Computer aided design of nano-structured materials with tailored ionic conductivities

Abstract

We show, using simulation techniques, that the high ionic conductivity in BaF₂/CaF₂ heterolayers is because the interfaces reduce the activation energy barriers to mobility and increase the number of charge carriers.

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The movement of charged atoms (ions) through a solid, referred to as ionic conduction, plays a central role in many solid-state reactions such as catalysis, fuel and solar cell technologies, high temperature batteries and sensors.¹ For example, the solid oxide fuel cell (SOFC) is composed of two electrodes with an oxide conducting ceramic between them.² Fuel is passed over the anode and reacts with oxygen, which is transported through ionic conduction from the cathode as negative ions. The release of electrons from the reaction with the fuel provides the electrical current for powering external devices. However, to achieve conductivities that are sufficiently high for commercial viability currently requires high operating temperatures: in the case of SOFC around 1000 °C. These temperatures are expensive to maintain and result in more rapid breakdown of the materials. Accordingly, there has been significant effort directed towards developing materials with high ionic conductivity at lower temperatures.

In one such study, Sata et al. prepared materials, which comprised alternating layers of BaF₂ and CaF₂, both of which are known to be ionic conductors at high temperature.³ The authors showed that the conductivity of the heterolayer system was not only much higher compared with the conductivity in the parent materials but also increased when the thickness of the BaF₂/CaF₂ heterolayers was reduced. This study demonstrated the capability to fabricate systems with tuneable conductivities. Clearly, a (simulation) tool that can correlate microstructure with properties would be a valuable and inexpensive aid to the design of systems with tailored conductivities.

In this study, molecular dynamics (MD) simulation⁴ was used to construct models of multi-layered BaF₂/CaF₂ thin films based upon those fabricated experimentally by Sata et al. Within these atomistic models various microstructural features, such as point defects, dislocations and grain-boundaries, were introduced. Finally, the conductivity of each system was calculated. The microstructural features were then correlated with changes in the conductivity to ascertain those structural features, which influence the conductivity.

Multilayer BaF₂/CaF₂ heterointerfaces were synthesised using an amorphisation and recrystallisation strategy.⁵ In this procedure, the system is forced to undergo an amorphous transition, which is achieved by performing MD simulation at, for example, high pressures and temperatures. To recrystallise the system, long-duration MD simulation is continued at reduced temperatures and pressures. Systems comprising different microstructural features can be generated by modifying the simulation conditions (i.e. temperature, pressure etc.). For example, tension induced amorphisation can lead to the evolution of polycrystalline films.⁶ In contrast to many previous approaches, which define the basic structure of the system before any simulation is performed, this strategy allow microstructural features to evolve during the course of the simulation (recrystallisation) influenced by the interactions between the component BaF₂ and CaF₂ thin films.

In this study, four BaF₂/CaF₂ heterolayer systems, each comprising a different microstructure, were generated. Graphical images of the atom positions for each of the four BaF₂/CaF₂ systems are presented in Fig. 1(a)–(d) together with (outline) microstructural attributes in Table 1.

Fig. 1 Sphere model representations of the atom positions comprising the four multiplayer BaF₂/CaF₂ heterointerface systems each with different microstructures: (a) BaF₂(111)/CaF₂(111); (b) BaF₂(100)/CaF₂(100); (c) BaF₂/CaF₂ with miss-oriented tubular BaF₂/CaF₂ grain inclusion, coloured grey, that traverses through the system; (d) nanopolycrystalline BaF₂/CaF₂. Calcium ions are coloured blue, barium ions are yellow and fluoride, red. In Figs. ‘(c)’ and particularly ‘(d)’ surface rendering is used to depict more clearly the different crystal grains.

Table 1 Outline microstructural features comprising the four multilayer BaF₂/CaF₂ systems. H: heterointerfaces (BaF₂/CaF₂), D: dislocations, I: intermixing of Ba/F across boundary planes, GB: grain-boundaries (BaF₂/BaF₂ and CaF₂/CaF₂), NP: nanopolycrystalline. The calculated activation energies (in eV) for fluoride ion diffusion, E_ACT, are presented in the final column

System	Microstructure	Graphic	E ACT
BaF2(bulk)	—	—	1.1
CaF2(bulk)	—	—	1.2
BaF2(111)/CaF2(111)	H, D	(a)	0.5
BaF2(100)/CaF(100)	H, D, I	(b)	0.5
BaF2/CaF2	H, D, I, GB	(c)	0.5
Nano-polycrystalline	H, D, I, GB, NP	(d)	0.5

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The ionic conductivity of each system was then calculated by running long duration MD simulation at a range of temperatures. Ionic conductivities can be obtained from this simulation data by monitoring the diffusion of the moving ions during the simulation. In this case the fluoride ions diffuse rapidly through the structure as shown by the mean square deviation (MSD) in Fig. 2. Fig. 2 also shows cation mobility, which corresponds to diffusion of some Ba and Ca ions located at grain-boundary regions. The diffusion coefficient is obtained as 1/6 of the slope of the MSD, which can be related directly to the ionic conductivity through the Nernst–Einstein equation. Fig. 3 shows the temperature, T, dependent ionic conductivity, σ, (ln σ vs. 1/T) for each system.

Fig. 2 Mean square displacements (MSD), units Ų, of fluoride ions (left ordinate; black trace) and barium and calcium ions (right ordinate; dark grey and light grey traces, respectively) comprising the nanopolycrystalline BaF₂/CaF₂ system ‘(d)’ calculated at 1000 K.

Fig. 3 Conductivities, calculated as a function of temperature, for each of the four BaF₂/CaF₂ systems.

The conductivity (Fig. 3) can now be correlated directly with the microstructure (Fig. 1; Table 1): Clearly, different microstructures give rise to very different conductivities. Indeed, the conductivity for the nano-polycrystalline system, Fig. 1(d), is over three-times greater than the (111) orientated system, Fig. 1(a). In general, we find that the conductivity increases with the interface density, which agrees with the findings of Sata et al. In addition the conductivity plot is not a straight line; rather, at higher temperatures, the gradient increases (this is rather subtle but can be seen more clearly if one places a straight edge along each conductivity trace (Fig. 3)). This unusual behaviour was also observed experimentally by Sata et al.

One question that is asked when optimising ionic conductivity is what is the mechanism of enhancement? Two central factors control ionic conductivity: the activation energy and the number of charge carriers (ions in this case). If the activation energy is reduced, the conductivity will increase as a consequence of more rapid diffusion of the ions. If the number of charge carriers increases, then the conductivity increases because of the greater movement of charge (even though each ion moves at the same rate). The calculations allow us to access this information. The activation energy can be obtained from the slope of the ln σ vs. 1/T plot (Fig. 3) and the resulting activation energies are shown in Table 1. For each of the multilayer systems the activation energy is the same (0.5 eV). However this does represent a marked decrease from that observed in bulk CaF₂ and BaF₂ indicating that the thin film systems do give rise to enhanced diffusion and hence conductivity due to a reduction in the activation energy. Conversely, this change in the activation energy does not explain the wide range of conductivities observed within the different multilayer systems. The source of the enhancement of one multilayer system over another must therefore be an increase in the effective number of charge carriers. To corroborate this, graphical techniques were used to observe the diffusion of the fluoride ions. The animations revealed that the fastest ions move along the boundary regions rather than traversing through bulk crystalline regions in the BaF₂/CaF₂ system. Clearly, it is easier for the ions to move through the ‘open’ channels offered in grain-boundary regions compared with regions within the grains themselves (assuming the channels are not blocked).

The conductivity is therefore dominated by diffusion through these boundary regions and the effective number of charge carriers becomes those ions within these regions. One way of evaluating this quantity is by extracting the excess volume associated with crystal boundary regions. For example, microstructural features, such as grain boundaries, heterointerfaces, dislocations, point defects and dopants (intermixing) are associated with a volume change compared with the ‘perfect’ parent material.⁷ Here, we define the excess volume as the volume of each system, Fig. 1(a)–(d)minus the volume of the (perfect) parent BaF₂ and CaF₂ comprising each system. Calculated excess volumes for each system are shown in Fig. 4. We note that the excess volumes were found to change very little with changes in temperature.

Fig. 4 Excess volumes, calculated for each of the four BaF₂/CaF₂ systems.

Correlating the conductivity, Fig. 3, with the microstructure (Fig. 1, Table 1 and ‘excess volume’, Fig. 4) reveals a compelling link between the conductivity of the system and the interface concentration (grain-boundary and heterointerface). Coupled with the fact that the activation energies are equivalent for all four microstructures, the correlation between conductivity and excess volume confirms an increase in the number of effective charge carriers. In addition, this explains the upward curve of the conductivity plots. Specifically, at very high temperatures movement of the ions through the bulk will start to make significant contributions with the higher activation energy causing the increase in slope at high temperatures.

Fthat the thinIn this study, it was important to establish whether the fluoride ion diffusion would lead to macroscopic conduction. For example, if the ions only move, albeit fast, within a limited region, perhaps with a circular motion, this behaviour would not facilitate conduction. For conduction to occur, the ions would need to navigate the whole of the material. Accordingly, movies of the diffusing ions were generated using molecular graphical techniques. For the nanopolycrystalline model, Fig. 1(d), the fastest moving fluoride ions were observed to traverse several grains in all directions, which we suggest will give rise to conduction. Indeed, the average fluoride ion MSD is 44 Å, Fig. 2, even though most of the fluoride ions will be located in ‘bulk’ crystalline regions and therefore much slower moving compared with fluoride ions located within boundary regions.

In summary, we have shown, using simulation techniques, that the high ionic conductivities in microstructural BaF₂/CaF₂ results from two essential attributes. First the microstructure facilitates a reduction in the activation energy for fluoride ion conduction compared with bulk crystalline BaF₂ and CaF₂ and second, the number of effective charge carriers increases with increasing interfacial density. In particular, we propose that it is the combination of these two factors that give rise to the remarkable properties of these systems.

This study demonstrates that by modifying the computational synthetic procedure to fabricate the desired microstructure (this may also be modified to include, for example, different materials, additional dopant species) and simulating the conductivity, one can begin to explore the influence of complex microstructures on technologically important processes. In particular, for the multilayer BaF₂/CaF₂ system, we predict that polycrystalline films with nanosized grains would exhibit high conductivities. This is achieved by reducing the activation energy for mobility compared with the component BaF₂ and CaF₂, whilst increasing the number of effective charge carriers. Ultimately, the strategy provides a framework for the computational materials design of systems with tailored ionic conductivities.

Acknowledgements

We acknowledge Cambridge–Cranfield HPC facility, JREI (JR99BAPAEQ) and Compaq for a 20 processor SC cluster and EPSRC (GR/S48431/1, GR/S48448/01) for funding.

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