Impedance%20spectroscopy%20of%20composite%20polymeric%20electrolytes%20-%20from%20experiment%20to%20computer%20modeling.

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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_at_soliton.ch.pw.edu.pl, tel
() 48 601 26 26 00, fax () 48 22 628 27 41
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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
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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.

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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

Cg
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Activation energy analysis

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

s(T) n(T)µ(T)ez s0exp(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.
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Almond West Formalism

• The application of Almond-West formalism to
  composite polymeric electrolyte
• is realized in the following steps
• application of Jonshers universal power law of
  dielectric response

s(?) sDC A?n

• calculation of wp for different temperatures

?p (sDC/A)(1/n)

• calculation of activation energy of migration
  from Arrhenius type equation

wp ?e exp (-Em/kT)

• calculation of effective charge carriers
  concentration

K sDCT/?p

• calculation of activation energy of charge
  carrier creation

Ec Ea - Em
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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

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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

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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

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

• Ii (Ui - U) / Ri
• S Ii S (Ui - U)/ Ri 0

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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 Ohms law.

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An example of the iteration progress
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Changes of node current during iteration
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Current flow around the single grain

• Vertical cross-section
• Horizontal cross-section

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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

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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

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Path approach results

• Results of the path oriented approach
  calculations for samples containing grains of
  different diameters, t/R1.0 and with different
  amounts of additive

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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
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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
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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.

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Acknowledgements

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