5. Ion Migration In BAM
5.1 Introduction
The structure prototype of BAM (BaMgAl10O17) is β-alumina (NaAl11O17).1 The structure of β-alumina can be described as oxygen close-packed spinel blocks separated by sodium-oxygen planes.2 It has the space group symmetry of P63/mmc that requires two spinel blocks and two Na-O planes in one primitive cell. On changing the material from β-alumina to BAM, magnesium ions are substituted for two aluminum ions on tetrahedral sites in order to compensate the charge generated by the Ba•Na substitution.
Although it has been found that there are two structures, with different Mg distributions, having nearly the same lattice energy, only the structure with the lower lattice energy (configuration I) was investigated in this chapter to check the migration property of europium ions (see Fig. 5.1).
The aluminum sites in BAM can be classified into four asymmetric positions labeled as Al(1)-Al(4), in which Al(1) and Al(4) are octahedral positions and the other two are tetrahedral positions. Magnesium has been found to occupy the Al(2) position. The Mg distribution of the structure investigated here has the two Mg ions in different spinel blocks and retains the 2-fold screw axis symmetry. This kind of distribution destroys the mirror symmetry at the barium-oxygen plane in between the two adjacent spinel blocks along the c-axis. The barium-oxygen plane is referred to as a conduction plane instead of a mirror plane because of the above reason, and because of the high mobility of barium in this plane.
In the conduction plane (Fig. 5.2) there are three different positions, labeled as BR, anti-BR and mO sites, named after the work of Beevers and Ross.3 Although BR and anti-BR sites are both surrounded by three oxygen ions in the conduction plane, their neighboring ions above and below them in the c direction are different. The BR site is in the center of an octahedron with three oxygen ions above and another three below. For the anti-BR site, there are oxygen ions immediately above and below it, and the three ions form a straight line parallel to the c-axis.
It has been found that Eu2+ ion is incorporated into the structure at the BR site, while Eu3+ ion prefers to substitute on the Al(2) site in the middle of spinel block from our earlier work (Chapter 4). As one would imagine, Eu3+ ion is quite likely to form at the BR site, through oxidation of the Eu2+ at that site. It has been found that there is about 4.5eV difference in the defect reaction enthalpy for Eu3+ entering into these two positions.
'' 3
2 1/2
2 /
1 Eu O →EuBa• +BaO+ Oi ∆H = 5.08eV and 1/2Eu2O3 →EuAl+1/2Al2O3 ∆H = 0.49eV.
Although the enthalpy difference is quite large, it is not clear whether it is large enough for the Eu3+ ion to move through the oxygen close-packed layers. Therefore, it is necessary to investigate the possibility of europium migration. Molecular dynamics (MD) simulation provides a useful tool to study the migration of ions at different temperature, and was used to investigate the migration of Eu in our study.
5.1.1 Molecular Dynamics Simulation
As its name implies, molecular dynamics simulation models the movement of particles: ions, atoms and molecules. In ionic crystals, ions are under the influence of all the other ions, long-range and short-range. They will move according to the summation of all the influences. Normally, ions just oscillate at their equilibrium position in an equilibrated material, and jump randomly with small jump frequency unless a gradient is established somehow (thermal, electrical, chemical etc.). If the temperature of the material is high enough, the frequency of an ion obtaining a kinetic energy large enough to overcome the migration obstacle formed by all other ions becomes larger. Actually, any kind of migration can occur at any temperature above 0K, but with different probability. The higher the temperature is, the larger the probability.
The first thing in setting up a MD simulation is to describe the system. For the ionic material of BAM, the system consists of many individual ions with charges determined by their valence state. Boundary conditions can be periodical or restrictive (nonperiodic in any of the x, y and z directions), depending on the simulation requirement. Periodic boundary conditions are used frequently to simulate bulk materials. The system size can be altered to adjust some species’ concentration, such as dopants and defects. Normally,
the core-shell model used in energy minimization simulations would not be adopted in a MD simulation, because the frequency of the core-shell vibration is very high which requires the simulation to work at that infinitesimal time-step and the time of the simulation becomes unacceptably long.4
Secondly, the interaction between ions in the system is defined. The interaction includes long-range Coulombic and short-range non-Coulombic interactions. The potential energy of one pair of ions is described as
( )
ij = i j/ ij+ ijexp(
− ij/ ij)
− ij ij−6ij r zz r A r C r
V ρ . (1)
In the above equation, the non-Coulombic potential is in Buckingham form for the consistency with earlier work.
For summation of the long-range potentials, the Ewald approach is used for periodic systems; the direct Coulombic sum can be used for periodic or non-periodic systems, but with long calculation time. The Ewald sum calculates the long-range potential in two steps. First, a spherical Gaussian cloud of opposite charge centered on each ion is superimposed on the system; this changes the long-range interaction to short-range and then the summation converges quickly. Second, another set of Gaussian clouds of the same charge as each ion is superimposed, so that the total effect of the two superimpositions is zero. The second set of Gaussian clouds can be summed quickly in reciprocal space. Therefore, the Ewald sum replaces an infinite sum in real space into two infinite sums: one in real space and the other in reciprocal space, but both converge quickly.5
The force acting on each ion can be calculated by differentiating the potential at that ion with respect to its coordinates. Ions will move under those forces for an infinitesimal period of time and then the forces have to be recalculated because the potential at each ion has changed after the ions’ positions have changed. Normally, the scale of the time- step is about 10−3 picosecond. It can not be too large or the calculation will become unrealistic; because the frequency of phonon motion is about 1013Hz, and thus, the simulation time-step must be far smaller than 10-13s. The small scale of the time-step limits the capability of MD simulation because it can not simulate in real time scale;
one-second simulation requires about 1014 time-steps for which the calculation will last
“forever”. For example, the calculation of one time-step for a one-thousand-atom
system lasts 0.1 second (based on our calculation), a simulation of 1014 time-steps will last 1013 seconds which is definitely unacceptable.
Normally MD simulations use the Verlet leapfrog scheme as standard to calculate the positions and velocities of ions in a system at each time-step, in the microcanonical (NVE) ensemble in which the total energy of the system is conserved.5 The temperature of the system may vary in a small range. This kind of algorithm is also used in our studies. Constant temperature MD simulation uses other algorithms to calculate the trajectories; the system energy may be conservative or not depending on the actual algorithm used. 5