# Stochastic 3D Modeling of Three-Phase Microstructures for Predicting Transport Properties: A Case Study

- 143 Downloads

## Abstract

We compare two conceptually different stochastic microstructure models, i.e., a graph-based model and a pluri-Gaussian model, that have been introduced to model the transport properties of three-phase microstructures occurring, e.g., in solid oxide fuel cell electrodes. Besides comparing both models, we present new results regarding the relationship between model parameters and certain microstructure characteristics. In particular, an analytical expression is obtained for the expected length of triple phase boundary per unit volume in the pluri-Gaussian model. As a case study, we consider 3D image data which show a representative cutout of a solid oxide fuel cell anode obtained by FIB-SEM tomography. The two models are fitted to image data and compared in terms of morphological characteristics (like mean geodesic tortuosity and constrictivity) as well as in terms of effective transport properties. The Stokes flow in the pore phase and effective conductivities in the solid phases are computed numerically for realizations of the two models as well as for the 3D image data using Fourier methods. The local and effective physical responses of the model realizations are compared to those obtained from 3D image data. Finally, we assess the accuracy of the two methods to predict permeability as well as electronic and ionic conductivities of the anode.

## Keywords

Stochastic microstructure modeling Effective conductivity Permeability Solid oxide fuel cells 3D image data## Nomenclature

- \(\beta _1, \beta _2, \beta _3\)
Constrictivities of the three phases

- \(\varepsilon _1, \varepsilon _2, \varepsilon _3\)
Volume fractions of the three phases

- \(\widehat{\varepsilon }\)
Estimator for the volume fraction of a stationary random closed sets

- \(\widehat{\varepsilon }^{\star }\)
Estimator for the volume fraction in the graph-based microstructure model

- \(\gamma _1, \gamma _2, \gamma _3\)
Parameters of the distance measure used for the graph-based microstructure model

- \(\varGamma \)
Pore–solid interface

- \(\kappa \,{(}\mathrm {m}^2{)}\)
Permeability

- \(\widehat{\kappa }~{(}\mathrm {m}^2{)}\)
Geometrical predictor of permeability

- \(\lambda _1, \lambda _2, \lambda _3~{(}\mathrm {m}^{-3}{)}\)
Intensities of the Poisson point processes

- \(\mu _f~{(}{{\mathrm {kg}}}\cdot \mathrm {m}^{-1} \cdot \mathrm {s}^{-1}{)}\)
Viscosity of an incompressible Newtonian fluid

- \(\nu _3\)
3-dimensional Lebesgue measure

- \(\varPhi \)
probability distribution function of the standard normal distribution

- \(\phi ~{(}{\mathrm {kg}}\cdot \mathrm {m}^2 \cdot \mathrm {s}^{-3} \cdot \mathrm {A}^{-1}{)}\)
Electrical potential (or ionic concentration)

- \(\rho _Y, \rho _Z\)
Covariance functions of the Gaussian random fields

*Y*and*Z*- \(\sigma ~{(}{\mathrm {kg}}^{-1}\cdot \mathrm {m}^{-2} \cdot \mathrm {s}^{3} \cdot \mathrm {A}^{2}{)}\)
Effective conductivity

- \(\sigma _{\mathrm {sol}}~{(}{\mathrm {kg}}^{-1}\cdot \mathrm {m}^{-2} \cdot \mathrm {s}^{3} \cdot \mathrm {A}^{2}{)}\)
Intrinsic conductivity

- \(\tau _1, \tau _2, \tau _3\)
Mean geodesic tortuosities of the three phases

- \(\varTheta \)
Parameter space

- \(\theta _{ij}~{(}\mathrm {m}^{-1}{)}\)
Parameters for modeling two-point coverage probability functions, \(i,j \in \lbrace 1,2 \rbrace \)

- \(\vartheta _0~{(}\mathrm {m}^{-2}{)}, \vartheta _1~{(}\mathrm {m}^{-1}{)}\)
Intensities of point processes related to the triple phase boundary

- \(\varXi _1, \varXi _2, \varXi _3\)
Random closed sets denoting the three different phases

- \(b_1, b_2, b_3\)
Parameters of the beta-skeletons

- \(C_1, C_2, C_3\)
Two-point coverage probability functions of the three phases

*d*(*x*,*A*)Euclidean distance between a point \(x \in \mathbb {R}^3\) and a set \(A \subset \mathbb {R}^3\)

- \(d_\gamma (x,A)\)
Distance measure with parameter \(\gamma \) between a point \(x \in \mathbb {R}^3\) and a set \(A \subset \mathbb {R}^3\)

- \(\mathbf {E}~{(}{\mathrm {kg}}\cdot \mathrm {m} \cdot \mathrm {s}^{-3} \cdot \mathrm {A}^{-1}{)}\)
Electrical vector field (or opposite gradient of ionic concentration)

- \(\mathbf {G}~{(}{\mathrm {kg}}\cdot \mathrm {m}^{-2} \cdot \mathrm {s}^{-2}{)}\)
Macroscopic pressure gradient

- \(\mathcal {G}_1, \mathcal {G}_2, \mathcal {G}_3\)
Beta-skeletons of the three phases

*h*Function used to estimate the volume fraction in the graph-based model

- \(\mathcal {H}_{k}\)
*k*-dimensional Hausdorff measure for \(k \in \lbrace 1,2,3 \rbrace \)- \(\mathbf {J}~{(}A{)}\)
Electrical current (or particle current)

- \(L_{\mathrm {TPB}}~{(}{\mathrm {m}}^{-2}{)}\)
Expected length of the triple phase boundary per unit volume

*M**M*-factor, i.e., the ratio of effective and intrinsic conductivity- \(\widehat{M}\)
Geometrical predictor of the

*M*-factor*o*Origin in the 3-dimensional Euclidean space

- \(p~{(}{\mathrm {kg}}\cdot \mathrm {m}^{-1} \cdot \mathrm {s}^{-2}{)}\)
Pressure field

- \(\mathbb {R}^3\)
3-dimensional Euclidean space

- \(R^2\)
Coefficient of determination

- \(r_{\mathrm {max}}~{(}\mathrm {m}{)}\)
Median of the volume equivalent particle radius distribution

- \(r_{\mathrm {min}}~{(}\mathrm {m}{)}\)
Median radius of the characteristic bottleneck in a microstructure

- \({\mathcal {S}}\)
Conductive phase

- \(S_1, S_2, S_3\)
Specific surface area of the three phases

- \(s_{\mathrm {GBM}}~{(}\mathrm {m}{)}\)
Smoothing parameter of the graph-based microstructure model

- \(s_{\mathrm {PGM}}~{(}\mathrm {m}{)}\)
Smoothing parameter of the pluri-Gaussian microstructure model

- \(u_Y, u_Z\)
Thresholds defining the excursion sets of the Gaussian random fields

*Y*and*Z*- \(\mathbf {v}~{(}\mathrm {m} \cdot \mathrm {s}^{-1}{)}\)
Velocity of an incompressible Newtonian fluid

- \(X_1, X_2, X_3\)
Homogeneous Poisson point processes

*Y*,*Z*Gaussian random fields

- \(\varDelta \)
Laplacian operator

- \(\nabla \)
Gradient operator

- \(\partial A\)
Boundary of a set \(A \subset \mathbb {R}^3\)

## Notes

## Supplementary material

## References

- Abdallah, B., Willot, F., Jeulin, D.: Stokes flow through a Boolean model of spheres: representative volume element. Transp. Porous Media
**109**(3), 711–726 (2015)CrossRefGoogle Scholar - Abdallah, B., Willot, F., Jeulin, D.: Morphological modelling of three-phase microstructures of anode layers using SEM images. J. Microsc.
**263**(1), 51–63 (2016)CrossRefGoogle Scholar - Adler, R.J.: The Geometry of Random Fields. Wiley, Chichester (1981)Google Scholar
- Ballani, F., Kabluchko, Z., Schlather, M.: Random marked sets. Adv. Appl. Probab.
**44**(3), 603–616 (2012)CrossRefGoogle Scholar - Bertei, A., Chueh, C.C., Pharoah, J.G., Nicolella, C.: Modified collective rearrangement sphere-assembly algorithm for random packings of nonspherical particles: towards engineering applications. Powder Technol.
**253**, 311–324 (2014)CrossRefGoogle Scholar - Brabec, C.J., Heeney, M., McCulloch, I., Nelson, J.: Influence of blend microstructure on bulk heterojunction organic photovoltaic performance. Chem. Soc. Rev.
**40**(3), 1185–1199 (2011)CrossRefGoogle Scholar - Cai, Q., Adjiman, C.S., Brandon, N.P.: Modelling the 3D microstructure and performance of solid oxide fuel cell electrodes: computational parameters. Electrochim. Acta
**56**(16), 5804–5814 (2011)CrossRefGoogle Scholar - Chiu, S.N., Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry and Its Applications, 3rd edn. Wiley, Chichester (2013)CrossRefGoogle Scholar
- Hirsch, C., Neuhäuser, D., Schmidt, V.: Connectivity of random geometric graphs related to minimal spanning forests. Adv. Appl. Probab.
**45**(1), 20–36 (2013)CrossRefGoogle Scholar - Holzer, L., Pecho, O., Schumacher, J., Marmet, P., Stenzel, O., Büchi, F.N., Lamibrac, A., Münch, B.: Microstructure-property relationships in a gas diffusion layer (GDL) for polymer electrolyte fuel cells, part I: effect of compression and anisotropy of dry GDL. Electrochim. Acta
**227**, 419–434 (2017)CrossRefGoogle Scholar - Holzer, L., Wiedenmann, D., Münch, B., Keller, L., Prestat, M., Gasser, P., Robertson, I., Grobéty, B.: The influence of constrictivity on the effective transport properties of porous layers in electrolysis and fuel cells. J. Mater. Sci.
**48**, 2934–2952 (2013)CrossRefGoogle Scholar - Kanit, T., Forest, S., Galliet, I., Mounoury, V., Jeulin, D.: Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Solids Struct.
**40**, 3647–3679 (2003)CrossRefGoogle Scholar - Kenney, B., Valdmanis, M., Baker, C., Pharoah, J.G., Karan, K.: Computation of tpb length, surface area and pore size from numerical reconstruction of composite solid oxide fuel cell electrodes. J. Power Sour.
**189**(2), 1051–1059 (2009)CrossRefGoogle Scholar - Kirkpatrick, D.R., Radke, J.D.: A framework for computational morphology. In: Toussaint, G.T. (ed.) Computational Geometry, pp. 217–248. North-Holland, Amsterdam (1985)CrossRefGoogle Scholar
- Lantuéjoul, C.: Geostatistical Simulation: Models and Algorithms. Springer, Berlin (2013)Google Scholar
- Matheron, G.: Random Sets and Integral Geometry. Wiley, New York (1975)Google Scholar
- MATLAB 2015b, The MathWorks. www.matlab.com (2015)
- Mayer, J., Schmidt, V., Schweiggert, F.: A unified simulation framework for spatial stochastic models. Simulation Modelling Practice and Theory
**12**(5), 307–326 (2004)CrossRefGoogle Scholar - Molchanov, I.: Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley, Chichester (1997)Google Scholar
- Møller, J., Waagepetersen, R.P.: Statistical Inference and Simulation for Spatial Point Processes. Chapman & Hall/CRC, Boca Raton (2004)Google Scholar
- Moulinec, H., Suquet, P.: A fast numerical method for computing the linear and non linear mechanical properties of the composites. Comptes Rendus de l’Académie des Sciences Série
**II**(318), 1417–1423 (1994)Google Scholar - Moussaoui, H., Laurencin, J., Gavet, Y., Delette, G., Hubert, M., Cloetens, P., Le Bihan, T., Debayle, J.: Stochastic geometrical modeling of solid oxide cells electrodes validated on 3D reconstructions. Comput. Mater. Sci.
**143**, 262–276 (2018)CrossRefGoogle Scholar - Nelder, J.A., Mead, R.: A simplex method for function minimization. Comput. J.
**7**, 308–313 (1965)CrossRefGoogle Scholar - Neumann, M., Hirsch, C., Staněk, J., Beneš, V., Schmidt, V.: Estimation of geodesic tortuosity and constrictivity in stationary random closed sets. Scand. J. Stat. (2019). https://doi.org/10.1111/sjos.12375
- Neumann, M., Staněk, J., Pecho, O.M., Holzer, L., Beneš, V., Schmidt, V.: Stochastic 3D modeling of complex three-phase microstructures in SOFC-electrodes with completely connected phases. Comput. Mater. Sci.
**118**, 353–364 (2016)CrossRefGoogle Scholar - Ohser, J., Schladitz, K.: 3D Images of Materials Structures: Processing and Analysis. Wiley, Weinheim (2009)CrossRefGoogle Scholar
- Pecho, O.M., Stenzel, O., Gasser, P., Neumann, M., Schmidt, V., Hocker, T., Flatt, R.J., Holzer, L.: 3D microstructure effects in Ni-YSZ anodes: prediction of effective transport properties and optimization of redox-stability. Materials
**8**(9), 5554–5585 (2015)CrossRefGoogle Scholar - Prakash, B.S., Kumar, S.S., Aruna, S.T.: Properties and development of Ni/YSZ as an anode material in solid oxide fuel cell: a review. Renew. Sustain. Energy Rev.
**36**, 149–179 (2014)CrossRefGoogle Scholar - Scholz, C., Wirner, F., Klatt, M.A., Hirneise, D., Schröder-Turk, G.E., Mecke, K., Bechinger, C.: Direct relations between morphology and transport in Boolean models. Phys. Rev. E
**92**(4), 043023 (2015)CrossRefGoogle Scholar - Stenzel, O., Neumann, M., Pecho, O.M., Holzer, L., Schmidt, V.: Big data for microstructure-property relationships: a case study of predicting effective conductivities. AIChE J.
**63**(9), 4224–4232 (2017)CrossRefGoogle Scholar - Stenzel, O., Pecho, O.M., Holzer, L., Neumann, M., Schmidt, V.: Predicting effective conductivities based on geometric microstructure characteristics. AIChE J.
**62**, 1834–1843 (2016)CrossRefGoogle Scholar - Suzuki, T., Hasan, Z., Funahashi, Y., Yamaguchi, T., Fujishiro, Y., Awano, M.: Impact of anode microstructure on solid oxide fuel cells. Science
**325**(5942), 852–855 (2009)CrossRefGoogle Scholar - Tjaden, B., Brett, D.J.L., Shearing, P.R.: Tortuosity in electrochemical devices: a review of calculation approaches. Int. Mater. Rev.
**63**(2), 47–67 (2018)CrossRefGoogle Scholar - Torquato, S.: Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer, New York (2013)Google Scholar
- Westhoff, D., Van Franeker, J.J., Brereton, T., Kroese, D.P., Janssen, R.A.J., Schmidt, V.: Stochastic modeling and predictive simulations for the microstructure of organic semiconductor films processed with different spin coating velocities. Modell. Simul. Mater. Sci. Eng.
**23**(4), 045003 (2015)CrossRefGoogle Scholar - Wiegmann, A.: Computation of the permeability of porous materials from their microstructure by FFF-Stokes (2007). http://kluedo.ub.uni-kl.de/files/1984/bericht129.pdf. Accessed 22 July 2015
- Willot, F., Abdallah, B., Pellegrini, Y.P.: Fourier-based schemes with modified green operator for computing the electrical response of heterogeneous media with accurate local fields. Int. J. Numer. Methods Eng.
**98**, 518–533 (2014)CrossRefGoogle Scholar