Title:

Simulation of impedance spectroscopy in electroceramics using a finite element method

Currently the electronic industry has a market demand for over a billion multi layer ceramic capacitors per annum. Electrical characterisation of the electroceramic component of these devices is required for optimisation of existing materials and to aid material discovery. Impedance spectroscopy is a technique that is commonly used to characterise the electrical properties of electroceramics. Experimental data is analysed using an equivalent circuit (usually some combination of resistors and capacitors connected in series and/or in parallel) to extract resistances and capacitances for specific components of a microstructure, e.g. bulk (grains), coreshell grains and grain boundaries. The ability to extract this information depends on the use of an appropriate equivalent circuit and on how to analyse the impedance data. Here an investigation of how the physical microstructure of an electroceramic can affect its impedance response using finite element modelling (FEM) is presented. By using a simulationbased approach the simulator can use the same methodology that would be used experimentally to obtain information on different microstructural components with prior knowledge of what the values should be, since the simulator has defined them. By comparing the values extracted to those originally inputted into the simulation allows the accuracy of the data analysis methods used to extract information to be evaluated and under what conditions these methods can be applied. The results presented in this thesis (chapters four to six) are divided into three studies. Chapter four considers the characterisation of coreshell grain microstructures by estimating core and shell volume fractions from the core to shell capacitance ratio. FEM simulation of the impedance response of a coreshell microstructure allows the capacitance ratio of the core and shell to be obtained from the electric modulus formalism. Several microstructures were considered: a nested cube; nested truncated octahedra; and a series layer model (SLM). The first two microstructures are approximations for a coreshell grain and were simulated using FEM. The layer model is an idealised case that can be solved analytically and with FEM for validation purposes. Here the relative permittivity of the core and shell regions is fixed at a value of 100 and the core has a conductivity three orders of magnitude greater than the shell. As the core volume fraction decreases, the core volume fraction extracted from the SLM is always accurate but becomes increasing inaccurate for the other models. This discrepancy agrees with the results of effective medium theory proving that our conclusions are physically reasonable. Plots of the electrical microstructure using a stream tracer method to view current flow showed increased heterogeneity in the current density in the core and shell. A quantitative study of the electrical microstructure showed the formation of conduction pathways through the parallel shell and increased curvature of the pathways through the core as the core volume fraction decreased. The electrical microstructure no longer resembled the physical microstructure, making extraction of volume ratios increasingly unreliable. Only for core volume fractions of 0.7 or greater could the core volume fraction be extracted from capacitance ratios with errors of less than 25%. Chapter five also considers the extraction of volume fractions from coreshell grains and other idealised microstructures. Here the conductivity of the core and shell regions is fixed and the permittivity of the core is greater than the shell. The impedance responses of an encased model, SLM and a parallel layer model (PLM) are simulated. The response of the encased model is shown to be more similar to the SLM than the PLM, implying serial connectivity in the encased model. Due to the difference in permittivity in the core and shell regions, the core volume fraction could not be obtained from capacitance ratios but only from resistance ratios obtained from the impedance formalism. The coreshell volume fractions of the encased model and SLM were varied and then extracted using resistance ratios. Similar trends to chapter four were observed, in chapter five, where the volume fraction could be accurately obtained for the SLM from resistance ratios for all input volume fractions. For the encased model, the error when extracting the core volume from resistance ratios increased as the core volume fraction decreased. Again, this error was in excess of 25% when the core volume fraction was less than 0.7. Finally, a stream tracer investigation of electrical microstructure revealed heterogeneous current density in the encased model caused by the formation of capacitive pathways through the microstructure. Chapter six examines the case where the microstructure is fixed and the material properties are varied. An encased model with a core volume fraction of 0.8 was chosen as it had been shown in the previous chapters that larger core volume fractions minimised the effects of conduction and capacitive pathways through the parallel shell but was still comparable to the volume fractions of coreshell microstructures in the literature. The core conductivity and relative permittivity was fixed at 0.1 mSm1 and 2000, respectively. The shell conductivity was varied from 0.1 mSm1 to 0.1 μSm1 and the relative permittivity from 2000 to 10. One hundred combinations over a range of shell properties was simulated. The resultant spectra were then fitted with three equivalent circuits where the fits were compared to find the best equivalent circuit using all four impedance formalisms. The first circuit was based upon a SLM with the same material properties and volume fractions inputted into the encased model. The second was called the series brick layer model (SBLM) and based on the encased model but neglecting the contribution of the parallel shell region. The third circuit was called the parallel brick layer model (PBLM) which included a separate resistor capacitor branch for the parallel shell region. The SLM provided a poor fit for all encased simulations with errors between ±34 to ±163%. The SBLM and PBLM provided better fits to the encased simualtions with errors from ±0.7 to ±20% and from ±0.55 to ±20%, respectively. Analysis showed that the SBLM provided the best fit when both the conductivty and the permittivity values of the core and shell were more than an order of magnitude different. The PBLM was best when either the shell conductivty or permititvty was within an order of magnitude of the core values. Finally, the best equivalent circuit for a given set of shell material properties was used to extract values of conductivity and permittivity (for both the core and shell) in all four impedance formalisms. The accuracy of the extracted values was calculated with respect to the input values for the simulation. This allowed the most reliable form of data analysis (i.e. formalism) for extracting conductivity and permittivity values for a given combination of material properties to be established. The accuracy of the most reliable formalism was mapped out for every material property combination. This optimal methodology was used to show the best case accuracy that could be achieved for extracting intrisic material properties from a core shell microstructure as the shell properties were systematically varied.
