Theory of critical phenomena in liquids

A theory of critical phenomena in liquids is constructed based on postulates of classical statistical mechanics (i.e., without invoking hypotheses underlying scaling theory). This theory allows for both long-and short-wave fluctuations, arising in the vicinity of a critical point. Equations for evaluating the parameters of the critical point (critical density, critical temperature, and critical pressure) from the given interaction potential are derived. An equation of state is obtained for the critical isotherm. From this equation it follows that in the vicinity of the critical point, the critical indices have classical values; scaling indices arise away from the critical point.     

Critical phenomena in liquids (theory)

It is shown that the Ornstein-Zernike equation has two solutions, classic (analytic) and critical (not analytic). Closure equations of the HNC, PY, etc. types well known in the theory of liquids correspond to the first solution. The second solution presupposes that the bridge functional of a system has the form of the sum B = B ^rg + B ^cr, where B ^rg is a regular (analytic) function of density and B ^cr is a critical (not analytic) function. It is shown the classic solution determines the coordinates of the critical point, critical amplitudes, and the other parameters of critical phenomena depending on the individual liquid characteristics. The critical solution determines the critical indices and the equations relating them. The regions in which classic and critical solutions are valid are separated on the phase plane by a line of singular points, at which the second derivatives of pressure experience discontinuity.     

The significant structure theory applied to halobenzenes

Abstract

Just as molecular structure, as revealed by X-ray diffraction, can be interpreted in terms of intuitive models, so liquid structure can be interpreted in terms of a model which leads to a partition function giving the Helmholtz free energy in terms of volume, temperature, and composition. From this explicit expression for the Helmholtz free energy all thermodynamic properties are calculable and can be compared with experiment. Absolute Rate Theory permits the prediction of transport properties from this same model, providing still further insight into liquid structure. Here, Significant Liquid Structure Theory has been applied to twelve substituted benzenes and the results compared with experiment. A single equation is derived for the twelve substances differing in ten of the cases only in three parameters having to do with the solid-like part of the liquid. For simple liquids these properties are those of the solid at the melting point. These properties are the energy of sublimation, molar volume of the solid, and the Einstein characteristic temperature, θ. Hindered rotation is explained in terms of a barrier to rotation of one tenth the energy of sublimation     

Fluctuation theory of critical phenomena in fluids

It is assumed that critical phenomena are generated by density wave fluctuations carrying a certain kinetic energy. It is noted that all coupling equations for critical indices are obtained within the context of this hypothesis. Critical indices are evaluated for 15 liquids more accurately than when using the current theory of critical phenomena.     

Application of significant structures theory to amorphous polyethylene

The thermodynamic properties of amorphous polyethylene are calculated from a model based on the method of significant structures. The motion of a molecule as a whole is described by the motion of segments, each segment moving independently of all others. It is assumed that on melting, holes appear in the solid lattice and the segments can move into these vacancies, obtaining some gaslike degrees of freedom. The complete frequency distribution for polyethylene is used for the solidlike degrees of freedom, while a corrected classical partition function is used for the gaslike degrees of freedom. The calculated thermodynamic properties are in reasonable agreement with experimentally determined values, assuming each gaslike segment to consist of 20 CH₂ groups.     

Two solutions for the Ornstein-Zernike equation and critical phenomena in liquids

It is shown that the Ornstein-Zernike equation, the equivalent of the Gibbs distribution, has two simultaneous solutions: analytical and nonanalytical. The analytical solution disappears at a critical point and only the nonanalytical solution remains (which, however, is not zero as we move away from the critical point). It is found that pressure and isothermal compressibility also have two components away from critical point: analytical and nonanalytical.     

Molecular theory of critical phenomena in aliphatic carboxylic acid monolayers

A statistical mechanical theory is presented for the behavior of monolayers of aliphatic carboxylic acids at the air/water interface. A lattice fluid model is introduced with monomer states to represent molecules in the all-trans conformation directed perpendicular to the interface, dimer states to represent rotational isomers, and vacant sites or “holes.” Using a Flory–Huggins-type approximation, the temperature and chain length dependences of the monolayer behavior are satisfactorily reproduced for a certain range of parameters. These parameters are compared with those intermolecular energies calculated using the methods of computational quantum chemistry. The agreement thus obtained is fairly satisfactory indicating that the higher-order nature of fatty acid monolayer transitions is due to the onset of the rotational ordering of pairs of chains about their longitudinal axes, while the sensitive temperature dependence is attributed to the flexible nature of fatty acid molecules. Some possible effects due to the presence of the substrate are discussed.     

Analysis of experimental data within the statistical theory of critical phenomena

An approach devised earlier is generalized for a binodal line and used to analyze experiments on a broad set of fluids from He⁴ to organic compounds. Experimental binodal lines and isothermal compressibility are described with precision for the above substances. The error characterizing the theory’s deviation from the experimental data is several times lower than in a similar analysis using the Ising model.     

The Ornstein-Zernike equation, critical phenomena, and the equation of state for liquids and gases

It is shown that the Ornstein-Zernike (OZ) equation has two solutions: the standard one, which depends explicitly on the interaction potential, and a second universal one, resulting from the infinity point of the partition function. It is stressed that there are two pressure components: the standard one and a universal one that is valid over the whole of the phase plane. It is concluded that the universal solution parameters depend in general on definite integrals of functions dependent on the interaction potential. In the vicinity of the critical point, however, the dependence on the interaction potential vanishes; i.e., the solution becomes fully universal. It is shown that in this range of the phase diagram, all results of the theory of critical phenomena (scaling theory) follow from the OZ equation.     

Application of magnetically perturbed time-dependent density functional theory to magnetic circular dichroism. II. Calculation of A terms

The magnetically perturbed time-dependent density functional theory is used to derive equations for the magnetic circular dichroism (MCD) of degenerate transitions of closed shell molecules. The MCD of this type of transition can be divided into two contributions. The dominant contribution is usually that from A terms that arise because of the breaking of the degeneracy of the excited state in the presence of the magnetic field. The second contribution comes from B terms that arise because of the perturbation of the transition dipole by the magnetic field. The formalism is applied to ten tetrahedral d(0) transition metal oxy- and thioanions. The MCD parameters of these systems are reproduced quite well by the calculations. Simulated spectra derived from the calculated parameters are in good agreement with the observed spectra.     

Density functional theory of inhomogeneous liquids. IV. Squared-gradient approximation and classical nucleation theory

The squared-gradient approximation to the modified-core Van der Waals density functional theory model is developed. A simple, explicit expression for the SGA coefficient involving only the bulk equation of state and the interaction potential is given. The model is solved for planar interfaces and spherical clusters and is shown to be quantitatively accurate in comparison to computer simulations. An approximate technique for solving the SGA based on piecewise-linear density profiles is introduced and is shown to give reasonable zeroth-order approximations to the numerical solution of the model. The piecewise-linear models of spherical clusters are shown to be a natural extension of classical nucleation theory and serve to clarify some of the nonclassical effects previously observed in liquid-vapor nucleation. Nucleation pathways are investigated using both constrained energy-minimization and steepest-descent techniques.     

Stochastic structure determination for conformationally flexible heterogenous molecular clusters: Application to ionic liquids

We present a novel method that enables accurate and efficient computational determination of conformationally flexible clusters, “Kick³” This method uses stochastically generated structures in combination with fast quantum mechanical methods. We demonstrate the power of this method by elucidating the structure of ionic liquid (IL) ([xMIM⁺][ clusters (x = E, B, D, n = 1–10,15). Dispersion‐corrected, third‐order self‐consistent‐charge density‐functional tight‐binding (DFTB3) is shown to be a computationally efficient, yet reliable approximation to density functional theory for predicting and understanding IL structure and stability. The presented approach, therefore, enables the accurate and efficient screening of ILs with high potential toward practical applications, without recourse to more expensive quantum chemical methods. © 2013 Wiley Periodicals, Inc.     

On the theory of vibronic structure of linear aggregates. Application to pseudoisocyanin (PIC)

A self-consistent theory for Frenkel excitons coupled to optical phonons is presented and tested by comparison with numerical calculations. As an example the spectrum of PIC is determined. It is found that the mean-field approach fails to predict accurately vibronic excitations at the upper band edges of the exciton zero- and one-phonon bands. However, a rather simple model which accounts for the spread of the lattice distortion via configuration interaction can describe these transitions quite well.     

Thermodynamic perturbation theory of simple liquids

It was proven that after averaging over the canonical Gibbs ensemble, the mean perturbation energy was singled out of the classical partition function before the expansion in a series of perturbation theory. Therefore, the term that formally coincides with first order perturbation theory in a decomposition of the Helmholtz free energy bears no relationship to perturbation theory. Then the proper series of the thermodynamic perturbation theory always starts with a second order infinitesimal. Therefore, the wellknown condition of applicability of the thermodynamic perturbation theory, “...the requirement that the perturbation energy per particle be small compared with T...” (L. D. Landau and E. M. Livshits, Statistical Physics, Vol. V, Pt. I), can be substantially weakened. The most important factor for applicability of thermodynamic perturbation theory is the value of many-particle correlations in an unperturbed system, but not the smallness of the perturbation potential.     

Local field cascading in third-order non-linear optical phenomena of liquids

Contributions to the third-order non-linear polarization response of molecular liquids due to cascaded second-order processes are discussed. The cascading occurs entirely because of the presence of a reaction field to the second-order polarization of non-centrosymmetric molecules. Application of a simple dielectric continuum model is used to investigate relative magnitudes, polarization anisotropy and resonance behavior.     

Application of density functional theory to capillary phenomena in cylindrical mesopores with radial and longitudinal density distributions

In this paper, we applied a version of the nonlocal density functional theory (NLDFT) accounting radial and longitudinal density distributions to study the adsorption and desorption of argon in finite as well as infinite cylindrical nanopores at 87.3 K. Features that have not been observed before with one-dimensional NLDFT are observed in the analysis of an inhomogeneous fluid along the axis of a finite cylindrical pore using the two-dimensional version of the NLDFT. The phase transition in pore is not strictly vapor-liquid transition as assumed and observed in the conventional version, but rather it exhibits a much elaborated feature with phase transition being complicated by the formation of solid phase. Depending on the pore size, there are more than one phase transition in the adsorption-desorption isotherm. The solid formation in finite pore has been found to be initiated by the presence of the meniscus. Details of the analysis of the extended version of NLDFT will be discussed in the paper.