Approximation and characteristic times in precipitation modelling

A new approximation to the equations describing Classical Nucleation and Growth Theories, is proposed providing quick, and intuitive insight. It gives a prediction of the mean precipitate radius and number density development under quasi-isothermal conditions. Current “mean-radius”, and “multi-class” approaches to modelling classical nucleation and growth theory for precipitation, require considerable computation times. An analytical approximation is proposed to solve the equations, and its results are compared to numerical simulations for quasi-isothermal precipitation. From the approximation a start and end time for the nucleation stage is predicted, as well as a time at which growth occurs and when the coarsening stage starts. Ultimately, these times, outline the numerical solution to the precipitation trajectory, providing key insight before performing numerical simulations. This insight can be used to more efficiently simulate precipitate development, as time scales at which the various stages in precipitate development occur can be predicted for individual precipitates. When these time scales are known a numerical simulation can be used for a specific goal, for instance to only simulate nucleation and growth, thus saving computational time. Moreover, for a first indication of the precipitate development in a composition under a particular heat treatment a numerical simulation is no longer necessary. This is also useful for process control as consequences of changes in treatment can be assessed on-line. Using these approximate analytical results an estimate can be made for the matrix concentration of precipitate forming elements. Additionally some dimensionless parameters are established to provide intuitive details to the precipitation trajectory