Unification of dynamic density functional theory for colloidal fluids to include inertia and hydrodynamic interactions: derivation and numerical experiments

Abstract

Abstract. Starting from the Kramers equation for the phase-space dynamics of the N -body probability distribution, we derive a dynamical density functional theory (DDFT) for colloidal fluids including the effects of inertia and hydrodynamic interactions (HI). We compare the resulting theory to extensive Langevin dynamics simulations for both hard rod systems and three-dimensional hard sphere systems with radially-symmetric external potentials. As well as demonstrating the accuracy of the new DDFT, by comparing with previous DDFTs which neglect inertia, HI, or both, we also scrutinise the significance of including these effects. Close to local equilibrium we derive a continuum equation from the microscopic dynamics which is a generalized Navier-Stokes-like equation with additional non-local terms governing the effects of HI. In the overdamped limit we recover analogues of existing configuration-space DDFTs but with a novel diffusion tensor.

Wetting of prototypical one- and two-dimensional systems: Thermodynamics and density functional theory

“…The unstable character of these states can be readily confirmed by going beyond the equilibrium theory and considering the dynamics of the system. 8,35,[49][50][51] We also note the existence of a metastable film adsorbed on the walls of the pore. The interval of ∆µ where we find the film phase is rather narrow and has a width ∆ µ ≈ 7.6 × 10 −3 .…”

“…Through continuation, we are also able to uncover all solutions to the Euler-Lagrange equation and eliminate the unstable ones by analyzing the convexity of the corresponding branches of the grand potential Ω(µ). For a deeper understanding of the stability of the various fluid states obtained here, one makes use of dynamic DFT approaches, such as those developed recently by Goddard et al 8,50 Throughout this work, we emphasized that a comprehensive understanding of wetting in a capped capillary is facilitated by identifying links and common features with wetting in a slit pore and on a planar wall. In capped capillaries, condensation follows either the route of condensation in the associated slit pore (first-order transition, for T ≤ T cw ) or a route similar to complete wall wetting (continuous transition, for T > T cw ).…”

“…From a theoretical point of view, a fluid confined inside a small pore is a statisticalmechanical system with a rich phase behavior since the parameters of the fluid-substrate potential and the characteristic dimensions of the pore act as thermodynamic degrees of freedom. In applications, enhancing our understanding of adsorption in small pores is essential in new and rapidly developing branches of engineering and science such as microand nano-fluidics, [1][2][3][4][5] biomimetics, 6,7 colloidal science, [8][9][10] and the design and operation of lab-on-a chip devices. 11,12 Investigations of wetting in prototypical nano-sized pores also provide a foundation for understanding adsorption on patterned substrates and the phenomenon of superhydrophobicity and superspreading.…”

Theory, simulations and the design of functionalized nanoparticles for biomedical applications: A Soft Matter Perspective

“…115,116 These equations can be obtained by noting that the Szleifer model can be thought of as a mean-field DFT model and the dynamical equations derived within the framework of dynamic DFT. 118,119 (A more heuristic derivation in the context of a generalised diffusion equation can also be found in ref. 106) In essence, the underlying assumption is that the system obeys Brownian motion.…”

Density functional study of condensation in capped capillaries

“…The surface tension of a planar interface of area A is given by [45] γA = Ω + P AL ⊥ , (C. 2) where Ω is the grand potential (8), P is the pressure (17), and L ⊥ is the dimension of the system in the direction perpendicular to the interface. The surface tensions γ wv and γ wl are obtained from (C.2) by setting P = P sat of saturation in (17) and using the density profiles ρ vap (y) and ρ liq (y) of the saturated vapor and liquid in contact with the planar wall, with ρ vap (y) → ρ vap and ρ liq (y) → ρ liq as y → ∞, respectively. The liquid-vapor surface tension γ lv is obtained in the same manner, by setting V (r) ≡ 0 in (14) and solving for the density profile ρ lv (y) of the free liquid-vapor interface, taking ρ lv (y) → ρ vap and ρ lv (y) → ρ liq as y → ±∞, respectively.…”