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Maciej S.Siekierski 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 Polymer Ionics Research Group

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Modeling of conductivity in Composite Polymeric Electrolytes with Phase Scale Models

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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

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

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R t Effective Medium Theory

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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.

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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.

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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. V c = V 2 / Y V 2 – volume fraction of the filler V c = 1 and = max when V 2 = Y If V 2 > 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.

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Effective Medium Theory - results

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Effective Medium Theory- results of the simulation

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Effective Medium Theory – smimulation vs. experiment

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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 T g value can be extracted for real samples from the DSC experiments. For composite system a dependence of T g can be fitted with the empiric equation. K 0 is related to the salt influence on T g without the filler addition K 1 represent the filler polymer interaction K 2 represents polymer – filler – salt interactions

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Effective Medium Theory – model improvement

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Effective Medium Theory – model improvement

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Effective Medium Theory – model improvement

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Effective Medium Theory – thermal dependence

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Effective Medium Theory – simulated Meyer-Neldel

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Effective Medium Theory – a.c. approach For a.c. conduction the parameters in all equations were replaced with complex conductance parameters expressed according to the following equation: j 2 = -1 – angular frequency – dielectric constant

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Effective Medium Theory – a.c. approach

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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.

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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

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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

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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

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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

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Model creation – stage 4 Finally, the three dimensional impedance network is created as a sample numerical representation

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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

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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

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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.

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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. U2U2 Z2Z2 U3U3 U4U4 U5U5 UlUl U6U6 U Z3Z3 Z4Z4 ZlZl Z6Z6 Z5Z5 I i = (U i - U )/ R i Σ I i = Σ [(U i - U)/ R i ] = 0 I = (1/R) * U I ΔU U U U UiUi UiUi UiUi ΔU = U i - U ΔU= 0 ΔU > 0 ΔU < 0 inflow outflow I =Σ I i

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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.

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An example of the iteration progress

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Changes of node current during iteration

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Current flow around the single grain Vertical cross-section Horizontal cross-section

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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

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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=0.75 m (3 units), - d=1.25 m (5 units), - d=1.75 m (7 units), - d=2.25 m (9 units), - d=2.75 m (11 units).Virtual matrix size 900x900x900 units.

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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 R t

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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 t

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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 3.0 m and different statistical distributions of t/R R t - t/R = 0 - t/R = 0.2 - t/R = 0.4 - t/R = 0.6 - t/R = 0.8 - t/R = 1.00

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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 m % v/v

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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 m. Uniform conductivity distribution in the shell.

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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

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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 m. Gaussian conductivity distribution in the shell.

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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 m. Uniform conductivity distribution in the shell.

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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 m. Gaussian conductivity distribution in the shell.

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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.

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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.

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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 m.

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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 m. Shell thickness varies

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AC approach – results 3/3 The phase composition of the simulated composite electrolyte sample as the function of the volume contents of the filler. Grain diameter is equal to 7 m. Shell thickness is equal to 5 m. 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 m. Shell thickness is equal to 5 m.

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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 m.

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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 m. Shell thickness varies

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The phase composition of the simulated composite electrolyte sample as the function of the volume contents of the filler. Grain diameter is equal to 7 m. Shell thickness is equal to 5 m. 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 m. Shell thickness is equal to 5 m.

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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 = 0 as a reference): - diameter of the semi-circle, - zero point of the semi-circle

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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: = 0 exp (-E a /RT) o = 9.62x 10 14 (S cm -1 ), E a = 126 (kJ mol -1 ) and VTF dependence for th shell = AT -0.5 exp (- B (T - T 0 )) T 0 = 195 K B =1204 K A = 26.97 (SK 0.5 cm -1 )

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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: = 0 exp (-E a /RT) o = 9.62x 10 14 (S cm -1 ), E a = 126 (kJ mol -1 ) and VTF dependence for th shell = AT -0.5 exp (- B (T - T 0 )) T 0 = 195 K B =1204 K A = 26.97 (SK 0.5 cm -1 )

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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: = 0 exp (-E a /RT) o = 9.62x 10 14 (S cm -1 ), E a = 126 (kJ mol -1 ) and VTF dependence for th shell = AT -0.5 exp (- B (T - T 0 )) T 0 = 195 K B =1204 K A = 26.97 (SK 0.5 cm -1 )

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EMT vs RRN comparison Comparison of experimental results ( ) of ionic conductivity of (PEO) 10 LiCIO 4 -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 m and d=5 m) (solid line) and RRN calculations (t=5 m and d=9 m) (dashed line).

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EMT vs RRN comparison A comparison of the experimental data (logarithmic scale) for (PEO) 10 NaI SiO 2 system (with simulations obtained by the EMT and RRN models. – experimental data, - basic EMT model, - EMT model with various t/R, - basic RRN model.

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EMT vs RRN comparison A comparison of the experimental data (logarithmic scale) for (PEO) 10 NaI SiO 2 system with simulations obtained by the EMT and RRN models. – experimental data, - corrected EMT model, - corrected RRN model.

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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.

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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

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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

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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). Conclusions

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