arXiv subjects

B. D. Goddard

Publications and source records attributed to B. D. Goddard.

9 recordsLinked to original sources

arXiv2025

Interactions between droplets in immiscible liquid suspensions and the influence of surfactants

We develop a general method for determining the effective interaction potential between two or more droplets suspended within a fluid phase. Our approach is based on classical density functional theory. Here, we apply the method to determine the interaction potential between oil droplets suspended in water and also consider the influence of adding a third species, alcohol. This ternary mixture is that found in the ouzo beverage. The ouzo system exhibits spontaneous emulsification when the neat spirit is mixed with water. The oil emulsion that forms has been observed to be surprisingly long-lived. Here we show that the alcohol in the system does indeed play a role in making the droplets more stable, by decreasing the oil-water interfacial tension and therefore also the strength of the attractive interactions between droplets. Within our theory, the surfactant nature of the alcohol can be enhanced without changing the bulk fluid thermodynamics. In fact, our theory can be used to model surfactant mixtures. In this model, the effective interaction between pairs of oil droplets can become repulsive, with a free-energy barrier to droplets merging, thus making them stable.

cond-mat.soft↗

arXiv2023

Stability of nanoparticle laden aerosol liquid droplets

We develop a model for the thermodynamics and evaporation dynamics of aerosol droplets of a liquid such as water, surrounded by the gas. When the temperature and the chemical potential (or equivalently the humidity) are such that the vapour phase is the thermodynamic equilibrium state, then of course droplets of the pure liquid evaporate over a relatively short time. However, if the droplets also contain nanoparticles or any other non-volatile solute, then the droplets can become thermodynamically stable. We show that the equilibrium droplet size depends strongly on the amount and solubility of the nanoparticles within, i.e. on the nature of the particle interactions with the liquid, and of course also on the vapour temperature and chemical potential. We develop a simple thermodynamic model for such droplets and compare predictions with results from a lattice density functional theory that takes as input the same particle interaction properties, finding very good agreement. We also use dynamical density functional theory to study the evaporation/condensation dynamics of liquid from/to droplets as they equilibrate with the vapour, thereby demonstrating droplet stability.

cond-mat.soft↗

arXiv2023

On the study of slow-fast dynamics, when the fast process has multiple invariant measures

Motivated by applications to mathematical biology, we study the averaging problem for slow-fast systems, {\em in the case in which the fast dynamics is a stochastic process with multiple invariant measures}. We consider both the case in which the fast process is decoupled from the slow process and the case in which the two components are fully coupled. We work in the setting in which the slow process evolves according to an Ordinary Differential Equation (ODE) and the fast process is a continuous time Markov Process with finite state space and show that, in this setting, the limiting (averaged) dynamics can be described as a random ODE (that is, an ODE with random coefficients.) Keywords. Multiscale methods, Processes with multiple equilibria, Averaging, Collective Navigation, Interacting Piecewise Deterministic Markov Processes.

math.PR↗

arXiv2022

The Mean Field Fokker-Planck Equation with Nonlinear No-flux Boundary Conditions

We consider the mean field Fokker-Planck equation subject to nonlinear no-flux boundary conditions, which necessarily arise when subjecting a system of Brownian particles interacting via a pair potential in a bounded domain. With the additional presence of an external potential $V_1$, we show, by analysing the linearised Fokker-Planck operator, that the spectral properties of the equilibrium densities can differ considerably when compared with previous studies, e.g., with periodic boundary conditions. Amongst other mean field models of complex many-body particle systems, we present numerical experiments encompassing in a wide range of physical applications, including: generalised exponential models (Gaussian, Morse); a Kuramoto model, for noisy coupled oscillators; and an Onsager model for liquid crystals. We showcase our results by using the numerical methods developed in the pseudospectral collocation scheme 2DChebClass.

math.NA↗

arXiv2021

Well-Posedness and Equilibrium Behaviour of Overdamped Dynamic Density Functional Theory

We establish the global well-posedness of overdamped dynamic density functional theory (DDFT): a nonlinear, nonlocal integro-partial differential equation used in statistical mechanical models of colloidal fluids, and other applications including nonlinear reaction-diffusion systems and opinion dynamics. With nonlinear no-flux boundary conditions, we determine the existence and uniqueness of the weak density and flux, subject to two-body hydrodynamic interactions (HI). We also show that the density is Lyapunov stable with respect to the usual (Helmholtz) free energy functional. Principally, this is done by rewriting the dynamics for the density in an implicit gradient flow form, resembling the classical Smoluchowski equation but with spatially inhomogeneous diffusion and advection tensors. We also rigorously show that the stationary density is independent of the HI tensors, and prove exponentially fast convergence to equilibrium.

math.AP↗

arXiv2020

The Singular Hydrodynamic Interactions Between Two Spheres In Stokes Flow

We study exact solutions for the slow viscous flow of an infinite liquid caused by two rigid spheres approaching each either along or parallel to their line of centres, valid at all separations. This goes beyond the applicable range of existing solutions for singular hydrodynamic interactions (HIs) which, for practical applications, are limited to the near-contact or far field region of the flow. For the normal component of the HI, by use of a bipolar coordinate system, we derive the stream function for the flow as $Re\to 0$ and a formula for the singular (squeeze) force between the spheres as an infinite series. We also obtain the asymptotic behaviour of the forces as the nondimensional separation between the spheres goes to zero and infinity, rigorously confirming and improving upon known results relevant to a widely accepted lubrication theory. Additionally, we recover the force on a sphere moving perpendicularly to a plane as a special case. For the tangential component, again by using a bipolar coordinate system, we obtain the corresponding infinite series expression of the (shear) singular force between the spheres. All results hold for retreating spheres, consistent with the reversibility of Stokes flow. We demonstrate substantial differences in numerical simulations of colloidal fluids when using the present theory compared with existing multipole methods. Furthermore, we show that the present theory preserves positive definiteness of the resistance matrix $\boldsymbol{R}$ in a number of situations in which positivity is destroyed for multipole/perturbative methods.

physics.flu-dyn↗

arXiv2020

Modelling inelastic granular media using Dynamical Density Functional Theory

We construct a new mesoscopic model for granular media using Dynamical Density Functional Theory (DDFT). The model includes both a collision operator to incorporate inelasticity and the Helmholtz free energy functional to account for external potentials, interparticle interactions and volume exclusion. We use statistical data from event-driven microscopic simulations to determine the parameters not given analytically by the closure relations used to derive the DDFT. We numerically demonstrate the crucial effects of each term in the DDFT, and the importance of including an accurately parametrised pair correlation function.

cond-mat.stat-mech↗

arXiv2019

A derivation of the Liouville equation for hard particle dynamics with non-conservative interactions

The Liouville equation is of fundamental importance in the derivation of continuum models for physical systems which are approximated by interacting particles. However, when particles undergo instantaneous interactions such as collisions, the derivation of the Liouville equation must be adapted to exclude non-physical particle positions, and include the effect of instantaneous interactions. We present the weak formulation of the Liouville equation for interacting particles with general particle dynamics and interactions, and discuss the results using an example.

cond-mat.stat-mech↗

arXiv2018

Non-adiabatic transitions in multiple dimensions

We consider non-adiabatic transitions in multiple dimensions, which occur when the Born-Oppenheimer approximation breaks down. We present a general, multi-dimensional algorithm which can be used to accurately and efficiently compute the transmitted wavepacket at an avoided crossing. The algorithm requires only one-level Born-Oppenheimer dynamics and local knowledge of the potential surfaces. Crucially, in contrast to standard methods in the literature, we compute the whole wavepacket, including its phase, rather than simply the transition probability. We demonstrate the excellent agreement with full quantum dynamics for a range of examples in two dimensions. We also demonstrate surprisingly good agreement for a system with a full conical intersection.

physics.chem-ph↗