New developments in classical density functional theory

The majority of Journal of Physics: Condensed Matter readers will be familiar with the basics of electronic density functional theory (DFT), developed by Hohenberg and Kohn (HK) [1] and Kohn and Sham [2] in 1964–5, and will be aware of the importance of its applications in solid state physics, quantum chemistry and in computational materials science; for recent overviews with historical perspective see [3, 4]. Fewer readers will be familiar with classical DFT, a formalism designed to tackle the statistical mechanics of inhomogeneous fluids. Whereas in electronic DFT the fundamental variable is the electron density in classical DFT this is the average one-body ‘particle’ density where the particles refer to atoms or ions, tackled at the Angstrom scale, or to colloidal particles, tackled at the micron length scale. Classical DFT has also been developed for polymeric systems and for liquid crystals. The equilibrium phenomena investigated using classical DFT range from the properties of fluid interfaces, including interfacial structure, surface tension, physics of adsorption and interfacial phase transitions such as wetting, to fluids subject to confining geometries as occur in porous materials and at structured substrates, and to the chemical physics of solvation. The theory of freezing, where the crystal is regarded as a particularly relevant case of a very inhomogeneous fluid, is also addressed within DFT. In recent years dynamical DFT (DDFT) has become increasingly important as a theory to treat the non-equilibrium physics in overdamped systems and this approach has found many applications in the study of properties away from equilibrium in colloidal science. The development of classical DFT has similarities with that of electronic DFT. If one chooses to argue that Thomas–Fermi–Dirac theory is a precursor of HK theory, then one might also argue that van der Waals’ 1893 treatment of the liquid–gas interface [5], which used the particle density as a basic variable, and Onsager’s treatment of the isotropic to nematic transition in hard-rod model fluids, using particle position and orientation as fundamental variables [6], were early examples of classical DFT. However, without the fundamental variational principle of HK the basis of both electronic and classical DFT would be questionable. Thus the Guest Editors associate the origins of what is now termed classical DFT with papers from the late seventies where various authors built upon ideas from HK. More specifically these papers built upon the less-well recognized work of Mermin [7] whose beautiful three page paper in 1965 extended HK to non-zero temperature and developed a variational principle for the grand potential as a functional of the electron density. The Mermin treatment translates straightforwardly to classical (Boltzmann) statistics appropriate for most liquids; one has a rigorous variational principle for the grand potential as a functional of the average particle density. The first application, using an approximate functional based on a partial summation of the gradient expansion due to HK, was reported in 1976 by Ebner et al [8] (see also [9]) who studied the surface tension of the liquid–gas interface and the density profile of a repulsive wall-liquid interface for a simple Lennard–Jones fluid. It is interesting that in the same year Yang et al [10], not knowing about the HK-Mermin formalism for electrons, published a formal derivation of square gradient theory for a fluid interface. In Appendix A of their paper a Legendre transformation from external potential as variable to particle density as variable is introduced1—which we now recognize as a key ingredient of DFT. The article by Evans published in 1979 [12] showed that earlier formal developments in the classical statistical mechanics of R Evans et al