Transcript Slajd 1 - Polymer Ionics Research Group

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
Thermodynamical 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
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.
Initial concept – helices and cations
also being very new one  (P.G. Bruce)
SPE as an liquid crystal polymer smectic versus
nematic alligment
SPE as an liquid crystal polymer smectic versus
nematic alligment
A sketch of the smectic short-chain system showing the disconiuity
of the helices and ion-pairing at the surface.
Free Volume Approach
•
•
•
•
•
M. H. Cohen & D. Turnbull J.Chem. Phys. 31 (1959) 1164
Diffusion of charged species is not thermally activated.
It is a result of redistribution of free volume within a liquid like amorphous phase.
The charged species are trapped in cages, except when a hole is opened being large enough for a
molecule to diffuse through.
The conductivity increases with the increase of the free volume with temperature:
Vf – free volume
T – temperature
•
vf = vg(0.025 + a (T-Tg)
vg – molar volume
Tg – glass transition temperature
a – thermal expansivity
Finally the thermal dependence of conductivity is descirbed by Vogel-Tamman-Fulcher (VTF) or
equivalent William-Landel-Ferry (WLF) equation
s = so exp (-B/(T- To)
s – conductivity
so – preexpotential factor
To – thermodynamical equilibrium glass transition temperature To = Tg – 50
•
•
Model is valid for monophase amorphous systems only.
In polymeric electrolytes which are often semicrystalline with noncomplete slat
dissociation cannot be applied.
Configurational Entropy Theory
•
•
•
•
G. Adam, J. H. Gibs J. Chem. Phys. 43 (1965) 139
Extension tot he Free Volume Approach
Charge Carrier Movement occurs by group cooperative rearangements
Conductivity is related to the propability of the rearangement
W = A exp (-Dmsc*/kTSc)
sc* - minimal configurational entropy needed for rearangement
Sc – configurational entropy for a temperature T
Dm – energy barrier for the rearangement process
•
•
•
Two parameter model
Leads to the VTF conductivity dependence
Similar limitation to free volume approach
Dynamic Bond Percolation Theory
•
•
•
S. D. Druger, M. A. Ratner, A.Nitzan Solid State Ionics 9&10 (1983) 1115
A first microscopic approach
The master equation for a static percolation approach:
Pi = S (PjWij – PiWji)
Pi – propability of finding carrier at site i
W – the frequency (hopping rate) for a carrier between sites
W = 0 with propability (1-f) and 1 with propabilty f
f – fraction of bonds (links between sites, not chemical bonds) which are open
(1-f) - fraction of bonds (links between sites, not chemical bonds) which are closed
•
•
•
This assumptions are valid for an ordered system
For polymers (disordered systems) an additional parameter is needed to
descibe the renewal of the lattice l = 1/tr where tr is a renewal time
Finally the correlation between static and dynamic diffusion coefficient can be
sketched:
Ddy (w) = Dst(w – il)
w – hopping frequency of charge carriers
Meyer Neldel Rule 1/3
•
•
D. P. Almond & A.R. West Solid State Ionics 23 (1987) 27
For semicrystalline polymeric elecrtolytes an Arrhenius dependence of
conductivity is observed:
s = so exp (-Ea/kT)
s – conductivity
Ea – activation energy
•
so – preexpotential factor
T - temperature
For a wide range of polymeric ionic conductors the magnitudes of the
preexpotential factor and the activation energies of conduction are connected by
the equation
ln so = aEa + b
so = K n0 exp (DSm/k)
K – correlation therm
n0 - ionic oscillation frequency
k – Boltzman’s constant
DSm- entropy of ion migration
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
Meyer Neldel Rule 2/3
•
For a range of materials the entropy of ion migration and the enthalpy of
activation are related with the order-disorder transition temperature (TD)
according to the following equation:
Ea / TD = DSm
•
•
•
•
•
•
For the polymeric electrolytes the TD temperature can be attributed to the
melting of the crystalline phase of the polymeric host
As an example of the applicability of the Meyer-Neldel rule to composite
polymeric electrolytes a PEO-NaI-Q-Al2O3 “mixed-phase” system can be used.
Fillers of different grain size were used as additive.
W. Wieczorek, K. Such, H. Wyciślik, J. Płocharski, Solid State Ionics 36 (1989)
255
The calculated value of TD is equal to 358 K being significantly higher than the
melting temperature of the pure crystalline PEO phase.
The values of Ea and Sa are rapidly growing with an increase of filler grain size.
Thus, even after exceeding the melting temperature the properties of the
amorphous phase present in the system are still affected by the presence of the
inorganic filler.
Meyer Neldel Rule 3/3
Plots of logarithm of conductivity
preexpotential factor against
activation energy for:
a) PEO-PMMA-LiClO4
b) PEO-PMMA-NaI
blend based polymeric electrolytes
of various blends composition
Plot of logarithm of conductivity
preexpotential factor against
activation energy for:
(PEO)10NaI – Q-Al2O3 10 wt% of
the filler for fillers of different
grain sizes
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
Impedance spectrum of the composite electrolyte
– real system
R
b
Cdl
Qg
ZCPE= -1/(jwC)n
w = 2Pf
n = 1.0 Df=90o
n = 0.5 Df=45o
n = 0.0 Df=0o
The dependency of the high frequency semi-arc on the value of the n parameter.
Coming to real systems leads to change of capacity Df=90o to the Constant Phase
Element df<90o and frequency independent.
Impedance spectrum of the composite electrolyte
– real system
R
b
Cdl
Qg
ZCPE= -1/(jwC)n
w = 2Pf
n = 1.0 Df=90o
n = 0.5 Df=45o
n = 0.0 Df=0o
The dependency of the high frequency semi-arc on the value of the n parameter.
Coming to real systems leads to change of capacity Df=90o to the Constant Phase
Element df<90o and frequency independent.
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.
Jonshers Universal Power Law of Dielectric
Response
σRe(ω) = σDC + Aωn
σRe(ω) - σDC = Aωn
ln(σRe(ω) – σDC) = ln A + n lnω
σRe – real part of the complex conductivity
σDC – DC conductivity of the sample
A,n – material parameters
Calculation of wp for a set of impedance spectra registered in different
Temperatures for the same sample
ωp = (σDC/A)(1/n)
Analysis of the impedance spectra according to the
Jonsher’s law of the Universal Dielectric Response
ln(σRe(ω) – σDC) = ln A + n lnω
σDC
Conductivity for (PEO)10NaI + 20% Q-Al2O3
as a function of the frequency.
Almond – West Formalism
The application of Almond-West formalism to composite polymeric electrolyte
Allows to divide the overall activation energy of the conduction process
to parts related to charge carrier migration and creation.
•calculation of activation energy of conductivity from Arrhenius type equation
s = so exp (-Ea/kT)
•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
Arrhenius plots of conductivity and hoping frequency
for a polymeric electrolyte
Thermal dependence of the effective concentration
of charge carriers and power exponent n
Thermal dependence of the effective concentration
of charge carriers
Thermal dependence of the effective concentration
of charge carriers
Activation energy of conduction and migration for a pristine
and composite polymeric electrolyte as a function of the
filler contents
PEO-NaI Ea
PEO-NaI Em
PEO-Triflate Ea
PEO - Triflate Em
1,75E+02
Energy kJ/mol
1,50E+02
1,25E+02
1,00E+02
7,50E+01
5,00E+01
2,50E+01
0
10
20
Filler concentation wt%
30
40
Activation energy of conduction, migration and
creation for a composite polymeric electrolytes as
a function of the filler contents
Wt.% of SiC in
sample
1mm. filler
7mm. filler
Ea
Em.
Ec
Ea
Em
Ec
0
89.2
44.0
45.2
89.2
44.0
45.2
5
82.4
35.3
47.1
70.3
28.1
42.2
10
67.6
26.1
41.5
72.1
24.3
47.8
15
85.0
36.7
48.3
82.9
37.3
45.6
20
107.9
55.6
52.3
118.2
65.3
54.7
25
140.8
82.1
58.7
145.9
82.5
63.4
30
140.4
79.3
61.1
145.3
82.8
62.5
40
135.8
68.5
57.3
139.8
81.3
58.5
Concept of mismatch and relaxation
• K. Funke, D. Wilmer, Solid State Ionics 136-137 (2000) 1329-1333
• Conductivity data were collected in a very wide frequency range combining
classical impedance spectroscopy measurements, microwave spectroscopy and far
infrared.
• The isea of the concept is to correlate the spectral data with the mobile ion
dynamics in the samples.
•A jump relaxation model was built over the CMR basis.
•After each hop of the mobile ion a mismatch is created between its own position
and the arrangement of the neighbours.
•The reduction of the mismatch is possible either through neighbours
rearrangement or through the hop back of the ion.
•In amorphous materials such as conducting glasses, ions encounter different kinds
of site and the model must be modified accordingly.
•One can assume that for very wide frequency range the conductivity vs frequency
plot reveals three different regions which are:
•Low frequence plateau
•Medium frequency power law region
•High frequency plateau
•Both high frequency and low frequency conductivites obey Arrhenius law with
different activation energies.
Concept of mismatch and relaxation
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 Theory-results 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.
ALISTORE
Sixth Framework Programme
Advanced lithium energy storage systems based on the u
se of
nano-powders and nano composite electrodes/electrolyte
s