The polygonal finite element method (PFEM) is proposed as a fast and accurate technique to simulate the impedance spectroscopy (IS) of polycrystalline materials. While conventional finite element method (FEM) requires explicit meshing of the grains and grain boundaries, in PFEM each region can be treated as an element. We demonstrate that the number of degrees of freedom in PFEM can be lower by a factor of 30 when compared to FEM, thus speeding up simulations by a factor of 3.5. A simple example demonstrates the use of PFEM to generate IS on samples with various grain boundary widths. © 2022 The Author(s)

Natarajan, Sundararajan

Ooi, Ean Tat

Swaminathan, Narasimhan

English

Federation ResearchOnline

Journal of The ElectrochemicalSociety     OPEN ACCESSCommunication—A Fast and Accurate Numerical Technique forImpedance Spectroscopy of MicrostructuresTo cite this article: Narasimhan Swaminathan et al 2022 J. Electrochem. Soc. 169 020543 View the article online for updates and enhancements.This content was downloaded from IP address 141.132.5.10 on 09/03/2023 at 00:55Communication—A Fast and Accurate Numerical Technique forImpedance Spectroscopy of MicrostructuresNarasimhan Swaminathan,1,z Sundararajan Natarajan,1 and Ean Tat Ooi21Department of Mechanical Engineering, Indian Institute of Technology, Madras, Chennai, 600036, India2School of Engineering, Information Technology and Physical Sciences, Federation University, Ballarat, AustraliaThe polygonal finite element method (PFEM) is proposed as a fast and accurate technique to simulate the impedance spectroscopy(IS) of polycrystalline materials. While conventional finite element method (FEM) requires explicit meshing of the grains and grainboundaries, in PFEM each region can be treated as an element. We demonstrate that the number of degrees of freedom in PFEMcan be lower by a factor of 30 when compared to FEM, thus speeding up simulations by a factor of 3.5. A simple exampledemonstrates the use of PFEM to generate IS on samples with various grain boundary widths.© 2022 The Author(s). Published on behalf of The Electrochemical Society by IOP Publishing Limited. This is an open accessarticle distributed under the terms of the Creative Commons Attribution Non-Commercial No Derivatives 4.0 License (CC BY-NC-ND, http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial reuse, distribution, and reproductionin any medium, provided the original work is not changed in any way and is properly cited. For permission for commercial reuse,please email: permissions@ioppublishing.org. [DOI: 10.1149/1945-7111/ac51a2]Manuscript submitted November 6, 2021; revised manuscript received January 24, 2022. Published February 16, 2022.Supplementary material for this article is available onlineImpedance spectroscopy (IS) is an experimental technique formaterials characterization. IS is used to quantify the contributions ofvarious constituents of a given heterogeneous system to its electricalbehavior1 and to understand mechanisms of surface reactions,kinetics of reactions and mass transport.2 In materials science thecontributions of the grain (bulk), grain boundaries, precipitates orother second phase particles, to the overall electrical response of agiven polycrystalline sample3–5 is often determined using IS. TheNyquist plot generated can provide useful information on the overallresistance or capacitance values. With some understanding of theactual physical processes that occur in the system, meaningfulequivalent circuits are generated that reproduce the Nyquist plot.When the electrical response of the system can be idealized as anequivalent circuit with a resistor and a capacitor in parallel, theNyquist plot is a semi-circle. When such an idealization is notpossible, the Nyquist plot may show additional features.6In the context of solid electrolytes, bricklayer models have beenused to quantify the conductivities of the grain boundaries, grain-bulk and the role of electrode-current collectors.7,8 These modelsneglect actual grain shapes and other inhomogeneities and are usefulonly for micro-sized grains and homogeneous grain boundary types.9Molecular dynamics simulations can predict the grain-core andgrain-boundary conductivities of ion conducting electroceramics.10,11These conductivities are often anisotropic and depend on temperature ordoping. Grain boundary conductivities also depend on the type (high-angle vs low-angle)12 and other impurities which may segregate there.Hence, the overall response of the material depends strongly on themicrostructure. It has now become necessary to predict the averageelectrical response of polycrystalline samples when conductivities ofvarious elements of the microstructure are known. This capability willenable design of electroceramics with microstructural features such asgrain orientations, grain boundary thickness and types, which canprovide favorable electrical response depending on the application.The favorable microstructure may have a certain grain size distribution,or orientation. The grain-widths, grain-type or other dopants may alsocontribute to the average response of the material.13 In such cases,obtaining IS data (experimentally) by varying each design parameter isdifficult.Computational methods can be used to obtain IS and evaluateaverage material responses. Conventional finite element method (FEM)has been used to solve spatio-temporal Maxwell’s equations to generatethe IS of polycrystalline samples.14 Reference 15 uses FEM to predictthe current density distribution within the sample to identify regionscontributing to high impedance. Similar ideas were used in16 tounderstand the role of porosity on the electrical properties of BaTiO3.In Ref. 17 the authors show that an increase in amplitude of roughnessof the ceramic/electrode interface in multilayer ceramic composites canresult in four times increase in electric field strength.While FEM is often the method of choice for many computationalstudies, it requires a properly constructed and conforming mesh of thedomain.18 Typical element shapes include triangles/quadrilaterals in 2Dand tetrahedra/hexahedra in 3D. Besides requiring a discretization of thegrain, a fine mesh is often needed in regions of suspected high gradients(such as grain boundaries) to accurately capture the solution. In thecontext of solving differential equations over microstructures with largegrains and comparatively smaller grain boundaries, a significant effortmight be needed to ensure that the element sizes are small enough tocapture the gradients in the current densities or the electric field. Often,this requirement increases the number of degrees of freedom. Forinstance, to model a cubic sample of dimension of 1 μm, the authorsused 250,000 tetrahedron prism elements in Ref. 14.The polygonal finite element method (PFEM),19–21 has evolvedas a versatile method to solve differential equations. One of the mainadvantages of PFEM over FEM is that elements of arbitrary shapescan be used to discretize the domain. This feature is particularlyadvantages for computational samples of polycrystalline materials asany polygonal region (like the grain) can be treated as one singleelement (polygonal). This aspect of treating one grain as a singleelement is expected to substantially decrease the number of degreesof freedom with only a marginal sacrifice in accuracy.We demonstrate the applicability of PFEM to generate IS forpolycrystalline samples. The accuracy and speed of PFEM isdemonstrated by comparing the results with FEM. We then showusing a simple example how PFEM can be used to examine thevariation of IS with grain boundary widths. A more involvedexample which uses a polycrystalline sample with grain boundarieshaving anisotropic conductivities is demonstrated in theSupplementary Material.Theoretical FormulationWe follow Ref. 14 and consider the differential form ofMaxwell’s continuity equation:ρ ρ∂∂+ ∇· = ∂∂+ ∇·( + ) = [ ]tJtJ J 0 1c dwhere J is the total current density and ρ the charge density.σ=J E,c is the differential form of Ohm’s law, =∂∂JdDtis thezE-mail: n.swaminathan@iitm.ac.inJournal of The Electrochemical Society, 2022 169 020543displacement current density, σ is the conductivity, E is the electricfield and D is the electric displacement.E and the electric potential ϕ are related by:ϕ= −∇ [ ]E 2D is related to E throughε( ) = ( ) ( ) [ ]D x t x E x t, , 3As no time dispersion is considered, the electric permittivity ε( )xis only a function of space. Using Eqs. 2–3, we transform Eq. 1 to⎡⎣⎢⎤⎦⎥σ ϕ ε ϕ−∇· ( )∇ ( ) + ( )∂∂∇ ( ) = [ ]x x t xtx t, , 0 4The corresponding weak form of Eq. 4 is: Find ϕ ∈ U such that∫∫∫δϕ δϕσ ϕδϕε ϕδϕσ ϕ∀ ∈ ∇ ( )∇ ( ) Ω+ ∇ ( ) ∂∂∇ ( ) Ω= ( )∇ ( ) Ω [ ]ΩΩ∂ΩV x x t dxtx t dx x t d, ,,, 5whereU and V are the trial and test spaces that include linear fields.In a routine way, the domain may be partitioned into non-overlapping elements, ϕh and on using shape functions Na that spanat least the linear space, we substitute the trial and test functions∑ϕ ϕ= Nha a aand ∑δϕ δϕ= N ,hb b brespectively, into Eq. 5 toobtain the space discretized weak form:∫∫∫δϕ δϕ σ ϕδϕ ε ϕδϕ σ ϕ∀ ∈ ∇ ( )∇ ( ) Ω+ ∇ ( ) ∂∂∇ ( ) Ω× ( )∇ ( ) Ω [ ]ΩΩ∂ΩV x x t dxtx t dx x t d, ,,, 6h h hh hh hIn FEM, the grains/grain boundaries must be discretized andmaterial properties (σ), should be assigned to each element. In thePFEM approach, each grain core and grain boundary regions areeach idealized as one single finite element, naturally reducing thenumber of unknowns in the problem.Polygonal finite element method—Overview.—PFEM and FEMdiffer in the manner the trial (or test) functions are constructed.PFEM is a generalization of FEM that allows for arbitrary elementshapes and sizes. The test functions in PFEM must satisfy partitionof unity, interpolation, linear completeness and non-negativity.There is no unique way to represent the trial functions with theseproperties over arbitrary polytopes.22–29 Wachspress,22 mean-valuecoordinates,24 Laplace interpolants,23,25 Harmonic coordinates,26–28are a few approaches to generate such functions (See Ref. 29).Mean-value coordinates (Floater24) are used here as they workefficiently for both convex and concave regions. The mean valuecoordinates for a point, ( )P x in an arbitrary polygon is given by:∑( ) = ( )( )[ ]=N xw xw x7aIJnJ1whereα α( ) = ( ) + ( )∨ − ∨[ ]−w xx xtan 2 tan 28II II1where n is the number of nodes in an element, xI is the coordinate ofthe point PI and αI’s are the internal angles (See Fig. 1 of Ref. 24). Naover arbitrary shaped polygons are rational functions and integrationrules and evaluation of derivatives to calculate Eq. 5 are not as clearas evidenced by growing literature on this aspect.30–33 A sub-triangulation of the polygonal domain is employed in this work, andintegrals are evaluated at Gauss-quadrature points of these triangles.Simulation DetailsAll simulations were conducted using MATLAB 2020a.Simulations to compare FEM and PFEM were conducted on quadcore Intel i7-@2.8 GHz with 16 GB RAM, with identical paralle-lization for the assembly of the stiffness matrix.Details of the microstructure.—A square domain with side l =300 nm was used to create a two-dimensional microstructure. Eachgrain was generated from circles of radius ≅ 2 nm amounting toabout 80 grains. Four grain boundary widths,ΔW = 1, 2, 3 and 4 nmwere considered for the study. The top boundary was extended by3 nm and this region was assumed to be highly conductive materialto simulate a metallic contact. The microstructure was generated bymodifying the microstructpy34 package. This microstructure is thenaltered by shrinking the grains by an appropriate amount related toFigure 1. Polycrystalline domain (a) finite element mesh with 8128 elements and 4181 nodes and (b) PFEM with 516 elements and 574 nodes including grainand grain boundary regions.Journal of The Electrochemical Society, 2022 169 020543ΔW to create grain boundary domains. The created grain boundarydomains are further meshed for FEM or PFEM.Material properties.—The conductivities of the grain core σ ,gcgrain boundary σgb and the metallic region σm were 100× 10−2 S m−1,100 × 10−6 S m−1 and 10 × 103 S m−1, respectively. The permittivityof all regions is assumed to be a constant with a relative permittivityεr = 100. The impedances were sampled in the range 0.001 Hz to1 GHz.Finite element mesh.—For FEM simulations two meshes withtriangular elements having 4,181 nodes and 16,489 nodes (Fig. 1a),were considered. For PFEM, the vertices of the grains or grainboundaries were the only nodes required (See Fig. 1b).Simulation procedure.—Five cycles of a sinusoidal Voltage (V)of magnitude 100 V were applied at the metallic end of themicrostructure, while the other sides were insulated. Ordinarydifferential equations arising from the FEM or PFEM werenumerically integrated in time using an implicit scheme with atime step Δ =ωt ,120where ω is the frequency in Hz. At each timestep, the current (I) at the metallic ends was calculated andaccumulated. For each ω, the phase difference φΔ between V andI is obtained using Fast Fourier Transforms. The imaginary (Z″) andreal (Z′) parts of the impedance are computed as Rsin(Δφ) and Rcos(Δφ), respectively, where R is the ratio of the amplitudes of V and I.The left and right sides were insulated and the potential φ at themetallic end is measured with respect to the lower side of thesample.Results and DiscussionComparison with the regular finite element method (FEM).—Figure 2a shows the Nyquist plot and the inset shows the highfrequency response. Figures S2a–S2d (available online at stacks.iop.org/JES/169/020543/mmedia) compares the current densities be-tween the two methods for two frequencies. It is very clear that theresults obtained from PFEM and that of FEM (with 16,489 nodes)match perfectly, in both high and low frequency regimes. For lowfrequencies, the coarser finite element mesh with nearly eight timesthe nodes as that of PFEM shows an inferior performance.Furthermore, the time taken for completing the simulation forPFEM (574 nodes) was ≅20 s, while that for the FEM (16,489nodes) was ≅70 s. Clearly, PFEM is an effective method to studyimpedance spectroscopy of microstructures.Examples.—To demonstrate the use of the proposed PFEM, weconsider a simple example of determining the average resistance/capacitance of a microstructure with different grain boundary widthsΔW . Figure 2b shows the Nyquist plots for four different grainboundary widths. The average resistance increases with ΔW , asexpected. A small change in ΔW enhances the resistance to a largeextent. An additional example that involves role of anisotropicconductivity of the grains and grain boundaries is provided in theSupplementary Material.ConclusionsSpeed, accuracy and robustness of the PFEM over FEM forcomputational simulation of IS is demonstrated. With the propertiesof the grain boundaries and grain cores specified, IS simulations canreveal a wealth of information on the average electrical behavior ofthe microstructure. While samples in this work were taken to be300 nm, microstructures have several hundreds to thousands ofgrains with typical dimensions in the order of several hundredmicrons and each grain being 1–10 μm in diameter. Application ofFEM to such a real microstructure with even a reasonable mesh sizewould be computationally expensive, time consuming and henceprohibitive. PFEM can treat each grain and grain boundary region asa single element and extract useful information on a realisticmicrostructure with reasonable accuracy in a fraction of the time.With such a tool, the effect of various features of the microstructurescan be quickly studied as demonstrated in this work. Such a tool willbe very effective in designing/tailoring microstructures within ashort time to suit specific applications.ORCIDNarasimhan Swaminathan https://orcid.org/0000-0001-8572-3766References1. A. Bard and L. Faulkner, Electrochemical Methods: Fundamentals andApplications (Wiley, New York, NY) (2000).2. N. O. Laschuk, E. B. Easton, and O. V. Zenkina, RSC Adv., 11, 27925 (2021).3. I. Hodge, M. Ingram, and A. West, J. Electroanal. Chem. Interf. Electrochem., 74,125 (1976).4. K. Srinivas, P. Sarah, and S. Suryanarayana, Bull. Mater. Sci., 26, 247 (2003).5. D. T. Swamy, K. E. Babu, and V. Veeraiah, Bull. Mater. Sci., 19, 1115 (2013).6. E. Abram, D. Sinclair, and A. West, J. Electroceramics, 10, 165 (2003).7. T. Van Dijk and A. Burggraaf, Phys. Status Solidi A, 63, 229 (1981).8. X. Guo and R. Waser, Prog. Mater Sci., 51, 151 (2006).9. N. Kidner, Z. Homrighaus, B. Ingram, T. Mason, and E. Garboczi,J. Electroceramics, 14, 283 (2005).10. L. Momenzadeh, I. V. Belova, and G. E. Murch, Comput. Condens. Matter, 28,e00583 (2021).11. A. R. Symington, M. Molinari, J. A. Dawson, J. M. Statham, J. Purton, P. Canepa,and S. C. Parker, J. Mater. Chem. A, 9, 6487 (2021).12. A. A. Riet, J. A. Van Orman, and D. J. Lacks, Geochim. Cosmochim. Acta, 292, 203(2021).13. E. Van Der Giessen et al., Model. Simul. Mater. Sci. Eng., 28, 043001 (2020).14. J. S. Dean, J. H. Harding, and D. C. Sinclair, J. Am. Ceram. Soc., 97, 885 (2014).15. J. P. Heath, J. S. Dean, J. H. Harding, and D. C. Sinclair, J. Am. Ceram. Soc., 98,1925 (2015).Figure 2. (a) Nyquist plot for the sample in Fig. 1 comparing FEM and PFEM results. The numbers within ( ) indicate the number of nodes in the sample (b):Nyquist plots for various grain boundary widths. 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Rimoli, SoftwareX, 12, 100595 (2020).Journal of The Electrochemical Society, 2022 169 020543

Communication-a-fast and accurate numerical technique for impedance spectroscopy of microstructures

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Communication-a-fast and accurate numerical technique for impedance spectroscopy of microstructures

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