The Effect of Solid-Phase and Melt Synthesis Methods on Dipole Ordering and Ion Conductivity of the Polar α-Phase of Na₃Fe₂(PO₄)₃ Polycrystals

Abstract

The article investigates the dielectric and conductive properties of the polar α-phase of Na₃Fe₂(PO₄)₃ polycrystals synthesized by solid-phase (sample type 1), melt (type 2), and melt-quenching (type 3) methods. To enable a rapid assessment of the dielectric properties of the polar α-phase of Na₃Fe₂(PO₄)₃, the thermo-polarization mobility parameter μ_Tp(T, E(ω)) was introduced. By studying the dielectric properties, it was concluded that the polar α-phase of type 1 samples consists of large and small dipoles and ordered sodium cations, which possess low values of μ_Tp(T, E(ω)), indicating the presence of strong interaction forces between the crystal lattice and the cationic part of the polycrystal. Additional studies of the samples’ conductivity confirm this conclusion. Studies of the polar α-phase of Na₃Fe₂(PO₄)₃ in type 2 samples have established that their structure contains dipoles and sodium cations with higher values of μ_Tr(T, E(ω)), and also exhibits higher conductivity than Type 1 samples. These data indicate a weakening of the interaction forces between the cationic and anionic components in type 2 polycrystals due to a partial increase in crystal symmetry. The results of studies of the polar α-phase of type 3 samples show that their structure contains dipoles and sodium cations with higher values of μ_Tr(T, E(ω)), and also exhibits higher conductivity than type 2 samples. It is concluded that the structure of type 3 samples is characterized by weak interaction forces between the cationic and anionic parts as a result of an increase in the symmetry of the polar α-phase of Na₃Fe₂(PO₄)₃, caused by sharply graded temperature conditions during the synthesis of polycrystals. By studying the dielectric properties of cathode materials, it is possible to obtain information on the extent of interactions between the cationic and anionic components in polycrystals. It is, therefore, appropriate to use this approach when investigating a wide range of new dielectric and ion-conducting materials.

1. Introduction

The high energy density and long service life of LIBs are key factors driving their popularity. Due to the rapid development of sectors such as portable electronics, drones, electric vehicles, robotics, and alternative energy sources, there is a growing demand for efficient LIBs [1,2,3,4,5,6].

The high cost of lithium necessitates the development of cheaper sodium-ion batteries (SIBs) [7,8,9]. However, SIBs have lower energy densities than LIBs, which is why the production of such commercial power sources has so far been held back [10,11]. The growing demand for renewable energy sources such as wind and solar power, coupled with their intermittent nature, necessitates the development and construction of energy-intensive, large-scale power SIB [12,13,14,15,16].

The efficiency of batteries depends on cathode materials, which must ensure high redox potential, electrical conductivity, electrochemical activity, and structural stability during cycling [17,18]. The cathode material α-Na₃Fe₂(PO₄)₃ can be used to test new methods for producing effective materials for SIBs, as it possesses the necessary ionic conductivity, structural stability, and redox potential [19,20,21,22]. The structure of Na₃Fe₂(PO₄)₃ belongs to the NASICON-type rhombohedral crystal lattice and exhibits three polymorphic forms: α, β, and γ [23,24,25,26].

The anionic crystal framework of this type contains extensive cavities of types (M(1)) and (M(2)), in which sodium cations are located [27,28]. A visual representation of the monoclinic distorted crystal framework α-Na₃Fe₂(PO₄)₃ is shown in Figure 1 [29].

The structure of the α-phase of Na₃Fe₂(PO₄)₃ has a superstructural unit cell and is characterised by antiferroelectric dipole ordering (AFE) [24,30]. After the α→β phase transition, the structure of the β-Na₃Fe₂(PO₄)₃ crystal becomes rhombohedral with space group R3c, although the unit cell retains weak superstructural reflections [24]. During the phase transition, the anion sublattice undergoes minor structural shifts, but the cation sublattice undergoes significant changes in the distribution of sodium cations [31,32,33].

The framework method of constructing the anion crystal structure {[Fe₂(PO₄)₃]³⁻}_3∞ and the presence of two types of cavities (M(1)) and (M(2)), as well as ‘windows’ between the cavities, contribute to the formation of three-dimensional ‘conductivity channels’ in the crystal framework [25,26,27,28,29].

The structure of the polycrystal in the α-phase Na₃Fe₂(PO₄)₃ is monoclinicallydistorted(pr. gr. C2/m), which leads to a shift from the equilibrium centres of statistically arranged sodium cations relative to the anionic framework and the formation of a dipole-type AFE, contributing to a decrease in ionic conductivity [30,31]. The patterns of the ordered distribution of sodium cations within the (M(1)) and (M(2)) cavities of the crystal lattice have been studied in [32,33,34].

The conductivity and electrical capacitance properties of the α-Na₃Fe₂(PO₄)₃ polycrystal depend on the choice of solid-phase or melt synthesis technology, i.e., on the thermodynamic conditions of sample synthesis [35,36,37]. The polycrystal α-Na₃Fe₂(PO₄)₃ is used as a cathode material at room temperature; it is necessary to determine the influence of various thermodynamic factors of synthesis on the roles of the cationic and anionic parts of the crystal lattice in the formation of dipole ordering and ionic conductivity.

This work aims to determine the influence of the process conditions for solid-state and melt synthesis methods on the degree of dipole ordering in the polar phase of the α-Na₃Fe₂(PO₄)₃ polycrystal, as well as on its dielectric and ionic conductivity properties.

2. Materials and Methods

Na₃Fe₂(PO₄)₃ polycrystals were obtained using the following reagents: Na₂CO₃, Fe₂O₃, and NH₄H₂PO₄. (All reagents were of the ‘OSCH’ brand, Labhimprom, Republic of Kazakhstan, Almaty).

To achieve the set goal, Na₃Fe₂(PO₄)₃ polycrystals were obtained using solid-phase, melt, and melt-quenching synthesis methods.

Structural studies of polycrystalline Na₃Fe₂(PO₄)₃ samples were carried out using a diffractometer (CuKα radiation from Bruker, Karlsruhe, Germany). Particle sizes were determined using an optical method. Sample density studies were conducted using a pycnometer method.

The polarization of the samples was carried out by studying the dependence of the dielectric permittivity ε(T, ω) in the range of T = 300–450 K and frequencies v = 2 kHz–1 MHz using E7-30. (MNIPI, Minsk, Belarus).

The ionic conductivity of the samples was determined using the impedance method with VM 507 and VM-538 devices, analyzing the impedance of the Na₃Fe₂(PO₄)₃ polycrystal as a function of temperature in the temperature range T = 300–450 K.

3. Results and Discussion

3.1. Synthesis of Samples

The synthesis of samples began with mechanical grinding of the initial mixtures in a planetary mill, pressing the resulting mixture into tablets, and firing for 1.5 h at a temperature of 350 °C. Then, the heat-treated mixture of reagents was ground, pressed into tablets, and fired.

The diagram of the ceramic technology for the solid-phase synthesis of type 1 Na₃Fe₂(PO₄)₃ polycrystals is shown in Figure 2, and the technological modes of sample synthesis are given in

Table 1. Technological modes of synthesis of polycrystalline samples of Na₃Fe₂(PO₄)₃ for three types.

Samples	1 Type		2 Type		3 Type
Na3Fe2(PO4)3	1st annealing	2nd annealing	Melting	Annealing	Melting	Annealing
Firing temperatures T, °C	600	820	980	820	980	820
Melting time, s			180		50
Firing time t, h	7	7		7		2
Cooling time t, s to T = 25 °C		0.5	300		60
Temperature cooling rate v = T/t, °C/s		48	3.26		16.3	48

.

The diagram of the melt-based synthesis technology for type 2 Na₃Fe₂(PO₄)₃ polycrystals is shown in Figure 2, and the technological modes of sample synthesis are given in Table 1.

The reagents were melted using an URN-2-ZP radiation heating unit [38]. After cooling the melt in air, the precursors were obtained. The melting and cooling processes were carried out in air. Type 2 polycrystals were obtained after isothermal calcination of the precursors at T = 820 °C.

In the melt-quenching method, the precursors were synthesized from a mixture of heat-treated reagents using a melting apparatus that heated and melted the samples using heat flow and IR radiation. The melt droplets were then rapidly cooled in a quenching device to obtain precursors in the form of droplets, balls, and wires. To produce type 3 polycrystals from the precursors, they were ground, pressed, and sintered at 820 °C for 2 h. The sequence of operations for the synthesis of polycrystals using the melt-quenching method is shown in Figure 2, and the process parameters for the synthesis are given in Table 1.

As a result of the synthesis, three types of Na₃Fe₂(PO₄)₃ polycrystals were obtained, which were dark brown in colour and in the form of tablets (10 mm in diameter and 1 mm thick).

3.2. Results of Structural Studies of Na₃Fe₂(PO₄)₃ Polycrystals Obtained by Solid-Phase and Melt Methods

From the diffractograms of three types of Na₃Fe₂(PO₄)₃ polycrystals shown in Figure 3, it can be seen that all three samples have clearly defined peaks at the same angles, but with different peak intensities. Based on the diffractograms shown in Figure 3, it can be concluded that the type 3 and type 2 polycrystals crystallized better and have a more distinct crystal grain structure than the type 1 samples, as they exhibit more intense peaks.

The diffractograms of the obtained samples were indexed using the Origin program, and the calculated data are given in

Table 2. Density and parameters of the unit cell of the three types of Na₃Fe₂(PO₄)₃ polycrystals.

Polycrystals	Space Group	Relative Density	Unit Cell Parameters
Types of Na3Fe2(PO4)3 Samples		%	a, Å	b, Å	c, Å	α0	β0	γ0
1—type	C2/m	82	15.1230	8.7168	21.5963	90.00	90.37	90.00
2—type	C2/m	90	15.1287	8.7106	21.5534	90.00	90.25	90.00
3—type	C2/m	96	15.1444	8.6840	21.5768	90.00	90.30	90.00

. It was also established that the three types of Na₃Fe₂(PO₄)₃ polycrystals have a monoclinic distorted crystal structure of pr. gr.

3.2.1. Dielectric Properties of Polycrystals of the α-Phase of Na₃Fe₂(PO₄)₃ Type 1

Since the α-phase of Na₃Fe₂(PO₄)₃ is polar, it is advisable to study its dielectric properties, as dielectric spectroscopy can provide additional information about the interaction between the cationic and anionic parts of the crystal lattice, which is necessary for analyzing the conductive properties of the sample.

The temperature-frequency dependencies of the dielectric permittivity (ε(T, ω)) of the Na₃Fe₂(PO₄)₃ polycrystal obtained by the solid-phase method (sample type 1) demonstrate the dispersion of ε parameters from frequency and temperature in the dipolarly ordered α and ion-conducting β phases (see Figure 4).

The dependence of the dielectric permittivities of α-Na₃Fe₂(PO₄)₃ on temperature is weak at low frequencies of the electric field (at ω = 200 kHz) and virtually absent at high frequencies, indicating the presence of a set of poorly mobile, dipole-ordered and ordered sodium cations. Consequently, the influence of thermal energy and an external electric field on the polar state of the α-Na₃Fe₂(PO₄)₃ structure, formed by spontaneous AFE-type dipole ordering, has virtually no effect on the strong bonds between the cationic and anionic parts of the polycrystal.

The increase in the dielectric permittivity (ε_i) of the polycrystal with rising temperature (T) and electric field (E(ω)) at frequency (ω_i) can be expressed by the thermo-polarization mobility (μ_Tp(T, E(ω)), which characterizes the ability of charged particles to polarize when the temperature changes by 1 degree (∆T = 1 K) under the influence of an external electric field E(ω),expressed as follows:

$${\mathsf{\mu }}_{\mathrm{T}\mathrm{p}}\left(\mathrm{T},\mathrm{E}\left({\mathsf{\omega }}_{\mathrm{i}}\right)\right)={\displaystyle \frac{1}{\mathrm{E}\left({\mathsf{\omega }}_{\mathrm{i}}\right)}}{\displaystyle \frac{{\mathsf{\epsilon }}_2\left(\mathrm{T},\mathrm{E}\left({\mathsf{\omega }}_{\mathrm{i}}\right)\right)-{\mathsf{\epsilon }}_1\left(\mathrm{T},\mathrm{E}\left({\mathsf{\omega }}_{\mathrm{i}}\right)\right)}{{\mathrm{T}}_2-{\mathrm{T}}_1}}={\displaystyle \frac{1}{\mathrm{E}\left({\mathsf{\omega }}_{\mathrm{i}}\right)}}{\displaystyle \frac{\mathsf{\Delta }\mathsf{\epsilon }\left(\mathrm{T},{\mathsf{\omega }}_{\mathrm{i}}\right)}{\mathsf{\Delta }\mathrm{T}}}$$

(1)

When a single electrical signal E(ω) = 1 is applied to the sample, Formula (1) is simplified:

$${\mathsf{\mu }}_{\mathrm{T}\mathrm{p}}\left(\mathrm{T},\mathrm{E}\left({\mathsf{\omega }}_{\mathrm{i}}\right)\right)={\displaystyle \frac{{\mathsf{\epsilon }}_2\left(\mathrm{T},{\mathsf{\omega }}_{\mathrm{i}}\right)-{\mathsf{\epsilon }}_1\left(\mathrm{T},{\mathsf{\omega }}_{\mathrm{i}}\right)}{{\mathrm{T}}_2-{\mathrm{T}}_1}}={\displaystyle \frac{\mathsf{\Delta }\mathsf{\epsilon }\left(\mathrm{T},{\mathsf{\omega }}_{\mathrm{i}}\right)}{\mathsf{\Delta }\mathrm{T}}}$$

(2)

In this case, the thermo-polarisation mobility of the samples at a specific frequency ω_i, μ_Tp(T, E(ω_i)), is determined by the slope of the ε(T, ω) curve.

Judging by the dependence ε(T,ω) (Figure 4a), it can be assumed that in a unit volume of α-Na₃Fe₂(PO₄)₃ at frequencies above ω = 1 MHz, there is one third of unpolarised but ordered sodium cations [27,28]. As the symmetry of the polycrystal in the β-phase of Na₃Fe₂(PO₄)₃ increases, a further rise in the ε values is observed, due to the interaction of the dipole-disordered subsystem of mobile or ‘free’ sodium cations with an alternating electric field.

The transition from the dielectric (α) to the ion-conducting (β) phase in Na₃Fe₂(PO₄)₃ at a temperature of T_α→β = 368 K is accompanied by a sharp increase in dielectric permittivities (see

Table 3. Parameters characterizing the dielectric properties of the three types of samples.

Types of Samples	Circular Frequency ωi 103, Hz	Dielectric Permittivity εi 102 at Frequency ωi at T = 300 K	Dielectric Permittivity ε at Frequency ωi at T = 340 K	Thermal Polarization Mobilities of Charges μTp ((T,E(ω)), ms/VK	Comparative Coefficient ${\mathbf{K}}_{\mathbf{V}} = \frac{1}{\mathsf{\epsilon }\left(\mathbf{r}\right)}$$\mathbf{Potential} \mathbf{Difference} \mathbf{V}\left(\overrightarrow{{\mathbf{r}}_{\mathbf{i}\mathbf{j}}}\right)$	Absolute Change in Permittivity ∆ε at Tc
1 Type	2	0.99	1.25	0.012		1.0
	1 × 103	0.925	0.927	0.010	1.08 × 10−2	0.56
2 Type	2	0.97	0.975	0.023		0.032
	1 × 103	0.91	0.9125	0.0138	1.09 × 10−2	0.04
3 Type	2	1.53	1.55	0.032		0.09
	1 × 103	1.425	1.430	0.024	6.91 × 10−3	0.02

), as thedipole-ordered subsystem is rearranged into a partially disordered one due to a change in their symmetry from c/2 to 3Rc [27,28,29].

3.2.2. Dielectric Properties of Na₃Fe₂(PO₄)₃ Polycrystals of Type 2

Figure 5 shows the dependencies ε(T, ω) of the type 2 Na₃Fe₂(PO₄)₃ polycrystal.The dependence ε(T, ω) shows that the polar α-phase of Na₃Fe₂(PO₄)₃ is characterized by a slight increase in dielectric permittivity with rising temperature, indicating a slight increase in the value of μ_Tr(T, E(ω)) for dipole-ordered sodium cations.

The increase in the thermo-polarization mobility of dipoles and sodium cations within the cavities of the crystal lattice indicates their ability to undergo vibrational motion under the influence of thermal energy, and to become further polarized (∆μ_Tp) under the influence of an electric field, increasing the dielectric permittivity ε to a value of (ε + ∆ε) due to the partial weakening of the interaction forces between the cationic and anionic crystal frameworks.

The increase in the value of μ_Tr(T, E(ω)) in type 2 polycrystals may be attributed to the fact that these samples were formed by isothermal sintering from precursors, thestructure of which was subjected to intense hydrostatic compressive forces, as they were obtained by rapid cooling of the melt (see Table 1). It is possible that during the sintering of the precursors, under the action of residual compressive forces, a partial removal of the monoclinic distortion of the crystal lattice occurred, which is equivalent to a partial increase in the symmetry of type 2 polycrystals.

It should be noted that with an increase in the electric field frequency, a decrease in dielectric permittivity is observed in the ε(T,ω) dependence (Figure 5). This change in the dielectric permittivity of the samples is probably due to the interaction of ‘small’ dipoles with high electric field frequencies. These data indicate that the polarized state of the α-phase of Na₃Fe₂(PO₄)₃ is formed by clearly ordered ‘large’ and ‘small’ sodium dipoles. It can be assumed that at frequencies above 1 MHz, unpolarized but ordered sodium cations may be present.

It should be noted that in type 2 samples, there is a slight jump in dielectric permittivity (∆ε) compared to type 1 samples near the T_α→β transition temperature (see Table 3 and Figure 5). These changes indicate that Type 2 samples were formed under moderate hydrostatic pressure, which contributed to an increase in the polarization of dipoles and sodium cations in the α-Na₃Fe₂(PO₄)₃ polycrystals as a result of a partial increase in symmetry within the monoclinic-distorted structure of the α-phase. Conversely, hydrostatic pressure leads to a decrease in the polarization of particles in the β-phase due to a partial reduction in the symmetry of its rhombohedral structure, caused by isotropic compressive forces during the preparation of type 2 samples.

3.2.3. Dielectric Properties of Na₃Fe₂(PO₄)₃ Polycrystals Type 3

Figure 6 shows the temperature-frequency dependence of the dielectric permittivity (ε(T, ω)) of a type 3 Na₃Fe₂(PO₄)₃ polycrystal. The ε(T, ω) curves show that the polar α-phase of the type 3 Na₃Fe₂(PO₄)₃ polycrystal is characterized by higher values of ε than in type 2 samples, which may be attributed to an increase in the concentration of dipole-ordered particles due to the higher density of samples of this type (see Table 2).

The noticeable increase in ε in the ε(T) plot for the α-phase of the Na₃Fe₂(PO₄)₃ type 3 polycrystal indicates the presence of more thermally polarized mobile sodium particles than in type 2 samples. The increase in ε in the ε(T) curve may be attributed to the fact that sodium cations undergo greater vibrations under the influence of thermal energy, whilst the external electric field creates additional induced polarization on top of the existing spontaneous polarization in α-Na₃Fe₂(PO₄)₃, leading to greater particle polarization, i.e., an increase in the sample’s dielectric constant. The noticeable increase in μ_Tp(T, E(ω)) or the thermo-polarization mobility of sodium particles in type 3 samples may be attributed to a significant increase in the symmetry of the monoclinically distorted crystal lattice of α-Na₃Fe₂(PO₄)₃, since the polycrystals were synthesized from precursors whose structure underwent greater compressive deformation due to the melt quenching procedure (i.e., rapid cooling of the melt) than in the preparation of Type 2 samples (see Table 1).

In general, the patterns of change in dielectric permittivity with electric field frequency in type 3 samples remain the same as in type 1 and type 2 samples.

In type 3 samples, a small jump ∆ε is observed at temperature T_α→β (see Table 3), comparable to type 2 samples (Figure 6), so the change in ∆ε can be interpreted in the same way as in the case of type 2 samples.

Furthermore, a qualitative assessment of the interaction between cations and anions in the α-phase of Na₃Fe₂(PO₄)₃ samples can be carried out using the lattice gas model [39]. According to this model, in an ionic conductor crystal, the total two-particle potential of the electrostatic interaction between the equilibrium-distributed ions in the crystal can be expressed, according to [39], as follows:

$$\mathrm{V}\left(\overrightarrow{\mathrm{r}}\right)={\displaystyle \frac{{\mathrm{q}}^2}{\mathsf{\epsilon }\left(\overrightarrow{\mathrm{r}}\right)\overrightarrow{\mathrm{r}}}}$$

(3)

From a microscopic point of view, it is necessary to take into account the interaction between the charges (q) of the i-th cationic and j-th anionic sublattices of the Na₃Fe₂(PO₄)₃ crystal; therefore, Equation (5) can be rewritten as follows:

$$\mathrm{V}\left(\overrightarrow{{\mathrm{r}}_{\mathrm{i}\mathrm{j}}}\right)={\displaystyle \frac{1}{\mathsf{\epsilon }\left(\overrightarrow{\mathrm{r}}\right)}}{\sum }_{\begin{array}{c}i=1\\ j=1\end{array}}^{\mathrm{n}}{\displaystyle \frac{{\mathrm{q}}_{\mathrm{i}}{\mathrm{q}}_{\mathrm{j}}}{\overrightarrow{{\mathrm{r}}_{\mathrm{i}\mathrm{j}}}}}$$

(4)

where $\mathsf{\epsilon }\left(\overrightarrow{\mathrm{r}}\right)$ is the absolute dielectric constant of a medium;

$\overrightarrow{{\mathrm{r}}_{\mathrm{i}\mathrm{j}}}$ is the distance between the centers of the i-th and j-th charges of the anionic and cationic sublattices of the crystal.

It follows from Equation (3) that the potential $\mathrm{V}\left(\overrightarrow{{\mathrm{r}}_{\mathrm{i}\mathrm{j}}}\right)$ is proportional to the inverse of the sample’s dielectric permeability $\frac{1}{\mathsf{\epsilon }\left(\overrightarrow{\mathrm{r}}\right)}$. It follows from Equation (3) that the potential is proportional to the inverse of the sample’s dielectric constant.

Therefore, for a comparative assessment of the electrostatic interaction potential of the total charges between the three types of samples, it is sufficient to determine and specify the value of $K_V={\displaystyle \frac{1}{\mathsf{\epsilon }\left(\overrightarrow{\mathrm{r}}\right)}}$K in Table 3, which is a coefficient serving as a comparative measure of the interaction potential $\mathrm{V}\left(\overrightarrow{{\mathrm{r}}_{\mathrm{i}\mathrm{j}}}\right)$ of charges in polycrystals, since the charges and structural units of these samples are similar to one another.

According to (3), the electrostatic interaction potential of the total charges between the anionic and cationic parts of Na₃Fe₂(PO₄)₃ is weaker in type 3 samples compared to type 1 and 2 samples, since the ε of type 3 samples is greater than that of type 2 samples, and the distancesbetween charges in the three sample types are comparable.

Distinctive features of the parameters characterizing the dielectric properties of the three types of samples are presented in Table 3.

According to Table 3, the thermo-polarization mobility of type 3 samples is the highest, whilst, based on an assessment of the qualitative total interaction potential between charges, they have the lowest value among the other sample types, indicating a weaker bond between the cationic and anionic parts of the crystal lattice than in samples of types 1 and 2. Thus, type 3 samples can be classified as dielectric-ionic conductors.

3.3. Conductive Properties of Na₃Fe₂(PO₄)₃ Polycrystals Obtained by Solid-Phase and Melt Methods

The ionic conductivities of Na₃Fe₂(PO₄)₃ polycrystals were determined usingthe impedance method. To establish the influence of synthesis methods on the ionic conductivity of α- and β-Na₃Fe₂(PO₄)₃ polycrystals, it is advisable to determine the temperature dependencies σ(T) of these samples. Figure 7 shows that the nature of the change in conductivity dependencies σ(T) for three samples within the α- and β-phases obeys the Arrhenius law:

$${\mathsf{\sigma }}_{\mathsf{\alpha },\mathsf{\beta }}={\mathsf{\sigma }}_{0\mathsf{\alpha },\mathsf{\beta }} \mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{\mathsf{\Delta }{\mathrm{E}}_{\mathsf{\alpha },\mathsf{\beta }}}{\mathrm{k}\mathrm{T}}}\right)$$

(5)

where ${\mathsf{\sigma }}_{0\mathsf{\alpha }}$σ and ${\mathsf{\sigma }}_{0\mathsf{\beta }}$σ are the conductivities of the α-phase at T = 300 K and the β-phase at T = 400 K; ∆E is the activation energy of conductivity; k is Boltzmann’s constant; and T is the absolute temperature.

It is likely that the ionic conductivity of α-Na₃Fe₂(PO₄)₃ type 1 samples can be attributed to unpolarised ordered sodium cations. According to the hopping model, the ionic conductivity (σ) of α-Na₃Fe₂(PO₄)₃ polycrystals can be represented as the hopping of sodium cations from cavity M(2) to cavity M(1) through the conductivity ‘window’ of the three-dimensional crystal lattice along the conductivity channel under the action of an electric field. Ionic conductivity (σ) is proportional to the amount of charge carriers (q), their concentration (n), and mobility (μ) and is expressed by the formula:

$$\mathsf{\sigma }=\mathrm{q}\mathrm{n}\mathsf{\mu }$$

(6)

The low conductivity of type 1 samples (Figure 7) may be due to theirlow concentration of sodium cations per unit volume, since their density is not high. Also, ordered sodium cations have low mobility and high activation energy values (see the dependence ε(T, ω) shown in Figure 7 and

Table 4. Conductivity parameters of three types of samples.

Parameters	Phases	Na3Fe2(PO4)3
		Type 1	Type 2	Type 3
Ionic conductivity σ, (Ohm∙cm)−1	α (300 K)	4.5 × 10−7	5.5 × 10−7	1.4 × 10−6
	β (373 K)	5.6 × 10−5	6.7 × 10−5	2.3 × 10−4
Activation energy ΔE, eV	α	0.63	0.62	0.59
	β	0.46	0.45	0.43

). The electrostatic interaction between the cationic and anionic parts is high due to lower ε values than insample 3 (see Table 3).

The conductivity of type 2 samples is slightly higher than that of type 1 samples, since the concentration of sodium cations per unit volume (density) and their mobility are higher compared to type 1 samples. The conductivity of type 3 samples is higher than that of type 2 samples, since their density, charge carrier concentration and mobility are higher than those of type 2 samples (see Figure 5 and Table 2).

Analysing the conductivity parameters of the three types of samples shown in Table 4, it can be concluded that type 2 samples have higher conductivity compared to type 1 samples, but lower than type 3 samples. The activation energy of type 2 samples is lower than that of type 1 samples, but higher than that oftype 3 samples.

3.4. Influence of the Thermodynamic Synthesis Regime on Dipole Ordering and Ionic Conductivity of α-Na₃Fe₂(PO₄)₃ Polycrystals

The dielectric and conductive properties of three types of Na₃Fe₂(PO₄)₃ polycrystals differ from each other, since the samples were synthesized under different thermodynamic conditions. The state of the system is characterized by Gibbs energy (G) during sample synthesis. According to [40], nucleation and crystallization processes are possible in systems that are out of equilibrium, when supercooling (∆T) occurs at temperatures (T) lower than the melting point (T_e):

$$\mathsf{\Delta }\mathrm{T}={\mathrm{T}}_{\mathrm{e}}-\mathrm{T},$$

(7)

and the change in Gibbs energy in the system ΔG(T) < 0:

$$\mathsf{\Delta }\mathrm{G}=\mathsf{\Delta }\mathrm{T}\mathsf{\Delta }\mathrm{S}<0,$$

(8)

where ∆S is the change in entropy during crystallization, i.e., the difference between the entropy of the crystal and the entropy of the amorphous substance.

Regardless of the synthesis method used, nucleation and crystallization must occur under the conditions specified in (8). Since the entropy S of the system cannot be negative, the supercooling must be negative (∆T < 0) to satisfy the conditions in (8).

Type 1 samples are obtained by a solid-phase method during isothermal firing of the charge. During isothermal heating at 600 °C, a solid-phase reaction between the reagents may occur. Prolonged heating (7 h) likely brings the system into a metastable state, when, under the influence of temperature fluctuations, i.e., at ∆T < 0 (for example, ∆T < 0 during automatic shutdown and startup of the muffle furnace), nucleation is possible. For further nucleation and crystallization, heating for 7 h at 820 °C is required. The growth of crystallites will occur through the prolonged diffusion of constituent particles from the sample towards the nuclei. However, the transition from a metastable state to a solid-phase crystalline structure takes place via a phase transformation. The sequence of technological processes in the synthesis of type 1 polycrystals is shown in Figure 8.

It is likely that, in the melt synthesis method for Type 2 samples (Figure 9), the chemical reactions leading to the formation of the substance occur during the rapid melting of the charge, whilst the conditions favorable for the nucleation of crystallites arise under conditions of temperature-gradient super cooling (∆T) of the melt, i.e., when the melt temperature T = 980 °C rapidly decreases to the ambient room temperature (T_r) through heat dissipation into the atmosphere.

Under conditions of moderate gradient-temperature undercooling (∆T) of the melt, the structures of the forming precursors may undergo significant compressive deformation (ε), whilst the crystallite nuclei within the precursors are subjected to moderate mechanical stresses (σ) and compressive forces [41]. With further isothermal heating of the precursors for 7 h, the mechanical stresses will decrease, but residual stresses and strains will remain in the structure of the resulting polycrystals.

It is likely that, during the synthesis of Type 3 samples by rapid cooling of the melt, the nucleation of numerous crystallites in the precursors occurs during the rapid cooling of the melt (ΔG(T) < 0), i.e., at high cooling rates (16.3 °C/s) (see Table 1). Under these conditions, enormous compressive deformations and mechanical stresses must be created in the forming precursors [41,42,43]. It is likely that isothermal sintering of the precursors at T = 820 °C for 2 h partially relieves the compressive deformations and mechanical stresses in the forming polycrystals; however, the action of residual stresses may contribute to the partial relief of monoclinic distortions in the crystalline framework of the crystallites within the α-Na₃Fe₂(PO₄)₃. This process can be described as a partial ‘symmetrization’ of the monoclinic structure of α-Na₃Fe₂(PO₄)₃.

The sequence of technological processes for the synthesisof type 3 samples is shown in Figure 10.

The increase in the thermally polarized mobility of dipoles and sodium cations in the polar α-phase of the Na₃Fe₂(PO₄)₃ polycrystal is likely due to a partial ‘symmetrization’ of the monoclinic unit cell in samples of types 2 and 3.

Thus, in fusion-based methods for synthesising polycrystals, the cooling rate of the precursors is a decisive factor influencing the formation of the dielectric and conductive properties of α-Na₃Fe₂(PO₄)₃.

4. Conclusions

• The solid-phase synthesis of Na₃Fe₂(PO₄)₃ polycrystals (Sample Type 1) proceeds over a prolonged period under isothermal conditions, leading to the formation of a polar α-phase with pronounced dielectric properties, characterized by low density and conductivity. The polar α-phase of Na₃Fe₂(PO₄)₃ is characterized by the presence of ‘large’ and ‘small’ polarized dipoles, and a small number of ‘free’ but ordered sodium cations, which possess low thermo-polarization mobility μ_Tp(T, E(ω)), indicating strong interactions between the cationic and anionic parts of the crystal lattice, caused by significant distortion of the rhombohedral structure of Na₃Fe₂(PO₄)₃ polycrystals. The dielectric nature of the polar α-phase of Na₃Fe₂(PO₄)₃ is confirmed by a sharp increase in dielectric permeability during the phase transition to the ion-conducting β-phase.
• The melt synthesis method for Na₃Fe₂(PO₄)₃ polycrystals (Type 2 samples) is carried out by rapidly melting the charge and cooling it at a rate of 3.26 °C/s; consequently, the polycrystals of the polar α-phase exhibit a less monoclinically distorted structure, better faceting, and higher density than Type 1 samples. The synthesis time is reduced by a factor of 2 compared to solid-phase synthesis. It has been established that the polar α-phase is moderately dielectric, exhibits a higher μ_Tp(T, E(ω)) than in Type 1 samples, and is characterized by the presence of both polarized and unpolarized sodium cations. These data may be associated with a partial reduction in the interaction between the cationic and anionic parts of the crystal lattice, resulting from a partial increase in the symmetry of the polycrystalline structure.
• The melt-quenching method for synthesizing Na₃Fe₂(PO₄)₃ polycrystals (3 sample types) involves rapidly melting the charge and quenching it at a rate of 16.3 °C/s. This method reduces the synthesis time by a factor of 7 compared to solid-state synthesis, yielding polycrystals of the polar α-phase with a more perfect grain structure, higher density, higher conductivity, and a less monoclinically distorted structure than type 2 samples. The polar phase of α-Na₃Fe₂(PO₄)₃ in type 3 samples is ion-conducting and is characterized by higher values of μ_Tp(T, E(ω)) than in type 2 samples. These data indicate a weaker bond between the cationic and anionic parts of the crystal lattice, associated with greater ‘‘symmetrisation” of the structure than in type 2 samples.
• An investigation intothe dielectric properties of the polar phase in three types of polycrystals has enabled an assessment of the degree of interconnection between the cationic and anionic components of their crystal frameworks. The obtained data facilitate interpretation of the ionic conductivity properties of the samples; therefore, dielectric spectroscopy is recommended when investigating new cathode materials.