Fundamental Measure Density Functional Theory 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 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.

Highly confined mixtures of parallel hard squares: A density-functional-theory study.

Using the fundamental-measure density-functional theory, we study theoretically the phase behavior of extremely confined mixtures of parallel hard squares in slit geometry. The pore width is chosen such that configurations consisting of two consecutive big squares, or three small squares, in the transverse direction, perpendicular to the walls, are forbidden. We analyze two different mixtures with edge lengths of species selected so as to allow or forbid one big plus one small square to fit into the channel. For the first mixture we obtain first-order transitions between symmetric and asymmetric packings of particles: Small and big squares are preferentially adsorbed at different walls. Asymmetric configurations are shown to lead to more efficient packing at finite pressures. We argue that the stability region of the asymmetric phase in the pressure-composition plane is bounded so that the symmetric phase is stable at low and very high pressure. For the second mixture, we observe strong demixing between phases which are rich in different species. Demixing occurs in the lateral direction, i.e., the dividing interface is perpendicular to the walls, and phases exhibit symmetric density profiles. The possible experimental realization of this behavior (which in practical terms is precluded by jamming) in strictly two-dimensional systems is discussed. Finally, the phase behavior of a mixture with periodic boundary conditions is analyzed and the differences and similarities between the latter and the confined system are discussed. We claim that, although exact calculations exclude the existence of true phase transitions in (1+ε)-dimensional systems, density-functional theory is still successful in describing packing properties of large clusters of particles.

Phase diagram of two-dimensional hard rods from fundamental mixed measure density functional theory.

A density functional theory for the bulk phase diagram of two-dimensional orientable hard rods is proposed and tested against Monte Carlo computer simulation data. In detail, an explicit density functional is derived from fundamental mixed measure theory and freely minimized numerically for hard discorectangles. The phase diagram, which involves stable isotropic, nematic, smectic, and crystalline phases, is obtained and shows good agreement with the simulation data. Our functional is valid for a multicomponent mixture of hard particles with arbitrary convex shapes and provides a reliable starting point to explore various inhomogeneous situations of two-dimensional hard rods and their Brownian dynamics.

Scaling behavior of thin films on chemically heterogeneous walls.

We study the adsorption of a fluid in the grand canonical ensemble occurring at a planar heterogeneous wall which is decorated with a chemical stripe of width L. We suppose that the material of the stripe strongly preferentially adsorbs the liquid in contrast to the outer material which is only partially wet. This competition leads to the nucleation of a droplet of liquid on the stripe, the height h_{m} and shape of which (at bulk two-phase coexistence) has been predicted previously using mesoscopic interfacial Hamiltonian theory. We test these predictions using a microscopic Fundamental Measure Density Functional Theory which incorporates short-ranged fluid-fluid and fully long-ranged wall-fluid interactions. Our model functional accurately describes packing effects not captured by the interfacial Hamiltonian but still we show that there is excellent agreement with the predictions h_{m}≈L^{1/2} and for the scaled circular shape of the drop even for L as small as 50 molecular diameters. For smaller stripes the droplet height is considerably lower than that predicted by the mesoscopic interfacial theory. Phase transitions for droplet configurations occurring on substrates with multiple stripes are also discussed.

Density functional theory and simulations of colloidal triangular prisms.

Nanopolyhedra form a versatile toolbox to investigate the effect of particle shape on self-assembly. Here we consider rod-like triangular prisms to gauge the effect of the cross section of the rods on liquid crystal phase behavior. We also take this opportunity to implement and test a previously proposed version of fundamental measure density functional theory (0D-FMT). Additionally, we perform Monte Carlocomputer simulations and we employ a simpler Onsager theory with a Parsons-Lee correction. Surprisingly and disappointingly, 0D-FMT does not perform better than the Tarazona and Rosenfeld's version of fundamental measure theory (TR-FMT). Both versions of FMT perform somewhat better than the Parsons-Lee theory. In addition, we find that the stability regime of the smectic phase is larger for triangular prisms than for spherocylinders and square prisms.

Observation of isotropic-isotropic demixing in colloidal platelet-sphere mixtures.

Mixtures of colloids with different sizes and shapes are ubiquitous in nature and industry. The possible existence of isotropic-isotropic (I1-I2) demixing of platelets and spheres remains an open question. Here we present direct experimental evidence of I1-I2 demixing using platelets with a very small thickness-to-diameter ratio mixed with silica spheres at the size ratio q = R(sphere)/R(disk) = 0.0901 ± 0.0004. The platelets cause the isotropic-to-nematic phase transition at a very low volume fraction because of their highly anisometric shape. The presence of silica spheres in the suspension accelerates the phase transition and packs the nematic phase more densely via depletion interaction. Increasing the sphere volume fraction to 0.0014, a tri-phase region emerges. This direct observation of I1-I2 demixing seems to validate the free-volume scaled particle theory and indicates the need for refinement of the fundamental measure density functional theory.

Free Energy Calculations of Crystalline Hard Sphere Complexes Using Density Functional Theory.

Recently developed fundamental measure density functional theory (FMT) is used to study binary hard sphere (HS) complexes in crystalline phases. By comparing the excess free energy, pressure, and phase diagram, we show that the fundamental measure functional yields good agreements to the available simulation results of AB, AB2, and AB13 crystals. Furthermore, we use this functional to study the HS models of five binary crystals, Cu5Zr(C15b), Cu51Zr14(β), Cu10Zr7(ϕ), CuZr(B2), and CuZr2(C11b), which are observed in the Cu-Zr system. The FMT functional gives a well-behaved minimum for most of the hard sphere crystal complexes in the two-dimensional Gaussian parameter space, namely a crystalline phase. However, the current version of FMT functional (White Bear) fails to give a stable minimum for the structure Cu10Zr7(ϕ). We argue that the observed solid phases for the HS models of the Cu-Zr system are true thermodynamic stable phases and can be used as a reference system in perturbation calculations.

Theoretical calculation of the melting curve of Cu-Zr binary alloys.

Helmholtz free energies of the dominant binary crystalline solids found in the Cu-Zr system at high temperatures close to the melting curve are calculated. Our theoretical approach combines fundamental measure density functional theory (applied to the hard-sphere reference system) and a perturbative approach to include the attractive interactions. The studied crystalline solids are Cu(fcc), Cu_{51}Zr_{14}(β), CuZr(B2), CuZr_{2}(C11b), Zr(hcp), and Zr(bcc). The calculated Helmholtz free energies of crystalline solids are in good agreement with results from molecular-dynamics (MD) simulations. Using the same perturbation approach, the liquid phase free energies are calculated as a function of composition and temperature, from which the melting curve of the entire composition range of this system can be obtained. Phase diagrams are determined in this way for two leading embedded atom method potentials, and the results are compared with experimental data. Theoretical melting temperatures are compared both with experimental values and with values obtained directly from MD simulations at several compositions.

Does surface roughness amplify wetting?

Any solid surface is intrinsically rough on the microscopic scale. In this paper, we study the effect of this roughness on the wetting properties of hydrophilic substrates. Macroscopic arguments, such as those leading to the well-known Wenzel's law, predict that surface roughness should amplify the wetting properties of such adsorbents. We use a fundamental measure density functional theory to demonstrate the opposite effect from roughness for microscopically corrugated surfaces, i.e., wetting is hindered. Based on three independent analyses we show that microscopic surface corrugation increases the wetting temperature or even makes the surface hydrophobic. Since for macroscopically corrugated surfaces the solid texture does indeed amplify wetting there must exist a crossover between two length-scale regimes that are distinguished by opposite response on surface roughening. This demonstrates how deceptive can be efforts to extend the thermodynamical laws beyond their macroscopic territory.

Condensation and evaporation transitions in deep capillary grooves

We study the order of capillary condensation and evaporation transitions of a simple fluid adsorbed in a deep capillary groove using a fundamental measure density functional theory (DFT). The walls of the capillary interact with the fluid particles via long-ranged, dispersion, forces while the fluid-fluid interaction is modelled as a truncated Lennard–Jones-like potential. We find that below the wetting temperature Tw condensation is first-order and evaporation is continuous with the metastability of the condensation being well described by the complementary Kelvin equation. In contrast above Tw both phase transitions are continuous and their critical singularities are determined. In addition we show that for the evaporation transition above Tw there is an elegant mapping, or covariance, with the complete wetting transition occurring at a planar wall. Our numerical DFT studies are complemented by analytical slab model calculations which explain how the asymmetry between condensation and evaporation arises out of the combination of long-ranged forces and substrate geometry.

Complete wetting near an edge of a rectangular-shaped substrate

We consider fluid adsorption near a rectangular edge of a solid substrate that interacts with the fluid atoms via long range (dispersion) forces. The curved geometry of the liquid–vapour interface dictates that the local height of the interface above the edge ℓE must remain finite at any subcritical temperature, even when a macroscopically thick film is formed far from the edge. Using an interfacial Hamiltonian theory and a more microscopic fundamental measure density functional theory (DFT), we study the complete wetting near a single edge and show that , as the chemical potential departure from the bulk coexistence δμ = μs(T) − μ tends to zero. The exponent depends on the range of the molecular forces and in particular for three-dimensional systems with van der Waals forces. We further show that for a substrate model that is characterised by a finite linear dimension L, the height of the interface deviates from the one at the infinite substrate as δℓE(L) ∼ L−1 in the limit of large L. Both predictions are supported by numerical solutions of the DFT.

Phase behaviour of liquid-crystal monolayers of rod-like and plate-like particles.

Orientational and positional ordering properties of liquid crystal monolayers are examined by means of Fundamental-Measure Density Functional Theory. Particles forming the monolayer are modeled as hard parallelepipeds of square section of size σ and length L. Their shapes are controlled by the aspect ratio κ = L/σ (>1 for prolate and <1 for oblate shapes). The particle centers of mass are restricted to a flat surface and three possible and mutually perpendicular orientations (in-plane and along the layer normal) of their uniaxial axes are allowed. We find that the structure of the monolayer depends strongly on particle shape and density. In the case of rod-like shapes, particles align along the layer normal in order to achieve the lowest possible occupied area per particle. This phase is a uniaxial nematic even at very low densities. In contrast, for plate-like particles, the lowest occupied area can be achieved by random in-plane ordering in the monolayer, i.e., planar nematic ordering takes place even at vanishing densities. It is found that the random in-plane ordering is not favorable at higher densities and the system undergoes an in-plane ordering transition forming a biaxial nematic phase or crystallizes. For certain values of the aspect ratio, the uniaxial-biaxial nematic phase transition is observed for both rod-like and plate-like shapes. The stability region of the biaxial nematic phase enhances with decreasing aspect ratios for plate-like particles, while the rod-like particles exhibit a reentrant phenomenon, i.e., a sequence of uniaxial-biaxial-uniaxial nematic ordering with increasing density if the aspect ratio is larger than 21.34. In addition to this, packing fraction inversion is observed with increasing surface pressure due to the alignment along the layers normal. At very high densities the nematic phase destabilizes to a nonuniform phases (columnar, smectic, or crystalline phases) for both shapes.

Fundamental measure density functional theory study of liquid-vapor interface of dipolar and quadrupolar fluids

We have studied interfacial structure and properties of liquid-vapor interfaces of dipolar fluids and quadrupolar fluids, respectively, using the classical density functional theory (DFT). Towards this end, we employ the fundamental measure DFT for a reference hard-sphere (HS) part of free energy and the modified mean field approximation for the correlation function of dipolar or quadrupolar fluid. At low temperatures we find that both the liquid-vapor interfacial density profile and orientational order parameter profile exhibit weakly damped oscillatory decay into the bulk liquid. At high temperatures the decay of interfacial density and order parameter profiles is entirely monotonic. The scaled temperature τ = 1 - T/T(c) that separates the two qualitatively different interfacial structures is in the range 0.10-0.15. At a given (dimensionless) temperature, increasing the dipolar or quadrupolar moment enhances the density oscillations. Application of an electric field (normal to the interface) will damp the oscillations. Likewise, at the given temperature, increasing the strength of any multipolar moment also increases the surface tensions while increasing the strength of the applied electric field will reduce the surface tensions. The results are compared with those based on the local-density approximations (LDA) for the reference HS part of free energy as well as with results of numerical experiments.

Density functional study of complete, first-order and critical wedge filling transitions

We present numerical studies of complete, first-order and critical wedge filling transitions, at a right angle corner, using a microscopic fundamental measure density functional theory. We consider systems with short-ranged, cut-off Lennard-Jones, fluid–fluid forces and two types of wall–fluid potential: a purely repulsive hard wall and also a long-ranged potential with three different strengths. For each of these systems we first determine the wetting properties occurring at a planar wall, including any wetting transition and the dependence of the contact angle on temperature. The hard wall corner is completely filled by vapour on approaching bulk coexistence and the numerical results for the growth of the meniscus thickness are in excellent agreement with effective Hamiltonian predictions for the critical exponents and amplitudes, at leading and next-to-leading order. In the presence of the attractive wall–fluid interaction, the corresponding planar wall–fluid interface exhibits a first-order wetting transition for each of the interaction strengths considered. In the right angle wedge geometry the two strongest interactions produce first-order filling transitions while for the weakest interaction strength, for which wetting and filling occur closest to the bulk critical point, the filling transition is second-order. For this continuous transition the critical exponent describing the divergence of the meniscus thickness is found to be in good agreement with effective Hamiltonian predictions.

Bulk fluid phase behaviour of colloidal platelet–sphere and platelet–polymer mixtures

Using a geometry-based fundamental measure density functional theory, we calculate bulk fluid phase diagrams of colloidal mixtures of vanishingly thin hard circular platelets and hard spheres. We find isotropic-nematic phase separation, with strong broadening of the biphasic region, upon increasing the pressure. In mixtures with large size ratio of platelet and sphere diameters, there is also demixing between two nematic phases with differing platelet concentrations. We formulate a fundamental measure density functional for mixtures of colloidal platelets and freely overlapping spheres, which represent ideal polymers, and use it to obtain phase diagrams. We find that, for low platelet-polymer size ratio, in addition to isotropic-nematic and nematic-nematic phase coexistence, platelet-polymer mixtures also display isotropic-isotropic demixing. By contrast, we do not find isotropic-isotropic demixing in hard-core platelet-sphere mixtures for the size ratios considered.

Scalar fundamental measure theory for hard spheres in three dimensions: Application to hydrophobic solvation

Hard-sphere mixtures provide one a solvable reference system that can be used to improve the density functional theory of realistic molecular fluids. We show how the Kierlik-Rosinberg's scalar version of the fundamental measure density functional theory of hard spheres [E. Kierlik and M. L. Rosinberg, Phys. Rev. A 42, 3382 (1990)], which presents computational advantages with respect to the original Rosenfeld's vectorial formulation or its extensions, can be implemented and minimized in three dimensions to describe fluid mixtures in complex environments. This implementation is used as a basis for defining a molecular density functional theory of water around molecular hydrophobic solutes of arbitrary shape.

Tension and Stiffness of the Hard Sphere Crystal-Fluid Interface

A combination of fundamental measure density functional theory and Monte Carlo computer simulation is used to determine the orientation-resolved interfacial tension and stiffness for the equilibrium hard-sphere crystal-fluid interface. Microscopic density functional theory is in quantitative agreement with simulations and predicts a tension of 0.66k(B)T/σ(2) with a small anisotropy of about 0.025k(B)T and stiffnesses with, e.g., 0.53k(B)T/σ(2) for the (001) orientation and 1.03k(B)T/σ(2) for the (111) orientation. Here k(B)T is denoting the thermal energy and σ the hard-sphere diameter. We compare our results with existing experimental findings.

Freezing of parallel hard cubes with rounded edges

The freezing transition in a classical three-dimensional system of rounded hard cubes with fixed, equal orientations is studied by computer simulation and fundamental-measure density functional theory. By switching the rounding parameter s from zero to one, one can smoothly interpolate between cubes with sharp edges and hard spheres. The equilibrium phase diagram of rounded parallel hard cubes is computed as a function of their volume fraction and the rounding parameter s. The second order freezing transition known for oriented cubes at s = 0 is found to be persistent up to s = 0.65. The fluid freezes into a simple-cubic crystal which exhibits a large vacancy concentration. Upon a further increase of s, the continuous freezing is replaced by a first-order transition into either a sheared simple cubic lattice or a deformed face-centered cubic lattice with two possible unit cells: body-centered orthorhombic or base-centered monoclinic. In principle, a system of parallel cubes could be realized in experiments on colloids using advanced synthesis techniques and a combination of external fields.

Contact values of highly asymmetric hard sphere mixtures from Fundamental Measure Density Functional Theory

The White Bear version of Fundamental Measure Theory (FMT-WB) has been tested in binary mixtures of hard spheres in the vicinity of the colloidal limit, where the size ratio of the two species is exceedingly large and the large sphere mole fraction is infinitely low. Contact values of large–large sphere radial distribution functions have been calculated and compared with molecular dynamics simulations and previously proposed theoretical formulas. In contrast to the failure of BMCSL (Boublik, Mansoori, Carnahan, Starling, Leland equation of state) predictions, FMT-WB gives good agreement with simulation for a range of species size ratios and mole fractions. The performance of BMCSL is qualitatively related to one of its model parameters, which could indicate the reliability of the BMCSL result. Our results confirm the accuracy of FMT-WB in the colloidal limit for the first time and suggest that BMCSL contact values must be applied carefully to account for chain connectivity when studying certain cases with classical Density Functional Theories.