Barker-Henderson Research Articles

Thermodynamic properties of fluids with Lennard–Jones–Gauss potential from computer simulation and the coupling parameter series expansion

ABSTRACTMonte Carlo NpT simulations have been performed to obtain the pressure, excess enthalpy, excess internal energy, isothermal compressibility coefficient, thermal expansion coefficient and constant-pressure excess heat capacity of fluids with several kinds of Lennard–Jones–Gauss potential models from subcritical to supercritical and from vapour to liquid states. These data have been used to assess the performance of a perturbation theory, namely the coupling parameter series expansion (CPSE) of the Helmholtz free energy, for continuous potential models without hard-sphere core. The main findings are summarised as follows: (1) The analysis of the results for the constant-pressure excess heat capacity shows that the CPSE is convergent until seventh order. (2) The seventh-order CPSE, when combined with the simple Barker–Henderson (BH) effective hard-sphere diameter, predicts rather accurately the thermodynamic properties, except at high densities and low temperatures. (2) The loss of accuracy in these situations is attributed to the inadequacy of the BH prescription that does not account for the density dependence of the effective hard-sphere diameter. (3) The CPSE performance is rather insensitive to the details of the potential model considered.

Thermodynamic properties of diamond and wurtzite model fluids from computer simulation and thermodynamic perturbation theory

Monte Carlo NVT simulations have been performed to obtain the thermodynamic and structural properties and perturbation coefficients up to third order in the inverse temperature expansion of the Helmholtz free energy of fluids with potential models proposed in the literature for diamond and wurtzite lattices. These data are used to analyze performance of a coupling parameter series expansion (CPSE). The main findings are summarized as follows, (1) The CPSE provides accurate predictions of the first three coefficient in the inverse temperature expansion of Helmholtz free energy for the potential models considered and the thermodynamic properties of these fluids are predicted more accurately when the CPSE is truncated at second or third order. (2) The Barker–Henderson (BH) recipe is appropriate for determining the effective hard sphere diameter for strongly repulsive potential cores, but its performance worsens with increasing the softness of the potential core. (3) For some thermodynamic properties the first-order CPSE works better for the diamond potential, whose tail is dominated by repulsive interactions, than for the potential, whose tail is dominated by attractive interactions. However, the first-order CPSE provides unsatisfactory results for the excess internal energy and constant-volume excess heat capacity for the two potential models.

A critical evaluation of perturbation theories by Monte Carlo simulation of the first four perturbation terms in a Helmholtz energy expansion for the Lennard-Jones fluid.

The Helmholtz energy of a fluid interacting by a Lennard-Jones pair potential is expanded in a perturbation series. Both the methods of Barker-Henderson (BH) and of Weeks-Chandler-Andersen (WCA) are evaluated for the division of the intermolecular potential into reference and perturbation parts. The first four perturbation terms are evaluated for various densities and temperatures (in the ranges ρ*=0-1.5 and T*=0.5-12) using Monte Carlo simulations in the canonical ensemble. The simulation results are used to test several approximate theoretical methods for describing perturbation terms or for developing an approximate infinite order perturbation series. Additionally, the simulations serve as a basis for developing fully analytical third order BH and WCA perturbation theories. The development of analytical theories allows (1) a careful comparison between the BH and WCA formalisms, and (2) a systematic examination of the effect of higher-order perturbation terms on calculated thermodynamic properties of fluids. Properties included in the comparison are supercritical thermodynamic properties (pressure, internal energy, and chemical potential), vapor-liquid phase equilibria, second virial coefficients, and heat capacities. For all properties studied, we find a systematically improved description upon using a higher-order perturbation theory. A result of particular relevance is that a third order perturbation theory is capable of providing a quantitative description of second virial coefficients to temperatures as low as the triple-point of the Lennard-Jones fluid. We find no reason to prefer the WCA formalism over the BH formalism.

Comparison between the statistical associating fluid theory for variable range potentials (SAFT-VR) and the first-order mean spherical approximation (FMSA) equations of state for Mie (α, 6) fluids

The respective accuracies of three recent analytical equations of state (EOS) (i.e. analytical representations of the Helmholtz free energy and compressibility factor Z) for the Mie(α,6) monomer fluids are evaluated by comparing the residual Helmholtz free energy, its Barker-Henderson (BH) perturbation terms and Z with the corresponding numerical simulations for the Mie(8,6), Mie(12,6), Mie(20,6) and Mie(30,6) fluids. Two of these EOS stem from the recent enhanced statistical associating fluid theory with variable range potentials (SAFT-VR Mie 2013), with two proposed parameterizations for the free energy first-order BH perturbation term of the Sutherland potentials building up the Mie potential. The third EOS uses a new mapping between the Mie(α,6) and a two-Yukawa potential, for which the first-order mean spherical approximation (FMSA) can be used to obtain analytically the free energy and pressure of the Mie(α,6) fluids up to second order.

Theoretical study of phase behaviour of DLVO model for lysozyme and γ-crystalline aqueous electrolyte solutions

Mean spherical approximation (MSA), second-order Barker-Henderson (BH) perturbation theory and thermodynamic perturbation theory (TPT) for associating fluids in combination with BH perturbation theory are applied to the study of the structural properties and phase behaviour of the Derjaguin-Landau-Verwey-Overbeek (DLVO) model of lysozyme and $\gamma$-cristalline aqueous electrolyte solutions. Predictions of the MSA for the structure factors are in good agreement with the corresponding computer simulation predictions. The agreement between theoretical results for the liquid-gas phase diagram and the corresponding results of the experiment and computer simulation is less satisfactory, with predictions of the combined BH-TPT approach being the most accurate.

What is liquid in random porous media: the Barker-Henderson perturbation theory

We apply the Barker-Henderson (BH) perturbation theory to the study of a Lennard-Jones fluid confined in a random porous matrix formed by hard sphere particles. In order to describe the reference system needed in this perturbation scheme, the extension of the scaled particle theory (SPT) is used. The recent progress in the development of SPT approach for a hard sphere fluid in a hard sphere matrix allows us to obtain very accurate results for thermodynamic properties in such a system. Hence, we combine the BH perturbation theory with the SPT approach to derive expressions for the chemical potential and the pressure of a confined fluid. Using the obtained expressions, the liquid-vapour phase diagrams of a LJ fluid in HS matrix are built from the phase equilibrium conditions. Therefore, the effect of matrix porosity and a size of matrix particles is considered. It is shown that a decrease of matrix porosity lowers both the critical temperature and the critical density, while the phase diagram becomes narrower. An increase of a size of matrix particles leads to an increase of the critical temperature. From the comparison it is observed that the results obtained from the theory are in agreement with computer simulations. The approach proposed in the present study can be extended to the case of anisotropic fluid particles in HS matrices.

Investigation of vapour—liquid nucleation properties for spherical and chain-like fluids by density functional theory

The excess Helmholtz free energy functional for nonpolar chain-like molecules is formulated in terms of a weighted density approximation (WDA) for short-range interactions and a Weaks–Chandler–Andersen (WCA) approximation and a Barker–Henderson (BH) theory for long-range attraction. Within the framework of density functional theory (DFT), vapour–liquid interfacial properties including density profile and surface tension, and vapour–liquid nucleation properties including density profile, work of formation and number of particles are investigated for spherical and chainlike molecules. The obtained vapour–liquid surface tension and the number of particles in critical nucleus for Lennard–Jones (LJ) fluids are consistent with the simulation results. The influences of supersaturation, temperature and chain length on vapour–liquid nucleation properties are discussed.

Simple Analytic Expression with High Precision for theBarker-Henderson Diameter

A fitting procedure is proposed to establish two analytic approximateexpressions for the complicated Barker-Henderson (BH) diameter withLennard-Jones potential in two temperature ranges. Considering that thedifferentiation is a process enlarging the errors and the integrationdecreasing the errors and that the derivative of BH diameter isimportant in the calculation of internal energy, we propose to fit thederivative directly, instead of usually fitting the original functionand subsequently deriving its derivative. The simplicity and precisionof two expressions developed are superior to the extensively usedexpressions in literature. The one with following form only has anaverage fitting 0.0063% in the reduced temperature range (0.4⩽kT/ε⩽15), and can be extrapolated to a widertemperature range (0.4⩽kT/ε⩽50) with anaverage error 0.13%, which is d/σ = 1.1755 + 0.02878ln τ-0.2072τ1/4 + 0.00463τ3/4.

Role of the Barker–Henderson diameter in thermodynamics

Sensitivity of thermodynamics to the Barker–Henderson (BH) diameter for the Lennard-Jones (LJ) potential is discussed, which covers both its approximation in calculation and improvement in rationality. With regarding to the approximation, pressure and internal energy for the LJ fluid, LJ chains and LJ chain mixtures are investigated. It is found that internal energy is much more sensitive to an approximation to the diameter than pressure for pure fluids, and both pressure and internal energy are very sensitive to the diameter for mixtures. It is also found that the approximating expression given by Cotterman et al. (1986) covers the widest range of temperatures. The rationality of the BH diameter itself at very high temperatures and densities is also analyzed. Through a functional expansion of Helmholtz free energy, we conclude that a density-dependent BH diameter is fundamentally more appropriate. The proposed diameter yields almost the same results as the original BH diameter at normal conditions and remedies its deficiencies at extreme ones. The density-dependent diameter provides a convenient way to study the LJ systems undergoing gas-solid phase transition or freezing.

Application of density functional theory for predicting the surface tension of pure polar and associating fluids

A thermodynamic method has been developed based on the density functional theory (DFT) to predict the surface tension of polar and associating fluids by the authors. The Barker–Henderson (BH) perturbation theory and statistical associating fluid theory (SAFT) are used to establish the equation of state (EOS). The hard sphere repulsion, dispersion, chain formation, and dipole–dipole or association interactions are taken into account. The parameters m, σ and ε/ k for non-associating polar fluids and m, σ, ε/ k, ε AB/ k and κ AB for associating fluids are correlated by simultaneously fitting the saturated vapor pressure and the liquid density data with the EOS. The surface region of a pure liquid is divided into many extreme thin layers. The chemical potential in every layer of the surface leads to a constant by optimizing the surface thickness. The density profile is obtained from the optimized surface thickness and the hyperbolic tangent function obtained from molecular simulation. By use of the obtained density profile and the regressed parameters in EOS, the surface tensions for four pure non-associating polar fluids and 11 associating fluids in wide temperature range are predicted satisfactorily.

Density Functional Theory and the Capillary Evaporation of a Liquid in a Slit

Density functional theory is applied to a Lennard-Jones fluid near a single hard wall and in a slit formed by two walls. We use some simplified versions of the Weeks–Chandler–Andersen (WCA) and the Barker–Henderson (BH) theories. Only the most crude mean field version of the WCA theory, in which the hard-sphere correlation function is set equal to unity for all distances, seems useful. Use of the full WCA approximation is impractical because the effective hard-sphere diameter is density dependent. Generally, the best results are obtained using the BH macroscopic compressibility approximation. Our earlier study of “evaporation” of Lennard-Jones molecules in a slit is extended to other densities using the mean field theory.

Aziz fermions and liquid 3He.

Starting from the well-known quantum-field-theory low-density expansion for the ground-state energy of a strongly interacting many-fermion assembly, we rearrange the series into a quantum-thermodynamic perturbation theory that is then applied to liquid $^{3}\mathrm{He}$. Extrapolation to physical densities is implemented order by order via Pad\'e and related approximants. We employ both the hard-core square-well potential and the Aziz pair potential with the Barker-Henderson (BH) and Weeks-Chandler-Andersen (WCA) potential-decomposition schemes familiar from classical fluid studies. The binding energy, saturation density, and sound velocity are calculated, with no adjustable parameters, and compared with Green-function Monte Carlo (GFMC) computer simulations, experimental data, and recent variational calculations. Our binding-energy and saturation-density results for the Aziz-WCA case, and the sound velocity result for the Aziz-BH case, are found to be within 2% of the GFMC data.

Ground-state Lennard-Jones and Aziz boson liquids: Perturbation theory and computer experiment.

The exact, low-density many-boson expansion for the ground-state energy is rearranged in a quantum-thermodynamic perturbation series. Extrapolation to physical densities is implemented order by order via Pad\'e-type approximants. Several ways of splitting the pair Lennard-Jones (LJ) or Aziz potentials are considered, including the Barker-Henderson (BH) and Weeks-Chandler-Andersen (WCA) schemes familiar from classical fluid studies. Results obtained using the LJ interaction with WCA splitting, when compared with benchmark simulation data, agree excellently in energy and density, but only moderately so in sound velocity. For the Aziz potential, however---perhaps because of a softer core and/or its ability to support a two-particle bound state---, neither potential splitting (BH or WCA) gave results of the LJ quality, but a method intermediate between BH and WCA proves to be superior. Our results are also compared with other recent alternative calculations, both variational and perturbative.

MC test of perturbation theories based on site–site potentials

The extension of the Barker–Henderson (BH) theory to diatomic potentials is examined by comparison of the predicted thermodynamic properties with ‘‘exact’’ MC results. Two choices of the potential separation have been considered; corresponding to the BH scheme and that of Weeks–Chandler–Andersen (WCA). Attention has been paid to the effect of the usual approximations to the exact properties of the hard reference system (equation of state and pair correlation functions) and to the convergence rate of the expansion. Conclusions can also be drawn for the extension of the WCA-blip function theory to these systems. It is shown that the behavior of the perturbation expansions exhibits the same features observed in atomic fluids. Nevertheless, the presence of the molecular anisotropy leads to a rather different assessment of the relative usefulness of each theory. Incidentally, small inaccuracies of the well-known Boublik and Nezbeda equation for the compressibility factor of dumbbells have been discovered at very high densities (outside the range previously reported).

On the accuracy of the liquid theory approximate methods for the description of liquid metal thermodynamics

The accuracy of the various liquid theory methods for calculating the thermodynamic properties of liquid metals is investigated. The calculations have been made for liquid Na, Rb, Mg and Al over wide ranges of temperature T and density n using (i) the variational method and hard-sphere reference system (the HSV method), (ii) the Barker-Henderson (BH) method, (iii) the Weeks-Chandler-Andersen (WCA) method and (iv) the expressions for energy and pressure in terms of the radial distribution function found in the Percus-Yevick approximation (the PYEP method). The results of the calculations are compared with each other and with experiment. The pseudopotential model is used which describes all the atomic properties of the alkali metals to a high precision, those of Mg rather well, and of Al fairly well. It has been shown that the simple HSV approximation suitable for a semi-quantitative description of the liquid metal properties is insufficient for quantitative calculations of differential characteristics, such as melting temperatures, entropies of fusion, entropies of liquid alloy mixing, etc. The PYEP and BH methods allow the authors to calculate some thermodynamic potentials rather precisely but the computational difficulties prevent them from being used for calculating other thermodynamic properties. The WCA method has been found to combine high accuracy with sufficient simplicity of calculations and seems, therefore, to be the most promising one for quantitative calculations of liquid metal thermodynamics. They also discuss the problems of applicability of the methods considered at various T and n (particularly in the vicinity of the critical point), of the dependence of their accuracy on the metal valency and the sensitivity of the results to the type of pseudopotential model used.

On the accuracy of the liquid theory approximate methods for calculating the structure factors in liquid metals

The accuracy with which the liquid theory approximate methods describe the structure factors S(k) in liquid metals is investigated. The results of the Barker-Henderson (BH) and Weeks-Chandler-Anderson (WCA) perturbation theories and the Percus-Yevick (PY) and hyper-netted-chain (HNC) equations are examined. Values of S(k)=Si(k) calculated by these methods for liquid metals are compared with the available computer simulation data using the same inter-ionic potentials phi (r). Si(k) for Na and Rb calculated with the previously described reliable phi (r), are also compared with experiment. It is concluded from the comparisons that the approximations SBH(k), SWCA(k) and SHNC(k) are not very accurate in liquid metals, particularly near the melting point. SHNC(k) underestimates the heights of the S(k) peaks while SWCA(k) overestimates them. Possibilities of using the conclusions for the experimental verification of theoretical phi (r) and in thermodynamic calculations for liquid metals are discussed. An effective iterative method for the solution of the PY and HNC equations is proposed.

A note on the perturbation theory and the triangular well potential

The integrals which appear in the first order term and the macroscopic compressibility (mc) and the local compressibility (lc) approximation for the second order term in the Barker–Henderson (BH) perturbation theory of liquids are evaluated analytically for the triangular-well potential. The compressibility factors so calculated are compared with the Monte Carlo calculations.

Perturbation Theory and the Triangular Well Potential

Accurate calculations of the second order term in the free energy and the first order term in the radial distribution function in the Barker–Henderson (BH) perturbation theory are presented for the triangular well potential. The BH theory is found to be fully satisfactory for this system. Thus, the conclusions of Card and Walkley regarding the accuracy of the BH theory are erroneous.

Barker–Henderson theory as an accurate theory of simple mixtures

The properties and excess properties of some simple binary mixtures have been calculated from the Lennard-Jones potential and the Barker–Henderson (BH) theory including the second order terms. In these calculations the radial distribution function was approximated by the Percus–Yevick g(R) value combined with the corrections devised by Verlet and Weis and Lee and Levesque. The results show that these corrections eliminate most of the major differences between the Monte Carlo values and the BH theory values obtained previously but in some cases lead to small increases in errors for then excess properties of some mixtures. The comparisons with experimental data while generally qualitatively correct showed greater uncertainties as the interaction potentials and their mixing rules are unspecified for these systems.