Dielectric spectroscopy

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 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.

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 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 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.

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

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 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 ${\displaystyle \eta =0}$.

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 as well as the charge transfer resistance. 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:

${\displaystyle Z_{\text{dl}}(\omega )={\frac {1}{{\text{i}}\,\omega \,C_{\text{dl}}}}}$

where ${\displaystyle \omega }$ is the angular frequency of a sinusoidal signal (rad/s), and ${\displaystyle \scriptstyle {{\text{i}}^{2}=-1}}$.

It is obtained:

${\displaystyle Z(\omega )={\frac {R_{\text{t}}}{1+R_{\text{t}}\,C_{\text{dl}}\,{\text{i}}\,\omega }}}$

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

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.

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]