Dr S S Bhoga

Complex Complex Impedance Impedance Spectroscopy Spectroscopy

S. S. Bhoga Department of Physics RTM Nagpur University, Nagpur

What is Complex Impedance Spectroscopy ? Involves measurement of Real and imaginary parts of impedance/ admittance Over wide frequency range (10-3 to 1014 Hz)

Convey information Microscopic ion dynamics Ionic bulk conductivity Electrode polarisation Activation enthalpy for ion migration Micro-heterogeneities Dielectric constant Electric polarization Etc.

What is Complex Impedance Spectroscopy ? σ (ω ) is Fourier transform of autocorrelation function of current density, i

V σ (ω ) = 3k β T here

∞ ∫ < i (0).i (t ) > exp(− jω t )dt, 0

1 N i (t ) = ∑ qi vi (t ), V 1

V - volume of the sample qi - Charge on mobile specie vi - Velocities ω - Angular frequency

It resolves elementary hopping processes It is a very powerful microscope in time.

Why Complex Impedance Spectroscopy ? Mass transport in solids is vital for many technological applications Mobility of ions under influences Chemical gradient - Diffusion Electric potential gradient Conductivity

ac Conductivity Avoids polarization

Single frequency First hand information (screening test)

m)

Polarization of ions introduces error

σ ( S/c

Disadvantage of dc conductivity

σ

No instrument is = Bulk capable to 0 determine σ ∞ −electronic Time (Min)

Fig:- Variation of dc conductivity with time

σ is time/frequency dependent

Why Complex Impedance Spectroscopy ? Bulk conductivity – dc Only due to ion migration Excludes polarization

electrode

Dielectric loss Grain-boundary contribution

Why Complex Impedance Spectroscopy ? Provides straight-forward determination of electrolyte resistance irrespective of degree of electrode polarisation. Does not necessarily require use of reversible electrodes or complicated cell geometry Determine dielectric properties of material Ease of studying electrode polarisation

Mathematical Formulation and Electrical Equivalent Models System characterization

Experiment

Theory

Mathematical model and Analyses

Plausible Physical model

Equivalent circuit

CIS – Basics/Electrical Equivalence Sine waves are use Input

∼

I(t) = Im exp(jω t φ )

Input and output waveforms are same Z in time domain obeys Ohms law Z (ω ) = |Z| exp (-jω φ )

Output

V(t) = Vm exp(jω t )

CIS – Basics/Electrical Equivalence Real and Imaginary parts

B ωR 2 C Z = 2 = i G + B 2 1 + (ωRC ) 2

Z”

G R Z = 2 = , r G + B 2 1 + (ω RC ) 2

ω 0 (R/2, R/2)

(0,0) Electrical Equivalent

Eliminate ω and rearrange (Zr -R/2)2 + Zi2 = (R/2)2 .

Eq. of circle with radius R/2

Z’

(R,0)

Complex impedance response

CIS – Basics/Electrical Equivalence Real and Imaginary parts

Zi =

1 + 2(ω τ0 )1−α sin(α π/ 2) + (ω τ0 ) R{(ω τ0τ )

1−α

2 (1 − α

)

Z”

Zr =

R{1 + (ω τ)1−α sin(α π/ 2)} CPE

cos(α π/ 2)}

1 + 2(ω τ0 )1−α sin(α π/ 2) + (ω τ0 )

(0,0) 2 (1 − α

Eliminate ω n and rearrange

)

Z’

Electrical Equivalent Complex impedance response

Depressed by angle α Centre (R/2,[R tan{α π /2}]/2)

(Zr -R/2 ) + ( Zi -{[R tan (α π /2)] 2

ω 0 (R/2, [R cos(ap/2)]/ [2(1+sin(ap/2]

1/2

}) = r 2

Top - (R/2, [R cos(α π /2)]/[2(1+ sin . (α π /2])

2

Radius r

CIS – Basics/Electrical Equivalence Solid ion conductor

Each physical processes give separate semicircle provided their values of τ are widely different

Z”

Metal electrodes

R1

Bulk

Z’

R1 + R2

Grain-boundaries

R1

R2

C1

C2

Metal electrode

R3

Cdl

Zw

As first approximation, each semicircles may be considered as response of R-C combination.

Electrical Equivalence Complications C1 C2 R1

(a’)

Obtaining equivalent circuit to simulate phenomenon of electrode

Z”

Z”

Z”=(jω C2)-1

ω R1C1=1

Z’

(a)

R1

Z’

Z”

Z”

(b)

Z’

(c)

Z’

(d)

Highly combination semicircle

of

convoluted line and

Experimental - Measurement

Impedance data during cooling cycle

Dwell time 30 min Measurement at end of dwell time

T1

te1 ts2 te2

T2

ts → Start time te → End time ts3 te3

T3 Temperature

Cooling 2oC per min

ts1

ts4 te4 Dwell time, 30 min .

T4

ts5 te5

T5 T6 40

100

160 Time

220

280

Experimental – Set up HP4192A Mass flow meters

Display

Key pad

parameters

Low High

cylinder O2 gas CO2/SO cylinder Ar gas 2 gas HP 16048 test leads

G P I B

compatible PC ntium processor

Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6

Furnace

Six samples 5 to 13mm dia

Keithley 7001 R.F. Switch Temperature Programmer

Gas partial pressure 100ppm to 10%

Temperature - R.T. to 700oC

Frequency

5 to 13x106 Hz

Experimental – Set up Frequency 10x10-6 to 1x106 Hz

Temperature

Solartron 1287

G P I B

Mass flow meters

Solartron 1255B FRA

R.T. to 700oC Scribner Z-plot and Zview

cylinder O2 gas CO2/SO cylinder Ar gas 2 gas

IBM compatible PC (Pentium processor

Sample Holder Furnace

Temperatur e programme r

Data Fitting/Circuit Simulation

4

Fitted (CNLSF) data to

2

0 2

4

6

8

10

12

Ζ”(k Ω)

0

Z(ω) = Z(∞) +

20

Z(0) − Z(∞)

α

∗ 1+  jωτ   

Sum of squares is minimized by unity weighting

10

0 0

10

20

30

Z’ (kΩ)

40

50

60

[

S i = ( ∆R i ) + ( ∆I i ) 2

]

2 1/ 2

Data Fitting/Circuit Simulation

4

2

0 2

4

6

8

10

12

10

20

30

40

50

60

Ζ”(k Ω)

0

20

10

0 0

Z’ (kΩ)

Conduction Mechanism

 Ea  f p = fo exp   kT  Intragrain conduction modifies on doping But not grain-boundary