Transcript Slajd 1 - Warsaw University of Technology

Maciej S.Siekierski
Polymer Ionics Research Group
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
Modeling of conductivity in
Composite Polymeric
Electrolytes with Phase Scale
Models
Modeling of the conductivity in polymeric electrolytes:
Thermodynamical models (macroscopic and microscopic):
• Free Volume Approach
• Configurational Entropy Approach
• Dynamic Bond Percolation Theory
• Dielectric Response Analysis
Molecular scale models:
• Ab initio quantum mechanics (DFT and Hartree-Fock)
• Semi empirical quantum mechanics
• Molecular mechanics
• Molecular dynamics
Phase scale models:
• Effective medium approach
• Random resistor network approach
• Finite element approach
• Finite gradient approach
Phase scale model of the solid 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.
This units are randomly distributed
in the matrix
Effective Medium Theory
t
R
Effective Medium Theory
•Conductivity can be easily numerically simulated by means of the
Effective Medium Theory.
•The geometry of the composite unit consisting of a grain and
a highly conductive shell suggests the application of the
Maxwell-Garnett mixing rule for the calculation of composite grain conductivity.
•The value of effective conductivity can be easily calculated for conductivities
of the grain (almost equal to 0), the shell and volume of the dispersed phase
in a composite grain.
•Later, the composite electrolyte can be treated as a quasi two-phase mixture
consisting of the pristine matrix and composite grains.
•Landauer and Bruggemanequations are valid only
for composite unit concentrations lower than 0.1.
Effective Medium Theory
•The obtained set of equations allows to predict conductivity of the composite
in all filler concentration ranges.
•Three characteristic volume fractions are defined for the system studied.
The first is the continuous percolation threshold where the composite grains
start to form a cluster.
•The second one is the volume fraction of the filler at which the cluster
of composite grains fills all the sample volume.
•The third one observed at very high filler concentrations, can be attributed to
conductor to insulator transition occurring when the polymer matrix loses its
continuity.
•These values can be attributed to the phenomena observed in the sample,
i.e. abrupt conductivity increase, conductivity maximum and, later, conductivity
deterioration, respectively.
•In real systems this value is much higher and thus the equation must be improved
by the corrections developed by Nan and Nakamura.
•System consists of pristine electrolyte and growing ammount of composite grains.
Vc = V2 / Y
V2 – volume fraction of the filler
Vc = 1 and s = max when V2 = Y
If V2 > Y then a different situation is observed. System consists of composite grains
and diluting them bare filler grains. A different set of equations must be used.
Effective Medium Theory - results
Effective Medium Theoryresults of the simulation
Effective Medium Theory –
smimulation vs. experiment
Effective Medium Theory –
model improvement
•A stiffening effect of the hard filler is observed for the amorphous shell.
•A conductivity decrease is observed.
•The conductivity of the amorphous phase is dependent on the filler volume ratio.
•As the shell is amorphous a VTF type equation can be applied.
•The Tg value can be extracted for real samples from the DSC experiments.
•For composite system a dependence of Tg can be fitted with the empiric equation.
•K0 is related to the salt influence on Tg without the filler addition
•K1 represent the filler polymer interaction
•K2 represents polymer – filler – salt interactions
Effective Medium Theory –
model improvement
Effective Medium Theory –
model improvement
Effective Medium Theory – model improvement
Effective Medium Theory – thermal dependence
Effective Medium Theory – simulated Meyer-Neldel
Effective Medium Theory – a.c. approach
•For a.c. conduction the s parameters in all equations were replaced with complex
conductance parameters expressed according to the following equation:
j2 = -1 w – angular frequency e – dielectric constant
Effective Medium Theory – a.c. approach
Effective Medium Theory – a.c. approach
Disadvantages of the EMT approach
•
•
•
•
•
Assumption that all grains are identical in
respect to their shape and size.
A need for a new mixing rule for each
particular grain shape.
A need of percolation threshold determination
for each particular grain shape.
Assumption that each grain generate shell of
the same thickness.
Assumption that the shell is uniform and no
changes in conductivity are observed within it.
Basics of the RRN 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
• The general impedance of the network represents
the value characteristic for the macroscopic sample
Model creation – summary
Common stages:
1. Virtual sample generaiton
2. Sample discretization
3. Conversion into resistivities
4. 3-D resistor/impedance network ready
Path approach:
5. Sample scaning for continous paths
DC approach:
5. Test potential added
6. Iteration procedure
AC approach:
5. Impedancies of elements are calculated for a
particular frequency
6. Test voltage added. Voltage must be real at
„electrodes” and can be complex inside the sample
7. Iteration procedure
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
•The effective value of the inter-nodal impedance
is calculated for each branch in the network
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.
•
All (not necessarily shortest) paths of percolation are taken into count.
•
The test potential is located along the z direction of the matrix.
•
The search starts from plane z=0
•
It goes to plane z=(n–1) , where n is the matrix size.
U
Model creation - stage 5 – path approach
•
Independently of the x,y coordinates of the start point
•
A target of the search lie on a plane with coordinate z equal to n–1 in a point
characterized by unrestricted coordinates x’, y’.
•
The preferred direction is a point (x, y, z+1), where a charge carrier moves
according to the direction of electrical field’s vector.
•
This movement is only possible when a highly conductive path lies between
these two points.
U
Model creation - stage 5 – path approach
•
If not another directions (x+1, y, z), (x, y+1, z), (x–1, y, z) and (x, y–1, z) are
tested for possible paths.
•
Finally (x, y, z–1) are analyzed in next order.
•
This algorithm works in loop until it attains the (x, y, n-1) point (we just have
found a percolation path) or when the buffer of history of movements will be
exhausted (there is no path of fast conductivity in the system for this particular
start point).
•
The procedure is repeated until the pool of unsigned clusters of shields will be
exhausted on the plain z=0.
Model creation – DC/AC current approach 1/2
•
•
•
•
Current coming through each node is calculated.
The current flow is calculated as a sum of all branch currents for a particular
node.
The branch current is calculated on the basis of the potentail difference in
the branches.
The quality of the fit is related with the number of nodes achieving zero
current state.
ΔU = Ui - U
I = (1/R) * U
U
I
inflow
outflow
U2
ΔU
U
U
Ui ΔU= 0
Ui
ΔU > 0
Ui
ΔU < 0
Zl
Z3
U
Ul
U4
Z6
U6
Ii = (Ui - U )/ Ri
I =Σ Ii
Σ Ii = Σ [(Ui - U)/ Ri] = 0
U3
Z2
Z4
Z5
U5
Model creation – DC/AC current approach 2/2
•
In each iteration step the voltage value of each node is changed as a
function of voltage values of neighbouring nodes in the way leading to the
fullfilment of the zero current condition for each node present in the network.
•
The iteration progress can be estimated either by the calculation of the
percentage of the nodes which are in the zero current stationary state or by
the analysis of average current differences for all nodes in the subsequent
iterations.
•
The current differences seem to be better test parameters in comparison
with the nodes count leading to much quicker iteration stop with similarily
small error.
•
When the stationary 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
Patch approach – results 1/4
Average number of poles needed for one filler grain insertion into
the virtual matrix (with maintaining the continuity rules) for different
grain sizes and different ammount of filler.
- d=0.75mm (3 units), - d=1.25mm (5 units),
- d=1.75mm (7 units), l- d=2.25mm (9 units),
r- d=2.75mm (11 units).Virtual matrix size 900x900x900 units.
Path approach – results 2/4
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 3/4
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
Path approach – results 4/4
The dependency of the number of percolation paths in the matrix
as a function of the grains volume fraction for constant t/R value equal to 1.25,
constant grain size equal to 3.0mm and different statistical distributions of t/R
R
t
 - st/R = 0
p - st/R = 0.2
 - st/R = 0.4
l - st/R = 0.6
 - st/R = 0.8
 - st/R = 1.00
Current approach – results 1/7
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
DC approach – results 2/7
The dependence of the sample conductivity on the shell thickness
and the filler amount for the constant filler grain size equal to 5 mm.
Uniform conductivity distribution in the shell.
DC approach – The idea of inhomogenous
distribution of the conductivity within the shell
•
•
•
•
•
In real system a stiffening of the amorphous phase in
close viscinity of the grain is observed.
To model this phenomenon a gaussian distribution of
the conductivity within the shell was applied
On the grain boundary the strongest stiffening is
observed leading to the local lowering of the shell
conductivity
In the middle of the shell the stiffening is no more
observed but the amorphisation is still present – the
conductivity acheived local maximum
At the outer border of the shell a step decrease of
the amorphisation is observed – the conductivity
reaches the value typical for the pristine polymer-salt
matrix
DC approach – results 3/7
The dependence of the sample conductivity on the shell thickness
and the filler amount for the constant filler grain size equal to 5 mm.
Gaussian conductivity distribution in the shell.
DC approach – results 4/7
The dependence of the sample conductivity on the filler grain size
and the filler amount for the constant shell thickness equal to 5 mm.
Uniform conductivity distribution in the shell.
DC approach – results 5/7
The dependence of the sample conductivity on the filler grain size
and the filler amount for the constant shell thickness equal to 5 mm.
Gaussian conductivity distribution in the shell.
DC approach – results 6/7
The dependence of the sample conductivity on the filler amount
for different filler grain size distributions.
Uniform conductivity distribution in the shell.
Current approach – results 7/7
The dependence of the maximal conductivity of the samples set
with varying filler amount on the filler grain diameter.
For varying shell thickness.
AC approach – results 1/3
The simulated impedance spectra (high frequency part) for a composite
electrolyte with different volume contents of the filler. Both the grain
diameter and the shell thickness are equal to 5 mm.
Z'' [Ohm]
2,00E+06
0%
1,50E+06
5%
10%
15%
1,00E+06
20%
25%
5,00E+05
0,00E+00
0,00E+00
30%
1,00E+06
2,00E+06
Z' [Ohm]
3,00E+06
4,00E+06
AC approach – results 2/3
The simulated impedance spectra (high frequency part) for a composite
electrolyte with the constant volume contents of the filler (10%).The grain
diameter is equal to 5 mm. Shell thickness varies
2,50E+06
Z'' [Ohm]
2,00E+06
1,50E+06
d5t3
d5t5
d5t7
1,00E+06
5,00E+05
0,00E+00
0,00E+00
1,00E+06
2,00E+06
3,00E+06
Z' [Ohm]
4,00E+06
5,00E+06
AC approach – results 3/3
100
grain
Phase composition
90
shell
80
basic polymer
70
60
50
40
30
20
10
0
0
5
10
15
20
25
30
35
40
Filler volume contents %
The phase composition of
the simulated composite
electrolyte sample as the
function of the volume
contents of the filler. Grain
diameter is equal to 7mm.
Shell thickness is equal to
5mm.
8,00E-05
Sigma [S/cm]
6,00E-05
4,00E-05
2,00E-05
0,00E+00
0
5
10
15
20
25
Filler volume contents %
30
35
The DC conductivity calculated
from the simulated impedance
spectra (high frequency part)
for a composite
electrolyte as a function of the
volume contents of the
filler.The grain
diameter is equal to 7 mm.
Shell thickness is equal to
5mm.
A set of high frequency simulated impedance
spectra calculated for a virtual sample
characterized by d = 7 mm and t = 5 mm. Filler
fraction equal to: - 0, -1, p-2,
-5, -10, -20, -30 vol%
Z" / Ohm
1,00E+10
1,00E+07
1,00E+04
1,00E+04
1,00E+07
Z' / Ohm
1,00E+10
A set of high frequency simulated impedance
spectra calculated for a virtual sample
characterized by d = 3 mm and t = 5 mm. Filler
fraction equal to: - 1, -2, -3,
-5, -10, , p-30 vol%
7,50E+06
ImZ/Ohm
5,00E+06
2,50E+06
0,00E+00
0,00E+00
2,50E+06
5,00E+06
ReZ/Ohm
7,50E+06
The simulated impedance spectra (high frequency part)
for a composite electrolyte with different volume
contents of the filler. Both the grain diameter
and the shell thickness are equal to 5 mm.
Z'' [Ohm]
2,00E+06
0%
1,50E+06
5%
10%
15%
1,00E+06
20%
25%
5,00E+05
0,00E+00
0,00E+00
30%
1,00E+06
2,00E+06
Z' [Ohm]
3,00E+06
4,00E+06
The simulated impedance spectra (high frequency part)
for a composite electrolyte with the constant volume
contents of the filler (10%).The grain
diameter is equal to 5 mm. Shell thickness varies
2,50E+06
Z'' [Ohm]
2,00E+06
1,50E+06
d5t3
d5t5
d5t7
1,00E+06
5,00E+05
0,00E+00
0,00E+00
1,00E+06
2,00E+06
3,00E+06
Z' [Ohm]
4,00E+06
5,00E+06
100
grain
Phase composition
90
shell
80
basic polymer
70
60
50
40
30
20
10
0
0
5
10
15
20
25
30
35
40
Filler volume contents %
The phase composition of
the simulated composite
electrolyte sample as the
function of the volume
contents of the filler. Grain
diameter is equal to 7mm.
Shell thickness is equal to
5mm.
8,00E-05
Sigma [S/cm]
6,00E-05
4,00E-05
2,00E-05
0,00E+00
0
5
10
15
20
25
Filler volume contents %
30
35
The DC conductivity calculated
from the simulated impedance
spectra (high frequency part)
for a composite
electrolyte as a function of the
volume contents of the
filler.The grain
diameter is equal to 7 mm.
Shell thickness is equal to
5mm.
Simulated d.c. conductivity values found from
the a.c. spectra of the virtual sample as a
function of filler concentration. Calculations
performed in various manners: extrapolation of the Z’ to w = 0, diameter of the semi-circle, r- zero point
of the semi-circle
Conductivity S/cm
9,00E-05
6,00E-05
3,00E-05
0,00E+00
0
10
20
Filler concentration vol. %
30
Relative errors of the simulated d.c. conductivity values found
from the a.c. spectra of the virtual sample as a function
of filler concentration. Calculations performed in various
manners (extrapolation of the Z’ to w = 0 as a reference):
 - diameter of the semi-circle,
r - zero point of the semi-circle
Relative error (%)
4,50
3,00
1,50
0,00
0
5
10
15
20
Filler concentration vol. %
25
30
DC approach – thermal dependencies 1/3
Finally the thermal dependence of the composite can be simulated.
To do so we use Arrhenius for the PEO matrix:
s= s0 exp (-Ea/RT) so= 9.62x1014 (S cm-1), Ea= 126 (kJ mol-1)
and VTF dependence for th shell
s = AT-0.5exp (- B (T - T0 )) T0 = 195 K B =1204 K A = 26.97 (SK0.5cm-1)
1,80E-04
sigma/S*cm-1
1,60E-04
1,40E-04
0C
1,20E-04
10C
20C
1,00E-04
30C
8,00E-05
40C
6,00E-05
50C
4,00E-05
60C
2,00E-05
0,00E+00
0
5
10
15
20
25
Filler contents / %vol.
30
35
DC approach – thermal dependencies 2/3
Finally the thermal dependence of the composite can be simulated.
To do so we use Arrhenius for the PEO matrix:
s= s0 exp (-Ea/RT) so= 9.62x1014 (S cm-1), Ea= 126 (kJ mol-1)
and VTF dependence for th shell
s = AT-0.5exp (- B (T - T0 )) T0 = 195 K B =1204 K A = 26.97 (SK0.5cm-1)
1,00E-03
1,00E-04
log(sigma)
0C
1,00E-05
10C
1,00E-06
20C
30C
1,00E-07
40C
1,00E-08
50C
60C
1,00E-09
1,00E-10
0
5
10
15
20
25
Filler contents / %vol
30
35
DC approach – thermal dependencies 3/3
Finally the thermal dependence of the composite can be simulated.
To do so we use Arrhenius for the PEO matrix:
s= s0 exp (-Ea/RT) so= 9.62x1014 (S cm-1), Ea= 126 (kJ mol-1)
and VTF dependence for th shell
s = AT-0.5exp (- B (T - T0 )) T0 = 195 K B =1204 K A = 26.97 (SK0.5cm-1)
Temperature dependence of coductivity for spearate
phases
1,0E-03
log(sigma)
1,0E-04
1,0E-05
1,0E-06
matrix
1,0E-07
shell
1,0E-08
1,0E-09
1,0E-10
0
20
40
T/C
60
EMT vs RRN comparison
Conductivity S/cm
8,00E-05
4,00E-05
0,00E+00
0
15
30
45
Filler concnetration vol %
Comparison of experimental results () of ionic conductivity of (PEO)10
LiCIO4-PAAM- system measured for various PAAM volume concentration
and theoretical values obtained from calculations for the same system by
means of the EMT () (t/R ratio is equal to 0.6), and RRN (t=3 mm and d=5
mm) (solid line) and RRN calculations (t=5 mm and d=9 mm) (dashed line).
EMT vs RRN comparison
Conductivity S/cm
-5,5
A comparison of the experimental data
(logarithmic scale) for (PEO)10 NaI
SiO2 system (with simulations obtained
by the EMT and RRN models.  –
experimental data,  - basic EMT
model,  - EMT model with various
t/R,  - basic RRN model.
-7,5
-9,5
0
10
20
Filler content vol%
30
EMT vs RRN comparison
-4
A comparison of the experimental data
(logarithmic scale) for (PEO)10 NaI
SiO2 system with simulations obtained
by the EMT and RRN models.  –
experimental data,  - corrected EMT
model,  - corrected RRN model.
Conductivity S/cm
-6
-8
-10
0
20
10
Filler content vol%
30
Conclusions
•
Random Resistor Network Approach is a valuable tool for computer
simulation of conductivity in composite polymeric electrolytes.
•
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.
•
Finally, it must be underlined that in general RRN models provide only
slightly worse fit to the experimental data in comparison with the EMT
approach utilizing a much smaller set of the empirical parameters.
Conclusions
• One of the main differences between Effective
Medium Theory and Random Resistor Network
lies in the fact that the first model uses only a
single parameter (t/R) to define spatial relations
in the studied sample whereas for the second
one can independently define the grain diameter
(d=2R) and the thickness of the amorphous shell
t. The additional feature of the RRN model lies in
the fact that a more realistic image of the sample
can be built by the introduction of the grain
diameter dispersion.
• .
Conclusions
• The discrepancy between the EMT corrected
model results and RRN inhomogeneous shell
model results can be additionally corrected by
the change of the assumption concerning the
total stiffening of the material belonging to shells
of more than one of the grains. An additive
function cumulating interaction of all grains in
the neighborhood could be used instead the
strongest possible interaction coming form only
one of the grains.
Conclusions
• Observations of the a.c. behavior suggest that
according to the EMT model composite
polymeric electrolytes behave as an ideal
dielectric if a sufficiently high frequency is used.
In contrast for the RRN simulations results show
a two time constant image for some impedance
spectra for samples characterized with the filler
amount values around the percolation threshold.
In this case the combination of the two types of
the conductivity can be observed with a
separable contribution originating from both bulk
of the sample and the highly conductive but still
non-percolating shells).