## 1 Introduction and Background

Impedance spectroscopy (IS) is widely employed to deconvolute the intrinsic (bulk) and/or extrinsic (grain boundary, electrode effects, etc.) contributions to the electrical properties of electroceramics by measuring the impedance response over a frequency spectrum,[1, 2] commonly from mHz to MHz. Since the late 1960s, extracting information from IS data, such as grain core (bulk) and grain-boundary capacitance and resistance values, has been done using some form of appropriate equivalent electrical circuit.[3] This usually consists of an arrangement of resistors and capacitors, connected in series and/or in parallel to model the IS response of the polycrystalline ceramic under investigation and to provide insight into the intrinsic and extrinsic properties. Identification of the correct form of the equivalent circuit is required for meaningful analysis of the system.[1, 4] This is based on the likely physical processes that occur in the material and often requires some level of intuition. In many cases, to a first approximation, the grain-core (bulk) response is described in an equivalent circuit by a parallel combination of a resistor and a capacitor (RC). This combination results in an ideal arc in the complex impedance and electric modulus plane plots, *Z** and *M**, respectively, and an ideal Debye peak in spectroscopic plots of the imaginary components of impedance, *Z*″, and electric modulus, *M*″, with a full-width half maximum (FWHM) of 1.14 decades on a logarithmic frequency scale.[5] Due to heterogeneities associated with defects, impurities, and complex conduction processes, such an ideal response for the grain core is seldom obtained. This leads to a nonideal Debye-like response, i.e., a depressed arc in *Z** and *M** plots and a nonideal Debye response in *Z*″ and *M*″ spectra with a FWHM >1.14 decades. Such responses cannot be treated accurately using a simple RC circuit[4, 6] and normally requires the addition of a constant phase element (CPE) to the equivalent circuit.

One of the first attempts to correlate the microstructure of an electroceramic with a mathematical combination of resistors and capacitors was proposed in the late 1960s by Bauerle.[3] This attributed the response of the ceramic to two parallel RC elements connected in series; one assigned to the grain core and the other to the grain boundary. This successfully represented an ion-conducting ceramic and modeled the dual-arc *Z** plots obtained from experimental IS data. This simple model was further developed into a three-dimensional resistive boundary layer model[7] and later into the well-known brick layer model (BLM) in the early 1980s.[8, 9] The BLM is a general representation of a ceramic using the analogy of bricks surrounded by mortar to represent grain cores surrounded by grain boundaries. Nafe[10] in the mid-1980s developed this further to allow the possibility of current flow around the grain core, through the grain core or a combination of both, by summing pathways (where appropriate) in parallel. Using these approximations it is possible to convert bulk data such as resistances and capacitances into intrinsic material properties such as conductivity and permittivity for the grain core. However, due to the unknown geometry of the grain boundaries, this method is generally considered to be unreliable to extract grain-boundary conductivity and permittivity values.

The BLM method has also been incorporated into a finite difference pixel-based simulation to calculate the current distribution.[11-13] Here, a pixel consists of six orthogonal nested cubes, each being assigned an RC element with the properties of a grain core or grain boundary. These pixels form the points on which the conduction path can be calculated with the nested cubes allowing a 3D interconnectivity of the microstructure to be constructed that the previous BLM methods could not provide. This not only allows the treatment of current pathways, thus replicating IS data, but also permits the BLM to be used for grain-core volume fractions from zero to unity with no breakdown of the calculation. These BLM methods, however, all have two intrinsic limitations. First, they simulate grains as cubes or regular shapes. Studies by Kidner *et al*.[13] varying the imposed shape of simulated grains have shown that the cubic grain approximation is only applicable to micrometer-sized grains in ceramics and is no longer valid for nano-sized grains. Second, the pathway the models predict through the sample is dependent on the nested cube connections. As this is limited to six per pixel it cannot fully represent the complex conduction paths that are possible in a ceramics with irregular grain shapes.

An alternative approach to simulate the electrical response of an electroceramic is to use effective medium theory.[14, 15] This is based on Maxwell's concept of an effective medium describing the ceramic as a collection of similarly shaped, coated spheres. Each one represents a grain core, which is coated with a shell to describe the grain boundary. The spheres are packed, either filling or partially filling an effective medium, which is then given the material properties of the grain core and grain boundary. The system can then be solved for the conductivity as a function of volume fraction of the grain cores. This method has been successfully used to determine the electrical response of heterogeneous ceramics and to extract values for grain-core and grain-boundary conductivity and permittivity. One major drawback is the model does not resemble the real microstructure of a ceramic.[13]

Simulating the IS response of an electroceramic using finite element modeling can overcome the deficiencies associated with the methods discussed earlier. The finite element method (FEM) is a powerful tool widely used for numerical modeling and simulation in many areas of science and engineering. The idea of using the FEM to model IS data is not new, and has been successful in describing highly resistive grain boundaries[16-18]; however, this approach has been limited to two-dimensional models and a comprehensive treatment of granular three-dimensional samples is still lacking.

Here, we present a finite element package, developed in-house to simulate IS data for electroceramics using realistic microstructures. We then apply this FEM to various electrical microstructures to simulate IS spectra. Using an appropriate equivalent circuit to extract the resistance and capacitance of the electroactive grain and grain-boundary components, we apply a bricklayer method to estimate the corresponding conductivity and permittivity of the components. These values are then compared with the input data of the simulations to highlight the appropriate and limiting conditions of this method for data analysis.