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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: [email protected],
tel (+) 48 601 26 26 00, fax (+) 48 22 628 27 41
Model of the composite polymeric electrolyte
t
R
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
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.
Impedance spectrum of the composite electrolyte
Equivalent circuit of the composite polymeric electrolyte measured in blocking electrodes
system consists of:
Rb
Bulk resistivity of the material Rb
Geometric capacitance Cg
Double layer capacitance Cdl
Cdl
Cg
log omega
0
-3
Z”
-3.5
w
-4
log sigma re
•
•
•
-4.5
-5
-5.5
Z’
-6
-6.5
0.5
1
1.5
2
2.5
3
3.5
4
4.5
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.
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
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
Ii = (Ui - U) / Ri
U3
Z2
Zl
Z3
U
Ul
U4
Z6
Σ Ii = Σ [(Ui - U)/ Ri] = 0
U6
Z4
Z5
U5
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 #
2
10
20
50
100
200
300
400
500
600
700
800
900
1000
1500
2000
2500
2820
Imax
154,174
150,938
148,090
141,927
135,169
127,841
124,483
122,912
122,173
121,825
121,660
121,584
121,550
121,538
121,526
121,516
121,511
121,509
Imin
102,530
105,884
106,999
109,488
113,027
117,293
119,354
120,415
120,942
121,203
121,332
121,398
121,433
121,452
121,485
121,496
121,501
121,503
Iav
126,553
125,227
124,612
123,712
122,895
121,094
121,765
121,623
121,560
121,530
121,517
121,511
121,508
121,507
121,506
121,506
121,506
121,506
DI
40,81
35,98
32,98
26,22
18,02
8,64
4,21
2,05
1,01
0,51
0,27
0,15
0,10
0,07
0,03
0,02
0,01
0,00
Nodes %
1,21
1,49
1,51
1,76
1,71
1,53
1,40
1,47
3,86
5,67
12,53
24,59
56,39
73,52
94,55
97,84
99,12
99,36
Changes of node current during iteration
I 160
150
140
130
120
110
100
90
0
100
200
300
400
500
600
700
800
900
iteration #
Maximal current
Minimal current
Average current
1000
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
R
250000
Numer of paths
200000
8/0.25
8/0.5
150000
8/0.75
8/1.0
100000
8/1.25
8/1.5
50000
0
0
50
100
150
200
Additive ‰ v/v
250
300
350
400
t
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
variable2R, t/R =1.0
180000
160000
Number of paths
140000
120000
4/1.0
100000
80000
6/1.0
8/1.0
60000
10/1.0
12/1.0
40000
20000
0
0
100
200
Additive ‰ v/v
300
400
t
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
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
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.
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