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Composite materials as electrolytes for solid oxide fuel cells: simulation of microstructure and electrical properties
Solid State Ionics 152 – 153 (2002) 509 – 515 www.elsevier.com/locate/ssi
Composite materials as electrolytes for solid oxide fuel cells: simulation of microstructure and electrical properties G. Dotelli a,*, I. Natali Sora b, C. Schmid c, C.M. Mari d a
Dipartimento di Chimica, Materiali e Ingegneria Chimica ‘‘Giulio Natta,’’ Politecnico di Milano, P.za L. da Vinci 32, 20133 Milan, Italy b Dipartimento d’Ingegneria Meccanica, Universita` di Brescia, via Branze 38, 25123 Brescia, Italy c Dipartimento d’Ingegneria dei Materiali e Chimica Applicata, Universita` di Trieste, via Valerio 2, 34127 Trieste, Italy d Dipartimento di Scienza dei Materiali, Universita` degli Studi di Milano-Bicocca, via Cozzi 53, 20125 Milan, Italy Accepted 14 February 2002
Abstract The simulated and the experimental electrical conductivity of ionic conductor composites (Al2O3/yttria-stabilized zirconia) either containing different amounts (5 and 50 wt.%) of alumina or having the same amount of insulating phase (10 wt.%) with different grain sizes are presented and compared. A digital image-based modelling procedure to simulate the electrical behaviour of the composites was used here. The method works, generating, by the Voronoi tessellation technique in which a genetic algorithm is used, two-phase polyhedral microstructures and then converting them into a random electrical network. The real and imaginary part of the electrical network impedance was computed by the transfer matrix method. The model is able to reproduce the experimental results well. D 2002 Elsevier Science B.V. All rights reserved. Keywords: SOFC; Composite materials; Electrical properties; Simulation; Random electrical networks
1. Introduction Yttria (8 wt.%)-stabilized cubic zirconia (YSZ), as it is well known, is the most widely used solid electrolyte in high-temperature solid oxide fuel cells (SOFCs) [1]. Such material suffers from relatively poor bending strength that reduces its performances, particularly in planar SOFCs [2– 7]. The addition of a-Al2O3, as a secondary phase, was suggested to overcome this drawback. In fact, this oxide hinders the grain growth process and allows the formation of a fine-grained *
Corresponding author. Tel.: +39-2-2399-3232; fax: +39-27063-8173. E-mail address: [email protected] (G. Dotelli).
ceramic, whose mechanical properties are enhanced. Because the presence of alumina particles decreases the electrical conductivity, a proper amount of this oxide has to be added to obtain a good compromise between electrical conductivity and mechanical properties. For such reason, the electrical properties of YSZ/Al2O3 composites as a function of the alumina amount were investigated [8– 15]. Unfortunately, the experimental data are slightly conflicting and inhomogenous. The lack of a systematic analysis of the microstructures on pure YSZ and on YSZ/Al2O3 composites makes the comparison among the results quite difficult and the explanation of the behaviour rather tricky. For this reason, a simulation procedure able to take into account the influence of the microstructure
0167-2738/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 3 8 ( 0 2 ) 0 0 3 8 0 - 6
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G. Dotelli et al. / Solid State Ionics 152 – 153 (2002) 509–515
on the electrical properties of pure YSZ and Al2O3/ 8YSZ composites might serve as a tool to better understand the ionic conduction behaviour of such materials. The problem of simulating the electrical behaviour of polycrystalline materials (both pure and composites) can be managed in a rigorous way by solving the classical Laplace’s equation [16]. In principle, the resulting differential problem is numerically solved with any desired accuracy, for instance by the finite element method (FEM). However, this route is not yet fully exploited because of its complexity. In the field of ionic conductors, this technique was recently used to outline the grain boundary influence on impedance spectra of polycrystalline solid electrolytes [17 –21].
The discrete medium models (DMMs) represent a possible alternative to the FEM approach. They simulate the electrical behaviour converting the material into a 2D or 3D network of discrete electrical elements and solving a set of algebraic equations. The main differences among these DMMs can be ascribed either to the way of arranging the discrete electrical elements inside the electrical networks or to the choice of the representative circuits. Originally used to calculate the parameters of the percolation equation [22,23], the DMMs were later applied as self-standing approaches to simulate the electrical properties of dispersed ionic conductors [24 – 28], of cement-based materials [29], of Ni –YSZ cermets [30 – 32] and of ionic conductor composites [33 –38].
Fig. 1. Schematic representation of a 2D Voronoi tessellation. Pixels labelled with even and odd integers stand for phases A and B, respectively. The dots are used to evidence the grain boundaries.
G. Dotelli et al. / Solid State Ionics 152 – 153 (2002) 509–515
The aim of the present paper is to test the forecasting qualities of a digital image-based DMM by comparing theoretical and experimental results performed on Al2O3/8YSZ composites. The original version of the computational protocol that gave some preliminary encouraging results is reported elsewhere [34,38]. An improvement concerning the microstructure generation is here used.
2. Experimental The samples with alumina (Sumitomo Chemical) contents up to 50 wt.% were prepared as previously reported [10,14]. The samples containing fixed amounts of alumina of different size [15] were prepared by mixing 10 wt.% Al2O3 powders (ALCOA 17) and fully stabilised (7.54 mol% Y2O3) zirconia powder (YSZ) (Unitec Ceramics, England, frequency particle size parameters d50=0.65 Am and d95=1.33 Am). The microstructural features were investigated by SEM and image analysis. Electrical measurements by impedance spectroscopy technique were carried out in the temperature range 350– 800 jC in air.
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lating the nuclei distance; NA and NB nuclei of phases A and B are randomly distributed in a fixed area (or volume), then they are grown at two different rates. Adopting here the pixel as reference unit of volume, the result of the growth process is a pseudo 3D digital image (having just one pixel in thickness) of the simulated microstructure that can be easily converted into a 2D array of figures. All the pixels of each grain are labelled with the same integer (see Fig. 1). This procedure makes easy the statistical analysis of grain geometrical properties: area and perimeter (in 2D) or volume and surface (in 3D). The algorithm that converts the digital image into the numerical matrix is designed in such a way that the grain boundaries and the bulks are clearly distinguished. Each pixel is converted into a discrete electrical circuit, so that the overall image becomes an electrical network (EN). The discrete electrical circuit is constituted of six impedances all connected to the centre of the pixel (see Fig. 2). Between two adjacent pixels, the connection of the impedances is ideal. As a result, one obtains a three-dimensional EN with a node at the centre of each pixel.
3. Modelling and simulation In the original model [34,36 – 38], the computer simulation starts by reproducing the microstructure of polycrystalline materials using the Voronoi tessellation technique [39]. The nuclei of N grains were randomly distributed in a surface or in a volume (2D or 3D) and successively the Voronoi hulls or polygons increased in size until the space was completely filled. In particular, the simulated structure is just the same as that resulting from a growth process that takes place from randomly distributed latent nuclei at constant speed. Because the representation of two-phase polyhedral systems is complicated in the previous version of the model, the task was carried out in two steps: a single-phase polyhedral structure was generated by the Voronoi tessellation, then spherical particles (a-Al2O3) were inserted among the convex hulls [34]. The approach is now refined and a true two-phase polyhedral microstructure is realised. The Voronoi tessellation [40] is here obtained by a genetic algorithm, instead of calcu-
Fig. 2. Conversion of a pixel into an electrical circuit. Each parallelepiped represents any possible combination of discrete electrical element (i.e., single resistance or capacitance, resistance – capacitance parallel circuit and so on).
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Different electrical circuits are assigned to bulk and grain boundary pixel as well as the value of single electrical element (resistance and capacitance). In particular, insulating behaviour was assigned to the pixels describing the Al2O3 particles (experimental alumina resistivity value) while the actual experimental values carried out on pure YSZ were given to the pixels representing the bulk and grain boundary of YSZ, respectively. The real and imaginary part of the EN impedance was computed by the transfer matrix method, as in statistical mechanics. In the present case, nonlinear matrix recursion relations are involved [41]. Such a procedure has the advantage of producing exact solutions (in contrast with relaxation methods which may have convergence problems) [42] and of requiring much less storage memory than systematic node elimination process, which, by the way, cannot be practically applied in 3D systems [43]. However, the transfer-matrix approach is more time-consuming than relaxation methods, particularly for very large networks.
4. Results and discussion The microstructural simulations (see Fig. 1) were performed creating digital images (150150 pixel2) and the electrical computations were carried out in its core (130130 pixel2) in order to avoid any spurious effects due to the finite size of the representation. The thickness of the microstructure was assumed to be one pixel. The amount of each individual phase, their average grain size and the conversion factor between pixel and microns are required to simulate the microstructure simulation of the pure YSZ and of the composites. As one can remark, the model is not only able to well mimic the polycrystalline microstructure but also to reproduce some detailed features. A typical simulated and experimental (obtained by quantitative image analysis of SEM micrographs) statistical analysis (performed on 10 wt.% alumina composite) is reported in Fig. 3. The slight discrepancies observed in the frequency distribution curves are probably due to the lack of well monodispersed alumina powders experimentally used. In principle, the genetic algorithm is also able to take into account more complex
Fig. 3. Simulated (E) and experimental (n) statistical analysis performed on 10 wt.% alumina composite: grain size distribution of YSZ (a) and Al2O3 (b).
particles distribution, but at the present less attention was paid to this aspect. The electrical forecasting qualities of the model were tested evaluating the resistivity of YSZ and of composites either containing different amounts (5 and 50 wt.%) of alumina or having the same amount of insulating phase (10 wt.%) with different grain sizes (1.0, 1.4 and 1.7 Am). Calculations were performed using the experimental bulk and grain boundary conductivity values of pure YSZ [10,14] at different temperatures and the constant resistivity value (1015 V cm) [44] of Al2O3 at 25 jC. The two contributions, i.e. bulk and grain boundary, were obtained by equivalent circuit analysis of the two arcs in the impedance spectrum, whenever possible, otherwise by difference, if only one arc was present (for instance at high temperature).
G. Dotelli et al. / Solid State Ionics 152 – 153 (2002) 509–515
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The calculated and the experimental total (bulk+ grain boundary) conductivity of YSZ and of composites with different amounts of alumina are reported as a function of 1/T in Fig. 4. As one can observe, a very good agreement exists between experimental and simulated electrical properties in the case of YSZ. This result attests the reliability of the model. The comparison between the two sets of data of each composite is good, particularly at the higher temperatures. Where the difference between experimental and simulated values are larger (lower temperatures) the discrepancy, however, is never greater than about 50%. Such behaviour is probably due to an underestimation of the grain boundary contribution. In fact, even if the different total grain boundary extension of each composition was considered in the simulations, the agreement between experimental and calculated values is not completely satisfactory, especially where the grain boundary contribution is
Fig. 5. Experimental (e) and simulated (s) total (grain boundary+ bulk) resistivity vs. 1/T of YSZ/Al2O3 composites having the same amount of insulating material (10 wt.%) with different grain sizes: 1.7 (n=e, 5=s), 1.4 (.=e, o=s) and 1.0 Am (E=e, D=s).
not negligible (lower temperatures). On the other hand, it is well known that grain boundary resistivity depends strongly on composition, impurities concentration and preparation method [45 – 47]. Simulations performed on composites containing the same amount of insulating phase (10 wt.%) with different grain sizes (ranging from 1 to 2 Am) show that the model also forecasts the negligible variation of the total conductivity as a function of the Al2O3 grain sizes, as experimentally observed (see Fig. 5).
5. Conclusions
Fig. 4. Experimental (e) and simulated (s) total (grain boundary+ bulk) resistivity vs. 1/T of YSZ (E=e, n=s) and YSZ/Al2O3 composites: 5 wt.% (.=e, x=s) and 50 wt.% (+=e, q=s).
The results here reported show that the proposed DMM approach is able to generate both the microstructure feature and the electrical behaviour of the ionic conductor composites. It has revealed high
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flexibility in the microstructure simulation so that any grain size distributions of a two-phase system can be easily generated. The forecasting capability in defining the electrical behaviours are less satisfactory even if the results appear good. On the other hand, the great difficulty in defining the input necessary to describe accurately the grain boundary electrical properties may be hardly overcome and a more complicated approach could only increase the overall procedure complexity without producing appreciable gains, at the up-to-date state of the art. The availability of more suitable experimental data will give the opportunity of well defining the actual quality of the model and it might also supply some other suggestions to eventually improve the modelling procedure.
Acknowledgements The research was partially supported by the Ministero dell’Universita` e della Ricerca Scientifica through the project COFIN 2000– 2001. We also thank Andrea Limido and Roberto Zanchettin for their help in the numerical treatment of data.
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[11] A.J. Feighery, J.T.S. Irvine, Solid State Ionics 121 (1999) 209. [12] Yuzaki, A. Kishimoto, Solid State Ionics 116 (1999) 47. [13] Y. Ji, J. Liu, Z. Lu¨, X. Zhao, T. He, W. Su, Solid State Ionics 126 (1999) 277. [14] I. Natali Sora, C. Schmid, G. Dotelli, R. Ruffo, C.M. Mari, J. Eur. Ceram. Soc. 22 (2002) 1645. [15] I. Natali Sora, C. Schmid, G. Dotelli, C.M. Mari, J. Eur. Ceram. Soc., submitted for publication. [16] C.M. Mari, G. Dotelli, J. Mater. Sci. 36 (2001) 1141. [17] J. Fleig, J. Maier, J. Electroceram. 1 (1997) 73. [18] J. Fleig, J. Maier, J. Electrochem. Soc. 145 (1998) 2081. [19] J. Fleig, J. Maier, J. Eur. Ceram. Soc. 19 (1999) 693. [20] J. Fleig, J. Maier, Phys. Chem. Chem. Phys. 1 (1999) 3315. [21] J. Fleig, J. Maier, J. Am. Ceram. Soc. 82 (1999) 3485. [22] S. Kirkpatrick, Rev. Mod. Phys. 45 (1973) 574. [23] M. Sahimi, B.D. Hughes, L.E. Scriven, H.T. Davis, J. Phys. C., Solid State Phys. 16 (1983) L521. [24] E. Roman, A. Bunde, W. Dieterich, in: F.W. Poulsen, N. Hessel Andersen, K. Clausen, S. Skaarup, O. Toft Sørensen (Eds.), Proceedings of the 6th Risø International Symposium on Metallurgy and Materials Science, ‘‘Transport – Structure Relations In Fast Ion and Mixed Conductors’’, Risø National Laboratory, Roskilde, 1985, p. 165. [25] E. Roman, A. Bunde, W. Dieterich, Phys. Rev., B 34 (1986) 3439. [26] A. Bunde, W. Dieterich, E. Roman, Solid State Ionics 18 – 19 (1986) 147. [27] R. Blender, W. Dieterich, J. Phys. C. Solid State Phys. 20 (1987) 6113. [28] R. Blender, W. Dieterich, Solid State Ionics 28 – 30 (1988) 82. [29] R.T. Coverdale, E.J. Garboczi, H.M. Jennings, B.J. Christensen, T.O. Mason, J. Am. Ceram. Soc. 76 (1993) 1513. [30] S. Sunde, J. Electrochem. Soc. 142 (1995) L50. [31] S. Sunde, J. Electrochem. Soc. 143 (1996) 1930. [32] J. Abel, A.A. Kornyshev, W. Lenhert, J. Electrochem. Soc. 144 (1997) 4253. [33] J.-M. Debierre, P. Knauth, G. Albinet, Appl. Phys. Lett. 71 (1997) 1335. [34] G. Dotelli, R. Volpe, I. Natali Sora, C.M. Mari, Solid State Ionics 113 – 115 (1998) 325. [35] P. Knauth, J.-M. Debierre, G. Albinet, Solid State Ionics 121 (1999) 101. [36] G. Dotelli, F. Casartelli, I. Natali Sora, C.M. Mari, in: P. Vincenzini, A. Degli Esposti (Eds.), 9th Cimtec-World Forum On New Materials—Symposium I—Computational Modelling and Simulation of Materials, Techna, Faenza, 1999, p. 417. [37] G. Dotelli, F. Casartelli, I. Natali Sora, C. Schmid, C.M. Mari, New Materials For Batteries and Fuel Cells, Mrs Symposium Proceedings, vol. 575, Mrs, Warrendale, 2000, p. 331. [38] C.M. Mari, G. Dotelli, Solid State Ionics 136 – 137 (2000) 1315. [39] M. de Berg, M. van Kreveld, M. Overmars, O. Schwarrzkopf, Computational Geometry, Springer-Verlag, Berlin, 1997, p. 145. [40] P. Tzionas, A. Thainalakis, Ph. Tsalides, Image Vis. Comput. 15 (1997) 35.
G. Dotelli et al. / Solid State Ionics 152 – 153 (2002) 509–515 [41] B. Derrida, J.G. Zabolitzky, J. Vannimenus, D. Stauffer, J. Stat. Phys. 36 (1984) 31. [42] R.T. Coverdale, H.M. Jennings, E.J. Garboczi, Comput. Mater. Sci. 3 (1995) 465. [43] D.J. Frank, C.J. Lobb, Phys. Rev., B 37 (1988) 302. [44] J.P. Schafer, A. Saxena, S.D. Antolovich, T.H. Sanders, S.B. Warner, The Science and Design of Engineering Materials, McGraw-Hill, Taipei, 1999, p. 430.
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[45] M. Miyayama, H. Inoue, H. Yanagida, J. Am. Ceram. Soc. 66 (1983) C-164. [46] M. Miyayama, H. Yanagida, J. Am. Ceram. Soc. 67 (1984) C194. [47] J.-H. Lee, T. Mori, J.-G. Li, T. Ikegami, S. Takenouchi, J. Eur. Ceram. Soc. 21 (2001) 13.
Composite materials as electrolytes for solid oxide fuel cells: simulation of microstructure and electrical properties G. Dotelli a,*, I. Natali Sora b, C. Schmid c, C.M. Mari d a
Dipartimento di Chimica, Materiali e Ingegneria Chimica ‘‘Giulio Natta,’’ Politecnico di Milano, P.za L. da Vinci 32, 20133 Milan, Italy b Dipartimento d’Ingegneria Meccanica, Universita` di Brescia, via Branze 38, 25123 Brescia, Italy c Dipartimento d’Ingegneria dei Materiali e Chimica Applicata, Universita` di Trieste, via Valerio 2, 34127 Trieste, Italy d Dipartimento di Scienza dei Materiali, Universita` degli Studi di Milano-Bicocca, via Cozzi 53, 20125 Milan, Italy Accepted 14 February 2002
Abstract The simulated and the experimental electrical conductivity of ionic conductor composites (Al2O3/yttria-stabilized zirconia) either containing different amounts (5 and 50 wt.%) of alumina or having the same amount of insulating phase (10 wt.%) with different grain sizes are presented and compared. A digital image-based modelling procedure to simulate the electrical behaviour of the composites was used here. The method works, generating, by the Voronoi tessellation technique in which a genetic algorithm is used, two-phase polyhedral microstructures and then converting them into a random electrical network. The real and imaginary part of the electrical network impedance was computed by the transfer matrix method. The model is able to reproduce the experimental results well. D 2002 Elsevier Science B.V. All rights reserved. Keywords: SOFC; Composite materials; Electrical properties; Simulation; Random electrical networks
1. Introduction Yttria (8 wt.%)-stabilized cubic zirconia (YSZ), as it is well known, is the most widely used solid electrolyte in high-temperature solid oxide fuel cells (SOFCs) [1]. Such material suffers from relatively poor bending strength that reduces its performances, particularly in planar SOFCs [2– 7]. The addition of a-Al2O3, as a secondary phase, was suggested to overcome this drawback. In fact, this oxide hinders the grain growth process and allows the formation of a fine-grained *
Corresponding author. Tel.: +39-2-2399-3232; fax: +39-27063-8173. E-mail address: [email protected] (G. Dotelli).
ceramic, whose mechanical properties are enhanced. Because the presence of alumina particles decreases the electrical conductivity, a proper amount of this oxide has to be added to obtain a good compromise between electrical conductivity and mechanical properties. For such reason, the electrical properties of YSZ/Al2O3 composites as a function of the alumina amount were investigated [8– 15]. Unfortunately, the experimental data are slightly conflicting and inhomogenous. The lack of a systematic analysis of the microstructures on pure YSZ and on YSZ/Al2O3 composites makes the comparison among the results quite difficult and the explanation of the behaviour rather tricky. For this reason, a simulation procedure able to take into account the influence of the microstructure
0167-2738/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 3 8 ( 0 2 ) 0 0 3 8 0 - 6
510
G. Dotelli et al. / Solid State Ionics 152 – 153 (2002) 509–515
on the electrical properties of pure YSZ and Al2O3/ 8YSZ composites might serve as a tool to better understand the ionic conduction behaviour of such materials. The problem of simulating the electrical behaviour of polycrystalline materials (both pure and composites) can be managed in a rigorous way by solving the classical Laplace’s equation [16]. In principle, the resulting differential problem is numerically solved with any desired accuracy, for instance by the finite element method (FEM). However, this route is not yet fully exploited because of its complexity. In the field of ionic conductors, this technique was recently used to outline the grain boundary influence on impedance spectra of polycrystalline solid electrolytes [17 –21].
The discrete medium models (DMMs) represent a possible alternative to the FEM approach. They simulate the electrical behaviour converting the material into a 2D or 3D network of discrete electrical elements and solving a set of algebraic equations. The main differences among these DMMs can be ascribed either to the way of arranging the discrete electrical elements inside the electrical networks or to the choice of the representative circuits. Originally used to calculate the parameters of the percolation equation [22,23], the DMMs were later applied as self-standing approaches to simulate the electrical properties of dispersed ionic conductors [24 – 28], of cement-based materials [29], of Ni –YSZ cermets [30 – 32] and of ionic conductor composites [33 –38].
Fig. 1. Schematic representation of a 2D Voronoi tessellation. Pixels labelled with even and odd integers stand for phases A and B, respectively. The dots are used to evidence the grain boundaries.
G. Dotelli et al. / Solid State Ionics 152 – 153 (2002) 509–515
The aim of the present paper is to test the forecasting qualities of a digital image-based DMM by comparing theoretical and experimental results performed on Al2O3/8YSZ composites. The original version of the computational protocol that gave some preliminary encouraging results is reported elsewhere [34,38]. An improvement concerning the microstructure generation is here used.
2. Experimental The samples with alumina (Sumitomo Chemical) contents up to 50 wt.% were prepared as previously reported [10,14]. The samples containing fixed amounts of alumina of different size [15] were prepared by mixing 10 wt.% Al2O3 powders (ALCOA 17) and fully stabilised (7.54 mol% Y2O3) zirconia powder (YSZ) (Unitec Ceramics, England, frequency particle size parameters d50=0.65 Am and d95=1.33 Am). The microstructural features were investigated by SEM and image analysis. Electrical measurements by impedance spectroscopy technique were carried out in the temperature range 350– 800 jC in air.
511
lating the nuclei distance; NA and NB nuclei of phases A and B are randomly distributed in a fixed area (or volume), then they are grown at two different rates. Adopting here the pixel as reference unit of volume, the result of the growth process is a pseudo 3D digital image (having just one pixel in thickness) of the simulated microstructure that can be easily converted into a 2D array of figures. All the pixels of each grain are labelled with the same integer (see Fig. 1). This procedure makes easy the statistical analysis of grain geometrical properties: area and perimeter (in 2D) or volume and surface (in 3D). The algorithm that converts the digital image into the numerical matrix is designed in such a way that the grain boundaries and the bulks are clearly distinguished. Each pixel is converted into a discrete electrical circuit, so that the overall image becomes an electrical network (EN). The discrete electrical circuit is constituted of six impedances all connected to the centre of the pixel (see Fig. 2). Between two adjacent pixels, the connection of the impedances is ideal. As a result, one obtains a three-dimensional EN with a node at the centre of each pixel.
3. Modelling and simulation In the original model [34,36 – 38], the computer simulation starts by reproducing the microstructure of polycrystalline materials using the Voronoi tessellation technique [39]. The nuclei of N grains were randomly distributed in a surface or in a volume (2D or 3D) and successively the Voronoi hulls or polygons increased in size until the space was completely filled. In particular, the simulated structure is just the same as that resulting from a growth process that takes place from randomly distributed latent nuclei at constant speed. Because the representation of two-phase polyhedral systems is complicated in the previous version of the model, the task was carried out in two steps: a single-phase polyhedral structure was generated by the Voronoi tessellation, then spherical particles (a-Al2O3) were inserted among the convex hulls [34]. The approach is now refined and a true two-phase polyhedral microstructure is realised. The Voronoi tessellation [40] is here obtained by a genetic algorithm, instead of calcu-
Fig. 2. Conversion of a pixel into an electrical circuit. Each parallelepiped represents any possible combination of discrete electrical element (i.e., single resistance or capacitance, resistance – capacitance parallel circuit and so on).
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G. Dotelli et al. / Solid State Ionics 152 – 153 (2002) 509–515
Different electrical circuits are assigned to bulk and grain boundary pixel as well as the value of single electrical element (resistance and capacitance). In particular, insulating behaviour was assigned to the pixels describing the Al2O3 particles (experimental alumina resistivity value) while the actual experimental values carried out on pure YSZ were given to the pixels representing the bulk and grain boundary of YSZ, respectively. The real and imaginary part of the EN impedance was computed by the transfer matrix method, as in statistical mechanics. In the present case, nonlinear matrix recursion relations are involved [41]. Such a procedure has the advantage of producing exact solutions (in contrast with relaxation methods which may have convergence problems) [42] and of requiring much less storage memory than systematic node elimination process, which, by the way, cannot be practically applied in 3D systems [43]. However, the transfer-matrix approach is more time-consuming than relaxation methods, particularly for very large networks.
4. Results and discussion The microstructural simulations (see Fig. 1) were performed creating digital images (150150 pixel2) and the electrical computations were carried out in its core (130130 pixel2) in order to avoid any spurious effects due to the finite size of the representation. The thickness of the microstructure was assumed to be one pixel. The amount of each individual phase, their average grain size and the conversion factor between pixel and microns are required to simulate the microstructure simulation of the pure YSZ and of the composites. As one can remark, the model is not only able to well mimic the polycrystalline microstructure but also to reproduce some detailed features. A typical simulated and experimental (obtained by quantitative image analysis of SEM micrographs) statistical analysis (performed on 10 wt.% alumina composite) is reported in Fig. 3. The slight discrepancies observed in the frequency distribution curves are probably due to the lack of well monodispersed alumina powders experimentally used. In principle, the genetic algorithm is also able to take into account more complex
Fig. 3. Simulated (E) and experimental (n) statistical analysis performed on 10 wt.% alumina composite: grain size distribution of YSZ (a) and Al2O3 (b).
particles distribution, but at the present less attention was paid to this aspect. The electrical forecasting qualities of the model were tested evaluating the resistivity of YSZ and of composites either containing different amounts (5 and 50 wt.%) of alumina or having the same amount of insulating phase (10 wt.%) with different grain sizes (1.0, 1.4 and 1.7 Am). Calculations were performed using the experimental bulk and grain boundary conductivity values of pure YSZ [10,14] at different temperatures and the constant resistivity value (1015 V cm) [44] of Al2O3 at 25 jC. The two contributions, i.e. bulk and grain boundary, were obtained by equivalent circuit analysis of the two arcs in the impedance spectrum, whenever possible, otherwise by difference, if only one arc was present (for instance at high temperature).
G. Dotelli et al. / Solid State Ionics 152 – 153 (2002) 509–515
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The calculated and the experimental total (bulk+ grain boundary) conductivity of YSZ and of composites with different amounts of alumina are reported as a function of 1/T in Fig. 4. As one can observe, a very good agreement exists between experimental and simulated electrical properties in the case of YSZ. This result attests the reliability of the model. The comparison between the two sets of data of each composite is good, particularly at the higher temperatures. Where the difference between experimental and simulated values are larger (lower temperatures) the discrepancy, however, is never greater than about 50%. Such behaviour is probably due to an underestimation of the grain boundary contribution. In fact, even if the different total grain boundary extension of each composition was considered in the simulations, the agreement between experimental and calculated values is not completely satisfactory, especially where the grain boundary contribution is
Fig. 5. Experimental (e) and simulated (s) total (grain boundary+ bulk) resistivity vs. 1/T of YSZ/Al2O3 composites having the same amount of insulating material (10 wt.%) with different grain sizes: 1.7 (n=e, 5=s), 1.4 (.=e, o=s) and 1.0 Am (E=e, D=s).
not negligible (lower temperatures). On the other hand, it is well known that grain boundary resistivity depends strongly on composition, impurities concentration and preparation method [45 – 47]. Simulations performed on composites containing the same amount of insulating phase (10 wt.%) with different grain sizes (ranging from 1 to 2 Am) show that the model also forecasts the negligible variation of the total conductivity as a function of the Al2O3 grain sizes, as experimentally observed (see Fig. 5).
5. Conclusions
Fig. 4. Experimental (e) and simulated (s) total (grain boundary+ bulk) resistivity vs. 1/T of YSZ (E=e, n=s) and YSZ/Al2O3 composites: 5 wt.% (.=e, x=s) and 50 wt.% (+=e, q=s).
The results here reported show that the proposed DMM approach is able to generate both the microstructure feature and the electrical behaviour of the ionic conductor composites. It has revealed high
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flexibility in the microstructure simulation so that any grain size distributions of a two-phase system can be easily generated. The forecasting capability in defining the electrical behaviours are less satisfactory even if the results appear good. On the other hand, the great difficulty in defining the input necessary to describe accurately the grain boundary electrical properties may be hardly overcome and a more complicated approach could only increase the overall procedure complexity without producing appreciable gains, at the up-to-date state of the art. The availability of more suitable experimental data will give the opportunity of well defining the actual quality of the model and it might also supply some other suggestions to eventually improve the modelling procedure.
Acknowledgements The research was partially supported by the Ministero dell’Universita` e della Ricerca Scientifica through the project COFIN 2000– 2001. We also thank Andrea Limido and Roberto Zanchettin for their help in the numerical treatment of data.
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