A unified theory of free energy functionals and applications to diffusion

Abstract

Significance The free energy functional is a central component of continuum dynamical models used to describe phase transitions, microstructural evolution, and pattern formation. However, despite the success of these models in many areas of physics, chemistry, and biology, the standard free energy frameworks are frequently characterized by physically opaque parameters and incorporate assumptions that are difficult to assess. Here, we introduce a mathematical formalism that provides a unifying umbrella for constructing free energy functionals. We show that Ginzburg–Landau framework is a special case of this umbrella and derive a generalization of the widely employed Cahn–Hilliard equation. More broadly, we expect the framework will also be useful for generalizing higher-order theories, establishing formal connections to microscopic physics, and coarse graining.

Wetting on silicone surfaces

“…Maxwell/Kelvin-Voigt bodies). Consideration of the (free) energy field gives rise to the Allen-Cahn/Cahn-Hilliard equations, 274,275 gradient dynamics, 276 or the heat equation. 277 The spatiotemporal distribution of the mean-field quantities is expressed with partial differential equations (PDE).…”

Perspective: New directions in dynamical density functional theory

“…Recently, some work has been done in this direction. This includes a microscopic extension of the active PFC model toward mixtures [220], the development of a framework for obtaining gradient-based free energies from more general expressions [288], and in particular a systematic assessment of the derivation of PFC models from DDFT by Archer et al [34], who argued that the order parameter ψ of PFC models should be interpreted not as the dimensionless deviation of the density from a reference value, but as the logarithm of the density.…”