Hierarchical Reference Theory Research Articles

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.

Theory and computer simulation of hard-core Yukawa mixtures: thermodynamical, structural and phase coexistence properties

We report extensive calculations, based on the modified hypernetted chain (MHNC) theory, on the hierarchical reference theory (HRT), and on Monte Carlo simulations, of thermodynamical, structural and phase coexistence properties of symmetric binary hard-core Yukawa mixtures (HCYM) with attractive interactions at equal species concentration. The obtained results are throughout compared with those available in the literature for the same systems. It turns out that the MHNC predictions for thermodynamic and structural quantities are quite accurate in comparison with the MC data. The HRT is equally accurate for thermodynamics, and slightly less accurate for structure. Liquid-vapor (LV) and liquid–liquid (LL) consolute coexistence conditions as emerging from simulations, are also highly satisfactorily reproduced by both the MHNC and HRT for relatively long ranged potentials. When the potential range reduces, the MHNC faces problems in determining the LV binodal line; however, the LL consolute line and the critical end point (CEP) temperature and density turn out to be still satisfactorily predicted within this theory. The HRT also predicts with good accuracy the CEP position.The possibility of employing liquid state theories HCYM for the purpose of reliably determining phase equilibria in multicomponent colloidal fluids of current technological interest, is discussed.

Solvent-mediated forces in critical fluids.

The effective interaction between two planar walls immersed in a fluid is investigated by use of density functional theory in the supercritical region of the phase diagram. A hard core Yukawa model of fluid is studied with special attention to the critical region. To achieve this goal a formulation of the weighted density approximation coupled with the hierarchical reference theory, able to deal with critical long wavelength fluctuations, is put forward and compared with other approaches. The effective interaction between the walls is seen to change character on lowering the temperature: The strong oscillations induced by layering of the molecules, typical of the depletion mechanism in hard core systems, are gradually smoothed and, close to the critical point, a long range attractive tail emerges leading to a scaling form which agrees with the expectations based on the critical Casimir effect. Strong corrections to scaling are seen to affect the results up to very small reduced temperatures. By use of the Derjaguin approximation, this investigation has natural implications for the aggregation of colloidal particles in critical solvents.

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.

Recent developments of the hierarchical reference theory of fluids and its relation to the renormalization group

The Hierarchical Reference Theory (HRT) of fluids is a general framework for the description of phase transitions in microscopic models of classical and quantum statistical physics. The foundations of HRT are briefly reviewed in a self-consistent formulation which includes both the original sharp cut-off procedure and the smooth cut-off implementation, which has been recently investigated. The critical properties of HRT are summarized, together with the behaviour of the theory at first-order phase transitions. However, the emphasis of this presentation is on the close relationship between HRT and non-perturbative renormalization group methods, as well as on recent generalizations of HRT to microscopic models of interest in soft matter and quantum many body physics.

Analysis of the critical region of the hierarchical reference theory

The critical region of the hierarchical reference theory (HRT) is investigated both by analytical and numerical methods. This is closely related to a recent work by one of us where the HRT was unified with another accurate theory, the self-consistent Ornstein–Zernike theory (SCOZA). It was found that somehow the properties of HRT alone would be closely related to the critical properties of the unified problem and vice versa. Our investigation is facilitated by the discovery of a situation where HRT and SCOZA give identical results. For the more general HRT situation we then find that an additional intermediate term appears, and the generalized scaling of SCOZA is replaced with standard scaling. Further we find that the numerical results can be accurately expressed by simple analytic expressions in the critical region. With an interplay between the leading scaling function and subleading contributions, simple rational numbers are found for the critical indices.

Hierarchical reference theory of critical fluids in disordered porous media

We consider the equilibrium behaviour of fluids imbibed in disordered mesoporous media, including their gas–liquid critical point when present. Our starting points are on the one hand a description of the fluid/solid-matrix system as a quenched-annealed mixture and on the other hand the Hierarchical Reference Theory (HRT) developed by A. Parola and L. Reatto to cope with density fluctuations on all length scales. The formalism combines liquid-state statistical mechanics and the theory of systems in the presence of quenched disorder. A straightforward implementation of the HRT to the quenched-annealed mixture is shown to lead to unsatisfactory results, while indicating that the critical behaviour of the system is in the same universality class as that of the random-field Ising model. After a detour via the field-theoretical renormalization group approach of the latter model, we finally lay out the foundations for a proper HRT of fluids in a disordered porous material.

Link between the hierarchical reference theory of liquids and a new version of the non-perturbative renormalization group in statistical field theory

I propose a new derivation of the Hierarchical Reference Theory of liquids. Two formalisms, one in the grand canonical ensemble, the other in the framework of statistical field theory are given in parallel. In the latter the theory is an avatar of a new version of the non-perturbative renormalization group [J.-M. Caillol, J. Phys. A : Math. Gen. 42, 225004 (2009)]. The flow of the Wilsonian action as well as that of the effective average action of Wetterich are derived and a simple relation between the two functionals is established. The standard Hierarchical Reference Theory for liquids [A. Parola and L. Reatto, Adv. Phys. 44, 211 (1995)] is recovered for a sharp infra-red cut-off of the propagator.

Liquid–vapour transition in screened Coulomb binary mixtures

The smooth cut-off hierarchical reference theory (HRT) is applied to a simple model of charged colloid mixture: an equimolar binary fluid of hard-core particles with a Yukawa potential which is repulsive between particles of the same species, and attractive between particles of different species. The critical behaviour of this Yukawa restricted primitive model (YRPM) is determined, and shown to belong to the Ising universality class for every value of the inverse Yukawa screening length z. This remains true even in the limit z → 0 of unscreened Coulomb potential, when the YRPM reduces to the restrictive primitive model (RPM) of ionic fluids, thereby providing a microscopic justification to the occurrence of Ising-like criticality in the RPM. A simple closure for the pair correlations based on the generalized mean spherical approximation (GMSA) is adopted. Within such an approximation scheme, the fluid–fluid phase diagram and critical point are obtained for two representative values of the inverse screening length: z = 2 and z = 0.1.

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.

An investigation of critical and noncritical correlations in model colloidal fluids

We present a study of the correlation lengthξ of a hard-core fluid with three different tail interactions, namely, the Asakura–Oosawa potential,an attractive Yukawa tail, and a double Yukawa tail which is attractive at short distance andrepulsive at long distance. The correlation length is obtained along the critical isochore in thewhole range of supercritical temperatures by determining the leading pole of the structure factorS(k) in the complexplane, where S(k) is obtained from the hierarchical reference theory. We focus on howξ changes from its Ornstein–Zernike form in the critical region to its high-temperature limit,and find instances of both the Fisher–Widom and Kirkwood scenarios for themonotonic to oscillatory crossover of the decay of correlations at long distance.

Liquid-Vapor Transition from a Microscopic Theory: Beyond the Maxwell Construction

A smooth cutoff formulation of the hierarchical reference theory (HRT) is developed and applied to a Yukawa fluid. The HRT equations are derived and numerically solved leading to the expected renormalization group structure in the critical region, nonclassical critical exponents and scaling laws, a convex free energy in the whole phase diagram (including the two-phase region), finite compressibility at coexistence, together with a fully satisfactory comparison with available numerical simulations. This theory, which also guarantees the correct short range behavior of two body correlations, represents a major improvement over the existing liquid-state theories.

Nonuniversal Routes to Universality: Critical Phenomena in Colloidal Dispersions

We investigate critical phenomena in colloids by means of the renormalization-group based hierarchical reference theory of fluids. We focus on three experimentally relevant model systems: namely, the Asakura-Oosawa model of a colloidal dispersion under the influence of polymer-induced attractive depletion forces; fluids with competing short-range attractive and longer-range repulsive interactions; solutions of star polymers whose pair potential presents both an attractive well and an ultrasoft repulsion at shorter distance. Our results show that the ability to tune the effective interactions between colloidal particles allows one to generate a variety of crossovers to the asymptotic critical behavior, which are not observed in atomic fluids.

Smooth cutoff formulation of hierarchical reference theory for a scalarϕ4field theory

The phi4 scalar field theory in three dimensions, prototype for the study of phase transitions, is investigated by means of the hierarchical reference theory (HRT) in its smooth cutoff formulation. The critical behavior is described by scaling laws and critical exponents which compare favorably with the known values of the Ising universality class. The inverse susceptibility vanishes identically inside the coexistence curve, providing a first principle implementation of the Maxwell construction, and shows the expected discontinuity across the phase boundary, at variance with the usual sharp cutoff implementation of HRT. The correct description of first and second order phase transitions within a microscopic, nonperturbative approach is thus achieved in the smooth cutoff HRT.

Liquid–vapour transition of the long range Yukawa fluid

Two liquid state theories, the self-consistent Ornstein–Zernike equation (SCOZA) and the hierarchical reference theory (HRT) are shown, by comparison with Monte Carlo simulations, to perform extremely well in predicting the liquid–vapour coexistence of the hard-core Yukawa (HCY) fluid when the interaction is long range. The long range of the potential is treated in the simulations using both an Ewald sum and hyperspherical boundary conditions. In addition, we present an analytical optimized mean field theory which is exact in the limit of an infinitely long-range interaction. The work extends a previous one by C. Caccamo, G. Pellicane, D. Costa, D. Pini, and G. Stell, Phys. Rev. E 60, 5533 (1999) for short-range interactions.

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.

Improvement on macroscopic compressibility approximation and beyond

A numerical procedure is proposed to extend the thermodynamic perturbation expansion (TPE) to a higher order. It is shown that the present second order term is superior to that due to a macroscopic compressibility approximation (MCA), a local compressibility approximation, and a superposition approximation by Barker and Henderson [Rev. Mod. Phys. 48, 587 (1976)]. Extensive model calculation and comparison with simulation data available in literature and supplied in the present report indicate that the present third order TPE is superior to a previous second order TPE based on the MCA, two previous perturbation theories, which are respectively based on an analytical mean spherical approximation for an Ornstein-Zernike equation, and an assumed explicit functional form for the Laplace transform of radial distribution function multiplied by radial distance, and a recent generalized van der Waals theory. The present critical temperature for a hard core attractive Yukawa fluid of varying range is in very good agreement with that due to a hierarchical reference theory. The present third order TPE is computationally far more modest than the self-consistent integral equation theory, and therefore is a viable alternative to use of the latter.

Critical behavior in colloid-polymer mixtures: Theory and simulation

We extensively investigated the critical behavior of mixtures of colloids and polymers via the two-component Asakura-Oosawa model and its reduction to a one-component colloidal fluid using accurate theoretical and simulation techniques. In particular the theoretical approach, hierarchical reference theory [A. Parola and L. Reatto, Adv. Phys. 44, 211 (1995)], incorporates realistically the effects of long-range fluctuations on phase separation giving exponents which differ strongly from their mean-field values, and are in good agreement with those of the three-dimensional Ising model. Computer simulations combined with finite-size scaling analysis confirm the Ising universality and the accuracy of the theory, although some discrepancy in the location of the critical point between one-component and full-mixture description remains. To assess the limit of the pair-interaction description, we compare one-component and two-component results.