Macroscopic Compressibility Approximation Research Articles

Second virial coefficient, Boyle temperature and equation of state of van Hove fluids with a downward concavity attractive parabolic-well

The second virial coefficient, the Boyle temperature and the equation of state of van Hove fluids with a relatively short ranged attractive parabolic-well of downward concavity are considered. The analytic second virial coefficient for this fluid is obtained explicitly and it is used to compute the Boyle temperature of the fluid as a function of the range of the potential. Further, an equation of state is derived using the second-order thermodynamic perturbation theory of Barker and Henderson in the macroscopic compressibility approximation, with the hard-sphere fluid being the reference fluid. For this latter we profit from the fully analytical expression of the radial distribution function, consistent with the Carnahan-Starling equation state, derived within the so-called rational function approximation method up to a range twice the size of the hard-core diameter. The results for the reduced pressure of the fluid as a function of the packing fraction and two values of the range of the potential well at different temperatures are compared with Monte Carlo simulation data. Estimates of the values of the critical temperature are also provided.

Thermodynamic Properties of the Parabolic-Well Fluid

The thermodynamic properties of the parabolic-well fluid are considered. The intermolecular interaction potential of this model, which belongs to the class of the so-called van Hove potentials, shares with the square-well and the triangular well potentials the inclusion of a hard-core and an attractive well of relatively short range. The analytic second virial coefficient for this fluid is computed explicitly and an equation of state is derived with the aid of the second-order thermodynamic perturbation theory in the macroscopic compressibility approximation and taking the hard-sphere fluid as the reference system. For this latter, the fully analytical expression of the radial distribution function, consistent with the Carnahan-Starling equation of state as derived within the rational function approximation method, is employed. The results for the reduced pressure of the parabolic-well fluid as a function of the packing fraction and two values of the range of the parabolic-well potential at different temperatures are compared with Monte Carlo and Event‐driven molecular dynamics simulation data. Estimates of the values of the critical temperature are also provided.

Analytic Equation of State for the Square-well Plus Sutherland Fluid from Perturbation Theory

Analytic expressions were derived for the compressibility factor and residual internal energy of the square-well plus Sutherland fluid. In this derivation, we used the second order Barker-Henderson perturbation theory based on the macroscopic compressibility approximation together with an analytical expression for radial distribution function of the reference hard sphere fluid. These properties are expressed in terms of density, temperature, and the potential parameters. Derived equations successfully applied to hard-core Lennard-Jones fluid. It is found that the agreement between theory and simulation is quite good for both the compressibility factor and the residual internal energy for a wide range of densities and temperature.

Unified SAFT-VR theory for simple and chain fluids formed of square-well, triangular-well, Sutherland and Mie segments

The thermodynamic properties of simple and chain fluids, whose molecules interact through the three following basic hard-core intermolecular potentials: the square-well (SW), the triangular-well (TW) and the Sutherland (SU) [−ε(d/r)γ] potentials are presented in a unified mathematical framework. In order to achieve this goal we use the Barker–Henderson perturbation theory with the macroscopic compressibility approximation (BH-MCA) of monomer fluids combined to a simple, analytic expression of the hard-sphere radial distribution function (RDF) derived by Sun [Can. J. Phys. 83 (2005) 55], which allows the first-order perturbation term of the Helmholtz free energy, f1, to be obtained in terms of the same generic analytic function for the three potentials. This function depends on three parameters which assume different values according to the potential considered. Other thermodynamic quantities such as the compressibility factors can be derived from f1, which thus plays a pivotal role in the equation of state (EOS) obtained from perturbation theory. Numerical values of f1 obtained from this generic function are in good agreement with recent Monte Carlo simulations for the three aforementioned basic potentials, as well as with the various parameterizations of f1 in terms of an effective packing fraction ηeff for the square-well and Sutherland potentials commonly used in the statistical associating fluid theory for potentials of variable range (SAFT-VR). For the corresponding chain fluids, Wertheim first-order thermodynamic perturbation theory (TPT1) is used. It requires the SAFT-VR expressions of the potentials RDF at the hard-core contact d, g(d), and its derivative with respect to the packing fraction, which can be expressed in terms of the generic function giving f1. Numerical values of g(d) are also in agreement with the corresponding Monte Carlo results. The equations of state (EOS) of the SW, TW and SU chain fluids are then obtained in simple, analytical forms. The above three basic potentials can then be combined to give more complicated molecular interactions. One important example of combination is studied. It involves the linear combination of two Sutherland potentials with γ=α, β to obtain the Mie(α,β) class of interactions, of which the Lennard–Jones LJ(12,6) or Mie(12,6) is a particular example. Numerical values of: a) the Mie(12,6) first-order perturbation term of the residual Helmholtz free energy, f1Mie, b) the radial distribution function at contact, gMie(σ), [uMie(σ)=0], and c) the reduced pressure or compressibility factor of the monomer, dimer and tetramer Mie(12,6) fluids, obtained from the generic function, f1, corresponding to the combined Sutherland potentials, are compared to Monte Carlo and molecular dynamics simulations and to the recent enhanced SAFT-VR Mie 20013 EOS of Lafitte et al. [J. Chem. Phys. 139 (2013) 154504], whose improvements with respect to the previous SAFT-VR Mie 2006 EOS are highlighted in detail. The results obtained from the generic function f1 are slightly more accurate for the Mie(12,6) monomer fluid, while the results from Lafitte et al., which uses a new f1 parameterization of ηeff for the Sutherland potentials, valid for 5<γ≤100, are more accurate for the Mie(12,6) dimer and tetramer 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.

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.

Improved analytical thermodynamic properties of the triangular-well fluid from perturbation theory

In this short communication, it is shown that the triangular-well fluid analytic equation of state (EOS) recently proposed by Betancourt-Cárdenas et al. [Mol. Phys. 105 (2007) 2987] within the framework of the Barker–Henderson perturbation theory of fluids with the macroscopic compressibility approximation (BH-MCA) can be improved in two ways. First it is cast into a more compact form, which avoids the calculations of seventy-five numerical coefficients. Secondly, by using a more accurate expression of the hard-sphere radial distribution function (HS-RDF), gHS(r), the accuracy of the analytical expression of the free energy second-order term is improved, being in agreement with its BH-MCA numerical evaluation. For short-range triangular-well potentials, the present EOS also leads to second-order residual internal energies and reduced pressures in better agreement with the molecular simulations than the Betancourt-Cárdenas et al. results.

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.

Simple analytic equations of state for the generalized hard-core Mie(α, β) and Mie(α, β) fluids from perturbation theory

New, simple and analytic perturbation theory equations of state for generalized hard-core Mie HCMie(α, β) and Mie(α, β) fluids are proposed. They are based on the second-order Barker–Henderson perturbation theory in the macroscopic compressibility approximation and the new analytical expression of the radial distribution function of hard spheres, g HS(r), developed by Sun in terms of a polynomial expansion of base functions adapted to the square-well and Sutherland potentials [Can. J. Phys. 83 (2005) 55], the combination of which yields the HCMie(α, β) and Mie(α, β) functions. The compressibility factors, the residual internal energies and the radial distribution function at contact with the hard core are then obtained from this equation of state for the HCLJ(12, 6) potential, which is a particular case of the HCMie(α, β) potentials with α = 12 and β = 6. The results are in good agreement with the existing Monte Carlo (MC) simulation data, and compare favorably with those obtained from five other equations of state, three of which contain numerical coefficients fitted to the Monte Carlo results. For the Mie(α, 6) (α = 8, 10, 12), fluids, the present equation of state is a good representation of recent molecular dynamics (MD) simulations of the pressure and internal energy. It is more accurate than the statistical associating fluid theory of variable range (SAFT-VR Mie(n, 6)) theory for n = 8, and 10, while for n = 12 the SAFT-VR theory is best. For the Mie(14, 7) fluid, which is outside the range of application of the SAFT-VR theory, the results for the pressure are in good agreement with the analytical equation of state obtained from the MC simulation data.

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.

Theoretical Investigation about the Possible Consequence of Artificial Discontinuity in Pair Potential Function on Overall Phase Behavior

On the basis of a traditional hard sphere square well (HSSW) model, we first test the performance of a recently proposed thermodynamic perturbation theory (TPT) originating from a coupling parameter expansion. It is concluded that (1) the coupling parameter expansion-TPT predicts a second-order term of the perturbation expansion far more accurately than several previous approximations including a well-known macroscopic compressibility approximation; (2) the fifth-order TPT is generally a little more accurate than the third-order TPT. Next, global phase diagrams of the HSSW model with several representative potential range parameter values are calculated by the fifth-order TPT for the fluid phase combined with a first-order TPT for the solid phase, and found in satisfactory agreement with those as found from available computer simulation. Once more, we theoretically compute the fluid phase critical curve, solid phase isostructural critical curve, and triple curve as a function of the potential range parameter of the HSSW model and an additional hard core attractive Yukawa (HCAY) model, respectively. It is found that these phase lines of the HSSW model gauging the outlines of phase diagrams display abnormity compared with those of the HCAY model; particularly a huge abnormity of the HSSW case is always related to participation of the solid phase. We explain that the abnormity originates from a combination of a discontinuity of the SW tail and a highly discrete situation of the particles in the solid phase. The present investigation eliminates the possibility of the HSSW potential as a feasible model of real fluids; the opportunity of the HSSW is perhaps narrowed as sample potential for testing theories harshly.

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.

Third-order thermodynamic perturbation theory for effective potentials that model complex fluids

We have performed Monte Carlo simulations to obtain the thermodynamic properties of fluids with two kinds of hard-core plus attractive-tail or oscillatory potentials. One of them is the square-well potential with small well width. The other is a model potential with oscillatory and decaying tail. Both model potentials are suitable for modeling the effective potential arising in complex fluids and fluid mixtures with extremely-large-size asymmetry, as is the case of the solvent-induced depletion interactions in colloidal dispersions. For the former potential, the compressibility factor, the excess energy, the constant-volume excess heat capacity, and the chemical potential have been obtained. For the second model potential only the first two of these quantities have been obtained. The simulations cover the whole density range for the fluid phase and several temperatures. These simulation data have been used to test the performance of a third-order thermodynamic perturbation theory (TPT) recently developed by one of us [S. Zhou, Phys. Rev. E 74, 031119 (2006)] as compared with the well-known second-order TPT based on the macroscopic compressibility approximation due to Barker and Henderson. It is found that the first of these theories provides much better accuracy than the second one for all thermodynamic properties analyzed for the two effective potential models.

Fifth-order thermodynamic perturbation theory of uniform and nonuniform fluids

A recently proposed numerical third-order thermodynamic perturbation theory (TPT) is extended to its fifth-order counterpart. Extensive performance evaluation based on several model potentials indicates that the fifth-order version is generally superior to both the third-order version and a well-known second-order macroscopic compressibility approximation TPT, and is at least comparable to an accurate self-consistent Ornstein-Zernike integral equation approximation (SCOZA) with regard to predictions of varying bulk thermodynamic properties. A bulk second-order direct correlation function (DCF) free of numerical solution of a bulk OZ integral equation, is proposed, which, in combination with the framework of classical density functional theory (DFT) and a thermodynamic consistency condition, extends the uniform fifth-order TPT to nonuniform case. Grand canonical ensemble Monte Carlo simulation is carried out to produce density profile of a core-softened fluid confined in a hard spherical cavity, the resultant density profile is employed to evaluate the performance of the present nonuniform fifth-order TPT and a recently proposed third +second-order perturbation DFT. It is found that the present nonuniform fifth-order TPT is generally more accurate than the third+second-order perturbation DFT. Additional advantages of the present nonuniform fifth-order TPT over the third +second-order perturbation DFT is discussed.

The effect of steepness of soft-core square-well potential model on some fluid properties

The effect of repulsive steepness of the soft-core square well (SCSW) potential model on the second virial coefficient, critical behaviour (two- phase region and the position of critical point), and coordination number are investigated. The soft-core thermodynamic perturbation theory (TPT) presented by Weeks-Chandler-Anderson (WCA) recently developed by Ben-Amotz and Stell (BAS) has been used for the reference system, and the Barker-Henderson TPT for the perturbed system. The Barker-Henderson macroscopic compressibility approximation has been used for all order perturbation terms in which the second-order one is improved by assuming that the molecules in every two neighbouring shells are correlated upon the original assumption. By using the hard-sphere isothermal compressibility consistency for the radial distribution function (RDF), an analytical closed expression has been derived for the Helmholtz free energy function contained effective hard-sphere diameter. The accuracy of the model has been examined for the hard-core system, and an appropriate range found for the attractive width of the potential well (R), then the effect of steepness parameter on the critical quantities, coordination number, and the inversion temperature of the second virial coefficient, has been investigated qualitatively. The predicted results are in good agreement with the computer simulation data for the critical constants, and coordination number at the limit of the hard-core square-well potential model at least qualitatively, and for the attractive range 1.55 ≤ R ≤ 1.7, quantitatively. It was found that the steepness of the potential model has a marginal effect on the critical behaviour, and also every thermodynamic quantity at low and medium temperatures for which the molecular penetration is negligible, but since the penetration at high temperatures is significant, the role of the steepness of potential on the inversion temperature of the second virial coefficient and coordination number is highlighted.

Can the second virial coefficient be a predictor for the critical temperature?

A recently proposed 3rd-order thermodynamic perturbation theory (TPT) is employed to calculate critical parameters of several model fluids. It is found that no matter how dis-continuous or continuous attractive perturbation part of whole interaction potential is, the 3rd-order TPT is accurate reliably for prediction of the critical parameters only on condition that the interaction potential is long-ranged or intermediate-ranged. Even when the 2nd-order macroscopic compressibility approximation TPT (2nd-order MCA-TPT) completely fails, the 3rd-order TPT still performs well. The 3rd-order TPT is then used to investiagte in detail validity and range of a recently emerging empirical rule. It is found that the empirical rule only applies for potentials consisting of a hard sphere core and a continuous and attractive perturbation tail, whether the attractive perturbation tail is differentiable with respect to particle separation or not does not matter. But for potentials with dis-continuous perturbation tail and those diaplaying two fluid–fluid critical points, the empirical rule fails completely.

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.

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.

Thermodynamic perturbation theory in fluid statistical mechanics

A methodology is proposed that pushes the thermodynamic perturbation theory (TPT) from first order to higher order. The second-order correction is superior to a macroscopic compressibility (MC) approximation of Barker and Henderson. The present third-order TPT performs far better than the original first-order TPT and second-order TPT based on the MC approximation for many subfields in fluid statistical mechanics, such as predicting excess Helmholtz free energy, excess chemical potential, bulk pressure, gas-liquid coexistence, and solid-liquid equilibrium of very short-range potential fluids. A nonuniform version of the TPT is proposed; it is also shown that the nonuniform third-order TPT performs far better than the nonuniform first-order TPT in predicting density profile of fluids in critical region. The present report indicates that the TPT still can be a "universal" and accurate theoretical tool that has general applicability in fluid statistical mechanics, especially in soft-matter physics.