From atomistic to continuums modelling of nucleation

The goal of the project is in pursuing a combined theoretical and experimental effort for the understanding of the nucleation process of small precipitates for materials of technological interest. Continuum theory is very successful in modelling materials properties for complex multi-component materials with a large number of particles. A corresponding numerical model has been developed recently. However, continuum mechanical approaches reach their limitation when the size of particles becomes smaller than about 10nm, since precipitates of this size contain only a small number of atoms and continuum mechanics is inappropriate in this case. For small precipitate dimensions, clearly an atomistic description is needed which on one side is able to provide accurate results for bonding energies and related quantities, purely based on the basic laws of quantum physics without the need of any empirical parameters. Within the framework of density functional theory such calculations can be done with high quality for ordered compounds with up to ~100 atoms in a unit cell. On the other hand, a procedure must be designed to go far beyond the limitations of such rather small number of atoms but still maintaining the quantum accuracy. This can be achieved with the cluster expansion method, by which a lattice is decomposed into small basic building elements bound together by interaction energies, which can be found from fitting the cluster expansion to a suitably selected set of density functional calculations. Such a strategy proved very fruitful for binary compounds even allowing to derive shape and sizes of precipitates by making use of Monte Carlo techniques. In our project, studying Cu precipitates and Al-Ni precipitates in Fe, the cluster expansion and MonteCarlo techniques have to be extended to ternary systems. This is a very challenging part of the project but we believe that due to the strong national and international cooperation with leading experts we will succeed. The results of the atomistic modelling and the results from a comprehensive experimental program (EF-TEM, HR- TEM, APFIM, SANS) will be used to develop a model, which bridges the continuum mechanical world to the world of single atoms. The accompanying experiments on well-defined samples of the above mentioned materials will provide a reliable basis on which the theoretical results can be tested. This combination -from Schrödinger`s equation via continuum modelling to applied materials science is a unique effort in Austria and will hopefully stimulate more and stronger links between basic research and technology.

The main idea of the project was the study of the precipitation process in the metal alloys with a focus on the precipitate nucleation stage. Existing models derived on the continuum mechanics are a standard too used for the simulation, but their reliability is put in question for the very small precipitates consisting of a few tens or hundreds of atoms. On the other hand, studying the precipitation process with the atomistic scale methods (first principles calculations) is not feasible, due to the required computational cost. The project goal was to establish a link between the existing atomistic-scale simulation methods and the continuum models in order to describe the precipitation process starting from the creation of nuclei up to the precipitates of micrometer size. Combining the density functional theory (DFT), phonon spectra calculation, and cluster expansion (CE) technique, the system energy was modeled on the atomistic level. This energy model was an input for the kinetic Monte Carlo simulation, where the actual precipitation kinetics was modeled. The parameters of continuum model were modified to reproduce the Monte Carlo results describing the initial part of the precipitate development and the meaning of the modifications was critically analyzed. The modeling predictions were to be validated by the precipitation characteristics obtained in the experiments performed in the Fe-Cu and Fe-Ni-Al systems using various transmission electron microscopy (TEM) techniques, atom probe tomography (APT), small-angle neutron scattering (SANS) and the measurements of the sample hardness. The tasks were divided between the group of R. Podloucky at the University of Vienna (first principles calculation), E. Kozeschnik at the Graz University of Technology/ Vienna University of Technology (precipitation kinetics simulation on atomistic and continuum scale) and H. Leitner at the University of Leoben (experimental measurements). In the Fe-Cu system, the Cu-rich precipitates with the bcc-structure were studied which are responsible for the hardness increase of these alloys. One of the unresolved issues in this system is the chemical composition of the precipitates, which seems to be different, depending on the experimental method used - APT method shows significant Fe content while SANS shows precipitates consisting of almost pure Cu. Our experimental results indicated a high Fe-content (about 50 at.%) in these precipitates obtained with APT measurement. Moreover, the interpretation of SANS measurement gave the same composition when the magnetic moments of the precipitates, found by first principles calculations, were included in the evaluation of the results (this was never done before). First principles calculations gave also the energy model which produced the phase diagram in agreement with the thermodynamic predictions. As the number of parameters in this model was too large for the precipitation kinetics simulation (due to the computational cost), some simplifications had to be taken. The new model was based on the pair-bond energies, the value of which was dependent on the chemical composition around the bonded atoms. This model was still able to reproduce the phase diagram obtained with first principles technique. The precipitation kinetics simulation gave the insight to the chemical composition of the precipitates and the energy of the precipitate/matrix interface which were the input parameters for the continuum model. The comparison of the experimental and calculation results for the precipitation characteristics showed a good agreement with some discrepancies about the precipitate composition noted. In the Fe-Ni-Al system, the formation of the NiAl precipitates was studied. Also here, the magnetic moment of precipitates was considered in the result interpretation and an agreement between the results of different measurement methods was observed. As this system is more complex, the construction of the energy model from first principles calculations took more time. Again, the obtained phase diagram agrees with very well with the thermodynamic predictions. The precipitation kinetics simulations are still running. The results obtained in this project clearly show, that the linking of the various scale models can be successful. This combination allows to interpret some of the experimental results in the new light. Although the tailoring of the models is far from being trivial, the further studies seems to be very perspective. Once the modeling of precipitation process in the many component systems is mastered, the manufacturing process of the alloys can be optimized and the design of the new materials will be much easier.