1. Impedance spectroscopy of composite polymeric electrolytes - from experiment to computer modeling.
Maciej Siekierski
Warsaw University of Technology, Faculty of Chemistry,
ul. Noakowskiego 3, Warsaw, POLAND
tel (+) , fax (+)

2. Model of the composite polymeric electrolyte
Sample consists of three different phases:
Original polymeric electrolyte – matrix
Grains
Amorphous grain shells
Last two form so called composite
grain characterized with the t/R ratio

3. Experimental determination of the material parameters:
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
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
d.c. conductivity value
diffusion process study
transport properties of the electrolyte-electrode border area
determination of a transference number of a charge carriers.

4. Impedance spectrum of the composite electrolyte
Equivalent 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
Cdl Cg Z’ Z” w

5. 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.

6. Almond – West Formalism
The application of Almond-West formalism to composite polymeric electrolyte
is realized in the following steps:
application of Jonsher’s universal power law of dielectric response
σ(ω) = σDC + Aωn
calculation of wp for different temperatures
ωp = (σDC/A)(1/n)
calculation of activation energy of migration from Arrhenius type equation
wp = ωe exp (-Em/kT)
calculation of effective charge carriers concentration
K = σDCT/ωp
calculation of activation energy of charge carrier creation
Ec = Ea - Em

7. 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

8. Model creation, stages 1,2 Grain Shell 1 Shell 2 Matrix
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

9. 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

10. Model creation, stage 4
Finally, the three dimensional impedance network is created
as a sample numerical representation

11. Model creation, stage 5 Ii = (Ui - U) / Ri Σ Ii = Σ [(Ui - U)/ Ri] = 0
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 Z2 U3 U4 U5 Ul U6 U Z3 Z4 Zl Z6 Z5
Ii = (Ui - U) / Ri
Σ Ii = Σ [(Ui - U)/ Ri] = 0

12. 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.

13. An example of the iteration progress

14. Changes of node current during iteration

15. Current flow around the single grain
Vertical cross-section
Horizontal cross-section

16. 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

17. Path approach results t R
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 R t

18. 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 R t

19. 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 mm
% v/v

20. 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 mm
% v/v

21. 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.

22. Acknowledgements Professor Władysław Wieczorek
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.