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.     

Van der Waals model and equation of state of nitrobenzene-alkanes solutions near the critical consolute temperature

On the basis of the fluctuation theory of phase transitions a system close to the critical point is the ideal gas of order parameter fluctuations. An extended equation of state for binary solutions close to the critical consolute temperature has been proposed taking into account the properties of a real Van der Waals gas in this model. This equation has been used to analyze the temperature dependences of the concentrations of a series of nitrobenzene + alkane binary solutions in terms of different order parameters. It has been shown that the molar concentration of the solution should be used as an order parameter of the analyzed systems. It has been determined that the parameters of the extended equation of state are linear functions of (a) the number of carbon atoms in alkanes and (b) the compressibility factor of the solution components.     

Vapor-liquid equilibrium of hexadecapolar fluids from a perturbation-based equation of state

In this work a numerically tractable expression for the interaction potential between two point hexadecapoles with octahedral symmetry and a molecular-based equation of state derived by perturbation theory for hexadecapolar fluids are presented. The polar system is modeled by square-well particles with a point hexadecapole with octahedral symmetry at their centers. This equation of state is analytical in the state variables and in the potential parameters and allows us to study the effects of the hexadecapolar moment strength on the thermodynamic properties and liquid-vapor phase diagram. The equation presented here is applied to the thermodynamics of sulfur hexafluoride and gives very good predictions for the saturation pressures and the vapor-liquid phase diagram.     

Universal behavior of the enthalpy of vaporization: an empirical equation

Values of the enthalpy of vaporization from the critical to the triple point are correlated by an empirical equation. The equation contains parameters which characterize each substance: the critical and triple point temperatures and the enthalpy of vaporization at the triple point, and for all substances the same universal critical ratio. This work suggests that a wide class of fluids with the exception of quantal liquids shows an universal behavior along the coexistence curve.     

A theoretical investigation on the honeycomb potential fluid

A local self-consistent Ornstein-Zernike (OZ) integral equation theory (IET) is proposed to provide a rapid route for obtaining thermodynamic and structural information for any thermodynamically stable or metastable state points in the bulk phase diagram without recourse to traditional thermodynamic integration, and extensive NVT-Monte Carlo simulations are performed on a recently proposed honeycomb potential in three dimensions to test the theory's reliability. The simulated quantities include radial distribution function (rdf) and excess internal energy, pressure, excess chemical potential, and excess Helmholtz free energy. It is demonstrated that (i) the theory reproduces the rdf very satisfactorily only if the bulk state does not enter deep into a two phases coexistence region; (ii) the excess internal energy is the only one of the four thermodynamic quantities investigated amenable to the most accurate prediction by the present theory, and the simulated pressure is somewhat overestimated by the theoretical calculations, but the deviation tends to vanish along with rising of the temperature; (iii) using the structural functions from the present local self-consistent OZ IET, a previously derived local expression, due to the present author, achieves even a higher accuracy in calculating for the excess chemical potential than the exact virial pressure formula for the pressure, and the resulting excess Helmholtz free energy is in surprisingly same with the simulation results due to offset of the errors. Based on the above observations, it is suggested that it may be a good procedure to integrate the theoretical excess internal energy along the isochors to get the excess Helmholtz free energy, which is then fitted to a polynomial to be used for calculation of all of other thermodynamic quantities in the framework of the OZ IET.     

Global fluid phase equilibria and critical phenomena of selected mixtures using the crossover soft-SAFT equation

We present here the extension of the crossover soft-statistical associating fluid theory (soft-SAFT) equation of state to mixtures, as well as some illustrative applications of the methodology to mixtures of particular scientific and technological interest. The procedure is based on White's work (White, J. A. Fluid Phase Equilib. 1992, 75, 53) from the renormalization group theory, as for the pure fluids, with the isomorphism assumption applied to the mixtures. The equation is applied to three groups of mixtures: selected mixtures of n-alkanes, the CO2/n-alkane homologous series, and the CO2/1-alkanol homologous series. The crossover equation is first applied to the pure components of the mixtures, CO2 and the 1-alkanol family, while an available correlation is used for the molecular parameters of the n-alkane series (Llovell et al. J. Chem. Phys 2004, 121, 10715). A set of transferable molecular parameters is provided for the 1-alkanols series; these are accurate for the whole range of thermodynamic conditions. The crossover soft-SAFT equation is able to accurately describe these compounds near to and far from the critical point. The theory is then used to represent the phase behavior and the critical phenomena of the selected mixtures. We use binary interaction parameters xi and eta for dissimilar mixtures. These parameters are fitted at some particular conditions (one subcritical temperature or binary critical data) and used to predict the behavior of the mixture at different conditions (other subcritical conditions and/or critical conditions). The equation is able to capture the continuous change in the critical behavior of the CO2/n-alkane and the CO2/1-alkanol homologous series as the chain length of the second compound increases. Excellent agreement with experimental data is obtained, even in the most nonideal cases. The new equation is proved to be a powerful tool to study the global phase behavior of complex systems, as well as other thermodynamic properties of very challenging mixtures.     

A perturbed hard-sphere-chain equation of state for mixtures: Prediction from critical point constants

A modified perturbed hard-sphere-chain equation of state by Eslami [H. Eslami, Fluid Phase Equilibr. 216 (2004) 21-26], is extended to mixtures. The resulting equation of state for mixtures consists of two temperature-dependent parameters as well as an additional parameter, reflecting the segment size for pure components. The temperature-dependent parameters of the equation of state are correlated as universal functions of the reduced temperature. It is shown that knowing just the critical constants of pure components is sufficient to calculate the temperature-dependent parameters. The equation of state for mixtures is checked against the experimental pressure-volume-temperature data for a large number of mixtures, having varieties of molecular sizes and shapes. It is shown that no interaction parameter is needed to describe the behavior of fluid mixtures. Among about 3500 data points for mixtures, the average absolute deviation, compared to the experimental data, is about 0.93%.     

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.     

Solutions of the Klein–Gordon equation with the improved Tietz potential energy model

We present the relativistic rotation–vibrational energy equation of a diatomic molecule which moves under the improved Tietz potential energy model in higher spatial dimensions. The nonrelativistic limits of the bound state solutions of the Klein–Gordon equation are the bound state solutions of the Schrödinger equation with the same potential energy function. Numerical analysis results show that there exists a critical point around which the solution behaviors bifurcate into two extreme cases. Below the critical point, the behavior of the relativistic vibrational energies for the ground electronic state of carbon monoxide in higher dimensions keeps similar to that of the three-dimensional system, while this symmetry phenomenon breaks and the Klein–Gordon equation has no stability solution upon the critical point.     

Corridor of existence of thermodynamically consistent solution of the Ornstein-Zernike equation

We obtain the exact equation for a correction to the Ornstein-Zernike (OZ) equation based on the assumption of the uniqueness of thermodynamical functions. We show that this equation is reduced to a differential equation with one arbitrary parameter for the hard sphere model. The compressibility factor within narrow limits of this parameter variation can either coincide with one of the formulas obtained on the basis of analytical solutions of the OZ equation or assume all intermediate values lying in a corridor between these solutions. In particular, we find the value of this parameter when the thermodynamically consistent compressibility factor corresponds to the Carnahan-Stirling formula.     

Solution of the compressibility equation of the adhesive hard-sphere model for mixtures

The solution of the generalized equations of Percus and Yevick is obtained for mixtures of molecules interacting via the adhesive hard-sphere potential. It is shown that the possibility of a discontinuous phase or fluid-fluid transition appears only in such binary systems where the adhesion acts between like particles of at least one of the components. The found distribution functions are used to obtain from the compressibility equation the expression for the pressure of a mixture consisting of v components in arbitrary concentration. The pressure, chemical potential and other thermodynamic properties are calculated explicitly in the limit when several components (the solutes) are dilute and one component (the solvent) is in its “liquid” phase. The solubility of substances of small molecules is shown to increase when the temperature T rises. The same regularity is found in the case when the interaction between solvent and solute consists only of a hard-sphere potential. In the general case of the presence of adhesion between solvent and solute molecules and for arbitrary ratio of particles sizes a minimum appears on the solubility cuves versus T. If the attraction is sufficiently big the drop in the solubility may be very sharp and reach the value zero at a certain temperature. The results obtained are qualitatively supported by examples from experiment.     

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.     

Casimir amplitudes and capillary condensation of near-critical fluids between parallel plates: renormalized local functional theory

We investigate the critical behavior of a near-critical fluid confined between two parallel plates in contact with a reservoir by calculating the order parameter profile and the Casimir amplitudes (for the force density and for the grand potential). Our results are applicable to one-component fluids and binary mixtures. We assume that the walls absorb one of the fluid components selectively for binary mixtures. We propose a renormalized local functional theory accounting for the fluctuation effects. Analysis is performed in the plane of the temperature T and the order parameter in the reservoir ψ(∞). Our theory is universal if the physical quantities are scaled appropriately. If the component favored by the walls is slightly poor in the reservoir, there appears a line of first-order phase transition of capillary condensation outside the bulk coexistence curve. The excess adsorption changes discontinuously between condensed and noncondensed states at the transition. With increasing T, the transition line ends at a capillary critical point T=T(c) (ca) slightly lower than the bulk critical temperature T(c) for the upper critical solution temperature. The Casimir amplitudes are larger than their critical point values by 10-100 times at off-critical compositions near the capillary condensation line.     

‘Exact’ integral equation theory and local formulation for excess thermodynamic properties of hard spheres

A hybrid hard sphere bridge function is proposed, which, in combination with the standard Ornstein–Zernike integral equation, can predict extremely accurately hard sphere compressibility, virial pressure, and correlation function. Second, a local formulation for determination of excess chemical potential is derived out, which, in combination with the present hybrid hard sphere bridge function and OZ integral equation, can predict the excess chemical potential also extremely accurately. The resultant excess entropy is in excellent agreement with that from the Carnahan–Starling equation of state. The present formalism performs excellently over the whole density range, i.e. from zero to freezing density, and is largely superior to a formalism available in the literature.     

Steady-state nucleation rate and flux of composite nucleus at saddle point

The steady-state nucleation rate and flux of composite nucleus at the saddle point is studied by extending the theory of binary nucleation. The Fokker-Planck equation that describes the nucleation flux is derived using the Master equation for the growth of the composite nucleus, which consists of the core of the final stable phase surrounded by a wetting layer of the intermediate metastable phase nucleated from a metastable parent phase recently evaluated by Iwamatsu [J. Chem. Phys. 134, 164508 (2011)]. The Fokker-Planck equation is similar to that used in the theory of binary nucleation, but the non-diagonal elements exist in the reaction rate matrix. First, the general solution for the steady-state nucleation rate and the direction of nucleation flux is derived. Next, this information is then used to study the nucleation of composite nucleus at the saddle point. The dependence of steady-state nucleation rate as well as the direction of nucleation flux on the reaction rate in addition to the free-energy surface is studied using a model free-energy surface. The direction of nucleation current deviates from the steepest-descent direction of the free-energy surface. The results show the importance of two reaction rate constants: one from the metastable environment to the intermediate metastable phase and the other from the metastable intermediate phase to the stable new phase. On the other hand, the gradient of the potential Φ or the Kramers crossover function (the commitment or splitting probability) is relatively insensitive to reaction rates or free-energy surface.     

A theoretical form of the Martin-Hou equation of state

A new equation of state is derived from the Barker-Henderson hard-sphere perturbation theory. It has the form similar to the Martin-Hou equation of state. The numerical values of the characteristic constants in the equation can be calculated by the method of Martin and Hou. The equation can be used to predict P-V-T properties accurately for fluids when the critical parameters (T_c, P_c and V_c) and one point on the vapor pressure cure are given. By using the functional relationships between the characteristic constants and the microscopic parameters, the molecular microscopic parameters of the substance can be obtained.     

Effect of electrolytes on the critical temperature of separation and physicochemical properties of binary liquid systems

The effect ions have on the equilibrium and kinetic properties of solutions near the critical temperature of separation is studied. From an analysis of the experimental data obtained in the work and from the literature it is shown that adding ions to a solution increases the correlation length of the system and changes the magnitude of the fluctuation part of the thermodynamic potential and the forces of interaction among the order parameter fluctuations in the vicinity of the critical point. This changes the parameters of the extended equation of state, increases the viscosity of substances, and lowers the coefficient of volume expansion of the system.     

A semi-empirical equation of state applied to vapor-liquid equilibria calculations

A five-parameter equation of state is proposed to calculate the vapor-liquid equilibria of compounds in binary and multicomponent mixtures. This equation is closely related to a previous equation of state proposed by the author, the main modification being in the entropic term where the parameter m assumes a constant value for all compounds. It is shown that the van der Waals conditions at the critical point and the Morbidelli-Carra' algorithm enable the calculation of three other constants. Rules are given to calculate the remaining constant K which pertains to the enthalpic term. The proposed method only requires knowledge of the critical constants and of the normal boiling temperature as input parameters. A wide application of the new equation to both polar and non-polar binary systems indicates the following: the proposed method is predictive for ideal or nearly ideal mixtures; the correlation of mixtures of hydrocarbons having very different molar volumes can be obtained by optimizing only the binary interaction parameter linked to the enthalpic term; the new equation also correlates well with strongly non-ideal systems which exhibit a miscibility gap; the prediction of multicomponent vapor-liquid equilibria from the binary data alone is also reliable for both polar and non-polar mixtures.