TPT2 and SAFTD equations of state for mixtures of hard chain copolymers

We present the second-order thermodynamic perturbation theory (TPT2) and the dimer statistical associating fluid theory (SAFTD) equations of state for mixtures consisting of hetero-nuclear hard chain molecules based on extensions of Wertheim's theory for associating fluids. The second-order perturbation theory, TPT2, is based on the hard sphere mixture reference fluid. SAFTD is an extension of TPT1 (= SAFT) and is based on the non-spherical (hard disphere mixture) reference fluid. The TPT2 equation of state requires only the contact values of the hard sphere mixture site-site correlation functions, while the SAFTD equation of state requires the contact values of site-site correlation functions of both hard sphere and hard disphere mixtures. We test several approximations for site-site correlation functions of hard disphere mixtures and use these in the SAFTD equation of state to predict the compressibility factor of copolymers. Since simulation data are available only for a few pure copolymer systems, theoretical predictions are compared with molecular simulation results for the compressibility factor of pure hard chain copolymer systems. Our comparisons show a very good performance of TPT2, which is found to be more accurate than TPT1 (= SAFT). Using a modified Percus-Yevick site-site correlation function SAFTD is found to represent a significant improvement over SAFT and is slightly more accurate than TPT2. Comparison of SAFTD with generalized Flory dimer (GFD) theory shows that both are equivalent at intermediate to high densities for the compressibility factor of copolymer systems investigated here.     

Statistical fluid theory for associating fluids containing alternating heteronuclear chain molecules

Statistical associating fluid theory of homonuclear dimerized chain fluids and homonuclear monomer-dimer mixture chain fluids are extended to fluids containing alternating heteronuclear chain molecules separately. The proposed models account for the appropriate site-site correlation functions at contact. The modified equations of state show a good agreement with generalized Flory dimer theory and MD simulation data for small and medium size ratio of hard sphere diameters.     

SAFT-D theory for hard sphere and hard disc chain fluids

SAFT-dimer (SAFT-D) theory is reformulated to yield an improved equation of state for the hard sphere chain fluid. Two sets of the equation of state are proposed by employing Chiew's expressions for the contact values of the m hard sphere site-site correlation function g(σ). Comparison with molecular simulation data shows that the improved SAFT-D equation of state predicts the compressibility factor more accurately than Ghonasgi and Chapman's equation of state. It has been shown that SAFT-dimer theory can be applied readily to fused hard sphere chain fluids by considering the correct value of the effective chain length (m*). SAFT-dimer theory is also extended to the 2-dimensional tangent and fused hard disc chain fluids. For the fused hard disc dimer fluid, the SAFT equation of state is found to be more accurate than the Boublik hard disc dimer equation of state. For tangent hard disc chain fluids, the results obtained from SAFT-dimer theory are compared with Monte Carlo results for 5-mers and with GFD theory for 4-mers, 8-mers and 16-mers.     

The structure of two-dimensional dimerizing fluids from Wertheim’s Ornstein-Zernike equation

A dimerizing fluid of hard discs is studied using two-dimensional (2D) Wertheim’s Ornstein-Zernike (WOZ) equation and associative Percus-Yevick (APY) closure. Dimerization takes place due to site-site associative interactions. The dependences of the association constant on disc density at different association energies are obtained. We calculate the compressibility and the virial equations of state (EOS) using the solution of the WOZ equation. Theoretical structure and thermodynamics is compared with Monte Carlo computer simulation data. Extension of our solution for polymerizing models is of special interest for the development of EOS for 2D chain fluids. This work was supported in part by Cray Research of Mexico under University Research and Development grant program and by KBN of Poland under the Grant No. 3T09A06210.     

MC simulation results for a hard core model of carbon tetrachloride

Monte Carlo simulations on a hard tetrahedron fluid (hard core model of CCl₄) have been performed. The average site-site correlation functions, their generalized (1, 0, 0) spherical harmonic expansion coefficients, equation of state, and virial coefficients have been calculated and compared with theoretical methods currently available. The RISM equation is less accurate for the model studied than for simpler models considered so far. For the equation of state the best results are obtained from the Boublik-Nezbeda equation which agrees with the simulation results throughout the density range considered.     

Self-consistent equations of state for hard body mixtures

The self-consistency (S-C) constraints on the solute chemical potential and equation of state are stated and employed to find corrections to thermodynamic functions in the colloidal limit for the most often used equations of state. It is shown that the S-C approach and Henderson's expression for the contact radial distribution functions yield the same correction term in the case of the Boublik—Mansoori—Carnahan—Starling—Leland (BMCSL) equation of state for hard spheres. For hard sphere (and hard convex body) mixtures a new variant of the equation of state and Helmholtz energy is proposed that fulfils better the self-consistency constraints than the frequently used equations. It is shown that the correction term for Δ_{μ 2} in hard convex body mixtures described by improved scaled particle theory differs from that for BMCSL only by the non-sphericity parameter. For the Kolafa—Boublik and modified scaled particle theory versions the correction terms are more complex.     

热力参数的对比变换与流体工质热力性质的通用对比方程

本文提出了判别流体热力学性质的通用对比方程的规则和选择作为通用对比方程的热力学参数对比交换的方法;具体给出了工质饱和液体焓和温度的合适对比变换式和通用对比方程。经检验,该通用方程的计算值与２５种常用流体工质的推荐值的平均绝对偏差为０．５％,算术平均偏差为－０．０４％。本文的理论纠正了传统的只用临界参数为对比值的片面认识,为建立在无通用对比方程而仅利用合适的参数变换,从一种已知物性物质推算另一种物质未知物性的一般方法和其它物性的通用对比方程奠定了基础。     

Kinetic theory of dense fluids. II. The generalized Chapman-Enskog solution

In the second paper of this series we solve the kinetic equation proposed in the previous paper by a method following the spirit of Chapman and Enskog (generalized Chapman-Enskog method). The zeroth-order solution to the kinetic equation leads to the Euler equations in hydrodynamics for real fluids, and the first-order solution to the Navier-Stokes equations for real fluids. General formulas for transport coefficients such as viscosity and heat-conductivity coefficients are obtained for dense fluids, which are given in terms of time-correlation functions of fluxes conjugate to the thermodynamic forces. The results have the same formal structures as the time-correlation functions in linear response theory except for the collision operator appearing in place of the Liouville operator in the evolution operator for the system.     

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.     

Simple approach to approximate predictions of the vapor–liquid equilibrium curve near the critical point and its application to Lennard-Jones fluids

We propose a simple analytical form of the vapor–liquid equilibrium curve near the critical point for Lennard-Jones fluids. Coexistence densities curves and vapor pressure have been determined using the Van der Waals and Dieterici equation of state. In described method the Bernoulli differential equations, critical exponent theory and some type of Maxwell?s criterion have been used. Presented approach has not yet been used to determine analytical form of phase curves as done in this Letter. Lennard-Jones fluids have been considered for analysis. Comparison with experimental data is done. The accuracy of the method is described.     

Irreversible thermodynamics of fluids

Irreversible thermodynamics of fluids is formulated based on a set of postulates. The theory thus constructed generalizes thermostatics and linear irreversible thermodynamics into the realm of nonlinear irreversible processes. In this theory the extended Gibbs relation and the entropy balance equation appear as a pair of mutually consistent equations under the postulates made. An equivalent theory is also formulated by replacing one of the postulates with another that is basically a variational principle. The variational principle yields the evolution equations for fluxes as the Euler equations that extremize the variational functional postulated. The local form of the extremized variational functional is the entropy balance equation for the irreversible processes in the system. Some further consequences of the theory are also considered. For example, nonequilibrium specific heats are shown to be at least quadratic functions of fluxes and reduce to the equilibrium specific heats in the limit of vanishing fluxes. In order to illustrate an example of possible applications, we have considered nonlinear transport processes in fluids. The connections of the present theory with other theories are discussed.     

Correlation functions for diatomic symmetric molecules from the RISM equation

Using the solution of the RISM equation for diatomic symmetric molecules outlined in a previous paper, the site-site radial distribution function (RDF) is calculated and compared with the Monte Carlo results and the numerical RDF of Lowden and Chandler. The RDF calculated here and the numerical RDF of Lowden and Chandler agree well at intermediate and high densities. At low density, however, both have systematic errors. The agreement between the RDF calculated here and the Monte Carlo results suggests that a simplified formulation of the RISM solution may serve well as a reference system in a perturbation theory for diatomic fluids.     

A comprehensive approach to an equation of state for hard spheres and Lennard-Jones fluids

We present a simple method of obtaining various equations of state for hard sphere fluid in a simple unifying way.We will guess equations of state by using suitable axiomatic functional forms (n=1,2,3,4,5) for surface tension S n m (r),r ≥ d/2 with intermolecular separation r as a variable,where m is an arbitrary real number (pole).Among the equations of state obtained in this way are Percus-Yevick,scaled particle theory and Carnahan-Starling equations of state.In addition,we have found a simple equation of state for the hard sphere fluid in the region that represents the simulation data accurately.It is found that for both hard sphere fluids as well as Lennard-Jones fluids,with m=3/4 the derived equation of state (EOS) gives results which are in good agreement with computer simulation results.Furthermore,this equation of state gives the Percus-Yevick (pressure) EOS for the m=0,the Carnahan-Starling EOS for m=4/5,while for the value of m=1 it corresponds to a scaled particle theory EOS.     

An equation of state for the isotropic phase of linear,partially flexible and fully flexible tangent hard-sphere chain fluids

A new equation of state is developed that accurately describes the isotropic phase behaviour of linear, partially flexible and fully flexible tangent hard-sphere chain fluids and their mixtures. The equation of state is based on the equation of state of Liu and Hu [H. Liu and Y. Hu, Fluid Phase Equilibr. 122, 75 (1996)] for fully flexible chain fluids. The effect of molecular flexibility is described by a pure-component parameter that is introduced in the theory at the level of the cavity correlation function of next-to-nearest neighbour segments in a chain molecule. The equation of state contains a total of three adjustable model constants. The extension to partially flexible- and linear chain fluids is based on a refitting of the first model constant to numerical data of the second virial coefficient of partially flexible and linear tangent hard-sphere chain fluids. The numerical data were obtained from an analytical approximation for the pair-excluded volume. The other two parameters were adjusted to molecular simulation data for the pressure of linear tangent hard-sphere chain fluids. For both, pure component systems and mixtures of chains of variable flexibility, the pressure and second virial coefficient obtained from the equation of state, are in excellent agreement with the results from Monte Carlo simulations. A significant improvement to TPT1, TPT2, generalised Flory-dimer theory and scaled particle theory is observed.     

Variational wave equations of two fermions interacting via scalar,pseudoscalar, vector,pseudovector and tensor fields

We consider a method for deriving relativistic two-body wave equations for fermions in the coordinate representation. The Lagrangian of the theory is reformulated by eliminating the mediating fields by means of covariant Green's functions. Then, the nonlocal interaction terms in the Lagrangian are reduced to local expressions which take into account retardation effects approximately. We construct the Hamiltonian and two-fermion states of the quantized theory, employing an unconventional “empty” vacuum state, and derive relativistic two-fermion wave equations. These equations are a generalization of the Breit equation for systems with scalar, pseudoscalar, vector, pseudovector and tensor coupling.     

Generic van der Waals equation of state for polymers, modified free volume theory, and the self-diffusion coefficient of polymeric liquids

In this paper, a molecular theory of self-diffusion coefficient is developed for polymeric liquids (melts) on the basis of the integral equation theory for site-site pair correlation functions, the generic van der Waals equation of state, and the modified free volume theory of diffusion. The integral equations supply the pair correlation functions necessary for the generic van der Waals equation of state, which in turn makes it possible to calculate the self-diffusion coefficient on the basis of the modified free volume theory of diffusion. A random distribution is assumed for minimum free volumes for monomers along the chain in the melt. More specifically, a stretched exponential is taken for the distribution function. If the exponents of the distribution function for minimum free volumes for monomers are chosen suitably for linear polymer melts of N monomers, the N dependence of the self-diffusion coefficient is N⁻¹ for the small values of N, an exponent predicted by the Rouse theory, whereas in the range of 2.3?lnN?4.5 the N dependence smoothly crosses over to N⁻², which is reminiscent of the exponent by the reptation theory. However, for lnN?4.5 the N dependence of the self-diffusion coefficient differs from N⁻², but gives an N dependence, N^−2−δ(0<δ<1), consistent with experiment on polymer melts in the range. For polyethylene δ≈0.48 for the parameters chosen for the stretched exponential. Because the stretched exponential function contains undetermined parameters, the N dependence of diffusion becomes semiempirical, but once the parameters are chosen such that the N dependence of D can be successfully given for a polymer melt, the temperature dependence of the self-diffusion coefficient can be well predicted in comparison with experiment. The theory is satisfactorily tested against experimental and simulation data on the temperature dependence of D for polyethylene and polystyrene melts.     

Analytical expressions for the correlation function of a hard sphere dimer fluid

A closed form expression is given for the correlation function of a hard sphere dimer fluid. A set of integral equations is obtained from Wertheim's multidensity Ornstein-Zernike integral equation theory with Percus-Yevick approximation. Applying the Laplace transformation method to the integral equations and then solving the resulting equations algebraically, the Laplace transforms of the individual correlation functions are obtained. By the inverse Laplace transformation, the radial distribution function (RDF) is obtained in closed form out to 3D (D is the segment diameter). The analytical expression for the RDF of the hard dimer should be useful in developing the perturbation theory of dimer fluids.     

Statistical approach to the kinetics of nonuniform fluids

We present a new formalism in Fourier space for the study of spatially nonuniform fluids in nonequilibrium states which generalizes previous work on uniform fluids. Starting from the Liouville equation we obtain a hierarchy of equations for the reduced distribution functions which gives their rate of change at any given order of the system mean density as a sum of a finite number of terms. Using a finite-ranged repulsive interaction potential we derive, as a first application of the formalism, the Boltzmann integrodifferential equation for an infinite system which is initially in a “weakly” inhomogeneous state. This is accomplished introducing an initial statistical assumption, namely initial molecular chaos; this condition is seen to hold during the time evolution described by the resulting kinetic equation.     

A self-consistent approach for modelling the interfacial properties and phase diagrams of Yukawa,Lennard-Jones and square-well fluids

A self-consistent density-functional approach is presented for describing the phase behaviour and interfacial tensions of van der Waals fluids represented by the hard-core Yukawa (HCY), Lennard-Jones (LJ) and square-well (SW) potentials. The excess Helmholtz energy functional is formulated in terms of a modified fundamental measure theory (MFMT) for the short-ranged repulsion and a density-gradient expansion for the van der Waals attractions. Analytical expressions for the direct correlation functions of uniform fluids are utilized to take into account the effect of van der Waals’ attraction on intermolecular correlations. For bulk phases, the density functional theory is reduced to an equation of state (EOS) that provides accurate saturation pressures and vapour–liquid phase diagrams. Near the critical region, the long-range fluctuations can be corrected by using the renormalization group (RG) theory. With the same set of molecular parameters, the theory also yields satisfactory surface tensions and interfacial density profiles at all relevant temperatures.