Impedance Spectroscopy

The electromagnetic radiation interacts with matter and its study is known as spectroscopy. Due to a wide spectrum of electromagnetism not a single tool can justify the whole interaction. There are lot of tools like: electron spectroscopy, atomic spectroscopy

Impedance spectroscopy (Dielectric Spectroscopy) is widely used tool in probing the dynamics of electron, ions, molecule motion inside the material. Even if it is a complete solid, fluid or semi-solid, impedance analysis gives a good information. Since the battery cell has different components with different resistance and reactance (from capacitance elements), the interference to the passage of electric current vector is been studied to obtain valuable information within the cell.

The information is obtained from dynamics of charge transport and dipole reorientation in the material.

How does this work?

A sine wave is being given as an input signal and the changes occur in the sine wave is measured. It measures dielectric properties as a function of frequency. For an AC sine wave with different frequency, we get data for capacitance, dielectric loss, Impedance and phase shift. From, these parameters we can plot Nyquist diagram to determine charge transfer resistance, double layer capacitance and ohmic resistance.

There are different Dielectric mechanisms, like electronic polarization, atomic polarization, dipole relaxation and ionic relaxation. As the size of the responding particles increases the frequency needed to respond decreases. Ions and dipoles which are very larger than electrons cant vibrate at higher frequencies while electron can easily vibrate. Therefore at low frequencies the dielectric effect is from all the these mechanisms while at higher frequencies is purely from electronic.

Representation of electrochemical reaction with an electronic circuit (Randles Circuit) is the main advantage of EIS. The electronic model consists combinations of resistors, capacitors, inductors constant phase elements etc., which is useful for understanding the system in a simpler way.

Plots like Nyquist, Bode, or Randels give an effective way to represent the data.

Formula's

For a parallel plate capacitor, the capacitance

 C= k ε₀(A/d)

where k is dielectric constant, ε₀ is permitivity of free space (=10⁷/4πC² = 8.854 x 10^-12 F/m), A is area of the electrode and d is the thickness of the capacitor.
 Using this we obtain

k = C. d/(A.ε₀)

The dielectric constant can also be termed as real permittivity and denoted by ε'

ε' = C. d/(A.ε₀)

While the imaginary component of permittivity is given by ε"

ε" = ε' * Dielectric Loss

Beside capacitance the interested parameters from the impedance analysis are Dielectric Loss (D) Impedance(Z) and Phase angle(θ). When an AC signal is applied to the sample there is a phase difference between the applied voltage signal and the observed current signal as illustrated in the phasor diagram (figure below). The current leads the voltage when a capacitive component of reactance is dominant over inductive reactance as depicted in the phasor diagram.

The potential V(t) and current I(t) are given as

V(t) = V₀ e^iωt

I(t) = I₀e^i(ωt-θ)

where V₀ and I₀ are amplitude of voltage and current and θ is the phase angle, ω = 2πf is the angular frequency and "i" is the complex number and its value is √-1. The impedance can be calculated as

Z = V(t)/I(t) 

Z = V₀ e^iωt / I₀ e^i(ωt-θ) = V₀ / I₀ e^iθ = Z₀ e^iθ

where e^iθ = cosθ + i sinθ (from Euler's relation) 

Z =Z₀ (cosθ + i sinθ)

Therefore the impedance is implemented in real and imaginary component as 

Z' = Z Cosθ

Z" = Z Sinθ

Also, these parameters are interrelated as

 θ = tan^-1(Z''/Z')

and |Z| = √((Z')²+(Z")²)

AC Conductivity is given by

σ = 2πf ε₀ε" 

Permitivity and impedance plots are enough for describing many characteristics or phenomenon in materials but few require some other variables like electric modulus...

M'  = 2πf ε₀ (A/d)Z"

M" = 2πf ε₀ (A/d)Z'

In some frequency plots we may obtain peaks which could shift with change in temperature, these are termed as Dielectric relaxation and the relaxation time is obtained using Arrhenius relation given by

τ = τ₀exp(-E_a/K_BT)

where τ₀ is the pre-exponential factor, E_a is the activation energy, K_B is the Boltzmann constant and T is the absolute temperature.

The required frequency range for investigating complete battery spectra from internal resistance to ionic diffusion is 1 MHz to 100 mHz.

 In general, the spectra consist of one or more semicircle’s in the high frequency range and sloping line in low frequency ranges.

 The ionic movement in liquid essentially has the sloping line without any semicircles which indicate diffusion. Semicircles are regarded for finite resistance which comes into play when solids obstruct the diffusion, more than one effect present in the system will give rise to each semicircle independently or clubbed to form a curvy pattern. Often the graph starts away from the origin, at high frequencies (1 MHz-10 KHz) the pattern intersects x-axis while away from origin, this intercept is regarded as the internal resistance of the cell formed from contact resistance.

Each semicircle is represented by an equivalent electronic element "RC" where R is resistance and C is capacitance. A material can have more than one semicircle separated significantly or overlapped. Apart from simple elements R & C, inductive elements can also be observed and it can be easily identified with evidence of negative capacitance. An element Q, which can be termed as lossy capacitor is generally used in this analysis.

The impedance plot for liquid electrolyte, the ionic motion in without any interference can be represented as

The ease of fitting the impedance plots with equivalent electronic circuits and getting information of inside the electrochemical cell is quite very handy-tool, which is useful to study the cell dynamics along-with the measurable VI characteristics. Since, this study is quite a fast and easy to get information of the cell inside, this method is used for battery managements system which gives measure for state of charge, battery life, etc
The diffusion component of the ion obtained from the oblique slope in the plot can be quantified using the equation

D = 1/2.(RT/n²F²Acσ)²

where R is gas constant, T is absolute temperature, n is number of electrons in the oxidation/reduction process, F is Faraday constant, A is the area, C is the molar density of ions at the electrode, σ is conductivity obtained from the oblique slope (plot Z' vs ω^-1/2)