Gas-liquid Transition in Charged Fluids

Abstract

The connexion between the equation of state of a classical fluid of non-polarizable ions, the character of the screening, and the appearance of long range oscillations in the chargetharge radial distribution function is examined. While considerations of stability lead to the usual inequalities for the inverse static dielectric function and the compressibility of the charged fluid, the square of the inverse screening length k ₃, does not need to be positive for thermodynamic stability. Through a study of an approximate equation of state for a twocomponent fluid of charged hard spheres, the regions of negative and positive k ² in the pressuredensity plane are related to a liquid phase and to an ionized-gas phase, respectively. The model fluid displays a gas-liquid critical point, above which the transition between the two types of screening is continuous. This behaviour of the charged-hard-spheres fluid is contrasted with the transition of a real ionic liquid to the molecular gaseous phase.     

A Green's function approach to the nonrelativistic radial wave equation of hydrogen atom

We report an efficient and consistent method for the solution of Schrödinger wave equation using the Green's function technique and its successful application to the nonrelativistic radial wave equation of hydrogen atom. For the radial wave equation, the Green's function is worked out analytically by means of Laplace transform method and the wave function is proposed under the boundary conditions. Computationally the product of potential term, the proposed wave function and the Green's function are integrated iteratively to get the nonrelativistic radial wave function. The resultant wave after each iteration is normalized and plotted against the standard nonrelativistic radial wave function. The solution converges to the standard wave with the increasing number of iterations. Results are verified for the first 15 states of hydrogen atom. The method adopted here can be extended to many‐body problem and hope that it can enhance our knowledge about complex systems.     

Structure of Liquid Metals

Abstract

The structure factors and radial distribution functions of liquid sodium and aluminium were calculated using the Hypernetted chain equation and the Machin-Woodhead-Chihara (MWC) integral equation. Various oscillatory potentials suggested for these metals were considered in an attempt to determine the applicability of these integral equations for these potentials. The calculated results are compared with molecular dynamic simulation results. These results indicate that the HNC equation underestimates the main peak in S(k). When the Friedel oscillations are absent then MWC theory gives good results for S(k). But when Friedel oscillations are present then MWC equation reproduces simulation results beyond the main diffraction peak.     

Statistical Spectral Analysis of liquid X-ray Diffraction Data

Abstract

The experimental procedure of determining the structure of a liquid by diffraction techniques is reformulated herein as a stochastic experiment subject to the data analysis formalism of statistical spectral analysis. Observed in such an experiment are averaged local microscopic fluctuations from the bulk density. The intensity function then represents a stochastic spectrum and it becomes necessary to statistically estimate a minimum bias, minimum variance covariance function which is the net radial distribution function. A low-pass tapered data window produces such an optimum estimate.     

Exact solution of multidimensional hyper‐radial Schrödinger equation for many‐electron quantum systems

In quantum theory, solving Schrödinger equation analytically for larger atomic and molecular systems with cluster of electrons and nuclei persists to be a tortuous challenge. Here, we consider, Schrödinger equation in arbitrary N‐dimensional space corresponding to inverse‐power law potential function originating from a multitude of interactions participating in a many‐electron quantum system for exact solution within the framework of Frobenius method via the formulation of an ansatz to the hyper‐radial wave function. Analytical expressions for energy spectra, and hyper‐radial wave functions in terms of known coefficients of inverse‐power potential function, and wave function parameters have been obtained. A generalized two‐term recurrence relation for power series expansion coefficients has been established. © 2016 Wiley Periodicals, Inc.     

The role of soft-core repulsions in the structure of liquids

A reformulation of the Ornstein-Zernike equation due to Baxter is used in connection with the Percus-Yevick closure for numerical solutions of the radial distribution function. It is shown that this method is useful for interaction potentials of finite range. The Lennard-Jones potential, truncated at its zero, is taken as an example, and compared with the hard sphere potential. The introduction of a finite repulsion lowers the peak of the radial distribution function appreciably, and damps its oscillations at larger distances.     

Solving the Schrodinger equation of an electron in a periodic crystal potential through elliptic functions

In this study, the Schrodinger equation of a valence electron in a periodic crystal potential is formulated and solved using the elliptic function formalism. The method allows double-periodic lattice planes to be represented in the Gauss plane. The reality of the obtained eigenfunctions and the structure of the valence and conduction bands are also investigated.

     

从硬球微扰理论推导马丁-侯状态方程

从Ｂａｒｈｅｒ－Ｈｅｎｄｅｒｓｏｎ的硬球微扰理论出发，运用幂级数形式的径向分布函数和分段的势能函数导出马丁－侯状态方程，推导过程中导出了一种理论式，计算Ｐ－Ｖ－Ｔ性质的精确度与马丁－侯原式相当，理论式的常数与分子微观参数有确定的函数关系。     

Comment on “a new approach to solve the Schrödinger equation with an anharmonic sextic potential”

We analyze a recent application of the Nikiforov-Uvarov (NU) method to an N -dimensional anharmonic oscillator with a central-field sextic potential-energy function. We show that most of the equations derived by the author exhibit errors (or typos) and that his interpretation of the results may not be correct. By means of the Frobenius (power-series) method we derive exact particular solutions to the Schrödinger equation and compare them with those coming from the NU method.

     

Some remarks to the derivation of the “generalized Lippmann equation”

In the present communication, an attempt is made to demonstrate (once again) some of the problems with the derivation of the “generalized Lippmann equation” considered to be valid by many researchers for solid electrodes and to address the problems in the framework of the Gibbs model of the interface by using only the basic principles of thermodynamics. By surveying the relevant literature, it has been shown that during the derivation of the equation, it was completely ignored that the Gibbs-Duhem equation (i.e., the electrocapillary equation) is a mathematical consequence which follows directly from the homogeneous degree one property of the corresponding thermodynamic potential function; consequently, the resulting expression cannot be correct. Some alternative approaches have also been considered. The adequacy of the open system and the partly closed system approach has been critically discussed, together with the possibility of introducing new thermodynamic potential functions.

     

Structural precursor to freezing: an integral equation study

Recent simulation studies have drawn attention to the shoulder which forms in the second peak of the radial distribution function of hard spheres at densities close to freezing and which is associated with local crystalline ordering in the dense fluid. We address this structural precursor to freezing using an inhomogeneous integral equation theory capable of describing local packing constraints to a high level of accuracy. The addition of a short-range attractive interaction leads to a well known broadening of the fluid-solid coexistence region as a function of attraction strength. The appearance of a shoulder in our calculated radial distribution functions is found to be consistent with the broadened coexistence region for a simple model potential, thus demonstrating that the shoulder is not exclusively a high density packing effect.     

Revisiting the generalized pseudospectral method: Radial expectation values,fine structure,and hyperfine splitting of confined atom

This paper revisits the generalized pseudospectral (GPS) method on the calculation of various radial expectation values of atomic systems, especially on the spatially confined hydrogen atom and harmonic oscillator. As one of the collocation methods based on global functions, the powerfulness and robustness of the GPS method has been well established in solving the radial Schrödinger equation with high accuracy. However, in our recent work, it was found that the previous calculations based on the GPS method for the radial expectation values of confined systems show significant discrepancies with other theoretical methods. In this work we have tackled such a problem by tracing its source to the GPS method and found that the method itself may not be able to obtain the system wave function at the origin. Combined with an extrapolation method developed here, the GPS method can fully reproduce the radial quantities obtained by other theoretical methods, but with more flexibility, efficiency, and accuracy. We apply the GPS-extrapolation method to investigate the relatievistic fine structure and hyperfine splitting of confined hydrogen atom in s-wave states where the zero-point wave function dominates. Good agreement with previous predictions is obtained for confined hydrogen in low-lying states, and benchmark results are obtained for high-lying excited states. The perturbation treatment of the fine and hyperfine interactions is validated in the confining environment.     

An effective quasirelativistic hamiltonian

Several ways of introducing effective relativistic potentials into the Schrödinger equation are discussed. A detailed study of a potential, which leads to the radial electron density exhibiting the same asymptotic behaviour as the Dirac density is performed. This potential, which is a j-dependent correction to the radial Schrödinger equation and which retains exactly the analytical form of the equation, may be easily implemented in atomic and molecular calculations.     

A general method to obtain Sturmian functions for continuum and bound state problems with Coulomb interactions

In this article, we investigate discretization schemes to represent Sturmian functions for both positive and negative energies in the presence of a long range Coulomb potential. We explore two methods to obtain Sturmian functions for positive energy. The first one involves the expansion of the radial wave function in a L² finite basis set, whereas the second one introduces the discretization of the radial coordinate domain of the Hamiltonian or, alternatively, the Green function. We apply them to find the bound states and scattering phase shift for ‐electron atoms close to the critical charge. Both methods are able to describe bound states near threshold, as well as continuum states with very good convergence properties. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Zur Kinetik der Oxydation von bestrahltem Polypropylen

Zusammenfassung Bei der Oxydation von bestrahltem Polypropylen l?uft eine radikalische Kettenreaktion ab. Ihre Kinetik wird mit Hilfe einer am Poly?thylenoxid entwickelten Modellvorstellung untersucht. Diese Modellvorstellung führt zu einer Gleichung, die die Sauerstoffaufnahme bestrahlter Proben in Abh?ngigkeit von der Abnahme der Radikalkonzentration beschreibt. Aus den experimentellen Ergebnissen folgt, da? diese Gleichung auch beim Polypropylen gilt. Es wird gezeigt, da? eine früher angegebene Gleichung die l. N?herung für den hier diskutierten Ansatz liefert.

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Summary The oxidation of irradiated polypropylene follows a radical chain reaction. A model for this reaction is discussed, which was originally developed for polyethyleneoxide. It leads to an equation, which describes the oxygen consumption of irradiated samples as a function of the decrease of the radical concentration. The experimental results follow this equation also for polypropylene. An earlier calculation is shown to be the first order approximation of the new equation.

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Ich danke Herrn Professor Dr. K.-H. Hellwege und Herrn Privatdozent Dr. H. Fischer für die Diskussion und die F?rderung der Arbeit.     

Derivation of Martin-Hou Equation of State from Hard-particle Perturbation Theory

IntroductionIn the mid 1950's, an empirical equation of state was proposed by Martin and HouLI] abbreviated to the M--H equation. It is a more accurate equation of state applicable to real gasesincluding polar compounds, such as ammonia and water over a wide range of density and temperature. In 1981, the M--H equation was extended further to liquid domain by Hou et al. [2),and later, it was used to calculate the vapor--liquid equilibrium, liquid--liquid equilibrium andother thermodynamic p…     

A completely analytic equation of state for the square-well chain fluid of variable well width

A completely analytic perturbation theory equation of state for the freely-jointed square-well chain fluid of variable well width (1 ≤ λ ≤ 2) is developed and tested against Monte Carlo simulation data. The equation of state is based on second-order Barker and Henderson perturbation theory to calculate the thermodynamic properties of the reference monomer fluid, and on first-order Wertheim thermodynamic perturbation theory to account for the connectivity of monomers to form chains. By using a recently developed real function expression for the radial distribution function of hard spheres in perturbation theory, we obtain analytic, closed form expressions for the Helmholtz free energy and the radial distribution function of square-well monomers of any well width. This information is used as the reference fluid in the perturbation theory of Wertheim to obtain an analytic equation of state, without adjustable parameters, that leads to good predictions of the compressibility factors and residual internal energies for 4-mer, 8-mer and 16-mer square-well fluids when compared with the simulation results. Further, very good results are obtained when this equation of state with temperature-independent parameters is used to correlate the vapor pressures and critical points of the linear alkanes from methane to n-decane.     

Non relativistic continuum wave functions for two coulomb centres

We give the continuum wave function solutions to the Schrödinger equation for an electron moving field of two point nuclei, as an expansion in terms of one centre Coulomb wave functions in a prolate elliptical coordinate system. These solutions may be chosen to have a convenient asymptotic behaviour, and tend to the conventional solutions of the Helmholtz equation in the limit that the nuclear charge goes to zero. In symmetric systems, where both nuclei have the same charge the angular wave functions are found to be identical with those occurring in the free case, and the expansion coefficients for the corresponding radial solutions are given for selected values of electron energy and nuclear separation.[/p]     

Application of a renormalization-group treatment to the statistical associating fluid theory for potentials of variable range (SAFT-VR)

An accurate prediction of phase behavior at conditions far and close to criticality cannot be accomplished by mean-field based theories that do not incorporate long-range density fluctuations. A treatment based on renormalization-group (RG) theory as developed by White and co-workers has proven to be very successful in improving the predictions of the critical region with different equations of state. The basis of the method is an iterative procedure to account for contributions to the free energy of density fluctuations of increasing wavelengths. The RG method has been combined with a number of versions of the statistical associating fluid theory (SAFT), by implementing White's earliest ideas with the improvements of Prausnitz and co-workers. Typically, this treatment involves two adjustable parameters: a cutoff wavelength L for density fluctuations and an average gradient of the wavelet function Φ. In this work, the SAFT-VR (variable range) equation of state is extended with a similar crossover treatment which, however, follows closely the most recent improvements introduced by White. The interpretation of White's latter developments allows us to establish a straightforward method which enables Φ to be evaluated; only the cutoff wavelength L then needs to be adjusted. The approach used here begins with an initial free energy incorporating only contributions from short-wavelength fluctuations, which are treated locally. The contribution from long-wavelength fluctuations is incorporated through an iterative procedure based on attractive interactions which incorporate the structure of the fluid following the ideas of perturbation theories and using a mapping that allows integration of the radial distribution function. Good agreement close and far from the critical region is obtained using a unique fitted parameter L that can be easily related to the range of the potential. In this way the thermodynamic properties of a square-well (SW) fluid are given by the same number of independent intermolecular model parameters as in the classical equation. Far from the critical region the approach provides the correct limiting behavior reducing to the classical equation (SAFT-VR). In the critical region the β critical exponent is calculated and is found to take values close to the universal value. In SAFT-VR the free energy of an associating chain fluid is obtained following the thermodynamic perturbation theory of Wertheim from the knowledge of the free energy and radial distribution function of a reference monomer fluid. By determining L for SW fluids of varying well width a unique equation of state is obtained for chain and associating systems without further adjustment of critical parameters. We use computer simulation data of the phase behavior of chain and associating SW fluids to test the accuracy of the new equation.     

A four-step exponentially fitted method for the numerical solution of the Schrödinger equation

In this paper, we present an exponentially fitted four-step method for the numerical solution of the radial Schr?dinger equation. More specifically we present a method that integrates exactly the functions {exp ( ±w x) , x {exp ( ±w x)}. We illustrated the efficiency of our newly produced scheme against well known methods, with excellent results. The numerical results showed that our method is considerably more efficient compared to well known methods used for the numerical solution of resonance problem of the radial Schr?dinger equation.