1

Introduction to EIS and Conversion of CPE into C

Dr. Zack Qin

The Blind Men and the Elephant

… Is very like a wall!… Is very like a spear!… Is very like a snake!… Is very like a tree!… Is very like a fan!… Is very like a rope!…….Not one of them has seen!

J. G. Saxe (1816-1887)

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Z is a transfer function

ExcitationE(t)

TransferFunction

ResponseI(t)

E(jω) = Z(jω) x I(jω)

Fourier transform

Z is defined in the frequency domain that relates to the time domain through Fourier/Laplace Transform.The equation can be considered as a generalized Ohm’s law.

3

Z is a complex function

'

"

' 2 " 2

"

'

( ) Re[ ( )]( ) Im[ ( )]

( ) ( ) ( )

( )( ) arctan( )

Z Z jZ Z j

Z Z Z

ZZ

ω ω

ω ω

ω ω ω

ωθ ωω

=

=

= +

⎛ ⎞= −⎜ ⎟

⎝ ⎠

' " ( )( ) ( ) ( ) ( ) jZ Z jZ Z e θ ωω ω ω ω −= + =

Z is described by pair-functions, Z’ and Z”, or |Z| and θ.Z’(ω), Z”(ω), |Z(ω)|, and θ(ω) are real functions.Other related immittance functions:

Admittance Modulus Dielectric constant

Y = Z-1 M = jωC0

Z ε

= (jωC0

Z)-1

Z”

Z’

4

Element impedances

/ RR E I Z R= ⇒ =

Storage of charge (Coulomb’s law): I leads E. For a parallel-plate capacitor: C = εε0 A/d

( ) Ld EL Z j LdI dI dt

ωΦ= = ⇒ =

Resistance (Ω)

Dissipation of energy (Ohm’s law): I in phase with EFor a uniform area resistor: R = ρ L/A

Capacitance (F) 1( ) C

dQ IC ZdE dE dt j Cω

= = ⇒ =

Inductance (H)

Storage of magnetic energy (Faraday’s law): I lags E.Inductive behaviours are not normally observed in electrochemical systems, but can be attributed to adsorption phenomenon.

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Kirchhoff’s

rules

1 21 2

E EZ Z ZI+

= = +

1 2

1 2

1 1 1I IZ E Z Z

+= = +

Impedances in series:

Z1 Z2

E1 IE2 I

Z1

Z2

I1

I2

E

Impedances in parallel:

Conservation of energy -- the sum of potential differences across each element around any closed circuit loop must be zero.Conservation of charge -- the sum of the currents entering any junction must equal the sum of the currents leaving that junction.

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Randles

circuit

Rs

Cdl

Rct

2

2

2

2 22

'1 ( )

"1 ( )

' ( ")2 2

cts

dl ct

dl ct

dl ct

ct cts

RZ RC R

C RZC R

R RZ R Z

ω

ωω

= ++

= −+

⎛ ⎞ ⎛ ⎞− − + =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

0

lim

lim lim

log log ( 1)

s

ct

dl

R Z

R Z Z

C Z

ω

ω ω

ω

→∞

→ →∞

=

= −

= − =

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Time constantThe time constant (relaxation time), τ, characterizes the response of a first order linear time-invariant system. It represents the time required to reach (1-1/e) of its asymptotic value in a step response, or 1/e of its initial value in a impulse response.The time constant will remain the same for the same system regardless of the starting conditions.RC circuit: τ = RC, and RL circuit: τ = L/R, regardless of series circuits or parallel circuits.

0( 0) 0, ( 0)0( ) (1 )in in

tV t V t Vin C C RCC

V V dVC V t V eR dt

−= = > =−= ⎯⎯⎯⎯⎯⎯⎯⎯→ = −

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Characteristic frequency

ωc is the physical property of a RC circuit.ωc is defined in the frequency domain, while τ is defined in the time domain.ωc is the resonance frequency at which –Z” is maximum.ωc is the cutoff frequency at which energy entering the system begins to be attenuated or reflected instead of transmitted.

11c R C

ω τ= =

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Other frequencies

The inflection frequency, ωθ, is the frequency at which the absolute value of the phase angle θ reaches maximum.

The breakpoint frequencies, ω1 and ω2, are frequencies at θ=-45°.

While ωc is the property of a RC circuit, ωθ and ω1,2 are the properties of Randles circuit.

2

1,2 2

1

1 1 4 42

ctc

s

c s s

ct ct

RR

R RR R

θω ω

ωω

= +

⎛ ⎞= ± − −⎜ ⎟

⎜ ⎟⎝ ⎠ ωθ

Relationships to ωc

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Constant phase element

( )10 2[ ( ) ]CPEZ Y j α απθω − == ⋅

α 1 0 -1 0.5

Y0 C 1/R 1/L 1/√2σ

CPE is a phenomenological term first used by Brug in 1984. It is an empirical impedance that its phase angle is independent of frequency.CPE is often related to frequency dispersion attributed to surface inhomogeneities and distributed time constants.CPE obeys Kramers-Kronig relations provided that |α| ≤ 1. The physical justification of CPE is not obvious except on the following circumstances:

= p/2 -

θ

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Conversion of CPE into

C

While the use of CPE usually increases the “goodness of the fit”, the physical meaning of the CPE should be discussed.Avoid using CPE. If this is inevitable, the causes of the non-idealism should be identified.CPE is often used to describe non-ideal capacitive behaviour. However, the amplitude Y0 is not a capacitance.The dimension of Y0 is secα/Ω, while that of C is F or sec/Ω.Small deviations of the exponent α from 1 can lead to large computational errors of capacitance.To obtain a trustworthy value of α, the testing frequencies should be at least a decade lower than ωc.For conversion into C, the exponent α must be in the range of 0.8-1. Otherwise, CPE won’t represent a capacitor.The conversion of Y0 into C has been controversial. At least three distinct equations have been proposed.

CPE is not a capacitor, and Y0

is not capacitance

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Brug

conversion

It applies to Randles circuit with no more than one τ.It depends on the solution conductivity. This dispersion is largest when Rs dominates over Rct.It obtained by comparing impedance (admittance) with relaxation analysis.

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0

1 1

s ct

YR R

Cαα −

+⎡ ⎤⎛ ⎞

= ⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

G.J. Brug, et al, J Electroanal. Chem. 176 (1984), 275-295.

0

0

1

1

1 11 ( )

1 1 ( )1 exp( )

s s ct s

s s ct

YR R R R Y j

Y F s dsR R R j s

αω

ωτ

∞

−∞

−

−

⎡ ⎤= ⎢ ⎥+ +⎣ ⎦

⎡ ⎤= ⎢ ⎥+ +⎣ ⎦

∫

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Mansfeld

conversion

The derivation was based on the fact that Z’ is independent of α at ωc, which itself is also independent of α. Therefore,

10 cC Y αω −=

C.H. Hsu and F. Mansfeld, Corrosion, 57 (2001), 747-748.

0

22 2 2 2

0

1 1( )

1 1c c

c c

Y j j C

ZY C

α

α

ω ω

ω ω

=

= =10-3 10-2 10-1 100 101 102 103 1040

2000

4000

6000

8000

10000

12000

Frequency (Hz)

Z'

a=0.8.za=0.9.za=1.z

-6000

-5000

-4000

-3000

-2000

-1000

0

Z''

It applies to an individual R||C, regardless how many branches are nested in the circuit.No dependency on the solution conductivity.

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Extension of Mansfeld

With Warburg

Nested R||C

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Westing/Mertens

conversion

Ilevbare and Scully proposed the use of the equation, but Hsu and Mansfeld claimed the equation is incorrect.

10

sin( 2)YC

αθω

απ

−

=

First described in the PhD thesis of Westing (1992), and later modified by Mertens (1997).The derivation assumed that ZCPE(ωθ) = ZC(ωθ), where ωθ is the inflection frequency.

0

1 1 ( )( )

cos sin2 2

Y j j C

j j

θα

α

ω ωω ω

απ απ−

= =

⎛ ⎞ ⎛ ⎞= − + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

S.F. Mertens, et al, Corrosion, 53 (1997), 381-388.

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CPE2C program

CPE2C is a lab-built code for the conversion of CPE into C, based on the Mansfeld or Brug equations.It is available to members of Shoesmith and Wren groups.Feedback and suggestions are welcome.

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Use of CPE2C

Fit EIS spectrum with an appropriate equivalent circuit according the spectrum and the system under investigation.Identify the origin of the CPE (capacitive, diffusive, porous, acombination of them, or even a wrong circuit). For non-ideal capacitive behaviour, select the conversion method:Mansfeld

•

The ωc

is the characteristic frequency associated with that R||C to be converted. Note the angular frequency ω

= 2πf.•

The calculated C must satisfy the condition when comparing with the measured ωc

, 1/Tolerance < RCωc

< Tolerance•

Independent of solution conductivity at all.Brug

•

Apply to a Randles

circuit, a blocking electrode, and geometry-induced frequency dispersion.

•

Conversion depends on solution conductivity.The exponent α must be in the range of 0.8-1.

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Example

CPE2C

Mansfeld R (Ω) ωc

(rad/s) Y0 α C (F)

CPE-1 100 5384.4 10-5 0.8 1.794e-6

CPE-2 250 4.6572 0.001 0.9 8.574e-4

Brug Rs

(Ω) R1 Y0 α C (F)

A 10 100 10-5 0.8 9.764e-7

B 1 100 10-5 0.8 5.609e-7