Shiqi Zhou

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.

Thermodynamics and phase behavior of a triangle-well model and density-dependent variety.

A hard sphere+triangle-well potential is employed to test a recently proposed thermodynamic perturbation theory (TPT) based on a coupling parameter expansion. It is found that the second-order term of the coupling parameter expansion surpasses by far that of a high temperature series expansion under a macroscopic compressibility approximation and several varieties. It is also found that the fifth-order version displays best among all of the numerically accessible versions with dissimilar truncation orders. Particularly, the superiority of the fifth-order TPT from other available liquid state theories is exhibited the most incisively when the temperature of interest obviously falls. We investigate the modification of the phase behavior of the hard sphere+triangle-well fluid resulting from a density dependence imposed on the original potential function. It is shown that (1) the density dependence induces polymorphism of fluid phase, particularly liquid-liquid transition in metastable supercooled region, and (2) along with enhanced decaying of the potential function as a function of bulk density, both the liquid-liquid transition and vapor-liquid transition tend to be situated at the domain of lower temperature, somewhat similar to a previously disclosed thumb rule that the fluid phase transition tends to metastable with respect to the fluid-solid transition as the range of the attraction part of a density-independence potential is sufficiently short compared to the range of the repulsion part of the same density-independence potential.

Comprehensive investigation about the second order term of thermodynamic perturbation expansion

Monte Carlo simulations are carried out for the second order term in the thermodynamic perturbation expansion around a hard sphere reference fluid. The sample potentials considered cover a wide spectrum: From two frequently employed, namely hard sphere plus square well potential and hard core attractive Yukawa potential, to two kinds of repulsive potentials, namely hard sphere plus square shoulder potential and hard sphere plus triangle shoulder potential; the investigated potential range also extends from extremely short range to rather long range. The obtained simulation data are used to evaluate performance of two theoretical approaches, i.e., a traditional macroscopic compressibility approximation (MCA) and a recent coupling parameter expansion. Extensive comparison shows that the coupling parameter expansion provides a reliable method for accurately calculating the second order term of the high temperature series expansion, while the widely accepted MCA fails quantitatively or even qualitatively for most of the situations investigated.

New free energy density functional and application to core-softened fluid

A new free energy density functional is advanced for general nonhard sphere potentials characterized by a repulsive core with a singular point at zero separation. The present functional is characterized by several features. (i) It does not involve with dividing the potentials into hard-sphere-like contribution and tail contribution in sharp contrast with usual effective hard sphere model+mean field approximation for tail contribution. (ii) It has no recourse to the use of weighted density and is computationally modest; it also does not resort to an equation of state and/or an excess Helmholtz free energy of bulk fluid over a range of density as input. Consequently, all of input information can be obtained by numerical solution of a bulk Ornstein-Zernike integral equation theory (OZ IET). Correspondingly, despite the use of bulk second-order direct correlation function (DCF) as input, the functional is applicable to the subcritical region. (iii) There is no any adjustable parameter associated with the present functional, and an effective hard sphere diameter entering the functional can be determined self-consistently and analytically once the input information, i.e., the second-order DCF and pressure of the coexistence bulk fluid, are obtained by the OZ IET. The present functional is applied to a core-softened fluid subject to varying external fields, and the density distributions predicted by the present functional are more self-consistent with available simulation results than a previous third-order+second-order perturbation density functional theory.

Inquiry into thermodynamic behavior of hard sphere plus repulsive barrier of finite height

A bridge function approximation is proposed to close the Ornstein-Zernike (OZ) integral equation for fluids with purely repulsive potentials. The performance of the bridge function approximation is then tested by applying the approximation to two kinds of repulsive potentials, namely, the square shoulder potential and the triangle shoulder potential. An extensive comparison between simulation and the OZ approach is performed over a wide density range for the fluid phase and several temperatures. It is found that the agreement between the two routes is excellent for not too low temperatures and satisfactory for extremely low temperatures. Then, this globally trustworthy OZ approach is used to investigate the possible existence or not of a liquid anomaly, i.e., a liquid-liquid phase transition at low temperatures and negative values of the thermal expansion coefficient in certain region of the phase diagram. While the existence of the liquid anomaly in the square shoulder potential has been previously predicted by a traditional first-order thermodynamic perturbation theory (TPT), the present investigation indicates that the liquid-liquid phase transition disappears in the OZ approach, so that its prediction by the first-order TPT is only an artifact originating from the low temperature inadequacy of the first-order TPT. However, the OZ approach indeed predicts negative thermal expansion coefficients. The present bridge function approximation, free of adjustable parameters, is suitable to be used within the context of a recently proposed nonhard sphere perturbation scheme.

A theoretical investigation on the honeycomb potential fluid

A local self-consistent Ornstein-Zernike (OZ) integral equation theory (IET) is proposed to provide a rapid route for obtaining thermodynamic and structural information for any thermodynamically stable or metastable state points in the bulk phase diagram without recourse to traditional thermodynamic integration, and extensive NVT-Monte Carlo simulations are performed on a recently proposed honeycomb potential in three dimensions to test the theorys reliability. The simulated quantities include radial distribution function (rdf) and excess internal energy, pressure, excess chemical potential, and excess Helmholtz free energy. It is demonstrated that (i) the theory reproduces the rdf very satisfactorily only if the bulk state does not enter deep into a two phases coexistence region; (ii) the excess internal energy is the only one of the four thermodynamic quantities investigated amenable to the most accurate prediction by the present theory, and the simulated pressure is somewhat overestimated by the theoretical calculations, but the deviation tends to vanish along with rising of the temperature; (iii) using the structural functions from the present local self-consistent OZ IET, a previously derived local expression, due to the present author, achieves even a higher accuracy in calculating for the excess chemical potential than the exact virial pressure formula for the pressure, and the resulting excess Helmholtz free energy is in surprisingly same with the simulation results due to offset of the errors. Based on the above observations, it is suggested that it may be a good procedure to integrate the theoretical excess internal energy along the isochors to get the excess Helmholtz free energy, which is then fitted to a polynomial to be used for calculation of all of other thermodynamic quantities in the framework of the OZ IET.

Non-hard sphere thermodynamic perturbation theory.

A non-hard sphere (HS) perturbation scheme, recently advanced by the present author, is elaborated for several technical matters, which are key mathematical details for implementation of the non-HS perturbation scheme in a coupling parameter expansion (CPE) thermodynamic perturbation framework. NVT-Monte Carlo simulation is carried out for a generalized Lennard-Jones (LJ) 2n-n potential to obtain routine thermodynamic quantities such as excess internal energy, pressure, excess chemical potential, excess Helmholtz free energy, and excess constant volume heat capacity. Then, these new simulation data, and available simulation data in literatures about a hard core attractive Yukawa fluid and a Sutherland fluid, are used to test the non-HS CPE 3rd-order thermodynamic perturbation theory (TPT) and give a comparison between the non-HS CPE 3rd-order TPT and other theoretical approaches. It is indicated that the non-HS CPE 3rd-order TPT is superior to other traditional TPT such as van der Waals/HS (vdW/HS), perturbation theory 2 (PT2)/HS, and vdW/Yukawa (vdW/Y) theory or analytical equation of state such as mean spherical approximation (MSA)-equation of state and is at least comparable to several currently the most accurate Ornstein-Zernike integral equation theories. It is discovered that three technical issues, i.e., opening up new bridge function approximation for the reference potential, choosing proper reference potential, and/or using proper thermodynamic route for calculation of f(ex-ref), chiefly decide the quality of the non-HS CPE TPT. Considering that the non-HS perturbation scheme applies for a wide variety of model fluids, and its implementation in the CPE thermodynamic perturbation framework is amenable to high-order truncation, the non-HS CPE 3rd-order or higher order TPT will be more promising once the above-mentioned three technological advances are established.

Excellence of numerical differentiation method in calculating the coefficients of high temperature series expansion of the free energy and convergence problem of the expansion

In this paper, it is shown that the numerical differentiation method in performing the coupling parameter series expansion [S. Zhou, J. Chem. Phys. 125, 144518 (2006); AIP Adv. 1, 040703 (2011)] excels at calculating the coefficients ai of hard sphere high temperature series expansion (HS-HTSE) of the free energy. Both canonical ensemble and isothermal-isobaric ensemble Monte Carlo simulations for fluid interacting through a hard sphere attractive Yukawa (HSAY) potential with extremely short ranges and at very low temperatures are performed, and the resulting two sets of data of thermodynamic properties are in excellent agreement with each other, and well qualified to be used for assessing convergence of the HS-HTSE for the HSAY fluid. Results of valuation are that (i) by referring to the results of a hard sphere square well fluid [S. Zhou, J. Chem. Phys. 139, 124111 (2013)], it is found that existence of partial sum limit of the high temperature series expansion series and consistency between the limit value and the true solution depend on both the potential shapes and temperatures considered. (ii) For the extremely short range HSAY potential, the HS-HTSE coefficients ai falls rapidly with the order i, and the HS-HTSE converges from fourth order; however, it does not converge exactly to the true solution at reduced temperatures lower than 0.5, wherein difference between the partial sum limit of the HS-HTSE series and the simulation result tends to become more evident. Something worth mentioning is that before the convergence order is reached, the preceding truncation is always improved by the succeeding one, and the fourth- and higher-order truncations give the most dependable and qualitatively always correct thermodynamic results for the HSAY fluid even at low reduced temperatures to 0.25.

A new scheme for perturbation contribution in density functional theory and application to solvation force and critical fluctuations

To surpass a traditional mean field density functional approximation for a perturbation term of interparticle potential function in liquid state, a correlation term is introduced by using weighted density approximation to deal with the perturbation free energy beyond the mean field one. Consequently, a free energy density functional approximation is advanced by combining the mean field term and correlation term with a hard sphere term treated with a Lagrangian theorem-based density functional approximation in the present work. The present free energy density functional approximation is applied in the framework of classical density functional theory (DFT) to a hard core attractive Yukawa (HCAY) fluid subject to external fields; comparison of the resulted predictions for density profiles with available simulation data is favorable for the present DFT approach as a highly accurate predictive approach. Then, the DFT approach is employed to investigate influencing factors for solvation forces between two infinite planar surfaces immersed in an intervening solvent with the HCAY potential function. It is found that (i) critical fluctuations induce negative adsorptions and long-ranged solvation forces; (ii) for narrow slit, the effect of external potential range is kept down; instead strength of the external field contact potential plays dominating role; (iii) state point in the bulk phase diagram, where the most remarkable critical effects are displayed, is the one with a bulk density a little higher than the critical density; remnants of critical fluctuations remain close to the bulk gas-liquid coexistence curve.

Phase behavior of density-dependent pair potentials

Phase diagram is calculated by a recently proposed third-order thermodynamic perturbation theory (TPT) for fluid phase and a recently proposed first-order TPT for solid phases; the underlying interparticle potential consists of a hard sphere repulsion and a perturbation tail of an attractive inverse power law type or Yukawa type whose range varies with bulk densities. It is found that besides usual phase transitions associated with density-independent potentials, the density dependence of the perturbation tail evokes some additional novel phase transitions including isostructural solid-solid transition and liquid-liquid transition. Novel triple points are also exhibited which includes stable fluid (vapor or liquid)-face-centered cubic(fcc)-fcc and liquid-liquid-fcc, metastable liquid-body-centered cubic(bcc)-bcc. It also is found that the phase diagram sensitively depends on the density dependence and the concrete mathematical form of the underlying potentials. Some of the disclosed novel transitions has been observed experimentally in complex fluids and molecular liquids, while others still remain to be experimentally verified.