Dynamic density functional theory with hydrodynamic interactions: Theoretical development and application in the study of phase separation in gas-liquid systems

Abstract

Building on recent developments in dynamic density functional theory, we have developed a version of the theory that includes hydrodynamic interactions. This is achieved by combining the continuity and momentum equations eliminating velocity fields, so the resulting model equation contains only terms related to the fluid density and its time and spatial derivatives. The new model satisfies simultaneously continuity and momentum equations under the assumptions of constant dynamic or kinematic viscosity and small velocities and/or density gradients. We present applications of the theory to spinodal decomposition of subcritical temperatures for one-dimensional and three-dimensional density perturbations for both a van der Waals fluid and for a lattice gas model in mean field theory. In the latter case, the theory provides a hydrodynamic extension to the recently studied dynamic mean field theory. We find that the theory correctly describes the transition from diffusive phase separation at short times to hydrodynamic behaviour at long times.

Nucleation and Post-Nucleation Growth in Diffusion-Controlled and Hydrodynamic Theory of Solidification

“…Since numerical solution of Equations (2-6) in 3D proved computationally rather demanding, we propose here a simplified HPFC model (sHPFC) that relies on linearized hydrodynamics. We assume that the velocity and density gradients are small, the advection term is omitted, and then the time derivative of the continuity equation is inserted into the divergence of the equation for the momentum transport, which is the way we obtain a specific form of the DDFT/HI model by Kikkinides and Monson that was successfully applied for describing capillary waves on the nanoscale [40]: [30]). Besides the 3D crystalline phases, see the presence of 2D periodic phases such as the triangular rod phase and the lamellar phase.…”

Classical dynamical density functional theory: from fundamentals to applications

“…The inertia of particles is included in hydrodynamic DDFTs in Refs. [317,319,320,[400][401][402]. Donev and Vanden-Eijnden [238] obtained a hydrodynamic DDFT that also includes the fluctuations (averaging then recovers the result from Rex and Löwen [41]).…”

“…(Historically, however, they also formed a basis for the derivation of overdamped DDFT [11], see Section 3.3.1.) Eliminating the velocity or momentum density field from the coupled equations gives a second-order equation of motion for the one-body density [402,504].…”

Density functional study of non-isothermal hard sphere fluids

“…Broadly speaking, DDFT has been developed following two distinct approaches. The starting point for the first approach [7][8][9][10][11][12][13][14][15][16][17] is the lowest order equation in the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy [18], which dictates the time evolution of the single particle distribution function f 1 (r, v, t), expressing the probability density of finding a particle at r with velocity v at time t. The equations of motion for the density, momentum, and energy (or temperature) fields are obtained by averaging proper microscopic expressions for these quantities with respect to f 1 (r, v, t). This is essentially the Irving-Kirkwood procedure [19] and gives rise to integrals involving the pair distribution function f 2 (r, v, r , v , t), the probability density of finding a pair of particles, one at (r, v) and the other at (r , v ) at time t. To evaluate these integrals, one must introduce various approximations typically informed by recent techniques from the kinetic theory [20][21][22][23].…”