Conductivity studies of some diphosphates with the general formula A^I₂B^IIP₂O₇ by impedance spectroscopy

2.1 Synthesis

The crystalline diphosphates Na₂CaP₂O₇, Na₂CuP₂O₇ and K₂MnP₂O₇ were obtained in a powder state from starting materials with high purity grade (>99%) and using a solid state technique. Reactions occurred according to the following equations: $${\text{Na}}_2{\text{CO}}_3+{\text{CaCO}}_3+2{\text{NH}}_4{\text{H}}_2{\text{PO}}_4\Rightarrow {\text{Na}}_2{\text{CaP}}_2{\text{O}}_7+2{\text{NH}}_3+3{\text{H}}_2\text{O}+2{\text{CO}}_2$$ $${\text{Na}}_2{\text{CO}}_3+\text{CuO}+2{\text{NH}}_4{\text{H}}_2{\text{PO}}_4\Rightarrow {\text{Na}}_2{\text{CuP}}_2{\text{O}}_7+2{\text{NH}}_3+3{\text{H}}_2\text{O}+{\text{CO}}_2$$ $${\text{K}}_2{\text{CO}}_3+{\text{MnCO}}_3+2{\text{NH}}_4{\text{H}}_2{\text{PO}}_4\Rightarrow {\text{K}}_2{\text{MnP}}_2{\text{O}}_7+2{\text{NH}}_3+3{\text{H}}_2\text{O}+2{\text{CO}}_2$$

For each diphosphate, the starting materials were mixed and ground together in an agate mortar and heated progressively in porcelain crucible with intermittent cooling and regrinding. The synthesis temperatures were 950 K for Na₂CaP₂O₇ and 873 K for Na₂CuP₂O₇ and K₂MnP₂O₇.

The powders were analyzed by X-ray powder diffraction, using a Siemens D-5000 diffractometer equipped with a copper anticathode (λ_Kα = 1.5406 ). All diffractograms showed good agreement with the respective single crystal data (Bennazha et al., 1999; Elmaadi et al., 1994; Erragh et al., 1995).

2.2 Electrical properties

The conductivity measurements have been carried out using powder samples pressed to form pellets, with thickness 1 mm, and diameter 8 mm, sintered under the same operating conditions as for synthesis. The Compactness was close to 90% for all the samples. In the case of vacuum evaporation, gold has been used as the electrode material. The Au//diphosphate sample//Au devices were placed in a quartz measurement cell, degassed at 473 K for 2 h and filled with dry argon to avoid eventual oxidation of cell electrodes. The electrical properties were determined by complex impedance method (Bauerle, 1969), using a frequency response analyzer Solartron 1260. The frequency range was 10⁻¹–10⁶ Hz over the thermal interval 300–700 K in several temperature cycles. The electrical measurements were carried out automatically for temperature steps equal to 10°. For each temperature measurement, the sample was preserved for 30 min in order to reach its thermal equilibrium.

3.1 Impedance hodographs

Impedance diagrams of Z′′(Ω) as a function of Z′(Ω) i.e. Cole–Cole diagrams (Almond and West, 1983) are shown in Fig. 2a–c for Na₂CaP₂O₇ at 573, 593 and 613 K, Na₂CuP₂O₇ at 593, 613 and 633 K and for K₂MnP₂O₇ at 559, 605 and 638 K, respectively.

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Figure 2

Complex impedance Cole–Cole plots of Z′′(Ω) s. Z′ (Ω) at various temperature for the three compounds (a) Na₂CaP₂O₇, (b) Na₂CuP₂O₇ and (c) K₂MnP₂O₇.

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The spectra can be classified into two groups:

1. Spectra relative to Na₂CaP₂O₇, which offer two depressed and distorted semi-circles at higher frequencies with a spike at lower frequencies. The presence of the inclined spike on the low frequency side of the semi-circles is characteristic of a polarization at the electrode-electrolyte interface.

2. Spectra relative to Na₂CuP₂O₇ and K₂MnP₂O₇, which consist of a single arc, approximately a distorted semi-circle independent of the temperature. Absence of an inclined spike on the low frequency side of the semi-circles in the spectra of K₂MnP₂O₇ is attributed to the mixed ionic and electronic conductor.

3.2 Equivalent electric circuits

An equivalent circuit containing two, three or more elements may be modified in various combinations but still yields the same overall ac response. This leads to the notion that there is no unique equivalent circuit for a particular system. Thus, the challenge is the selection of the type of equivalent circuit to be used for the analysis and interpretation of the electrical behavior of a system. Usually, the equivalent circuit is selected based on the following:

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  Intuition as to what kind of impedances is expected to be present in the studied system and whether they are connected in series or in parallel.

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  Examination of the experimental data to determine whether the response is consistent with the proposed circuit.

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  Inspection of R and C values obtained in order to check if they are realistic and their temperature dependence, if any, is reasonable.

An equivalent circuit widely used to represent bulk and grain boundary phenomena in polycrystalline materials consists of two parallel R_bC_b and R_gbC_gb elements connected in series (Hodge et al., 1976). The determination of different electrical components of the circuit is best achieved using a combination of impedance and electrical modulus formalism since each parallel RC element gives rise to a semi-circle in the complex plan (Z′′ vs. Z′ ; M′′ vs. M′) or a Debye peak in the spectroscopic plots of the imaginary functions (Z′′; M′′ vs. log(f)) Sinclair, 1995.

The response in the Z^∗ plane for a single parallel RC element has the form of a semi-circular arc passing through the origin. Its low frequency intercept on the real axis corresponds to the resistance R of the element. In practice, semi-circles associated with bulk relaxation processes in the Z^∗ plots of many conducting materials are found to be non ideal. Depressed semi-circles are obtained with their centers displaced below the real axis. There are two main reasons for such non-ideal behavior:

1. The presence of a distribution in relaxation times within the bulk response (Howell et al., 1974).

2. Distortion due to other relaxations, e. g. grain boundary relaxations, whose time constants are within two orders of magnitude of that of the bulk (Sinclair, 1995; Ming et al., 1995).

3.2.1

3.2.1 Na₂CaP₂O₇ sample

The Z^∗ hodographs obtained for the diphosphate Na₂CaP₂O₇ studied are consistent with the equivalent electric circuit shown in Fig. 2a. Two semi-circumferences of the Z′′ = f(Z′) curve, which appear successively at decreasing frequency (Fig. 1a), re associated with relaxation processes characteristic of intragranular (R_bC_b) and intergranular (R_gbC_gb), respectively. The polarization phenomena at the electrode–electrolyte interface are represented by the C element.

For a given temperature, the values of R_b and R_gb parameters have been, at first, estimated from experimental spectra at low frequency intercepts of each semi-circle with the real axis. For each relaxation process, the values of C_b and C_gb have been estimated from experimental points corresponding to the frequency ω_max of the Z′′ maximum. Indeed, ω_max, R and C for a parallel RC element are bound by the (ω_max. RC = 1) relationship. Recall that ω (ω = 2.π.f) was the angular frequency.

The dispersive properties of different relaxation processes have been taken into account in the simulation of impedance spectra by including the complex capacitance C_i of each R_iC_i element in the form (Jonscher, 1983): $$C_{\text{i}}=B_{\text{i}}·(j\omega )ni-1(0<n_{\text{i}}<1)$$ A fitting approach for each temperature has been conducted by using complex non-linear least-squares fitting of both the real and the imaginary parts of Z^∗(ω). LEVEM version 7.0 has been used for this research project (Macdonald, 1997). Typical fitting results of Na₂CaP₂O₇ sample at T = 240, 300 and 340 °C are illustrated in Fig. 3a, where the relative residuals are defined by the relationship (Ming et al., 1995): $${\mathrm{\Delta }}_{\text{re}}=\frac{Z_{\text{mes}.}^{\prime }-Z_{\text{cal}}^{\prime }}{Z_{\text{mes}.}^{\prime }}\text{and}{\mathrm{\Delta }}_{\text{im}}=\frac{Z_{\text{mes}.}^{″}-Z_{\text{cal}}^{″}}{Z_{\text{mes}.}^{″}}$$ An optimum fit is obtained as the relative residuals were randomly distributed around the frequency axis. The relative errors were less than 5% in the measurement frequency range (Fig. 3a), which were in agreement with the experimental error. Therefore, an excellent agreement between experimental and theoretical calculation data has been established in a wide frequency range, confirming the validity of the equivalent electric circuit proposed.

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Figure 3

Equivalent circuit used to represent the electrical behavior of (a) Na₂CaP₂O₇, (b) Na₂CuP₂O₇ and (c) K₂MnP₂O₇.

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Values of different electric parameters, at different temperatures, of the equivalent circuit determined after fitting were illustrated in Table 1. The R_i parameters vary with temperature according to Arrhenius-type laws, while the B_i and the n_i parameters were relatively independent of temperature.

Table 1 Electrical values of the equivalent circuit parameters calculated for the compound Na₂CaP₂O_7.

T (°C)	R1 (Ω)10+5	C1 (F) 10−12	n1	R2 (Ω)10+5	C2 (F) 10−11	n2	C3 (F) 10−7	n3
200	37.2	1.34	0.8	30.2	8.02	0.78	9.00	0.8
205	30.1	1.03	0.8	30.0	8.02	0.78	9.00	0.8
210	24.7	1.03	0.8	22.0	8.02	0.78	9.00	0.8
215	20.5	1.03	0.8	20.0	1.00	0.78	9.00	0.8
220	17.1	1.03	0.8	14.1	1.00	0.78	9.00	0.8
225	14.2	1.03	0.8	13.0	1.00	0.78	9.00	0.8
230	12.0	1.34	0.8	12.4	1.00	0.78	9.00	0.8
235	10.4	1.34	0.8	9.80	1.00	0.8	9.00	0.8
240	87.9	1.03	0.8	6.22	1.00	0.8	9.00	0.8
245	75.3	1.03	0.8	5.50	1.00	0.8	9.00	0.8
250	6.49	1.03	0.8	4.37	1.00	0.8	9.00	0.8
260	4.76	2.30	0.8	2.60	1.00	0.8	9.00	0.8
265	4.06	2.30	0.8	4.00	1.00	0.8	9.00	0.8
270	3.43	2.30	0.8	2.00	1.00	0.8	9.00	0.8
280	2.33	2.30	0.8	1.47	1.00	0.8	9.00	0.8
285	1.94	2.30	0.8	1.70	1.00	0.8	9.00	0.8
290	1.60	2.30	0.8	1.00	1.00	0.8	9.00	0.8
300	1.09	2.30	0.8	1.00	1.00	0.8	9.00	0.8
310	0.750	3.30	0.8	0.490	1.00	0.8	9.00	0.8
320	0.521	3.30	0.8	0.511	1.00	0.8	9.00	0.8
330	0.372	3.30	0.8	0.352	1.00	0.8	9.00	0.8
340	0.266	3.30	0.8	0.240	1.00	0.8	9.00	0.8
350	0.194	3.30	0.8	0.174	1.00	0.8	9.00	0.8
360	0.144	3.30	0.8	0.134	1.00	0.8	9.00	0.8

3.3 Temperature dependence of electrical properties in Na₂CaP₂O₇, Na₂CuP₂O₇ and K₂MnP₂O₇

The temperature dependence of conductivity σ deduced from R_b is shown in Figs. 4a–c for Na₂CaP₂O₇, Na₂CuP₂O₇ and K₂MnP₂O₇ for a temperature range from 473 K to 673 K. The graphs consist of two linear segments, each of which involving a behavior of Arrhenius-type, σ.T = σ₀exp(−ΔEσ/kT). The activation energies relative to both temperature intervals for each diphosphate, (ΔEσ)_I and (ΔEσ)_II are illustrated in Table 4. For the two compounds Na₂CaP₂O₇ and Na₂CuP₂O₇, the crossing at increasing temperature from low temperature regime to high temperature one is accompanied by a small increase of activation energy. In contrast, in the case of K₂MnP₂O₇, the activation energy decreases from the low temperature regime to high temperature.

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Figure 4a

Impedance complex plane plot measured and calculated and relative residuals of the diphosphate Na₂CaP₂O₇ at T = 513 K, T = 573 K and T = 613 K, using the equivalent circuit of Fig. 2a.

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Figure 4b

Impedance complex plane plot measured and calculated and relative residuals of the diphosphate Na₂CuP₂O₇ at T = 533, 573 and 613 K, using the equivalent circuit of Fig. 2b.

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Figure 4c

Impedance complex plane plot measured and calculated and relative residuals of the diphosphate K₂MnP₂O₇ at T = 558, 608 and 638 K, using the equivalent circuit of Fig. 2c.

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Using the complex modulus formalism M∗ = 1/ɛ∗ = j(ωC₀)Z∗; where C₀ is the vacuum capacitance of the empty measuring cell, an analysis of ac impedance data of Na₂CaP₂O₇, Na₂CuP₂O₇ and K₂MnP₂O₇ was carried out in order to determine the conductivity relaxation parameters. Considering an equivalent electric circuit composed of two parallel RC elements associated in series, the modulus plot gives emphasis to the element with the smallest capacitance where as the impedance plot highlights that with the largest resistance. Consequently, the M^∗ formalism can determine grain boundary phenomena, electrode polarization and other interfacial effects in solid electrolytes (Bauerle, 1969; Sinclair, 1995). Frequency dependence of the normalized imaginary part of M^∗, M′′/M′′_max, for Na₂CaP₂O₇, Na₂CuP₂O₇ and K₂MnP₂O₇ at different temperature, is shown in Fig. 5. The non-symmetric of curves M′′/M′′_max is in agreement with non exponential behavior of the electrical function, which is well described by the Kohlrausch function φ(t) = exp[−(t/_σ)^β] with 0 < β < 1 (Ngai and Martin, 1989; Howell et al., 1974). The _σ and β parameters of the stretched exponential function are the conductivity relaxation time and the Kohlrauch exponent respectively. The smallest is the value of β, and the largest is attributed to the deviation of the relaxation with respect to a Debye-type relaxation (β = 1). The frequency range, where the peak occurs, is indicative of the transition from short-range to long-range mobility at decreasing frequency and is defined by the condition ωτ_σ = 1 where τ_σ is the most probable ion relaxation time (Patel and Martin, 1992). The energy values ΔE_f were extracted from _P frequency (f_P = 1/2πτ_σ), data relative to M′′_max. The full width at half-height (FWHH) of the M′′/M′′_max spectrum is clearly wider than the breadth of Debye peak (1.14 decades) resulting in a value β = 1.14/FWHH for the Kolrausch exponent. For each material, β may be considered as temperature-independent in the studied temperature range. The β values obtained for the three materials under investigation are reported in Table 4. The values of β relative to the compounds Na₂CaP₂O₇ and Na₂CuP₂O₇ are relatively equal and are smaller than those corresponding to K₂MnP₂O₇. As the temperature increases, modulus peak maxima relative to each material under investigation shift to higher frequencies. The temperature dependence of the f_p frequency relative to M′′_max for the Na₂CaP₂O₇, Na₂CuP₂O₇ and K₂MnP₂O₇ is reported in Figs. 4a–c. The activation energies determined from the impedance (ΔE_σ) and modulus (ΔE_f) spectra are close enough to suggest that the charge carriers in the three phases studied are probably due to a hopping mechanism (Chowdari and Gopalakrishnan, 1987) (See Fig. 6).

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Figure 5

Temperature dependence of logσ versus reciprocal temperature for the three compounds Na₂CaP₂O₇, Na₂CuP₂O₇ and K₂MnP₂O₇.

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Figure 6

Normalized modulus, M′′/M′′_max versus log f at various temperature for of the three compounds Na₂CaP₂O₇, Na₂CuP₂O₇ and K₂MnP₂O₇.

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3.4 Correlation between electrical and structural properties in Na₂CaP₂O₇, Na₂CuP₂O₇ and K₂MnP₂O₇

The conductivity properties inside the two phases Na₂CaP₂O₇ and Na₂CuP₂O₇ can be considered as ionic. The Differential Scanning Calorimetry (DSC) analysis of the two compounds Na₂CaP₂O₇ and Na₂CuP₂O₇ has been conducted over the temperature range 298–673 K (temperature range studied by the ac conductivity). DSC thermograms do not present any phase transition or other thermal phenomenon. The results suggest that the phenomenon of the change in the slope observed in the plot of log σT against reciprocal temperature, for the two compounds Na₂CaP₂O₇ and Na₂CuP₂O₇, is not due to a phase transition but probably due to the crystallographic structure. Indeed, the transport properties inside the two phases Na₂CaP₂O₇ and Na₂CuP₂O₇ are respectively associated with the displacement of Na⁺ ions in the tunnels seen according to the[1 0 0] direction and in the space of interlayer seen according to[0 1 0] and[1 0 0] directions. For Na₂CaP₂O₇, the sodium ions occupy two crystallographic sites Na⁺(1) and Na⁺(2). Thus, the first segment of weak activation energy is attributed to motion of Na⁺(1) ions which are free in the center of their respective tunnels. Whereas the second segment of high activation energy is due, in addition to Na⁺(1) ions, to the contribution of Na⁺(2) ions which are rather wedged with polyhedra CaO₆ and P₂O₇ and require more thermal energy to become mobile. The activation energy measured over the temperature 533 K is the average between the hopping energy of ions Na⁺(1) and Na⁺(2). For the compound Na₂CuP₂O_7, we have one kind of sodium cation according to its single crystallographic site. Each cation is surrounded by six oxygen atoms. The thermal activation can cause a motion of the P₂O₇ groups which influence the movement of the cations Na⁺ neighbors and modify the behavior of ionic conductivity. The conductivity in the compound K₂MnP₂O₇ is rather of electronic nature. This electronic conductivity found may result from a charge carrier supported by the existence of the Mn element which has several oxidation states.

This compound is characterized by two mechanisms of ionic conduction. One with low temperatures, and an activation energy close to 0.71 eV and one with high temperatures, which energy is much lower and of the order of 0.21 eV. Such a phenomenon is also encountered in materials with tysonite structure type (El Omari et al., 1998a, 1998). These materials have three fluorine subnets. For low temperatures, only one sub-lattice of the three is mobile. In the contrary for high temperatures an exchange between the three networks is established.