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Simulation of the Impedance Response of Materials with More Than One Electrical Path

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Simulation of the Impedance Response of Materials with More Than One Electrical Path Simulation of the Impedance Response of Materials with More than One Electrical Path Rosario A. Gerhardt* and Youngho Jin School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA *Corresponding author: [email protected] Abstract: Materials and devices can often build up or other effects[3]. However, because contain multiple interfaces that may or may not the properties of the boundaries are often very participate in the active function of a given different than that of the bulk grains, it is useful material or device. However, they are often only to consider treating them as if they were made up evaluated at the specific frequency of application of two different phases. Therefore, concepts or in dc mode, which essentially disregards the developed to explain composite materials, such rich information that the full frequency spectrum as effective media, connectivity of phases and can contain. This causes a lot of useful percolation [4] may be able to be used to derive information, which could be used to help better understanding of the properties of understand their electrical behaviour, either for polycrystalline materials as a function of grain quality control, life prediction, reliability or basic size, defect density or the presence of inadvertent understanding, to be lost. In this article, we impurities. utilize a combination of COMSOL Multiphysics Such a modeling effort becomes even more simulations and equivalent circuit modelling to important in the case of nanocrystalline materials demonstrate the advantages of considering which have comparable volume fraction of grain broadband ac measurements as a way to develop boundaries and bulk grains due to the high more detailed understanding of materials with surface areas these materials possess[5]. Such more than one electrical path. materials have been shown to display tremendous variations in their mechanical behavior[6], electrical properties[7] and Keywords: COMSOL Multiphysics, impedance spectroscopy, polycrystalline sinterability as a function of grain size, heat materials, grain size effect, percolation treatment or other external forces[8]. The finite element model being used here 1. Introduction was first developed to represent an ordered insulator-conductor composite with a segregated network microstructure[9]. Using such a model Polycrystalline single phase materials often allows varying the grain size and grain boundary display electrical properties that are a function of thickness easily. Additionally, this also permits their grain size. One of the best ways to assigning them different properties to evaluate characterize the properties of the bulk grains and their effect on the resultant response. the grain boundaries is to use an alternating current technique, known as impedance or ® dielectric spectroscopy[1]. This method is ideal 2. Use of COMSOL Multiphysics for detecting the presence of more than one current path. The reasons for the different In this study, we use a finite element response from the grains and the grain approach to solve the electric potential in the AC boundaries is that electrically, the region around environment for an idealized two phase the grain boundaries is necessarily different due microstructure, as depicted in Figure 1. The to the presence of defects, inadvertent impurities faceted grains represent the main material phase and segregation of certain materials towards or and the boundary region has finite thickness and away from the boundary. It was originally distinct electrical properties that may or may not thought that that it was necessary to have a percolate with itself[9]. A numerical calculation completely different phase located at the of the impedance, capacitance, and related boundaries[2], but minor compositional changes quantities naturally requires the solution of the and/or the presence of defects can also give rise electric fields in the material under an applied to different current patterns due to space charge electric potential or electric current. In the case 1 Excerpt from the Proceedings of the 2015 COMSOL Conference in Boston of impedance measurements the potential/current is usually time harmonic[10]. Maxwell’s equations, in terms of electric potential, can be simplified as equation (1), assuming no externally generated current. ∂ +⋅∇ εεσ V =∇⋅ 0 (1) r 0 ∂t Equation (1) can be rewritten as equation (2) by replacing time derivative and electric Figure 1. Schematic of the geometric model used for potential by /1 jω operator and potential phasor, the simulations. The parameter α was used to modify respectively, since time harmonic analysis is the size of the grains and width of boundaries as well appropriate for the impedance calculations as the electrical conductivities of the two regions. σ ~ ~~ 3. Impedance spectroscopy and ε '−⋅∇ Vj ε ∇⋅∇=∇⋅ V = 0 (2) ω equivalent circuits Figure 2 illustrates simulated equivalent where ε = − j /' ωσε is complex permittivity and circuit complex impedance spectra for a series of ~ V is the potential phasor amplitude. The electric cases where the grain boundary resistance, Rgb, is field can be obtained by evaluation the gradient greater, equal to or smaller than the bulk grains, of potential phasor amplitude[10]. which have resistance Rbulk. The simulations 3 utilized Cgb > 10 Cbulk . The equivalent circuit is The steps used in the simulation include: shown below. (1) Selecting time harmonic-electric currents Since most materials can be represented by a solver in the AC/DC module in COMSOL combination of a resistance R and a capacitance Multiphysics®, (2) Defining the electrical C in parallel, it has long been accepted that two properties inside the grains and the grain distinguishable electrical paths in series can boundaries, (3) Solving and finding the electric produce two semicircles if the RC time constants field distributions and (4) Using post-processing are separable [1,2]. It is clear that changes in the capabilities in COMSOL Multiphysics® to conductivity of the main grains may or may not determine the impedance response.). An electric be detected, depending on whether the grain shielding boundary condition was applied to boundaries are more or less conducting than the demonstrate thin grain boundary domains which matrix grains. can reduce the number of mesh elements. A forced voltage port electrode was applied to the top edge as a Dirichlet boundary condition, while a ground electrode was applied to the bottom edge of the geometric model. The electrical insulation was applied to the side edges as a Neumann boundary condition Jn =⋅ 0 . An electric shielding boundary condition was applied to demonstrate thin grain boundary domains, which can reduce the number of mesh elements. 2 Excerpt from the Proceedings of the 2015 COMSOL Conference in Boston -7.5e6 FitResult -2.0e6 FitResult One can see in Figure 3 that the overall impedance began to decrease in this case. -1.5e6 R =R -5.0e6 R >R gb bulk gb bulk Furthermore, once 1 S/m is reached, the Z'' Z'' -1.0e6 contributions from the grain boundaries cannot -2.5e6 be distinguished separately. -5.0e5 It is to be noted that the trends observed here do not follow the well accepted 2 parallel RC 0 0 0 2.5e6 5.0e6 7.5e6 0 5.0e5 1.0e6 1.5e6 2.0e6 Z' Z' circuit in series described earlier. The reasons -1.5e7 FitResult -1.5e8 for this discrepancy have to do with the fact that FitResult R <R the percolated path contains conductive channels gb bulk R >>R -1.0e7 bulk gb -1.0e8 that occur in the direction of the field and or viceversa! Z'' perpendicular to it so that neither semicircle Z'' -5.0e6 observed in the impedance spectrum is directly -5.0e7 related to the initial values of Rbulk, Rgb, Cbulk or C . Transformation of the impedance into other 0 gb 0 5.0e6 1.0e7 1.5e7 0 Z' 0 5.0e7 1.0e8 1.5e8 dielectric functions such as admittance, electric Z' modulus or permittivity would help to separate these different effects[11], but this is beyond the Figure 2. Simulated complex impedance spectra when the electrical response of the boundaries is scope of this short paper. equal, larger or smaller than the main phase grains. Modified from ref[1]. 4. Results The FEA results were obtained for four different percolated cases with 0.15 vol% occupied by the grain boundaries and one unpercolated case containing the same fraction of grain boundaries. The majority of the simulations were obtained by keeping the size of the bulk grains Figure 3. Complex impedance graph for a system µ constant at 100 µm and 60 nm for the grain with 100 m size grains and 60 nm thick grain boundaries that percolate. Effect of changing the boundary width. Some evaluations of changing grain boundary conductivity is demonstrated. the bulk grain size and the width of the grain boundaries were also conducted. In contrast, changing the bulk grain 4.1 Effect of bulk grain and grain properties, while keeping the grain boundary boundary conductivity properties constant at 10-3 S/m, causes much greater changes to be observed. In order to more Figure 3 displays the impedance response for easily demonstrate the several order of the case where the bulk grain conductivity was magnitude changes, we show the impedance kept constant at 10-3 S/m and the conductivity of magnitude vector in Figure 4(a) and the phase the grain boundaries was allowed to vary from angle in Figure 4(b). These are frequency 10-5 S/m up to 1 S/m. For such a situation, we explicit plots that display how a four order of found that the impedance is mostly dominated by magnitude change in the bulk conductivity from -4 the bulk grain properties until the conductivity of σbulk = 10 S/m to 1 S/m causes a four order of the grain boundaries exceeded the bulk grain magnitude change in the impedance vector value.
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