Self-consistent Ornstein-Zernike Approximation Research Articles

Self-consistent Ornstein-Zernike approximation for fluids

Accurate results for fluids at thermal equilibrium, valid both outside and inside the critical region, are difficult to obtain due to the singular nature of this region. However, for simple models like interacting hard spheres and the Ising model high accuracy has been obtained by use of SCOZA (self-consistent Ornstein-Zernike approximation). Here this approach is extended to one component fluids more generally. Outside the critical region good analytic approximations may be obtained by use of statistical mechanical results that are adjusted to experimental data and/or computer simulations. However, such approximations typically will be of mean field type in the sense that the usual singular behavior in the critical region and in the transition to it, is not present.Here the SCOZA is generalized such that approximations accurate outside the critical region are utilized as basic input in the SCOZA problem. The singular behavior is connected to correlations of increasing range that develop as the critical point is approached. These correlations contribute both to the internal energy and compressibility of the fluid, and they give another contribution to the SCOZA partial differential equation. Accurate results in the critical region are expected to be obtained by adjusting one or a few parameters.

Generalized equation of state for fluids: From molecular liquids to colloidal dispersions

In this work, a new parameterization for the Statistical Association Fluid Theory for potentials of Variable Range (SAFT-VR) is coupled to the discrete potential theory to represent the thermodynamic properties of several fluids, ranging from molecular liquids to colloidal-like dispersions. In this way, this version of the SAFT-VR approach can be straightforwardly applied to any kind of either simple or complex fluid. In particular, two interaction potentials, namely, the Lennard-Jones and the hard-core attractive Yukawa potentials, are discretized to study the vapor-liquid equilibrium properties of both molecular and complex liquids, respectively. Our results are assessed with Monte Carlo computer simulations and available and accurate theoretical results based on the self-consistent Ornstein-Zernike approximation.

Critical properties of the hierarchical reference theory: Further investigations

The critical region of the hierarchical reference theory (HRT) is investigated further. This extends an earlier work by us where the critical properties of the HRT were concluded indirectly via another accurate but somewhat different theory, the self-consistent Ornstein-Zernike approximation (SCOZA), and numerical work. In the present work we perform our analysis directly upon the HRT partial differential equation to establish ordinary differential equations for the subleading scaling contributions. Again we find that the HRT critical indices in three dimensions are simple rational numbers.

Prediction of fluid-phase behavior of symmetrical binary Yukawa fluids using transition matrix Monte Carlo

Binary fluid mixtures display a wide array of fluid phase behavior ranging from simple vapor-liquid equilibrium diagrams to more complicated topology like azeotropy and heteroazeotropy. As a first step towards understanding the phenomena of equilibrium of binary molecular systems, the properties of binary monoatomic fluids have been studied in this work. An additional simplification made in this work is that of a symmetrical binary system where all similar molecules interact via the hard-core Yukawa (HCY) potential, u(r), while the dissimilar molecules interact via a potential, δu(r) where δ is a scalar parameter. The value of δ here is 0.75, which leads to the dissimilar molecules showing a tendency to dislike one another and the resulting phase diagram is such that the mixing-demixing line intersects the vapor-liquid equilibrium curve away from the vapor-liquid critical point. The effect of the variable potential range parameter of the HCY potential on the topology of the phase diagrams is investigated using grand canonical transition matrix Monte Carlo simulations and the conditions of temperature and pressure at which the system exhibits azeotropy and heteroazeotropy are ascertained. We find that the densities of the mixtures, as predicted by our simulation at equimolar concentrations, are in close agreement with the self-consistent Ornstein-Zernike approximation results of Schöll-Paschinger and co-workers.

Thermodynamic mechanism of the density anomaly of liquid water

Although density anomaly of liquid water has long been studied by many different authors up to now, it is not still cleared what thermodynamic mechanism induces the anomaly. The thermodynamic properties of substances are determined by interparticle interactions. We analyze what characteristics of pair potential cause the density anomaly on the basis of statistical mechanics and thermodynamics using a thermodynamically self-consistent Ornstein-Zernike approximation (SCOZA). We consider a fluid of spherical particles with a pair potential given by a hard-core repulsion plus a soft-repulsion and an attraction. We show that the density anomaly occurs when the value of the soft-repulsive potential at hard-core contact is in some proper range, and the range depends on the attraction. Further, we show that the behavior of the excess internal energy plays an essential role in the density anomaly and the behavior is mainly determined by the values of the soft-repulsive potential, especially near the hard core contact. Our results show that most of ideas put forward up to now are not the direct causes of the density anomaly of liquid water.

Interparticle interactions between water molecules

We determined many functional representations of interparticle interactions between water molecules, all of which reproduce the experimentally measured density-temperature relation at 1 bar with an accuracy better than obtained by previous models. Numerous similar descriptions of pair interactions will be discovered increasingly in the coming years, which will help us to understand why solid water has polymorphic structures and why liquid water has a large number of anomalies. We used a self-consistent Ornstein-Zernike approximation (SCOZA) with a potential given by multi-Yukawa terms. Because any smooth potential function can be fitted by multi-Yukawa terms, the method can be applied to various types of fluids. We also present a new simple fitting technique that makes the application of the SCOZA to any type of liquid much easier compared to a conventional Yukawa fit. Our new SCOZA fitting technique is among the most useful methods for determining the pair interaction between molecules of any liquid, and the potential will be helpful in improving realistic models.

Critical region of D-dimensional spins: extension and analysis of the hierarchical reference theory

The hierarchical reference theory (HRT) is generalised to spins of dimensionality D. Then its properties are investigated by both analytical and numerical evaluations for supercritical temperatures. The HRT is closely related to the self-consistent Ornstein–Zernike approximation (SCOZA) that was developed earlier for arbitrary D. Like the D = 1 case we studied earlier, our investigation is facilitated by a situation where both HRT and SCOZA give identical results with a mean spherical model behaviour (i.e. D = ∞). However, for the more general situation we find that an additional intermediate term appears. With an interplay between leading and subleading contributions, simple rational numbers, independent of D (<∞), are found for the critical indices.

Phase behavior of the modified-Yukawa fluid and its sticky limit

Simple model systems with short-range attractive potentials have turned out to play a crucial role in determining theoretically the phase behavior of proteins or colloids. However, as pointed out by D. Gazzillo [J. Chem. Phys. 134, 124504 (2011)], one of these widely used model potentials, namely, the attractive hard-core Yukawa potential, shows an unphysical behavior when one approaches its sticky limit, since the second virial coefficient is diverging. However, it is exactly this second virial coefficient that is typically used to depict the experimental phase diagram for a large variety of complex fluids and that, in addition, plays an important role in the Noro-Frenkel scaling law [J. Chem. Phys. 113, 2941 (2000)], which is thus not applicable to the Yukawa fluid. To overcome this deficiency of the attractive Yukawa potential, D. Gazzillo has proposed the so-called modified hard-core attractive Yukawa fluid, which allows one to correctly obtain the second and third virial coefficients of adhesive hard-spheres starting from a system with an attractive logarithmic Yukawa-like interaction. In this work we present liquid-vapor coexistence curves for this system and investigate its behavior close to the sticky limit. Results have been obtained with the self-consistent Ornstein-Zernike approximation (SCOZA) for values of the reduced inverse screening length parameter up to 18. The accuracy of SCOZA has been assessed by comparison with Monte Carlo simulations.

Self-Consistent Ornstein-Zernike Approximation (SCOZA) and exact second virial coefficients and their relationship with critical temperature for colloidal or protein suspensions with short-ranged attractive interactions

We focus on the second virial coefficient B2 of fluids with molecules interacting through hard-sphere potentials plus very short-ranged attractions, namely, with a range of attraction smaller than half hard-sphere diameter. This kind of interactions is found in colloidal or protein suspensions, while the interest in B2 stems from the relation between this quantity and some other properties of these fluid systems. Since the SCOZA (Self-Consistent Ornstein-Zernike Approximation) integral equation is known to yield accurate thermodynamic and structural predictions even near phase transitions and in the critical region, we investigate B2(SCOZA) and compare it with B2(exact), for some typical potential models. The aim of the paper is however twofold. First, by expanding in powers of density the condition of thermodynamic consistency included in the SCOZA integral equation, a general analytic expression for B2(SCOZA) is derived. For a given potential model, a comparison between B2(SCOZA) and B2(exact) may help to estimate the regimes where the SCOZA closure is reliable. Second, following the Vliegenthart-Lekkerkerker (VL) and Noro-Frenkel suggestions, the relationship between the critical B2 and the critical temperature Tc is discussed in detail for two prototype models: the square-well (SW) potential and the hard-sphere attractive Yukawa (HSY) one. The known simulation data for the SW model are revisited, while for the HSY model new SCOZA results have been generated. Although B2(HSY) at the critical temperature is found to be a slowly varying function of the range of Yukawa attraction ΔY over a wide interval of ΔY, it turns out to diverge as ΔY vanishes. For fluids with very short-ranged attractions, such a behavior contrasts with the VL assumption that B2 at the critical temperature should be nearly independent of the range of attraction. A very simple analytic representation is found for the available Monte Carlo data for Tc(HSY) and B2(HSY) as functions of the range of attraction, for ΔY smaller than half hard-sphere diameter.

Phase behavior of colloids and proteins in aqueous suspensions: Theory and computer simulations

The fluid phase behavior of colloidal suspensions with short-range attractive interactions is studied by means of Monte Carlo computer simulations and two theoretical approximations, namely, the discrete perturbation theory and the so-called self-consistent Ornstein-Zernike approximation. The suspensions are modeled as hard-core attractive Yukawa (HCAY) and Asakura-Oosawa (AO) fluids. A detailed comparison of the liquid-vapor phase diagrams obtained through different routes is presented. We confirm Noro-Frenkel's extended law of scaling according to which the properties of a short-ranged fluid at a given temperature and density are independent of the detailed form of the interaction, but just depend on the value of the second virial coefficient. By mapping the HCAY and AO fluids onto an equivalent square-well fluid of appropriate range at the critical point we show that the critical temperature as a function of the effective range is independent of the interaction potential, i.e., all curves fall in a master curve. Our findings are corroborated with recent experimental data for lysozyme proteins.

An investigation of the SCOZA for narrow square-well potentials and in the sticky limit

We present a study of the self-consistent Ornstein–Zernike approximation (SCOZA) for square-well (SW) potentials of narrow width δ. The main purpose of this investigation is to elucidate whether, in the limit δ → 0, the SCOZA predicts a finite value for the second virial coefficient at the critical temperature B 2(T c), and whether this theory can lead to an improvement of the approximate Percus–Yevick solution of the sticky hard-sphere (SHS) model due to Baxter [J. Chem. Phys. 49, 2770 (1968)]. For the SW of non-vanishing δ, the difficulties due to the influence of the boundary condition at high density, already encountered in an earlier investigation by Schöll-Paschinger et al. [J. Chem. Phys. 123, 234513 (2005)], prevented us from obtaining reliable results for δ <0.1. In the sticky limit, this difficulty can be circumvented, but then the SCOZA fails to predict a liquid–vapor transition. The picture that emerges from this study is that, for δ → 0, the SCOZA does not fulfill the expected prediction of a constant B 2(T c) [J. Chem. Phys. 113, 2941 (2000)], and that, for thermodynamic consistency to be usefully exploited in this regime, one should probably go beyond the Ornstein–Zernike ansatz.

A self-consistent Ornstein–Zernike approximation for a fluid with a screened power series interaction

We present a thermodynamically self-consistent Ornstein-Zernike approximation (SCOZA) for a fluid of spherical particles with a pair potential given by a hard-core repulsion and screened power series (SPS) tails. We take advantage of the known analytic properties of the solution of the Ornstein-Zernike equation for the case in which the direct correlation function outside the repulsive core is given by the SPS tails [M. Yasutomi, J. Phys.: Condens. Matter 13, L255 (2001)]: c(r)=∑(n=1) (N)exp(-z(n)r)∑(τ=-1) (L(n) )K((n,τ))z(n) (τ+1)r(τ) r>1. The analytic properties are rewritten so as to be optimally suited to the numerical computations. The SCOZA is known to provide very good overall thermodynamics, remarkably accurate critical point, and coexistence curve. In this paper, we present some numerical results for parameters in c(r) which are chosen to fit the Lennard-Jones potential. We show that both the energy and the compressibility paths lead to the same thermodynamics with high accuracy due to the thermodynamic consistency condition that has been enforced. The present method will be applicable to fluids with a large variety of smooth, realistic isotropic potentials where the pair potentials can be fitted by the SPS tails. The fitting procedure is superior to that by multi-Yukawa tails which is the only method presented so far.

Thermodynamic properties of short-range attractive Yukawa fluid: Simulation and theory

Coexistence properties of the hard-core attractive Yukawa potential with inverse-range parameter kappa=9, 10, 12, and 15 are calculated by applying canonical Monte Carlo simulation. As previously shown for longer ranges, we show that also for the ranges considered here the coexistence curves scaled by the critical density and temperature obey the law of corresponding states, and that a linear relationship between the critical density and the reciprocal of the critical temperature holds. The simulation results are compared to the predictions of the self-consistent Ornstein-Zernike approximation, and a good agreement is found for both the critical points and the coexistence curves, although some slight discrepancies are present.

The smooth cut-off hierarchical reference theory of fluids

We provide a comprehensive presentation of the Hierarchical Reference Theory (HRT) in the smooth cut-off formulation. A simple and self-consistent derivation of the hierarchy of differential equations is supplemented by a comparison with the known sharp cut-off HRT. The theory is then applied to a hard core Yukawa fluid (HCYF): a closure, based on a mean spherical approximation ansatz, is studied in detail and its intriguing relationship to the self-consistent Ornstein–Zernike approximation is discussed. The asymptotic properties close to the critical point are investigated and compared with the renormalization group results both above and below the critical temperature. The HRT free energy is always a convex function of the density, leading to flat isotherms in the two-phase region with a finite compressibility at coexistence. This makes HRT the sole liquid-state theory able to obtain fluid–fluid phase equilibrium in homogeneous systems without resorting to the Maxwell construction. The way the mean field free energy is modified due to the inclusion of density fluctuations suggests how to identify the spinodal curve. Thermodynamic properties and correlation functions of the HCYF are investigated for three values of the inverse Yukawa range, z = 1.8, z = 4 and z = 7, where Monte Carlo simulations are available. The stability of the liquid–vapor critical point with respect to freezing is also studied.

Unification of hierarchical reference theory and self-consistent Ornstein-Zernike approximation: Analysis of the critical region for fluids and lattice gases

The hierarchical reference theory (HRT) and the self-consistent Ornstein-Zernike approximation (SCOZA) are two liquid state theories that both yield a largely satisfactory description of the critical region as well as the phase coexistence and equation of state in general. In two previous works, unification of these theories has been considered and general equations were established. Further it was shown that the solution of the mean spherical model and a generalized version of it can be obtained in this way. In the present work, analysis of the critical region for fluids and lattice gases is performed. A key result of our HRT-SCOZA approximation is that for the standard three-dimensional fluid, lattice gas, or Ising model, the critical index for the critical isotherm is delta=5 and for the curve of coexistence it is beta=13 provided full scaling is assumed. More generally we find beta=2(delta+1) .

How to make thermodynamic perturbation theory to be suitable for low temperature?

Low temperature unsuitability is a problem plaguing thermodynamic perturbation theory (TPT) for years. Present investigation indicates that the low temperature predicament can be overcome by employing as reference system a nonhard sphere potential which incorporates one part of the attractive ingredient in a potential function of interest. In combination with a recently proposed TPT [S. Zhou, J. Chem. Phys. 125, 144518 (2006)] based on a lambda expansion (lambda being coupling parameter), the new perturbation strategy is employed to predict for several model potentials. It is shown that the new perturbation strategy can very accurately predict various thermodynamic properties even if the potential range is extremely short and hence the temperature of interest is very low and current theoretical formalisms seriously deteriorate or critically fail to predict even the existence of the critical point. Extensive comparison with existing liquid state theories and available computer simulation data discloses a superiority of the present TPT to two Ornstein-Zernike-type integral equation theories, i.e., hierarchical reference theory and self-consistent Ornstein-Zernike approximation.

Fifth-order thermodynamic perturbation theory of uniform and nonuniform fluids

A recently proposed numerical third-order thermodynamic perturbation theory (TPT) is extended to its fifth-order counterpart. Extensive performance evaluation based on several model potentials indicates that the fifth-order version is generally superior to both the third-order version and a well-known second-order macroscopic compressibility approximation TPT, and is at least comparable to an accurate self-consistent Ornstein-Zernike integral equation approximation (SCOZA) with regard to predictions of varying bulk thermodynamic properties. A bulk second-order direct correlation function (DCF) free of numerical solution of a bulk OZ integral equation, is proposed, which, in combination with the framework of classical density functional theory (DFT) and a thermodynamic consistency condition, extends the uniform fifth-order TPT to nonuniform case. Grand canonical ensemble Monte Carlo simulation is carried out to produce density profile of a core-softened fluid confined in a hard spherical cavity, the resultant density profile is employed to evaluate the performance of the present nonuniform fifth-order TPT and a recently proposed third +second-order perturbation DFT. It is found that the present nonuniform fifth-order TPT is generally more accurate than the third+second-order perturbation DFT. Additional advantages of the present nonuniform fifth-order TPT over the third +second-order perturbation DFT is discussed.

Self-consistent Ornstein–Zernike approximation for the Yukawa fluid with improved direct correlation function

Thermodynamic consistency of the mean spherical approximation as well as the self-consistent Ornstein-Zernike approximation (SCOZA) with the virial route to thermodynamics is analyzed in terms of renormalized gamma-ordering. For continuum fluids, this suggests the addition of a short-ranged contribution to the usual SCOZA direct correlation function, and the shift of the adjustable parameter from the potential term to this new term. The range of this contribution is fixed by imposing consistency with the virial route at the critical point. Comparison of the results of our theory for the hard-core Yukawa potential with the simulation data show very good agreement for cases where the liquid-vapor transition is stable or not too far into the metastable region with respect to the solid state. In the latter case for extremely short-ranged interactions discrepancies arise.

Phase diagram and surface tension of the hard-core attractive Yukawa model of variable range: Monte Carlo simulations

The liquid-vapor phase diagram and surface tension for hard-core Yukawa potential with 4<or=kappa<or=7 are calculated by applying canonical Monte Carlo simulation. The authors' new simulation results are more precise than those reported before, which allows them to affirm the accuracy of the previously reported self-consistent Ornstein-Zernike approximation calculations for kappa=5 and 7.

Towards a unification of hierarchical reference theory and self-consistent Ornstein-Zernike approximation: Analysis of exactly solvable mean-spherical and generalized mean-spherical models

The hierarchical reference theory (HRT) and the self-consistent Ornstein-Zernike approximation (SCOZA) are two liquid state theories that both furnish a largely satisfactory description of the critical region as well as the phase coexistence and equation of state in general. Furthermore, there are a number of similarities that suggest the possibility of a unification of both theories. Earlier in this respect we have studied consistency between the internal energy and free energy routes. As a next step toward this goal we here consider consistency with the compressibility route too, but we restrict explicit evaluations to a model whose exact solution is known showing that a unification works in that case. The model in question is the mean spherical model (MSM) which we here extend to a generalized MSM. For this case, we show that the correct solutions can be recovered from suitable boundary conditions through either SCOZA or HRT alone as well as by the combined theory. Furthermore, the relation between the HRT-SCOZA equations and those of SCOZA and HRT becomes transparent.