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Johnson Matthey Technol. Rev., 2019, 63, (4), 247

doi:10.1595/205651319x15598975874659

对萤石结构材料晶界结构预测

界面是影响材料特性的一种扩展缺陷。 随着材料纳米结构的日益普遍,晶界(GB)等界面的影响变得更加重要。 晶界计算建模对于我们加深对这些缺陷的了解至关重要,因为它使我们能够分离特定结构并了解其产生的特性。 晶界精确建模的第一步是生成精确的结构描述。 本文针对萤石结构材料(氟化钙和二氧化铈)提出了低角度镜面倾斜晶界结构。 我们比较了计算生成的特定晶界结构和实验已知的结构,它们表现出了惊人的一致性。 我们所提出的预测这些结构的方法具有极高的精度,可用于预测尚未经实验验证的界面,也可应用于异质界面。

Predicting the Structure of Grain Boundaries in Fluorite-Structured Materials

Understanding the impact of defects in crystalline materials

  • Aoife K. Lucid*, Aoife C. Plunkett, Graeme W. Watson**
  • School of Chemistry and Centre for Research on Adaptive Nanostuctures and Nanodevices (CRANN), Trinity College Dublin, The University of Dublin, College Green, Dublin 2, Ireland
  • Email: *lucida@tcd.ie **watsong@tcd.ie
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Article Synopsis

Interfaces are a type of extended defect which govern the properties of materials. As the nanostructuring of materials becomes more prevalent the impact of interfaces such as grain boundaries (GBs) becomes more important. Computational modelling of GBs is vital to the improvement of our understanding of these defects as it allows us to isolate specific structures and understand resulting properties. The first step to accurately modelling GBs is to generate accurate descriptions of the structures. In this paper, we present low angle mirror tilt GB structures for fluorite structured materials (calcium fluoride and ceria). We compare specific GB structures which are generated computationally to experimentally known structures, wherein we see excellent agreement. The high accuracy of the method which we present for predicting these structures can be used in the future to predict interfaces which have not already been experimentally identified and can also be applied to heterointerfaces.

1. Introduction

When considering the properties of crystalline materials, the impact of defects is essential. Point defects such as vacancies and dopants are the defects most commonly considered in both computational and experimental studies of material properties. Furthermore, the modelling of materials at an atomic level is often confined to bulk systems which contain these point defects (14). Considerably less is known about extended defects which appear in polycrystalline systems such as surfaces, dislocations and GBs. As nanostructuring of materials is becoming more prevalent, the behaviour of these extended defects is becoming significantly more important (58). GBs give rise to structural discontinuities within materials which result in specific structures and potential non-stoichiometry and can lead to the segregation of point defects to varying degrees, depending on the specific structure (913). This can significantly affect the macroscopic properties, for example: ionic conductivity, electronic conductivity, thermal conductivity, thermal expansion, elasticity and strength – all of which are crucial for many applications. Therefore, the understanding of interfaces in these materials is key to optimising their performance. Despite this, relatively little is known about the structure and even less of the effects of these interfaces on material properties due to the inherent complexity of the issues.

An example of the importance of interfaces and polycrystallinity is in the fluorite structured fast oxide ion conductors (14), such as yttria stabilised zirconia (YSZ or ZrO2-Y2O3) or trivalently doped ceria (CeO2), used in solid oxide cells, oxygen membranes and oxygen sensors (6, 15, 16). It is reported that the ionic conductivity within the GBs of these materials is several orders of magnitude lower than the bulk (1720) with the effect attributed to a wide range of causes including impurities, dopant segregation, defect cluster formations and space charge layers (7, 8, 2025). In contradiction it has been observed that other materials, such as Bi2O3 (26) and nanostructured YSZ, that the ionic conductivity is enhanced (27, 28). Much of the experimental data is based on average effects observed in impedence spectroscopy, where all GBs are treated equally as an average effect (29, 30). In fact, GBs can take on specific structures, an example of this is shown in Figure 1. That is, the atomistic description of what is happening at the GB is incomplete when obtained from macroscopic observations. Another issue which may arise when only considering average effects is that it is likely that different specifically defined interfaces will behave in different ways. As it is difficult to isolate and study the effects of GBs experimentally, computational studies are invaluable to further our knowledge of these defects and their impact on material properties.

Fig. 1.

(a) Schematic illustration of a polycrystalline material; (b) the specific structure of a GB

(a) Schematic illustration of a polycrystalline material; (b) the specific structure of a GB

The failure to understand the basis of material properties in polycrystalline samples is a significant impediment to the development of new materials and the application of inexpensive processing methods to existing materials. An enhanced understanding of the impact of GBs and polycrystallinity on the properties of materials would allow us to explore alternative routes to optimise their properties and ultimately enhance devices. In order to model the properties of these interfaces we first require a method for accurate prediction of interfacial structures. In this paper we present a computational method for accurately predicting the structure of low angle mirror tilt GBs which can be applied to other interfaces and even heterointerfaces. This method utilises both atomistic simulation and classical molecular dynamic simulation with sophisticated, polarisable force fields derived from ab initio data. Previous theoretical studies of GB structures generally utilise static lattice simulations with empirical force fields and structures based on experimental structures (31, 32). These results often have to be validated using first principles due to the quality and limitations of the force field. In this work the structures are predicted and validated using high-quality force fields derived from ab initio data, this is discussed further in the methodology.

Two fluorite-structured materials are investigated in this study: calcium fluoride (CaF2) and CeO2. CaF2 is the prototypical fluorite material which is a super-ionic conductor at high temperatures (>1100 K) (33). CeO2 (usually doped) is a highly technologically significant material which is both an ionic and electronic conductor with a wide range of applications including catalysis, solid oxide fuel and electrolysis cells and oxygen sensing (6, 15, 16). We compare their predicted GB structures to experimental structures from the literature obtained via transmission electron microscopy (TEM).

2. Grain Boundary Structures: Generation and Definition

All GBs simulated here were generated using the minimum energy techniques applied to dislocation, interface and surface energies code (METADISE) (34). The most stable GB structures were found by carrying out optimisation scans of the GB. Surfaces with specific Miller indices were first cut and then reflected to form an interface. A potential energy surface (PES) was then calculated using a forcefield by scanning one surface relative to the other. From this scan, a two-dimensional (2D) PES for the boundary was calculated which allowed the minimum energy structure to be identified. The minimum energy GBs were then optimised and the most stable boundary was selected to investigate using molecular dynamics. The 2D potential energy scan along with the GB structure (before and after optimisation) for the Σ9(221) GB in CeO2 is shown in Figure 2.

Fig. 2.

(a) The PES scan used to identify the minimum energy Σ9(221) GB; (b) the Σ9(221) GB in pure CeO2 before optimisation; (c) the Σ9(221) GB in pure CeO2 after optimisation. Cerium atoms are shown in green with oxygen atoms in red

(a) The PES scan used to identify the minimum energy Σ9(221) GB; (b) the Σ9(221) GB in pure CeO2 before optimisation; (c) the Σ9(221) GB in pure CeO2 after optimisation. Cerium atoms are shown in green with oxygen atoms in red

GBs are defined by a number of parameters: the crystallographic directions of the axes of the two grains which come together to form the interface (hi, ki, li ), the rotation axis o = (ho, ko, lo ), the misorientation angle θ around the axis o and the normal axis to the GB plane n. When n is parallel to o the boundary is defined as a twist GB and when n is perpendicular to o the boundary is defined as a tilt boundary. The GBs which are studied in this work are high-angle mirror tilt GBs (n ⊥ o) and the rotation axis is (001).

The geometric definition of the GBs used in this work is the coincidence site lattice model (35). A coincidence lattice site can be defined when there exists a finite fraction of coinciding lattice sites between the two lattices (grains). This model is based on the assumption that when the energy of the GB is low, the coincidence of the atomic sites between the two grains is high, i.e. there are few bonds which are broken across the boundary. The reciprocal density of coincidence lattice sites is known as Σ and is used to characterise the geometry of the GB, as given in Equation (i):

 

(i)

 

For cubic lattices, the Σ value can be given by the sum of the squares of the Miller indices of the symmetrical tilt boundary, given by Equation (ii):

 

(ii)

 

where δ = 1 if is odd and δ = 0.5 if is even, thus in cubic systems Σ is always an odd number (35, 36). For example, the Σ9(221) GB shown above is defined by the (221) Miller index of the surfaces which are scanned to give this boundary, i.e. (22+22+12) = 9, which is odd so δ = 1 and thus this is written as Σ9(221). The other GBs studied here are defined in the same way.

3. Methodology

Initial GB structures were generated as outlined above using METADISE with shell model interaction potentials for both CaF2 (37, 38) and CeO2 (39). These structures were then expanded to at least 30 Ångström (Å) in the x-direction, 22 Å in the y-direction (parallel to the GB) and 76 Å in the z-direction (perpendicular to the GB). Each simulation cell contained two identical GBs as illustrated in Figure 3, with each grain having a depth of at least ~35 Å.

Fig. 3.

Schematic illustration of a GB cell used for simulations

Schematic illustration of a GB cell used for simulations

Molecular dynamics simulations were then carried out to determine the average GB structures. The interaction potential used for the molecular dynamics simulations is known as the dipole polarisable ion model (DIPPIM) (40), implemented in the polarisable ion model aspherical ion model (PIMAIM) code (41). The DIPPIM consists of four elements: charge-charge interactions, short-range repulsion, dispersion interactions and polarisation. This is a highly accurate, polarisable, potential, in which the dipoles are solved self consistently at each molecular dynamics step. This leads to a highly accurate description of the dipoles on ions in the simulation which is of particular importance when simulating highly polarisable ions such as F and O2–. The data used to fit the DIPPIM potentials used in this work were calculated using ab initio methods (2, 42, 43). The use of ab initio data allows for non-equilibrium details on the PES to be accounted for which leads to a highly accurate, transferable interatomic potential.

Often interatomic potentials for fluorite materials are derived from equilibrium experimental data or are formed using interatomic potentials from a range of different sources resulting in inconsistent, non-transferable potentials which may have difficulties taking effects of different coordination environments into account, i.e. surfaces and interfaces. Such interatomic potentials are usually better suited to static lattice simulations as opposed to molecular dynamics simulations. In previous work on the surfaces of CeO2 (44) we have shown that the DIPPIM provides an accurate description of extended defects and the effect of such defects on ionic transport.

The simulation cells were heated to 1473 K for 500 ps in order to simulate annealing of the GB structures, they were then cooled to 573 K for 500 ps and finally simulated at 300 K for 500 ps. Temperature scaling was carried out (at all three temperatures) every 0.025 ps before data collection for analysis began. Final GB structures were generated by averaging over the frames of the trajectory at 300 K. The DIPPIM potential parameters used for CaF2 were previously derived by Pyper and Wilson et al. (42, 43) and those for CeO2 were obtained by Burbano et al. (2). All steps of the simulation were carried out using the isothermal-isobaric ensemble (NPT). CaF2 simulations utilised a timestep of 5 fs and a short-range cut-off of 14 Å and in the case of CeO2 simulations had a timestep of 4 fs and a short-range cut-off of 11 Å. The GBs which were simulated for CaF2 were Σ3(111), Σ5(210), Σ5(310), Σ9(221), Σ11(332), Σ13(320) and Σ13(510); and for CeO2 are Σ3(111), Σ5(210) and Σ9(221).

4. Results and Discussion

Here we present the average predicted structures obtained for GBs in fluorite structured materials and compare these with TEM images obtained from experimental studies. As the F and O2– ions present in CaF2 and CeO2 are difficult to image due to their low atomic masses we only compare the cation structures obtained with the aforementioned TEM images. First, we discuss the CaF2 structures followed by those found for CeO2. To the authors’ knowledge there are no experimental studies of GB structures in CaF2 so those found in this study are compared to those of other fluorite materials (CeO2, ZrO2 and YSZ).

4.1 Calcium Fluoride Grain Boundaries

The average cation structure of the Σ3(111) GB in CaF2 is shown in Figure 4, alongside the structure identified by Feng et al. for CeO2 using high-angle annular dark-field (HAADF) scanning transmission electron microscopy (STEM) (45, 46). The structure obtained from our predictive method presented here shows excellent agreement with the experimental structure. Other studies of the Σ3(111) GB in fluorite structured materials (ZrO2, YSZ, CeO2, uranium dioxide (UO2)) show similar levels of agreement with our predicted structure (12, 4749).

Fig. 4.

(a) The average structure of the Σ3(111) GB in CaF2 obtained in this work; (b) a HAADF STEM image of the Σ3(111) GB in CeO2 identified by Feng et al. (45, 46). Only cations are shown. TEM image reproduced from (46) under the Creative Commons license

(a) The average structure of the Σ3(111) GB in CaF2 obtained in this work; (b) a HAADF STEM image of the Σ3(111) GB in CeO2 identified by Feng et al. (45, 46). Only cations are shown. TEM image reproduced from (46) under the Creative Commons license

In Figure 5 the average cation structure of the Σ5(210) GB in CaF2 is presented with the HAADF STEM image of the CeO2 identified by Feng et al. and Hojo et al. (46, 50). The agreement seen here is less striking than that observed for the Σ3(111) GB. Other examples of the Σ5(210) GB in CeO2 (51, 52), UO2 (48) and YSZ (10, 13, 31) show very similar structures which are also comparable to those predicted here.

Fig. 5.

(a) The average structure of the Σ5(210) GB in CaF2 obtained in this work; (b) a HAADF STEM image of the Σ5(210) GB in CeO2 identified by Feng et al. and Hojo et al. (46, 50). Only cations are shown. TEM image reproduced from (46) under the Creative Commons license

(a) The average structure of the Σ5(210) GB in CaF2 obtained in this work; (b) a HAADF STEM image of the Σ5(210) GB in CeO2 identified by Feng et al. and Hojo et al.  (46, 50). Only cations are shown. TEM image reproduced from (46) under the Creative Commons license

The Σ5(310) GB structure is compared to a HAADF STEM image of the Σ5(310) GB in CeO2 in Figure 6. The STEM image was obtained by Tong et al. (52). Again, the structure is extremely comparable with the experimental structure shown here as well as those appearing in the literature for UO2 (32, 48), YSZ (10, 31, 5355) and other studies of CeO2 (49).

Fig. 6.

(a) The average structure of the Σ5(310) GB in CaF2 obtained in this work; (b) a HAADF STEM image of the Σ5(310) GB in CeO2 identified by Tong et al. (52). Only cations are shown. TEM image reproduced from (52) with permission from Elsevier

(a) The average structure of the Σ5(310) GB in CaF2 obtained in this work; (b) a HAADF STEM image of the Σ5(310) GB in CeO2 identified by Tong et al. (52). Only cations are shown. TEM image reproduced from (52) with permission from Elsevier

The Σ9(221) GB in CaF2 is given in Figure 7. This is compared to the HAADF STEM image of the Σ9(221) in CeO2 studied by Feng et al. (46). The comparison between our predicted structure and that of Feng is excellent. Other studies have identified this GB in fluorite materials (YSZ (10, 47), UO2 (48), CeO2 (56)) which give the same level of agreement.

Fig. 7.

(a) The average structure of the Σ9(221) GB in CaF2 obtained in this work; (b) a HAADF STEM image of the Σ9(221) GB in CeO2 identified by Feng et al. (46). Only cations are shown. TEM image reproduced from (46) under the Creative Commons license

(a) The average structure of the Σ9(221) GB in CaF2 obtained in this work; (b) a HAADF STEM image of the Σ9(221) GB in CeO2 identified by Feng et al. (46). Only cations are shown. TEM image reproduced from (46) under the Creative Commons license

Studies of the Σ11(332) GB are far less common than others studied here with the only available comparison being that of Feng et al. ’s CeO2 structure (shown in Figure 8), which displays a high level of agreement with our predicted structure (46).

Fig. 8.

(a) The average structure of the Σ11(332) GB in CaF2 obtained in this work; (b) a HAADF STEM image of the Σ11(332) GB in CeO2 identified by Feng et al. (46). Only cations are shown. TEM image reproduced from (46) under the Creative Commons license

(a) The average structure of the Σ11(332) GB in CaF2 obtained in this work; (b) a HAADF STEM image of the Σ11(332) GB in CeO2 identified by Feng et al. (46). Only cations are shown. TEM image reproduced from (46) under the Creative Commons license

The final structure studied for CaF2 was the Σ13(510) GB. In Figure 9 our predicted structure is compared with that of Dickey et al., whose Σ13(510) GB in ZrO2 was observed using high Z-contrast STEM (57). As for the previous GBs studied here the level of agreement is extremely good. In addition to the structure from Dickey et al. other fluorite materials (YSZ (10, 58) and CeO2 (39, 45)) are equally comparable to that shown here.

Fig. 9.

(a) The average structure of the Σ13(510) GB in CaF2 obtained in this work; (b) a STEM image of the Σ13(510) GB in ZrO2 identified by Dickey et al. (57). Only cations are shown. TEM image reproduced from (57) with permission from John Wiley and Sons

(a) The average structure of the Σ13(510) GB in CaF2 obtained in this work; (b) a STEM image of the Σ13(510) GB in ZrO2 identified by Dickey et al. (57). Only cations are shown. TEM image reproduced from (57) with permission from John Wiley and Sons

4.2 Ceria Grain Boundaries

In the case of CeO2 three GBs were investigated: the Σ3(111), Σ5(210) and Σ9(221). These three GBs were selected as they span a range of stabilities and therefore will be important going forward to study dynamic properties of these interfaces and because there are TEM images of these GBs in CeO2 available for comparison (9). The levels of agreement observed for CaF2 are also seen for CeO2 in Figure 10, Figure 11 and Figure 12. The primary difference is that for CeO2 the structures are being directly compared to experimental results for CeO2, which likely accounts for the improved agreement observed for the Σ5(210) GB over that seen for CaF2.

Fig. 10.

(a) The average structure of the Σ3(111) GB in CeO2 obtained in this work; (b) a HAADF STEM image of the Σ3(111) GB CeO2 identified by Feng et al. (45, 46). Only cations are shown. TEM image reproduced from (46) under the Creative Commons license

(a) The average structure of the Σ3(111) GB in CeO2 obtained in this work; (b) a HAADF STEM image of the Σ3(111) GB CeO2 identified by Feng et al. (45, 46). Only cations are shown. TEM image reproduced from (46) under the Creative Commons license

Fig. 11.

(a) The average structure of the Σ5(210) GB in CeO2 obtained in this work; (b) a HAADF STEM image of the Σ5(210) GB CeO2 identified by Feng et al. and Hojo et al. (46, 50). Only cations are shown. TEM image reproduced from (46) under the Creative Commons license

(a) The average structure of the Σ5(210) GB in CeO2 obtained in this work; (b) a HAADF STEM image of the Σ5(210) GB CeO2 identified by Feng et al. and Hojo et al. (46, 50). Only cations are shown. TEM image reproduced from (46) under the Creative Commons license

Fig. 12.

(a) The average structure of the Σ9(221) GB in CeO2 obtained in this work; (b) a HAADF STEM image of the Σ9(221) GB in CeO2 identified by Feng et al. (46). Only cations are shown. TEM image reproduced from (46) under the Creative Commons license

(a) The average structure of the Σ9(221) GB in CeO2 obtained in this work; (b) a HAADF STEM image of the Σ9(221) GB in CeO2 identified by Feng et al. (46). Only cations are shown. TEM image reproduced from (46) under the Creative Commons license

The GBs which were studied for both CaF2 and CeO2 (Σ3(111), Σ5(210) and Σ9(221)) showed largely similar structures to one another which were in line with structures observed in the literature for both previous computational and experimental studies. This provides significant validation for the method we have presented here for the prediction of interfacial structures in materials.

5. Conclusions

We have presented a computational method for the prediction of the structure of mirror tilt GBs in fluorite structured materials. This method utilises interatomic potentials which are derived from first-principles data meaning the process is entirely predictive. The excellent level of agreement with existing experimental data on the structures of fluorite GBs highlights the power of the method. The ability to accurately predict these structures is an important first step into the computational investigation of the properties of these materials, which is key to future materials and device optimisation. The method presented here can be extended to the prediction of interfaces in different materials, interfaces of different types (i.e. twist GBs) and even heterointerfaces.

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Acknowledgements

This research was supported by Science Foundation Ireland (SFI) through the Investigators Programme (Grant No. 12/IA/1414). All calculations were performed using the Kelvin (funded through grants from the Higher Education Authority, through its PRTLI program), Lonsdale (funded through a grant from SFI – 06/IN.1/I92/EC07) and Pople (funded by SFI – 12/IA/1414) supercomputers maintained by the Research IT at Trinity College Dublin and the Fionn supercomputer, maintained by ICHEC (tcche054b).

The Authors


Aoife K. Lucid graduated from University College Cork, Ireland, in 2013 with a BSc in Chemical Physics. In 2018 she graduated with her PhD from Trinity College Dublin, Ireland, with a thesis entitled ‘Computational Modelling of Solid Oxide Electrolytes and their Interfaces for Energy Applications’. Her research interests include using first principles and classical computational methods to investigate the impact of dopants and interfaces in energy materials. She is currently a postdoctoral researcher in the Materials Theory Group at Tyndall National Institute, Cork, Ireland.


Aoife C. Plunkett obtained a BA (Mod) in Nanoscience, Physics and Chemistry of Advanced Materials at Trinity College Dublin in 2015. In 2017 she completed an MSc by research titled ‘Diffusion Within Fluorite Structured Materials and the Effect of Defects’ in the group of Professor Graeme Watson also at Trinity College Dublin.


Graeme W. Watson is a Professor of Theoretical Chemistry at Trinity College Dublin. His research interests include solid state materials and the effect of point defects, dislocations, surfaces and grain boundaries on their properties. These include reactivity, oxide and proton diffusion, electronic conductivity and thermal conductivity which are all important in a range of functional materials.

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