Routes to the Density Profile and Structural Inconsistency.

The swift progression and expansion of machine learning (ML) have not gone unnoticed within the realm of statistical mechanics. In particular, ML techniques have attracted attention by the classical density-functional theory (DFT) community, as they enable automatic discovery of free-energy functionals to determine the equilibrium-density profile of a many-particle system. Within classical DFT, the external potential accounts for the interaction of the many-particle system with an external field, thus, affecting the density distribution. In this context, we introduce a statistical-learning framework to infer the external potential exerted on a classical many-particle system. We combine a Bayesian inference approach with the classical DFT apparatus to reconstruct the external potential, yielding a probabilistic description of the external-potential functional form with inherent uncertainty quantification. Our framework is exemplified with a grand-canonical one-dimensional classical particle ensemble with excluded volume interactions in a confined geometry. The required training dataset is generated using a Monte Carlo (MC) simulation where the external potential is applied to the grand-canonical ensemble. The resulting particle coordinates from the MC simulation are fed into the learning framework to uncover the external potential. This eventually allows us to characterize the equilibrium density profile of the system by using the tools of DFT. Our approach benchmarks the inferred density against the exact one calculated through the DFT formulation with the true external potential. The proposed Bayesian procedure accurately infers the external potential and the density profile. We also highlight the external-potential uncertainty quantification conditioned on the amount of available simulated data. The seemingly simple case study introduced in this work might serve as a prototype for studying a wide variety of applications, including adsorption, wetting, and capillarity, to name a few.

Classical density functional theory (DFT) provides a versatile framework to study the polymers with complex topological structure. Generally, a classical DFT describes the excess Helmholtz free energy of nonbonded chain connectivity due to excluded-volume effects and electrostatic correlations using the first-order thermodynamic perturbation theory (referred to as DFT-TPT1). Beyond first-order perturbation, the second-order TPT (TPT2) captures not only the correlations between neighboring monomers but also the interactions within three consecutive monomers, playing a crucial role in describing the polymer topology. However, the numerical implementation of TPT2 is limited by the lack of an effective triple correlation function (CF), especially for charged systems. Here, we propose an effective triple CF and incorporate it into DFT using TPT2 (referred to as DFT-eTPT2) to describe the nonbonded chain connectivity due to excluded-volume effects and electrostatic correlations. Using the data from molecular dynamics simulation as a benchmark, DFT-eTPT2 shows a clear improvement over DFT-TPT1 in predicting the density profiles of both neutral and charged branched polymer brushes, accurately capturing key structural features, such as the significant peaks near the branching point in the density profiles. In short, this work provides a precise and efficient theoretical tool for revealing molecular-level insights into branched polymers and their brushes.

A system of soft ellipsoid molecules confined between two planar walls is studied using classical density-functional theory. Both the isotropic and nematic phases are considered. The excess free energy is evaluated using two different Ansätze and the intermolecular interaction is incorporated using two different direct correlation functions (DCF's). The first is a numerical DCF obtained from simulations of bulk soft ellipsoid fluids and the second is taken from the Parsons-Lee theory. In both the isotropic and nematic phases the numerical DCF gives density and order parameter profiles in reasonable agreement with simulation. The Parsons-Lee DCF also gives reasonable agreement in the isotropic phase but poor agreement in the nematic phase.

A new three-parameter (valency, ion size, and charge asymmetry) model, the asymmetric restricted primitive model (ARPM) of ionic liquids, has recently been proposed. Given that ionic liquids generally are composed of monovalent species, the ARPM effectively reduces to a two-parameter model. Monte Carlo (MC) simulations have demonstrated that the ARPM is able to reproduce key properties of room temperature ionic liquids (RTILs) in bulk and at charged surfaces. The relatively modest complexity of the model raises the possibility, which is explored here, that a classical density functional theory (DFT) could resolve its properties. This is relevant because it might generate great improvements in terms of both numerical efficiency and understanding in the continued research of RTILs and their applications. In this report, a DFT for rod-like molecules is proposed as an approximate theoretical tool for an ARPM fluid. Borrowing data on the ion pair fraction from a single bulk simulation, the ARPM is modelled as a mixture of dissociated ions and connected ion pairs. We have specifically studied an ARPM where the hard-sphere diameter is 5 Å, with the charge located 1 Å from the hard-sphere centre. We focus on fluid structure and electrochemical behaviour of this ARPM fluid, into which a model electrode is immersed. The latter is modelled as a perfect conductor, and surface polarization is handled by the method of image charges. Approximate methods, which were developed in an earlier study, to take image interactions into account, are also incorporated in the DFT. We make direct numerical comparisons between DFT predictions and corresponding simulation data. The DFT theory is implemented both in the normal mean field form with respect to the electrostatic interactions and in a correlated form based on hole formation by both steric repulsions and ion-ion Coulomb interactions. The results clearly show that ion-ion correlations play a very important role in the screening of the charged surfaces by our ARPM ionic liquid. We have studied electrostatic potentials and ion density profiles as well the differential capacitance. The mean-field DFT fails to reproduce these properties, but the inclusion of ion-ion correlation by a simple approximate treatment yields quite reasonable agreement with the corresponding simulation results. An interesting finding is that there appears to be a surface phase transition at relatively low surface charge which is readily explored by DFT, but seen also in the MC simulations at somewhat higher asymmetry.

Most theoretical studies of an electrical double layer, which is formed by an electrolyte in contact with a charged electrode, employ a primitive model in which the solvent is represented by a dielectric continuum. This implicit-solvent model is convenient because computations are comparatively simple. However, it suppresses oscillations in the density profiles of ionic species that result from the discreteness of the solvent molecules. Furthermore, the implicit-solvent model yields poor results for the capacitance. In comparison with experiment at fixed electrode charge density, it predicts a too small electrode potential, and the resultant capacitance is too large. This latter discrepancy can be compensated in part by postulating the existence of an often fictitious inner layer whose properties are parametrized to agree best with experiment. The use of an implicit solvent model and an inner layer helps in correlating experimental results but rests on a faulty microscopic picture. Unfortunately, explicit consideration of solvent molecules poses both theoretical and numerical difficulties and, as a result, studies using an explicit solvent model have been few and far between. In this study, we consider a simple nonprimitive or explicit solvent model in which each solvent molecule is represented by a dimer composed of touching positive and negative hard spheres, with a resulting dipole moment that is equal to that of a water molecule, and the ions are represented by charged hard spheres. The density profiles and charge-potential relationship of this model are examined using the classical density functional theory. We find that the introduction of an explicit solvent increases the electrode potential, at fixed electrode charge, without the need to postulate a parametrized inner layer. Because of the solvent polarity, the ion profiles become strong oscillatory and show local charge inversion near a highly charged electrode surface at all ion concentrations.

This study demonstrates the capability of the density functional theory (DFT) formalism to predict contact angles and density profiles of model fluids and of real substances in good quantitative agreement with molecular simulations and experimental data. The DFT problem is written in cylindrical coordinates, and the solid-fluid interactions are defined as external potentials toward the fluid phase. Monte Carlo (MC) molecular simulations are conducted in order to assess the density profiles resulting from the Helmholtz energy functional used in the DFT formalism. Good quantitative agreement between DFT predictions and MC results for Lennard-Jones and ethane nanodroplets is observed, both for density profiles and for contact angles. That comparison suggests, first, that the Helmholtz energy functional proposed in a previous study [ Sauer , E. ; Gross , J. Ind. Eng. Chem. Res. 56 , 2017 , 4119 - 4135 ] is suitable for three-phase contact lines and, second, that Lagrange multipliers can be used to constrain the number of molecules, similar to a canonical ensemble. Experiments of sessile droplets on solid surfaces are performed to assess whether a real solid with its microscopic roughness can be described through a simple model potential. Comparison of DFT results to experimental data is done for a Teflon surface because Teflon can be regarded as a substrate exhibiting only attractive interactions of van der Waals type. It is shown that the real solid can be described as a perfectly planar solid with effective solvent-to-solid interactions, defined through a single adjustable parameter for the solid. Subsequent predictions for the contact angle of eight solvents, including polar components such as water, are found in very good agreement to experimental data using simple Berthelot-Lorentz combining rules. For the eight investigated solvents, we find mean absolute deviations of 3.77°.

This article reviews the recent progress on predicting the adsorption properties of metal-organic framework by using classical density functional theory and focused on the application of the classical density functional theory to the high-throughput screening, which is accelerated by fast Fourier Transform. Comparing to the conventional molecular simulations, the advantage of the accelerated classical density functional theory is the calculation speed, especially for simple small molecule systems, which makes the high-throughput screening on MOF materials feasible. However, it appears that there is a lack of efficient method to deal with the complicated molecules. How to construct a reasonable free energy functional of complicated fluid is the main challenge to state of art classical density functional theory. In a word, the improvement of CDFT theory and the combination of CDFT and molecular simulation are the two main ways for CDFT to predict gas adsorption in MOF.

We report a theoretical and simulation study of the drying and wetting phase transitions of a truncated Lennard-Jones fluid at a flat structureless wall. Binding potential calculations predict that the nature of these transitions depends on whether the wall-fluid attraction has a long ranged (LR) power law decay or is instead truncated, rendering it short ranged (SR). Using grand canonical Monte Carlo simulation and classical density functional theory, we examine both cases in detail. We find that for the LR case wetting is first order, while drying is continuous (critical) and occurs exactly at zero attractive wall strength, i.e., in the limit of a hard wall. In the SR case, drying is also critical but the order of the wetting transition depends on the truncation range of the wall-fluid potential. We characterize the approach to critical drying and wetting in terms of the density and local compressibility profiles and via the finite-size scaling properties of the probability distribution of the overall density. For the LR case, where the drying point is known exactly, this analysis allows us to estimate the exponent ν∥, which controls the parallel correlation length, i.e., the extent of vapor bubbles at the wall. Surprisingly, the value we obtain is over twice that predicted by mean field and renormalization group calculations, despite the fact that our three dimensional system is at the upper critical dimension where mean field theory for critical exponents is expected to hold. Possible reasons for this discrepancy are discussed in the light of fresh insights into the nature of near critical finite-size effects.

The ongoing scientific interest in the properties and structure of electric double layers (EDLs) stems from their pivotal role in (super)capacitive energy storage, energy harvesting, and water treatment technologies. Classical density functional theory (DFT) is a promising framework for the study of the in- and out-of-plane structural properties of double layers. Supported by molecular dynamics simulations, we demonstrate the adequate performance of DFT for analyzing charge layering in the EDL perpendicular to the electrodes. We discuss charge storage and capacitance of the EDL and the impact of screening due to dielectric solvents. We further calculate, for the first time, the in-plane structure of the EDL within the framework of DFT. While our out-of-plane results already hint at structural in-plane transitions inside the EDL, which have been observed recently in simulations and experiments, our DFT approach performs poorly in predicting in-plane structure in comparison to simulations. However, our findings isolate fundamental issues in the theoretical description of the EDL within the primitive model and point towards limitations in the performance of DFT in describing the out-of-plane structure of the EDL at high concentrations and potentials.

Inhomogeneous solvation theory (IST) and classical density functional theory (DFT) each provide a framework for relating distribution functions of solutions to their thermodynamic properties. As reviewed in this work, both IST and DFT can be formulated in a way that use two "end point" simulations, one of the pure solvent and the other of the solution, to determine the solute chemical potential and other thermodynamic properties of the solution and of subvolumes in regions local to the solute containing hydrating waters. In contrast to IST, where expressions for the excess energy and entropy of solution are the object of analysis, in the DFT end point formulation of the problem, the solute-solvent potential of mean force (PMF) plays a central role. The indirect part of the PMF corresponds to the lowest order (1-body) truncation of the IST expression. Because the PMF is a free energy function, powerful numerical methods can be used to estimate it. We show that the DFT expressions for the solute excess chemical potential can be written in a form which is local, involving integrals only over regions proximate to the solute. The DFT end point route to estimating solvation free energies provides an alternative path to that of IST for analyzing solvation effects on molecular recognition and conformational changes in solution, which can lead to new insights. In order to illustrate the kind of information that is contained in the solute-solvent PMF, we have carried out simulations of β-cyclodextrin in water. This solute is a well studied "host" molecule to which "guest" molecules bind; host-guest systems serve as models for molecular recognition. We illustrate the range of values the direct and indirect parts of the solute-solvent PMF can have as a water molecule is brought to the interface of β-cyclodextrin from the bulk; we discuss the "competition" between these two terms, and the role it plays in molecular recognition.

In classical density functional theory (DFT), the part of the Helmholtz free energy functional arising from attractive inter-particle interactions is often treated in a mean-field or van der Waals approximation. On the face of it, this is a somewhat crude treatment as the resulting functional generates the simple random phase approximation (RPA) for the bulk fluid pair direct correlation function. We explain why using standard mean-field DFT to describe inhomogeneous fluid structure and thermodynamics is more accurate than one might expect based on this observation. By considering the pair correlation function g(x) and structure factor S(k) of a one-dimensional model fluid, for which exact results are available, we show that the mean-field DFT, employed within the test-particle procedure, yields results much superior to those from the RPA closure of the bulk Ornstein-Zernike equation. We argue that one should not judge the quality of a DFT based solely on the approximation it generates for the bulk pair direct correlation function.

On the study of the vapor-liquid interface of associating fluids with classical density functional theory

Stability, structure, compressibility factor, excess free energy and density profile of hard prolate pear shape fluids

In vitro antifungal activity, cytotoxicity and binding analysis of imidazolium based ionic liquids with fluconazole: DFT and spectroscopic study

This study extends entropy scaling to inhomogeneous fluids by using the classical density functional theory together with a new viscosity reference that takes into account the influence of solid–fluid interactions on the fluid viscosity. The density functional theory uses a Helmholtz energy functional based on the perturbed-chain statistical associating fluid theory; the local residual entropy per particle is determined from the temperature derivative of the Helmholtz energy functional in combination with an appropriate weighted density profile. The weighted density calculation requires a single transferable parameter, which is adjusted to a reference molecular dynamics simulation. In particular, local viscosity values for fluids under nanoconfinement near solid–fluid interfaces are predicted using the same entropy scaling parameters as for homogeneous fluids. We validate the model by comparing viscosity and velocity profiles with results from non-equilibrium molecular dynamics simulations of a Couette flow in a slit pore. Good agreement is found between the entropy scaling model and the non-equilibrium molecular dynamics results for both the viscosity and velocity profiles of the Lennard–Jones truncated and shifted fluid. The proposed model extrapolates well to systems with different temperatures, fluid densities, and shear forces as well as to systems with different wetting behaviors. These results demonstrate that entropy scaling can be generalized to inhomogeneous fluids using an appropriate combination of residual entropy profile and viscosity reference.