Oscillatory Density Profile Research Articles

Density functional study of non-isothermal hard sphere fluids

ABSTRACT Using a version of dynamical density functional theory, we explore a microscopic structure of hard-sphere fluids in the presence of a temperature gradient. When combined with the assumption of local equilibrium, this approach predicts the density profile in confinement and in bulk both in very good overall agreement with the results from the reverse non-equilibrium molecular dynamics simulation, which we used to impose a temperature gradient. Thus, the assumption of local equilibrium is found to be surprisingly accurate down to a microscopic scale even under a large temperature gradient. An oscillatory density profile, indicating the layering of particles, was observed in bulk as well as in confinement. This behaviour in the latter is well known and results from packing of hard spheres next to a wall. In the former, the oscillation is seen to emanate from the point of sharp change in temperature. However, our theoretical predictions greatly exaggerate the amplitude of oscillation when the temperature exhibits a sharp change over a very small distance.

Layering and phase transition of liquid aluminum confined by different substrates

Molecular dynamics simulation has been performed to study the layering in the nano-confined liquid aluminum, a magic phenomenon characterized by oscillatory density profile across the molecular thin films. Carbon and copper walls are served as the confinement substrates, which can bring about discrepancy in the layering behavior, coordination number and diffusion coefficient. We highlight the abnormal changes of potential energy and volume when the new layer forms, accompanying the distinctive changes of the average coordination number and diffusion coefficient, which is a typical character of phase transition. The role of pressure and temperature on the layering is also reported. Specifically, the pressure-confinement and temperature-confinement diagrams are plotted to further analyze the layering transition.

Simple theory for oscillatory charge profile in ionic liquids near a charged wall

The mesoscopic field theory for ionic systems (Ciach and Stell, 2000) is extended to the system with charged boundaries. A very simple expression for the excess grand potential functional of the charge density is developed. The size of hard-cores of ions is taken into account in the expression for the internal energy. The functional is suitable for a description of a distribution of ions in ionic liquids and ionic liquid mixtures with neutral components near a weakly charged wall. The Euler-Lagrange equation is obtained, and solved for a flat confining surface. An exponentially damped oscillatory charge density profile is obtained. The electrostatic potential for the restricted primitive model agrees with the simulation results on a semiquantitative level.

Influence of the fluid structure on the binding potential: Comparing liquid drop profiles from density functional theory with results from mesoscopic theory.

For a film of liquid on a solid surface, the binding potential g(h) gives the free energy as a function of the film thickness h and also the closely related (structural) disjoining pressure Π=-∂g/∂h. The wetting behaviour of the liquid is encoded in the binding potential and the equilibrium film thickness corresponds to the value at the minimum of g(h). Here, the method we developed in the work of Hughes et al. [J. Chem. Phys. 142, 074702 (2015)], and applied with a simple discrete lattice-gas model, is used with continuum density functional theory (DFT) to calculate the binding potential for a Lennard-Jones fluid and other simple liquids. The DFT used is based on fundamental measure theory and so incorporates the influence of the layered packing of molecules at the surface and the corresponding oscillatory density profile. The binding potential is frequently input in mesoscale models from which liquid drop shapes and even dynamics can be calculated. Here we show that the equilibrium droplet profiles calculated using the mesoscale theory are in good agreement with the profiles calculated directly from the microscopic DFT. For liquids composed of particles where the range of the attraction is much less than the diameter of the particles, we find that at low temperatures g(h) decays in an oscillatory fashion with increasing h, leading to highly structured terraced liquid droplets.

Simulating confined particles with a flat density profile.

Particle simulations confined by sharp walls usually develop an oscillatory density profile. For some applications, most notably soft matter liquids, this behavior is often unrealistic and one expects a monotonic density climb instead. To reconcile simulations with experiments, we propose mirror-and-shift boundary conditions where each interface is mapped to a distant part of itself. The main result is that the particle density increases almost monotonically from zero to bulk, over a short distance of about one particle diameter. The method is applied to simulate a polymer brush in explicit solvent, grafted on a flat silicon substrate. The simulated density profile agrees favorably with neutron reflectometry measurements and self-consistent field theory results.

Temperature dependence of local solubility of hydrophobic molecules in the liquid-vapor interface of water.

One important aspect of the hydrophobic effect is that solubility of small, nonpolar molecules in liquid water decreases with increasing temperature. We investigate here how the characteristic temperature dependence in liquid water persists or changes in the vicinity of the liquid-vapor interface. From the molecular dynamics simulation and the test-particle insertion method, the local solubility Σ of methane in the liquid-vapor interface of water as well as Σ of nonpolar solutes in the interface of simple liquids are calculated as a function of the distance z from the interface. We then examine the temperature dependence of Σ under two conditions: variation of Σ at fixed position z and that at fixed local solvent density around the solute molecule. It is found that the temperature dependence of Σ at fixed z depends on the position z and the system, whereas Σ at fixed local density decreases with increasing temperature for all the model solutions at any fixed density between vapor and liquid phases. The monotonic decrease of Σ under the fixed-density condition in the liquid-vapor interface is in accord with what we know for the solubility of nonpolar molecules in bulk liquid water under the fixed-volume condition but it is much robust since the solvent density to be fixed can be anything between the coexisting vapor and liquid phases. A unique feature found in the water interface is that there is a minimum in the local solubility profile Σ(z) on the liquid side of the interface. We find that with decreasing temperature the minimum of Σ grows and at the same time the first peak in the oscillatory density profile of water develops. It is likely that the minimum of Σ is due to the layering structure of the free interface of water.

Electrical double layers and differential capacitance in molten salts from density functional theory.

Classical density functional theory (DFT) is used to calculate the structure of the electrical double layer and the differential capacitance of model molten salts. The DFT is shown to give good qualitative agreement with Monte Carlo simulations in the molten salt regime. The DFT is then applied to three common molten salts, KCl, LiCl, and LiKCl, modeled as charged hard spheres near a planar charged surface. The DFT predicts strong layering of the ions near the surface, with the oscillatory density profiles extending to larger distances for larger electrostatic interactions resulting from either lower temperature or lower dielectric constant. Overall the differential capacitance is found to be bell-shaped, in agreement with recent theories and simulations for ionic liquids and molten salts, but contrary to the results of the classical Gouy-Chapman theory.

Nonequilibrium molecular dynamics simulation of water transport through carbon nanotube membranes at low pressure

Nonequilibrium molecular dynamics (NEMD) simulations are used to investigate pressure-driven water flow passing through carbon nanotube (CNT) membranes at low pressures (5.0 MPa) typical of real nanofiltration (NF) systems. The CNT membrane is modeled as a simplified NF membrane with smooth surfaces, and uniform straight pores of typical NF pore sizes. A NEMD simulation system is constructed to study the effects of the membrane structure (pores size and membrane thickness) on the pure water transport properties. All simulations are run under operating conditions (temperature and pressure difference) similar to a real NF processes. Simulation results are analyzed to obtain water flux, density, and velocity distributions along both the flow and radial directions. Results show that water flow through a CNT membrane under a pressure difference has the unique transport properties of very fast flow and a non-parabolic radial distribution of velocities which cannot be represented by the Hagen-Poiseuille or Navier-Stokes equations. Density distributions along radial and flow directions show that water molecules in the CNT form layers with an oscillatory density profile, and have a lower average density than in the bulk flow. The NEMD simulations provide direct access to dynamic aspects of water flow through a CNT membrane and give a view of the pressure-driven transport phenomena on a molecular scale.

SUBCRITICAL PATTERN FORMATION IN GRANULAR FLOW

We study a minimal model for the flow of granular material on a conveyor belt consisting of a staircase-like array of K vertically vibrated compartments. Applying a steady inflow rate Q to the top compartment, we determine the maximum value Q cr (K) for which a continuous flow through the system is possible. Beyond Q cr (K), which depends on the vibration strength and the dimensions of the system, a dense cluster forms in one of the first compartments and obstructs the flow. We find that the formation of this cluster is already announced belowQ cr (K) by the appearance of an oscillatory density profile along the entire length of the conveyor belt, with a distinct two-compartment wavelength. These model predictions concerning the breakdown of the granular flow admit an elegant explanation in terms of bifurcation theory. In particular, the subcritical oscillatory pattern is shown to be a side effect of the period doubling bifurcation by which the uniform density profile (associated with a smooth particle flow) becomes unstable. The effect turns out to be robust enough to survive the presence of a reasonable amount of noise and even certain qualitative modifications to the flux model. The density oscillations may therefore well be of practical value and provide a warning signal for imminent clustering on actual conveyor belts.

Orbital free ab initio simulation of surface freezing in a dilute Ga-Tl alloy

We have used the orbital-free ab initio molecular dynamics method to study the phenomenon of surface freezing in a liquid Ga-Tl alloy (nominal Tl concentration of 0.14). Several thermodynamic states are considered from a high temperature of 675 K to a low temperature of 350 K. At the higher temperature the Tl atoms segregate to the surface and form a liquid layer. Upon cooling the layer eventually crystallizes into a close-packed hexagonal solid layer. These results agree with experiments for a dilute liquid alloy of Tl in Ga, except for some overestimation of the surface freezing temperature. X-ray reflectivity data are also well reproduced. The wavelength of the oscillatory density profile coincides with that deduced from experiments and with that obtained in earlier computer simulation results. We also analyze the consequences of surface freezing on the structure of the liquid in contact with the solid layer, finding no influence whatsoever, as the structure basically coincides with the bulk liquid one.

Structure of free surface crystallization of Hg: a first attempt

Oscillatory density profiles (DP) are a well-known feature occurring at the free surfaces of liquid metals. We analyse first the layered structure of Hg at or below the melting temperature with a simple interaction model [J.-M. Bomont and J.-L. Bretonnet, J. Chem. Phys. 124 , 054504 (2006)], by molecular dynamics simulation in slab geometry. Then, by investigating the in-plane structure, some state transitions are observed. In particular, crystalline planes, whose axes are tilted by several degrees from the surface normal, appear and seem to be signalized by flat DP.

Computer simulation calculations of the free liquid surface of mercury

Oscillatory density profiles are a well-known feature occurring at the free surfaces of liquid metals. Recently, the thermophysical properties of expanded liquid mercury have been successfully investigated with an atypic isotropic effective interaction Bomont et al., J. Chem. Phys. 124, 054504 2006 . We use this model to investigate the liquid-vapor interface properties taking explicitly into account the density dependence of the potential related to the metal-nonmetal transition along the interface. Ionic and electronic density distributions along the normal of the interface display strong stratification and the calculated reflectivities compare well with experimental data.

Surface tension and scaling of critical nuclei in diatomic and triatomic fluids

Density functional theory has been used to investigate surface tension and scaling of critical clusters in fluids consisting of diatomic and rigid triatomic molecules. The atomic sites are hard spheres with attractive interactions obtained from the tail part of the Lennard-Jones potential. Asymmetry in attractive interactions between the atomic sites has been introduced to cause molecular orientation and oscillatory density profiles at liquid-vapor interfaces. The radial dependence of cluster surface tension in fluids showing modest orientation in unimolecular layer at the interface or no orientation at all resembles the surface tension behavior of clusters in simple monoatomic fluids, although the surface tension maximum becomes more pronounced with increasing chain length of the molecule. Surface tension of clusters having multiple oscillatory layers at the interface shows a prominent maximum at small cluster sizes; however, the surface tension of large clusters is lower than the planar value. The scaling relation for the number of molecules in the critical cluster and the nucleation barrier height developed by McGraw and Laaksonen [Phys. Rev. Lett. 76, 2754 (1996)] are well obeyed for fluids with little structure at liquid-vapor interface. However, fluids having enhanced interfacial structure show some deviation from the particle number scaling, and the barrier height scaling breaks up seriously.

Observation of Surface Layering in a Nonmetallic Liquid

Oscillatory density profiles (layers) have previously been observed at the free surfaces of liquid metals but not in other isotropic liquids. We have used x-ray reflectivity to study a molecular liquid, tetrakis(2-ethylhexoxy)silane. When cooled to T/Tc approximately 0.25 (well above the freezing point for this liquid), density oscillations appear at the surface. Lateral order within the layers is liquidlike. Our results confirm theoretical predictions that a surface-layered state will appear even in dielectric liquids at sufficiently low temperatures, if not preempted by freezing.

Density-functional theory of spherical electric double layers and zeta potentials of colloidal particles in restricted-primitive-model electrolyte solutions.

A density-functional theory is proposed to describe the density profiles of small ions around an isolated colloidal particle in the framework of the restricted primitive model where the small ions have uniform size and the solvent is represented by a dielectric continuum. The excess Helmholtz energy functional is derived from a modified fundamental measure theory for the hard-sphere repulsion and a quadratic functional Taylor expansion for the electrostatic interactions. The theoretical predictions are in good agreement with the results from Monte Carlo simulations and from previous investigations using integral-equation theory for the ionic density profiles and the zeta potentials of spherical particles at a variety of solution conditions. Like the integral-equation approaches, the density-functional theory is able to capture the oscillatory density profiles of small ions and the charge inversion (overcharging) phenomena for particles with elevated charge density. In particular, our density-functional theory predicts the formation of a second counterion layer near the surface of highly charged spherical particle. Conversely, the nonlinear Poisson-Boltzmann theory and its variations are unable to represent the oscillatory behavior of small ion distributions and charge inversion. Finally, our density-functional theory predicts charge inversion even in a 1:1 electrolyte solution as long as the salt concentration is sufficiently high.

Statistical mechanics of inhomogeneous model colloid—polymer mixtures

We describe two strategies for tackling the equilibrium statistical mechanics of inhomogeneous colloid—polymer mixtures treated in terms of the simple Asakura—Oosawa—Vrij (AO) model, in which colloid—colloid and colloid—polymer interactions are hard-sphere like, whereas the polymer—polymer interaction is zero (perfectly interpenetrating polymer spheres). The first strategy is based upon integrating out the degrees of freedom of the polymer spheres to obtain an effective one-component Hamiltonian for the colloids. This is particularly effective for small size ratios q = σp/σc < 0.1547, where σp and σc are the diameters of colloid and polymer spheres, respectively, since in this regime three and higher body contributions to the effective Hamiltonian vanish. In the second strategy we employ a geometry based density functional theory (DFT), specifically designed for the AO model but based on Rosenfeld's fundamental measure DFT for additive mixtures of hard-spheres, that treats colloid and polymer on an equal footing and which accounts for the fluid-fluid phase separation occurring for larger values of q. Using the DFT we investigate the properties of the ‘free’ interface between colloid-rich (liquid) and colloid-poor (gas) fluid phases and adsorption phenomena at the interface between the AO mixture and a hard-wall, for a wide range of size ratios. In particular, for q = 0.6 to 1.0, we find rich interfacial phenomena, including oscillatory density profiles at the free interface and novel wetting and layering phase transitions at the hard-wall-colloid gas interface. Where appropriate we compare our DFT results with those from computer simulations and experiment. We outline several very recent extensions of the basic AO model for which geometry based DFTs have also been developed. These include a model in which the effective polymer sphere—polymer sphere interaction is treated as a repulsive step function rather than ideal and one in which the depletant is a fluid of infinitely thin rods (needles) with orientational degrees of freedom rather than (non-interacting) polymer spheres. We comment on the differences between results obtained from these extensions and those of the basic AO model. Whilst the interfacial properties of the AO model share features in common with the those of simple (atomic) fluids, with the polymer reservoir density replacing the inverse temperature, we emphasize that there are important differences which are related to the many-body character of the effective one-component Hamiltonian.

Density functional theory for random sequential adsorption11This work is dedicatedto the memory of Yasha Rosenfeld.

We treat the non-equilibrium process of random sequential adsorption of hardparticles onto a solid substrate by means of a geometry-based density functionaltheory. As a prerequisite we solve the zero-dimensional case exactly and use it toconstruct density functionals in higher dimensions, permitting the treatment ofadsorption onto arbitrary spatially inhomogeneous substrates. As applications westudy the influence of a hard boundary of the adsorption region in theone-dimensional car-parking problem and for colloidal deposition on atwo-dimensional solid substrate. Comparing to our computer simulationresults, we find that the respective density functionals correctly predictthe oscillatory density profiles near the boundary, with amplitudes thatare considerably smaller than in the corresponding equilibrium models.

Molecular-scale density oscillations in water adjacent to a mica surface.

High-resolution specular x-ray reflectivity of the mica(001)-water interface under ambient conditions reveals oscillations in water oxygen density in the surface-normal direction, giving evidence of interfacial water ordering. The spacings between neighboring water layers in the near-surface, strongly oscillatory region are 2.5(2)-2.7(2) A, approximately the size of the water molecule. The density oscillations extend to about 10 A above the surface and do not strictly maintain a solvent-size periodicity as that in interfacial liquid metal and hard-sphere molecular liquids. We interpret this oscillatory density profile of the interfacial water as due to the "hard-wall" effect of the molecularly smooth mica surface.

Atom-laser coherence length and atomic standing waves

We consider the dynamical Bragg reflection of an atomic matter wave from a finite optical lattice; the resulting energy dependence of the transmittivity allows us to filter out the velocities of the incident atoms. In particular, we show how the coherence length of the incident beam can be inferred from the sharpness of the transmittivity as a function of the lattice intensity. For incident frequencies well inside the reflecting window, the interference of incident and reflected matter waves gives rise to an oscillatory density profile in front of the lattice which can be observed by means of light diffraction. The angular width of the diffracted peaks also provides information on the coherence length of the incident atomic beam.

Surface structure of liquid metals and the effect of capillary waves: X-ray studies on liquid indium

We report x-ray reflectivity (XR) and small angle off-specular diffuse scattering (DS) measurements from the surface of liquid Indium close to its melting point of $156^\circ$C. From the XR measurements we extract the surface structure factor convolved with fluctuations in the height of the liquid surface. We present a model to describe DS that takes into account the surface structure factor, thermally excited capillary waves and the experimental resolution. The experimentally determined DS follows this model with no adjustable parameters, allowing the surface structure factor to be deconvolved from the thermally excited height fluctuations. The resulting local electron density profile displays exponentially decaying surface induced layering similar to that previously reported for Ga and Hg. We compare the details of the local electron density profiles of liquid In, which is a nearly free electron metal, and liquid Ga, which is considerably more covalent and shows directional bonding in the melt. The oscillatory density profiles have comparable amplitudes in both metals, but surface layering decays over a length scale of $3.5\pm 0.6$ \AA for In and $5.5\pm 0.4$ \AA for Ga. Upon controlled exposure to oxygen, no oxide monolayer is formed on the liquid In surface, unlike the passivating film formed on liquid Gallium.