Equilibrium properties of hard non-sphere fluids

Derivation of the thermodynamic properties of fluids of hard non-spherical molecules of arbitrary symmetry is based on the decoupling approximation. Theoretical expressions are given and calculations made for the equation of state and virial coefficients for hard ellipsoids. These results are compared with Monte Carlo values and show fair agreement in all cases. The theoretical predictions for the equation of state for binary mixtures are compared with the Monte Carlo results for hard spheres and hard prolate spherocylinders. Theoretical expressions for the first order quantum correction to the free energy, pressure and virial coefficients are also given. The quantum effects increase with increase of density and with increase of anisotropy parameter.     

Square-well fluid based decoupling approximation for system of hard non-spherical particles with spherical square-wells

A new theory based on the virial equation has been developed to study a system of hard non-spherical particles embedded into square-wells. The theory relies on a mapping of the pair correlation function of the system of interest onto that of spherical square-wells. This is a generalization of the Parsons—Lee theory of pure hard convex bodies. Comparisons show that this theory is superior to previous theories and incorporates them into a common framework. Competition between the excluded volume effect of repulsive forces and the condensation effect of attractive forces has been demonstrated. We have found three types of phase diagrams, among which one exhibits a nematic—nematic phase transition.     

An equation of state for hard convex body chains

The thermodynamic perturbation theory of hard sphere chains is generalized to derive an equation of state for hard convex body chains. The hard convex body chain equation of state contains two parameters that are related directly and rigorously to the geometry of the hard convex body. The compressibility factors and second virial coefficients of chains composed of prolate spherocylinders, oblate spherocylinders and doublecones are calculated and compared with hard sphere chain calculations. The comparison indicates that the nature of the hard convex body has a profound influence on the properties of the chain.     

Virial coefficients and equations of state for mixtures of hard discs,hard spheres and hard hyperspheres

The composition-independent virial coefficients of a d-dimensional binary mixture of (additive) hard hyperspheres following from a recent proposal for the equation of state of the mixture (Santos, A., Yuste, S. B., and López de Haro, M., 1999, Molec. Phys., 96, 1) are examined. Good agreement between theoretical estimates and available exact or numerical results is found for d = 2, 3, 4 and 5, except for mixtures whose components are very disparate in size. A slight modification that remedies this deficiency is introduced and the resummation of the associated virial series is carried out, leading to a new proposal for the equation of state. The case of binary hard sphere mixtures (d = 3) is analysed in some detail.     

Equation of State for Parallel Rigid Spherocylinders

The pair distribution function of monodisperse rigid spherocylinders is calculated by Shinomoto’s method, which was originally proposed for hard spheres. The equation of state is derived by two different routes: Shinomoto’s original route, in which a hard wall is introduced to estimate the pressure exerted on it, and the virial route. The pressure from Shinomoto’s original route is valid only when the length-to-width ratio is less than or equal to 0.25 (i.e., when the spherocylinders are nearly spherical). The virial equation of state is shown to agree very well with the results of numerical simulations of spherocylinders with length-to-width ratio greater than or equal to 2.     

Equation of state for the hard sphere model in the third virial coefficient approximation

Exactly solvable systems of infinite-dimensional integral equations are investigated. These systems approximate the hard sphere model and the Ising model for antiferromagnetism. An equation of state for the hard sphere model in the approximation of the third virial coefficient is obtained.     

Phase diagram of hard colloidal platelets: a theoretical account

We construct the complete liquid crystal phase diagram of hard plate-like cylinders for variable aspect ratio using Onsager's second virial theory and employing the Parsons–Lee decoupling approximation to account for higher-body interactions in the isotropic and nematic fluid phases. The stability of the solid (columnar) state at high packing fraction is included by invoking a simple equation of state based on a Lennard–Jones–Devonshire cell model which has proven to be quantitatively reliable over a large range of packing fractions. By employing an asymptotic analysis based on the Gaussian approximation we are able to show that the nematic–columnar transition is universal and independent of particle shape. The predicted phase diagram is in qualitative agreement with simulation results.     

Explicit and implicit finiteness corrections of virial coefficients

The equation of state of finite systems deviates from thermodynamic limit. The corresponding finiteness corrections of virial coefficients are studied. Most calculations are based on the canonical ensemble with periodic boundary conditions.Explicit andimplicit finiteness corrections occur. They are displayed up to the eighth virial coefficient. These results are applied to hard spheres in one, two, and three dimensions.     

Critical properties of non-spherical molecule fluids from the virial expansion

Critical constants of pure fluids (as important reference data in constructing vapour-liquid phase diagrams and basic input of various estimation methods) were determined for systems of non-spherical Kihara molecules; values of the critical temperature, density, compression factor and pressure of fluids composed of prolate and oblate molecules were evaluated from the fourth-order virial expansion. The second and third virial coefficients of the Kihara molecules were determined by applying the recently proposed method in which the effect of molecular core geometry and functional dependence of a pair interaction on the surface-surface distance are factorized and the former contribution determined from a formula for the corresponding hard convex body virial coefficient. The virial expansion for non-spherical Kihara molecules is applied to determine the critical constants of n-alkanes (methane to octane) and cyclic hydrocarbons (cyclopentane, cyclohexane, benzene and naphthalene); a fair agreement with experimental data was found.     

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.     

Perturbation expansions and series acceleration procedures: Part-II. Extrapolation techniques

Three new procedures for the extrapolation of series coefficients from a given power series expansion are proposed. They are based on (i) a novel resummation identity, (ii) parametrised Euler transformation (pet) and (iii) a modifiedpet. Several examples taken from the Ising model series expansions, ferrimagnetic systems, etc., are illustrated. Apart from these applications, the higher order virial coefficients for hard spheres and hard discs have also been evaluated using the new techniques and these are compared with the estimates obtained by other methods. A satisfactory agreement is revealed between the two.     

Spontaneous chiral symmetry-breaking and partial differential equations

Using a modified normal-product quantization as suggested by Lowenstein and Zimmermann which incorporates B. Lee's resummation problem from the very beginning into the BPHZ renormalization, we discuss the derivation of partial differential equations of the Callan-Symanzik type. In addition to the renormalization group and the Callan-Symanzik equation which are the appropriate differential-equations for the high energy limit (the overall zero-mass preasymptotic theory), we find a third partial differential equation which allows discussion of the spontaneous broken symmetry limit m_π→0 to all orders of the new perturbation theory. In this treatment there is no necessity for loop summation; all the interesting questions are delegated to ordinary perturbation problems.     

Monte-Carlo integration for virial coefficients re-visited: hard convex bodies,spheres with a square-well potential and mixtures of hard spheres

Techniques to adapt the hit-and-miss Monte-Carlo numerical integration are proposed with the aim to determine virial coefficients up to eighth order in fluids of hard convex bodies, hard spheres with an attractive square-well potential and a two-component mixture of hard spheres. These algorithms make use of look-up tables of all the blocks contributing to the coefficients. Each type of block is represented in the tables by several entries. These correspond to all possible topologically equivalent graphs that can be generated by the Monte-Carlo process. This rendered the Monte-Carlo method statistically more efficient. In the case of a two-component system the look-up tables had to have representations of blocks having two sorts of vertices. The reported data are: improved values of the seventh and eighth virial coefficients for hard spheres, the sixth, seventh and eighth coefficients of spheroids, spherocylinders and cutspheres, fifth virial coefficient of spheres with a square-well potential of relative range 1.25; 1.5; 1.75 and 2.0 and the partial contributions of the sixth virial coefficient for a mixture of hard spheres with the size ratio 0.1.     

Fluid suspensions of colloidal ellipsoids: direct structural measurements

A fluid of spheroids, ellipsoids of revolution, is among the simplest models of the disordered matter, where positional and rotational degrees of freedom of the constituent particles are coupled. However, while highly anisometric rods, and hard spheres, were intensively studied in the last decades, the structure of a fluid of spheroids is still unknown. We reconstruct the structure of a simple fluid of spheroids, employing direct confocal imaging of colloids, in three dimensions. The ratio t between the polar axis and the equatorial diameter for both our prolate and oblate spheroids is not far from unity, which gives rise to a delicate interplay between rotations and translations. Strikingly, the measured positional interparticle correlations are significantly stronger than theoretically predicted, indicating that further theoretical attention is required, to fully understand the coupling between translations and rotations in these fundamental fluids.     

由实验数据计算氢气的维里系数问题

我们对计算气体物态方程中维里系数的方法作了全面的探讨,提出了如何判定物态方程中项数的四个原则。这四个原则是:(1)用∑v²的大小的变化来判断项的数目应如何选取,其中v为pV的计算值减去其实验值。(2)多项式要满足逐项在数值上减少的级数性质。(3)多项式的末一项大小要与实验误差大小相同。(4)误差v的正负号数目要大致相等。我们建议在计算中要用A₀+B₀+C₀+…=1条件,但不要用A=RT条件。此外还讨论了用压强展开和用密度展开的问题,级数展开式中省去奇数幂次项问题,维里系数的有效位数问题,若干实验数据的修正问题等。我们根据上述原则和建议,重新计算了氢气的维里系数,结果与各原作者所得的有相当的不同。在同一温度用不同实验窒的数据所得的维里系数比各原作者所得的要互相接近些,一般在两倍标准误差内一致。     

Molar volume for mixtures of methane with krypton,argon, ethane and carbon monoxide using the ISM equation of state

In this paper, an analytical equation of state (EoS) proposed by Ihm–Song–Mason (ISM) was employed to calculate the molar volume of mixtures of methane with krypton, argon, ethane and carbon monoxide. The best available pair interaction potential energy has been used to evaluate the second virial coefficients required by the ISM EoS. The calculated values of second virial coefficients were applied to the ISM EOS to predict the molar volumes for the above mentioned mixtures. Agreement with the experiment is excellent for all systems.