Functional Theory For Inhomogeneous Fluids Research Articles

Neural functional theory for inhomogeneous fluids: Fundamentals and applications

We present a hybrid scheme based on classical density functional theory and machine learning for determining the equilibrium structure and thermodynamics of inhomogeneous fluids. The exact functional map from the density profile to the one-body direct correlation function is represented locally by a deep neural network. We substantiate the general framework for the hard sphere fluid and use grand canonical Monte Carlo simulation data of systems in randomized external environments during training and as reference. Functional calculus is implemented on the basis of the neural network to access higher-order correlation functions via automatic differentiation and the free energy via functional line integration. Thermal Noether sum rules are validated explicitly. We demonstrate the use of the neural functional in the self-consistent calculation of density profiles. The results outperform those from state-of-the-art fundamental measure density functional theory. The low cost of solving an associated Euler–Lagrange equation allows to bridge the gap from the system size of the original training data to macroscopic predictions upon maintaining near-simulation microscopic precision. These results establish the machine learning of functionals as an effective tool in the multiscale description of soft matter.

A density functional approach to ferrogels

Ferrogels consist of magnetic colloidal particles embedded in an elastic polymer matrix. As a consequence, their structural and rheological properties are governed by a competition between magnetic particle–particle interactions and mechanical matrix elasticity. Typically, the particles are permanently fixed within the matrix, which makes them distinguishable by their positions. Over time, particle neighbors do not change due to the fixation by the matrix. Here we present a classical density functional approach for such ferrogels. We map the elastic matrix-induced interactions between neighboring colloidal particles distinguishable by their positions onto effective pairwise interactions between indistinguishable particles similar to a ‘pairwise pseudopotential’. Using Monte-Carlo computer simulations, we demonstrate for one-dimensional dipole-spring models of ferrogels that this mapping is justified. We then use the pseudopotential as an input into classical density functional theory of inhomogeneous fluids and predict the bulk elastic modulus of the ferrogel under various conditions. In addition, we propose the use of an ‘external pseudopotential’ when one switches from the viewpoint of a one-dimensional dipole-spring object to a one-dimensional chain embedded in an infinitely extended bulk matrix. Our mapping approach paves the way to describe various inhomogeneous situations of ferrogels using classical density functional concepts of inhomogeneous fluids.

The method of collective variables: a link with the density functional theory

Recently, based on the method of collective variables the statistical field theory for multicomponent inhomogeneous systems was formulated [O. Patsahan, I. Mryglod, J.-M. Caillol, Journal of Physical Studies, 2007, 11, 133]. In this letter we establish a link between this approach and the classical density functional theory for inhomogeneous fluids.

Density functional theory of inhomogeneous liquids. I. The liquid-vapor interface in Lennard-Jones fluids

A simple model is proposed for the direct correlation function (DCF) for simple fluids consisting of a hard-core contribution, a simple parametrized core correction, and a mean-field tail. The model requires as input only the free energy of the homogeneous fluid, obtained, e.g., from thermodynamic perturbation theory. Comparison to the DCF obtained from simulation of a Lennard-Jones fluid shows this to be a surprisingly good approximation for a wide range of densities. The model is used to construct a density functional theory for inhomogeneous fluids which is applied to the problem of calculating the surface tension of the liquid-vapor interface. The numerical values found are in good agreement with simulation.

Density functional theory of fluids in the isothermal-isobaric ensemble

We present a density functional theory for inhomogeneous fluids at constant external pressure. The theory is formulated for a volume-dependent density, n(r,V), defined as the conjugate variable of a generalized external potential, nu(r,V), that conveys the information on the pressure. An exact expression for the isothermal-isobaric free-energy density functional is obtained in terms of the corresponding canonical ensemble functional. As an application we consider a hard-sphere system in a spherical pore with fluctuating radius. In general we obtain very good agreement with simulation. However, in some situations a peak develops in the center of the cavity and the agreement between theory and simulation becomes worse. This happens for systems where the number of particles is close to the magic numbers N=13, 55, and 147.

New Weighted Density Functional Theory Based on Perturbative Approach

A new weighted density functional theory for inhomogeneous fluids is developed by combining the functional Taylor expansion of the perturbative approach with the nonperturbative weighted density concept. In this approach, the functional Taylor expansion of the one-particle direct correation function (DCF) is truncated at second order and the effect of a subset of the higher order terms is taken into account by evaluating the third-order DCF at a suitable weighted density, determined by demanding the fourth order DCF to be reproduced from the proposed expansion in the homogeneous limit. The present approach uses as input the two-particle DCF of the corresponding uniform fluid along with an approximation to the three-particle DCF. The proposed theory is applied to hard sphere as well as Lennard-Jones fluids in confined geometries, and the calculated density profiles for both the systems are shown to be in very good overall agreement with the available simulation data.

Renormalization-Group Calculation of the Dependence on Gravity of the Surface Tension and Bending Rigidity of a Fluid Interface

The surface tension and the bending rigidity of a planar liquid-vapor interface in the presence of vanishing gravity are analyzed using the renormalization group. Based on the density functional theory of inhomogeneous fluids, we show that a term, quartic in the density fluctuations, can be added to the classical capillary-wave model so that a renormalization-group calculation can be performed. By comparing the outcome of such a calculation with rigorous results relating the direct correlation function with surface tension and bending rigidity, we find the scaling dependence of the latter on gravity. The results agree with the expected fact that the interface should become unstable as gravity vanishes.

Two- and Three-Dimensional Nonlocal Density Functional Theory for Inhomogeneous Fluids: II. Solvated Polymers as a Benchmark Problem

In a previous companion paper, we presented the details of our algorithms for performing nonlocal density functional theory calculations in complex two- and three-dimensional geometries. We discussed scaling and parallelization, but did not discuss other issues of performance. In this paper, we detail the precision of our methods with respect to changes in the mesh spacing. This is a complex issue because given a Cartesian mesh, changes in mesh spacing will result in changes in surface geometry. We discuss these issues using a series of rigid solvated polymer models including square rod polymers, cylindrical polymers, and bead–chain polymers. In comparing the results of the various models, it becomes clear that surface curvature or roughness plays an important role in determining the strength of structural solvation forces between interacting solvated polymers. The results in this paper serve as benchmarks for future application of these algorithms to complex fluid systems.

Two- and Three-Dimensional Nonlocal Density Functional Theory for Inhomogeneous Fluids: I. Algorithms and Parallelization

Fluids adsorbed near surfaces, near macromolecules, and in porous materials are inhomogeneous, exhibiting spatially varying density distributions. This inhomogeneity in the fluid plays an important role in controlling a wide variety of complex physical phenomena including wetting, self-assembly, corrosion, and molecular recognition. One of the key methods for studying the properties of inhomogeneous fluids in simple geometries has been density functional theory (DFT). However, there has been a conspicuous lack of calculations in complex two- and three-dimensional geometries. The computational difficulty arises from the need to perform nested integrals that are due to nonlocal terms in the free energy functional. These integral equations are expensive both in evaluation time and in memory requirements; however, the expense can be mitigated by intelligent algorithms and the use of parallel computers. This paper details our efforts to develop efficient numerical algorithms so that nolocal DFT calculations in complex geometries that require two or three dimensions can be performed. The success of this implementation will enable the study of solvation effects at heterogeneous surfaces, in zeolites, in solvated (bio)polymers, and in colloidal suspensions.

Effective Hamiltonian for liquid-vapor interfaces.

Starting from a density functional theory for inhomogeneous fluids we derive an effective Hamiltonian for liquid-vapor interfaces of simple fluids which goes beyond the common phenomenological capillary-wave description. In contrast to other approaches we take into account the long-ranged power-law decay of the dispersion forces between the fluid particles which changes the functional form of the wave-vector-dependent surface tension qualitatively. In particular, we find two different forms of the bending rigidity for the capillary waves, a negative one for small wave vectors determined by the long-ranged dispersion forces and a positive rigidity for large wave vectors due to the distortions of the intrinsic density profile in the vicinity of the locally curved interface. The differences to the standard capillary-wave theory and the relevance of these results for the interpretation of scattering experiments are discussed.

Bulk and interfacial properties of polar fluids

AbstractBased on a suitable density functional theory for inhomogeneous fluids we investigate both the bulk phase diagram and interfacial structures of polar fluids in general and of Stockmayer fluids in particular. We derive a scheme which allows us to analyze one‐component fluids composed of molecules with strong permanent dipole moments which are shown to affect significantly the bulk phase diagram. The results for the bulk phase diagram are in good agreement with published Monte Carlo data. Furthermore we investigate the profiles of the number density and of the orientational order at the liquid‐vapor interface as well as the corresponding surface tension. We systematically analyze the dependences of these quantities on the temperature and on the strength of the dipole moment of the molecules. This reveals power laws and, close to criticality, scaling behavior. Our approach, which is reliable even for large dipole moments, allows us to determine separately those contributions to the surface tension which are due to the orientational degrees of freedom.

Density profiles of concentrated colloidal suspensions in sedimentation equilibrium

Sedimentation equilibria in concentrated colloidal suspensions are studied within the framework of density functional theory for inhomogeneous fluids. The density profile in the gravitational field is calculated exactly for the one-dimensional model of hard rods. The predictions of the local density approximation agree very well with coarse-grained density profiles derived from extensive Monte Carlo simulations of three-dimensional systems of hard spheres and charge-stabilized colloidal particles. The simulations show that the shape of the density profiles is very sensitive to the nature of the interactions between colloidal particles, and allow a direct test of a simple inversion procedure, which extracts the osmotic equation of state of colloidal suspensions from a measurement of their density profile. The method is sufficiently sensitive to allow the crystal-fluid tie line to be located at high colloid concentrations. The local density approximation is generalized to calculate density profiles of bidisperse hard sphere suspensions; the competition between gravity and entropy leads to nonmonotonic density profiles which are sensitive to the mass ratio, for a given size ratio. The density functional approach and Monte Carlo simulations are finally extended to study the effect of interparticle attractions on the density profiles, which exhibit a region of rapid variation corresponding to the ‘‘liquid-gas’’ interface for sufficiently strong attractive interactions.

The role of attractive intermolecular forces in the density functional theory of inhomogeneous fluids

The density functional theory of Meister and Kroll for nonuniform simple fluids [Phys. Rev. A 31, 4055, (1985)] is modified in order to take into account pair correlations in the attractive energy term. The theory is applied for the calculation of density profiles of argon in the case of the free liquid surface and in the case of adsorption on ‘‘solid carbon dioxide.’’ For the gas–liquid interface the theory yields dew and bubble densities, which are in good agreement with computer simulations. For the adsorption of argon on solid CO2, the first-order thin-film to thick-film transition is confirmed.

Density functional theory for inhomogeneous fluids

Using the method developed by Meister and Kroll (1985, Phys. Rev. A, 31, 4055), we have proposed an extension of the density functional theory for singlet distribution function of a fluid in contact with a solid by taking into account triple coupling function. The proposed equation is then solved numerically for hard spheres in contact with a hard wall and with a soft wall interacting through a Lennard-Jones (9, 3) potential and the results obtained are compared with the previous theoretical predictions and with results of Monte Carlo simulations, performed on Cyber 205 supercomputer. The numerical calculations indicate that the inclusion of the third-order term improves the accuracy of the density functional calculations.

The role of capillary wave fluctuations in determining the liquid-vapour interface

We investigate the role played by capillary-wave like fluctuations in the determination of the equilibrium density profile and the intermolecular correlation functions of various models of a planar liquid-vapour interface in an external gravitational field. We show that the interfacial profile calculated from the van der Waals model, which is the simplest (squaregradient) approximation in a density functional theory of inhomogeneous fluids, is stable against capillary-wave fluctuations and therefore cannot be used to describe the ‘bare’ interface in a capillary wave model. The density-density correlation function of the van der Waals model exhibits long-ranged correlations in the interface. These are of precisely the same form as those predicted by Wertheim and can be associated with capillary-wave like fluctuations of the Gibbs dividing surface. These fluctuations do not lead to a ‘diffuse’ interfacial profile in the limits of infinite interfacial area and zero gravity; the interfacial thickness remains ...