**7. Conclusion**

With the rapid development of computational capabilities and new experimental techniques we are moving closer to understanding the full role of microstructure on the materials performance. Not only are advanced approaches for simulating microstructures being used but also models of as-measured structures are actively being developed. Among the former, tools such as the presented 2D and 3D Voronoi tessellations can be used whereas for the latter, experimental techniques such as the X-ray diffraction contrast tomography which enable 3D characterization of grains are indispensable. Basics of these approaches are covered here. Both approaches share a common difficult task of creating a finite element model in terms of both appropriate meshes and model sizes. Surface reconstruction issues and complex geometry make the process more difficult. The presented approach effectively deals with some of these issues. The others, however, remain and are subject to further work and research.

The demonstration of the approach is presented on several cases of 3D Voronoi tessellations and two cases of a 400 *μ*m diameter stainless steel wire. In all cases the grain boundaries are explicitly modeled using the cohesive zone approach with zero physical thickness finite elements. Grain boundaries are classified into resistant and susceptible grain boundaries, depending upon the crystallographic orientation of the neighboring grains. Grain boundary damage initialization and early development is then computed for a stainless steel case for several constitutive laws, ranging from isotropic elasticity up to crystal plasticity for the bulk grain material. Since isotropic elasticity approach disregards the crystallographic orientations it should not be used in these cases. Little differences were observed between the anisotropic elasticity and anisotropic elasticity+crystal plasticity approaches, with the latter resulting in more than twice as long computation times.

In all cases almost uniform degradation of the grain boundaries is observed. This is attributed to a) a missing link between the grain boundary load and rate at which the corrosion penetrates the grain boundaries, b) uniform grain boundary stiffnesses whereas stiffness distribution based upon the properties of adjacent grains is expected (Coffman & Sethna, 2008) and c) the selected corrosive environment penetration approach does not account for the topology of the grain boundary network. These issues are the subject of further work.

The numerical stability of the simulation including damage is reasonable, with slightly better convergence for the anisotropic elasticity+crystal plasticity approach. The degradation of a cohesive element is linked to the stability of a simulation. If this degradation process occurs within one computational time increment, the convergence is degraded, even more so when this occurs simultaneously in several cohesive elements. Small computational time increments should therefore be used. Similarly, the *δ*<sup>0</sup> *<sup>n</sup>* and *δδ <sup>f</sup> <sup>n</sup>* values should not be very small since at already small load increment the resulting separation of the cohesive element faces could be

high enough to instantaneously completely damage the element, again causing convergence issues.

#### **8. Acknowledgments**

22 Polycrystalline Materials

boundaries was at its highest, resulting in stress redistribution from the failed areas of grain boundaries to the neighboring areas of grains. The last row therefore displays only the Mises stress for the AE+CP model. Redestribution of the Mises stress can be observed due to the degradation of the susceptible, tensile-loaded grain boundaries which decreases the amount of stress transferred from one side of the boundary to the other. Stress is redistributed in the neighboring areas of grain boundaries and grains, increasing the compressive loading. This

The presented approach is, however, not without its deficiencies. First, all tensile-loaded grain boundaries degrade at the same rate. In reality, higher degradation rates might be expected for the susceptible grain boundaries with higher tensile load. Also, the initial grain boundary stiffness is taken to be uniform whereas stiffness distribution based upon the properties of adjacent grains is expected (Coffman & Sethna, 2008). Lastly, the selected corrosive environment penetration approach does not account for the topology of the grain boundary network. These issues will therefore need to be addressed in future work. Slightly better convergence of the AE+CP model was observed. However, the computational times for

> Wallclock time [s] 252 005 531 133 Memory for min I/O ≈40 GB ≈80 GB Number of elements 903 199 903 199

With the rapid development of computational capabilities and new experimental techniques we are moving closer to understanding the full role of microstructure on the materials performance. Not only are advanced approaches for simulating microstructures being used but also models of as-measured structures are actively being developed. Among the former, tools such as the presented 2D and 3D Voronoi tessellations can be used whereas for the latter, experimental techniques such as the X-ray diffraction contrast tomography which enable 3D characterization of grains are indispensable. Basics of these approaches are covered here. Both approaches share a common difficult task of creating a finite element model in terms of both appropriate meshes and model sizes. Surface reconstruction issues and complex geometry make the process more difficult. The presented approach effectively deals with some of these

The demonstration of the approach is presented on several cases of 3D Voronoi tessellations and two cases of a 400 *μ*m diameter stainless steel wire. In all cases the grain boundaries are explicitly modeled using the cohesive zone approach with zero physical thickness finite elements. Grain boundaries are classified into resistant and susceptible grain boundaries, depending upon the crystallographic orientation of the neighboring grains. Grain boundary damage initialization and early development is then computed for a stainless steel case for several constitutive laws, ranging from isotropic elasticity up to crystal plasticity for the bulk grain material. Since isotropic elasticity approach disregards the crystallographic orientations it should not be used in these cases. Little differences were observed between the anisotropic

issues. The others, however, remain and are subject to further work and research.

AE AE+crystal plasticity

the AE+CP model were more than twice those for the AE model, see Table 3.

30K case, 60 processor cores used

Table 3. Model performance data. 10.0 *μ*m element size.

**7. Conclusion**

can be seen as dark patches in the last row.

The authors would like to thank Prof. James Marrow, (formerly of the University of Manchester, now University of Oxford) and Dr. Andrew King from the European Synchrotron Radiation Facility for kindly providing the results of DCT measurements and also useful discussions. The authors also gratefully acknowledge the financial support from the Slovenian research agency through research programmes P2-0026 and J2-9168.

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**Part 2** 

**Methods of Synthesis, Structural Properties** 

**Characterization and Applications of Some** 

**Polycrystalline Materials** 


**Part 2** 

**Methods of Synthesis, Structural Properties Characterization and Applications of Some Polycrystalline Materials** 

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**4** 

*India*

**NASICON Materials:** 

Lakshmi Vijayan and G. Govindaraj

*Pondicherry University, R. V. Nagar, Kalapet* 

**Structure and Electrical Properties** 

*Department of Physics, School of Physical, Chemical and Applied Sciences,* 

Solid electrolytes are one of the functional materials, practically applied in industries because of its high ion conducting property. It provides scientific support for wide variety of advanced electrochemical devices such as fuel cells, batteries, gas separation membranes, chemical sensors and in the last few years, ionic switches. NASICON type ion conductors have been tested widely in energy applications for instance in electric vehicles. High ion conductivity and stability of phosphate units are advantages of NASICON over other electrolyte materials (Hong, 1976). Among the batteries those based on lithium show the

In NASICON frame-work, AxBy(PO4)3, A is an alkali metal ion and B is a multivalent metal ion. The charge compensating A cations occupy two types of sites, M1 and M2 (1:3 multiplicity), in the interconnected channels formed by corner sharing PO4 tetrahedra and BO6 octahedra. M1 sites are surrounded by six oxygens and located at an inversion center and M2 sites are symmetrically distributed around three-fold axis of the structure with tenfold oxygen coordination. In three-dimensional frame-work of NASICON, numerous ionic substitutions are allowed at various lattice sites. Generally, NASICON structures crystallize in thermally stable rhombohedral symmetry. But, members of A3M2(PO4)3 family (where A=Li, Na and M=Cr, Fe) crystallize in monoclinic modification of Fe2(SO4)3-type structure and show reversible structural phase transitions at high temperatures (d'Yvoire et al.,1983). NASICON based phosphates are widely studied in past decades. But LiTi2(PO4)3 is an interesting system because of its high ion conductivity at room temperature. The Na3Cr2(PO4)3 and Li3Fe2(PO4)3 are intriguing due to its structural peculiarity. These materials crystallize in structurally unstable phase by conventional synthesis technique. Since, Na3Cr2(PO4)3 and Li3Fe2(PO4) systems are not stable at the room temperature phase, a chemical synthesis technique of solution combustion is explored. In the present work we have achieved a stable phase through solution combustion technique and electrical properties are investigated and results are reported. The LiTi2(PO4)3 and Li3Fe2(PO4)3 systems used as electrolytes in solid state batteries and Na3Cr2(PO4)3 system used in is sodium sensors. High energy ball milling technique can control the crystallite size through milling duration. In LiTi2(PO4)3 system, milling is performed for various duration to study

**1. Introduction** 

best performance.

the effect of crystallite size on electrical conductivity.
