Dielectric spectroscopy

Dielectric spectroscopy (which falls in a subcategory of the impedance spectroscopy) measures the dielectric properties of a medium as a function of frequency.^[2][3][4][5] It is based on the interaction of an external field with the electric dipole moment of the sample, often expressed by permittivity.

A dielectric permittivity spectrum over a wide range of frequencies. The real and imaginary parts of permittivity are shown, and various processes are depicted: ionic and dipolar relaxation, and atomic and electronic resonances at higher energies.^[1]

It is also an experimental method of characterizing electrochemical systems. This technique measures the impedance of a system over a range of frequencies, and therefore the frequency response of the system, including the energy storage and dissipation properties, is revealed. Often, data obtained by electrochemical impedance spectroscopy (EIS) is expressed graphically in a Bode plot or a Nyquist plot.

Impedance is the opposition to the flow of alternating current (AC) in a complex system. A passive complex electrical system comprises both energy dissipater (resistor) and energy storage (capacitor) elements. If the system is purely resistive, then the opposition to AC or direct current (DC) is simply resistance. Materials or systems exhibiting multiple phases (such as composites or heterogeneous materials) commonly show a universal dielectric response, whereby dielectric spectroscopy reveals a power law relationship between the impedance (or the inverse term, admittance) and the frequency, ω, of the applied AC field.

Almost any physico-chemical system, such as electrochemical cells, mass-beam oscillators, and even biological tissue possesses energy storage and dissipation properties. EIS examines them.

This technique has grown tremendously in stature over the past few years and is now being widely employed in a wide variety of scientific fields such as fuel cell testing, biomolecular interaction, and microstructural characterization. Often, EIS reveals information about the reaction mechanism of an electrochemical process: different reaction steps will dominate at certain frequencies, and the frequency response shown by EIS can help identify the rate limiting step.

Dielectric mechanisms

Dielectrics spectroscopy machine

There are a number of different dielectric mechanisms, connected to the way a studied medium reacts to the applied field (see the figure illustration). Each dielectric mechanism is centered around its characteristic frequency, which is the reciprocal of the characteristic time of the process. In general, dielectric mechanisms can be divided into relaxation and resonance processes. The most common, starting from high frequencies, are:

Electronic polarization

This resonant process occurs in a neutral atom when the electric field displaces the electron density relative to the nucleus it surrounds.

This displacement occurs due to the equilibrium between restoration and electric forces. Electronic polarization may be understood by assuming an atom as a point nucleus surrounded by spherical electron cloud of uniform charge density.

Atomic polarization

Atomic polarization is observed when the nucleus of the atom reorients in response to the electric field. This is a resonant process. Atomic polarization is intrinsic to the nature of the atom and is a consequence of an applied field. Electronic polarization refers to the electron density and is a consequence of an applied field. Atomic polarization is usually small compared to electronic polarization.

Dipole relaxation

This originates from permanent and induced dipoles aligning to an electric field. Their orientation polarisation is disturbed by thermal noise (which mis-aligns the dipole vectors from the direction of the field), and the time needed for dipoles to relax is determined by the local viscosity. These two facts make dipole relaxation heavily dependent on temperature, pressure,^[6] and chemical surrounding.

Ionic relaxation

Ionic relaxation comprises ionic conductivity and interfacial and space charge relaxation. Ionic conductivity predominates at low frequencies and introduces only losses to the system. Interfacial relaxation occurs when charge carriers are trapped at interfaces of heterogeneous systems. A related effect is Maxwell-Wagner-Sillars polarization, where charge carriers blocked at inner dielectric boundary layers (on the mesoscopic scale) or external electrodes (on a macroscopic scale) lead to a separation of charges. The charges may be separated by a considerable distance and therefore make contributions to the dielectric loss that are orders of magnitude larger than the response due to molecular fluctuations.^[2]

Dielectric relaxation

Dielectric relaxation as a whole is the result of the movement of dipoles (dipole relaxation) and electric charges (ionic relaxation) due to an applied alternating field, and is usually observed in the frequency range 10²-10¹⁰ Hz. Relaxation mechanisms are relatively slow compared to resonant electronic transitions or molecular vibrations, which usually have frequencies above 10¹² Hz.

Principles

Steady-state

For a redox reaction R ${\displaystyle \leftrightarrow }$ O + e, without mass-transfer limitation, the relationship between the current density and the electrode overpotential is given by the Butler–Volmer equation:^[7] $${\displaystyle j_{\text{t}}=j_{0}\left(\exp(\alpha _{\text{o}}\,f\,\eta )-\exp(-\alpha _{\text{r}}\,f\,\eta )\right)}$$ with $${\displaystyle \eta =E-E_{\text{eq}},\;f=F/(R\,T),\;\alpha _{\text{o}}+\alpha _{\text{r}}=1.}$$ ${\displaystyle j_{0}}$ is the exchange current density and ${\displaystyle \alpha _{\text{o}}}$ and ${\displaystyle \alpha _{\text{r}}}$ are the symmetry factors.

Fig. 1: Steady-state current density vs. overpotential for a redox reaction

The curve ${\displaystyle j_{\text{t}}}$ vs. ${\displaystyle E}$ is not a straight line (Fig. 1), therefore a redox reaction is not a linear system.^[8]

Dynamic behavior

Faradaic impedance

Main article: Faradaic impedance

In an electrochemical cell the faradaic impedance of an electrolyte-electrode interface is the joint electrical resistance and capacitance at that interface.

Let us suppose that the Butler-Volmer relationship correctly describes the dynamic behavior of the redox reaction: $${\displaystyle j_{\text{t}}(t)=j_{\text{t}}(\eta (t))=j_{0}\left(\exp(\alpha _{\text{o}}\,f\,\eta (t))-\exp(-\alpha _{\text{r}}\,f\,\eta (t))\right)}$$

Dynamic behavior of the redox reaction is characterized by the so-called charge transfer resistance ${\displaystyle R_{\text{ct}}}$ defined by: $${\displaystyle R_{\text{ct}}={\frac {1}{\partial j_{\text{t}}/\partial \eta }}={\frac {1}{f\,j_{0}\,\left(\alpha _{\text{o}}\exp(\alpha _{\text{o}}\,f\,\eta )+\alpha _{\text{r}}\exp(-\alpha _{\text{r}}\,f\,\eta )\right)}}}$$

The value of the charge transfer resistance changes with the overpotential. For this simplest example the faradaic impedance is reduced to a resistance. It is worthwhile to notice that: $${\displaystyle R_{\text{ct}}={\frac {1}{f\,j_{0}}}}$$ for ${\displaystyle \eta =0}$.

Double-layer capacitance

Fig. 2: Equivalent circuit for a redoxin reaction without mass-transfer limitation

Fig. 3: Electrochemists Nyquist diagram of a RC parallel circuit. The arrow indicates increasing angular frequencies.

An electrode ${\displaystyle |}$ electrolyte interface behaves like a capacitance called electrochemical double-layer capacitance ${\displaystyle C_{\text{dl}}}$. The equivalent circuit for the redox reaction in Fig. 2 includes the double-layer capacitance ${\displaystyle C_{\text{dl}}}$ as well as the charge transfer resistance ${\displaystyle R_{\text{ct}}}$. Another analog circuit commonly used to model the electrochemical double-layer is called a constant phase element.

The electrical impedance of this circuit is easily obtained remembering the impedance of a capacitance which is given by: $${\displaystyle Z_{\text{dl}}(\omega )={\frac {1}{i\omega C_{\text{dl}}}}}$$ where ${\displaystyle \omega }$ is the angular frequency of a sinusoidal signal (rad/s), and ${\displaystyle i^{2}=-1}$.

It is obtained: $${\displaystyle Z(\omega )={\frac {R_{\text{t}}}{1+R_{\text{t}}C_{\text{dl}}i\omega }}}$$

Nyquist diagram of the impedance of the circuit shown in Fig. 3 is a semicircle with a diameter ${\displaystyle R_{\text{t}}}$ and an angular frequency at the apex equal to ${\displaystyle 1/(R_{\text{t}}\,C_{\text{dc}})}$ (Fig. 3). Other representations, Bode plots, or Black plans can be used.^[9]

Ohmic resistance

The ohmic resistance ${\displaystyle R_{\Omega }}$ appears in series with the electrode impedance of the reaction and the Nyquist diagram is translated to the right.

Universal dielectric response

Main article: Universal dielectric response

Under AC conditions with varying frequency ω, heterogeneous systems and composite materials exhibit a universal dielectric response, in which overall admittance exhibits a region of power law scaling with frequency. ${\displaystyle Y\propto \omega ^{\alpha }}$.^[10]

Measurement of the impedance parameters

Plotting the Nyquist diagram with a potentiostat^[11] and an impedance analyzer, most often included in modern potentiostats, allows the user to determine charge transfer resistance, double-layer capacitance and ohmic resistance. The exchange current density ${\displaystyle j_{0}}$ can be easily determined measuring the impedance of a redox reaction for ${\displaystyle \eta =0}$.

Nyquist diagrams are made of several arcs for reactions more complex than redox reactions and with mass-transfer limitations.

Applications

Electrochemical impedance spectroscopy is used in a wide range of applications.^[12]

In the paint and coatings industry, it is a useful tool to investigate the quality of coatings^[13][14] and to detect the presence of corrosion.^[15][16]

It is used in many biosensor systems as a label-free technique to measure bacterial concentration^[17] and to detect dangerous pathogens such as Escherichia coli O157:H7^[18] and Salmonella,^[19] and yeast cells.^[20][21]

Electrochemical impedance spectroscopy is also used to analyze and characterize different food products. Some examples are the assessment of food–package interactions,^[22] the analysis of milk composition,^[23] the characterization and the determination of the freezing end-point of ice-cream mixes,^[24][25] the measure of meat ageing,^[26] the investigation of ripeness and quality in fruits^[27][28][29] and the determination of free acidity in olive oil.^[30]

In the field of human health monitoring is better known as bioelectrical impedance analysis (BIA)^[31] and is used to estimate body composition^[32] as well as different parameters such as total body water and free fat mass.^[33]

Electrochemical impedance spectroscopy can be used to obtain the frequency response of batteries and electrocatalytic systems at relatively high temperatures.^[34][35][36]

Biomedical sensors working in the microwave range relies on dielectric spectroscopy to detect changes in the dielectric properties over a frequency range, such as non-invasive continuous blood glucose monitoring.^[37][38] The IFAC database can be used as a resource to get the dielectric properties for human body tissues.^[39]

For heterogenous mixtures like suspensions impedance spectroscopy can be used to monitor the particle sedimentation process.^[40]

See also

• Debye relaxation
• Dielectric absorption, ultra-low frequency changes
• Dielectric loss
• Electrochemistry
• Ellipsometry
• Green–Kubo relations
• Induced polarization (IP)
• Kramers–Kronig relations
• Linear response function
• Potentiostat
• Spectral induced polarisation (SIP)

References

1. From the Dielectric spectroscopy page of the research group of Dr. Kenneth A. Mauritz.
2. Kremer F., Schonhals A., Luck W. Broadband Dielectric Spectroscopy. – Springer-Verlag, 2002.
3. Sidorovich A. M., Dielectric Spectrum of Water. – Ukrainian Physical Journal, 1984, vol. 29, No 8, p. 1175-1181 (In Russian).
4. Hippel A. R. Dielectrics and Waves. – N. Y.: John Wiley & Sons, 1954.
5. Volkov A. A., Prokhorov A. S., Broadband Dielectric Spectroscopy of Solids. – Radiophysics and Quantum Electronics, 2003, vol. 46, Issue 8, p. 657–665.
6. Floudas G., Paluch, M., Grzybowski A., Ngai K. L. Molecular Dynamics of Glass-Forming Systems - Effects of Pressure. Springer-Verlag, 2011.
7. Okajima, Yoshinao; Shibuta, Yasushi; Suzuki, Toshio (2010). "A phase-field model for electrode reactions with Butler–Volmer kinetics". Computational Materials Science. 50 (1): 118–124. doi:10.1016/j.commatsci.2010.07.015.
8. Linear vs. non-linear systems in impedance measurements Archived December 5, 2008, at the Wayback Machine
9. "Potentiostat stability mystery explained" (PDF). Archived from the original (PDF) on 2013-10-23. Retrieved 2011-11-08.
10. Zhai, Chongpu; Hanaor, Dorian; Gan, Yixiang (2017). "Universality of the emergent scaling in finite random binary percolation networks". PLOS ONE. 12 (2): e0172298. Bibcode:2017PLoSO..1272298Z. doi:10.1371/journal.pone.0172298. PMC 5312937. PMID 28207872.
11. Impedance, admittance, Nyquist, Bode, Black, etc. Archived July 21, 2011, at the Wayback Machine
12. Lasia, A. Electrochemical Impedance Spectroscopy and Its Applications. In "Modern aspects of electrochemistry", volume 32. pp. 143–248.
13. McIntyre, J.M.; Pham, H.Q. (1996). "Electrochemical impedance spectroscopy; a tool for organic coatings optimizations". Progress in Organic Coatings. 27 (1–4): 201–207. doi:10.1016/0300-9440(95)00532-3.
14. Amirudin, A.; Thieny, D. (1995). "Application of electrochemical impedance spectroscopy to study the degradation of polymer-coated metals". Progress in Organic Coatings. 26 (1): 1–28. doi:10.1016/0300-9440(95)00581-1.
15. Bonora, P.L.; Deflorian, F.; Fedrizzi, L. (1996). "Electrochemical impedance spectroscopy as a tool for investigating underpaint corrosion". Electrochimica Acta. 41 (7–8): 1073–1082. doi:10.1016/0013-4686(95)00440-8.
16. Rammelt, U.; Reinhard, G. (1992). "Application of electrochemical impedance spectroscopy (EIS) for characterizing the corrosion-protective performance of organic coatings on metals". Progress in Organic Coatings. 21 (2–3): 205–226. doi:10.1016/0033-0655(92)87005-U.
17. Maalouf, R.; Fournier-Wirth, C.; Coste, J.; Chebib, H.; Saikali, Y.; Vittori, O.; Errachid, A.; Cloarec, J.P.; Martelet, C.; Jaffrezic-Renault, N. (2007). "Label-Free Detection of Bacteria by Electrochemical Impedance Spectroscopy: Comparison to Surface Plasmon Resonance". Analytical Chemistry. 79 (13): 4879–4886. doi:10.1021/ac070085n. PMID 17523594. S2CID 38589225.
18. Ruan, C.; Yang, L.; Li, Y. (2002). "Immunobiosensor Chips for Detection of Escherichia coli O157:H7 Using Electrochemical Impedance Spectroscopy". Analytical Chemistry. 74 (18): 4814–4820. doi:10.1021/ac025647b. PMID 12349988. S2CID 2068234.
19. Nandakumar, V.; La Belle, J.T.; Reed, J.; Shah, M.; Cochran, D.; Joshi, L.; Alford, T.L. (2008). "A methodology for rapid detection of Salmonella typhimurium using label-free electrochemical impedance spectroscopy". Biosensors & Bioelectronics. 24 (4): 1039–1042. doi:10.1016/j.bios.2008.06.036. PMID 18678481.
20. Soley, A.; Lecina, M.; Gamez, X.; Cairo, J.J.; Riu, P.; Rosell, X.; Bragos, R.; Godia, F. (2005). "On-line monitoring of yeast cell growth by impedance spectroscopy". Journal of Biotechnology. 118 (4): 398–405. doi:10.1016/j.jbiotec.2005.05.022. PMID 16026878.
21. Chen, H.; Heng, C.K.; Puiu, P.D.; Zhou, X.D.; Lee, A.C.; Lim, T.M.; Tan, S.N. (2005). "Detection of Saccharomyces cerevisiae immobilized on self-assembled monolayer (SAM) of alkanethiolate using electrochemical impedance spectroscopy". Analytica Chimica Acta. 554 (1–2): 52–59. doi:10.1016/j.aca.2005.08.086.
22. Hollaender, J. (2009). "Rapid assessment of food/package interactions by electrochemical impedance spectroscopy (EIS)". Food Additives & Contaminants. 14 (6–7): 617–626. doi:10.1080/02652039709374574. PMID 9373526.
23. Mabrook, M.F.; Petty, M.C. (2003). "Effect of composition on the electrical conductance of milk". Journal of Food Engineering. 60 (3): 321–325. doi:10.1016/S0260-8774(03)00054-2.
24. Grossi, Marco; Lanzoni, Massimo; Lazzarini, Roberto; Riccò, Bruno (August 2012). "Automatic ice-cream characterization by impedance measurements for optimal machine setting" (PDF). Measurement. 45 (7): 1747–1754. Bibcode:2012Meas...45.1747G. doi:10.1016/j.measurement.2012.04.009. S2CID 110783818.
25. Grossi, M.; Lazzarini, R.; Lanzoni, M.; Riccò, B. (October 2011). "A novel technique to control ice cream freezing by electrical characteristics analysis" (PDF). Journal of Food Engineering. 106 (4): 347–354. doi:10.1016/j.jfoodeng.2011.05.035.
26. Damez, J.L.; Clerion, S.; Abouelkaram, S.; Lepetit, J. (2008). "Beef meat electrical impedance spectroscopy and anisotropy sensing for non-invasive early assessment of meat ageing". Journal of Food Engineering. 85 (1): 116–122. doi:10.1016/j.jfoodeng.2007.07.026.
27. Rehman, M.; Abu Izneid, J.A.; Abdullha, M.Z.; Arshad, M.R. (2011). "Assessment of quality of fruits using impedance spectroscopy". International Journal of Food Science & Technology. 46 (6): 1303–1309. doi:10.1111/j.1365-2621.2011.02636.x. S2CID 23053716.
28. Harker, F.R.; Forbes, S.K. (1997). "Ripening and development of chilling injury in persimmon fruit: An electrical impedance study". New Zealand Journal of Crop and Horticultural Science. 25 (2): 149–157. doi:10.1080/01140671.1997.9514001.
29. Bauchot, A.D.; Harker, F.R.; Arnold, W.M. (2000). "). The use of electrical impedance spectroscopy to assess the physiological condition of kiwifruit". Postharvest Biology and Technology. 18 (1): 9–18. doi:10.1016/S0925-5214(99)00056-3.
30. Grossi, M.; Di Lecce, G.; Gallina Toschi, T.; Riccò, B. (December 2014). "A novel electrochemical method for olive oil acidity determination" (PDF). Microelectronics Journal. 45 (12): 1701–1707. doi:10.1016/j.mejo.2014.07.006. S2CID 13168066.
31. Kyle, U.G.; Bosaeus, I.; De Lorenzo, A.D.; Deurenberg, P.; Elia, M.; Gomez, J.M.; Heitmann, B.L.; Kent-Smith, L.; Melchior, J.C.; Pirlich, M.; Scharfetter, H.; Schols, A.; Pichard, C. (2004). "Bioelectrical impedance analysis—part I: review of principles and methods". Clinical Nutrition. 23 (5): 1226–1243. doi:10.1016/j.clnu.2004.06.004. PMID 15380917. S2CID 21000697.
32. Tengvall, M.; Ellegard, L.; Malmros, V.; Bosaeus, N.; Lissner, L.; Bosaeus, I. (2009). "Body composition in the elderly: Reference values and bioelectrical impedance spectroscopy to predict total body skeletal muscle mass". Clinical Nutrition. 28 (1): 52–58. doi:10.1016/j.clnu.2008.10.005. PMID 19010572.
33. Van Loan, M.D.; Withers, P.; Matthie, J.; Mayclin, P.L. Use of Bioimpedance Spectroscopy to Determine Extracellular Fluid, Intracellular Fluid, Total Body Water, and Fat-Free Mass. Chapter in Human Body Composition, Volume 60 of the series Basic Life Sciences. pp. 67–70.
34. M. Radtke, C. Hess, Easy-Made Setup for High-Temperature (Up to 1100 °C) Electrochemical Impedance Spectroscopy, J. of Materi Eng and Perform (2022). https://doi.org/10.1007/s11665-022-06653-3
35. Macdonald, Digby D. (2006). "Reflections on the history of electrochemical impedance spectroscopy". Electrochimica Acta. 51 (8–9): 1376–1388. doi:10.1016/j.electacta.2005.02.107.
36. Dokko, K.; Mohamedi, M.; Fujita, Y.; Itoh, T.; Nishizawa, M.; Umeda, M.; Uchida, I. (2001-05-01). "Kinetic Characterization of Single Particles of LiCoO2 by AC Impedance and Potential Step Methods". Journal of the Electrochemical Society. 148 (5): A422–A426. Bibcode:2001JElS..148A.422D. doi:10.1149/1.1359197. ISSN 0013-4651.
37. Choi, H.; Naylon, J.; Luzio, S.; Beutler, J.; Birchall, J.; Martin, C., and Porch, A., "Design and in-vitro interference test of microwave noninvasive blood glucose monitoring sensor," IEEE Trans. Microw. Theory Techn., vol. 63, no. 10, pp. 3016-3025, Oct. 2015. https://doi.org/10.1109/TMTT.2015.2472019
38. Choi, H.; Luzio, S.; Beutler, J.; and Porch, A., "Microwave noninvasive blood glucose monitoring sensor: Human clinical trial results," in 2017 IEEE MTT-S International Microwave Symposium (IMS) Proceedings, pp. 876-879, Honolulu, HI, USA, 4–9 June 2017. https://doi.org/10.1109/MWSYM.2017.8058721
39. D.Andreuccetti, R.Fossi and C.Petrucci (1997). "An Internet resource for the calculation of the dielectric properties of body tissues in the frequency range 10 Hz - 100 GHz". Based on data published by C.Gabriel et al. in 1996. IFAC-CNR, Florence (Italy).
40. Doppelhammer, Nikolaus; Pellens, Nick; Kirschhock, Christine E.A.; Jakoby, Bernhard; Reichel, Erwin K. (2020). "Using Moving Electrode Impedance Spectroscopy to Monitor Particle Sedimentation". IEEE Sensors Journal. 21 (8): 9636–9641. doi:10.1109/JSEN.2020.3004510. ISSN 1530-437X.