Fundamental Measure Theory Research Articles

Density functional for the lattice gas from fundamental measure theory.

We construct a density functional for the lattice gas or Ising model on square and cubic lattices based on lattice fundamental measure theory. To treat the nearest-neighbor attractions between the lattice gas particles, the model is mapped to a multicomponent model of hard particles with additional lattice polymers where effective attractions between particles arise from the depletion effect. The lattice polymers are further treated via the introduction of polymer clusters (labelled by the numbers of polymer they contain) such that the model becomes a multicomponent model of particles and polymer clusters with nonadditive hard interactions. The density functional for this nonadditive hard model is constructed with lattice fundamental measure theory. The resulting bulk phase diagram recovers the Bethe-Peierls approximation and planar interface tensions show a considerable improvement compared to the standard mean-field functional and are close to simulation results in three dimensions. We demonstrate the existence of planar interface solutions at chemical potentials away from coexistence when the equimolar interface position is constrained to arbitrary real values.

Solvation force and adsorption isotherm of a fluid mixture in nanopores of complex geometry based on fundamental measure theory

A novel method based on the fundamental measure theory is developed to calculate the solvation force and adsorption isotherm of a Lennard-Jones fluid mixture in complex geometries. Fast Fourier transform and 3D-voxel discretization are used for accurately computing the confined fluid densities in a closed pore of arbitrary geometry. Given the fluid densities, the solvation force distribution at the solid surface can be calculated using a new formulation from either mechanical or thermodynamic approach. Understanding the solvation force behavior, which depends on many factors such as pore geometry, confined density distribution, molecule size, is very important to analyze the pore deformation from a poromechanical point of view. Special attention in the numerical simulations is given to the adsorption problem of CH4 and CO2 gas mixture in ellipsoidal pore.

Hybrid density-potential functional theory of electric double layers

We present a field theoretic derivation for the free energy of the electrolyte solution. A reference system is introduced to describe non-electrostatic interactions, notably hard-sphere interactions, between charged particles in the electrolyte solution. The reference system is described by the Bikerman theory — a local density approximation, and the fundamental measure theory (FMT) — a nonlocal density approximation. Combining the classical part for charged particles in electrolyte solution and a quantum mechanical part for interacting electrons, we obtain a hybrid density-potential functional for the grand canonical potential of the electric double layer (EDL). Variation analysis of the hybrid density-potential functional leads to two controlling equations in terms of the electron density and the electric potential, respectively. Numerical implementation of the developed model is demonstrated for a simple EDL without specific adsorption. Particularly, oscillating density of counterions occurs near the metal surface when the FMT is used for the reference system.

Ionic Competition over Adsorption into Charged Spherical Cavities Affecting the Shape of Electric Double Layer Capacitance Curve and Zeta Potential: A Density Functional Theory Study

Modified fundamental measure theory in framework of restricted primitive model is employed to investigate the formation of the electric double layer at the concave wall of a charged spherical cavity when two positive ions, A + and C 2+, are in competition. Results indicate that, compared to A + at a bulk concentration 1M, the presence of even low concentrations (0.01 M) of C 2+ leads to a remarkable change in the capacitance curve’s shape. The concentration plays an important role in ionic adsorption at low values of electric potential, EP, while the ion charge has a dominant role at high enough EPs. The mole fraction of A + ions in the cavity, shows an initial increase with EP because of their higher concentration but they are later desorbed from the cavity. Hence, shows a maximum at EP at which their substitution with C 2+ ions start. This maximum shifts to lower EP with increasing C 2+ concentration and cavity curvature. This substitution of ions leads to a remarkable change in the capacitance curve’s shape, converting the symmetric capacitance curve to an asymmetric one. The zeta potential whose value is effectively changed by this substitution, is investigated along with its role in the diffuse capacitance.

Fundamental measure theory of inhomogeneous two-body correlation functions.

For the three-dimensional hard-sphere model we investigate the inhomogeneous two-body correlations predicted by Rosenfeld's fundamental measure theory. For the special cases in which the density has either planar or spherical symmetry we provide analytic formulas for the Hankel and Legendre transforms, respectively, of the inhomogeneous two-body direct correlation function as explicit functionals of the density. When combined with the Ornstein-Zernike relation our analytical results allow for rapid calculation of inhomogeneous hard-sphere density correlations in real space. These not only provide information about the packing structures of the hard-sphere system but also form an essential building-block for constructing perturbation theories of more realistic models.

Diffuse and Stern capacitances at the concave wall of spherical cavities by density functional theory

Abstract The dependence of Stern and diffuse capacitances on curvature for the electric double layer at the concave wall of spherical cavities is investigated using modified fundamental measure theory within the framework of restricted primitive model. As expected, both Stern and diffuse capacitances are found to increase with size. Moreover, the diffuse capacitance curve assumes a bell shape for small enough cavities while it exhibits a bell-camel transition with size. The Stern capacitance is observed to have a greater contribution to the total capacitance curve than does the diffuse one at low EPs for a (+1:−1) electrolyte irrespective of cavity size. At high electric potentials, the contributions of Stern and diffuse capacitances depend on cavity size. The diffuse potential is strongly affected by ion charge that, in turn, determines the shape of the capacitance curve. The sign of diffuse potential changes with electric potential depending on concentration for electrolytes with divalent counter ions. It causes that the diffuse potential passes from zero which leads to an infinite value for diffuse capacitances. Such a behavior has been observed with concentration for asymmetric electrolytes with divalent co ions. Finally, it is found that the diffuse potential can be used to predict the shape of the capacitance curve for symmetrical electrolytes.

Explicitly stable fundamental-measure-theory models for classical density functional theory.

The derivation of the state of the art tensorial versions of Fundamental Measure Theory (a form of classical Density Functional Theory for hard spheres) is reexamined in the light of the recently introduced concept of global stability of the density functional based on its boundedness [Lutsko and Lam, Phys. Rev. E 98, 012604 (2018)2470-004510.1103/PhysRevE.98.012604]. It is shown that within the present paradigm, explicit stability of the functional can be achieved only at the cost of giving up accuracy at low densities. It is argued that this is an acceptable trade-off since the main value of DFT lies in the study of dense systems. Explicit calculations for a wide variety of systems show that a proposed explicitly stable functional is competitive in all ways with the popular White Bear models while sharing some of their weaknesses when applied to non-close-packed solids.

The effect of electro-neutrality violation inside a charged spherical cavity on the capacitance curve shape in DFT approach and interpretation of mean electrostatic potential

The effect of electro-neutrality deviation in small charged cavities on the shape of the capacitance curve is investigated in this study. The data required are obtained using the modified fundamental measure theory and two models; namely, the restricted primitive model (RPM) and the primitive model (PM). The results of RPM investigations show that the capacitance curve will take on a bell shape when electro-neutrality parameter decreases with increasing surface charge, but that it exhibits a camel shape when electro-neutrality parameter shows a maximum versus surface charge. This behavior is essentially governed by both curvature and bulk concentration as it is observed that electro-neutrality parameter declines at high enough bulk concentrations and large enough curvatures. The PM study results indicate that an increase in the ratio of anion size to that of cation gives rise to higher positive values of the electric potential at zero surface charge. Moreover, it is seen that an increase in counter ion size causes more deviations from the electro-neutral condition while an increase in co-ion size relative to that of counter ions leads to the minimum value of electro-neutrality parameter versus surface charge but to a maximum for larger counter ions. Finally, it is found that, at the saturation point, the value for electro-neutrality parameter and, thereby, that for capacitance both become dependent on the size of counter ions. Base on the concept of the work in the thermodynamics and definition of mean electrostatic potential we interpreted all the terms present in the mean electrostatic potential in spherical cavity.

Heterogeneous surface charge confining an electrolyte solution.

The structure of dilute electrolyte solutions close to a surface carrying a spatially inhomogeneous surface charge distribution is investigated by means of classical density functional theory within the approach of fundamental measure theory. For electrolyte solutions, the influence of these inhomogeneities is particularly strong because the corresponding characteristic length scale is the Debye length, which is large compared to molecular sizes. Here, a fully three-dimensional investigation is performed, which accounts explicitly for the solvent particles, and thus provides insight into effects caused by ion-solvent coupling. The present study introduces a versatile framework to analyze a broad range of types of surface charge heterogeneities even beyond the linear response regime. This reveals a sensitive dependence of the number density profiles of the fluid components and of the electrostatic potential on the magnitude of the charge as well as on the details of the surface charge patterns at small scales.

Nonlocal density functional theory of water taking into account many-body dipole correlations: binodal and surface tension of ‘liquid–vapour’ interface

In this paper we formulate a nonlocal density functional theory of inhomogeneous water. We model a water molecule as a couple of oppositely charged sites. The negatively charged sites interact with each other through the Lennard–Jones potential (steric and dispersion interactions), square-well potential (short-range specific interactions due to electron charge transfer), and Coulomb potential, whereas the positively charged sites interact with all types of sites by applying the Coulomb potential only. Taking into account the nonlocal packing effects via the fundamental measure theory, dispersion and specific interactions in the mean-field approximation, and electrostatic interactions at the many-body level through the random phase approximation, we describe the liquid–vapour interface. We demonstrate that our model without explicit account of the association of water molecules due to hydrogen bonding and with explicit account of the electrostatic interactions at the many-body level is able to describe the liquid–vapour coexistence curve and the surface tension at the ambient pressures and temperatures. We obtain very good agreement with available in the literature MD simulation results for density profile of liquid–vapour interface at ambient state parameters. The formulated theory can be used as a theoretical background for describing of the capillary phenomena, occurring in micro- and mesoporous materials.

A tensorial fundamental measure density functional theory for the description of adsorption in substrates of arbitrary three-dimensional geometry

Fundamental measure theory (FMT) is commonly considered within classical density functional theory (DFT) to describe inhomogeneous hard-sphere (HS) fluids. As opposed to the original FMT of Rosenfeld [Phys. Rev. Lett. 63, 980 (1989)], the dimensional interpolation FMT (DI-FMT) is a specific version of FMT which is well adapted to accurately describe the freezing of HSs and adsorption in extreme confinements by including tensorial weighted densities. The computation of these weighted densities is generally performed analytically for specific simple scenarios (e.g., planar, cylindrical, or spherical geometries), and this method is challenging to apply to pores of generic geometry. On the other hand, numerical approaches, using fast Fourier transform (FFT) techniques, can be adapted to deal with arbitrary 3D geometries. Computations with tensorial weights are, however, generally not considered with these approaches. In our current work, the FFT computation of weighted densities is detailed for tensorial quantities. We present a DI-FMT in general 3D computational space, for an arbitrary pore geometry, to obtain density profiles of pure HS fluids or mixtures. The other thermodynamic quantities, such as surface tension or excess adsorption, can then be determined by using the standard DFT framework. As an example of the implementation of the method, we present the results for the adsorption on a hard-wall model, representative of the solid structure of an anisotropic zeolite cavity.

A new study of associating inhomogeneous fluids with classical density functional theory

ABSTRACT A new density functional for the study of associating inhomogeneous fluids based on Wertheim's first-order thermodynamic perturbation theory is presented and compared to the most currently used associating density functionals. This functional is developed using the weighted density approximation in the range of association of hard spheres. We implement this functional within the framework of classical density functional theory together with modified fundamental measure theory to account for volume exclusion of hard spheres. This approach is tested against molecular simulations from literature of pure associating hard spheres and mixtures of non-associationg and associating hard spheres with different number of bonding sites close to a hard uniform wall. Furthermore, we compare and review our results with the performance of associating functionals from literature, one based on fundamental measure theory and the inhomogeneous version of Wertheim's perturbation theory. Results obtained with classical DFT and the three functionals show excellent agreement with molecular simulations in systems with one hard wall. For the cases of small pores where only one or two layers of fluid are allowed discrepancies between results with classical DFT and molecular simulations were found.

A fast inverse Hankel Transform of first Order for computing vector-valued weight Functions appearing in Fundamental Measure Theory in cylindrical Coordinates

Application of classical density functional theory in cylindrical coordinates requires a fast Hankel transform algorithm of order zero and its inverse, when the involved convolution integrals are solved in Fourier space. Vector-valued fundamental measure theory requires a fast Hankel transform of first order and its inverse. Compared to naïve real space convolution, this not only reduces complexity of the required computer code, but also increases computational performance due to the efficiency of the fast Hankel transform. This study proposes a new approach to compute the inverse of the first order fast Hankel transform on equidistant grids as a combination of a modified inverse Abel transform and a fast sine transform. Equidistant grids have a significant advantage over alternative implementations that require logarithmic grid spacing, since most problems, such as pores or droplets, require a certain resolution in the outer region of the radial domain, which in the case of a logarithmic grid necessitates an overly high number of grid points in the inner region of the radial domain. The proposed algorithm for the modified inverse Abel transform is straightforward to implement, while for the fast sine transform off-the-shelf algorithms can be used.

Many-body correlations from integral geometry.

In a recent letter we presented a framework for predicting the concentrations of many-particle local structures inside the bulk liquid as a route to assessing changes in the liquid approaching dynamical arrest. Central to this framework was the morphometric approach, a synthesis of integral geometry and liquid-state theory, which has traditionally been derived from fundamental measure theory. We present the morphometric approach in a new context as a generalization of scaled-particle theory, and we derive several morphometric theories for hard spheres of fundamental and practical interest. Our central result is a new theory that is particularly suited to the treatment of many-body correlation functions in the hard-sphere liquid, which we demonstrate by numerical tests against simulation.

Effects of bispherical nanopore concavo-convex walls on fluid structure, tangential and normal pressure components, and the validity of well-established bulk regularities; a DFT approach

The modified fundamental measure theory is employed to investigate the effects of wall curvature and wall-fluid interactions on the structure tangential and normal pressure of fluids confined in bispherical nanopores. Results show that the values for density and tangential pressure at the concave wall of hard bispherical pores are larger than those at the convex wall. Opposite results are obtained when a strong attraction potential is applied to the convex wall while the concave wall is a hard one. It is possible to get symmetrical profiles of density and tangential pressures with respect to two walls of a pore by varying the curvature and wall-fluid interactions. Although these two factors drastically affect the local density, pressures, and their average values, the behavior of the average values of their thermodynamic properties does not depend on them. This indicates that certain well-established bulk regularities are valid for fluids confined in bispherical pores.

Guide to efficient solution of PC-SAFT classical Density Functional Theory in various Coordinate Systems using fast Fourier and similar Transforms

Classical density functional theory (DFT) is a powerful tool for studying solvation or problems where resolution of interfacial domains or interfacial properties among phases (or thin films) is required. Many interesting problems necessitate multi-dimensional modeling, which calls for robust and efficient algorithmic implementations of the Helmholtz energy functionals. A possible approach for achieving efficient numerical solutions is using the convolution theorem of the Fourier transform. This study is meant to facilitate research and application of DFT methods, by providing a detailed guide on solving DFT problems in multi-dimensional domains. Methods for efficiently solving the convolution integrals in Fourier space are presented for Cartesian, cylindrical, and spherical coordinates. For cylindrical and spherical coordinate systems, rotational and spherical symmetry is exploited, respectively. To enable easy implementation, our approach is based on fast Fourier, fast Hankel, fast sine and cosine transforms on equidistant grids, all of which can be applied using off-the-shelf algorithms. Subtle details for implementing algorithms in cylindrical and spherical coordinate systems are emphasized. The work covers functionals based on weighted densities exemplarily. Functionals according to fundamental measure theory (FMT) as well as a Helmholtz energy functional based on the perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state are worked out in detail (and given as Supporting Information).

Differential capacitance of uniformly charged hard-sphere ions in planar electric double layers

The theoretical model, which is based both on the modified fundamental-measure theory for the hard-sphere contribution and on the mean-spherical approximation for the cross correlation between the hard-sphere contribution and Coulomb interaction, has been developed to study the differential capacitance of uniformly charged hard-sphere ions. The charge distribution of ions, electronic valence, bulk concentration and ion size are affected on the differential capacitance and the camel-to-bell shape transition. The charge distribution of ions enhances the differential capacitance and tends to delay the camel-to-bell shape transition. The low bulk concentration predicts higher differential capacitance. The transition from a camel-shaped to bell-shaped curve occurs when the bulk concentration rises to an appropriate value. The camel-to-bell shape transition occurs at a large ion size with increasing the bulk concentration. The high electrical valence tends to delay the transition and raise the transition concentration. The increase of electronic valence shifts the maximum to the low surface charge density. The decrease of , after passing through the maximum , results from the packing effect due to the formation of counterion layers.

Time-dependent density functional theory for the freezing/melting transition in interfacial systems

We propose a three-dimensional time-dependent density functional theory (TDDFT) to investigate the freezing/melting of Lennard-Jones fluids in interfacial systems including flat interfaces and nano-droplets. The theory is based on a modified fundamental measure theory (MFMT) for the hard-sphere reference system plus mean field theory (MFT) for the attractive contributions plus an additional weighted density approximation (WDA) contribution to enforce that bulk liquid equation of state is that of modified Benedict, Webb and Rubin (MBWR). By using different initial states, our theory generated a series of equilibrium structures including crystals and polyhedral particles. The non-linear effect and the irreversibility of the freezing/melting process were captured. The profiles for free energy and the order parameter indicate that the free energy barrier existing in the freezing/melting process is caused by the break in symmetry. Additionally, the time-dependent properties were also examined, and provided insight into the mechanism of the freezing process.

Transfer Free Energies of Test Proteins Into Crowded Protein Solutions Have Simple Dependence on Crowder Concentration.

The effects of macromolecular crowding on the thermodynamic properties of test proteins are determined by the latter's transfer free energies from a dilute solution to a crowded solution. The transfer free energies in turn are determined by effective protein-crowder interactions. When these interactions are modeled at the all-atom level, the transfer free energies may defy simple predictions. Here we investigated the dependence of the transfer free energy (Δμ) on crowder concentration. We represented both the test protein and the crowder proteins atomistically, and used a general interaction potential consisting of hard-core repulsion, non-polar attraction, and solvent-screened electrostatic terms. The chemical potential was rigorously calculated by FMAP (Qin and Zhou, 2014), which entails expressing the protein-crowder interaction terms as correlation functions and evaluating them via fast Fourier transform (FFT). To high accuracy, the transfer free energy can be decomposed into an excluded-volume component (Δμe−v), arising from the hard-core repulsion, and a soft-attraction component (Δμs−a), arising from non-polar and electrostatic interactions. The decomposition provides physical insight into crowding effects, in particular why such effects are very modest on protein folding stability. Further decomposition of Δμs−a into non-polar and electrostatic components does not work, because these two types of interactions are highly correlated in contributing to Δμs−a. We found that Δμe−v fits well to the generalized fundamental measure theory (Qin and Zhou, 2010), which accounts for atomic details of the test protein but approximates the crowder proteins as spherical particles. Most interestingly, Δμs−a has a nearly linear dependence on crowder concentration. The latter result can be understood within a perturbed virial expansion of Δμ (in powers of crowder concentration), with Δμe−v as reference. Whereas the second virial coefficient deviates strongly from that of the reference system, higher virial coefficients are close to their reference counterparts, thus leaving the linear term to make the dominant contribution to Δμs−a.

Curvature dependence of the camel-bell curve transition on the capacitance curve of spherical electric double-layer in porous electrodes: Density Functional Theory

Using the Modified Fundamental Measure Theory as the theoretical framework, we investigate the effect of curvature on camel-bell curve transition on the capacitance curve of a super-capacitor formed inside the spherical cavity of a porous electrode. The results show that at a constant concentration, the capacitance curve has a camel shape for large enough cavities, while a transition is observed from camel to bell shape with increasing curvature. In fact, at constant concentration this transition happened with decreasing the cavity size while at constant cavity size it occurs with concentration. Although concentration of transition decreases with curvature, the packing parameter for the fluid in the cavity is about to 0.2, irrespective of the curvature. Also, the potential at which the saturation phenomenon happens increases with cavity size. Despite the increased capacitance with increasing cavity size, there is an optimum size beyond which capacitance shows no remarkable increase. Investigation of the effect of ion charge on capacitance shows that the position of maximum capacitance shifts toward lower electric potentials with increasing ion charge. Finally, it is found that, at high electric potentials and at constant concentrations, electrolytes with the same counter ions exhibit capacitances which is independent of co-ion charge due to saturation.