3.4 Model 4: parallel combinations R1C1, R2C2 connected in series with CPE

Parallel combinations R1C1, R2C2 connected in series with a constant phase angle element (CPE) are shown in Figure 6(a). The impedance of CPE is given as

$$\mathbf{Z\_{CPE}} = \left(\,^{\mathrm{Y}}\mathrm{P}\mathbf{E}\right)^{-1} = \left[\,^{\mathrm{A}}\mathrm{A}(j\omega)^{\Psi}\right]^{-1} = \frac{1}{\mathrm{A}\_{0}\omega^{\Psi}}\cos\left(\frac{\Psi\pi}{2}\right) - \mathrm{j}\,\frac{\mathbf{1}}{\mathrm{A}\_{0}\omega^{\Psi}}\sin\left(\frac{\Psi\pi}{2}\right) \tag{8}$$

Plot of imaginary part vs. real part of ZCPE is a straight line with slope tan(ψπ/2), which remains constant as ω varies (hence the name CPE). For ψ = 1, the real part becomes zero and ZCPE = 1/(jωA0) and the CPE behaves like an ideal capacitor having capacitance A0. For ψ = 0, the imaginary part becomes zero and ZCPE = 1/A0

4. Experimental setup, measurements and results

DOI: http://dx.doi.org/10.5772/intechopen.81398

Impedance Spectroscopy: A Powerful Technique for Study of Electronic Ceramics

using silver paste and cured at 600°C for 15 minutes.

The sample holder used to house the sample is schematically shown in Figure 7(a). The spectroscopic and complex plane plots for M<sup>0</sup> and M<sup>00</sup> are shown in Figure 7(b,c). The corresponding Z plots are not shown for brevity. The way an equivalent circuit model representing the data was developed is now described. A quick look at the M<sup>00</sup> vs. M<sup>0</sup> plot shown in Figure 7(b) reveals that there is no shift in the graph as well as no steeply rising high-frequency branch. Similar behaviour was seen in the Z plots also. Therefore, following the tips presented in Section 2, presence of series resistance or capacitance is ruled out. The plot is not a clear semicircular arc but is slightly depressed indicating the presence of more than one charge transfer processes in the system. As the sample contains two phases, we have a system comprising two types of grains, grain boundary and contact electrode interface. Therefore, an equivalent circuit model comprising four parallel RC's connected in series, where two RC's say R1C1 and R2C2 represent the two phases, R3C3 represents the grain boundary and R4C4 correspond to the sample-electrode interface, seems to be a plausible model. If we assume that R1C1 < R2C2 < R3C3 < R4C4 then, since electrode responses appear at low frequencies [1], R1C1 and R2C2 may be assigned to grains, R3C3 to grain boundary and R4C4 to electrode interface. The individual contributions from these RC's may be depicted by drawing tentative arcs (as shown by dotted lines in Figure 7(b)) making intercepts on the M<sup>0</sup> axis at C0/ C4, C0/C4 + C0/C3, C0/C4 + C0/C3 + C0/C2 and C0/C4 + C0/C3 + C0/C2 + C0/C1. As the plot is very symmetric, the intercepts and hence C's may be taken to be equal. By noting the values of intercepts from the graph, taking the value of C0 as 0.6832 � <sup>10</sup>�12, noting the frequencies where the tentative arcs would peak and using the relation ωRC = 1, the values of R' and C's are estimated as R1 = 362 Ω, R2 = 1087 <sup>Ω</sup>, R3 = 32,613 <sup>Ω</sup>, R4 = 9944 <sup>Ω</sup> and C1 = C2 = C3 = C4 = 4.88 � <sup>10</sup>�<sup>10</sup> F. By using these as initial guesses, the values of the components were obtained accurately by running the CNLS program IMPSPEC.BAS developed by one of the authors [22]

and being regularly used by us. These values were R1 = (0.77 � 0.01) kΩ, C1 = (0.38 � 0.01) nF, R2 = (3.43 � 0.02) kΩ, C2 = (0.44 � 0.01) nF, R3 = (10.87 � 0.09) kΩ, C3 = (1.08 � 0.06) nF, R4 = (15.36 � 0.26) kΩ and C4 = (5.81 � 0.09) nF. The M<sup>0</sup> and M<sup>00</sup> values corresponding to the fitted RC's are

Use of different models for impedance spectroscopy of few other ceramics is briefly described now. Analysis of data for SrTiO3 borosilicate glass ceramics having

also shown in Figure 7(c).

89

Development of equivalent circuit model for representing the ceramic system BaFexTi1-xO3 (x = 0.05), prepared in our laboratory, is now described as an illustration. Sample was prepared by solid state synthesis method by taking BaCO3 (Merck 99.5%), Fe2O3 (Merck 99.5%) and TiO2 (Merck 99.5%) in appropriate amounts, mixing in acetone medium for 6 hours and calcining at 1100°C for 6 hours. The calcined powder was mixed with small amount of PVA binder and pressed into disclike (dia 12 mm, thickness 1.5 mm) pellets using uniaxial hydraulic press with 60 kN pressure. These pellets were sintered in an electrical furnace (Lenton, made in Germany) where, first, binder was removed by raising the temperature to 500°C and holding for 2 hours and then increasing the temperature to 1250°C at 5°C/min and maintaining there for 10 hours followed by cooling it to room temperature. Xray diffraction analysis revealed that the sample contained tetragonal and hexagonal phases in equal amounts. Impedance measurements were carried out as function of frequency (20 Hz to 1 MHz) at temperatures from 300 to 650 K. For this, the pellets were polished using emery papers of grade 1/0 and 2/0 and electroded on both sides

Figure 6.

(a) Equivalent circuit model containing series combination of parallel R1C1, parallel R2C2 and CPE. Normalized (b) Z″/(R1 + R2) vs Z<sup>0</sup> /(R1 + R2), (c) M″/(C0/C1 + C0/C2 vs M<sup>0</sup> /(C0/C1 + C0/C2, (d) Y″/[1/ (R1 + R2)] vs Y<sup>0</sup> [1/(R1 + R2)] and (e) <sup>ɛ</sup>″/[(1/C0){C1(R1/(R1 + R2))<sup>2</sup> + C2(R2/(R1 + R2))<sup>2</sup> }] vs ɛ<sup>0</sup> /[(1/C0) {C1(R1/(R1 + R2))2 + C2(R2/(R1 + R2))<sup>2</sup> }] for R2/R1 = 2, R2C2/R1C1 = 1, 5, 10, 100, A0 = 6 � <sup>10</sup>–<sup>7</sup> and θ = 40.

and the CPE behaves like an ideal register of value 1/A0 [6]. The values of Z<sup>0</sup> and Z<sup>00</sup> for the model shown in Figure 6(a) are given as [9, 21].

$$Z' = \frac{\mathbf{R}\_1}{\mathbf{1} + \left(\alpha \mathbf{C}\_1 \mathbf{R}\_1\right)^2} + \frac{\mathbf{R}\_2}{\mathbf{1} + \left(\alpha \mathbf{C}\_2 \mathbf{R}\_2\right)^2} + \left(\frac{\mathbf{1}}{\mathbf{A}\_0 \alpha^\Psi}\right) \cos\left(\frac{\Psi \pi}{2}\right) \tag{9}$$

$$\mathbf{Z}^{\prime\prime} = \frac{\mathbf{o}\mathbf{C}\_1\mathbf{R}\_1\mathbf{R}\_1}{\mathbf{1} + \left(\mathbf{o}\mathbf{C}\_1\mathbf{R}\_1\right)^2} + \frac{\mathbf{o}\mathbf{C}\_2\mathbf{R}\_2\mathbf{R}\_2}{\mathbf{1} + \left(\mathbf{o}\mathbf{C}\_2\mathbf{R}\_2\right)^2} + \left(\frac{\mathbf{1}}{\mathbf{A}\_0\mathbf{o}\mathbb{V}}\right)\sin\left(\frac{\mathbb{V}\pi}{2}\right) \tag{10}$$

A model comprising series combination of parallel R1-CPE1 and parallel R2-CPE2 is one of the models very widely used to represent the behaviour of a ceramic when the impedance plots have two depressed arcs. The reader is referred to [9] where this has been discussed by the authors in detail.
