Publication by Reddy KS, Rajagopal A., Mechanics of Materials. 2026 Apr 13:105696.
A finite element framework designed to solve the Cahn–Hilliard equation for modelling "phase separation" in supersaturated bainitic steel.
Bainite is correctly defined as a diffusionless displacive transformation, and a model is created for precipitation within bainitic ferrite, to recreate the structure of lower bainite.
Of particular concern is the forced implementation of uphill diffusion of carbon to deal with the precipitation of cementite, thus contradicting the known thermodynamics of Fe-C solutions. This was done to fit the modelling method used. That these carbides form by a displacive mechanism is neglected.
Figure 6 of the publication represents simulated uphill diffusion of carbon in ferrite, leading to precipitation within the supersaturated bainitic ferrite phase, approximated as a diffusional phase separation process driven by the non-standard, uphill diffusion of carbon.
This incorrect representation of thermodynamics requires the chemical free energy of the system to obey a continuous double-well function (two minima):
The double-well potential is a mathematical approximation adopted as a modelling strategy to simplify the simulation of internal carbide precipitation. By forcing the system to separate into two target concentrations—\(c_1\) and \(c_2\)—the model effectively creates an image of precipitation based simply on carbon redistribution.
Hence, the complexity of the real precipitation process is avoided, but to what purpose?
The simulations lead to spherical particles when the interfacial energy and misfit are isotropic. The particles are in reality, plates, with shapes determined by the displacive growth mechanism.