Carnahan-Starling Equation Research Articles

Development of the semi-empirical equation of state for square-well chain fluid based on the Statistical Associating Fluid Theory (SAFT)

A semi-empirical equation of state for the freely jointed square-well chain fluid is developed. This equation of state is based on Wertheim’s thermodynamic perturbation theory (TPT) and the statistical associating fluid theory (SAFT). The compressibility factor and radial distribution function of square-well monomer are obtained from Monte Carlo simulations. These results are correlated using density expansion. In developing the equation of state the exact analytical expressions are adopted for the second and third virial coefficients for the compressibility factor and the first two terms of the radial distribution function, while the higher order coefficients are determined from regression using the simulation data. In the limit of infinite temperature, the present equation of state and the expression for the radial distribution function are represented by the Carnahan-Starling equation of state. This semi-empirical equation of state gives at least comparable accuracy with other empirical equation of state for the square-well monomer fluid. With the new SAFT equation of state from the accurate expressions for the monomer reference and covalent terms, we compare the prediction of the equation of state to the simulation results for the compressibility factor and radial distribution function of the square-well monomer and chain fluids. The predicted compressibility factors for square well chains are found to be in a good agreement with simulation data. The high accuracy of the present equation of state is ascribed to the fact that rigorous simulation results for the reference fluid are used, especially at low temperatures and low densities.

Assessment of S(0,φ) from multiply scattered light

Two-wavelength measurements of isotropic scattering coefficient in concentrated, multiply scattering colloidal suspensions are used to extract S(0,φ) values, using a small wave number expansion of the interparticle structure factor, S(q,φ). Owing to the small particle size (150 nm) and near-infrared wavelengths (670–820 nm) used in this study, the approximation is reasonable and allows accurate estimation of S(0,φ). The estimated values of S(0,φ) agree well with that predicted using the Carnahan–Starling equation of state.

Closed-loop critical curves in simple hard-sphere van der Waals-fluid models consistent with the packing fraction limit

Two new hard-sphere equations are proposed which, in combination with a van der Waals attraction term, lead to a biquadratic, respectively a cubic, equation of state. The new equations show the correct limiting behavior at low as well as at high densities; their poles are close to the physical packing fraction of hard spheres. Both equations of state were extended towards mixtures by one-fluid mixing rules, and their global phase behavior was investigated for the special case of equal-sized molecules. Both equations are able to predict closed-loop liquid–liquid immiscibility; the topology of the phenomenenon is the same as for the Carnahan–Starling equation. It appears the occurrence of closed-loop liquid–liquid immiscibility does not depend on the location of the pole nor on the degree of the equation of state used.

Osmotic pressures of aqueous bovine serum albumin solutions at high ionic strength

Using osmometers similar to those described by Vilker et al. [V.L. Vilker, C.K. Colton, K.A. Smith, J. Colloid and Interface Sci. 79 (1981) 548–566] and Amos et al. [D.A. Amos, C.J. Radke, S. Lynn, J. Phys. Chem. B 102 (1998) 2739–2753], osmotic pressures of aqueous bovine serum albumin (BSA) solutions were measured at three pH (4.5, 5.4, 7.4) and at different sodium chloride concentrations (from 1 to 5 M). Experimental data were compared with results calculated from two van der Waals-type models using a potential of mean force including hard-sphere and double-layer repulsions, and van der Waals attraction. In both models, the Carnahan–Starling equation of state represents the contribution of the hard-sphere repulsion to the osmotic pressure. Van der Waals attraction and double-layer repulsion are represented by their contributions to the osmotic second virial coefficient in the first model, or in the second, to the random-phase-approximation (RPA). Although both models give a semi-quantitative description of the osmotic pressures of BSA solutions at various conditions, the first model represents the data slightly better than the second, using the same number of adjustable parameters. Further improvement of modeling requires a more accurate potential of mean force for protein molecules in salt solution.

Pair and triplet correlation entropies based on the hard sphere solution of the Percus–Yevick equation

In this paper we have attempted to estimate the pair and triplet correlation entropies ( S (2) and S (3)) of the hard sphere fluid using the Percus–Yevick approximation. The pair correlation entropies are computed from the pair correlation functions, while the triplet correlation entropies are estimated by the structure factors with the help of an analytical formula for evaluation of a change in the structure factor on application of the pressure. The calculated pair and triplet correlation entropies are consistent with those obtained by the computer simulations due to Baranyai and Evans over a wide range of thermodynamic states. The sum of the pair and triplet correlation entropies is discussed in comparison with the configurational entropy of hard spheres calculated thermodynamically from the Carnahan–Starling equation of state.

PRELIMINARY COMMUNICATION Equation of state of a multicomponent d-dimensional hard-sphere fluid

A simple recipe to derive the compressibility factor of a multicomponent mixture of d-dimensional additive hard spheres in terms of that of the one-component system is proposed. The recipe is based (i) on an exact condition that has to be satisfied in the special limit where one of the components corresponds to point particles; and (ii) on the form of the radial distribution functions at contact as obtained from the Percus-Yevick equation in the three-dimensional system. The proposal is examined for hard discs and hard spheres by comparison with well-known equations of state for these systems and with simulation data. In the special case of d=3, our extension to mixtures of the Carnahan-Starling equation of state yields a better agreement with simulation than the already accurate Boublik-Mansoori-Carnahan-Starling-Leland equation of state.

Equation of state for polymer systems

We developed an equation of state for systems containing chain-like molecules. The equation consists of three contributions. As a reference, we first adopted the equation for hard-sphere-chain fluids (HSCF) derived through the r-particle cavity-correlation function of sticky hard spheres. The HSCF equation has a simple form similar to the Carnahan–Starling equation and has been satisfactorily tested by computer-simulation results for chain fluids with a wide range of chain length. Secondly, for the perturbation, we used the square-well (SW) equation. Thirdly, for molecules with specific oriented interactions such as hydrogen bonding, we introduced a contribution from chemical association (ASS). The corresponding expression for association comes naturally from the sticky model used in deriving the reference equation. The ASS term was well tested with computer-simulation results for associated molecules. The equation of state was widely tested by fitting experimental vapor–liquid equilibrium data. For liquid–liquid equilibria, we adopted a lattice model developed previously as a high-pressure limit that virtually serves as a mixing rule.

Liquid crystal formation in DNA fragment solutions

The critical volume fractions pertaining to the formation of DNA liquid crystals were obtained from polarization microscopy, 31P-nmr, and phase separation experiments. The DNA length (approximately one to two times the persistence length 50 nm), ionic strength, and counterion variety dependencies are reported. The cholesteric–isotropic transition is interpreted in terms of the coexistence equations, which are derived from the solution free energy including orientational entropy and excluded volume effects. With the wormlike chain as reference system, the electrostatic contribution to the free energy is evaluated as a thermodynamic perturbation in the second virial approximation with a Debye–Hückel potential of mean force. The hard core contribution has been evaluated with scaled particle theory and/or a simple generalization of the Carnahan–Starling equation of state for hard spheres. For sufficiently high ionic strengths, the agreement is almost quantitative. At lower amounts of added salt deviations are observed, which are tentatively attributed to counterion screening effects. The contour length dependence agrees with a DNA persistence length 50 nm. © 1998 John Wiley & Sons, Inc. Biopoly 46: 31–37, 1998

Liquid crystal formation in DNA fragment solutions.

The critical volume fractions pertaining to the formation of DNA liquid crystals were obtained from polarization microscopy, 31P-nmr, and phase separation experiments. The DNA length (approximately one to two times the persistence length 50 nm), ionic strength, and counterion variety dependencies are reported. The cholesteric-isotropic transition is interpreted in terms of the coexistence equations, which are derived from the solution free energy including orientational entropy and excluded volume effects. With the wormlike chain as reference system, the electrostatic contribution to the free energy is evaluated as a thermodynamic perturbation in the second virial approximation with a Debye-Hückel potential of mean force. The hard core contribution has been evaluated with scaled particle theory and/or a simple generalization of the Carnahan-Starling equation of state for hard spheres. For sufficiently high ionic strengths, the agreement is almost quantitative. At lower amounts of added salt deviations are observed, which are tentatively attributed to counterion screening effects. The contour length dependence agrees with a DNA persistence length 50 nm.

Equation of state and radial distribution functions of FCC particles in a CFB

AbstractKinetic theory of granular flow was verified experimentally for flow of fluid catalytic cracking particles in a vertical pipe. Measurements of particle pressure using a differential transducer and granular temperature with a digital camera as a function of bulk density, determined using an X‐ray densitometer, showed that a relation exists among pressure, temperature, and density, analogous to the ideal gas law. In the limit of zero solids volume fraction: (Solid Pressure)/[(Granular Temperature) × (Bulk Density)] = 1.0. Measurements of radial distribution functions using the digital camera showed that their peak values occur at particle contact and lie between the predictions from the Bagnold equation and Carnahan–Starling equation. The hard sphere model was corrected for a cohesive pressure using the minimum in the measured radial distribution function. The new model agrees with the pressure measurements in the dense regime.

Mean field curvature corrections to the surface tension

Expressions for the surface tension σ, Tolman length δ, and for the rigidity constants k and k̄ of a curved liquid-vapor interface in mean field approximation are presented. The local free energy comes from the Carnahan-Starling equation of state in mean field, and the usual square gradient term is replaced with a more general, integral nonlocal term. The surface tension and the Tolman length are calculated numerically, and the behavior as the critical temperature is approached is discussed; this behavior is compared with results from a molecular dynamics simulation. Two appendices are included; the first presents derivations of alternative expressions for σ and δ, and the second gives an analytical calculation of the value of δ at the critical point. The Tolman length is found to be negative and approximately 0.2 molecular diameters in magnitude for all temperatures investigated. It is also found to approach a constant value of −0.20263 molecular diameters at the critical point, in agreement with previous results.

A generalized mixing rule for hard-sphere equations of state of Percus–Yevick type

An analogical approach has been used to extend to mixtures the radial distribution function of Percus–Yevick type of equations of state for hard spheres. The proposed approach follows the same formalism employed by Mansoori et al. to extend to mixtures the Carnahan–Starling equation of state. The generality of the proposed method permits to extend to mixtures equations of state with different singularities. In this work, it is applied to an equation of state which meets the correct low density and high density limits, previously proposed by Khoshkbarchi and Vera. The results show that both the radial distribution function and the resulting equation of state for a mixture of hard spheres, developed in this study, can accurately represent the computer simulation data while satisfying the correct limit at close-packing.

Mixing rules for hardsphere chain mixtures and their extension to heteronuclear hardsphere polyatomic fluids and mixtures

Mixing rules are very important for the calculation of fluid properties using different equations of state. In order to find the theoretical lead of the mixing rule for the size parameter, a mixing rule [1] for hardsphere mixtures has been proposed on the basis of Carnahan-Starling equation and Boublik-Mansoori equation. As its extension, mixing rules for hardsphere chain mixtures are proposed in this work. A mixing rule for the segment number (or chain length) is derived on the limitation of the equality of segment diameters, from the first order thermodynamic perturbation theories (TPT1) for pure chain fluids and for chain mixtures. Meanwhile, the mixing rule for the segment diameter is the same as the mixing rule for hardsphere mixtures on the limitation of monomer mixtures. The two mixing rules are checked together over wide ranges of conditions for hardsphere chain mixtures and compared with the first order thermodynamic perturbation theory (TPT1) and also with simulation data available in literature. An another interesting usage of new mixing rules is to describe the heteronuclear hardsphere polyatomic pure fluids, which consist of hardspheres with different segment diameters as in methane and ethane in which carbon and hydrogen atoms are looked as bonded spheres, and heteronuclear hardsphere polyatomic mixtures. The comparison with simulation data shows the validity of the mixing rules.

Virial Approach to Hard-Sphere Demixing

We propose a perturbative modification of the Carnahan-Starling equation of state for a binary hard-sphere mixture which brings its virial coefficients into better agreement with recent numerical findings. The resulting analytic equation of state predicts a demixing transition whose spinodals and binodals are in qualitative agreement with recent numerical and experimental findings.

Thermodynamics and elastic moduli of fluids with steeply repulsive potentials

Analytic expressions for the thermodynamic properties and elastic moduli of molecular fluids interacting with steeply repulsive potentials are derived using Rowlinson’s hard-sphere perturbation treatment which employs a softness parameter, λ specifying the deviation from the hard-sphere potential. Generic potentials of this form might be used to represent the interactions between near-hard-sphere stabilized colloids. Analytic expressions for the equivalent hard-sphere diameter of inverse power [ε(σ/r)n where ε sets the energy scale and σ the distance scale] exponential and logarithmic potential forms are derived using the Barker–Henderson formula. The internal energies in the hard-sphere limit are predicted essentially exactly by the perturbation approach when compared against molecular dynamics simulation data using the same potentials. The elastic moduli are similarly accurately predicted in the hard-sphere limit, as they are trivially related to the internal energy. The compressibility factors from the perturbation expansion do not compare as favorably with simulation data, and in this case the Carnahan–Starling equation of state prediction using the analytic effective hard-sphere diameter would appear to be a preferable route for this thermodynamic property. A more refined state point dependent definition for the effective hard-sphere diameter is probably required for this property.

A mixing rule for the size parameter from Carnahan-Starling equation and Boublik-Mansoori equation

A density-dependent one-fluid mixing rule for the size parameter is proposed from Carnahan-Starling equation and Boublik-Mansoori equation using a numerical method. This mixing rule coincides with the mixing rules usually used at zero pressure and at infinite pressure. It also tends to Carnahan-Starling equations for pure fluids at limiting conditions, such as infinite dilute solution, diameter ratio and equal diameters. Its application to equations of state is also carried out for Lennard-Jones mixtures including binary and ternary mixtures. Compared with the van der Waals one-fluid mixing rule, distinct improvement is obtained, especially for predictions of PVT, chemical potential and vapor-liquid equilibrium data.

A simplified hard-sphere equation of state meeting the high and low density limits

A simplified hard-sphere equation of state has been developed, which meets the correct limit for close-packed conditions. It is shown that the proposed equation of state for hard spheres can represent accurately the computer simulation compressibility factor data and virial coefficients over a wide density range. The comparison of the results of the calculations using this equation, the Carnahan-Starling equation, and the two equations proposed by Iglesias-Silva and Hall, shows that the equation proposed here represents the compressibility factor data and the virial coefficients with better accuracy.

A simple and accurate mixture model

A simple mixture model based on the mixture pair correlation functions was developed and tested. For a Lennard-Jones fluid, the new model showed high accuracy when compared to simulation data. The model was compared to the van der Waals model, the dense fluid theory model and the more complex perturbation model of Weeks-Chandler-Anderson, Lee-Levesque and Grundke-Henderson (WCA-LL-GH); it showed a comparable accuracy to the latter. For hard sphere molecules, the model predictions are comparable to the accurate Mansoori-Carnahan-Starling-Leland equation for mixtures of hard spheres. However, at high densities, an abnormal behavior was observed for small concentrations of the larger component, when the model is used with the Carnahan-Starling equation of state for the pure hard spheres. This might point to phase separation or to a limitation of the new model under the conditions studied. A procedure for extending the model to non-spherical and polar molecules is suggested.

Computer simulation study of the approximations associated with the generalized Flory theories

The chain increment method and configurational bias Monte Carlo methods are used to test the approximations made in the derivation of the generalized Flory-Dimer (GF-D) theory for tangent hard sphere chains. Insertion probabilities and residual chemical potentials are calculated for hard chain fluids containing chains of length n=4, 8, 16, and 32 at monomer densities, ρM, up to 0.8. We find that the largest errors in the GF-D theory are those associated with assuming that the probability of inserting a monomer into a chain fluid is approximately equal to the probability of inserting a monomer into a monomer fluid, as predicted by the Carnahan–Starling equation of state. The errors in the incremental compressibility factor of the second segment associated with assuming that the conditional probability of inserting a second bead next to the first bead in a chain fluid is approximately equal to the probability of inserting a second bead next to the first bead in a dimer fluid as predicted by combining the Carnahan–Starling and Tildesley–Streett equations of state are relatively small. Consistent with the findings of Mooij and Frenkel, we find that these two approximations lead to an overprediction of the incremental contributions to the compressibility factor. Despite the overprediction of the incremental contributions to the compressibility factor of the first segment, the GF-D equation of state accurately predicts the compressibility of hard chains; this accuracy is traced to (1) the insensitivity of the compressibility factor to errors in the insertion probability and (2) cancellation of errors in the incremental compressibility factor of the first segment with small cumulative errors in the incremental compressibility factors of the third and subsequent segments.

Ultrafiltration of Colloidal Dispersions—A Theoretical Model of the Concentration Polarization Phenomena

A general thermodynamic model of the concentration polarization phenomena of colloidal particles at a membrane surface is presented. The model is based on the balance between a thermodynamic force, due to the osmotic pressure gradient, and a frictional force, due to the fluid flow around each particle. A cell model description is used to model the concentration dependence of the thermodynamic force as well as the flow properties in the concentrated colloidal solution. Equilibrium thermodynamics of the colloidal system can be used in the cell calculations since local equilibrium is assumed in the neighborhood of each colloidal particle (i.e., in each cell). This means that the concentration dependence of the osmotic pressure can be obtained, either from an experimental determination or from a theoretical model of the bulk properties of the colloidal system. To exemplify the usefulness of the model when establishing the influence of different operating parameters, such as the transmembrane pressure, the fluid shear, or different solution properties, such as concentration, particle size, pH, and ionic strength, a model system of charged spherical colloidal particles is used. The interaction between the particles is in the presented examples assumed to be a combination of electrostatic interactions, calculated from the Poisson–Boltzmann equation, dispersion forces, calculated as additive 1/r6interactions, and a hard sphere interaction calculated from the Carnahan–Starling equation.