Dielectric spectroscopy

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: $${\displaystyle j_{\text{t}}=j_0\left(\exp ({\alpha }_{\text{o}}f\eta )-\exp (-{\alpha }_{\text{r}}f\eta )\right)}$$ with $${\displaystyle \eta =E-E_{\text{eq}},f=F/(RT),{\alpha }_{\text{o}}+{\alpha }_{\text{r}}=1.}$$ ${\displaystyle j_0}$ is the exchange current density and ${\displaystyle {\alpha }_{\text{o}}}$ and ${\displaystyle {\alpha }_{\text{r}}}$ are the symmetry factors.

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.

Dynamic behavior

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: $${\displaystyle R_{\text{ct}}=\frac{1}{\mathrm{\partial }{j}_{\text{t}}/\mathrm{\partial }\eta }=\frac{1}{f{j}_0\left({\alpha }_{\text{o}}\exp ({\alpha }_{\text{o}}f\eta )+{\alpha }_{\text{r}}\exp (-{\alpha }_{\text{r}}f\eta )\right)}}$$

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

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 such as Bode plots can be used instead.

Ohmic resistance

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

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. ${\displaystyle Y\propto {\omega }^{\alpha }}$.

Measurement of the impedance parameters

Plotting the Nyquist diagram with a potentiostat 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.

In the paint and coatings industry, it is a useful tool to investigate the quality of coatings and to detect the presence of corrosion.

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

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

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

Electrochemical impedance spectroscopy can be used to obtain the frequency response of batteries and electrocatalytic systems at relatively high temperatures.

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. The IFAC database can be used as a resource to get the dielectric properties for human body tissues.

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