Computer simulations on an atomic or molecular scale give a direct link between the microscopic details of a system (atomic masses, molecular geometry, etc.) on the one hand, and macroscopic properties that are of experimental interest (energies, transport coefficients etc.) on the other. By comparing the outcome of the simulation with experimental results, understanding can be obtained of the (fundamental) phenomena that underlie the experimental results. But there is another practical advantage to computer simulations. Sometimes it is difficult (i.e. expensive) or even impossible to conduct experiments under extreme conditions – low or high pressures, low or high temperatures – while performing a computer simulation is very feasible.
The ceramic material yttria-stabilized zirconia (YSZ) is a so-called fast ion conductor: a material in which some ions are highly mobile and allow for the electrical (in fact: ionic) conduction. This is only possible due to the presence of empty places (vacancies) in the crystal lattice, to which adjacent mobile ions can jump.
YSZ is a solid solution of Y2O3 in ZrO2. If the zirconia contains more than 9 mole% Y2O3, the material has a cubic lattice structure at room temperature. At higher temperatures, this stabilisation into the cubic structure already occurs at lower Y2O3 contents. But this doping has more effects. The partly replacement of the Zr4+ ion by the Y3+ ion leads to the formation of oxygen vacancies – equal to half the number of Y3+ ions – in order to keep the material electrically neutral. At room temperature, the oxygen ions do not possess sufficient energy to move from their lattice positions, and remain vibrating around their original position. But if the temperature is sufficiently high – starting at about 1000 degrees Celsius for YSZ – these oxygen ions are mobile, and they can jump to an adjacent vacancy. In this way, transport of oxygen ions through the lattice occurs. An oxygen ion conductor can be applied in oxygen sensors and solid oxide fuel cells (SOFC).
Here the molecular dynamics (MD) technique is being used to investigate the diffusion process of oxygen ions in yttria-stabilized zirconia with an yttria content between 3.85 and 16.13 mole% and temperatures between 1125 and 1500 °C. Molecular dynamics is an example of a simulation technique operating on an atomic/molecular scale. It is a powerful tool to investigate migration of oxygen ions in yttria-stabilized zirconia. By following the particles during their movement, macroscopic quantities such as self-diffusion coefficients and ionic conductivities can be obtained.
In molecular dynamics computer simulations, the lattice positions and velocities of each of the hundreds of atoms – in this case zirconium, yttrium and oxygen ions – are being followed in a model system in time. For each of these particles, you can form Newton’s classical equation of motion, F = m x a, that is the force applied to each of the particles equals its mass times its acceleration. When integrating Newton’s equation one resp. two times in time, you can calculate the velocity resp. position of each particle at each moment. For this you will need a good representation of force F, and this equals the gradient of the potential energy of all particles in the entire model system. Hence the ions affect each other, and due to the very many interactions the positions and velocities of the particles are not calculated by hand, but in practice by using numerical algorithms.
During the investigation, the molecular dynamics software has been adapted to be able to handle ionic systems in a decent way. Especially to process the Coulombic interactions between the charged particles, the Ewald transformation has been added to the software package.
This computer simulation generates large numbers of data, namely the lattice positions and velocities of each of the hundreds of ions in the model system at different moments in time. These data are analysed and converted into macroscopic properties such as the (self) diffusion coefficient. Also here, the Fortran computer programming language is used. The best way to obtain the oxygen self-diffusion coefficient is by following the mean square displacements (MSD in the image below) of all oxygen ions in time. Values obtained in this way are much more accurate than e.g. by integrating the velocity autocorrelation function. The ionic conductivity that is calculated via the Nernst-Einstein relation is in good agreement with the experimental value.
A maximum value for the self-diffusion coefficient at 1500 °C is found for the composition (ZrO2)0.92(Y2O3)0.08; this is in good agreement with experimental values, where an optimum in ionic conductivity is observed for YSZ with about 10 mol% yttria. The temperature dependence of the self-diffusion coefficient (Eact = 70.1 kJ/mol) is in good agreement with experimental ionic conductivity data (Eact = 71.2 kJ/mol).
Oxygen migration occurs predominantly in the directions between tetrahedral sites; the octahedral sites do not play a role in the diffusion process. Only effective diffusive jumps play a role in the diffusion process; forward-backward movements between two nearest tetrahedral sites do not.
During his PhD research project at University of Twente between 1990 and 1994, Eddy Brinkman conducted computer simulations on the ionic transport within yttria-stabilized zirconia, in order to gain a better understanding of this transport. The results of this investigation, dated just before the ‘real’ digital area, are too valuable to not disclose any further. Hence, you can find them on this website. In the article ‘Molecular dynamics simulations of yttria-stabilized zirconia‘, published in Chemical Physics Letters in 1995, you can find the backgrounds and results of these computer simulations.