Effect of Core-Ion Potentials on Thermodynamic Properties of Liquid Alkali Metals and Alloys

Abstract

A pseudopotential perturbation scheme based on Gibbs-Bogoliubov variational technique is considered to investigate the structure and volume dependent contributions to the internal energies of liquid alkali metals. This has paved the way to analyse the effect of core-ion potentials on Helmholtz free energy, enthalpy, pairwise interactions and on entropy of liquids Na, K, Rb and Cs. Excess entropy of mixing for Na-K. Na-Cs and K-Rb are also calculated and are found very sensitive to the core-ion potentials.     

Thermodynamic perturbation theory of simple liquids. A hierarchy of equations for expansion coefficients

The properties of an expansion of the statistical sum of a simple liquid with respect to the potential in thermodynamic perturbation theory are analyzed. The coefficients of this expansion are determined by the unperturbed potential, depend on temperature and density, and can be calculated by means of mathematical modeling. It is shown here that the derivatives of these coefficients with respect to temperature and density are expressed through the higher expansion coefficient (these relations are usually called a hierarchy of equations). These coefficients determine the expansion of the Helmholtz free energy and RDF with respect to the perturbation potential. The thermodynamic characteristics of the system (entropy, internal energy, pressure) are expressed through both the differential relations for the Helmholtz free energy and the integral expressions containing RDF. It is found that the hierarchy of equations obtained in this work makes these different methods equivalent. This is important for the application of thermodynamic perturbation theory because it becomes unnecessary to model any other equilibrium properties of the system apart from the expansion coefficients.     

Determination of the solid-fluid coexistence of the n - 6 Lennard-Jones system from free energy calculations

The solid-fluid coexistence properties of the n - 6 Lennard-Jones system, n from 7 to 12, are reported. The procedure relies on determining Helmholtz free energy curves as a function of volume for each phase independently, from several NVT simulations, and then connecting it to points of known absolute free energy. For n = 12 this requires connecting the simulated points to states of very low densities on the liquid phase, and to a harmonic crystal for the solid phase, which involves many extra simulations for each temperature. For the reference points of the remaining systems, however, the free energy at a given density and temperature can be calculated relative to the n = 12 system. The method presented here involves a generalization of the multiple histogram method to combine simulations performed with different potentials, provided they visit overlapping regions of the phase space, and allows for a precise calculation of relative free energies. The densities, free energies, average potential energies, pressure, and chemical potential at coexistence are presented for up to T? = 5.0 and new estimations of the triple points are given for the n - 6 Lennard-Jones system.     

Modified PT-LJ-SAFT Density Functional Theory: I. Prediction of surface properties and phase equilibria of non-associating fluids

A new modified version of a Perturbation Density Functional Theory (PT-DFT) based on the Statistical Association Fluid Theory (SAFT) with a Lennard–Jones interaction potential is proposed to model the vapor–liquid phase equilibrium and to predict the interfacial behavior of non-associating hydrocarbon fluids. In the interaction model for the Helmholtz free energy functional the molecules are separated into m spherical segments interacting via a Lennard–Jones potential. The segments form chains of tangent spheres. In the perturbation approximation to Density Functional Theory the interaction potential is split according to WCA and the attractive term to the free energy functional consists of a suitable modification of the perturbation expression. This modification to PT-DFT yields surface tensions for the Lennard–Jones sphere fluid (m = 1.0) which are in perfect agreement with simulation data.The new PT-DFT model combines the high flexibility of the SAFT free energy functional with a modified density functional approach that enables to perform accurate calculations of interfacial properties. To take into account the contributions to surface tension resulting from mesoscale thermal fluctuations a semiempirical model is proposed that allows to correct the microscopic intrinsic surface tension.The model is used to describe the phase equilibrium of lower alkanes and aromatics. The results demonstrate the capability to fit vapor–liquid equilibrium data and to predict very accurately the surface properties of these fluids within the uncertainties of the experimental data.     

Solvent-polymer interaction: I. Molecular transport of some selected organic liquids in polymer membrane

Methods and microtechniques for determining solubility, diffusivity, thermodynamic properties, and kinetic parameters of 12 selected organic liquid solvents in polyurethane membrane by thermogravimetry (TG) are described. TG provides a simple, sensitive, rapid, and accurate microtechnique for measuring a minute change in weight (or mass) of a substance as a function of time at isothermal condition or as a function of temperature at dynamic manner. Thus from a single isothermal TG-desorption experiment, solubility and diffusivity of solvent molecules in polyurethane membrane were obtained simultaneously. Furthermore, by a dynamic TG-desorption run, kinetic parameter such as activation energy of desorption of solvent molecules from polyurethane membrane was determined. In addition, much other useful information such as equilibrium sorption constant, the changes in standard enthalpy and standard entropy of sorption, permeability, the activation energy of diffusion and so forth for solvents in polyurethane membrane are also evaluated and discussed. Finally, the correlation between the microscopic molecular structure and macroscopic properties of solvent molecules in polymer membrane is interpreted in terms of linear free energy relationships.     

Surface Tension Calculations for Liquid Metals

Abstract

We present a class of models for the surface of a liquid metal, which may be part of an electrochemical interface. The particles of the system, for the purpose of derivation of thermodynamic properties, are the charged ion cores, while the energy of the electrons is evaluated using the electron density functional formalism, previously principally applied to solids. An expression for the surface energy U^s , defined as the energy required to create unit area of surface by separation of a volume of homogeneous metal into two parts, is derived (Eqs. 18–20). The surface tension γ is obtained by differentiating the Helmholtz free energy with respect to the area of the system, keeping volume and particle number constant (Eqs. 27–37). The surface tension is also equal to the difference between the free energy of the system containing a surface and the free energy of a reference system. It thus defines a surface energy through the Gibbs-Helmholtz equation, and this surface energy is shown to be identical to U^s .

The expressions for U^s and γ are made explicit (Eqs. 45–57) by insertion of particular assumptions for the ion-density profile, the electron-density profile, the interionic interaction and pair distribution function, and the electronic energy. Only information about bulk liquid metal is used. The parameter in the electron-density profile is obtained by minimizing the surface energy. The simplest assumption for the interionic interaction, hard-sphere and Coulombic repulsions, requires a choice for the hard-sphere diameter, which is made such that the pressure of bulk metals is given correctly (52–55). For the alkali metals, the surface tension calculated from this model is about half the experimental value in each case, while calculated surface energies are too high (1/5 too high for Cs, but three times too high for Li). For the electrical potential difference between the inside and the outside of a metal, and for the electrochemical potential, agreement with experiment is good. The main reason for the disagreements in the other properties is traced to the simple form used for the ion pair distribution function.     

Vibrational relaxation of oxygen in an argon cage

The vibrational relaxation of oxygen embedded in an argon cage through vibrational to local translation, rotation, and argon phonon modes has been studied using semiclassical procedures. The collision model is based on the trapped molecule undergoing the restricted motions (local translation and hindered rotation) in a cage formed by its twelve nearest argon neighbors in a face-centered-cubic structure. At 85 K in the liquid argon temperature range, the deexcitation probability of O(2)(v=1) is 5.8 x 10(-12) and the relaxation rate constant with the collision frequency from local translation is 23 s(-1). The rate constant decreases to 5.1 s(-1) at 50 K and to 0.016 s(-1) at 10 K in the solid argon temperature range. Transfer of the vibrational energy to local translation, rotation (both hindered and free), and argon phonon modes is the relaxation pathway for the trapped oxygen molecule.     

Thermodynamic properties of nitrogen from a perturbation theory for ISM fluids

A first-order Barker-Henderson perturbation theory for interaction-site model (ISM) fluids has been applied to calculate the Helmholtz free energy, entropy and internal energy of liquid nitrogen. Comparison with experiment reinforces the idea that the theory is accurate over a wide range of temperatures and densities corresponding to the liquid state, except for the critical region.     

Prediction of critical transitions of ternary mixtures containing ammonia and n-alkanes

The analysis of the critical transitions that occur in ternary mixtures is important to describe their physical behavior. It also enables the phase behavior of multi-component mixtures at high temperatures to be inferred. The objective of this work was to identify the critical transitions that occur in ternary mixtures containing ammonia and n  -alkane. The mixture’s critical loci were obtained and tested for stability using thermodynamic criteria expressed in terms of the Helmholtz free energy. Two equations of state were used to represent the Helmholtz free energy: the Carnahan–Starling–Redlich–Kwong (CSRK) and the Simplified Perturbed Hard Chain Theory (SPHCT). In order to identify the existing critical transitions, profiles of the critical loci were calculated along constant compositional ratios χ=x₁/x₂$\chi =x_1/x_2$. Some of the curves depict higher order critical transitions between liquid–liquid and gas–liquid critical point regions, or two different liquid–liquid critical regions. One of the critical transitions found could be considered as a new sub-class within existing classifications for ternary mixtures proposed by Sadus [R.J. Sadus, J. Phys. Chem. 96 (1992) 5197–5202].     

Phase equilibria and surface tension of pure fluids using a molecular layer structure theory (MLST) model

This paper presents a new model based on thermodynamic and molecular interaction between molecules to describe the vapour–liquid phase equilibria and surface tension of pure component. The model assumes that the bulk fluid can be characterised as set of parallel layers. Because of this molecular structure, we coin the model as the molecular layer structure theory (MLST). Each layer has two energetic components. One is the interaction energy of one molecule of that layer with all surrounding layers. The other component is the intra-layer Helmholtz free energy, which accounts for the internal energy and the entropy of that layer. The equilibrium between two separating phases is derived from the minimum of the grand potential, and the surface tension is calculated as the excess of the Helmholtz energy of the system. We test this model with a number of components, argon, krypton, ethane, n-butane, iso-butane, ethylene and sulphur hexafluoride, and the results are very satisfactory.     

Absolute Helmholtz free energy of highly anharmonic crystals: theory vs Monte Carlo

We discuss the problem of the quantitative theoretical prediction of the absolute free energy for classical highly anharmonic solids. Helmholtz free energy of the Lennard-Jones (LJ) crystal is calculated accurately while accounting for both the anharmonicity of atomic vibrations and the pair and triple correlations in displacements of the atoms from their lattice sites. The comparison with most precise computer simulation data on sublimation and melting lines revealed that theoretical predictions are in excellent agreement with Monte Carlo simulation data in the whole range of temperatures and densities studied.     

Equation of state and phase behavior of a propane-tetrafluoroethane mixture

A general equation of state was derived for a propane-tetrafluoroethane mixture in a form representing the free Helmholtz energy of a mixture as the function of density, temperature, and composition. Coefficients of equation were determined according to data on the density of the mixture and assuming the equilibrium condition of phases. The equation describes the thermodynamic properties of vapor and liquid in the temperature range from 255 to 400 K under pressures up to 6 MPa. The thermodynamic behavior of the mixture at the saturation state was analyzed. It is established that the difference between the temperatures of saturated vapor and of liquid on isobars for four compositions did not exceed 1.8 K.     

Development of a new equation of state for mixed salt and mixed solvent systems,and application to vapour–liquid and solid (hydrate)–vapour–liquid equilibrium calculations

An equation of state that can be used for phase equilibrium and other thermodynamic property calculations at high pressures is developed for systems that contain aqueous solutions of strong electrolytes and molecular species. The proposed equation of state is based upon contributions to the Helmholtz free energy from a non-electrolyte term and three electrolyte terms. The non-electrolyte term comes from the Trebble–Bishnoi equation of state and the electrolyte terms consist of a Born energy term, a mean spherical approximation term and a newly developed hydration term. The application of the proposed equation of state to aqueous systems containing mixed salts and mixed solvents is illustrated by calculating the vapour–liquid equilibrium (VLE) and solid (Clathrate hydrate)–vapour–liquid equilibrium (SVLE) conditions for several systems. The solubility of CO₂ in salt water systems is examined at elevated pressures. As well, the new equation of state is used in conjunction with the model of van der Waals and Platteeuw to predict the SVLE conditions for gas hydrate forming systems in the presence of single salts, mixed salts and a mixture of aqueous salts and methanol. It is found that the new equation of state is able to accurately represent the experimental data over a wide range of pressure, temperature and salt concentration.     

Rapid calculation of partition functions and free energies of fluids

The partition function (Q) is a central quantity in statistical mechanics. All the thermodynamic properties can be derived from it. Here we show how the partition function of fluids can be calculated directly from simulations; this allows us to obtain the Helmholtz free energy (F) via F = -k(B)T ln Q. In our approach, we divide the density of states, assigning half of the configurations found in a simulation to a high-energy partition and half to a low-energy partition. By recursively dividing the low-energy partition into halves, we map out the complete density of states for a continuous system. The result allows free energy to be calculated directly as a function of temperature. We illustrate our method in the context of the free energy of water.     

A new method for tieline calculations in binary fluid systems

We describe a method to determine tielines for a mixture with a given volume V_t and a given mole fraction x_t at fixed temperature, which is based on the orientation of the tangent plane to the Helmholtz free energy surface. The method works equally well for gas-liquid separation as for liquid-liquid separation. The method is based on a rotation in coordinate space followed by a rotation in function space which establishes the correct direction and the end points of the tieline. The mathematics is described and the general outline of a computer program to implement the method is given. This is followed by an application to a special form of the van der Waals equation, which describes the influence of chain-molecules as proposed by Tompa. The example given shows a three phase separation and the influence of the chain-length thereupon.     

EFFECTS OF TEMPERATURE ON THE VISCOSITY OF NONIONIC MJCELLAR SOLUTION IN THE PRESENCE OF ADDED SALTS

ABSTRACT

The effects of temperature on the viscosity of nonionic surfactant, Triton X-100, in the presence of added salts are presented. It was found that the plots of logarithm of relative viscosity versus 1/T were obtained from the slopes of these plots. The activation free energies (ΔG?) of the viscous flow were obtained from the slopes of these plots. The activation enthalpies ( ΔH?) and the entropies ( ΔS?) for the viscous flow were calculated and the entropic contribution was found to be negligible. The ΔG? and ΔH? were highly dependent on the nature of the positive ions. The greater the effect on the energy of the sphere-to-rod transition, the more strongly ions are bound. The ΔG? is interpreted in terms of the energy required to activate the transition of micellar structure capable of being transformed into rod micelle.     

First‐principles calculations on thermodynamic properties of BaTiO3 rhombohedral phase

The calculations based on the linear combination of atomic orbitals have been performed for the low‐temperature phase of BaTiO₃ crystal. Structural and electronic properties, as well as phonon frequencies were obtained using hybrid PBE0 exchange–correlation functional. The calculated frequencies and total energies at different volumes have been used to determine the equation of state and thermal contribution to the Helmholtz free energy within the quasiharmonic approximation. For the first time, the bulk modulus, volume thermal expansion coefficient, heat capacity, and Grüneisen parameters in BaTiO₃ rhombohedral phase have been estimated at zero pressure and temperatures form 0 to 200 K, based on the results of first‐principles calculations. Empirical equation has been proposed to reproduce the temperature dependence of the calculated quantities. The agreement between the theoretical and experimental thermodynamic properties was found to be satisfactory. © 2012 Wiley Periodicals, Inc.