**4.3 Modeling of the electrical conductivity through the equivalent electrical circuit**

From the modeling performed by an electrical circuit during the study of intercalation according to [39, 40], we have learned some physical characteristics of this

**Figure 16.**

*Variation of the percentage of the relative difference in (a) grain boundary resistance and (b) grain at the different times of adsorption.*

*Experimental and Theoretical Study of the Adsorption Behavior of Nitrate Ions by Layered… DOI: http://dx.doi.org/10.5772/intechopen.90883*

material. The circuit consists of a resistor in parallel with a constant phase element (CPE) characterized by a pseudocapacitance and Tp (scattering coefficient).

The conductivity of this circuit is of the following form [40]:

$$
\sigma'\_{ac}(a) = \frac{e}{\mathcal{S}} \times \mathcal{y}'(a) \tag{10}
$$

where the real part of the admittance is

$$Y'(\alpha) = \frac{1}{R} \left( 1 + (\alpha \pi)^p \cos\left(\frac{p\pi}{2}\right) \right) \tag{11}$$

Using the simulation of Eq. 11, we have obtained the equation of the admittance of grain established by the following expression:

$$\mathbf{Y}'\_{\mathcal{g}}(\boldsymbol{\omega}) = \frac{\mathbf{1}}{R} \left[ \mathbf{1} + (\boldsymbol{\alpha} \mathbf{r})\_{\mathcal{g}}^{p\_{\mathcal{g}}} \cos \left( \frac{p\_{\mathcal{g}} \pi}{2} \right) \right] \tag{12}$$

Finally the conductivity of grain is shown in the following equation:

$$\begin{aligned} \sigma'\_{\rm ac}(o) &= \frac{k}{R} \left[ 1 + (o\tau)^{\rm pg}\_{\rm g} \cos\left(\frac{\rm pg\pi}{2}\right) \right] = \sigma\_{\rm dg} + \frac{k}{R} \times (o\tau)^{\rm pg}\_{\rm g} \cos\left(\frac{p\_{\rm g}\pi}{2}\right) \\\sigma'\_{\rm ac}(o) &= \sigma\_{\rm dg} \left( 1 + (\tau)^{p\_{\rm g}}\_{\rm g} \cos\left(\frac{p\_{\rm g}\pi}{2}\right) \times o^{p\_{\rm g}} \right) \\\sigma'\_{\rm ac}(o) &= \sigma\_{\rm dg} \left( 1 + \left( \frac{o}{o\nu\_{\rm g}} \right)^{\rm pg} \right) \end{aligned} \tag{13}$$

with *ωhg* ¼ *τ pg <sup>g</sup>* cos *pg<sup>π</sup>* 2 � � � � �<sup>1</sup> the pulsation of hopping

In the term of conductivity, plots of the lower frequency dispersion correspond to the presence in grain boundary; this region can be approximately modeled by the circuit similar to the one depicted in **Figure 17** (grain boundary).

By similar calculations, the expression of the grain seal conductivity follows the shape of the following expression:

$$\begin{aligned} \left(\sigma'\_{\rm ac}(o)\right)\_{\rm b\mathbf{g}} &= \frac{k}{R} \times \left[1 + (o\tau)^{p\_{\rm b\mathbf{g}}}\_{\rm b\mathbf{g}} \times \cos\left(\frac{p\_{\rm b\mathbf{g}}\pi}{2}\right)\right] \\ &= (\sigma\_{\rm dc})\_{\rm b\mathbf{g}} + \frac{k}{R} \times (o\tau)^{p\_{\rm b\mathbf{g}}} \cos\left(\frac{p\_{\rm b\mathbf{g}}\pi}{2}\right) \\ \left(\sigma'\_{\rm ac}(o\nu)\right)\_{\rm b\mathbf{g}} &= (\sigma\_{\rm dc})\_{\rm b\mathbf{g}} \times \left[1 + (\tau)^{p\_{\rm b\mathbf{g}}}\_{\rm b\mathbf{g}} \cos\left(\frac{p\_{\rm b\mathbf{g}}\pi}{2}\right)\right] \times o^{p\_{\rm b\mathbf{g}}}, \\ \left(\sigma'\_{\rm ac}(o\nu)\right)\_{\rm b\mathbf{g}} &= (\sigma\_{\rm dc})\_{\rm b\mathbf{g}} \times \left[1 + \left(\frac{o\nu}{o\nu\_{\rm b\mathbf{g}}}\right)^{p\_{\rm b\mathbf{g}}}\right] \end{aligned} \tag{14}$$

with *ωhbg* ¼ *τ pbg bg* cos *pbgπ* 2 � � � � �<sup>1</sup> is the pulsation of hopping

**Figure 17.** *Corresponding equivalent circuit used to fit by EIS experimental data of intercalation.*

The total conductivity of the sample is similar for the different adsorption times; the evolution is typical called the double power law of Jonscher [41].

From the modeling performed by an electrical circuit during the adsorption study, we learned some physical characteristics of this material through the equivalent circuit modeling of the grain boundaries according to literature reviews [42] (**Figure 18**).

**Figure 19** shows the electrical conductivity of the sample as a function of frequency. Two different regions can be distinguished. In Region 1, the conductivity is dominated by grains and in Region 2 is dominated by boundary grains [42], where the conductivity increases with increasing frequency. The electrical conduction of the sample follows a consecutive hopping mechanism. Whenever it is transferred to another site, the surrounding molecules respond to this perturbation with structural changes, and the electron or hole is temporarily trapped in the potential well leading to polarization. Another aspect of this charge hopping mechanism is that the electron or hole tends to associate with local defects [3, 43].

The dependence of the AC conductivity on frequency can be expressed by the following law:

$$
\sigma'\_{ac}(\alpha) = \sigma\_{\rm DC} + A\_{\rm g} \times \alpha^{p\_{\rm g}} + A\_{\rm bg} \times \alpha^{p\_{\rm g}} \tag{15}
$$

exponent decreases with increasing surface adsorption. This result is in clear agreement with the correlated barrier hopping (CBH) model, so the frequency dependence

*Experimental and Theoretical Study of the Adsorption Behavior of Nitrate Ions by Layered…*

Better fundamental understanding of the adsorption phenomenon is modeled by

This circuit has the expression of conductivity is following Eqs. 10. Use the fit by this circuit the results of the simulation are tabulated in **Table 5**. The values of the conductivity of grains and grain boundaries fitted and evaluated using the equiva-

All frequency sweeps from this experiment were analyzed using the model of double power law to give σbg values σ<sup>g</sup> grain relaxation time and grain boundary. In all cases the double power law provided an excellent fit to the data; **Figure 10a** and **b**

In the model hopping, we distinguish two different characteristics in measurement frequency range. The charge transport takes place via an infinite percolation path in intermediate frequency. At high frequencies when the conductivity

increases, the transport is dominated by a hopping contribution in finished areas of the system and is manifested in the variation of conductivity as a function of

Throughout in this study, the electrical properties of adsorption of nitrate by system LDH using the spectroscopy impedance as technical the investigation and monitoring of adsorption is important for the excellent result using the resistor of

In order to attributed the second semicircle to a feature of the system, it is

grain boundaries and consider the factors which control the magnitude of the grain

These results such as those shown in **Figure 21a-d** are useful for several reasons:

• To indicate whether the overall resistance of a material is dominated by bulk

• To assess the quality and electrical homogeneity in the monitoring of adsorption since there is generally a link between sintering/microstructure and AC response.

• To measure the values of the resistances and capacitances at different times

**Time (min) σ<sup>g</sup> (μS/m) σbg (μS/m) τbg (μS) τ<sup>g</sup> (ms) Cbg (nF) Cg (pF)** 8.685 1.620 1.10 3.89 1.01 0.634 2.770 0.7812 1.69 7.86 0.601 0.713 2.95 0.369 2.02 21.16 0.752 0.784 3.044 0.2493 2.58 28.26 0.983 0.704 3.761 0.34602 2.85 7.2 0.795 0.550 2.29 0.3125 3.35 3.86 0.895 0.121

*Values of the conductivity relaxation time and the capacitance of grains and grain boundaries.*

)ads-LDH with grains and

)ads-LDH.

of σ'ac can be explained in terms of this model [47–53].

lent electrical circuit are reported in **Table 5**.

*DOI: http://dx.doi.org/10.5772/intechopen.90883*

frequency by slope breaks such as pg and pbg.

of adsorption

**Table 5.**

**131**

grains and the resistor of boundaries grains (**Figure 20**).

essential to have a picture of an idealized Zn3Al-Cl-(NO3

boundary impedance. **Figure 2b** model represents a Zn3Al-Cl-(NO3

or grain boundary assured by the adsorption of nitrate

the equivalent circuit depicted in parentheses (**Figure 19**).

is a plot of σ'dc values as a function of frequency for the two regions.

**4.4 Correlation between kinetic and impedance spectroscopy studies**

where Ag is a pre-exponential factor and Abg is the frequency exponent [8, 44–46], which generally is less than or equal to 1. **Figures 7** and **8** show the frequency dependence of the AC electrical conductivity at different times of adsorption. It is clear from the plot that above a certain point, the conductivity increases linearly with frequency. In these figures it is also evident that the DC contribution is important at small frequencies and high frequencies, whereas the frequencydependent term dominates at high frequencies. Also in the low-frequency region, the conductivity depends on the time of adsorption. Such dependence may be described by the variable range hopping (VRH) mechanism also called hopping conduction mechanism. The value of Ag and Abg in Eq. (13) was extracted from the slope of the plot of Log(σ'ac) versus Log(f), and this value was used to explain the conduction mechanism in the sample. The capacitance adsorption is called double layer capacitance is the dependence of values was plotted, and it is seen that the frequency

**Figure 18.**

*Corresponding equivalent circuit to fit by EIS experimental data contribution of cluster (NO3 ─ H3O<sup>+</sup> ) adsorbed on the surface (grain boundary).*

#### **Figure 19.**

*Corresponding equivalent circuit used to fit the EIS experimental data of conductivity for sample study.*

*Experimental and Theoretical Study of the Adsorption Behavior of Nitrate Ions by Layered… DOI: http://dx.doi.org/10.5772/intechopen.90883*

exponent decreases with increasing surface adsorption. This result is in clear agreement with the correlated barrier hopping (CBH) model, so the frequency dependence of σ'ac can be explained in terms of this model [47–53].

Better fundamental understanding of the adsorption phenomenon is modeled by the equivalent circuit depicted in parentheses (**Figure 19**).

This circuit has the expression of conductivity is following Eqs. 10. Use the fit by this circuit the results of the simulation are tabulated in **Table 5**. The values of the conductivity of grains and grain boundaries fitted and evaluated using the equivalent electrical circuit are reported in **Table 5**.

All frequency sweeps from this experiment were analyzed using the model of double power law to give σbg values σ<sup>g</sup> grain relaxation time and grain boundary. In all cases the double power law provided an excellent fit to the data; **Figure 10a** and **b** is a plot of σ'dc values as a function of frequency for the two regions.

In the model hopping, we distinguish two different characteristics in measurement frequency range. The charge transport takes place via an infinite percolation path in intermediate frequency. At high frequencies when the conductivity increases, the transport is dominated by a hopping contribution in finished areas of the system and is manifested in the variation of conductivity as a function of frequency by slope breaks such as pg and pbg.

#### **4.4 Correlation between kinetic and impedance spectroscopy studies**

Throughout in this study, the electrical properties of adsorption of nitrate by system LDH using the spectroscopy impedance as technical the investigation and monitoring of adsorption is important for the excellent result using the resistor of grains and the resistor of boundaries grains (**Figure 20**).

In order to attributed the second semicircle to a feature of the system, it is essential to have a picture of an idealized Zn3Al-Cl-(NO3 )ads-LDH with grains and grain boundaries and consider the factors which control the magnitude of the grain boundary impedance. **Figure 2b** model represents a Zn3Al-Cl-(NO3 )ads-LDH.

These results such as those shown in **Figure 21a-d** are useful for several reasons:



• To measure the values of the resistances and capacitances at different times of adsorption

**Table 5.**

*Values of the conductivity relaxation time and the capacitance of grains and grain boundaries.*

visualized in **Figure 17** show the variation of imaginary part of permittivity at 1 kHz

*(a) Ratio dimensional between grain and grain boundaries as a function of adsorption time. (b) Variation the fractality of system versus adsorption time. (c) Evolution of the partition coefficient between absorbance and absorbance as a function of time. (d) Variation in a comparative study between impedance spectroscopy and*

*Experimental and Theoretical Study of the Adsorption Behavior of Nitrate Ions by Layered…*

process, has been successful in the FT-IR diagram. Dielectric response of Zn3-Al-Cl-LDH samples has been explained using the Cole-Cole presentation during the adsorption phenomenon. The resistor of the sample increased from 73,215 to 363,094 Kohms, and also the conductivity spectra exhibited high conductivity in high frequency according to two mechanisms of hopping conduction: one of the water molecules and the other of the nitrate ions adsorbed in the LDH. The mathematical fitting obtained using the equivalent circuit of these diagrams was carried out to obtain the conductivity following the double power law

The author would like to thank the reviewers for their insight and their

and LDH, which indicated to the intercalating

the frequency.

**Figure 21.**

**5. Conclusion**

of Jonscher.

painstaking.

**133**

**Acknowledgements**

Characteristic band of NO3

*the kinetic study indicating the adsorption equilibrium.*

*DOI: http://dx.doi.org/10.5772/intechopen.90883*

**Figure 20.** *Correlation figures between the kinetic and impedance spectroscopy studies of adsorption.*

The distinction between kinetic and complex impedance spectroscopy study later controls the magnitude of the grain and the grain boundary and with a typical bulk permittivity in the range 103 to 105 using the relation and

*Experimental and Theoretical Study of the Adsorption Behavior of Nitrate Ions by Layered… DOI: http://dx.doi.org/10.5772/intechopen.90883*

#### **Figure 21.**

*(a) Ratio dimensional between grain and grain boundaries as a function of adsorption time. (b) Variation the fractality of system versus adsorption time. (c) Evolution of the partition coefficient between absorbance and absorbance as a function of time. (d) Variation in a comparative study between impedance spectroscopy and the kinetic study indicating the adsorption equilibrium.*

visualized in **Figure 17** show the variation of imaginary part of permittivity at 1 kHz the frequency.
