Fundamental Measure Theory Research Articles

Lattice Fundamental Measure Theory Beyond 0D Cavities: Dimers on Square Lattices

Using classical density functional theory, we study the behavior of dimers, i.e. hard rods of length L=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L=2$$\\end{document}, on a two-dimensional cubic lattice. For deriving a free energy functional, we employ Levy’s prescription which is based on the minimization of a microscopic free energy with respect to the many-body probability under the constraint of a fixed density profile. Using that, we recover the "0D cavity" functional originally found by Lafuente and Cuesta and derive an extension by applying a more general "cluster density functional theory" method introduced by Lafuente and Cuesta as well. Moreover, we introduce a new free energy functional, which is based on approximated configurational probabilities. Both derived free energy functionals are exact on cavities that can hold at most two particles simultaneously. The first functional allows to improve the prediction of the free energy in bulk and both of them improve the prediction in highly confined systems, especially for high packing fractions, in comparison to the "0D cavity" functional.

Density functional theory for responsive hard-sphere fluids

We investigate the equilibrium properties of responsive hard-sphere fluids, where the particle size is a coarse-grained property that fluctuates within an internal free-energy landscape, responding to changes in concentration or the application of external fields. For this purpose, we employ a generalised density functional theory for responsive colloids based on the fundamental measure theory for polydisperse hard-sphere mixtures. We find that increasing particle softness yields a substantial reduction in mean particle size, volume fraction and pressure. We also examine the density profiles and size segregation of responsive hard-sphere fluids subjected to three representative external potentials: a planar hard wall, gravitational and Archimedes buoyant fields. We observe that increasing stiffness or particle concentration enhances density oscillations, converging to the behaviour of a monodisperse fluid. In a gravitational field, size segregation occurs, with smaller particles accumulating at the bottom due to sedimentation. This effect is more pronounced when the Archimedes force is included, causing larger colloids to accumulate at the top, leading to density oscillations in this region that intensify with particle softness, an effect not observed in monodisperse systems. Finally, we demonstrate that responsiveness leads to distinct behaviour compared to conventional polydisperse non-responsive fluids, in which particle compression is not allowed.

Fingerprints of ordered self-assembled structures in the liquid phase of a hard-core, square-shoulder system.

We investigate the phase ordering (pattern formation) of systems of two-dimensional core-shell particles using Monte Carlo (MC) computer simulations and classical density functional theory (DFT). The particles interact via a pair potential having a hard core and a repulsive square shoulder. Our simulations show that on cooling, the liquid state structure becomes increasingly characterized by long wavelength density modulations and on further cooling forms a variety of other phases, including clustered, striped, and other patterned phases. In DFT, the hard core part of the potential is treated using either fundamental measure theory or a simple local density approximation, whereas the soft shoulder is treated using the random phase approximation. The different DFTs are benchmarked using large-scale grand-canonical-MC and Gibbs-ensemble-MC simulations, demonstrating their predictive capabilities and shortcomings. We find that having the liquid state static structure factor S(k) for wavenumber k is sufficient to identify the Fourier modes governing both the liquid and solid phases. This allows us to identify from easier-to-obtain liquid state data the wavenumbers relevant to the periodic phases and to predict roughly where in the phase diagram these patterned phases arise.

Toward Modeling the Structure of Electrolytes at Charged Mineral Interfaces Using Classical Density Functional Theory.

The organization of water molecules and ions between charged mineral surfaces determines the stability of colloidal suspensions and the strength of phase-separated particulate gels. In this article, we assemble a density functional that measures the free energy due to the interaction of water molecules and ions in electric double layers. The model accounts for the finite size of the particles using fundamental measure theory, hydrogen-bonding between water molecules using Wertheim's statistical association theory, long-range dispersion interactions using Barker and Henderson's high-temperature expansion, electrostatic correlations using a functionalized mean-spherical approximation, and Coulomb forces through the Poisson equation. These contributions are shown to produce highly correlated structures, aptly rendering the layering of counterions and co-ions at highly charged surfaces and permitting the solvation of ions and surfaces to be measured by a combination of short-range associations and long-ranged attractions. The model is tested in a planar geometry near soft, charged surfaces to reproduce the structure of water near graphene and mica. For mica surfaces, explicitly representing the density of the outer oxygen layer of the exposed silica tetrahedra allows water molecules to hydrogen-bond to the solid. When electrostatic interactions are included, water molecules assume a hybrid character, being accounted for implicitly in the dielectric constant but explicitly otherwise. The disjoining pressure between approaching like-charged surfaces is calculated, demonstrating the model's ability to probe pressure oscillations that arise during the expulsion of ions and water layers from the interfacial gap and predict strong interattractive stresses that form at narrow interfacial spacing when the surface charge is overscreened. This interattractive stress arises not due to in-plane correlations under strong electrostatic coupling but due to the out-of-plane structuring of associating ions and water molecules.

Modeling the Electric Double Layer at the Liposome Vesicle via Classical Density Functional Theory: Solution of Poisson's Equations for Curved Membranes.

The electric double layer at the liposome vesicle membrane has been investigated by a modified fundamental-measure theory in the framework of the restricted primitive model. An analytical equation has been obtained for the mean electrostatic potential (MEP) by solving Poisson's equation for curved membranes. This study investigates the influence of vesicle size, membrane thickness, surface charges, and electrolyte concentration on the structure, composition, and width of electric double layers (EDLs) on the inner and outer membrane walls. Our findings indicate that a thin and denser layer of ions is formed at the concave wall of the membrane (inner wall) compared to that at the outer membrane. As expected, the width of the diffuse layer decreases with the concentration and surface charge. Also, when the surface charges on both concave and convex walls are the same, the absolute value of MEPs on the inner membrane, concave wall, is greater than that on the convex wall. We have also investigated the diffuse potential, which decreases with concentration, membrane thickness, and cavity size, whereas it increases with surface charges. As we expect, the contact density of counterions at the inner concave wall of the vesicle cavity is always greater than the corresponding value at the convex wall, whereas this trend reverses for co-ions. Also, the contact density of counterions (co-ions) at the inner wall decreases (increases) with cavity size, whereas it increases at the outer wall (decreases). Finally, depletion of co-ions occurs at the membrane walls with enhancement in surface charges.

Biaxial nematic order in fundamental measure theory.

Liquid crystals consisting of biaxial particles can exhibit a much richer phase behavior than their uniaxial counterparts. Usually, one has to rely on simulation results to understand the phase diagram of these systems since very few analytical results exist. In this work, we apply fundamental measure theory, which allows us to derive free energy functionals for hard particles from first principles and with high accuracy, to systems of hard cylinders, cones, and spherotriangles. We provide a general recipe for incorporating biaxial liquid crystal order parameters into fundamental measure theory and use this framework to obtain the phase boundaries for the emergence of orientational order in the considered systems. Our results provide insights into the phase behavior of biaxial nematic liquid crystals and, in particular, into methods for their analytical investigation.

Confinement Effects in Droplet Formation on a Solid Particle.

Formation of a droplet around a spherical solid particle in supersaturated vapor is considered. The number and stability of equilibrium solutions in a closed small system are studied in the canonical ensemble in comparison to an open system in the grand canonical ensemble. Depending on the system's parameters, two modes exist in the canonical ensemble: the first one with only one solution and the second one with three solutions; the presence of the third solution is due to confinement. The analysis is conducted first on a macroscopic thermodynamic level of description, and then the results are supported by studies within two versions of classical density functional theory: the square-gradient approximation with the Carnahan-Starling equation of state for hard spheres on a completely wettable particle and the random-phase approximation with the fundamental measure theory on a poorly wettable particle. In the latter case, a solution breaking the spherical symmetry is observed at a small total number of molecules.

Density Functional Theory with Fundamental Measure Theory for Stable Drops and Bubbles on Lyophilic and Lyophobic Nucleation Cores

Density Functional Theory with Fundamental Measure Theory for Stable Drops and Bubbles on Lyophilic and Lyophobic Nucleation Cores

Effect of solvent on electric double layer capacitance and over screening inside spherical cavity by density functional theory

In this article, the solvent primitive model (SPM) and the restricted primitive model (RPM) in the framework of modified fundamental measure theory are compared for the prediction of the electric double layer (EDL) at the contact surface of the concave wall of a charged spherical cavity. The structure of EDL, EDL capacitance, Stern and diffuse contributions, zeta potential, and over screening phenomenon are predicted and compared with the two models. Our results show that considering the solvent molecules as hard spheres according to SPM leads to an appearance of layering structure and oscillatory behavior of density profiles inside the cavity. Also, the amount of ion adsorption in SPM is greater than that of RPM at the same condition. As a result of more counter-ion adsorption in comparison to co-ion adsorption, the total net charge inside the cavity increases. That way, the predicted amount of EDL capacitance and its diffuse contribution by SPM are greater than the corresponding values by RPM. Note that the Stern contribution to the capacitance has the same value in both models at fixed and equal sizes of ions and solvent molecules. Finally, the mean electrostatic potential (MEP) in the cavity, which is strongly affected by more counter-ion adsorption in SPM, shows evidence of an over screening phenomenon in the stern region at a lower concentration in comparison to RPM. Also, the slope of MEP is higher for SPM (in the Stern region), which leads to lower absolute values for its zeta potential.

Particle trapping by charged spherical cavity and its effect on ions adsorption, mean electric potential, and EDL capacitance curve

A model has been presented to investigate the electric double layer at the wall of the spherical cavity with trapped charged mobile particle. The modified Fundamental Measure Theory has been employed to study the role of the trapped particle on the amount of net ion charge and electric double layer capacitance. Our results indicate that, in the presence of the mobile particle in somerange of low electric potentials, total net ion charges into the cavity and therefore EDL capacitance increase if the charges on the mobile particle and the cavity are of the opposite sign. This range decreases with increasing bulk electrolyte concentration and decreasing the cavity curvature. At high electric potentials the role of the excluded volume due to the particle leads to a decrease in the net ion charges. The behavior of the mean electric potential, which has been explained by the local volume charge density, is significantly influenced by the position of the particle intothe cavity, which is determined by its charge and size. Finally, the ions adsorption into the cavity at different bulk concentrations has been investigated to predict the amount of ions transfer from the cavity wall.

Classical density functional theory for interfacial properties of hydrogen, helium, deuterium, neon, and their mixtures.

We present a classical density functional theory (DFT) for fluid mixtures that is based on a third-order thermodynamic perturbation theory of Feynman-Hibbs-corrected Mie potentials. The DFT is developed to study the interfacial properties of hydrogen, helium, neon, deuterium, and their mixtures, i.e., fluids that are strongly influenced by quantum effects at low temperatures. White Bear fundamental measure theory is used for the hard-sphere contribution of the Helmholtz energy functional, and a weighted density approximation is used for the dispersion contribution. For mixtures, a contribution is included to account for non-additivity in the Lorentz-Berthelot combination rule. Predictions of the radial distribution function from DFT are in excellent agreement with results from molecular simulations, both for pure components and mixtures. Above the normal boiling point and 5% below the critical temperature, the DFT yields surface tensions of neon, hydrogen, and deuterium with average deviations from experiments of 7.5%, 4.4%, and 1.8%, respectively. The surface tensions of hydrogen/deuterium, para-hydrogen/helium, deuterium/helium, and hydrogen/neon mixtures are reproduced with a mean absolute error of 5.4%, 8.1%, 1.3%, and 7.5%, respectively. The surface tensions are predicted with an excellent accuracy at temperatures above 20K. The poor accuracy below 20K is due to the inability of Feynman-Hibbs-corrected Mie potentials to represent the real fluid behavior at these conditions, motivating the development of new intermolecular potentials. This DFT can be leveraged in the future to study confined fluids and assess the performance of porous materials for hydrogen storage and transport.

DISJOINING PRESSURE IN THIN SPHERICAL LIQUID FILMS AND VAPOR LAYERS WITH MOLECULAR CORRELATIONS INCLUDED

Based on the expression for a grand thermodynamic potential as a molecular density functional, disjoining pressures in thin liquid films around nanosized wettable spherical particles and in thin vapor layers around nonwettable particles are calculated depending the degree of lyophilicity, film thickness and particle size. A characteristic feature of the approach is the full consideration of hard-sphere molecular correlations according to the fundamental measure theory in the density functional method and finding the complete dependence of the grand thermodynamic potential of the system on stable droplet or bubble size. Although the obtained results show a qualitative agreement between the new calculated disjoining pressure dependences and those obtained by us earlier in the framework of a simpler gradient method of the molecular density functional, the new results differ significantly quantitatively. It is confirmed that the disjoining pressure in the liquid film around nanosized lyophilic particle grows with the particle radius and lyophilicity.

Soft fundamental measure theory functional for the Weeks-Chandler-Andersen repulsive potential.

We introduce a soft fundamental measure theory functional for the purely repulsive Weeks-Chandler-Andersen (WCA) fluid. This classical density functional could serve as a reference fluid for functionals created using thermodynamic perturbation theory instead of the hard-sphere fluid. Our functional incorporates temperature-dependent parameters describing the length scale and effective softness of the particle interaction, and which reproduce the second virial coefficient of the WCA fluid. We find that this approach is comparable in accuracy to the Barker-Henderson approach combined with the White Bear density functional for the hard-sphere fluid.

Using classical density functional theory to determine crystal-fluid surface tensions.

Classical density functional theory is used to determine the fluid-solid surface tensions for low-index faces of crystals of hard spheres and Lennard-Jones particles. The calculations make use of the recently introduced explicitly stable fundamental measure theory model for hard spheres, and we show that this gives state-of-the-art accuracy compared to simulation. For the Lennard-Jones system, results are presented for both solid-liquid and solid-vapor interfaces, and in both cases the FCC results compare favorably with existing results from the literature. We find that the BCC crystal has significantly lower solid-liquid surface tension than the FCC structure. For the solid-vapor interface, our results indicate that the BCC phase is unstable with respect to transition to the HCP structure, in agreement with various zero-temperature results in the literature.

Dynamic density functional theory for drying colloidal suspensions: Comparison of hard-sphere free-energy functionals.

Dynamic density functional theory (DDFT) is a promising approach for predicting the structural evolution of a drying suspension containing one or more types of colloidal particles. The assumed free-energy functional is a key component of DDFT that dictates the thermodynamics of the model and, in turn, the density flux due to a concentration gradient. In this work, we compare several commonly used free-energy functionals for drying hard-sphere suspensions, including local-density approximations based on the ideal-gas, virial, and Boublík-Mansoori-Carnahan-Starling-Leland (BMCSL) equations of state as well as a weighted-density approximation based on fundamental measure theory (FMT). To determine the accuracy of each functional, we model one- and two-component hard-sphere suspensions in a drying film with varied initial heights and compositions, and we compare the DDFT-predicted volume fraction profiles to particle-based Brownian dynamics (BD) simulations. FMT accurately predicts the structure of the one-component suspensions even at high concentrations and when significant density gradients develop, but the virial and BMCSL equations of state provide reasonable approximations for smaller concentrations at a reduced computational cost. In the two-component suspensions, FMT and BMCSL are similar to each other but modestly overpredict the extent of stratification by size compared to BD simulations. This work provides helpful guidance for selecting thermodynamic models for soft materials in nonequilibrium processes, such as solvent drying, solvent freezing, and sedimentation.

Structural properties of liquids in extreme confinement.

We simulate a hard-sphere liquid in confined geometry where the separation of the two parallel, hard walls is smaller than two particle diameters. By systematically reducing the wall separation we analyze the behavior of structural and thermodynamic properties, such as inhomogeneous density profiles, structure factors, and compressibilities when approaching the two-dimensional limit. In agreement with asymptotic predictions, we find for quasi-two-dimensional fluids that the density profile becomes parabolic and the structure factor converges toward its two-dimensional counterpart. To extract the compressibility in polydisperse samples a perturbative expression is used which qualitatively influences the observed nonmonotonic dependence of the compressibility with wall separation. We also present theoretical calculations based on fundamental-measure theory and integral-equationtheory, which are in very good agreement with the simulation results.

The primitive model in classical density functional theory: beyond the standard mean-field approximation

The primitive model describes ions by point charges with an additional hard-core interaction. In classical density-functional theory (DFT) the mean-field electrostatic contribution can be obtained from the first order of a functional perturbation of the pair potential for an uncharged reference system of hard spheres. This mean-field electrostatic term particularly contributes at particle separations that are forbidden due to hard-core overlap. In this work we modify the mean-field contribution such that the pair potential is constant for distances smaller than the contact distance of the ions. We motivate our modification by the underlying splitting of the potential, which is similar to the splitting of the Weeks–Chandler–Andersen potential and leads to higher-order terms in the respective expansion of the functional around the reference system. The resulting formalism involves weighted densities similar to the ones found in fundamental measure theory. To test our modifications, we analyze and compare density profiles, direct and total correlation functions, and the thermodynamic consistency of the functional via a widely established sum rule and the virial pressure formula for our modified functional, for established functionals, and for data from computer simulations. We found that our modifications clearly show improvements compared to the standard mean-field functional, especially when predicting layering effects and direct correlation functions in high concentration scenarios; for the latter we also find improved consistency when calculated via different thermodynamic routes. In conclusion, we demonstrate how modifications toward higher order corrections beyond mean-field functionals can be made and how they perform, by this providing a basis for systematic future improvements in classical DFT for the description of electrostatic interactions.

The Effect of Stern Layer Thickness on the Diffuse Capacitance for Size Asymmetric Electrolyte inside the Charged Spherical Cavities by Density Functional Theory

The Modified Fundamental Measure Theory was used within the Primitive Model to determine Stern layer thickness in charged spherical cavity. The different behaviors for mean electric potential in Stern region with those one in the diffuse region was considered as the best criteria to determine this thickness. Depending on concentration, ion charge and size, and cavity charge the cations or anions may be found at the closest distance from the cavity’s wall, which lead two values for Stern thickness. In the present case study in which anions are assumed to be larger than cations, at zero or low positive values of cavity surface charge, the cations are found at the cavity wall, although the anion’s contact density is larger. But, by increasing the surface charge to a specific level, the cations are repelled away from the wall so that Stern thickness increases and becomes equal to anion radius. It is shown that this specific surface charge decreases not only with anion size, concentration and cation charge but also with rising cavity curvature. Also, we studied that how the diffuse capacitance for size-asymmetric electrolytes is affected by variation in Stern layer thickness.

Effects of a tiny particle trapped in a spherical cavity on fluid structure, contact density, and capillary condensation (a DFT approach)

The Modified Fundamental Measure Theory is exploited to investigate the effects of a mobile tiny particle, TP, trapped in a spherical pore on fluid structure, contact density, and capillary condensation. The decreasing and oscillatory behaviors for contact densities versus mobile TP size are observed respectively for hard sphere and Lennard Jones fluids. The capillary condensation of a Lennard Jones confined fluid is investigated in the presence of a mobile TP and compared with those in cases in which the TP is either absent or fixed at the pore center. The configurational chemical potential at which capillary condensation occurs for a fixed TP is smaller than that for a mobile one while both values are still smaller than that for a fluid confined in a spherical pore with no TP. Increasing the size of the mobile TP leads to a decrease in the configurational chemical potential (μ∗) at which capillary condensation occurs and in the coexistence curve width.

Droplet condensation in the lattice gas with density functional theory.

A density functional for the lattice gas with next-neighbor attractions (Ising model) from fundamental measure theory is applied to the problem of droplet states in three-dimensional, finite systems. The density functional is constructed via an auxiliary model with hard lattice gas particles and lattice polymers to incorporate the attractions. Similar to previous simulation studies, the sequence of droplets changing to cylinders and to planar slabs is found upon increasing the average density ρ[over ¯] in the system. Owing to the discreteness of the lattice, additional effects in the state curve for the chemical potential μ(ρ[over ¯]) are seen upon lowering the temperature away from the critical temperature [oscillations in μ(ρ[over ¯]) in the slab portion and spiky undulations in μ(ρ[over ¯]) in the cylinder portion as well as an undulatory behavior of the radius of the surface of tension R_{s} in the droplet region]. This behavior in the cylinder and droplet region is related to washed-out layering transitions at the surface of liquid cylinders and droplets. The analysis of the large-radius behavior of the surface tension γ(R_{s}) gave a dominant contribution ∝1/R_{s}^{2}, although the consistency of γ(R_{s}) with the asymptotic behavior of the radius-dependent Tolman length seems to suggest a weak logarithmic contribution ∝lnR_{s}/R_{s}^{2} in γ(R_{s}). The coefficient of this logarithmic term is smaller than a universal value derived with field-theoretic methods.