Mean - field dynamical density functional theory

We examine the out-of-equilibrium dynamical evolution of density profiles of ultrasoft particles under time-varying external confining potentials in three spatial dimensions. The theoretical formalism employed is the dynamical density functional theory (DDFT) of Marini Bettolo Marconi and Tarazona [J. Chem. Phys. 110, 8032 (1999)], supplied by an equilibrium excess free energy functional that is essentially exact. We complement our theoretical analysis by carrying out extensive Brownian Dynamics simulations. We find excellent agreement between theory and simulations for the whole time evolution of density profiles, demonstrating thereby the validity of the DDFT when an accurate equilibrium free energy functional is employed. Density functional theory (DFT) is a very powerful tool for the quantitative description of the equilibrium states of many-body systems under arbitrary external fields. It rests on the exact statement that the Helmholtz free energy of the system, F [ρ], is a unique functional of the inhomogeneous one-particle density ρ(r). Moreover, the equilibrium profile ρ 0 (r) minimises F [ρ] under the constraint of fixed particle number N [1]. The task is then to approximate the unknown functional F [ρ] from which all equilibrium properties of the system follow. Much more challenging is the problem of studying out-of equilibrium dynamics of many-body systems, for which analogous uniqueness and minimisation principles are lacking. In this paper, we present results based on a recently-proposed dynamical density functional theory (DDFT) formalism and we demonstrate that the latter is capable of describing out-of-equilibrium diffusive processes in colloidal systems at the Brownian time scale. We are concerned with the dynamics of typical soft-matter systems, such as suspensions of mesoscopic spheres and polymer chains in a microscopic solvent [2]. The enormous difference in the masses of the suspended particles and the solvent molecules implies a corresponding separation in the relaxational time scales of the two. At times of the order of the Fokker-Planck scale, τ FP ∼ 10 −14 sec, the solvent coordinates are long relaxed to thermal equilibrium. On the Brownian diffusive time scale, τ B ∼ 10 −9 sec, the momentum coordinates of the solute particles relax to equilibrium with the heat bath of the solvent molecules and thus a statistical description involving only the positions of the colloids is feasible [3]. In this regime, the evolution of the coordinates