A broad range of systems undergoing change, such as the temperature-dependent eutectoid transformation in steels, crystallization in supersaturated solutions, the propagation of information in social systems, or the spread of infection, exhibit behavior similar to classical nucleation and growth. In all cases, a driving force for change exists due to a difference between some variable and its equilibrium value. For crystallization in melts, the driving force is a consequence of supersaturation that results from a change in temperature. In the case of crystallization from solution, the driving force can also be a consequence of supersaturation that results from a change in composition.
While detailed models exist for predicting the behavior of many such systems on a case by case basis, a broad view is beneficial to communicating (and understanding) the principles of nucleation and growth based transformations more generally, and to developing useful graphical representations that can aid with process design.
We present a means of generalizing the classical model for nucleation and growth in systems that are undergoing a phase transformation or analogous process. Our approach is based on the thermodynamics of nucleus formation and diffusion-limited reactant transport. We then expand the general model from 3-D nucleation and growth with one system variable to include phase transformations that are dependent on m arbitrary system variables. Furthermore, we have laid out the mathematics necessary to consider nucleation and growth in an arbitrary number of n spatial dimensions. This is particularly interesting for explaining fractal growth, and for abstracting of the model to fields outside of materials science, such as epidemiology.
We use (i) the formation of crystalline phases during the drying of protein solutions and (ii) the formation of a nematic phase from an isotropic solution as examples to illustrate the utility of the generalized model. We show that, at constant temperature, contours of constant transformed fraction plotted in the concentration-time plane have a C-shaped profile, similar to the well-known time-temperature-transformation (TTT) diagrams for the eutectoid transformation in steel. Furthermore, it follows that the contours of constant transformed fraction are cup-shaped when plotted in time-concentration-temperature space.
The lack of a concise representation of drying kinetics is a bottleneck in organic synthesis and pharmaceutical development. Prediction of such time-concentration-temperature-transformation (“TCTT”) diagrams will allow for improved process design in crystallization of proteins and other macromolecules.