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.