismuth trititanium oxide (Table 5) and the corresponding simple substances Bi (cr), Ti (cr), and O2 (g) [20, 21] were used to calculate the standard entropy of formation of the compound under study at 298.15 K, ΔfSo(298.15, Bi4Ti3O12, cr) = −1086.6 ± 0.8 J K−1 mol−1.

Differential scanning calorimetry and thermal deformation model

Joint application of X-ray diffraction studies depending on the temperature and differential thermal analysis made it possible to establish some peculiarities of processes taking place in the compound under investigation during heating. Figure 3 represents DTA curve of Bi4Ti3O12, where we can see reversible endothermic effect at 934 K. This effect corresponds to reversible second-order transitions associated with sharp changes in the symmetry of the material, which can be characterized by a Curie temperature (TC) (“normal” ferroelectric behavior) according to our research [4] and works [22, 23]. Melting compound corresponds to a second large endothermic effect at 1476 K.

Fig. 3
figure 3

DTA curve of Bi4Ti3O12

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In the next stage of our work, we decided to clarify the mechanisms of atomic displacements in the ferroelectric phase, which lead later to a second-order transition (TC). Our research has shown that most “mobile” atoms are O3, O4, O5 in slabs and Bi2 in blocks. The displacements of these atoms are shown by the arrows in Fig. 2b, c. The most significant displacement of the atoms is observed for the oxygen atoms in the outer octahedra of slabs (Fig. 2b). Bismuth atoms in the blocks move much less (Fig. 2c). All movements occur along the crystallographic c axis. We have noted that the direction of movement essentially depends on temperature. The distortion of octahedra in the slabs is observed at temperatures down to θ max (Fig. 4). On the contrary, at higher temperatures occurs gradually alignment of the bond lengths in the octahedra. This process leads to an increase in the symmetry of the structure and phase transition. Thus, the symmetrization of the octahedra in the slab is the main reason of transition for ferroelectric Aurivillius phases in the paraelectric phase.

Fig. 4
figure 4

Temperature dependence of bond lengths in crystal structure of Bi4Ti3O12 (1—Ti2–O4; 2—Ti2–O3; 2—average Bi2–O4)

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Conclusions

The temperature dependence of heat capacity of Bi4Ti3O12 has been measured in the range from 7 to 346 K by precision adiabatic vacuum calorimetry. The experimental data were used to calculate standard thermodynamic functions. We have studied the mechanism of thermal deformations in the Aurivillius phase.