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Conductivity Relaxation

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Conductivity Relaxation Powered By Docstoc
					Electrochemical
   Impedance
      Spectroscopy
                              University of Twente,
                              Dept. of Science &
                              Technology, Enschede,
                              The Netherlands


Bernard A. Boukamp

Nano-Electrocatalysis,
                              Research Institute
U. Leiden, 24-28 Nov. 2008.   for Nanotechnology
  EIS &
Nano-El-Cat          My „where abouts‟
 Nov. „08.



              E-mail: b.a.boukamp@utwente.nl

          Address:
          University of Twente
          Dept. of Science and Technology
          P.O.Box 217
          7500 AE Enschede
          The Netherlands
          www.ims.tnw.utwente.nl
  EIS &
Nano-El-Cat   Electrochemical techniques
 Nov. „08.


Time domain (incomplete!):
• Polarisation,      (V – I )                             steady state
• Potential Step, (V – I (t) )              Next slide     relaxation
• Cyclic Voltammetry, (V   f(t)-   I(V ) )                    dynamic
• Coulometric Titration, (V - I dt )                      relaxation
• Galvanostatic Intermittent Titration (Q – V (t) ) transient


Frequency domain:
• Electrochemical Impedance Spectroscopy perturbation of
     (EIS)                              equilibrium state
               EIS &
Nano-El-Cat                 Time or frequency domain?
 Nov. „08.

               1.E-04




               1.E-05
Current, [A]




               1.E-06




               1.E-07
                        0        1000                 2000   3000
                                        Time, [sec]
  EIS &
Nano-El-Cat      Advantages of EIS:
 Nov. „08.


System in thermodynamic equilibrium
Measurement is small perturbation (approximately linear)
Different processes have different time constants
Large frequency range, Hz to GHz (and up)
   • Generally analytical models available
   • Evaluation of model with „Complex Nonlinear Least
     Squares‟ (CNLS) analysis procedures (later).
   • Pre-analysis (subtraction procedure) leads to plausible
     model and starting values (also later)
Disadvantage:      rather expensive equipment,
                   low frequencies difficult to measure
  EIS &
Nano-El-Cat              So, what is EIS?
 Nov. „08.


  Probing an electrochemical system with a small
  ac-perturbation, V0ejt, over a range of frequencies.
  The impedance (resistance) is given by:                           = 2f
                      V () V0 e jt           V0                   j = -1
              Z ()             j ( t  )
                                               cos   j sin 
                      I () I 0 e              I0
  The magnitude and phase shift depend on frequency.

  Also: admittance (conductance), inverse of
  impedance:              j ( t  )
                        1    I0 e      I0
              Y ()             jt
                                       cos   j sin 
                      Z () V0 e       V0
                                          “real +j. imaginary”
  EIS &
Nano-El-Cat             Complex plane
 Nov. „08.



                                      Impedance  „resistance‟
                                      Admittance  „conductance‟:
                                                  1   Z re  jZim
                                        Y ()        2
                                                Z () Z re  Zim
                                                               2


                                      hence:

                                                   1   Yre  jYim
                                         Z ()        2 2
                                                 Y () Yre  Yim

 Representation of impedance value,
 Z = a +jb, in the complex plane
  EIS &
Nano-El-Cat
                   Adding impedances and
 Nov. „08.              admittances
 A linear arrangement of
 impedances can be added in
 the impedance representation:
      Ztotal  Z1  Z 2  Z3  ...   Z n
                                     n




                      A „ladder‟ arrangement of admittances
                      (inverse impedances) can be added in
                      the admittance representation :

                          Ytotal  Y1  Y2  Y3  ...   Yn
                                                       n
  EIS &
Nano-El-Cat         Simple elements
 Nov. „08.


  The most simple element
  is the resistance:
                                              1
                               Z R  R ; YR 
                                              R
  (e.g.: electronic- /ionic conductivity,
         charge transfer resistance)

  Other simple elements:
  • Capacitance: dielectric capacitance, double layer C,
                 adsorption C, „chemical C‟ (redox)
                                             See next page
  • Inductance: instrument problems, leads,
                „negative differential
                 capacitance‟ !
  EIS &
Nano-El-Cat               Capacitance?
 Nov. „08.


Take a look at the properties of a capacitor: C  A0
                                                             d
Charge stored (Coulombs):          Q  C V
Change of voltage results            dQ    dV
in current, I:                    I    C
                                     dt    dt

                                      dV0  e jt
Alternating voltage (ac): I (t )  C              j C V0  e jt
                                         dt
                                      V ()       1
Impedance:                 Z C           
                                      I () jC
Admittance:                    YC    Z ()  jC
                                              1
  EIS &
Nano-El-Cat        Combination of elements
 Nov. „08.


What is the impedance of an -R-C-
circuit?
                           1
              Z ()  R       R  j / C
                          jC

                                      1
Admittance?               Y ()             
                                  R  j / C
                            2C 2 R        C
                                    j
                          1  C R
                              2 2 2
                                       1  2C 2 R 2

                                  Semi-                „time constant‟:
                                  circle                     = RC
  EIS &
Nano-El-Cat   A parallel R-C combination
 Nov. „08.


   The parallel combination of a resistance and a
   capacitance, start in the admittance representation:

                        1
                 Y ()   jC                           R
                        R
   Transform to impedance representation:                    C

                        1        1      1/ R  jC
              Z ()                             
                      Y () 1/ R  jC 1/ R  jC
                     R  jR 2C    1  j
                                R
                     1  R C
                          2 2 2
                                   1  2 2
                                                 Plot next slide
   A semicircle in the impedance plane!
                EIS &
Nano-El-Cat                              Impedance plot (RC)
 Nov. „08.

                8.0E+04

                          fmax = 1/(6.3x310-9x105)=530 Hz

                6.0E+04                                                       R = 100 k
                                                 518 Hz
                                                                              C = 3 nF
-Zimag, [ohm]




                4.0E+04




                2.0E+04




                0.0E+00
                          1 MHz                                                       1 Hz
                     0.0E+00   2.0E+04     4.0E+04     6.0E+04      8.0E+04      1.0E+05     1.2E+05
                                                     Zreal, [ohm]
  EIS &
Nano-El-Cat               Limiting cases
 Nov. „08.


  What happens for  <<  and for  >>  ?


   <<  :
                        1  j
              Z ()  R            R  jR  R  jR 2C
                        1  2 2



   >>  : Z ()  R 1  j  R  j R            1
                                                       j
                                                           1
                       1  
                          2 2
                                 
                               2 2
                                               RC
                                                 2   2
                                                          C

  This is best observed in a so-called Bode plot
  log(Zre), log(Zim) vs. log(f ), or
  log|Z| and phase vs. log(f )                          Next slides
                       EIS &
Nano-El-Cat                                   Bode plot (Zre, Zim)
 Nov. „08.

                       1.E+05

                                                                                         Zreal
                       1.E+04                                                            Zimag

                       1.E+03
Zreal, -Zimag, [ohm]




                       1.E+02


                       1.E+01
                                                                                           -1

                       1.E+00                                                      -2
                       1.E-01


                       1.E-02
                            1.E+00   1.E+01    1.E+02       1.E+03        1.E+04     1.E+05      1.E+06
                                                        frequency, [Hz]
                EIS &
Nano-El-Cat                            Bode, abs(Z), phase
 Nov. „08.

                1.E+05                                                                       90



                                                                             abs(Z)          75
                                                                             Phase (°)

                1.E+04                                                                       60
abs(Z), [ohm]




                                                                                                  Phase (degr)
                                                                                             45



                1.E+03                                                                       30



                                                                                             15



                1.E+02                                                                        0
                     1.E+00   1.E+01    1.E+02       1.E+03        1.E+04   1.E+05       1.E+06
                                                 Frequency, [Hz]
  EIS &
Nano-El-Cat     Other representations
 Nov. „08.



 Capacitance: C(ω) = Y(ω) /jω                 for an (RC) circuit:
                                    1                     1
               C ()  Y () / j    jC  / j  C  j
                                    R                    R
 Dielectric:   ε(ω) = Y(ω) /jωC0              C0 = Aε0/d
                               d           ion      d: thickness
               ()  Y ()         j
                              A0          0       A: surf. area
 Modulus:      M(ω) = Z(ω) jω

                                    2CR 2  jR
               M ()  Z ()  j 
                                     1  2C 2 R 2
  EIS &
Nano-El-Cat                   Simple model
 Nov. „08.

Example: an ionically conducting solid,
   e.g. yttrium stabilized zirconia,

        Zr1-xYxO2-½x .
Apply two ionically blocking electrodes,       Schematic
                                               arrangement of sample
  in this case thick gold.                     and electrodes.
Measure the „resistance‟ (impedance)
as function of frequency:
                          1
     Z () 
                               1
               jCg 
                                  1
                        Rion  1
                               2 jCint   Equivalent circuit: (C[RC])
  EIS &
Nano-El-Cat      Low & high f - response
 Nov. „08.



  Low frequency regime,
  series combination Rion-Cint: Z ()  Rion  j / 1 Cint
                                                   2


              Straight vertical line in impedance plane.

  High frequency regime,
  parallel combination of Rion//Cgeom:
                            Rion           RionCgeom
                                              2

              Z ()                  j
                      1   RionCgeom
                           2 2   2
                                         1   RionCgeom
                                              2 2   2



              Semicircle through the origin.
  EIS &
Nano-El-Cat         „Debije‟ model:
 Nov. „08.

  An ionic conductor between two blocking electrodes:




                                                 1
                                       Y () 
                                               Z ()




 Impedance representation       Admittance representation
  EIS &
Nano-El-Cat      Other representations
 Nov. „08.




                                          Different Bode
                                  Zimag   representation



                                  Zreal




          „Bode‟ representation
   EIS &
 Nano-El-Cat   Diffusion, Warburg element
  Nov. „08.

Semi-infinite diffusion,
  Flux (current) : J   D C
  (Fick-1)                 x      x 0

                               RT
   Potential      : EE          ln C
                               nF
   ac-perturbation: C (t )  C  c(t )


   Fick-2          : C    2C
                        D 2
                     t   x
   Boundary                                Li-battery cathode
                                          Redox on inert
   condition       : C ( x, t )      C   electrode.
                                x 

    Solution through Laplace transform: next page
  EIS &
Nano-El-Cat      Warburg element, cont.
 Nov. „08.


 Laplace transform: c( x, t )  C ( x, p)

                                         2 C ( x, p )
 Transform of Fick-2: p  C ( x, p)  D
                                             x 2

 General solution:      C ( x, p)  A  cosh x p / D  B  sinh x p / D

                               RT
 Transform of V (t): E ( p)      C ( x, p )
                              nFC                        Boundary
                                   C ( x, p)            condition:
 Transform of I (t): I ( p)  nFD                       C ( x, p ) x  0
    (Fick-1)                          x x 0
  EIS &
Nano-El-Cat           Warburg impedance
 Nov. „08.

Define impedance in Laplace space!
           E ( p)        RT
  Z ( p)         
           I ( p) (nF )2 C D  p
Take the Laplace variable, p, complex:
 p = s + j . Steady state: s  0,
which yields the impedance:

               RT
Z ()               Z0 (1/ 2  j1/ 2 )
        (nF ) C jD
             2


with:                    In solution:
             RT                         RT     1      1    
  Z0                    Z 0  ( ) 2 2     *     *      
       (nF )2 C 2 D                 n F A 2  CO DO CR DR
                                            
                                                            
                                                            
  EIS &
Nano-El-Cat        Transmission line
 Nov. „08.


Real life Warburg,
                                         r
semi-infinite coax cable     ZW () 
with r /m and c F/m:                   jc




    Combination:
      • Electrolyte resistance, Re‟lyte
      • Double layer capacitance, Cdl        Equivalent
      • Charge transfer resistance, Rct       circuit
      • Warburg (diffusion) impedance, Wdiff
  EIS &
Nano-El-Cat   Equivalent Circuit Concept
 Nov. „08.




                                   



                                           45°
  EIS &
Nano-El-Cat           Instruments
 Nov. „08.




      Measurement methods
      Bulk, conductivity:
         • two electrodes
         • pseudo-four electrodes
         • true four electrodes

      Electrode properties:
         • three electrodes
  EIS &
Nano-El-Cat
                    Frequency Response
 Nov. „08.               Analyser
                                    Multiplier:
                                    Vx(t)sin(t) &
                                    Vx(t)cos(t)
                                    Integrator:
                                    integrates
                                    multiplied signals
                                    Display result:
                                    a + jb = Vsign/Vref

But be aware of the input
                                    Impedance:
impedance of the FRA!               Zsample = Rm (a + jb)
  EIS &
Nano-El-Cat   Potentiostat, electrodes
 Nov. „08.


                                   Vpwr.amp = A k Vk
                                   A= amplification
                                   Vwork – Vref =
                                      Vpol. + V3 + V4

                                   Current-voltage
                                   converter
                                   provides virtual
                                   ground for
                                   Work-electrode.

                                   Source of
General schematic                  inductive effects
  EIS &
Nano-El-Cat             Data validation
 Nov. „08.


Kramers-Kronig relations (old!)
    Real and imaginary parts are linked
    through the K-K transforms:

              Kramers-Kronig
              conditions:         Response only
                 • causality         Response
                                   due to input
                                     State of
                                  scales linearly
                 • linearity          signal
                                   system may
                                    with input
                 • stability        not change
                                      signal
                 • (finiteness)       during
                                  measurement
  EIS &
Nano-El-Cat    Putting ‘K-K’ in practice
 Nov. „08.


 Relations,
                                   
                              2 Z re ( x)  Z re ()
 Real  imaginary: Z im () 
                                  x2  2 dx
                                0
                                                not a singularity!
                                       
                                  2 xZim ( x)  Zim ()
 Imaginary  real: Z re ()  R                       dx
                                  0       x 
                                            2    2


          Problem:
             Finite frequency range: extrapolation
             of dispersion  assumption of a model.
                  [1] M. Urquidi-Macdonald, S.Real & D.D. Macdonald,
                      Electrochim.Acta, 35 (1990) 1559.
                  [2] B.A. Boukamp, Solid State Ionics, 62 (1993) 131.
  EIS &
Nano-El-Cat      Linear KK transform
 Nov. „08.


Linear set of parallel RC circuits:



                                                k = RkCk



   Create a set of  values: 1 = max-1 ; M = min-1
   with ~7 -values per decade (logarithmically spaced).

    If this circuit fits the data, the data must
    be K-K transformable!

                [3] B.A.Boukamp, J.Electrochem.Soc, 142 (1995) 1885
  EIS &
Nano-El-Cat                          Actual test
 Nov. „08.

 Fit function simultaneously to
 real and imaginary part:
                                          1  ji  k
                                              M
                   Z KK (i )  R   Rk
                                     k 1 1  i2  2
                                                     k

 Set of linear equations in Rk,
 only one matrix inversion!                                     It works like a
                                                                   „K-K compliant‟
                                                                       flexible curve
 Display relative residuals:

              Z re, i  Z KK , re (i )                Z im, i  Z KK , im (i )
    real                                ,  imag 
                        Zi                                       Zi
  EIS &
Nano-El-Cat      Example „K-K check‟
 Nov. „08.

Impedance of a sample, not in
equilibrium with the ambient.




                                   2KK = 0.9·10-4
                                   2CNLS = 1.4 ·10-4
  EIS &
Nano-El-Cat            Finite length diffusion
 Nov. „08.



 Particle flux at x=0:

                     ~ dC ( x, t )
          J (t )   D
                         dt x 0

 Fick‟s 2nd law:

          dC ( x, t ) ~ d 2C ( x, t )
                     D
            dt             dx 2
 But now a boundary
 condition at x = L.
 Activity of A is measured at the interface at x=0. with
 respect to a reference, e.g. Amet
  EIS &
Nano-El-Cat      Finite length diffusion
 Nov. „08.


                                                         Replace concentration
                                                         by its perturbation:
                                                         c( x, t )  C ( x, t )  C   0
Impermeable
                  dC ( x, t )
boundary at x =L:                         0 FSW
                        dx       x l


Ideal source/sink
with C = CL (=C0): C ( x, t )  Cl  C 0
                             x l
                                                       FLW


General expression
for permeable
boundary:
                   dC ( x, t )
                        dx
                                                 
                                          k C ( x, t ) xl  Cl       General!
                                  x l
  EIS &
Nano-El-Cat                FLD, continued
 Nov. „08.


 Voltage with respect to reference C0 (a0):
                       RT a x 0  RT  d ln a 
              E (t )     ln 0     0         c ( x, t ) x  0
                       nF    a   nFC  d ln C 

 Current through interface at x = 0:
                                                   ~ dc( x, t )
              I (t )  nF  S  J (t )  nF  S  D
                                                       dx x 0

                                       a      a 0  a       a  a
 Assumption: a << a0:              ln 0  ln           ln 1  0   0
                                                             a  a
                                                    0
                                      a           a
 Relation a  C from                 da a 0 d ln a a   a
 „titration curve‟:                     0           
                                     dC C d ln C C c( x, t ) x 0
  EIS &
Nano-El-Cat   Up to the Frequency Domain!
 Nov. „08.


 Laplace transformation of E(t) and I(t)
 gives the complex impedance (with p=j):              with:
                                                      R T Vm  d ln a 
             E (ω)           Z0         jω        Z0  2 2           
 FSW Z (ω)                   ~ coth l D~            n F S  d ln c 
             I (ω)           jωD                       Vm  d E 
                                                     
                                                       nFS  d δ 
                                                            

 FLW              E (ω)      Z0         jω
          Z (ω)              ~ tanh l D~
                  I (ω)      jωD


Laplace space                                 p            p
solution of Fick-2:         C ( p)  A cosh x ~  B sinh x ~
                                              D            D
  EIS &
Nano-El-Cat                            Dispersions
 Nov. „08.



                                         High frequencies:
                                               Z() = Z0 (j )-1/2
                                  1
                         d E 
          Cint 
                 Vs nF
                                       = Warburg diffusion
                  VM      dδ 

                                         Low frequency limit:
                                         FSW = capacitive
                                         FLW = dc-resistance


                                                Impedance representation of
                                                FSW and FLW.
  EIS &
Nano-El-Cat
                 General finite length
 Nov. „08.            diffusion

  Generic finite                Z0    jωD  coth l    jω
                                                           k
  length diffusion:   Z (ω)                          D

                                jωD k coth l   jω
                                               D
                                                     jωD

                                         If k =0 then
                                         blocking interface
                                          FSW
                                         If k =  then ideal
                                         passing interface
                                          FLW

                                         Plot for different
                                        values of k.
  EIS &
Nano-El-Cat       A Simple Example
 Nov. „08.


                       AgxNbS2 is a layered structure,
                       consisting of two-dimensional NbS2
                       layers. Insertion and extraction of Ag+
                       ions goes in an ideal manner (see graph).
  H.J.M. Bouwmeester
                       Isostatically pressed and sintered
                       sample. Some preferential orientation
                       (in the proper direction!) will occur.

                                                  Simple cell
                                                  design for EIS
                                                  measurements.
  EIS &
Nano-El-Cat     Circuit Description Code
 Nov. „08.


 The Circuit Description Code presents an
 unique way to define an equivalent circuit
 in terms suitable for computer processing.
     Elements: R, C, L, W
     Finite length diffusion:
          T = FSW = Tanhyp (Adm.)     Cothyp (Imp.)
          O = FLW = Cothyp (Adm.)     Tanhyp (Imp.)


     Constant Phase Element:
          Q = CPE = Y0(j )n (Adm.)   Z0(j )-n (Imp.)
  EIS &
Nano-El-Cat                  The CPE
 Nov. „08.


    Constant Phase Element:
          YCPE = Y0 n {cos(n /2) + j sin(n /2)}
           n=1            Capacitance: C = Y0
          • n=½           Warburg:      = Y0
          • n=0           Resistance: R = 1/Y0
          • n = -1        Inductance: L = 1/Y0
    All other values, „fractal?‟


    „Non-ideal capacitance‟, n < 1 (between 0.8 and 1?)
  EIS &
Nano-El-Cat         Non-ideal behaviour
 Nov. „08.


General observations:
• Semicircle (RC )  depressed
• vertical spur (C )  inclined
• Warburg             less than 45°


Deviation from „ideal‟ dispersion:
Constant Phase Element (CPE),
(symbol: Q )

                                n        n    n = 1, ½,
      YCPE     Y0 ( j)  Y0 cos  j sin 
                     n      n

                                  2        2    0, -1, ?
  EIS &
Nano-El-Cat       The Fractal Concept
 Nov. „08.




  How to explain this non-ideal behaviour?
  1980‟s: „Fractal behaviour‟ (Le Mehaut)
      = fractal dimensionality
  i.e.: „What is the length of the coast line of England?‟
   Depends on the size of the measuring stick!



                Self similarity 
  EIS &
Nano-El-Cat              Fractals
 Nov. „08.


          Fractal line




              Self similarity!


                                    „Sierpinski carpet‟
  EIS &
Nano-El-Cat          „Fractal electrode‟
 Nov. „08.


                               „Cantor bar‟
                               arrangement




Impedance of the network:
                 a       a
                         1
 (
Z ) R     a Z ()
Z  R              2     22             R   aR   a2R   a3R
 a         jC
            jaC Z ()  2
              C
                    Z ()
                  aR 
                    R
                               1
                                 22
                        jC  2
                           aC
                                 R
                               aaR  ...
  EIS &
Nano-El-Cat           Arriving at the „CPE‟
 Nov. „08.


                                
 Frequency scaling relation: Z    R 
                                           a Z ()
                               a       jC Z ()  2
 In the low frequency
 limit this reduces to:
                                           a
                                       Z    Z ()
                                         a 2
 Which is satisfied by
                                                            n
 the formula:                          Z ()  A( j)
 with n = 1 – ln2/lna
 Fractal dimension of Cantor bar, d = ln2/lna
 Hence: n = 1 –d
S.H. Liu and T. Kaplan, Solid State Ionics 18 & 19 (1986) 65-71.
  EIS &
Nano-El-Cat                   CDC
 Nov. „08.



 CDC = „instruction string‟ for response calculation
 Uses brackets:
     • [ … ] series combination, e.g.: [RC]


     • ( … ) parallel combination, e.g. (RC)




 Randles circuit: R(C[RW])
                                               W
  EIS &
Nano-El-Cat   Determining the CDC
 Nov. „08.




          (C[(Q[R(RQ)])(C[RQ])])
  EIS &
Nano-El-Cat               CNLS data analysis
 Nov. „08.


Model function, Z(,ak), or equivalent circuit.
Adjust circuit parameters, ak, to match data,
Minimise error function:
      n
S   wi Zre,i  Zre ( i )  Zim,i  Zim ( i )
            {                                                }
                                 2                       2

     i 1

with: wi  Zi
                  2                      2
                       Z ( i , a k )       (weight factor)

                       d
  for k = 1 ..M           S 0
                      dak
Non-linear, complex model function!
                                                                 Effect of minimisation
  EIS &
Nano-El-Cat             Non-linear systems
 Nov. „08.


Function Y (a1..aM) is not linear in its parameters, e.g.:

                           
              Z ()  Z 0  k   j   
                                       
                                              Z (, Z 0 , k , , )         („Gerischer‟)


Linearisation: Taylor development around „guess values‟, ajo:
                                      Y ( x, a1..aM )
 Y ( x, a1..aM )  Y ( x, a ..a )  
                              
                                                                         a j  ....
                                            a j
                           1   M
                                    j                              
                                                             a1 .. aM

Derivative of error sum with respect to aj :
S                                           Y ( xi , a1.. M )  Y ( xi , a1.. M )
      0  2 wi  yi  Y ( xi , a1.. M )  
                                  
                                                                ak 
a j        i                              k      ak                 a j
  EIS &
Nano-El-Cat                        NLLS-fit
 Nov. „08.


A set of M simultaneous equations, in matrix form:

          a =             ,        solution: a         = -1  =   

                         Y ( xi , a1.. M ) Y ( xi , a1.. M )
With:      j , k   wi                   
                      i       a j               ak
                                               Y ( xi , a1.. M )
and:       k   wi  yi  Y ( xi , a1.. M )
                    i                                ak
Derivatives are taken in point ao1..M.

Iteration process yields new, improved values: a‟j = aoj + aj.
  EIS &
Nano-El-Cat      Marquardt-Levenberg
 Nov. „08.


Analytical search: fast and accurate near true minimum
                   slow far from minimum
                   (and often erroneous)
Gradient search or steepest descent (diagonal terms only):
                   fast far from minimum
                   slow near minimum
Hence, combination!                   Successful iteration:
                                           Snew < Sold
Multiply diagonal terms with (1+).    decrease  (= /10).
                                       Otherwise increase
   •    << 1, analytical search            (= 10)
   •    >> 1, gradient search
Bottom line: good starting parameter estimates are essential!
  EIS &
Nano-El-Cat              Error estimates
 Nov. „08.


 For proper statistical analysis the weight factors, wi,
 should be established from experiment.
 Other (dangerous) method:
 Step 1: set weight factors, wi = g  i-2
 Step 2: assume variances can be replaced by parent
              distribution, hence 2  1   (with  = N –M – 1)
 Step 3:
           1  yi  Y ( xi , a1..M ) 1
                2                     2
                                              1      S
          
          2
                                      S              1
            i         i 2
                                        i wi  i g  
                                                 2



 Hence proportionality factor, g = S/.
  EIS &
Nano-El-Cat       Error analysis NLLS-fit
 Nov. „08.


 Based on this assumption we can derive
                                    2
                                                  
 the variances of the parameters:  a  g   k , k
                                             k
                                                        1
                                                              g   k ,k
 Error matrix, , also contains the covariance of the
 parameters:            g  
                         aj   ak      j ,k



 g   j,k  0, no correlation between aj and ak.
 g   j,k  1, strong correlation between aj and ak.

 Only acceptable for many data points AND
 random distribution of the „residuals‟
  EIS &
Nano-El-Cat
              Weight factors and error
 Nov. „08.           estimates
Errors in parameters:
• estimates from CNLS-fit procedure
• assumption: error distribution equal to „parent distribution‟
• only valid for random errors,
• no systematic errors allowed!
                                   Zre,i  Zre ( i )              Zim,i  Zim ( i )
      Residuals graph:     re                         ,  im 
                                        Z ( i )                        Z ( i )
Large error estimates:
strongly correlated
parameters (+ noise).
Option: modification of
weight factors.
  EIS &
Nano-El-Cat     Two different CNLS-fits
 Nov. „08.


Example of correct
error estimates:                                R(RC)(RC)



 CDC: R(RQ)(RQ)
 2 2.410-5
 R1 999         0.8%
 R2 4000        1.7%                      2 3.810-3
                       And of incorrect
 Q3 1.0310-9   7%     error estimates:   R1 1290      4%
 -n3 0.898      0.6%                      R2 4650      2.7%
 R4 8020        0.9%   CDC: R(RC)(RC)
                       Values seem O.K.   C3 2.3810-103.8%
 Q5 1.0310-7   3.6%   but look at the    R4 6580      2.6%
 -n5 0.697      0.7%   residuals!
                                          C5 6.0710-9 7.3%
  EIS &
Nano-El-Cat         Residuals plot!
 Nov. „08.


Systematic deviation,
„Trace‟, bad fit




                                  Good fit (not bad for
                                  a straight simulation!)
  EIS &
Nano-El-Cat                   „Fingerprinting‟
 Nov. „08.


Classification of capacitance
source                                                   approximate value

geometric                                                2-20 pF     (cm-1)
grain boundaries                                         1-10 nF     (cm-1)
double layer / space charge                              0.1-10 F/cm2
surface charge /”adsorbed species”                       0.2 mF/cm2
(closed) pores                                           1-100 F/cm3
“pseudo capacitances”
“stoichiometry” changes                                  large !!!!
Modified after: Peter Holtappels, TMR symposium „Alternative anodes...‟,
Jülich, March 2000.
  EIS &
Nano-El-Cat    Gas phase capacitance
 Nov. „08.



Capacitance of gas volume (e.g. O2):
                                                   PV=nRT
Capacitance: i  C
                  dE     or: C 
                                 i dt
                  dt             dE
O2 produced: i dt  4F dn

Nernst: dE 
              RT P  dP RT dP (RT )2 dn
                  ln                 
              4F       P      4F P      4F V

                              2        Example:
Combination: C   4F  V  P          air, 700°C, Vol. = 10 mm3
              ox     
                      RT             Cox = 0.456 F !
  EIS &
Nano-El-Cat      Conclusions on „fitting‟
 Nov. „08.


Many parameter, complex systems modelling:
    •   Use Marquardt-Levenberg when quality starting
        values are available
    •   Simplex (or Genetic Algorithm) for optimisation of
        „rough guess‟ starting values, as input for M-L NLSF
    •   Check residuals when calculating Error Estimates
    •   Look for systematic error contributions, remove if
        feasible.
    •   Provide error estimates in publications!

                        It‟s human to err, its dumb not to include
                        an error estimate with a number result
   EIS &
Nano-El-Cat           „Molecular Printboard‟
 Nov. „08.

ferrocenyl (Fc) decorated poly(propylene imine) dendrimer


                                                              β-cyclodextrin




Christian A. Nijhuis, B.A. Boukamp, B-J. Ravoo,
J. Huskens and D.N. Reinhoudt                J. Phys. Chem. C 111 (2007) 9799
  EIS &
Nano-El-Cat   Electrochemical response
 Nov. „08.

                              Impedance graphs of
                              an aqueous solution of 1
                              mM (in Fc
                              functionality) of G4-
                              PPI-(Fc)32-(β-CD)32
                              at a β-CD SAM.
                              (10 mM β-CD at pH = 2)




                              Potential: -0.15 V to 0.15 V
                              Frequency: 10 kHz to 10 mHz
                              KK-test:          < 10  10-6
  EIS &
Nano-El-Cat      Subtraction procedure
 Nov. „08.



    • Partial CNLS-fit of recognizable structure
         Semicircle
         Straight line (CPE, Cap., Ind.)
    • Subtract dispersion as series- or parallel component
    • Repeat steps until „garbage‟ is left
    • Be aware of „errors‟ due to consecutive subtractions
    • Sometimes restart and do a partial fit of a larger
      group of parameters
  EIS &
Nano-El-Cat                  Impedance G-4 at 0.105 V
 Nov. „08.


                        3.E+04



                                 SAM cap.//resist.
                                                              Randles response
       - Zimag, [ohm]




                        2.E+04




                        1.E+04




                        0.E+00
                             0.E+00                  1.E+04    2.E+04   3.E+04   4.E+04
                                                              Zreal, [ohm]
  EIS &
Nano-El-Cat                    Subtract Rel’lyte, CSAM
 Nov. „08.


                     3.E+04

                                   Fit of Randles circuit: R(C[RW])
    - Zimag, [ohm]




                     2.E+04




                     1.E+04




                     0.E+00
                          0.E+00   1.E+04   2.E+04    3.E+04   4.E+04
                                            Zreal, [ohm]
  EIS &
Nano-El-Cat                                                          First CNLS-result
 Nov. „08.

          -Z -im a g x 1 e 5


                   4
                        R e a l,I m a g -e r r o r x 1 e -2                                                                                       EqCWIN analysis
                   3

                   2
                                                                   Full circuit
                   1

                   0

                   -1

   0 .2            -2

                   -3

                   -4


                                                   1 e -1               1                 1e 1                            1e 2          1e 3
                                                                            Fre que ncy




                                                                                               R e a l,I m a g -e r r o r x 1 e -2



                                                                                                                                     Residuals plot(R[RC])
                                                                                          4
   0 .1                                                                                   3

                                                                                          2
                                                                                                                                      Circuit without
                                                                                          1

                                                                                          0

                                                                                          -1

                                                                                          -2

                                                                                          -3

                                                                                          -4


                                                                                                                          1 e -1         1                    1e 1   1e 2   1e 3
                                                                                                                                                Fre que ncy

     0
      0                                                     0 .1                                0 .2                                     0 .3                        0 .4
                                                                                                          Z -r e a l x 1 e 5
              G4-ferrocene, 0.105 V
  EIS &
Nano-El-Cat                     Subtract Rel’lyte, CSAM
 Nov. „08.


                      3.E+04

                                    Fit of Randles circuit: R(C[RW])
     - Zimag, [ohm]




                      2.E+04




                      1.E+04




                      0.E+00
                           0.E+00    1.E+04   2.E+04    3.E+04   4.E+04
                                              Zreal, [ohm]
  EIS &
Nano-El-Cat                    Subtract Randles
 Nov. „08.
     - Zimag, [ohm]




                      100




                        0
                        3800        3900            4000
                                     Zreal, [ohm]
  EIS &
Nano-El-Cat                                            Small difference, but …
 Nov. „08.

          -Z -im a g x 1 e 5


                   4
                        R e a l,I m a g -e r r o r x 1 e -2                                                                                     EqCWIN analysis
                   3

                   2
                                                                   Full circuit
                   1

                   0

                   -1

   0 .2            -2

                   -3

                   -4


                                                   1 e -1               1                 1e 1                            1e 2        1e 3
                                                                            Fre que ncy




                                                                                               R e a l,I m a g -e r r o r x 1 e -2
                                                                                          4
   0 .1                                                                                   3

                                                                                          2
                                                                                                                                     Circuit without (R[RC])
                                                                                          1

                                                                                          0

                                                                                          -1

                                                                                          -2

                                                                                          -3

                                                                                          -4


                                                                                                                          1 e -1       1                    1e 1   1e 2   1e 3
                                                                                                                                              Fre que ncy

     0
      0                                                     0 .1                                0 .2                                   0 .3                        0 .4
                                                                                                          Z -r e a l x 1 e 5
              G4-ferrocene, 0.105 V
  EIS &
Nano-El-Cat    Equivalent Circuit
 Nov. „08.




                              +
              Au
  EIS &
Nano-El-Cat   Consistency of Circuit!
 Nov. „08.




                     R2          CSAM


                R1
                                        C2




                             Tentative model
  EIS &
Nano-El-Cat          Modelling of diffusion
 Nov. „08.

                                           Modelling diffusion:
                                                                    1
                                           Qdiff  Q0
                                                           1           DO 
                                                        1      1 K    
                                                           K         DR 
                                                                             


                                           with: Q0  n 2F 2 A 2 0
                                                                CFc ,tot DO
                                                          RT
                                            for right hand side only.

Actually: Qdiff= Qconst + Qdiff(V)

Modelling of double-                                              nF
                                                                  RT
                                                                     (  )     0
                                                                e
layer capacitance:   Cdl  Cdl ,1  Cdl ,2 (1  ) , with:       nF
                                                                               ( 0 )
                                                                   1 e   RT
  EIS &
Nano-El-Cat                Diffusion & generation
 Nov. „08.

      Generation 1 to 4 measured at the same βCD SAM:
           ■ : G1
           ▼: G2
       100 ▲: G3

           ●: G4
   (F)




                                              RT       1     1    
                                          2 2           *      
                                           n F A 2  CO DO CR DR   
           10
                                                      *
                                                                  
                -0.02   0.00    0.02   0.04   0.06   0.08   0.10
                               Polarization (V)
  EIS &
Nano-El-Cat         Stokes-Einstein
 Nov. „08.



                                    Stokes-Einstein relation:
                                              kT
                                        D
                                             6r

                                    Qdiff  D
                                    and r  Generation nr.
                                    Hence: Qdiff  (Gx)-1/2


The Warburg admittance corrected for the
concentration of dendrimers and the number
of electrons involved per molecules plotted vs
the (square root of generation)-1.
  EIS &
Nano-El-Cat           Reaction Pathways
 Nov. „08.




Schematic of the potential-dependent surface coverage of the
dendrimers (left), and a scheme of adsorption and desorption kinetics
(right), Redsol = reduced dendrimers in solution, Redads = dendrimers
adsorbed at the βCD SAM, Oxsol = oxidized dendrimers in solution,
Oxads = oxidized dendrimers at the surface; kdr, kar, kdo and kao are
adsorption (a) and desorption (d) rates of oxidized (o) and reduced (r)
dendrimers; kb and kf are electrochemical rate constants.
  EIS &
Nano-El-Cat    Praise of the time domain …
 Nov. „08.

Intercalation cathode.
Change of potential = change
of aA at the interface, hence
A-diffusion:
                  dCA ( x, t )
    J (t )   DA
                    dx x 0
Voltage-activity relation:

              RT a A, x 0
     E (t )     ln 0
              nF    aA
Fick 1 & 2, boundary conditions        V ()     Z0         j
+ Laplace transform:           Z (ω)               coth l
                                       I ()     jD        D
   EIS &
Nano-El-Cat          Real cathode: LixCoO2
 Nov. „08.

                                        10.0
                                                     Measurement                           Rsl    Rct W
                                                     Simulation                     Re
                                                                                           Qsl     C dl
                                         7.5

                                                                       20Hz
                                                                                    3.78V




                              Z" [k]
                                         5.0
                                                                27Hz

                                                             36Hz
                                                                          3.69V
                                                                      3.84V
                                         2.5          63Hz
                                                             3.88V
                                                  110Hz
                                                      4.06V

                                         0.0
                                            0.0      2.5        5.0       7.5       10.0         12.5     15.0
                                                                          Z' [k

LiCoO2, RF film on silicon.             IS of a RF-film electrode: (○) „fresh‟;
                                        (□) charged; () intermediate SoC‟s.
Peter J. Bouwman, Thesis,               (+) CNLS-fit. Range: 0.01 Hz – 100
U.Twente 2002.                          kHz.
  EIS &
Nano-El-Cat         Diffusive part?
 Nov. „08.
                                                  10

 The lithium diffusion                             9

 process is found at lower                         8
                                                                   4.05V


 frequencies!                                      7
                                                                   4.10V




                               Current [A.cm ]
 Compare the potential-step




                               -2
                                                   6

 response time with lowest
                                                           4.00V   4.15V
                                                   5

 frequency of EIS                                  4               4.20V

 experiment:                                       3       3.95V


 teq. >> 3000 s (~ 0.3 mHz)
                                                   2
                                                           3.90V
                                                   1       3.85V

 fmin ~ 10 mHz                                     0
                                                           3.80V

                                                       0           1000          2000   3000
 MEASURE RESPONSE IN                                                       Time [s]


  THE TIME DOMAIN!             Current response of a 0.75m RF-
                               film to sequential 50mV potential
                               steps from 3.80V to 4.20V.
  EIS &
Nano-El-Cat        Fourier transform
 Nov. „08.

  Fourier transform of a temporal function X (t):
                               
                     X ()   X (t )  e  jt dt
                               0

                             V ()
   Impedance:        Z () 
                             I ()
                                               V0
   E.g. with a voltage step, V0:       V () 
                                               j

  Model function: Laplace transform of transport equations
  and boundary conditions, with p = s +j . Set s = 0: 
  impedance
   EIS &
Nano-El-Cat                    Fourier Transform
 Nov. „08.



Two problems with F-T:                                                     X(t )=at + b
• Data is discrete:
   approximate by summation (X =at + b)                                         Xi -1
• Data set is finite (next slide)                                                       Xi


Very Simple Summation Solution (VS3):                                           ti -1 ti
          N
             L
X ( )   X i sin ti  X i 1 sin ti 1 
             M
                                               a
                                                    b            gO
                                                 cos ti  cos ti 1  1 
                                                                  P
         i 1N                                                   Q
               L
               X
                N
          j  Mcos t  X       i 1 cos ti 1    b t  sin t g
                                                     a
                                                       sin
                                                                   O
                                                                   P      1

               Ni 1
                       i   i
                                                    
                                                           i
                                                                   Qi 1
  EIS &
Nano-El-Cat                   Simple exponential extension
 Nov. „08.


  Assume finite value, Q0, for t  ,
  this value can be subtracted before total FT.

  Fit exponential function to
  selected data set in end range: Q(t )  Q0  Q1 e  t /
  Full Fourier Transform:

                                    z                                          z
                                    tN                                         
                                                         Q0
                                                   jt
                        X ( )  [ X (t )  Q0 ] e dt  j  Q1 e  t /  e  jt dt
                                0
                                                             tN


  Analytical transform of exponential extension:

                                                      R cost   sin t                                       U
  z
                                                       1
                                                                                  cos t N   1 sin t N
Q1 e   t / 
                e    jt
                            dt  Q1  e   t N / 
                                                     S        N           N
                                                                               j                              V
 tN                                                   T    2    2
                                                                                          2   2            W
  EIS &
Nano-El-Cat        Fourier transformed data
 Nov. „08.


Simple discrete Fourier transform:
           tN

 X (ω)   X (t ) e  jωt dt 
           0
  N
      X (tk )  X (tk 1 )
  t t                    cos ωt  j sin ωt 
 k 1       k   k 1

Correction / simulation for t:
 X (t )  X 0  X 1e t / 
X 0 = leakage current.

                   V (t )
Impedance: Z () 
                   I (t )
  EIS &
Nano-El-Cat        V-step experiment
 Nov. „08.




 Sequence of 10 mV step Fourier transformed impedance spectra,
 from 3.65 V to 4.20 V at 50 mV intervals. Fmin = 0.1 mHz
  EIS &
Nano-El-Cat   CNLS-fit of FT-data
 Nov. „08.


              Circuit Description R1         :   550        0.5 %
              Code:               R2         :   49         10 %
                                    Q3, Y0   :   6.810-3   12 %
                     R(RQ)OT   *)
                                    ,, n     :   0.96        8 %
              Fit result:           O4, Y0   :   0.047      1.5 %
                                    ,, B     :   30         2.4 %
                2CNLS = 3.710-5   T5, Y0   :   0.028      2.9 %
                                    ,, B     :   5.9        2.9 %

              *) O   = „FLW‟
                T = „FSW‟
   EIS &
Nano-El-Cat                   Bode Graph
 Nov. „08.


                                                               Double
                                                               logarithmic
                                                               display
                                                               almost
                                                               always
                                                               gives
                                                               excellent
                                                               result !




„Bode plot‟, Zreal and Zimag versus frequency in double log plot
  EIS &
Nano-El-Cat            Conclusions
 Nov. „08.

 Electrochemical Impedance Spectroscopy:
     • Powerful analysis tool
     • Subtraction procedure reveals small contributions
     • Presents more „visual‟ information than time domain
     • Almost always analytical expressions available
     • Equivalent Circuit approach often useful
     • Data validation instrument available (KK transform)
     • Also applicable to time domain data
              (FT: ultra low frequencies possible)
     • Able to analyse complex systems
                Unfortunately, analysis requires experience!
  EIS &
Nano-El-Cat   Not just electrochemistry!
 Nov. „08.


Data analysis strategy is applicable to any system where:
   • a driving force
   • a flux
can be defined/measured.
Examples:
   • mechanical properties, e.g. polymers: G () or J () & 
   • catalysis, pressure & flux, e.g. adsorption
   • rheology
   • heat transfer, etc.
              No need to measure in the frequency domain!
  EIS &
Nano-El-Cat   Last slide
 Nov. „08.
  EIS &
Nano-El-Cat
 Nov. „08.
  EIS &
Nano-El-Cat                              Effect of truncation
 Nov. „08.
                  10
                            Delta-Real
                   8
                            Delta-Imag           tmax = 100   = 30 sec, ( max) ) = 3.6%
                                               tmax = 100 s, s, = 20 sec, Y Yt(tmax= 0.67%
                   6
                                                                                  Delta-Real
                   4                                                              Delta-Imag
  re, im, [%]




                   2
                                                tmax = 100 s,  = 40 sec, Y (tmax) = 8.2%
                   0

                   -2

                   -4

                   -6

                   -8

                  -10
                     0.01                0.1              1               10               100

                        Zimag ()  Zim,tr ()      Frequency, [Hz]            Z real ()  Z re,tr ()
  im                                                                 re 
                                Z ()                                                  Z ()
  EIS &
Nano-El-Cat        More Fourier transform
 Nov. „08.


Method Martijn Lankhorst:                                                                             piece wise
    fit polynomials to small sets                                                                   integration
                                            m
     of data points (sections): Pm (t ) t   Ak t k
                                        t                                  r

                                                                           q
                                                                                    k 0
    analytical transformation to frequency domain:
                                                           i 1 k         jt q          i 1 k         jt r
                    m i 1
                                      (i  1)!         t             e             t               e
       P ( ) t    Ai
              tr
                                                   
                                                           q                               r
                                                                                        k 1
               q
                   i  0 k 1     (i  1  k )!                                ( j )

More general extrapolation function (stretched exponential):
                                 t /  
       Q(t )  Q0  Q1  e                    , 0   1
(Fourier transform complicated, can be done numerically)
   EIS &
Nano-El-Cat               Non linear effects
 Nov. „08.


Electrode response based on                                         0.05


Butler-Vollmer:
                        RTF   (1a ) F  
                           a

               I  I 0 e       e RT                                 0




                                                     Current, [A]
                                             
When the voltage amplitude is                                       -0.05
                                                                                                  I0 = 1 mA
too large, the current response                                                                   αa = 0.4
                                                                                                  T = 23°C
will contain higher harmonics (i.e.                                  -0.1

is not linear with V).                                                   -0.2   -0.1          0
                                                                                       Polarisation, [V]
                                                                                                           0.1   0.2




Substituting
a = αaF/RT,                               a 22 a 33                      b 22 b33       
                      I  I 0 1  a                     ...  1  b              ...
b = (1-αc)F/RT                              2!       3!                      2!     3!      
and a serial
expression for                               (a 2  b 2 )2 (a 3  b3 )3        
exp(), we obtain:        I 0  ( a  b )                                  ...
                                                   2!                3!          
  EIS &
Nano-El-Cat             Higher-order terms
 Nov. „08.

At zero bias, with the perturbation voltage, Δ∙ejωt, this
equation yields:
                                 jt (a 2  b2 ) j 2t (a3  b3 ) j 3t  
        I (t )     I 0 (a  b)e              e              e  ...
                                          2!               3!            
This clearly shows the occurrence of higher-order terms.
When the polarization current is „symmetric‟ the even terms
will drop out as a = b. At a dc-polarization the response is
more complex:
                   
                                           (a3  b3 )2       jt
      I (t )  I 0  (a  b)  (a  b ) 
                                  2   2
                                                          ... e 
                   
                                                2!           
                a 2  b 2 (a3  b3 )  j 2 t  a 3  b3       j 3t
                                    ... e           ... e  ...   
                2!            2!                 3!          
  EIS &
Nano-El-Cat                    The derivatives!
 Nov. „08.




   Having the derivatives is essential!
        •    best method, calculate the derivatives on basis
             of the function: accuracy and speed.
        •    Second best: numerical evaluation* (for proper
             derivatives we have to calculate F(xi,a1..M)
             2M +1 times!!
                        F ( xi , a1.., a j  a j ,..aM )  F ( xi , a1.., a j  a j ,..aM )
     F ( xi , a1.. M ) 
a j                                                    2a j

                                               * This is actually an approximation