Benjamin D. Goddard

General Dynamical Density Functional Theory for Classical Fluids

We study the dynamics of a colloidal fluid in the full position-momentum phase space. The full underlying model consists of the Langevin equations including hydrodynamic interactions, which strongly influence the non-equilibrium properties of the system. For large systems, the number of degrees of freedom prohibit a direct solution of the Langevin equations and a reduced model is necessary, e.g. a projection of the dynamics to those of the reduced one-body distribution. We derive a generalized dynamical density functional theory (DDFT), the computational complexity of which is independent of the number of particles. We demonstrate that, in suitable limits, we recover existing DDFTs, which neglect either inertia, or hydrodynamic interactions, or both. In particular, in the overdamped limit we obtain a DDFT describing only the position distribution, and with a novel definition of the diffusion tensor. Futhermore, near equilibrium, our DDFT reduces to a Navier-Stokes-like equation but with additional non-local terms. We also demonstrate the very good agreement between the new DDFT and full stochastic calculations, as well as the large qualitative effects of inertia and hydrodynamic interactions.

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

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 scrutinize 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. For the overdamped limit we recover analogues of existing configuration-space DDFTs but with a novel diffusion tensor.

Fluid structure in the immediate vicinity of an equilibrium three-phase contact line and assessment of disjoining pressure models using density functional theory

We examine the nanoscale behavior of an equilibrium three-phase contact line in the presence of long-ranged intermolecular forces by employing a statistical mechanics of fluids approach, namely density functional theory (DFT) together with fundamental measure theory (FMT). This enables us to evaluate the predictive quality of effective Hamiltonian models in the vicinity of the contact line. In particular, we compare the results for mean field effective Hamiltonians with disjoining pressures defined through (I) the adsorption isotherm for a planar liquid film, and (II) the normal force balance at the contact line. We find that the height profile obtained using (I) shows good agreement with the adsorption film thickness of the DFT-FMT equilibrium density profile in terms of maximal curvature and the behavior at large film heights. In contrast, we observe that while the height profile obtained by using (II) satisfies basic sum rules, it shows little agreement with the adsorption film thickness of the DFT results. The results are verified for contact angles of 20, 40 and 60 degrees.

The overdamped limit of dynamic density functional theory: Rigorous results

Consider the overdamped limit for a system of interacting particles in the presence of hydrodynamic interactions. For two-body hydrodynamic interactions and one- and two-body poten- tials, a Smoluchowski-type evolution equation is rigorously derived for the one-particle distribution function. This new equation includes a novel definition of the diffusion tensor. A comparison with existing formulations of dynamic density functional theory is also made.

Multi-species dynamical density functional theory

We study the dynamics of a multi-species colloidal fluid in the full position-momentum phase space. We include both inertia and hydrodynamic interactions, which strongly influence the non-equilibrium properties of the system. Under minimal assumptions, we derive a dynamical density functional theory (DDFT), and, using an efficient numerical scheme based on spectral methods for integro-differential equations, demonstrate its excellent agreement with the full underlying Langevin equations. We utilise the DDFT formalism to elucidate the crucial effects of hydrodynamic interactions in multi-species systems.

Nanoscale fluid structure of liquid-solid-vapour contact lines for a wide range of contact angles

We study the nanoscale behaviour of the density of a simple fluid in the vicinity of an equilibrium contact line for a wide range of Young contact angles between 40 and 135 degrees. Cuts of the density profile at various positions along the contact line are presented, unravelling the apparent step-wise increase of the film height profile observed in contour plots of the density. The density profile is employed to compute the normal pressure acting on the substrate along the contact line. We observe that for the full range of contact angles, the maximal normal pressure cannot solely be predicted by the curvature of the adsorption film height, but is instead softened -- likely by the width of the liquid-vapour interface. Somewhat surprisingly however, the adsorption film height profile can be predicted to a very good accuracy by the Derjaguin-Frumkin disjoining pressure obtained from planar computations, as was first shown in [Nold et al., Phys. Fluids, 26, 072001, 2014] for contact angles less than 90 degrees, a result which here we show to be valid for the full range of contact angles. This suggests that while two-dimensional effects cannot be neglected for the computation of the normal pressure distribution along the substrate, one-dimensional planar computations of the Derjaguin-Frumkin disjoining pressure are sufficient to accurately predict the adsorption height profile.

Dynamical Density Functional Theory with Hydrodynamic Interactions in Confined Geometries

We study the dynamics of colloidal fluids in both unconfined geometries and when confined by a hard wall. Under minimal assumptions, we derive a dynamical density functional theory (DDFT) which includes hydrodynamic interactions (HI; bath-mediated forces). By using an efficient numerical scheme based on pseudospectral methods for integro-differential equations, we demonstrate its excellent agreement with the full underlying Langevin equations for systems of hard disks in partial confinement. We further use the derived DDFT formalism to elucidate the crucial effects of HI in confined systems.

ASYMPTOTICS-BASED CI MODELS FOR ATOMS: PROPERTIES, EXACT SOLUTION OF A MINIMAL MODEL FOR LI TO NE, AND APPLICATION TO ATOMIC SPECTRA ∗

Configuration-interaction (CI) models are approximations to the electronic Schro- dinger equation which are widely used for numerical electronic structure calculations in quantum chemistry. Based on our recent closed-form asymptotic results for the full atomic Schrodinger equa- tion in the limit of fixed electron number and large nuclear charge (SIAM J. Math. Anal., 41 (2009), pp. 631-664), we introduce a class of CI models for atoms which reproduce, at fixed finite model dimension, the correct Schrodinger eigenvalues and eigenstates in this limit. We solve exactly the ensuing minimal model for the second period atoms, Li to Ne, except for optimization of eigenvalues with respect to orbital dilation parameters, which is carried out numerically. The energy levels and eigenstates are in remarkably good agreement with experimental data (comparable to that of much larger scale numerical simulations in the literature) and facilitate a mathematical understanding of various spectral, chemical, and physical properties of small atoms.

EXPLICIT LARGE NUCLEAR CHARGE LIMIT OF ELECTRONIC GROUND STATES FOR Li, Be, B, C, N, O, F, Ne AND BASIC ASPECTS OF THE PERIODIC TABLE ∗

This paper is concerned with the Schrodinger equation for atoms and ions with N = 1 to 10 electrons. In the asymptotic limit of large nuclear charge Z, we determine explicitly the low-lying energy levels and eigenstates. The asymptotic energies and wavefunctions are in good quantitative agreement with experimental data for positive ions, and in excellent qualitative agree- ment even for neutral atoms (Z = N ). In particular, the predicted ground state spin and angular momentum quantum numbers ( 1 S for He, Be, Ne, 2 S for H and Li, 4 S for N, 2 P for B and F, and 3 P for C and O) agree with experiment in every case. The asymptotic Schrodinger ground states agree, up to small corrections, with the semiempirical hydrogen orbital configurations developed by Bohr, Hund, and Slater to explain the periodic table. In rare cases where our results deviate from this picture, such as the ordering of the lowest 1 D o and 3 S o states of the carbon isoelectronic sequence, experiment confirms our predictions and not Hunds.

Nonequilibrium molecular dynamics simulations of nanoconfined fluids at solid-liquid interfaces

We investigate the hydrodynamic properties of a Lennard-Jones fluid confined to a nanochannel using molecular dynamics simulations. For channels of different widths and hydrophilic-hydrophobic surface wetting properties, profiles of the fluid density, stress, and viscosity across the channel are obtained and analysed. In particular, we propose a linear relationship between the density and viscosity in confined and strongly inhomogeneous nanofluidic flows. The range of validity of this relationship is explored in the context of coarse grained models such as dynamic density functional-theory.