Modified Weighted-density Approximation Research Articles

A non-perturbative approach to freezing of superfuid 4He in density functional theory

Freezing of various classical liquids is successfully described by density functional theory (DFT). On the other hand, so far no report has been published that DFT describes the freezing of superfuid 4He correctly. In fact, DFT gives too stable solid phase and the superfuid phase does not exist at finite positive pressures within a second order perturbation. In this paper we try a non-perturbative version of DFT, that is modified weighted density approximation (MWDA) to go beyond second order perturbation for the freezing of superfuid 4He. Via an exact zero temperature quantum Monte-Carlo (QMC) method we have computed the equation of state and the compressibility of superfuid 4He. By utilizing a recently introduced analytic continuation method (the GIFT method), we have obtained also density response functions at different densities from QMC imaginary time correlation functions. Contrary to second order perturbation, by employing these QMC data as DFT input we find a too stable superfuid phase, preventing freezing around the experimentally observed freezing pressure. We find the same pathological behavior by using another model energy functional of superfuid 4He (Orsay-Trento model). We conclude that the straightforward MWDA calculation gives such a poor result when liquid-gas transition is present.

Density Functional Theory and Bose Statistics for the Freezing of Superfluid 4He

Density functional theory so far has experienced pathological results in the description of the freezing of superfluid 4He, even with the use of accurate inputs, namely the density response function, χ(q), and the equation of state of the superfluid. In fact, while the second order approximation of Ramakrishnan-Yussouff type gives a too much stable solid phase (Likos et al. in Phys. Rev. B 55:8867, 1997; Minoguchi and Galli in J. Law Temp. Phys. 162:160, 2011), a non-perturbative treatment called modified weighted density approximation (MWDA) gives, on the other hand, a too stable superfluid phase (Minoguchi et al. in J. Phys. Conf. Ser. 2012). We have preliminarily performed a MWDA calculation for a specific Gaussian distortion α=0.387 A−2 of liquid 4He. By using χ(q) and the equation of state of a system of liquid 4He in which Bose statistics has been suppressed, we have found that the calculation provides a lower energy. This indicates that the solid may be better described by a distorted normal liquid. The input data has been obtained via Path Integral Monte Carlo simulations of a system of distinguishable 4He atoms. The χ(q) has been obtained from the dynamic structure factors extracted, via the Genetic Inversion via Falsification of Theories (GIFT) method (Vitali et al. in Phys. Rev. B 82:174510, 2010) from imaginary time density–density correlation functions.

Studies of Liquid–Solid Transitions Using a Thermodynamic Perturbation Method with Modified Weighted Density Approximation

Despite intensive investigations based on statistical mechanics, liquid–solid transition is far from complete understanding. The reason for this difficulty lies in the fact that any conventional theory including the density functional theory cannot handle the attractive part that exists in any interaction potential. Thus far, a density functional theory incorporated with the modified weighted density approximation (MWDA) has been developed to predict the liquid–solid transition of a hard-sphere system. In this approach, the free energy and direct correlation function (DCF) in a liquid state were calculated using the analytical solution of the Percus–Yevick (PY) equation. The coexisting density of solid and liquid states given by this approach has agreed with those obtained by computer simulations within 0.7%. However, this method cannot be applied to realistic systems where constituents interact mutually through potential energy with an attractive part. This is because there are no analytical expressions available for the free energy and DCF. In this short note, we apply an alternative approach to analyzing liquid–solid transitions for atoms whose interaction potential has an attractive part. We employ a thermodynamic perturbation method developed by Rascon et al. The method has been applied successfully to the analysis of systems with nonmonotonic potential energy such as the Lennard-Jones, square well, and Dzugutov potential. The required information in this approach is only the potential function and the structure of a solid. Exploiting the theory developed by Weeks, Chandler, and Andersen (WCA), we first expand the free energy of the liquid and solid states. Namely, we split the potential energy into a hard-sphere reference potential and a perturbation potential. Thus, the free energy can be written as F1⁄2 ðrÞ 1⁄4 FHS1⁄2 ðrÞ þ 2 N 2 V Z 1

DENSITY FUNCTIONAL THEORY AND FREEZING OF HARD SPHERES, BEYOND THE PERCUS–YEVICK APPROXIMATION

The density functional theory for the freezing hard spheres is studied. We use a variety of the hard sphere direct correlation functions (DCFs) such as the one introduced by Roth et al. [J. Phys. Condens. Matter14, 12063 (2002)]; we call it RELK DCF, a new hard sphere DCF developed here by a combination of the RELK and the Percus–Yevick DCFs, and finally the generalized mean spherical approximation (GMSA). The structure factor, the freezing, and order parameters are calculated using these DCFs. The structure factor obtained by the new DCF is in good agreement with the Monte Carlo simulation. The best result for the freezing parameters in comparison with the Monte Carlo simulations is obtained by using our new expression for the DCF. Finally we obtain the Helmholtz free energy of the hard sphere FCC crystals using modified weighted density approximation (MWDA), and again the best results are obtained by using the new expression for the hard sphere DCF.

Thermodynamic Properties of Inhomogeneous Structures in Supercooled Liquids

The weighted density-functional theory is applied to investigate the free-energy landscape of dense supercooled liquids. Metastable states intermediate to the liquid and crystal phases are found, which can be identified with the supercooled states seen in computer simulations. These states are marked by a lower degree of mass localization as compared to the highly localized state termed as "hard-sphere glass" found in earlier studies. We evaluate the free energy using the modified weighted density approximation (MWDA), as formulated by Denton and Ashcroft (1989) Phys. Rev. A , 39 , 4701. The inhomogeneous density is parametrized in terms of Gaussian profiles centered around random lattice sites. The effects of heterogeneity coming from a fluctuation of the width of these Gaussian profiles show that the free energy of the system increases with increase in the fluctuations and, finally, the metastable minima disappear with growing fluctuations.

Crystallization of a Yukawa fluid via a modified weighted density approximation with a solid reference state

We investigate the freezing behavior of particles interacting with a Yukawa potential using extensions of the Denton and Ashcroft modified weighted density approximation (MWDA) model of density functional theory [A. R. Denton and N. W. Ashcroft, Phys. Rev. A 39, 470 (1989)]. An attempt is made to incorporate properties of the static solid into the fluid-based MWDA model via our previous model for the crystallization of inverse nth-power fluids [D. C. Wang and A. P. Gast, J. Chem. Phys. 110, 2522 (1999)], as well as a model that includes the Einstein vibrations of the localized particles. Both extensions yield improvements over the MWDA model in terms of coexisting densities and the ability to stabilize a body-centered cubic solid compared with computer simulation data. The fractional change in density upon freezing also compares favorably with results from available simulation studies and those for the inverse nth-power system. Reasons for the differences in results obtained for freezing properties of the Yukawa system among computer simulation data, theoretical approaches, and experimental studies are discussed.

Equation of state of the hard-sphere solid: The modified weighted-density approximation with a static solid reference state

We have investigated the high-density hard-sphere solid through density-functional theory by applying our recent formulation of the modified weighted-density approximation (MWDA) with a static solid reference state. Our model utilizes both information about the fluid state, modeled by the Percus-Yevick approximation, as well as physical insight into the static solid. We find very good agreement with computer simulation data for the equation of state, as well as for the ideal and excess components of the pressure, and are able to improve the results from the original MWDA model over a wide range of packing fractions. The degree of particle localization and the value of the Lindemann parameter for the high-density hard-sphere solid are also improved over the original MWDA theory. Finally, we comment on the result that the excess pressure is negative for most packing fractions, suggesting possible physical means for this observation.

Crystallization of power-law fluids: A modified weighted density approximation model with a solid reference state

We investigate the freezing behavior of particles interacting with an inverse nth power potential under the modified weighted density approximation (MWDA) formalism of Denton and Ashcroft [A. R. Denton and N. W. Ashcroft, Phys. Rev. A 39, 470 (1989)]. We model the liquid state with the perturbative hypernetted chain (PHNC) integral equation [H. S. Kang and F. H. Ree, J. Chem. Phys. 103, 3629 (1995)], chosen for its small computational time and high degree of accuracy. The deterioration of MWDA predictions of equilibrium properties with decreasing n is traced to its inability to accurately estimate the free energy in the static solid limit. An improvement in the MWDA theory is suggested by incorporating information on the static lattice into the model. This is done by moderating the direct correlation function via the Ornstein–Zernicke equation. It is found that this new model can drastically improve results for the coexisting densities for inverse nth power fluids. Other properties, such as the Lindemann parameter, are also improved in this new scheme.

Density functional theory for hard spherocylinders: phase transitions in the bulk and in the presence of external fields

The phase behaviour of hard spherocylinders is calculated using the classical density functional theory of freezing. In particular, we construct a modified weighted-density approximation and calculate the bulk phase diagram. While the stability regime of the isotropic, nematic and smectic-A phases is in reasonable agreement with computer simulation data, the theory fails to describe the correct stability of the crystalline phases. Furthermore, we investigate the phase behaviour in an external field coupling to the orientational degrees of freedom. As a result, bulk phases which have the same translational symmetry but a different orientational symmetry can be transformed continuously into each other above a critical strength of the external field.

Density-functional theory of quantum freezing: sensitivity to liquid-state structure and statistics

Density-functional theory is applied to compute the ground-state energies of quantum hard-sphere solids. The modified weighted-density approximation is used to map both the Bose and the Fermi solid onto a corresponding uniform Bose liquid, assuming negligible exchange for the Fermi solid. The required liquid-state input data are obtained from a paired-phonon analysis, combined with an enhanced hypernetted-chain integral equation, and the Feynman approximation, connecting the static structure factor and the linear response function. The Fermi liquid is treated by the Wu - Feenberg cluster expansion, which approximately accounts for the effects of antisymmetry. Liquid - solid transitions for both systems are obtained with no adjustment of input data, the Fermi liquid freezing at lower density than the Bose liquid because of the destabilizing influence of fermionic statistics. Predictions for these quantum systems are shown to be more sensitive to the accuracy of the input data than is the case for the classical counterpart. Limited quantitative agreement with simulation indicates a need for still further improvement of the liquid-state input, probably through practical alternatives to the Feynman approximation.

Density functional theory of crystal growth: Lennard-Jones fluids

We employ an extension of density functional theory to the dynamics of phase transitions in order to study the velocities of crystal growth and melting at planar undercooled and superheated crystal-melt interfaces. The free energy functional we use has a square-gradient form, with the parameters for a Lennard-Jones interaction potential determined by a modified weighted density approximation (MWDA) applied locally through the liquid–solid interface. We explore the role of the density change on freezing in crystal and melt growth, and discover a significant asymmetry between freezing and melting both close to and far from the equilibrium freezing point. The behavior of the superheated solid is governed by the close proximity of a spinodal, whereas in the undercooled liquid there is no evidence for a spinodal and the growth at large undercoolings is affected instead by the density deficit that appears in front of the growing interface. Comparisons are made with other density functional approaches and with computer simulations.

Thermodynamic Perturbation Approach to Freezing of the Classical One-Component Plasma

Systematic investigations are made of the thermodynamic perturbation approach to freezing, which has proved successful for systems interacting through soft-core or long-ranged potentials. The classical one-component plasma (OCP) is taken as the extreme case of such systems and the investigations focus on the reference systems employed in this approach and on the approximation schemes used to treat that system. It is confirmed that good results are obtained for the freezing properties of the OCP if a system with repulsive, short-ranged potential is properly chosen as the reference system, and both this system and the remaining part due to the long-range interaction are separately treated by the modified weighted-density approximation (MWDA) using accurate input data. However, if we adopt the hard spheres as the reference system in this approach and use either the MWDA or the generalized effective-liquid approximation (GELA) for this system, we cannot get similar results unless we use the input data in the ...

Freezing of the Hard Spheres: Re-Examination of the Weighted-Density-Functional Theories

Freezing of the hard spheres is re-examined using the modified weighted-density approximation (MWDA) of Denton and Ashcroft and the generalized effective-liquid approximation (GELA) of Lutsko and Baus. It is found that one owes part of success of these theories to the use of the Percus-Yevick (PY) direct correlation function and the corresponding equation of state of uniform fluids as the input data in these theories. In fact, if one uses virtually \lq\lqexact” input data in place of the PY ones, the free energies of the solid phase are somewhat lowered and predicted freezing properties worsen. It is argued that this unfavorable feature of the MWDA and the GELA becomes much more serious when these theories are applied to the reference hard spheres in the thermodynamic perturbation approach to freezing of systems with long-ranged potentials.

Nonperturbative density functional theory of solid-to-solid isostructural transitions

We examine the phase diagram of systems of hard, spherical particles with a short-range attraction. Recent simulations as well as theoretical investigations predict for such systems an fcc-to-fcc isostructural transition which terminates at a critical point, and the existence of a single fluid phase. We use a nonperturbative density functional approach, by making a mapping of the inhomogeneous system onto a uniform fluid by means of the modified weighted density approximation (MWDA). Previous approaches were instead based on the separation of the potential into a hard-sphere repulsion and a short-range attraction with the latter treated in a mean-field fashion. We obtain improved results for the critical temperature, the middle density at the triple point and the overall shape of the phase diagram, but a worsening of the triple temperature with increasing range of interaction.

Density-functional approach to the equation of state of a hard-sphere crystal.

The equation of state (pressure) of a hard-sphere fcc crystal is computed by means of a classical density-functional theory based on the modified weighted-density approximation and a simple Gaussian approximation for the density distribution. Predictions for the total pressure compare favorably with computer simulation data for packing fractions throughout the range 0.460.68 (i.e., from just below to well above the fluid-solid transition). The ideal-gas and excess contributions are computed individually and found to exhibit physically interesting variations with packing fraction. In particular, whereas the ideal-gas pressure is always positive and generally makes the largest contribution, the excess pressure is relatively small and, for \ensuremath{\eta}0.63, negative in sign, implying an effective attraction between neighboring hard spheres. Preliminary analysis of available simulation data for mean-square atomic displacements lends support to these predictions. Implications for a recently proposed heuristic model of hard-sphere crystal pressures are also discussed.

Density-functional theory of solid-to-solid isostructural transitions

We apply density-functional theory to study the expanded-FCC-to-condensed-FCC transition of a system of hard, spherical particles with a short-ranged attractive interaction, predicted recently by the simulations of Bolhuis and Frenkel (1994). Our approach is based on a non-perturbative treatment of the repulsive hard-core part of the potential, using the modified weighted density approximation (MWDA), and a mean-field approximation for the attractive part. We confirm by means of this simple theoretical treatment the existence of an isostructural solid-to-solid transition which terminates at a critical point, in quantitative agreement with the simulation data. We obtain, within this approximation, classical critical exponents for the continuous transition.

Freezing of Systems Interacting through Inverse-Power Potentials

A self-consistent theory of freezing developed by the author (J. Phys. Soc. Jpn. 62 (1993) 4316) is applied to systems interacting through inverse-power potentials. The theory is based on the idea of the thermodynamic perturbation approach and consists in a generalization of the modified weighted-density approximation of Denton and Ashcroft (Phys. Rev. A 39 (1989) 4701). The results of applications are in good agreement with the simulation data and usefulness of the theory is confirmed for systems interacting through soft-core or long-ranged potentials. Both theoretical and numerical comparisons between the present and other similar approaches are also made.

Self-Consistent Theory of Freezing of Systems Interacting through Long-Ranged Potentials

A self-consistent theory of freezing which is applicable to systems interacting through long-ranged potentials is presented. It consists of a new formulation of the modified weighted-density approximation (MWDA) based on the idea of the thermodynamic perturbation approach, in which the potential is split into a short-range (reference) part and a long-range part. The MWDA is used separately for the excess free energy of the reference system and the remaining contribution due to the long-range part of the potential. When applied to the one-component plasma, the theory yields results which are in reasonable agreement with the simulation data. It also provides a possible explanation for the reason why the usual approach, in which the solid is mapped onto a single effective liquid, fails for systems with long-ranged potentials.

Density-functional theory of nonuniform classical liquids: An extended modified weighted-density approximation

An extension of the modified weighted-density approximation (MWDA) is presented which retains the key features of the original MWDA in that it continues to describe the nonuniform system through the use of low-order correlation functions of the uniform counterpart. However, the approximate free energy functional is now exact up to third order in the functional expansion of the free energy, and therefore requires as input both the second- and third-order direct correlation functions of the uniform liquid, as well as its excess free energy per particle. The theory has been applied previously to the problem of isochoric freezing of the classical one-component plasma, and is here applied to the well-known problem of isobaric freezing of hard spheres. We use two different approaches to describe the less well-known third-order direct correlation function of the uniform liquid. The first approach is representative of a class of models for this function that are derived through three functional density differentiations of an approximate free energy model. The second is a factorization ansatz based on liquid-state diagrammatic expansions. The results are quite sensitive to these choices: The first leads to an improvement over the already satisfactory results of the original MWDA for the hard-sphere system, whereas the second does not lead to freezing at all. These differences are traced to the ways in which the approximations treat long-range and short-range potentials.

On the solid–fluid interface of adhesive spheres

The adhesive-sphere interaction potential provides a good model system to study the influence of the attractive well depth on phase behavior and interfacial phenomena. We investigate the solid–fluid phase behavior of adhesive spheres with the modified weighted density approximation (MWDA) of Denton and Ashcroft. We then apply a planar-averaged density functional approach (PWDA) to determine interfacial properties. We find both a narrowing of the interface between fluid and coexisting fcc solid and an increase in the interfacial energy with increasing attractive interaction strength in accord with the empirical relation γ≊ 0.47ΔHρ2/3s. In addition, we investigate metastable solid nucleation through calculation of metastable bcc solid–fluid interfacial tensions and find results suggesting the possibility of such a route to stable solid formation.