Density Functional Theories of Hard Particle Systems

Abstract

To the memory of our friend Yasha Rosenfeld, who discovered the Fundamental Measure Theory, making this chapter grow into a thick one.This chapter deals with the applications of the density functional (DF) formalism to the study of inhomogeneous systems with hard core interactions. It includes a brief tutorial on the fundamentals of the method, and the exact free energy DF for one-dimensional hard rods obtained by Percus. The development of DF approximations for the free energy of hard spheres (HS) is presented through its milestones in the weighted density approximation (WDA) and the fundamental measure theory (FMT). The extensions of these approaches to HS mixtures include the FMT treatment of polydisperse systems and the approximations for mixtures with non-additive core radii. The DF treatment of non-spherical hard core systems is presented within the generic context of the study of liquid crystals phases. The chapter is directed to the potential users of these theoretical techniques, with clear explanations of the practical implementation details of the most successful approximations. IntroductionThe density functional (DF) formalism for classical particles [1] was developed to find out the equilibrium density distribution ρ(r) of inhomogeneous systems at interfaces or in the presence of an external potential V (r). In most cases, like the layering of fluids against walls or liquids confined in nano-capillaries, the sharpest level of structure in ρ(r) comes from the effects of molecular packing, and hence the development of DF theories for hard-core models has been a

Bubble nucleation in polymer–CO2 mixtures

“…The effects of this equilibrium triple line on nucleation in generic binary mixtures and in model polymer-CO 2 mixtures have been previously elucidated. 6 However, in the CO 2 supersaturated metastable polymer-rich phase at ambient pressure, at the same respective chemical potential for CO 2 and polymer, there can also exist metastable CRV and CRL phases whose densities are determined by eqn (15). These metastable phases are lower in their grand potential density than the parent metastable phase (which is just the negative of the ambient pressure, i.e.…”

Mesoscopic structure prediction of nanoparticle assembly and coassembly: Theoretical foundation

“…40 Although the 1-dimensional result is not widely useful in itself, it is an important starting point for evaluating other hard-particle interactions. 41 The FMT method pioneered by Rosenfeld for hard-particle interactions in 3 dimensions is based on Percus' result. 39 In the present work, we utilize the FMT since it is the most advanced hard particle DFT for inhomogeneous hard body systems.…”

“…39 In the present work, we utilize the FMT since it is the most advanced hard particle DFT for inhomogeneous hard body systems. 41 The FMT was originally developed for inhomogeneous mixture systems of HSs. It supersedes other approaches in (i) providing flexibility for incorporating inhomogeneous hard particle mixtures, and in (ii) providing accurate predictions for dense packing behaviors, e.g.…”

“…liquid to solid transitions, of hard particles. [41][42] The excess free energy functional is given as ( )…”

Fundamental measure theory for non-spherical hard particles: predicting liquid crystal properties from the particle shape

“…A driving force for these efforts is the possibility that these experimental systems form structures that can be applied as novel materials. While these hard-particle systems were originally studied using computer simulations [2], the application of continuum theories is more natural for some long-wave-length or high-symmetry problems.Density functional theory (DFT) [3,4] is a continuum theory for systems that are inhomogeneous or anisotropic either due to applied external fields [3,5,6] or spontaneous symmetry breaking [7][8][9]. Hard spheres represent a classical and quite tractable system to which density functional theory has been applied in many studies.…”