Carnahan Starling Research Articles

Bubble characterizations on hydrophobic surface using lattice Boltzmann simulation with large density ratios

PurposeThe maintenance of the air–water interface is crucial for the drag reduction on hydrophobic surfaces. But the air bubbles become unstable and even washed away under high speed flow, causing the failure of surface hydrophobicity. Thereby, this paper aims to understand the relations between bubble behaviors and surface properties, flow conditions and to discover new methods to maintain the air–water interface.Design/methodology/approachBubble properties on hydrophobic surfaces were characterized using single-component multiphase lattice Boltzmann simulation. Three equations of state (EOSs), including the Peng–Robinson, Carnahan–Starling and modified Kaplun–Meshalkin EOSs, were incorporated to achieve high density ratios.FindingsBoth the static and dynamic properties of bubbles on hydrophobic surfaces were investigated and analyzed under different flow conditions, solid–liquid interactions and surface topology.Originality/valueBy revealing the properties of bubbles on hydrophobic surfaces, the effects of flow conditions and surface properties were characterized. The maintenance method of air–water interface can be proposed according to the bubble properties in the study.

A new method to obtain the Carnahan–Starling equation and its generalization

We obtain the Carnahan–Starling equation for a system of hard spheres using the Euler method of accelerated series convergence. For this purpose, the virial series is transformed into a new series with coefficients that differ slightly from each other, even when considering the eleven currently known virial coefficients. The method of accelerated convergence was applied to this series; it allowed us to obtain the Carnahan–Starling equation. In this work, this equation is derived for the first time using the method of accelerated convergence. It is generalized to accurately reproduce all of the known virial coefficients and the asymptotic behavior of the free energy at high densities. This also makes it possible to describe a metastable region with a high degree of accuracy and to obtain the equation of state for a homogeneous system of hard spheres with the accuracy of a computer experiment.

Salting-Out and Salting-In of Polyelectrolyte Solutions: A Liquid-State Theory Study

We study the phase behavior of polyelectrolyte (PE) solutions with salt using a simple liquid-state (LS) theory. This LS theory accounts for hard-core excluded volume repulsion by the Boublik–Mansoori–Carnahan–Starling–Leland equation of state, electrostatic correlation by the mean-spherical approximation, and chain connectivity by the first-order thermodynamic perturbation theory. We predict a closed-loop binodal curve in the PE concentration-salt concentration phase diagram when the Bjerrum length is smaller than the critical Bjerrum length in salt-free PE solution. The phase-separated region shrinks with decreasing Bjerrum length, and disappears below a critical Bjerrum length. These results are qualitatively consistent with experiments, but cannot be captured by the Voorn–Overbeek theory. On the basis of the closed-loop binodal curve and the lever rule, we predict three scenarios of salting-out and salting-in phenomena with addition of monovalent salt into an initially salt-free PE solution. The Galvani potential—the electric potential difference between the coexisting phases—is found to depend nonmonotonically on the overall salt concentration in some PE concentration range, which is related to the partition of the co-ions in the coexisting phases. Free energy analysis suggests that phase separation is driven by a gain in the electrostatic energy, at the expense of a large translational entropy penalty, due to significant counterion accumulation in the PE-rich phase.

Key thermodynamic characteristics of nucleation on charged and neutral cores of molecular sizes in terms of the gradient density functional theory

The gradient density functional theory and the Carnahan–Starling model formulated for describing the contribution of hard spheres have been used to calculate the profiles of condensate density in small critical droplets formed via homogeneous nucleation, as well as in stable and critical droplets formed via heterogeneous nucleation on solid charged and neutral condensation cores of molecular sizes. The calculations performed for water and argon at different values of condensate chemical potential have yielded the heights of the activation barriers for homoand heterogeneous nucleation as functions of vapor supersaturation at preset system temperatures. The interaction of condensate molecules with a solid core has been described by the resultant potential of molecular attractive forces. In the case of a charged core, the long-range Coulomb potential of electric forces has additionally been taken into account. Dielectric permittivities have been calculated as known functions of the local density of the fluid and temperature. The radius of the equimolecular droplet surface has been chosen as a variable describing the droplet size. Dependences of the chemical potential of condensate molecules in a droplet on its size have been plotted for water and argon with allowance for the action of capillary, electrostatic, and molecular forces. It has been shown that the role of the molecular force potential in heterogeneous nucleation increases with the size of condensation cores.

Chemical potential and entropy in monodisperse and polydisperse hard-sphere fluids using Widom's particle insertion method and a pore size distribution-based insertion probability.

We estimate the excess chemical potential Δμ and excess entropy per particle Δs of computer-generated, monodisperse and polydisperse, frictionless hard-sphere fluids. For this purpose, we utilize the Widom particle insertion method, which for hard-sphere systems relates Δμ to the probability to successfully (without intersections) insert a particle into a system. This insertion probability is evaluated directly for each configuration of hard spheres by extrapolating to infinity the pore radii (nearest-surface) distribution and integrating its tail. The estimates of Δμ and Δs are compared to (and comply well with) predictions from the Boublík-Mansoori-Carnahan-Starling-Leland equation of state. For polydisperse spheres, we employ log-normal particle radii distributions with polydispersities δ = 0.1, 0.2, and 0.3.

Correlation of the mean activity coefficient of aqueous electrolyte solutions using an equation of state

Accurate calculation of thermodynamic properties of electrolyte solution is essential in the design and optimization of many processes in chemical industries. A new electrolyte equation of state is developed for aqueous electrolyte solutions. The Carnahan–Starling repulsive model and an attractive term based on square-well potential are adopted to represent the short range interaction of ionic and molecular species in the new electrolyte EOS. The long range interaction of ionic species is expressed by a simplified version of Mean Spherical Approximation theory (MSA). The new equation of state also contains a Born term for charging free energy of ions. Three adjustable parameters of new eEOS per each electrolyte solution are size parameter, square-well potential depth and square-well potential interaction range. The new eEOS is applied for correlation of mean activity coefficient and prediction of osmotic coefficient of various strong aqueous electrolyte solutions at 25°C and 0.1MPa. In addition, the extension of the new eEOS for correlation of mean activity coefficient and solution density of a few aqueous electrolytes at temperature range of 0 to 100°C is carried out.

Dependence of the condensate chemical potential on droplet size in thermodynamics of heterogeneous nucleation within the gradient DFT

A scheme of computation of the condensate chemical potential per molecule as a function of the droplet equimolecular radius for stable and critical droplets on uncharged or charged spherical particle of molecular size at heterogeneous nucleation has been considered. The scheme is based of the gradient density functional theory (DFT) with the van der Waals (vdW) and Carnahan–Starling (СS) models for the hard-sphere contribution to intermolecular interaction in liquid and vapor phases and interfaces. The particle serving as a condensation center in the case of heterogeneous nucleation has been characterized by an attractive short-range molecular potential and the long-range electric Coulomb potential. The dielectric permittivities of the droplet–vapor systems have been taken as known functions of the local condensate density and temperature for polar and nonpolar fluids. Detailed numerical calculations of the density profiles in critical and stable equilibrium droplets at water or argon nucleation in presence of the capillary, electrostatic and molecular forces have been performed. Dependence of the condensate chemical potential in the droplet on the droplet equimolecular radius has been analyzed in the case of homogeneous and heterogeneous nucleation and compared for water and argon within the vdW and СS models for the hard-sphere part of the equation of state.

Mean-field approach in the multi-component gas of interacting particles applied to relativistic heavy-ion collisions

A generalized mean-field approach for the thermodynamic description of relativistic single- and multi-component gas in the grand canonical ensemble is formulated. In the framework of the proposed approach, different phenomenological excluded-volume procedures are presented and compared to the existing ones. The mean-field approach is then used to effectively include hard-core repulsion in hadron-resonance gas model for description of chemical freeze-out in heavy-ion collisions. We calculate the collision energy dependence of several quantities for different values of hard-core hadron radius and for different excluded-volume procedures such as the van der Waals and Carnahan–Starling models. It is shown that a choice of the excluded-volume model becomes important for large particle densities. For large enough values of hadron radii ( fm) there can be a sizable difference between different excluded-volume procedures used to describe the chemical freeze-out in heavy-ion collisions. At the same time, for the smaller and more commonly used values of hard-core hadron radii ( fm), the precision of the van der Waals excluded-volume procedure is shown to be sufficient.

Linear Yukawa Isotherm Regularity for dense fluids derived based on the perturbation theory

In the present work, the thermodynamic of dense fluids, both compressed liquids and dense supercritical fluids, has been modeled, solely, based on the contribution of attraction of effective pair potential. The intermolecular interaction is modeled by the hard-core Yukawa potential (HCY) as an effective pair potential (EPP) with temperature dependent hard-core diameter. Using this EPP in the exact thermodynamic relations, an equation of state (EoS) for the compressibility factor of dense fluid has been derived. This EoS shows that (Z − ZCS) as function of ρ1/3 must be linear for each isotherm of fluid where ZCS is the compressibility factor of the reference fluid (Carnahan-Starling (CS) EoS) with temperature-dependent hard-core diameter and Z is the experimental compressibility factor of fluid. To our knowledge, this is the first report on this new regularity for dense fluids only based on the perturbation theory in literature. The validity of this regularity has been tested for different fluids including Ar, Kr, Ne, H2, N2, NH3, NF3, CH4, C2H2, C3H8, C2H6, C2H4 and CH3OH. This regularity is valid for densities greater than ρB, where ρB is the Boyle density. Finally, using the proposed regularity, the values of the EPP parameters for the mentioned fluids were obtained at different temperatures.

Influence of steric interactions on the dielectric and electrokinetic properties in colloidal suspensions

One of the main assumptions of the standard electrokinetic model is that ions behave as point like entities. In this work we remove this assumption and analyze the main consequences of finite ionic size on the dielectric and electrokinetic properties of colloidal suspensions. We represent the steric interactions by means of the Bikerman and the Carnahan–Starling equations and solve numerically the standard linearized electrokinetic equations in the stationary and the frequency domains, for surface charge density and electrolyte solution concentration values typically encountered in colloidal suspensions. In all cases the steric interactions improve upon the predictions of the standard model since the surface potential, the electrophoretic mobility, and the conductivity and permittivity increments increase. However, the corrections introduced by the Bikerman equation are generally small: less than 10% as compared to the standard model. On the contrary, the Carnahan–Starling equation leads to corrections to the surface potential versus surface charge and the electrophoretic mobility values that easily surpass 10% and can attain values as high as 50%. Corrections to the conductivity and permittivity increments are smaller but still non negligible.

How to predict the ideal glass transition density in polydisperse hard-sphere packings.

The formula for the entropy s of the accessible volume of the phase space for frictionless hard spheres is combined with the Boublík-Mansoori-Carnahan-Starling-Leland (BMCSL) equation of state for polydisperse three-dimensional packings to obtain an analytical expression for s as a function of packing density φ. Polydisperse hard-sphere packings with log-normal, Gaussian, and Pareto particle diameter distributions are generated to estimate their ideal glass transition densities φg. The accessible entropy s at φg is almost the same for all investigated particle diameter distributions. We denote this entropy as sg and can predict φg for an arbitrary particle diameter distribution through an equation s(φ) = sg. If the BMCSL equation of state is used for s(φ), then φg is found to depend only on the first three moments of a particle diameter distribution.

Cubic and quartic hard-sphere and Lennard-Jones chain equations of state as foundations for complex fluid modeling

Hard-sphere compressibility factor Z and hard-dimer radial distribution function at contact gHD(σ) are fit to simple packing fraction η expressions and are used in the dimer version of Wertheim’s perturbation theory to obtain five cubic equations of state (EOS’s) for athermal hard-sphere chain fluids. The predicted Zm versus η for various chain lengths m and the reduced second-virial coefficient versus m are compared with simulation data and the original noncubic dimer perturbation theory EOS’s. A cubic analog of the Boublik–Mansoori–Carnahan–Starling (BMCS) EOS successfully represents the unlike pair correlation function g12(σ) and Z versus η for binary hard-sphere mixtures of moderate size difference in the components. The new CTPT-D models are extended to calculate Zm versus η for binary homonuclear hard-sphere chain mixtures. The choice of a suitable EOS dispersion term to simultaneously represent single phase PVT and phase equilibria is briefly explored. A similar TPT approach was used to derive a quartic EOS for Lennard-Jones chain fluids, and Zm versus η isotherms are determined and compared with simulation results for chains up to 100 segments long. The new cubic and quartic EOS’s provide a simple foundation for developing equations of state for complex fluids including multipolar and association effects.

The study of binary interaction parameters on VLE for alternative refrigerant mixtures

AbstractThe vapor–liquid equilibrium (VLE) data for hydrofluorocarbons (HFCs) + hydrocarbons (HCs) and HFCs + dimethyl ether (DME) are correlated by the Soave–Redlich–Kwong, Peng–Robinson, and Carnahan–Starling–DeSantis, respectively, with one adjustable binary interaction parameter kij, and each equation can represent the satisfactory results. For the three equations, the obtained optimum interaction parameter kij is almost the same for the binaries containing the same HFC component in the same class, and when the kij of the lack‐of‐data mixtures are needed, it just needs to know the kij of any binary in the same class on the basis of the phenomena and law. Moreover, a new equation for kij is developed that enables the three equations of state to predict the VLE properties for such mixtures within reasonable accuracy. © 2015 Curtin University of Technology and John Wiley & Sons, Ltd.

Electric double-layer structure in primitive model electrolytes: comparing molecular dynamics with local-density approximations.

We evaluate the accuracy of local-density approximations (LDAs) using explicit molecular dynamics simulations of binary electrolytes comprised of equisized ions in an implicit solvent. The Bikerman LDA, which considers ions to occupy a lattice, poorly captures excluded volume interactions between primitive model ions. Instead, LDAs based on the Carnahan-Starling (CS) hard-sphere equation of state capture simulated values of ideal and excess chemical potential profiles extremely well, as well as the relationship between surface charge density and electrostatic potential. Excellent agreement between the EDL capacitances predicted by CS-LDAs and computed in molecular simulations is found even in systems where ion correlations drive strong density and free charge oscillations within the EDL, despite the inability of LDAs to capture the oscillations in the detailed EDL profiles.

Lattice Boltzmann simulation of liquid–vapor system by incorporating a surface tension term**Project supported by the National Nature Science Foundation of China (Grant No. 51109178) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20116102120009).

In this study, we investigate the pseudopotential multiphase model of lattice Boltzmann method (LBM) and incorporate a surface tension term to implement the particle interaction force. By using the Carnahan–Starling (CS) equation of state (EOS) with a proper critical pressure–density ratio, a density ratio over 160000 is obtained with satisfactory numerical stability. The added surface tension term offers a flexible choice to adjust the surface tension strength. Numerical tests of the Laplace rule are conducted, proving that smaller spurious velocity and better numerical stability can be acquired as the surface tension becomes stronger. Moreover, by wall adhesion and heterogeneous cavitation tests, the surface tension term shows its practical application in dealing with problems in which the surface tension plays an important role.

Helmholtz energies from equations of state and their relative deviations

Extension of the hard-sphere equations of state to non spherical hard-body systems represents a formidable task. For an equation of state that is applicable to liquid and vapor states (and explicit for pressures), the corresponding characteristic function is the residual energy. The equations of state of Carnahan Starling, Boublik and R J Sadus are used to derive an expression for residual Helmholtz energies. Relative deviations of residual Helmholtz energies for these equations of state have been reviewed. The deviations of compressibility factor against packing fraction have been plotted for these equations of state. The agreement of these plots is found to be good for lower n-alkanes but not for longer n-alkanes. Thus it becomes obvious that the compressibility factor depends considerably on the shape of the studied molecules.

Development of a modified van der Waals-type equation of state for pure and mixture of ionic liquids

The main aim of the present study is to modify the attractive part of van der Waals-type EOS to model the volumetric ionic liquids and their mixtures. The present study uses the Carnahan–Starling (CS) repulsion term and also focuses on introducing a variable parameter in the attractive term of modified vdW-type EOS. Investigation of the results obtained revealed that this variable is a function of temperature. Our results also show that employment of CS repulsion term together with an appropriately chosen exponent in the attractive term of EOS will greatly improve the accuracies of the calculated densities of various ionic liquids. The average absolute deviation (AAD) of the calculated densities from the experimental ones for 848 data points of pure ionic liquids was found to be 0.36. Further, in the case of mixtures, for 1014 data points the AAD was 0.20.

Influence of the finite size and effective permittivity of ions on the equilibrium double layer around colloidal particles in aqueous electrolyte solution

The equilibrium properties of the electrical double layer surrounding a charged spherical colloidal particle immersed in an aqueous electrolyte solution are examined taking into account the finite ion size. This study includes the representation of the steric interactions among ions using both the Bikerman and the Carnahan–Starling models, an account of all the effects related to the representation of hydrated ions as dielectric spheres (dependence of the electrolyte solution permittivity, on the local ion concentration, and appearance of the Born and the dielectrophoretic forces acting on the ions), and solution of the problem for both high and low surface charge values. We find that the Carnahan–Starling model together with effective ion permittivity related effects appears to be able to provide an interpretation to the electrokinetic potential vs. surface charge dependence in the case of colloidal particles suspended in aqueous electrolyte solutions. On the contrary, for electrode–electrolyte systems, both the Bikerman and the Carnahan–Starling models might be able to explain this dependence.

Note: Equation of state and the freezing point in the hard-sphere model

The merits of different analytical equations of state for the hard-sphere system with respect to the recently computed high-accuracy value of the freezing-point packing fraction are assessed. It is found that the Carnahan–Starling–Kolafa and the branch-point approximant equations of state yield the best performance.

Equations of states for an ionic liquid under high pressure: A molecular dynamics simulation study

The high-pressure dependence of density given by empirical equation of states (EoS) for the ionic liquid 1-butyl-3-methylimidazolium trifluoromethanesulfonate (or triflate), [C4C1im][TfO], is compared with results obtained by molecular dynamics (MD) simulations. Two EoS proposed for [C4C1im][TfO] in the pressure range of tens of MPa, which give very different densities when extrapolated to pressures beyond the original experiments, are compared with a group contribution model (GCM). The MD simulations provide support that one of the empirical EoS and the GCM is valid in the pressure range of hundreds of MPa. As an alternative to these EoS that are based on modified Tait equations, it is shown that a perturbed hard-sphere EoS based on the Carnahan–Starling–van der Waals equation also fits the densities calculated by MD simulations of [C4C1im][TfO] up to ∼1.0GPa.