The phase field equations are derived from the free energy functional which is combination of two terms F(u) double well potential and gradient of phase field grad(u). The question is that why is this F(u) restricted to a double well potential function? Can we take any other potential function instead of double well potential function?
In principle you can, of course, take any physically reasonable free energy. However, the Cahn-Hillard equation is traditionally used to describe phase separation dynamics. For this, you need a free energy that admits two phases (that can then spatially separate) and therefore it has to have (at least) two minima. As any free energy has to be bounded from below you end up with a double well (or triple well, and so on, the double well certainly being the simplest and most useful case). If you use a free energy with a single minimum I am not sure I would still call it a Cahn-Hillard equation but of course it remains a perfectly fine equation and depending on initial conditions, it may be useful to study it.
Dear Muhammad Sohaib, In the free energy functional, there is actually no restriction in choosing the potential F(u) to be a double-well. It can be a double-obstacle like here: Article Multi-phase-field model for surface and phase-boundary diffusion
or a multi-well like here:Article Phase-Field Simulations of Crystal Growth
or even a top-hat like here:Article Phase-field models in materials science
The choice of double-well potential goes back to the free energy curves with unstable regions; it is actually mimicking the free energies for van der Waals gas or the regular solutions.
The key to understanding the 'potential' is that it is the energy penalty that balances the gradient energy, thus resulting in forming an interface with a finite width. In any potential curve, there is a 'bump', between the two low-energy states, over which the values of the order parameter (can be the phase-field or concentration field) are not thermodynamically favorable; so the potential pushes for narrowing the interface to zero. However, the gradient energies of such a sharp interface approach infinity! (see for instance here: Article A model for grain boundary thermodynamics
) In balance, the two potential and gradient energy terms compromise for given interface energy and width. So, in principle, as long as the interface energy and width of your choice can be reproduced you are free to choose the potential you want ---Of course, for any physical problem, there can be constraints that you might need to consider additionally.- Badges
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