Coupling Parameter Series Expansion Research Articles

Padé approximant for hard sphere ＋ square well and hard sphere ＋ square well ＋ square shoulder model fluids

The coupling parameter series expansion (CPSE) is used to calculate coefficientsai (i=1,2,3,…7) of the high temperature series expansion (HTSE) for two hard sphere ＋ square well models and two hard sphere ＋ square well ＋ square shoulder models, the four sets of coefficients are used to construct 28 Padé approximant variants Pn∕d for the four models. Based on the 28 Padé variants, six common thermodynamic functions for the four models are calculated and compared with corresponding results based on MC simulations reported in the literature and the HTSE truncated to seventh order in the present work, respectively. Several conclusions are generalized from the comparisons: (i) the commonly used rule of the minimum value for n−d to choose the best Padé approximation may not select the best one, which depends on the model and relevant parameters values and has to be determined in advance by reference to simulations. (ii) From the whole table of 28 Padé approximants, reliable (throughout the text, by “reliable” we mean the results are at least qualitatively correct) variants always can be singled out, which not only provides an improvement over the raw HTSE truncated at the optimal order but also is free of pole(s) and consequently applicable over reasonably wide phase space of interest; particularly, the improvement can be immense for most but not all of the models considered when the excess heat capacity is being considered. (iii) Among the reliable Pn∕dapproximations picked, some have high values of n+d indicating that the higher-order HTSE coefficients calculated by the CPSE provide important and reliable information for the unknown excess Helmholtz free energy, and the conventional Padé summation indeed provides an effective method to utilize the information. (iv) The Padé summation can be a resort in statistical mechanics perturbation theories of simple potential fluids because of its simplicity in the actual application and possibility in improving the raw HTSE; however, to ensure the use reliably, one comprehensive pre-research should be carried out.

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.

Thermodynamic properties of a hard-core Lennard-Jones fluid from computer simulation and the coupling parameter series expansion

ABSTRACTThe thermodynamic properties of a fluid with an interaction potential consisting in a hard-sphere core plus a Lennard-Jones tail have been obtained by Monte Carlo (MC) NVT simulation as a function of the density along several isotherms. In addition, the liquid–vapour coexistence has been determined by means of histogram-reweighting MC. These data have been used to analyse the performance of perturbation theory. To this end, the first three perturbation terms of the inverse temperature expansion of the Helmholtz free energy have been obtained by means of MC NVT simulations to test the convergence of the perturbation series and to compare with the predictions of the coupling parameter series expansion. Then, the predictions of the latter theory for the thermodynamic properties have been compared with the simulations, revealing the overall excellent performance of this perturbation theory for this model fluid, except in the vicinity of the critical point.

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.

On equivalence of high temperature series expansion and coupling parameter series expansion in thermodynamic perturbation theory of fluids

The coupling parameter series expansion and the high temperature series expansion in the thermodynamic perturbation theory of fluids are shown to be equivalent if the interaction potential is pairwise additive. As a consequence, for the class of fluids with the potential having a hardcore repulsion, if the hard-sphere fluid is chosen as reference system, the terms of coupling parameter series expansion for radial distribution function, direct correlation function, and Helmholtz free energy follow a scaling law with temperature. The scaling law is confirmed by application to square-well fluids.

Coupling parameter series expansion for fluid with square-well plus repulsive-square-barrier potential

Monte Carlo simulations in the canonical ensemble are performed for fluid with potential consisting of a square-well plus a square-barrier to obtain thermodynamic properties such as pressure, excess energy, constant volume excess heat capacity, and excess chemical potential, and structural property such as radial distribution function. The simulations cover a wide density range for the fluid phase, several temperatures, and different combinations of the parameters defining the potential. These simulation data have been used to test performances of a coupling parameter series expansion (CPSE) recently proposed by one of the authors [S. Zhou, Phys. Rev. E 74, 031119 (2006)], and a traditional 2nd-order high temperature series expansion (HTSE) based on a macroscopic compressibility approximation (MAC) used with confidence since its introduction in 1967. It is found that (i) the MCA-based 2nd-order HTSE unexpectedly and depressingly fails for most situations investigated, and the present simulation results can serve well as strict criteria for testing liquid state theories. (ii) The CPSE perturbation scheme is shown to be capable of predicting very accurately most of the thermodynamic properties simulated, but the most appropriate level of truncating the CPSE differs and depends on the range of the potential to be calculated; in particular, the shorter the potential range is, the higher the most appropriate truncating level can be, and along with rising of the potential range the performance of the CPSE perturbation scheme will decrease at higher truncating level. (iii) The CPSE perturbation scheme can calculate satisfactorily bulk fluid rdf, and such calculations can be done for all fluid states of the whole phase diagram. (iv) The CPSE is a convergent series at higher temperatures, but show attribute of asymptotic series at lower temperatures, and as a result, the surest asymptotic value occurs at lower-order truncation.

Convergence and low temperature adaptability analysis of the high temperature series expansion of the free energy

By appealing to the coupling parameter series expansion to calculate the first seven perturbation coefficients of the high temperature series expansion (HTSE) of the free energy, analysis of convergence and low temperature adaptability of the HTSE in calculating fluid thermodynamic properties is performed for the first time; the fluid thermodynamic properties considered include critical parameters, vapor-liquid coexistence curve, thermodynamic characteristic functions, chemical potential, pressure, and constant volume excess heat capacity. To proceed with the analysis, a well known square well model is used as sample; the well widths considered range over a wide interval, and the relevant temperatures amenable to simulation calculations (used as "exact" results to analyze the HTSE) can be both very high and very low. The main discoveries reached are summarized as follows: (1) The HTSE usually converges at the 4th-order truncation, but with decrease of the temperature considered, the lowest truncation order, which makes the HTSE to converge, tends to rise. As a conservative estimate, it is considered that the HTSE always converges for reduced temperature T* higher than 0.25, whereas for T* < 0.25 there appear signs indicating that the HTSE may diverge from the 7th-order truncation. (2) Within the temperature interval with T* ≥ 0.5, the HTSE converges approximately to the correct solution, and the HTSE can be reliably used to calculate the fluid thermodynamic properties, and within this temperature interval, the 4th-order truncation is enough; whereas for T* < 0.5, such as within the temperature interval with 0.275 ≤ T* ≤ 0.355, although the HTSE does converge, it does not converge to the correct solution, and the deviations between the HTSE calculations and MC simulations become an ever-prominent issue with the rising of the density, and the slopes of the thermodynamic properties over density are not satisfactorily represented. As a result, the HTSE is not suited for calculations for temperature interval T* < 0.5.

The First Three Coefficients in the High Temperature Series Expansion of Free Energy for Simple Potential Models with Hard-Sphere Cores and Continuous Tails

The first three coefficients of the high temperature series expansion (HTSE) of the Helmholtz free energy for a number of simple potential models with hard-sphere cores plus continuous tails are obtained for the first time from Monte Carlo simulations. The potential models considered include Square-well, Sutherland, attractive Yukawa, and triangle-well with different potential ranges, as well as a model potential qualitatively resembling the depletion potential in colloidal dispersions. The simulation data are used to evaluate performance of a recent coupling parameter series expansion (CPSE) in calculating for these coefficients, and a traditional macroscopic compressibility approximation (MCA) for the second-order coefficient only. A comprehensive comparison based on these coefficients from the two theoretical approaches and simulations enables one to conclude that (i) unlike one common experience that the widely used MCA usually underestimates the second-order coefficient, the MCA can both overestimate and underestimate the second-order coefficient, and worsens as the range of the potential decreases; and (ii) in contrast, the CPSE not only reproduce the trends in the density dependence of the perturbation coefficients, even the third one, observed in the simulations, but also the agreement is quantitative in most cases, and this clearly highlights the potential of the CPSE in providing accurate estimations for the higher-order coefficients, thus giving rise to an accurate higher-order HTSE.

Monte Carlo and theoretical calculations of the first four perturbation coefficients in the high temperature series expansion of the free energy for discrete and core-softened potential models

The first four perturbation coefficients in the expansion of the Helmholtz free energy in power series of the inverse of the reduced temperature for a number of potential models with hard-sphere cores plus core-softened and discontinuous tails are obtained from Monte Carlo simulations. The potential models considered include square-well, double square-well, and square-shoulder plus square-well, with different potential parameters. These simulation data are used to evaluate the performance of a traditional macroscopic compressibility approximation (MCA) for the second order coefficient and a recent coupling parameter series expansion (CPSE) for the first four coefficients. Comprehensive comparison indicates the incapability of the MCA for the second order coefficient in most non-stringent situations, and significance of the CPSE in accurately calculating these four coefficients.

Liquid theory with high accuracy and broad applicability: Coupling parameter series expansion and non hard sphere perturbation strategy

Thermodynamic and structural properties of liquids are of fundamental interest in physics, chemistry, and biology, and perturbation approach has been fundamental to liquid theoretical approaches since the dawn of modern statistical mechanics and remains so to this day. Although thermodynamic perturbation theory (TPT) is widely used in the chemical physics community, one of the most popular versions of the TPT, i.e. Zwanzig (Zwanzig, R. W. J. Chem. Phys. 1954, 22, 1420-1426) 1st-order high temperature series expansion (HTSE) TPT and its 2nd-order counterpart under a macroscopic compressibility approximation of Barker-Henderson (Barker, J. A.; Henderson, D. J. Chem. Phys. 1967, 47, 2856-2861), have some serious shortcomings: (i) the nth-order term of the HTSE is involved with reference fluid distribution functions of order up to 2n, and the higher-order terms hence progressively become more complicated and numerically inaccessible; (ii) the performance of the HTSE rapidly deteriorates and the calculated results become even qualitatively incorrect as the temperature of interest decreases. This account deals with the developments that we have made over the last five years or so to advance a coupling parameter series expansion (CPSE) and a non hard sphere (HS) perturbation strategy that has scored some of its greatest successes in overcoming the above-mentioned difficulties. In this account (i) we expatiate on implementation details of our schemes: how input information indispensable to high-order truncation of the CPSE in both the HS and non HS perturbation schemes is calculated by an Ornstein-Zernike integral equation theory; how high-order thermodynamic quantities, such as critical parameters and excess constant volume heat capacity, are extracted from the resulting excess Helmholtz free energy with irregular and inevitable numerical errors; how to select reference potential in the non HS perturbation scheme. (ii) We give a quantitative analysis on why convergence speed of the CPSE in both the HS and non HS perturbation schemes is certainly faster than that of the HTSE and the HS perturbation scheme. (iii) We illustrate applications of the CPSE TPT in both the HS and non HS perturbation schemes in calculating thermodynamic properties of various coarse-grained potential function models and as input information of other liquid state theories such as a classical density functional theory (DFT), and also discuss, in the framework of classical DFT, the potential of our CPSE scheme in several typical problems of chemical physics interest. (iv) Finally, we consider several topics which are possibly expected to be settled in the immediate future and possible integration with other liquid state theory frameworks aiming to solve problems in complex fluids in both bulk and inhomogeneous states.

Low temperature behavior of thermodynamic perturbation theory

Monte Carlo simulations have been carried out for a hard sphere square well model fluid with well widths of lambda = 1.01, 1.02 and 1.04 to obtain thermodynamic properties, such as pressure, excess Helmholtz free energy and internal energy, constant volume excess heat capacity, excess chemical potential, and excess enthalpy. The extremely narrow well widths considered allowed us to perform simulations at extremely low temperatures without reaching the liquid-vapour transition, which is metastable for these values of lambda. These simulation data have been used to explore the low temperature behavior of two thermodynamic perturbation theories (TPT), namely a high temperature series expansion truncated at second order within the local compressibility approximation (thereafter denoted as 2nd-order LCA-TPT) and a recently proposed coupling parameter series expansion truncated at 3rd-, 4th- and 5th-order (thereafter denoted as 3rd-order, 4th-order and 5th-order TPT, respectively). It has been found that the 2nd-order LCA-TPT is qualitatively incorrect in most of the cases analyzed, whereas the 3rd-order TPT is quantitatively correct in most of them. With increasing the well width, and consequently the temperatures considered, the 3rd-order TPT quickly becomes more accurate. Among the six thermodynamic quantities analyzed, the one most difficult to predict accurately is the constant volume excess heat capacity, for which the TPT based on the coupling parameter expansion provides satisfactory results only in the case of lambda = 1.04. For the cases studied, the performance of the 3rd-order, 4th-order, and 5th-order TPT are essentially equal; the change of the order of the truncation sometimes results in an improvement of some thermodynamic quantities within certain regions of the thermodynamic surface of states, while worsening in others.