Maciej S. Siekierski

1. Maciej S.Siekierski Polymer Ionics Research Group Warsaw University of Technology, Faculty of Chemistry, ul. Noakowskiego 3, 00-664 Warsaw, POLAND e-mail: alex@soliton.ch.pw.edu.pl, tel (+) 48 601 26 26 00, fax (+) 48 22 628 27 41

2. Modeling of conductivity inComposite Polymeric Electrolytes Thermodynamical models

3. Modeling of the conductivity in polymeric electrolytes: Thermodynamicalmodels (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

4. 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 • d.c. conductivity value • diffusion process study • transport properties of the electrolyte-electrode border area • determination of a transference number of a charge carriers. • 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

5. Initial concept – helices and cationsalso being very new one  (P.G. Bruce)

6. SPE as an liquid crystal polymer smectic versus nematic alligment

7. 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.

8. 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 = vg(0.025 + a (T-Tg) Vf – free volume vg – molar volume a – thermal expansivity T – temperature Tg – glass transition temperature • 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.

9. 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

10. 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

11. 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 – conductivity so – preexpotential factor Ea – activation energy T - temperature • For a wide range of polymeric ionic conductors the magnitudes of the preexpotential factorand the activation energies of conduction are connected by the equation ln so = aEa + b so = K n0 exp (DSm/k) K – correlation therm k – Boltzman’s constant n0 - ionic oscillation frequency DSm- entropy of ion migration s = so exp (-Ea/kT)

12. t R 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

13. 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.

14. Meyer Neldel Rule 3/3 • Plots of logarithm of conductivity • preexpotential factor against • activation energy for: • PEO-PMMA-LiClO4 • 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

15. Rb Cdl Z” w Z’ 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

16. Rb Qg Cdl Impedance spectrum of the composite electrolyte – real system 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.

17. Rb Qg Cdl Impedance spectrum of the composite electrolyte – real system 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.

18. 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.

19. 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)

20. 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

21. Conductivity for (PEO)10NaI + 20% Q-Al2O3 as a function of the frequency.

22. 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

23. Arrhenius plots of conductivity and hoping frequency for a polymeric electrolyte

24. Thermal dependence of the effective concentration of charge carriers and power exponent n

25. Thermal dependence of the effective concentration of charge carriers

26. Thermal dependence of the effective concentration of charge carriers

27. Activation energy of conduction and migration for a pristine and composite polymeric electrolyte as a function of the filler contents

28. Activation energy of conduction, migration and creation for a composite polymeric electrolytes as a function of the filler contents

29. 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 combiningclassical 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.

30. Concept of mismatch and relaxation

31. t R Solid composite polymeric electrolyte Last two form so called composite grain characterized with the t/R ratio. This units are randomly distributed in the matrix • Sample consists of three different phases: • Original polymeric electrolyte – matrix • Grains • Amorphous grain shells

32. t R Effective Medium Theory

33. 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.

34. 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.

35. 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.

36. Effective Medium Theory - results

37. Effective Medium Theory-results of the simulation

38. Effective Medium Theory – smimulation vs. experiment

39. 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

40. Effective Medium Theory – model improvement

41. Effective Medium Theory – model improvement

42. Effective Medium Theory – model improvement

43. Effective Medium Theory – thermal dependence

44. Effective Medium Theory – simulated Meyer-Neldel

45. 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

46. Effective Medium Theory – a.c. approach

47. Effective Medium Theory – a.c. approach

48. 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.

49. ALISTORE Sixth Framework Programme Advanced lithium energy storage systems based on the use of nano-powders and nano composite electrodes/electrolytes