Mie Potential Research Articles

Description of the geometry of crystals with a hexagonal close-packed structure based on pair interaction potentials

The pair force interaction potential that allows one to describe a deviation from spherical symmetry, which is typical for hexagonal close-packed structures, is constructed using the “spherically symmetric” Mie potential that depends only on the interatomic distance. The parameters of the considered potential, which ensure the stability of hexagonal close-packed lattices, are obtained for a wide range of metals, namely, beryllium, gadolinium, hafnium, holmium, dysprosium, yttrium, cobalt, lutetium, magnesium, osmium, rhenium, ruthenium, scandium, thallium, terbium, technetium, titanium, thulium, cerium, zirconium, and erbium. It is shown that for this pair interaction potential the hexagonal close-packed structure is energetically more favorable than the face-centered cubic structure. The proposed potential can be used to perform computational experiments and analytical investigations.

Mie Potentials for Phase Equilibria Calculations: Application to Alkanes and Perfluoroalkanes

Transferable united-atom force fields, based on n - 6 Lennard-Jones potentials, are presented for normal alkanes and perfluorocarbons. It is shown that by varying the repulsive exponent the range of the potential can be altered, leading to improved predictions of vapor pressures while also reproducing saturated liquid densities to high accuracy. Histogram-reweighting Monte Carlo simulations in the grand canonical ensemble are used to determine the vapor liquid coexistence curves, vapor pressures, heats of vaporization, and critical points for normal alkanes methane through tetradecane, and perfluorocarbons perfluoromethane through perfluorooctane. For all molecules studied, saturated liquid densities are reproduced to within 1% of experiment. Vapor pressures for normal alkanes and perfluorocarbons were predicted to within 3% and 6% of experiment, respectively. Calculations performed for binary mixture vapor-liquid equilibria for propane + pentane show excellent agreement with experiment, while slight deviations are observed for the ethane + perfluoroethane mixture.

On the limitations of the equation of state based on the Lennard-Jones potential

It is emphasized that any equation of state (EOS) based on the generalized Lennard-Jones potential or the Mie potential, suffers from two main shortcomings as pointed out by Stacey and Davis [2]. One of the shortcomings viz. the problem related to imaginary numbers for the exponents in the potential function, has been removed recently by Jiuxun [11] by using a relationship between the exponents. However, the modified EOS obtained by Jiuxun suffers from the second shortcoming viz. it gives lower values for −B 0 B″0, an important equation of state parameter related to the second pressure derivative of the bulk modulus. Values of B 0 B″0 obtained by Jiuxun are not consistent with those reported by Stacey and Davis.

New criteria for the equation of state development: Simple model fluids

Recently we have proposed (J. Chem. Phys. 128 (2008) 134508) a new rescaling of fluid density ρ by its critical value ρ c 2 / 3 to apply the corresponding states law for the attractive Yukawa fluids study. Analysis of precise simulation results allows us to generalize this concept to the case of simple fluids with different interparticle interactions, like Mie ( n , m ) and Sutherland pair potentials. It is shown, that there is a linear relationship between the critical pressure and critical temperature, as well as the critical density and inverse critical temperature for these frequently used pair potentials. As a consequence, the critical compressibility factor of these model fluids is close to its universal value measured experimentally for different real substances.

Polynomial solutions of the Mie-type potential in the D-dimensional Schrödinger equation

The polynomial solution of the D-dimensional Schrödinger equation for a special case of Mie potential is obtained with an arbitrary l ≠ 0 states. The exact bound state energies and their corresponding wave functions are calculated. The bound state (real) and positive (imaginary) cases are also investigated. In addition, we have simply obtained the results from the solution of the Coulomb potential by an appropriate transformation.

Application of perturbed chain equation-of-state to solid–liquid equilibria: I. Pure component

A perturbed chain equation-of-state was applied to the solid-fluid equilibria (SFE) of Lennard–Jones (LJ) chains and of n-alkanes with carbon number of 4–32. Results for the LJ chains were reasonable and consistent. The model was able to accurately correlate the SFE of the n-alkanes using parameters that represent segment size, chain length, and intermolecular interaction. Parameters regressed from the n-alkanes varied smoothly and consistently with carbon number. A repulsive exponent in Mie's potential of 40 was found to be optimal. This work sets-up extension of the pure-component model to mixtures.

Bound state solution of the Schrödinger equation for Mie potential

Exact solution of Schrodinger equation for the Mie potential is obtained for an arbitrary angular momentum. The energy eigenvalues and the corresponding wavefunctions are calculated by the use of the Nikiforov–Uvarov method. Wavefunctions are expressed in terms of Jacobi polynomials. The bound states are calculated numerically for some values of l and n with n ≤ 5. They are applied to several diatomic molecules.

A Comprehensive Description of Chemical Association Effects on Second Derivative Properties of Alcohols through a SAFT-VR Approach

A recently derived version of the statistical associating fluid theory (SAFT), denoted as SAFT-VR Mie, which incorporates the Mie potentials within the SAFT-VR framework to model the monomer segment interactions (Lafitte et al. J. Chem. Phys. 2006, 124, 024509), is used for the study of second-order derivative properties and phase equilibria of alcohols and 1-alcohol + n-alkane binary mixtures. For this purpose, a variable repulsive potential is used to induce nonconformal interactions in the reference nonbonded fluid. These features have a significant influence on the chain and association contributions through the contact value of the radial distribution function, and they enhance the SAFT theory performance in the application to associating substances. When dealing with pure alcohols and 1-alcohol + n-alkane binary mixtures, an accurate description of both phase equilibria and second-order derivatives is obtained with a single set of molecular parameters. To explore the predictive ability limit of the model we have particularly focused our attention on secondary derivative properties, which display singularities due to the formation of aggregates. With this approach, we have found that the model is able to reproduce accurately the complex behavior of the isobaric heat capacity of alcohols as, for instance, the maximum versus temperature in the compressed liquid region. Furthermore, in the case of 1-hexanol + n-hexane binary mixtures, the proposed equation is found to capture the association effects on the pressure and temperature dependence of the isobaric thermal expansivity. These two special features, which to our knowledge have never been described by a theoretical model, emphasize both the validity of the changes in the model proposed and the physical meaning of the molecular parameters obtained in this study.

Simultaneous estimation of phase behavior and second-derivative properties using the statistical associating fluid theory with variable range approach

A modified statistical associating fluid theory (SAFT) with variable range version is presented using the family of m-n Mie potentials. The use of this intermolecular potential for modeling repulsion-dispersion interactions between the monomer segments, together with a new method for optimizing the molecular parameters of the equation of state, is found to give a very accurate description of both vapor-liquid equilibria and compressed liquid bulk properties (volumetric and derivative properties) for long-chain n-alkanes. This new equation improves other SAFT-like equations of state which fail to describe derivative properties such as the isothermal compressibility and the speed of sound in the condensed liquid phase. Emphasis is placed on pointing out that the key for modeling the latter properties is the use of a variable repulsive term in the intermolecular potential. In the case of the n-alkanes series, a clear dependence of the characteristic molecular parameters on increasing chain length is obtained, demonstrating their sound physical meaning and the consistency of the new fitting procedure proposed. This systematic method for optimizing the model parameters includes data on the saturation line as well as densities and speed of sound data in the condensed liquid phase, and the results show undoubtedly that the model performance is enhanced and its range of applicability is now widened, keeping in any case a good balance between the accuracy of the different estimated properties.

Applicability of isothermal three-parameter equations of state of solids—a reappraisal

The objective of the present study, an extension of a recent one, is to inter-comparethe utility of the various isothermal three-parameter equations of state (EOSs)of solids, considered viable at different stages in the development of the EOSfield, spanning over a period of about a century now. In a recent paper we havecompared our isothermal three-parameter equation of state of solids with seventhree-parameter isothermal EOSs—five corresponding to the regression curves ofV/V0 onP, and twoto those of P on V/V0. In this study, we investigate the relative utility of 21, i.e., virtually all, of theviable three-parameter EOSs—for the purposes of smoothing and interpolationof pressure–volume data, and extraction of accurate values of isothermal bulkmodulus and its pressure derivative—corresponding to the regression curves ofP on V/V0. We have applied the EOSs, with no constraint on the parameters, to accurateand model-independent isotherms of nine solids, and assessed the goodnessof the fitting accuracy; goodness of the stability of the fit parametersB0,B0′,and B0′′ with variation in the pressure/compression ranges; and goodness of the agreement of the fit parametersB0 and B0′ with experiment. Further, an additional test of goodness of randomization of the datapoints about the fit curves, quantified in terms of the number of wiggles of the datadeviation curves about the fits, is also applied in the present study. The EOSs subjected tothese seven tests are the three-parameter extensions of the EOS models formulated byBridgman (1929), Murnaghan (1937), Birch (1938), Slater (1939), Davis and Gordon(1967), Macdonald (1969), Holzapfel (1991), Poirier and Tarantola (1998), and theso-called ‘universal EOS’ promoted by Vinet et al (1986). Also included for theinter-comparison purposes are the three-parameter EOSs proposed by Keane (1954), Mao(1970), Thomsen (1970), Huang and Chow (1974), Luban (1983), Freund andIngalls (1989), Kumari and Dass (1990), Hama and Suito (1996), and Bose Royand Bose Roy (1999), and also the EOS based on a modified Eulerian strain assuggested by Sushil et al (2004). Interestingly, the three-parameter Mie–GruneisenEOS, built on the century-old Mie potential (Partington 1957, Stacey and Davis2004), is also tested. The present study leads to some remarkable findings. Mostnotably, some of the old EOS models like those of Birch and of Keane, and also theMie–Gruneisen EOS, are observed to be in better agreement with experiment than most ofthe EOSs that appeared much later in the literature. An inter-comparison ofthe overall results inferred from all the 21 EOSs—based on the most stringentdiscrimination technique comprising seven important tests ever applied in theliterature—demonstrates that, while most of the EOSs marginally differ from each otherconstituting a narrow band of applicability, our model is decisively superior to all of them.

LIQUID–GAS PHASE TRANSITION TO FIRST ORDER OF AN ARGON-LIKE FLUID MODELED BY MIE POTENTIAL

In this paper, we analyze the liquid–gas phase transiton to first order of an Argon-like fluid modeled by Mie potential by the theory in Ref. 1. At the same time, we extend the theory to arbitrary D space dimensions.

고분자액체에 대한 포텐셜의 비교

고분자액체에 대한 이론들은 많은 경우 cell, hole, free volume 또는 lattice 등의 개념에 근거를 두고 있다. 여기에 van der Waals 포텐셜이나 Lennard-Jones 6-12포텐셜 또는 이를 개선한 형태의 포텐셜을 보통 사용되고 있다. 본 연구에서는 고분자 액체를 설명?求?성공적인 이론으로 알려져 있는 격자유체이론에서 불연속적 격자를 고전적으로 연속적으로 격자로까지 확장한 연속격자유체이론과 Flory의 상태방정식이론을 개선한 Dee-Walsch의 Cell 이론에 각각 Mie(p,6)포텐셜을 적용하여 반발포텐셜항의 지수 p에 따른 PVT 값을 계산하여 실험값과 비교를 하였다. 또한 격자유체이론, Flory의 상태방정식이론, 섭동법을 이용한 Cho-Sanchez 이론의 계산값과도 비교를 하였다. 계산결과를 통하여 고분자액체에 대한 포텐셜로 Van der Waals 포텐셜, Lennaed-Jones 6-12 포텐셜, Mie(p,6)포텐셜들을 비교하였다. Many theories for polymeric liquids are based on the concepts of cell, hole, free volume or lattice etc. In the theories, van der Waals potential, Lennard-Jones 6-12 potential and their modified potentials are commonly used.In this work, Mie(p, 6) potential was applied to the Continuous Lattice Fluid Theory (which extends the discrete lattices of Lattice Fluid Theory to classically continuous lattices) and Dee-Walsch's Cell Theory (which modifies Flory's Equa-tion of State Theory). Both of them are known to be successful theories for polymeric liquids. Thus, PVT values chang-ing with p (the exponent in the repulsion potential) were calculated and compared with experimental values. And, calculated values of Lattice Fluid Theory, Flory's Equation of State Theory and Cho-Sanchez Theory using pertubation method were also compared. Through the calculated results, van der Waals potential, Lennard-Jones 6-12 potential and Mie(p, 6) potential for polymeric liquids were compared with each other.

Mie Potential and Equation of State of Zr48Nb8Cu14Ni12Be18 Bulk Metallic Glass

The linear expansion of Zr48Nb8Cu14Ni12Be18 bulkmetallic glass (BMG) with excellent glass forming ability and high thermalstability is investigated by a dilatometry method. The average expansioncoefficient is αTG= 1.04×10-5 K-1 (300-656 K)for the BMG and αTC= 1.11×10-5 K-1(356~890 K) for the crystallized alloy. The Mie potential as well asthe equation of state of the BMG and its corresponding crystallized stateare determined from the thermal expansion and ultrasonic data, and thedifferences among them are phenomenologically explained.

Theoretical Density of Refractory and Ceramic Materials

The concept of the Mie potential and a new understanding of the nature of the forces acting between structural components of a substance (atoms, ions, molecules) are used to develop a model allowing for the effect of the thermal action of particles (the thermal field) on the chemical bonds in the substance. The new concepts on the chemical bond are used to derive a formula for the theoretical strength of materials in a wide temperature range. The calculated values of the theoretical strength of Al2O3, SiO2 (glass), and SiC at 293 K agree well with the actual strength of whiskers. The established temperature dependence of the theoretical strength of Al2O3 is adequate to the temperature dependence of the actual strength of whiskers. The theoretical strength is a physical parameter of a material and can serve as a comparative characteristic for choosing a material for specific service conditions.

General Form of the Cross-Energy Parameter of Equations of State

Phase equilibrium calculations with cubic equations of state are sensitive to mixing and combining rules employed. In this work, we present a suitable general form of the combining rule for the cross-energy parameter, often considered to be the key property in phase equilibrium calculations. The proposed rule is derived from molecular considerations, namely, the London−Mie theory. The typically used geometric mean (GM) and other combining rules can be deducted from this expression from different values of the parameter n, which is the attractive tail of the Mie potential. We show that using this n parameter as the variable instead of the commonly employed kij offers useful insight into the behavior of cubic equations of state for a large number of asymmetric systems including gas/alkanes, polymer solutions and blends, and alcohol/alkane and gas/solid systems.

An ab Initio Calculation of the Potential for the Interaction of a Hydrogen Atom with an Ethane Molecule

In this paper ab initio methods including PMP2/aug-cc-pVTZ, CCSD(T)/6-311++G(d,p), and G2 methods are used to calculate the interaction potential for a hydrogen atom approaching ethane along the carboncarbon axis of ethane. The potential shows a shallow minimum at about 4.0 A and a strong repulsive core. The position of the minimum moves to smaller distances as the size of the basis set increases. Concurrently, the strength of the core repulsion decreases and the well depth increases. PMP2 calculations show a slightly shallower well than do CCSD(T) calculations, but the repulsive potentials are virtually identical. The potential is well represented by a Morse potential. An exponential-6 potential, a simple exponential, an anti-Morse potential or a Kihara potential fit the data less well. A Lennard-Jones potential and a Mie potential show substantial deviations. A new exponentially damped exponential-6 potential fits the repulsive region slightly better than the Morse exponential-6, anti-Morse, and Lennard-Jones potentials. Potential parameters are included in the paper.

An Analytical Free Energy and the Temperature−Pressure Superposition Principle for Pure Polymeric Liquids

Recently, it was found that polymers follow the principle of temperature−pressure (T−P) superposition. This principle states the temperature insensitiveness of the shape of the configurational free energy. An analytical free energy for polymeric liquids is developed in the framework of the perturbation theory to provide a better understanding of the T−P superposition principle. The intermer potential is separated into the repulsive reference and the attractive perturbed parts. The hard sphere potential is taken as the repulsive reference. The attractive part of the Mie (p,6) potential is taken as the perturbation. The free energy for the reference system of hard chains is obtained from the integration of the hard chain equation of state from the Baxter−Chiew theory. The limiting case of the reference free energy at infinite dilution is derived from the direct evaluation of the configurational partition function. The average packing energy is added as a perturbation energy. The local packing in the nearest...

Formulations of the extended boundary condition method for three-dimensional scattering using the method of discrete sources

Novel formulations of the extended boundary condition method for three-dimensional scattering problems are derived by using a system of magnetic and electric dipoles, a system of ‘Mie potentials’ and a system of lowest-order multipoles as complete systems of functions. The key step in these approaches is to use discrete sources located on auxiliary supports to guarantee the null-field condition for the total electric field and to generate the surface current densities.

On the Mie potential and the acoustic nonlinear parameter B/A

The set of two coupled equations of Hartmann connecting the acoustical nonlinear parameter B/A with the reciprocal sound velocity 1/c, for liquids, through the parameters of the Mie potential, is replaced by a simple empirical equation linear and near quadratic in linear combinations of B/A and 1/c. The latter equation contains an exponent μ which is a function of the Mie exponents m and n. It is shown that this empirical relation closely approximates the pair solution over a significantly wider range of variables than the purely linear equation posited by Hartmann to explain the empirical Ballou’s rule.

Finite-temperature properties of materials from analytical statistical mechanics

Abstract An approximate and yet highly accurate analytical method for obtaining fundamental material properties over a wide range of temperatures from the interatomic potential without resorting to numerical simulation is presented. In this approach, classical statistical mechanics are combined with the local-harmonic method in order to obtain thermodynamic and mechanical properties of solids, including the free energy. Specifically, we present results for a general Mie (Lennard-Jones m, n) potential. The dependence of these properties on potential parameters is examined and a number of results are summarized. We find that the analytical theory yields remarkably accurate results for such thermodynamic quantities as volume, energy, free energy and bulk modulus to temperatures approaching the melting point.