On the definition of a Monte Carlo model for binary crystal growth

We show that consistency of the transition probabilities in a lattice Monte Carlo (MC) model for binary crystal growth with the thermodynamic properties of a system does not guarantee the MC simulations near equilibrium to be in agreement with the thermodynamic equilibrium phase diagram for that system. The deviations remain small for systems with small bond energies, but they can increase significantly for systems with large melting entropy, typical for molecular systems. These deviations are attributed to the surface kinetics, which is responsible for a metastable zone below the liquidus line where no growth occurs, even in the absence of a 2D nucleation barrier. Here we propose an extension of the MC model that introduces a freedom of choice in the transition probabilities while staying within the thermodynamic constraints. This freedom can be used to eliminate the discrepancy between the MC simulations and the thermodynamic equilibrium phase diagram. Agreement is achieved for that choice of the transition probabilities yielding the fastest decrease of the free energy (i.e., largest growth rate) of the system at a temperature slightly below the equilibrium temperature. An analytical model is developed, which reproduces quite well the MC results, enabling a straightforward determination of the optimal set of transition probabilities. Application of both the MC and analytical model to conditions well away from equilibrium, giving rise to kinetic phase diagrams, shows that the effect of kinetics on segregation is even stronger than that predicted by previous models.     

Thermodynamically consistent closure approximation for hard spheres systems

We present a new closure relation that is an extension of the Percus-Yevick approximation. In the proposed closure, we introduce an additional term and a mixing coefficient that can be determined by imposing a condition of thermodynamic self-consistency. Moreover, the mixing coefficient is calculated analytically within a linear approximation. In the case of a monodisperse system of hard spheres, we compare the results of our model to well-established thermodynamic expressions and also to the structural properties of fairly known closure approximations. In the second case, and using an equivalent scheme, the new closure relation is extended to the depletion potential between two large hard spheres immersed in a liquid of small hard spheres. In both cases, the results of our model are in good agreement with numerical simulations performed at intermediate concentrations.     

Cluster-variation approximation for a network-forming lattice-fluid model

We consider a three-dimensional lattice model of a network-forming fluid, which has been recently investigated by Girardi et al. by means of Monte Carlo simulations [J. Chem. Phys. 126, 064503 (2007)], with the aim of describing water anomalies. We develop an approximate semianalytical calculation, based on a cluster-variation technique, which turns out to reproduce almost quantitatively different thermodynamic properties and phase transitions determined by the Monte Carlo method. Nevertheless, our calculation points out the existence of two different phases characterized by long-range orientational order, and of critical transitions between them and to a high-temperature orientationally disordered phase. Also, the existence of such critical lines allows us to explain certain "kinks" in the isotherms and isobars determined by the Monte Carlo analysis. The picture of the phase diagram becomes much more complex and richer, though unfortunately less suitable to describe real water.     

Power laws for the establishment of a stationary concentration in a binary solution droplet growing in a vapor-gas medium

The kinetic laws are found for establishing a stationary concentration in solution droplets growing under diffusion and free-molecular regimes in a vapor-gas medium composed of two condensing vapors and a passive gas. Parameters of these laws are related to the thermodynamic and kinetic characteristics of condensing substances. Original Russian Text ? F.M. Kuni, A.A. Lezova, 2009, published in Kolloidnyi Zhurnal, 2009, Vol. 71, No. 4, pp. 563–565.     

Evaporation dynamics of a binary sessile droplet: Theory and comparison with experimental data on a droplet of a sulfuric-acid solution

Diffusion evaporation of a sessile binary droplet in an atmosphere of a noncondensable carrier gas has been considered. For a droplet consisting of two infinitely miscible liquids, a relation between the current values of solution concentration and volume of the droplet has been derived in an explicit form under the ideal solution approximation. It has been shown that the volume of a sessile binary droplet may, as well as the volume of a free binary droplet, vary nonmonotonically with time. The evaporation of a droplet of an aqueous sulfuric-acid solution has been considered in detail taking into account the nonideality of the solution. Time variations in the volume, base area, and contact angle have been experimentally measured for the sessile droplet of an aqueous sulfuric-acid solution on a hydrophobized substrate. The experimental data obtained at different initial humidities of water-vapor and droplet-solution concentrations have been analyzed within the theory of the stationary isothermal diffusion evaporation of a sessile binary droplet.     

A cell model of a liquid droplet

An expression for the free energy of a droplet composed of attracting hard spheres is found using a simple cell model. The hard-sphere repulsion is assumed to act only between molecules in the same cell, whereas attraction extends over many cells. A maximum term analysis gives rise to a mean-field free energy which includes terms proportional to the first and second power of the droplet radius R with coefficients which can be related to the planar surface tension and Tolman length. Certain Gaussian fluctuations about the maximum term are also considered, corresponding to droplet translation and capillary wave fluctuations. Inclusion of these fluctuations is necessary to ensure that the nucleation rate is proportional to the system volume. They also reduce the planar surface tension and introduce a logarithmic term, -4/3 ln R, into the free energy. The inclusion of other fluctuations and the relationship between these equations and those arising in density-functional theories of nucleation is discussed.     

The Born closure approximation for the scattering amplitude of an electron-molecule collision

When the fixed-nuclei (FN) approximation is applied to the calculation of electron scattering from a polar molecule, the resulting cross section diverges in the forward direction of scattering. This is due to the long-range nature of the interaction between the electron and the molecular dipole. To avoid this difficulty, a hybrid method is proposed for the calculation of the scattering amplitude. This method is based on the FN approximation for a close collision and the Born approximation for a distant collision. The present paper describes the detailed formulation of the method for practical applications. Furthermore the present approach is extended to other long-range interactions (due to quadrupole moment and/or polarization effect) and to a dipole-allowed vibrational excitation. In these cases, while no divergence occurs, it is often difficult to confirm the convergence of the partial-wave expansion. With the employment of the present approach, it is much easier to confirm the convergence and hence to obtain reliable cross sections. The formulas are given for diatomic molecules as well as for polyatomic ones. Received: 12 June 2000 / Accepted: 6 July 2000 / Published online: 24 October 2000     

Kinetics of binary condensation on the droplet growing in diffusion regime

A relaxation equation determining the regular tendency of the concentration of binary solution in the growing droplet to the stationary value, at which there is a self-similar solution to the problem of the condensation in a binary mixture, is derived. An analytical solution of the relaxation equation is obtained and it is demonstrated that the stationary value of concentration is achieved via the power law. The time interval that elapses from the emergence of the droplet until the diffusion regime of droplet growth and derived relaxation equation become effective is revealed. The stationary value of concentration is found for the model of ideal solution.     

Nonstationary vapor concentration fields near the growing droplet of binary solution

Nonstationary vapor concentration fields near the droplet of binary solution growing in the vapor—gas mixture are revealed using the concepts of similarity. The revealed fields are determined with the exact account of the motion of droplet surface and refer to the times at which the droplet reaches sizes that provide for the diffusion regime of droplet growth. To obtain the self-similar solution of the problem of binary condensation, it is necessary to ensure a constant (in time) concentration of binary solution in the growing droplet. The velocities of an increase in the number of molecules and the radius of two-component droplet with time are found with allowance for the equation ensuring this solution. The conditions for the transformation of the self-similar solution of the problem of the condensation of two-component mixture into the solution, which was derived previously for the condensation of one component, are elucidated.     

Improved binary parameters using GA for multi-component aromatic extraction: NRTL model without and with closure equations

Application of genetic algorithm (GA), which leads to globally optimal binary interaction parameters from multi-component liquid–liquid equilibrium data, has been recently demonstrated for some ternary, quaternary and quinary systems. The binary interaction parameters are related to each other through the closure equations. In this work, the binary interaction parameters based on non-random two liquid (NRTL) activity coefficient model have been estimated using GA, without and with closure equations for 65 multi-component aromatic extraction systems: 53 ternary, 9 quaternary and 3 quinary systems. Parameters that satisfy the closure equations exhibit better root mean square deviations than those that do not satisfy the closure equations. Root mean square deviation value without implementation of closure equations is 0–80% better than literature as compared to 0–90% better with implementation of closure equations.     

Analysis of the equilibrium droplet shape based on an ellipsoidal droplet model

The extent of a droplet's spreading over a flat, smooth solid substrate and its equilibrium height in the presence of gravity are determined approximately, without a numerical solution of the governing nonlinear differential equation, by assuming that the droplet takes on the shape of an oblate spheroidal cap and by minimizing the corresponding free energy. The comparison with the full numerical evaluations confirms that the introduced approximation and the obtained results are accurate for contact angles below about 120° and for droplet sizes on the order of the capillary length of the liquid. The flattening effect of gravity is to increase the contact radius and decrease the height of the droplet, with these being more pronounced for higher values of the Bond number.     

Mean spherical model approximation for the primitive model of sodium chloride

A calculation of the excess internal energy and the osmotic coefficient for a mixture of charged hard spheres of diameters equal to the ionic radii of NaCl, is done in the mean spherical approximation. The results are compared to the best available data, provided by the hypernetted chain theory. The agreement is better for the osmotic coefficient than for the internal energy, and improves at higher concentrations.     

On the solidification of a supercooled liquid droplet lying on a surface

We model the solidification and subsequent cooling of a supercooled liquid droplet that is lying on a cold solid substrate after impact. It is assumed that solidification occurs for a given fixed droplet shape. The shapes used by the model are a sphere, truncated spheres, and an experimentally registered droplet shape. The freezing process is conduction-dominant and is modeled as a one-phase Stefan problem. This moving boundary problem is reformulated with the enthalpy method and then solved numerically with an implicit finite-difference technique. The numerical results for the simple case of a spherical droplet touching a surface are similar to those of a freely freezing spherical droplet and are well confirmed by the 1D asymptotic analytical model of Feuillebois et al. (J. Colloid Interface Sci. 169 (1995) 90). A freezing water droplet is considered as an example. The numerical results for full freezing time, subsequent cooling time, and last freezing point coordinate for the various droplets shapes are fitted by analytical functions depending on supercooling, thermal resistance of the target surface (expressed by Biot number), and spreading parameter. These functions are proposed for direct application, thus avoiding the need to solve the full freezing and cooling problem.     

A thermodynamic approximation for the estimation of velocity correlation coefficients in binary liquid mixtures

An equation is derived using the relations of equilibrium thermodynamics to estimate limiting distinct diffusion coefficients, (or velocity cross-correlation coefficients) in binary non-electrolyte mixtures. It is tested against 13 mixtures. Good agreement is obtained with distinct diffusion coefficients derived experimentally in the case of near deal systems. For non-ideal mixtures the calculated values appear to follow the experimental coefficients to some extent. For this reason, the thermodynamic approximation is not as suitable for a reference equation as a structural approximation previously derived.     

Stationary concentration establishment in the growing or evaporating droplet of ideal binary solution

The establishment of stationary solution concentration in a growing or evaporating droplet of an ideal binary solution (binary droplet) placed in a vapor mixture of constituting substances and passive gas is described analytically. Relations defining time dependences of solution concentration in a droplet, the number of molecules of each constituting component, and droplet radius are derived at known parameters of the vapor-gas mixture and the initial composition of a binary droplet. The results of calculations of time dependences of aforementioned values are reported for several variants of the initial composition of a droplet and a vapor mixture.     

Thermal effects accompanying stationary binary condensation of vapors into overcritical droplet

A set of equations is derived to calculate the stationary temperature and concentration of a solution in a overcritical droplet with regard to the heat release accompanying the condensation of a binary mixture of vapors in a diffusion or free-molecular regime. In the approximation of an ideal solution, relations are found for the stationary temperature of droplet growing under the conditions of strong and weak thermal effects. For the general case and the cases of strong and weak thermal effects, the temperature and concentration of the droplet and the coefficient of the thermal deceleration of the droplet growth are calculated as functions of the density of a passive gas. The influence of the condensation heat values of the first and second components of the mixture on the stationary temperature and concentration of the solution in the growing droplet is investigated separately.     

Simple electrostatic model for the I — A approximation

Employing the approximation which replaces r _ij ^–1 by (r_i+r_j)^–1 we derive the relation 2/4l+5=I–A where is the orbital exponent of a Slater AO. This semiempirical relation appears to agree with experiment for the cases compared.     

On the distorted wave approximation for chemical reactions

Various procedures for constructing transition amplitudes for chemical reactions in the distorted wave approximation are demonstrated. Calculations are performed for the collinear reaction H+H₂(ν = 0) → H₂(ν = 0) + H at low energies. Comparison is made with exact results after unitarization is invoked through (1) exponential unitarization and (2) the Condon approximation. In the threshold region the Condon approximation gives quite reasonable agreement with exact results.     

On a scheme for the calculations of large molecules in the molecular orbital approximation

The possibility of decreasing the number of molecular integrals to be calculated by using the same basis of high symmetry for all molecules is discussed.     

On the approximation of potential-energy functions for two-center systems

Let E(R) be a potential-energy function for a neutral or ionic diatom in the Born-Oppenheimer approximation. Then the approximation of E(R) for 0 < R < ∞ starting from finite and typically small sets of given information is considered. The approach is based on the fact that the scaled potential curves F(R) = R²E(R) derive from an eigenvalue problem which depends linearly on R. The nature of the curves F(R) is examined in detail. The results include the discovery of various approximants, some of which display rigorous bounding properties and others, closely related, whose behavior with respect to the approximated function appears to be predictable.