Self-consistent Ornstein-Zernike Approach Research Articles

Thermodynamics of a long-range triangle-well fluid

The long-range triangle-well fluid has been studied using three different approaches: firstly, by an analytical equation of state obtained by a perturbation theory, secondly via a self-consistent integral equation theory, the so-called self-consistent Ornstein–Zernike approach (SCOZA) which is presently one of the most accurate liquid-state theories, and finally by Monte Carlo simulations. We present vapour–liquid phase diagrams and thermodynamic properties such as the internal energy and the pressure as a function of the density at different temperatures and for several values of the potential range. We assess the accuracy of the theoretical approaches by comparison with Monte Carlo simulations: the SCOZA method accurately predicts the thermodynamics of these systems and the first-order perturbation theory reproduces the overall thermodynamic behaviour for ranges greater than two molecular diameters except that it overestimates the critical point. The simplicity of the equation of state and the fact that it is analytical in the potential range makes it a good candidate to be used for calculating other thermodynamic properties and as an ingredient in more complex theoretical approaches.

Performance Evaluation of Third-Order Thermodynamic Perturbation Theory and Comparison with Existing Liquid State Theories

To evaluate the performance of a recently proposed third-order thermodynamic perturbation theory (TPT), we employ the third TPT for calculation of thermodynamic properties such as compressibility factor, internal energy, excess chemical potential, gas-liquid coexistence curve, and critical properties of several fluids. By comparing the third-order TPT results with corresponding simulation data available in literature and supplied in the present report and theoretical results from several other theoretical approaches, one concludes that the third-order TPT is, in general, more accurate than other approaches such as Barker-Henderson second-order TPT using a macroscopic compressibility approximation (MCA-TPT), self-consistent Ornstein-Zernike approach, Monte Carlo perturbation theory, and a specially devised equation of state. Specifically, the third-order TPT can predict quantitatively a double critical phenomena of gas-liquid transition and a low-density liquid (LDL)-high-density liquid (HDL) transition associated with a soft core (SC) potential fluid very satisfactorily, but the predictions for the LDL-HDL transition based on the second-order MCA-TPT are quantitatively very bad or qualitatively incorrect. The failure of the second-order MCA-TPT for the SC fluid can be ascribed to the facts that for the SC potential the second-order and third-order terms of the perturbation expansion are not small quantities and that the second-order term is underestimated by the MCA. It is concluded that the present third-order version of the TPT is reliable for varying model fluids.

Ornstein−Zernike Random-Walk Approach for Polymers

A new theory of the thermodynamics and structure of polymer-chain fluids is developed. It is based on relating, in a novel way, polymer-chain structure with random-walk structure in the context of Ornstein−Zernike equations. It incorporates much of the interaction-site description of correlation used in reference-system interaction site model (RISM) theory but introduces several simplifying approximations in connecting the RISM equation to random-walk theory that departs in a significant way from the extension of RISM used in polymer RISM (PRISM) theory. Both continuum-space and lattice-gas versions of the theory are summarized, and the extension of the theory to a self-consistent Ornstein−Zernike approach (SCOZA) for polymers is given.

Progress in thermodynamic perturbation theory and self-consistent Ornstein–Zernikeapproach relevant to structural-arrest problems

We discuss the derivation and use of a key feature of a new version of thermodynamicperturbation theory (TPT) that Ben-Amotz and Stell have developed recently—the choiceof a hard-core diameter to be used in modelling soft-core systems. We go on to show theway the new version of TPT can be used as input for the self-consistent Ornstein–Zernikeapproach (SCOZA) of Høye and Stell to obtain highly accurate static structure factors andthermodynamics for fluids and glasses. In obtaining quantitatively useful results for thestructural arrest and other dynamic features in colloidal systems with pair potentials thatinclude strong short-range attraction, accurate structure factors are a necessity.

A simple SCOZA for simple fluids

A simple version of the self-consistent Ornstein–Zernike approach (SCOZA) is investigated quantitatively. The simplification consists of retaining thermodynamic self-consistency between the compressibility and energy routes to thermodynamics from the pair distribution function g( r). However, we do not require that g( r) be identically zero within the hard repulsive core of the models we consider here. The resulting theory lends itself to analysis of Hamiltonian models to which the earlier SCOZA formulation cannot be readily applied as well as providing a convenient means of analyzing simpler models. We choose the hard-core Yukawa fluid as an illustrative example to calibrate the accuracy of the new version. The results are compared with our earlier SCOZA and simulation results, and a suggestion is given to systematically improve the accuracy of the theory in the applications that warrant so doing.

Self-consistent Ornstein–Zernike Approach to Polymer Fluids

In this work we study Flory–Huggins lattice polymers using a self-consistent Ornstein–Zernike approach. The results are compared with those derived from classical mean-field theory. A simple continuum-space (i.e. off-lattice) extension of the Flory–Huggins model is also proposed and studied.

Self-consistent Ornstein–Zernike approximation compared with Monte Carlo results for two-dimensional lattice gases

The self-consistent Ornstein–Zernike approach (SCOZA) is solved numerically for a lattice gas or Ising model on the simple square lattice in two dimensions. Interactions of varying range are considered, and the results are compared with corresponding simulation ones. We focus especially upon the location of the critical temperature Tc which is identified with the maximum of the specific heat. The maximum remains finite for the finite-sized simulation sample and also for SCOZA, which treats infinite lattices in two dimensions as though they were finite samples. We also investigate the influence of the precise form of the interaction, first using an interaction that extends the nearest-neighbor case in a simple way and then considering the square-well interactions used in the simulations. We find that the shift in Tc away from its mean-field value is governed primarily by the range of interaction. Other specific features of the interaction leave a smaller influence but are relevant to a quantitative comparison with simulations. The SCOZA yields accurate results, and the influence of the precise form of the attractive interaction plays a significant role in SCOZA’s success.

SCOZA critical exponents and scaling in three dimensions

The critical behavior of a self-consistent Ornstein–Zernike approach (SCOZA) that describes the pair correlation function and thermodynamics of a classical fluid, lattice gas, or Ising model is analyzed in three dimensions below the critical temperature, complementing our earlier analysis of the supercritical behavior. The SCOZA subcritical exponents describing the coexistence curve, susceptibility (compressibility), and specific heat are obtained analytically ( β=7/20, γ′=7/5, α′=−1/10 ). These are in remarkable agreement with the exact values ( β≈0.326, γ′≈1.24, α′≈0.11 ) considering that the SCOZA uses no renormalization group concepts. The scaling behavior that describes the singular parts of the thermodynamic functions as the critical point is approached is also analyzed. The subcritical scaling behavior in the SCOZA is somewhat less simple than that expected in an exact theory, involving two scaling functions rather than one.

Critical properties of the self-consistent Ornstein–Zernike approximation for three-dimensional lattice gases with varying range of interaction

The self-consistent Ornstein–Zernike approach (SCOZA) is solved numerically, and its properties in the critical region are investigated for the lattice gas or Ising model in three dimensions. We especially investigate how critical properties depend upon the inverse range of interaction. We find effective critical indices that depend upon this range. However, the SCOZA does not fulfill scaling. Nevertheless, comparing with experimental results for fluids and magnets we find good agreement. Away from the critical point we find that SCOZA yields deviations from scaling that seem similar to experiments.

SCOZA for continuous spins

Our earlier work applying the self-consistent Ornstein-Zernike approach (SCOZA) to the D-vector model is extended to a more general class of continuous-spin models for which the constraint of a single-spin magnitude is relaxed.

SCOZA for D-dimensional spins

The self-consistent Ornstein-Zernike approach (SCOZA) developed earlier by the authors for the lattice gas and Ising model as well as simple continuum fluids is extended to the D-vector spin model that includes the classical rotator model ( D = 2), the classical Heisenberg model ( D = 3), and the spherical-model limit ( D → ∞), in which the SCOZA becomes exact. The approach entails first formulating an exact treatment of the model that includes its response to the presence of a field H with one primary component H |) and a transverse component H ⊥.