Hard-sphere Fluid Mixture Research Articles

Calculation of interfacial free energy for binary hard sphere mixtures

In this work, we present a method to calculate interfacial free energy for a binary hard sphere fluid mixture in coexistence with disordered face centered cubic solid. Monte Carlo(MC) Simulations are performed at 0.97 size ratio of small to large particle diameters. Interfacial free energy is calculated across composition for (100),(110) and (111) orientations of the substitutionally disordered solid phase along the coexistence line. Our calculations are based on the concept of Gibbs equimolar dividing surface. First, we calculate the surface excess quantities using the thermodynamic properties obtained from MC simulations then we integrate rewritten equivalent form of Gibbs adsorption equation with respect to pressure using pure hard sphere system interfacial free energy as a reference to get binary hard sphere mixture interfacial free energy. In our simulations we also varied cross sectional areas of solid exposed to fluid mixtures by varying system sizes, the computed interfacial free energies are consistent within error limits and confirms to scaling laws.

Chemical potential of a test hard sphere of variable size in hard-sphere fluid mixtures.

A detailed comparison between the Boublík-Mansoori-Carnahan-Starling-Leland (BMCSL) equation of state of hard-sphere mixtures is made with Molecular Dynamics (MD) simulations of the same compositions. The Labík and Smith simulation technique [S. Labík and W. R. Smith, Mol. Simul. 12, 23-31 (1994)] was used to implement the Widom particle insertion method to calculate the excess chemical potential, βμ0ex, of a test particle of variable diameter, σ0, immersed in a hard-sphere fluid mixture with different compositions and values of the packing fraction, η. Use is made of the fact that the only polynomial representation of βμ0ex which is consistent with the limits σ0 → 0 and σ0 → ∞ has to be of the cubic form, i.e., c0(η)+c¯1(η)σ0/M1+c¯2(η)(σ0/M1)2+c¯3(η)(σ0/M1)3, where M1 is the first moment of the distribution. The first two coefficients, c0(η) and c¯1(η), are known analytically, while c¯2(η) and c¯3(η) were obtained by fitting the MD data to this expression. This in turn provides a method to determine the excess free energy per particle, βaex, in terms of c¯2, c¯3, and the compressibility factor, Z. Very good agreement between the BMCSL formulas and the MD data is found for βμ0ex, Z, and βaex for binary mixtures and continuous particle size distributions with the top-hat analytic form. However, the BMCSL theory typically slightly underestimates the simulation values, especially for Z, differences which the Boublík-Carnahan-Starling-Kolafa formulas and an interpolation between two Percus-Yevick routes capture well in different ranges of the system parameter space.

The isotropic-nematic and nematic-nematic phase transition of binary mixtures of tangent hard-sphere chain fluids: an analytical equation of state.

An analytical equation of state (EoS) is derived to describe the isotropic (I) and nematic (N) phase of linear- and partially flexible tangent hard-sphere chain fluids and their mixtures. The EoS is based on an extension of Onsager's second virial theory that was developed in our previous work [T. van Westen, B. Oyarzún, T. J. H. Vlugt, and J. Gross, J. Chem. Phys. 139, 034505 (2013)]. Higher virial coefficients are calculated using a Vega-Lago rescaling procedure, which is hereby generalized to mixtures. The EoS is used to study (1) the effect of length bidispersity on the I-N and N-N phase behavior of binary linear tangent hard-sphere chain fluid mixtures, (2) the effect of partial molecular flexibility on the binary phase diagram, and (3) the solubility of hard-sphere solutes in I- and N tangent hard-sphere chain fluids. By changing the length bidispersity, two types of phase diagrams were found. The first type is characterized by an I-N region at low pressure and a N-N demixed region at higher pressure that starts from an I-N-N triphase equilibrium. The second type does not show the I-N-N equilibrium. Instead, the N-N region starts from a lower critical point at a pressure above the I-N region. The results for the I-N region are in excellent agreement with the results from molecular simulations. It is shown that the N-N demixing is driven both by orientational and configurational/excluded volume entropy. By making the chains partially flexible, it is shown that the driving force resulting from the configurational entropy is reduced (due to a less anisotropic pair-excluded volume), resulting in a shift of the N-N demixed region to higher pressure. Compared to linear chains, no topological differences in the phase diagram were found. We show that the solubility of hard-sphere solutes decreases across the I-N phase transition. Furthermore, it is shown that by using a liquid crystal mixture as the solvent, the solubility difference can by maximized by tuning the composition. Theoretical results for the Henry's law constant of the hard-sphere solute are in good agreement with the results from molecular simulation.

Calculation of the interfacial free energy of a binary hard-sphere fluid at a planar hard wall

Using molecular-dynamics simulation and Gibbs-Cahn Integration, we calculate the interfacial free energy γ of a binary hard-sphere fluid mixture at a structureless, planar hard wall. The calculation is performed as a function of packing fraction (density) for several values of the diameter ratio α = σ2∕σ1, where σ1 and σ2 are the diameters of the two particle types in the mixture. Our results are compared to those obtained from the bulk version of the White Bear Mark II (WBII) classical density-functional theory, which is a modification of the Fundamental-Measure Theory of Rosenfeld. The WBII bulk theory is shown to be in very good agreement with the simulation results, with significant deviation only at the very highest packing fractions.

Phase separation of binary nonadditive hard sphere fluid mixture confined in random porous media

I analyze the fluid-fluid phase separation of nonadditive hard sphere fluid mixture absorbed in random porous media. An equation of state is derived by using the perturbation theory to this complex system with quenched disorders. The results of this theory are in good agreement with those obtained from semi-grand canonical ensemble Monte Carlo simulations. The contact value of the fluid-fluid radial distribution functions of the reference which is the key point of the perturbation process is derived as well, the comparison against Monte Carlo simulations shows that it has an excellent accuracy.

Demixing in liquids: approaches based on simple equations of state

The demixing transition in partially miscible liquid mixtures is investigated within the framework of the simple van der Waals (vdW)-type equations of state (EOS) approaches. Two approximations for the excluded volume free energy have been considered. One is a modified vdW expression (denoted as vdW here). The other is the well known Mansoori–Carnahan–Starling–Leland (MCSL) excess free energy for hard sphere fluid mixtures (MCSL). In order to adopt an universal representation, we suggest to rescale the physical quantities such as the temperature, pressure and density by the critical parameters of the liquid–vapour (L–V) transition of the monodisperse fluid. We compare phase diagrams from the two theories and the theoretical predictions with available simulation results of binary Lennard–Jones mixtures. For the systems studied, the two theoretical approaches give qualitatively similar L–V and liquid–liquid (L–L) phase diagrams. The MCSL EOS is able to predict in a semi-quantitative way the L–L phase diagram.

Class of consistent fundamental-measure free energies for hard-sphere mixtures

In fundamental-measure theories the bulk excess free-energy density of a hard-sphere fluid mixture is assumed to depend on the partial number densities {ρ(i)} only through the four scaled-particle-theory variables {ξ(α)}, i.e., Φ({ρ(i)})→Φ({ξ(α)}). By imposing consistency conditions, it is proven here that such a dependence must necessarily have the form Φ({ξ(α)})=-ξ(0)ln(1-ξ(3))+Ψ(y)ξ(1)ξ(2)/(1-ξ(3)), where y≡ξ(2)(2)/12πξ(1)(1-ξ(3)) is a scaled variable and Ψ(y) is an arbitrary dimensionless scaling function which can be determined from the free-energy density of the one-component system. Extension to the inhomogeneous case is achieved by standard replacements of the variables {ξ(α)} by the fundamental-measure (scalar, vector, and tensor) weighted densities {n(α)(r)}. Comparison with computer simulations shows the superiority of this bulk free energy over the White Bear one.

Equation of state and jamming density for equivalent bi- and polydisperse, smooth, hard sphere systems

We study bi- and polydisperse mixtures of hard sphere fluids with extreme size ratios up to 100. Simulation results are compared with previously found analytical equations of state by looking at the compressibility factor, Z, and agreement is found with much better than 1% deviation in the fluid regime. A slightly improved empirical correction to Z is proposed. When the density is further increased, excluded volume becomes important, but there is still a close relationship between many-component mixtures and their binary, two-component equivalents (which are defined on basis of the first three moments of the size distribution). Furthermore, we determine the size ratios for which the liquid-solid transition exhibits crystalline, amorphous or mixed system structure. Near the jamming density, Z is independent of the size distribution and follows a -1 power law as function of the difference from the jamming density (Z → ∞). In this limit, Z depends only on one free parameter, the jamming density itself, as reported for several different size distributions with a wide range of widths.

Nonadditive hard-sphere fluid mixtures: A simple analytical theory

We construct a nonperturbative fully analytical approximation for the thermodynamics and the structure of nonadditive hard-sphere fluid mixtures. The method essentially lies in a heuristic extension of the Percus-Yevick solution for additive hard spheres. Extensive comparison with Monte Carlo simulation data shows a generally good agreement, especially in the case of like-like radial distribution functions.

Phase coexistence in the hard-sphere Yukawa chain fluid with chain length polydispersity: High temperature approximation

High temperature approximation (HTA) is used to describe the phase behavior of polydisperse Yukawa hard-sphere chain fluid mixture with chain length polydispersity. It is demonstrated that in the frames of the HTA the model belongs to the class of ‘truncatable free energy models’, i.e. the models with thermodynamical properties defined by the finite number of generalized moments. Using this property we were able to calculate the complete phase diagram (i.e. cloud and shadow curves as well as binodals) and chain length distribution functions of the coexisting phases.

A semi-empirical hard-sphere chain equation of state: Pure and mixture

The thermodynamic perturbation dimer theory (TPT-D) is modified by scaling the site–site correlation function at contact for hard-dimers using the contact value of site-site correlation function for hard-spheres. With the exception of a constant, the resulting modified TPT-D is identical with the thermodynamic perturbation theory (TPT). The modified TPT-D, however, predicts the Monte Carlo (MC) compressibility factors for hard-sphere chains more accurately than TPT. The average absolute percent deviations (%AAD) was found to be 2.75 for the modified TPT-D and 6.24 for TPT. The modified TPT-D was then extended to mixtures of hard-sphere chain fluids. The modified TPT-D accurately predicts compressibility factors for binary mixtures of hard-sphere chains or mixtures containing hard-sphere chain and hard-sphere fluids. When the theories were used to predict compressibility factors for 89 mixtures containing hard-spheres and hard-sphere chains with different lengths and number of segments the %AAD was found to be 1.49 for the modified TPT-D and 2.78 for TPT.

Adsorption of a Binary Mixture of Adhesive Fluids in Planar Pores: A Monte Carlo Study

Grand canonical Monte Carlo simulation is used to study the adsorption of a binary sticky hard-sphere fluid mixture in planar pores. The wall-component 1 and wall-component 2 contact densities are determined to calculate the pressure as a function of the composition of the mixture and the separation between the walls. From these data dependence of the solvation force between the plates on pore width is estimated. The simulation results are compared with the predictions of the Percus-Yevick approximation for planar pores. The density profiles of particular components show interesting shapes stemming from the interplay between the steric effects and the competitive adhesion among all possible species pairs. It is shown that narrowing of the pore causes selective partitioning of individual components of the mixture between the bulk phase and the interior of the pore. The agreement between the two methods is better at wider pores and for the component comprised of weakly adhesive particles.

Phase coexistence in polydisperse multi-Yukawa hard-sphere fluid: High temperature approximation

High temperature approximation (HTA) is used to describe the phase behavior of polydisperse multi-Yukawa hard-sphere fluid mixtures. It is demonstrated that in the frames of the HTA the model belongs to the class of "truncatable free energy models," i.e., the models with thermodynamical properties (Helmholtz free energy, chemical potential, and pressure) defined by the finite number of generalized moments. Using this property we were able to calculate the complete phase diagram (i.e., cloud and shadow curves as well as binodals) and size distribution functions of the coexisting phases of several different models of polydisperse fluids. In particular, we consider polydisperse one-Yukawa hard-sphere mixture with factorizable Yukawa coefficients and polydisperse Lennard-Jones (LJ) mixture with interaction energy parameter and/or size polydispersity. To validate the accuracy of the HTA we compare theoretical results with previously published results of more advanced mean spherical approximation (MSA) for the one-Yukawa model and with the Monte Carlo (MC) computer simulation results of [Wilding et al. J. Chem. Phys. 121, 6887 (2004); Phys. Rev. Lett. 95, 155701 (2005)] for the LJ model. We find that overall predictions of the HTA are in reasonable agreement with predictions of the MSA and MC, with the accuracy range from semiquantitative (for the phase diagram) to quantitative (for the size distribution functions).

Viscosity of Alternative Refrigerants R407C and R407E in the Vapor Phase from 298.15 to 423.15 K

This paper presents new measurements of the viscosity of gaseous R407C (23 mass% HFC-32, 25 mass% HFC-125, 52 mass% HFC-143a) and R407E (25 mass% HFC-32, 15 mass% HFC-125, 60 mass% HFC-143a). The measurements were carried out with an oscillating-disk viscometer of the Maxwell type at temperatures from 298.15 to 423.15 K. The densities of these two fluid mixtures were calculated with the equation-of-state model in REFPROP. The viscosity at normal pressures was analyzed with the extended law of corresponding states developed by Kestin et al., and the scaling parameters needed in the analysis were obtained from our previous studies for the viscosity of the binary mixtures consisting of HFC-32, HFC-125, and HFC-134a. The modified Enskog theory developed by Vesovic and Wakeham (V-W method) was applied to predict the viscosity for the ternary gaseous HFC mixtures under pressure. As for the calculation of pseudo-radial distribution functions in mixtures, a method based on the equation of state for hard-sphere fluid mixtures proposed by Carnahan-Starling was applied. It was found that the V-W method can predict the viscosity of R407C and R407E without any additional parameters for the ternary mixture.

Solution of the mean spherical approximation for polydisperse multi-Yukawa hard-sphere fluid mixture using orthogonal polynomial expansions

The Blum-Hoye [J. Stat. Phys. 19 317 (1978)] solution of the mean spherical approximation for a multicomponent multi-Yukawa hard-sphere fluid is extended to a polydisperse multi-Yukawa hard-sphere fluid. Our extension is based on the application of the orthogonal polynomial expansion method of Lado [Phys. Rev. E 54, 4411 (1996)]. Closed form analytical expressions for the structural and thermodynamic properties of the model are presented. They are given in terms of the parameters that follow directly from the solution. By way of illustration the method of solution is applied to describe the thermodynamic properties of the one- and two-Yukawa versions of the model.

Deriving the equation of state of additive hard-sphere fluid mixtures from that of a pure hard-sphere fluid

We have analyzed the rate of convergence of a series expansion, in terms of the density, of the ratio of the excess compressibility factor of fluid mixtures of additive hard spheres to that of a pure hard-sphere fluid with the same reduced density. The terms in the series can be obtained from the virial coefficients. We have found that the series converges quickly, so that frequently the knowledge of the first two terms of the series, that can be obtained from the second and third virial coefficients which are known analytically, is sufficient to provide an accurate equation of state.

Mapping a hard-sphere fluid mixture onto a single component hard-sphere fluid

The possibility of obtaining the thermodynamic and structural properties of a binary additive hard-sphere fluid mixture on the basis of the corresponding properties of a suitable single-component hard-sphere fluid is analyzed. To this end, Monte Carlo simulations have been performed for binary mixtures of hard spheres for different densities, compositions and diameter ratios in order to obtain the compressibility factor Z and the partial radial distribution functions g ij ( r ) for pairs ij of the mixtures. These data are used to test the reliability of different proposals available in the literature for mapping the thermodynamic and structural properties of conformal mixtures onto those of a single-component fluid. It is found that, while the averaged radial distribution function and the equation of state of the mixture can be reasonably well reproduced by means of those of an equivalent single-component fluid, the partial radial distribution functions cannot be obtained with enough accuracy from the radial distribution function of the equivalent fluid. A possible explanation for this fact is suggested.

Computer simulation of chemical potentials of ternary hard-sphere fluid mixtures

A new version of the scaled particle Monte Carlo (SP-MC) computer simulation method is proposed. It is used to determine chemical potentials of components of ternary mixtures of additive hard-sphere fluids at several densities and compositions. The results are used to test the chemical potentials given by two literature equations of state (EOS). It has been found that the simulation and EOS’s results are in very good agreement with the new data in this work.

Viscosity of Gaseous Mixtures of HFC-134a (1,1,1,2-Tetrafluoroethane)+HFC-32 (Difluoromethane)

This paper reports experimental results for the viscosity of gaseous mixtures of HFC-134a (1,1,1,2-tetrafluoroethane)+HFC-32 (difluoromethane). The measurements were carried out with an oscillating-disk viscometer of the Maxwell type at temperatures from 298.15 to 423.15 K. The viscosity was measured for three mixtures containing 25.00, 52.40, and 74.98 mole% HFC-134a in HFC-32. Experimental results for the viscosity at normal pressures show a minimum as plotted against mole fraction in the higher temperature region, which may be the first experimental observation of the minima for dilute binary gaseous mixtures of HFCs. The viscosity at normal pressures was analyzed with the extended law of corresponding states developed by Kestin et al., and the scaling parameters were obtained for unlike-pair interactions between HFC-32 and HFC-134a. The modified Enskog theory developed by Vesovic and Wakeham was applied to predict the viscosity for the binary gaseous mixtures under pressure. As for the calculation of pseudo-radial distribution functions in mixtures, a method based on the equation of state for hard-sphere fluid mixtures proposed by Carnahan–Starling was applied.

A new efficient method for density functional theory calculations of inhomogeneous fluids

The accurate computation of the effects of solvation on chemical systems can be done using density functional theories (DFT) for inhomogeneous multicomponent fluids. The DFT models of interest are non-local theories which accurately treat hard-sphere fluid mixtures; attractive inter-particle potentials (Lennard–Jones) are added as perturbations. In this paper, we develop and demonstrate a new efficient method for an accurate non-local DFT. The method described here differs from previous work in the use of fast fourier transform (FFT) methods to carry out the convolutions. As with our previous real space work (J. Comput. Phys. 159(2) (2000) 407, 425), we demonstrate that the Fourier space approach can be solved with a Newton–GMRES approach; however, we now employ a very efficient matrix-free algorithm. A simple but effective preconditioner is presented. The method is demonstrated with calculations performed for one-, two-, and three-dimensional systems, including problems with single and multicomponent fluids. Timing comparisons with previous implementations are given.