Dynamic density functional theory with inertia and background flow.

We present dynamic density functional theory (DDFT) incorporating general inhomogeneous, incompressible, time-dependent background flows and inertia, describing externally driven passive colloidal systems out of equilibrium. We start by considering the underlying nonequilibrium Langevin dynamics, including the effect of the local velocity of the surrounding liquid bath, to obtain the nonlinear, nonlocal partial differential equations governing the evolution of the (coarse-grained) density and velocity fields describing the dynamics of colloids. In addition, we show both with heuristic arguments, and by numerical solution, that our equations and solutions agree with existing DDFTs in the overdamped (high friction) limit. We provide numerical solutions that model the flow of hard spheres, in both unbounded and confined domains, and compare with previously derived DDFTs with and without the background flow.

Surfactant Control of Coffee Ring Formation in Carbon Nanotube Suspensions.

The coffee ring effect regularly occurs during the evaporation of colloidal droplets and is often undesirable. Here we show that adding a specific concentration of a surfactant can mitigate this effect. We have conducted experiments on aqueous suspensions of carbon nanotubes that were prepared with cationic surfactant dodecyltrimethylammonium bromide added at 0.2, 0.5, 1, 2, 5, and 10 times the critical micelle concentration. Colloidal droplets were deposited on candidate substrates for printed electronics with varying wetting characteristics: glass, polyethylene terephthalate, fluoroethylene propylene copolymer, and polydimethylsiloxane. Following drying, four pattern types were observed in the final deposits: dot-like, uniform, coffee ring deposits, and combined patterns (coffee ring with a dot-like central deposit). Evaporation occurred predominantly in constant contact radius mode for most pattern types, except for some cases that led to uniform deposits in which early stage receding of the contact line occurred. Image analysis and profilometry yielded deposit thicknesses, allowing us to identify a coffee ring subfeature in all uniform deposits and to infer the percentage coverage in all cases. Importantly, a critical surfactant concentration was identified for the generation of highly uniform deposits across all substrates. This concentration resulted in visually uniform deposits consisting of a coffee ring subfeature with a densely packed center, generated from two distinct evaporative phases.

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 gas. When the temperature and the chemical potential (or equivalently the humidity) are such that the vapor phase is in 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 vapor 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 vapor, thereby demonstrating droplet stability.

Stochastic transitions: Paths over higher energy barriers can dominate in the early stages.

The time evolution of many physical, chemical, and biological systems can be modeled by stochastic transitions between the minima of the potential energy surface describing the system of interest. We show that in cases where there are two (or more) possible pathways that the system can take, the time available for the transition to occur is crucially important. The well-known results of the reaction rate theory for determining the rates of transitions apply in the long-time limit. However, at short times, the system can, instead, choose to pass over higher energy barriers with a much higher probability, as long as the distance to travel in phase space is shorter. We construct two simple models to illustrate this general phenomenon. We also apply a version of the geometric minimum action method algorithm of Vanden-Eijnden and Heymann [J. Chem. Phys. 128, 061103 (2008)] to determine the most likely path at both short and long times.

Density Distribution in Soft Matter Crystals and Quasicrystals.

The density distribution in solids is often represented as a sum of Gaussian peaks (or similar functions) centered on lattice sites or via a Fourier sum. Here, we argue that representing instead the logarithm of the density distribution via a Fourier sum is better. We show that truncating such a representation after only a few terms can be highly accurate for soft matter crystals. For quasicrystals, this sum does not truncate so easily, nonetheless, representing the density profile in this way is still of great use, enabling us to calculate the phase diagram for a three-dimensional quasicrystal-forming system using an accurate nonlocal density functional theory.

Which Wave Numbers Determine the Thermodynamic Stability of Soft Matter Quasicrystals?

For soft matter to form quasicrystals an important ingredient is to have two characteristic length scales in the interparticle interactions. To be more precise, for stable quasicrystals, periodic modulations of the local density distribution with two particular wave numbers should be favored, and the ratio of these wave numbers should be close to certain special values. So, for simple models, the answer to the title question is that only these two ingredients are needed. However, for more realistic models, where in principle all wave numbers can be involved, other wave numbers are also important, specifically those of the second and higher reciprocal lattice vectors. We identify features in the particle pair interaction potentials that can suppress or encourage density modes with wave numbers associated with one of the regular crystalline orderings that compete with quasicrystals, enabling either the enhancement or suppression of quasicrystals in a generic class of systems.

Dynamical Density Functional Theory for the Evaporation of Droplets of Nanoparticle Suspension.

We develop a lattice gas model for the drying of droplets of a nanoparticle suspension on a planar surface, using dynamical density functional theory (DDFT) to describe the time evolution of the solvent and nanoparticle density profiles. The DDFT assumes a diffusive dynamics but does not include the advective hydrodynamics of the solvent, so the model is relevant to highly viscous or near to equilibrium systems. Nonetheless, we see an equivalent of the coffee-ring stain effect, but in the present model it occurs for thermodynamic rather the fluid-mechanical reasons. The model incorporates the effect of phase separation and vertical density variations within the droplet and the consequence of these on the nanoparticle deposition pattern on the surface. We show how to include the effect of slip or no-slip at the surface and how this is related to the receding contact angle. We also determine how the equilibrium contact angle depends on the microscopic interaction parameters.

Three-Dimensional Icosahedral Phase Field Quasicrystal.

We investigate the formation and stability of icosahedral quasicrystalline structures using a dynamic phase field crystal model. Nonlinear interactions between density waves at two length scales stabilize three-dimensional quasicrystals. We determine the phase diagram and parameter values required for the quasicrystal to be the global minimum free energy state. We demonstrate that traits that promote the formation of two-dimensional quasicrystals are extant in three dimensions, and highlight the characteristics required for three-dimensional soft matter quasicrystal formation.

Quasicrystalline order and a crystal-liquid state in a soft-core fluid.

A two-dimensional system of soft particles interacting via a two-length-scale potential is studied. Density functional theory and Brownian dynamics simulations reveal a fluid phase and two crystalline phases with different lattice spacing. Of these the larger lattice spacing phase can form an exotic periodic state with a fraction of highly mobile particles: a crystal liquid. Near the transition between this phase and the smaller lattice spacing phase, quasicrystalline structures may be created by a competition between linear instability at one scale and nonlinear selection of the other.

Relationship between local molecular field theory and density functional theory for non-uniform liquids.

The local molecular field theory (LMF) developed by Weeks and co-workers has proved successful for treating the structure and thermodynamics of a variety of non-uniform liquids. By reformulating LMF in terms of one-body direct correlation functions we recast the theory in the framework of classical density functional theory (DFT). We show that the general LMF equation for the effective reference potential φ(R)(r) follows directly from the standard mean-field DFT treatment of attractive interatomic forces. Using an accurate (fundamental measures) DFT for the non-uniform hard-sphere reference fluid we determine φ(R)(r) for a hard-core Yukawa liquid adsorbed at a planar hard wall. In the approach to bulk liquid-gas coexistence we find the effective potentials exhibit rich structure that can include damped oscillations at large distances from the wall as well as the repulsive hump near the wall required to generate the low density "gas" layer characteristic of complete drying. We argue that it would be difficult to obtain the same level of detail from other (non-DFT based) implementations of LMF. LMF emphasizes the importance of making an intelligent division of the interatomic pair potential of the full system into a reference part and a remainder that can be treated in mean-field approximation. We investigate different divisions for an exactly solvable one-dimensional model where the pair potential has a hard-core plus a linear attractive tail. Results for the structure factor and the equation of state of the uniform fluid show that including a significant portion of the attraction in the reference system can be much more accurate than treating the full attractive tail in mean-field approximation. We discuss further aspects of the relationship between LMF and DFT.

Phase separation dynamics in a two-dimensional magnetic mixture.

Based on classical density functional theory (DFT), we investigate the demixing phase transition of a two-dimensional, binary Heisenberg fluid mixture. The particles in the mixture are modeled as Gaussian soft spheres, where one component is characterized by an additional classical spin-spin interaction of Heisenberg type. Within the DFT we treat the particle interactions using a mean-field approximation. For certain magnetic coupling strengths, we calculate phase diagrams in the density-concentration plane. For sufficiently large coupling strengths and densities, we find a demixing phase transition driven by the ferromagnetic interactions of the magnetic species. We also provide a microscopic description (i.e., density profiles) of the resulting non-magnetic/magnetic fluid-fluid interface. Finally, we investigate the phase separation using dynamical density functional theory, considering both nucleation processes and spinodal demixing.

Dynamical density functional theory for molecular and colloidal fluids: a microscopic approach to fluid mechanics.

In recent years, a number of dynamical density functional theories (DDFTs) have been developed for describing the dynamics of the one-body density of both colloidal and atomic fluids. In the colloidal case, the particles are assumed to have stochastic equations of motion and theories exist for both the case when the particle motion is overdamped and also in the regime where inertial effects are relevant. In this paper, we extend the theory and explore the connections between the microscopic DDFT and the equations of motion from continuum fluid mechanics. In particular, starting from the Kramers equation, which governs the dynamics of the phase space probability distribution function for the system, we show that one may obtain an approximate DDFT that is a generalization of the Euler equation. This DDFT is capable of describing the dynamics of the fluid density profile down to the scale of the individual particles. As with previous DDFTs, the dynamical equations require as input the Helmholtz free energy functional from equilibrium density functional theory (DFT). For an equilibrium system, the theory predicts the same fluid one-body density profile as one would obtain from DFT. Making further approximations, we show that the theory may be used to obtain the mode coupling theory that is widely used for describing the transition from a liquid to a glassy state.

Modelling approaches to the dewetting of evaporating thin films of nanoparticle suspensions.

We review recent experiments on dewetting thin films of evaporating colloidal nanoparticle suspensions (nanofluids) and discuss several theoretical approaches to describe the ongoing processes including coupled transport and phase changes. These approaches range from microscopic discrete stochastic theories to mesoscopic continuous deterministic descriptions. In particular, we describe (i) a microscopic kinetic Monte Carlo model, (ii) a dynamical density functional theory and (iii) a hydrodynamic thin film model. Models (i) and (ii) are employed to discuss the formation of polygonal networks, spinodal and branched structures resulting from the dewetting of an ultrathin 'postcursor film' that remains behind a mesoscopic dewetting front. We highlight, in particular, the presence of a transverse instability in the evaporative dewetting front, which results in highly branched fingering structures. The subtle interplay of decomposition in the film and contact line motion is discussed. Finally, we discuss a simple thin film model (iii) of the hydrodynamics on the mesoscale. We employ coupled evolution equations for the film thickness profile and mean particle concentration. The model is used to discuss the self-pinning and depinning of a contact line related to the 'coffee-stain' effect. In the course of the review we discuss the advantages and limitations of the different theories, as well as possible future developments and extensions.

Structural crossover in a model fluid exhibiting two length scales: Repercussions for quasicrystal formation.

We investigate the liquid state structure of the two-dimensional model introduced by Barkan et al. [Phys. Rev. Lett. 113, 098304 (2014)10.1103/PhysRevLett.113.098304], which exhibits quasicrystalline and other unusual solid phases, focusing on the radial distribution function g(r) and its asymptotic decay r→∞. For this particular model system, we find that as the density is increased there is a structural crossover from damped oscillatory asymptotic decay with one wavelength to damped oscillatory asymptotic decay with another distinct wavelength. The ratio of these wavelengths is ≈1.932. Following the locus in the phase diagram of this structural crossover leads directly to the region where quasicrystals are found. We argue that identifying and following such a crossover line in the phase diagram towards higher densities where the solid phase(s) occur is a good strategy for finding quasicrystals in a wide variety of systems. We also show how the pole analysis of the asymptotic decay of equilibrium fluid correlations is intimately connected with the nonequilibrium growth or decay of small-amplitude density fluctuations in a bulk fluid.

Density functional theory for the crystallization of two-dimensional dipolar colloidal alloys.

Two-dimensional mixtures of dipolar colloidal particles with different dipole moments exhibit extremely rich self-assembly behaviour and are relevant to a wide range of experimental systems, including charged and super-paramagnetic colloids at liquid interfaces. However, there is a gap in our understanding of the crystallization of these systems because existing theories such as integral equation theory and lattice sum methods can only be used to study the high temperature fluid phase and the zero-temperature crystal phase, respectively. In this paper we bridge this gap by developing a density functional theory (DFT), valid at intermediate temperatures, in order to study the crystallization of one and two-component dipolar colloidal monolayers. The theory employs a series expansion of the excess Helmholtz free energy functional, truncated at second order in the density, and taking as input highly accurate bulk fluid direct correlation functions from simulation. Although truncating the free energy at second order means that we cannot determine the freezing point accurately, our approach allows us to calculate ab initio both the density profiles of the different species and the symmetry of the final crystal structures. Our DFT predicts hexagonal crystal structures for one-component systems, and a variety of superlattice structures for two-component systems, including those with hexagonal and square symmetry, in excellent agreement with known results for these systems. The theory also provides new insights into the structure of two-component systems in the intermediate temperature regime where the small particles remain molten but the large particles are frozen on a regular lattice.

Modelling the evaporation of nanoparticle suspensions from heterogeneous surfaces.

We present a Monte Carlo (MC) grid-based model for the drying of drops of a nanoparticle suspension upon a heterogeneous surface. The model consists of a generalised lattice-gas in which the interaction parameters in the Hamiltonian can be varied to model different properties of the materials involved. We show how to correctly choose the interactions, to minimise the effects of the underlying grid so that hemispherical droplets form. We also include the effects of surface roughness to examine the effects of contact-line pinning on the dynamics. When there is a 'lid' above the system, which prevents evaporation, equilibrium drops form on the surface, which we use to determine the contact angle and how it varies as the parameters of the model are changed. This enables us to relate the interaction parameters to the materials used in applications. The model has also been applied to drying on heterogeneous surfaces, in particular to the case where the suspension is deposited on a surface consisting of a pair of hydrophilic conducting metal surfaces that are either side of a band of hydrophobic insulating polymer. This situation occurs when using inkjet printing to manufacture electrical connections between the metallic parts of the surface. The process is not always without problems, since the liquid can dewet from the hydrophobic part of the surface, breaking the bridge before the drying process is complete. The MC model reproduces the observed dewetting, allowing the parameters to be varied so that the conditions for the best connection can be established. We show that if the hydrophobic portion of the surface is located at a step below the height of the neighbouring metal, the chance of dewetting of the liquid during the drying process is significantly reduced.

Dynamical density functional theory analysis of the laning instability in sheared soft matter.

Using dynamical density functional theory (DDFT) methods we investigate the laning instability of a sheared colloidal suspension. The nonequilibrium ordering at the laning transition is driven by nonaffine particle motion arising from interparticle interactions. Starting from a DDFT which incorporates the nonaffine motion, we perform a linear stability analysis that enables identification of the regions of parameter space where lanes form. We illustrate our general approach by applying it to a simple one-component fluid of soft penetrable particles.

Soft core fluid in a quenched matrix of soft core particles: a mobile mixture in a model gel.

We present a density-functional study of a binary phase-separating mixture of soft core particles immersed in a random matrix of quenched soft core particles of larger size. This is a model for a binary polymer mixture immersed in a cross-linked rigid polymer network. Using the replica "trick" for quenched-annealed mixtures we derive an explicit density functional theory that treats the quenched species on the level of its one-body density distribution. The relation to a set of effective external potentials acting on the annealed components is discussed. We relate matrix-induced condensation in bulk to the behavior of the mixture around a single large particle. The interfacial properties of the binary mixture at a surface of the quenched matrix display a rich interplay between capillary condensation inside the bulk matrix and wetting phenomena at the matrix surface.

Screening in Yukawa fluid mixtures.

The effective pair potential between mesoscopic charged particles in a neutralizing background medium takes a Yukawa form exp(-lambda r)/r with screening length lambda(-1). We consider a dilute suspension of such Yukawa particles dispersed in a solvent with correlation length xi

Density functional theory for the freezing of soft-core fluids.

We present a simple density functional theory for the solid phases of systems of particles interacting via soft-core potentials. In particular, we apply the theory to particles interacting via repulsive point Yukawa and Gaussian pair potentials. We find qualitative agreement with the established phase diagrams for these systems. The theory is able to account for the bcc-fcc solid transitions of both systems and the re-entrant melting that the Gaussian system exhibits.