Impedance spectroscopy of composite polymeric

electrolytes - from experiment to computer modeling.

Maciej Siekierski

Warsaw University of Technology, Faculty of Chemistry, ul. Noakowskiego 3, 00-664 Warsaw, POLAND

e-mail: alex@soliton.ch.pw.edu.pl, tel (+) 48 601 26 26 00, fax (+) 48 22 628 27 41

Model of the composite polymeric electrolyte

Sample consists of three different phases:•Original polymeric electrolyte – matrix•Grains•Amorphous grain shells

Rt

Last two form so called composite grain characterized with the t/R ratio

Experimental determination of the material parameters:

• d.c. conductivity value

• diffusion process study

• transport properties of the electrolyte-electrode border area

• determination of a transference number of a charge carriers.

Variable experimental techniques are applied to composite polymeric electrolytes:•Molecular spectroscopy (FT-IR, Raman)•Thermal analysis•Scanning electron microscopy and XPS•NMR studies•Impedance spectroscopy

The studied system is complicated and its properties vary with both composition and temperature changes. These are mainly:•Contents of particular phases•Conductivity of particular phases•Ion associations•Ion transference number

Impedance spectrum of the composite electrolyteEquivalent circuit of the composite polymeric electrolyte measured in blocking electrodes

system consists of:

• Bulk resistivity of the material Rb

• Geometric capacitance Cg

• Double layer capacitance Cdl

Rb

Cg

Cdl

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

log omega

log

sig

ma

re

Z’

Z”

Activation energy analysis

For most of the semicrystalline systems studied the Arrhenius type of temperature conductivity dependence is observed:

σ(T) = n(T)μ(T)ez = σ0exp(–Ea/kT)

Where Ea is the activation energy of the conductivity process.

The changes of the conductivity value are related to the charge carriers:•mobility changes•concentration changes

Finally, the overall activation energy (Ea) can be divided into:•activation energy of the charge carriers mobility changes (Em)•activation energy of the charge carriers concentration changes (Ec)

Ea = Em + Ec

These two values can give us some information, which of two above mentioned processes is limiting for the conductivity.

The application of Almond-West formalism to composite polymeric electrolyteis realized in the following steps:•application of Jonsher’s universal power law of dielectric response

•calculation of p for different temperatures

•calculation of activation energy of migration from Arrhenius type equation

•calculation of effective charge carriers concentration

•calculation of activation energy of charge carrier creation

Almond – West Formalism

σ(ω) = σDC + Aωn

ωp = (σDC/A)(1/n)

K = σDCT/ωp

p = ωe exp (-Em/kT)

Ec = Ea - Em

Modeling of the conductivity in composites

• Ab initio quantum mechanics

• Semi empirical quantum mechanics

• Molecular mechanics / molecular dynamics

• Effective medium approach

• Random resistor network approach

•System is represented by three dimensional network •Each node of the network is related to an element with a single impedance value•Each phase present in the system has its characteristic impedance values•Each impedance is defined as a parallel RCPE connection

Model creation, stages 1,2

• Grains are located randomly in the matrix• Shells are added on the grains surface• Sample is divided into single uniform cells

Grain

Shell 1

Shell 2

Matrix

Model creation, stage 3

•The basic element of the model is the node where six impedance branches are connected

•The impedance elements of the branches are serially connected to the neighbouring ones

•For each node the potential difference towards one of the sample edges (electrodes) is defined

Model creation, stage 4

Finally, the three dimensional impedance network is created as a sample numerical representation

Model creation, stage 5

• Path approach: Sample is scanned for continuous percolation paths coming form one edge (electrode) to the opposite. Number of paths found gives us information about the sample conductivity.

• Current approach: Current coming through each node is calculated. Model is fitted by iteration algorithm. The iteration progress is related with the number of nodes achieving current equilibrium.

U2

Z2U3

U4

U5

Ul

U6

U Z3

Z4

Zl

Z6

Z5

Ii = (Ui - U) / Ri

Σ Ii = Σ [(Ui - U)/ Ri] = 0

Model creation, stage 6

• In each iteration step the voltage value of each node is changed as a function of voltage values of neighbouring nodes.

• The quality of the iteration can be tested by either the percent of the nodes which are in the equilibrium stage or by the analysis of current differences for node in the following iterations.

• The current differences seem to be better test parameters in comparison with the nodes count.

• When the equilibrium state is achieved the current flow between the layers (equal to the total sample current) can be easily calculated.

• Knowing the test voltage put on the sample edges one can easily calculate the impedance of the sample according to the Ohm’s law.

An example of the iteration progress

Step # Imax Imin IavDI Nodes %

2 154,174 102,530 126,553 40,81 1,2110 150,938 105,884 125,227 35,98 1,4920 148,090 106,999 124,612 32,98 1,5150 141,927 109,488 123,712 26,22 1,76

100 135,169 113,027 122,895 18,02 1,71200 127,841 117,293 121,094 8,64 1,53300 124,483 119,354 121,765 4,21 1,40400 122,912 120,415 121,623 2,05 1,47500 122,173 120,942 121,560 1,01 3,86600 121,825 121,203 121,530 0,51 5,67700 121,660 121,332 121,517 0,27 12,53800 121,584 121,398 121,511 0,15 24,59900 121,550 121,433 121,508 0,10 56,391000 121,538 121,452 121,507 0,07 73,521500 121,526 121,485 121,506 0,03 94,552000 121,516 121,496 121,506 0,02 97,842500 121,511 121,501 121,506 0,01 99,122820 121,509 121,503 121,506 0,00 99,36

Changes of node current during iteration

90

100

110

120

130

140

150

160

0 100 200 300 400 500 600 700 800 900 1000iteration #

I

Maximal current Minimal current Average current

Current flow around the single grain

• Vertical cross-section

• Horizontal cross-section

Some more nice pictures

• Voltage distribution around the single grain – vertical cross-section

• Current flow in randomly generated sample with 20 % v/v of grains – vertical cross-section

Path approach results Results of the path oriented approach calculations for samples containing grains

of 8 units diameter, different t/R values and with different amounts of additive

2R =8

0

50000

100000

150000

200000

250000

0 50 100 150 200 250 300 350 400

Additive ‰ v/v

Nu

mer

of p

ath

s

8/0.25

8/0.5

8/0.75

8/1.0

8/1.25

8/1.5

Rt

Path approach results

Results of the path oriented approach calculations for samples containing grains of different diameters, t/R=1.0 and with different amounts of additive

variable2R, t/R =1.0

0

20000

40000

60000

80000

100000

120000

140000

160000

180000

0 100 200 300 400

Additive ‰ v/v

Nu

mb

er

of

pa

ths

4/1.0

6/1.0

8/1.0

10/1.0

12/1.0

Rt

Current approach results

The dependence of the sample conductivity on the filler grain size and the filler amount for constant shell thickness equal to 3 m

% v/v

Current approach results

The dependence of the sample conductivity on the shell thickness and filler amount for the constant filler grain size equal to 5 m

% v/v

Conclusions• Random Resistor Network Approach is a valuable tool for computer simulation of

conductivity in composite polymeric electrolytes.

• Both approaches (current-oriented and path-oriented) give consistent results.

• Proposed model gives results which are in good agreement with both experimental data and Effective Medium Theory Approach.

• Appearing simulation errors come mainly from discretisation limits and can be easily reduced by increasing of the test matrix size.

• Model which was created for the bulk conductivity studies can be easily extended by the addition of the elements related to the surface effects and double layer existence.

• Various functions describing the space distribution of conductivity within the highly conductive shell can be introduced into the software.

• The model can be also extended by the addition of time dependent matrix property changes to simulate the aging of the material or passive layer growth.

Acknowledgements

Author would like to thank all his colleagues from the Solid State Technology Division.

Professor Władysław Wieczorek

was the person who introduced me into the composite polymeric electrolytes field and is the co-originator of the application of the Almond-West Formalism to the polymeric materials.

My students:

Piotr Rzeszotarski Katarzyna Nadara

realized in practice my ideas on Random Resistor Network Approach.