On the use of bi-variate population balance equations for modelling barium titanate nanoparticle precipitation

In this work barium titanate hydrothermal synthesis is studied from the modelling point of view. In fact, a mathematical model can be used to investigate the origin of the typical multi-crystalline particles obtained during this precipitation process. Their origin is not completely clear, since it is not understood which is the controlling mechanism (i.e., secondary nucleation and/or aggregation). Previous works on this subject tried to retrieve the kinetics underlying this particulate process but being based on mono-varied population balance equations failed to give a definitive answer. In this work a bi-variate model is presented, discussed and solved. Results clearly show that such an approach can overcome the limitations of previous modelling works and provides an useful tool for more detailed kinetic parameters estimation. Moreover the model shows that secondary nucleation is indeed very important but aggregation cannot be neglected.