# Capacitance–Resistive PEDOT:PSS Cotton Fabric Satisfied Jonscher’s Law with Index Exceeding One

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## Abstract

Jonscher’s law is investigated in the context of PEDOT:PSS impregnated conductive cotton fabric for frequencies from 10 Hz to 13 MHz and temperatures from 30°C to 100°C using complex impedance spectroscopy. The drop-casting and drying method was used to prepare samples of conductive cotton fabric with low and high concentrations of dopant. Argand plots of the ratio of AC–DC conductivities of the conductive fabric demonstrated the presence of reactance at high frequencies at each concentration of dopant. Regression analysis demonstrated that Jonscher’s power law was obeyed over a significant range of high frequencies. The hopping frequency and Jonscher index are found to depend on the concentration of dopant, but are insensitive to temperature over the range used in this study. By contrast with numerous experimental studies reporting that the Jonscher index is less than one, this experimental investigation found that Jonscher’s index exceeded one. We further investigated whether or not the hopping frequency identified by regression corresponded to a natural frequency arising in the Argand plot of the complex conductivity ratio. Considerations of curvature and phase angle identified two candidate frequencies, but neither was close to the hopping frequency identified by regression.

## Keywords

Cotton capacitance conductivity regression Argand## List of symbols

- \(\sigma _{\mathrm{AC}}\)
AC electrical conductivity

- \(\omega \)
Angular frequency

*Z*Electrical impedance

- \(Z_{\mathrm{I}}\)
Imaginary part of impedance

- \(\sigma _{\mathrm{I}}\)
Imaginary part of AC conductivity

*n*Jonscher’s power

*N*Size of data set

*a*Gradient of regression line

- \(\mu _x\)
Mean value of

*x*data- \(\sigma _{xx}\)
Variance of

*x*data- \(\sigma _{xy}\)
Covariance of

*x*and*y*data- \(R^2_{\mathrm {adj}}\)
Quality of fit adjusted for sample size

- RMS
Root mean squared integrated mean error

- \(\sigma _{\mathrm{DC}}\)
DC electrical conductivity

- \(\omega _c\)
Hopping frequency

- \(Z_{\mathrm{R}}\)
Real part of impedance

- \(\sigma _{\mathrm{R}}\)
Real part of AC conductivity

*A*Constant parameter

- \(\Phi (a,b)\)
Least squared target function

- \(b,\widehat{b}\)
Intercept on

*x*-axis and its estimate- \(\widehat{a}\)
Estimate of gradient of regression line

- \(\mu _y\)
Mean value of

*y*data- \(\sigma _{yy}\)
Variance of

*y*data- \(R^2\)
Measures quality of fit of regression

- \(L_1\)
Average integrated mean error

- \(\varepsilon \)
Arbitrary small parameter

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## Notes

### Acknowledgments

The author thanks Umm AL-Qura University for their support of this work.

## References

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