Hard-core Yukawa Research Articles

Infinite Compressibility States in the Hierarchical Reference Theory of Fluids. II. Numerical Evidence

Continuing our investigation into the Hierarchical Reference Theory of fluids for thermodynamic states of infinite isothermal compressibility κT we now turn to the available numerical evidence to elucidate the character of the partial differential equation: Of the three scenarios identified previously, only the assumption of the equations turning stiff when building up the divergence of κT allows for a satisfactory interpretation of the data. In addition to the asymptotic regime where the arguments of part I directly apply, a similar mechanism is identified that gives rise to transient stiffness at intermediate cutoff for low enough temperature. Heuristic arguments point to a connection between the form of the Fourier transform of the perturbational part of the interaction potential and the cutoff where finite difference approximations of the differential equation cease to be applicable, and they highlight the rather special standing of the hard-core Yukawa potential as regards the severity of the computational difficulties.

Sedimentation Equilibrium of Colloidal Suspensions in a Planar Pore Based on Density Functional Theory and the Hard-Core Attractive Yukawa Model

The sedimentation equilibrium of colloidal suspensions modeled by hard-core attractive Yukawa (HCAY) fluids in a planar pore is studied. The density profile of the HCAY fluid in a gravitational field and its distribution between the pore and uniform phases are investigated by a density functional theory (DFT) approach, which results from employing a recently proposed parameter-free version of the Lagrangian theorem-based density functional approximation (Zhou, S. Phys. Lett. A 2003, 319, 279) for hard-sphere fluids to the hard-core part of the HCAY fluid, and the second-order functional perturbation expansion approximation to the tail part as was done in a recent partitioned density functional approximation (Zhou, S. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2003, 68, 061201). The resultant DFT approach is, thus, the first adjustable parameter-free DFT for HCAY fluids. The validity of the present DFT for HCAY fluids of reduced range parameter z(red) = 1.8 under various external potentials is established in the first of the papers cited previously. The present DFT for HCAY fluids can predict the radial distribution function for the bulk HCAY fluid accurately in the colloidal limit (large value of z(red)), and in the hard-sphere limit, its prediction for the density profile of the hard-sphere fluid in a gravitational field is in very good agreement with the existing simulation data. The dependence of the density profile and distribution coefficient on the magnitude of the interparticle attraction, gravitational field, and degree of confinement is investigated in detail by the present DFT approach. Intuitive and qualitative analyses are also compared with the quantitative DFT calculational results.

Structure of Inhomogeneous Attractive and Repulsive Hard-Core Yukawa Fluid: Grand Canonical Monte Carlo Simulation and Density Functional Theory Study

A density functional theory is proposed for an inhomogeneous hard-core Yukawa (HCY) fluid based on Rosenfeld's perturbative method. The excess Helmholtz energy functional is derived from a modified fundamental measure theory for the hard-core repulsion and a quadratic functional Taylor expansion for the long-ranged attractive or repulsive interactions. To test the established theory, grand canonical ensemble Monte Carlo simulations are carried out to simulate the density profiles of attractive and repulsive HCY fluid near a wall. Comparison with the results from the Monte Carlo simulations shows that the present density functional theory gives accurate density profiles for both attractive and repulsive HCY fluid near a wall. Both the present theory and simulations suggest that there is depletion for attractive HCY fluid at low temperature, but no depletion is found for repulsive HCY fluid. The calculated results indicate that the present density functional theory is better than those of the modified version of the Lovett-Mou-Buff-Wertheim and other density functional theories. The present theory is simple in form and computationally efficient. It predicts accurate radial distribution functions of both attractive and repulsive HCY fluid except for the repulsive case at high density, where the theory overestimates the radial distribution function in the vicinity of contact.

Analysis of the validity of perturbation density functional theory: Based on extensive simulation for simple fluid at supercritical and subcritical temperature under various external potentials

Because of the scarcity of available simulation data for confined hard-core attractive Yukawa model fluid, extensive Monte Carlo (MC) simulation research for this fluid under the influence of various external potentials were carried out. The present MC simulation results were employed to test a performance of the third-order perturbation density functional theory (DFT) based on a high order direct correlation function (DCF) [S. Zhou and E. Ruckenstein, Phys. Rev. E. 61, 2704 (2000)]. It was found that the present perturbation DFT formalism is soundly structured only if the imported second-order DCF is reliable. In this case, the accuracy of the results can be satisfactory or even very high for various types of external potentials. Further, the associated adjustable parameter can be universal, i.e., independent of the particular external field responsible for the generation of a nonuniform density profile. Dependence of both the maintenance of the reliability of the formalism and holding of the universality of the adjustable parameter on the accuracy of the imported bulk second-order DCF can be strengthened by the large difference between the external field investigated and that caused by a single hard wall used for specification of the adjustable parameter. In case the gaseous density in the subcritical region is below the coexistence density, an excellent performance of the present formalism is observed even for the mean spherical approximation's second-order DCF as an input. This advantageous property, combined with the fact that the present formalism needs only the second-order DCF of fluid at the coexistence state as an input, enables the present formalism to be a very good theoretical tool for the investigations of wetting and prewetting transitions.

Phase diagram of a binary symmetric hard-core Yukawa mixture

We assess the accuracy of the self-consistent Ornstein-Zernike approximation for a binary symmetric hard-core Yukawa mixture by comparison with Monte Carlo simulations of the phase diagrams obtained for different choices of the ratio alpha of the unlike-to-like interactions. In particular, from the results obtained at alpha=0.75 we find evidence for a critical endpoint in contrast to recent studies based on integral equation and hierarchical reference theories. The variation of the phase diagrams with range of the Yukawa potential is investigated.

Effect of repulsive and attractive interactions on depletion forces in colloidal suspensions: A density functional theory treatment

The author employs density functional theory to study colloidal interactions in solution. Hardcore Yukawa potentials with soft tails, either repulsive or attractive, are used to model colloid-solvent and solvent-solvent interactions. We analyze the effect of these interactions on the solvent-mediated potential of mean force between two colloids in solution. Overall, theory is shown to be in good agreement with recent simulation data. We use the theory to study the density dependence of the colloid-colloid second virial coefficient.

A self-consistent approach for modelling the interfacial properties and phase diagrams of Yukawa, Lennard-Jones and square-well fluids

A self-consistent density-functional approach is presented for describing the phase behaviour and interfacial tensions of van der Waals fluids represented by the hard-core Yukawa (HCY), Lennard-Jones (LJ) and square-well (SW) potentials. The excess Helmholtz energy functional is formulated in terms of a modified fundamental measure theory (MFMT) for the short-ranged repulsion and a density-gradient expansion for the van der Waals attractions. Analytical expressions for the direct correlation functions of uniform fluids are utilized to take into account the effect of van der Waals’ attraction on intermolecular correlations. For bulk phases, the density functional theory is reduced to an equation of state (EOS) that provides accurate saturation pressures and vapour–liquid phase diagrams. Near the critical region, the long-range fluctuations can be corrected by using the renormalization group (RG) theory. With the same set of molecular parameters, the theory also yields satisfactory surface tensions and interfacial density profiles at all relevant temperatures.

Thermodynamic and structural properties of repulsive hard-core Yukawa fluid: integral equation theory, perturbation theory and Monte Carlo simulations.

The thermodynamic and structural properties of purely repulsive hard-core Yukawa particles in the fluid state are determined through Monte Carlo simulation and modeled using perturbation theory and integral equation theory in the mean spherical approximation (MSA). Systems of particles with Yukawa screening lengths of 1.8, 3.0, and 5.0 are examined with results compared to variations of MSA and perturbation theory. Thermodynamic properties were predicted well by both theories in the fluid region up to the fluid-solid phase boundary. Further, we found that a simplified exponential version of the MSA is the most accurate at predicting radial distribution function at contact. Radial distribution function of repulsive hard-core Yukawa particles are also reported. The results show that methods based on MSA and perturbation theory that are typically applied to the attractive hard-core Yukawa potential can also be extended to the purely repulsive hard-core Yukawa potential.

Application of Lagrangian theorem-based density-functional approximation free of adjustable parameters to nonhard-sphere fluid

A recently proposed parameter free version of a Lagrangian theorem-based density functional approximation (LTDFA) [S. Zhou, Phys. Lett. A 319, 279 (2003)] for hard-sphere fluid is applied to hard-core attractive Yukawa model fluid by dividing bulk second-order direct correlation function (DCF) of fluid under consideration into hard-core part and tail part. The former is treated by the parameter free version of the LTDFA, while the tail part is treated by second-order functional perturbation expansion approximation as done in a recent partitioned DFA [S. Zhou, Phys. Rev. E 68, 061201 (2003)]. Two versions of mean spherical approximation (MSA) for the bulk second-order DCF are employed as input, one is the less accurate plain MSA whose tail part of the second-order DCF is strictly independent of a density argument, the other is the more accurate inverse temperature expansion version of the MSA whose tail part is not strictly independent of the density argument. Calculational results indicate that prediction based on the plain MSA is far more accurate than that based on the inverse temperature expansion version of the MSA. The reason is considered to be that the partitioned DFA requires that the tail part is highly or completely independent of the density argument, the plain MSA, by assuming that the tail part is exactly the potential itself, embodies all of the nonlinearities into the hard-core part which can be treated satisfactorily by the parameter free version of the LTDFA. The present investigation results in a universal method for constructing DFA for nonuniform any nonhard-sphere interaction potential fluids.

Solid−Liquid Phase Transition of the Hard-Core Attractive Yukawa System and Its Colloidal Implication

Both a free volume approach for Helmholtz free energy and theoretically based fitted formulas for a radial distribution function of hard-sphere solid are combined to describe the Helmholtz free energy of a solid phase with the hard-core attractive Yukawa (HCAY) potential in the framework of first-order thermodynamic perturbation theory. The corresponding Helmholtz free energy of the fluid phase is calculated from a recently proposed analytical expression based on the inverse temperature expansion and mean spherical approximation. The predicted solid−liquid phase transition for the HCAY system is in far better agreement with simulation than a previous perturbation weighted density approximation. The present formalism is employed as a predictor of protein solidification and metastable liquid−liquid coexistence of crystallizing protein solution, in satisfactory agreement with the available experimental data.

A simple SCOZA for simple fluids

A simple version of the self-consistent Ornstein–Zernike approach (SCOZA) is investigated quantitatively. The simplification consists of retaining thermodynamic self-consistency between the compressibility and energy routes to thermodynamics from the pair distribution function g( r). However, we do not require that g( r) be identically zero within the hard repulsive core of the models we consider here. The resulting theory lends itself to analysis of Hamiltonian models to which the earlier SCOZA formulation cannot be readily applied as well as providing a convenient means of analyzing simpler models. We choose the hard-core Yukawa fluid as an illustrative example to calibrate the accuracy of the new version. The results are compared with our earlier SCOZA and simulation results, and a suggestion is given to systematically improve the accuracy of the theory in the applications that warrant so doing.

The influence of thermodynamic self-consistency on the phase behaviour of symmetric binary mixtures

We have investigated the phase behaviour of a symmetric binary mixture with particles interacting via hard-core Yukawa potentials. To calculate the thermodynamic properties we have used the mean spherical approximation (MSA), a conventional liquid state theory, and the closely related self-consistent Ornstein–Zernike approximation which is defined via an MSA-type closure relation, requiring, in addition, thermodynamic self-consistency between the compressibility and the energy-route. We investigate on a quantitative level the effect of the self-consistency requirement on the phase diagram and on the critical behaviour and confirm the existence of three archetypes of phase diagram, which originate from the competition between the first order liquid/vapour transition and the second order demixing transition.

Surface tension of associating fluids by Monte Carlo simulations.

Canonical Monte Carlo (NVT-MC) simulations were performed to obtain surface tension and coexistence densities at the liquid-vapor interface of one-site associating Lennard-Jones and hard-core Yukawa fluids, as functions of association strength and temperature. The method to obtain the components of the pressure tensor from NVT-MC simulations was validated by comparing the equation of state of the associative hard sphere system with that coming from isothermal-isobaric Monte Carlo simulations. Surface tension of the associative Lennard-Jones fluid determined from NVT-MC is compared with previously reported results obtained by molecular dynamics simulations of a pseudomixture model of monomers and dimers. A good agreement was found between both methods. Values of surface tension of associative hard-core Yukawa fluids are presented here for the first time.

Simple analytic equations of state for hard-core single and double Yukawa fluids and mixtures based on second-order Barker-Henderson perturbation theory.

A simple analytic expression with high precision for the radial distribution function of hard spheres is proposed. The form of the expression has been carefully selected to combine the well-known Camahan-Starling equation of state in it and satisfy the limit condition at low density, its simplicity and precision is superior to the well-known Percus-Yevick expression. The coefficients contained in the expression have been determined by fitting the Monte Carlo data for the first coordination shell, and by fitting both the Monte Carlo data and the numerical results of the Percus-Yevick expression for the second coordination shell. The expression has been applied to develop simple analytic equations of state for the hard-core single, double Yukawa fluids, and the hard-core Yukawa mixtures. The comparisons show that the agreement of our model with the computer simulation data is slightly better than the mean spherical approximation and other analytic models.

Phase diagram of hard-core repulsive Yukawa particles with a density-dependenttruncation: a simple model for charged colloids

Using computer simulations we study the phase behaviour of hard spheres with repulsiveYukawa interactions and with the repulsion set to zero at distances larger than adensity-dependent cut-off distance. Earlier studies based on experiments and computersimulations in colloidal suspensions have shown that the effective colloid–colloid pairinteraction that takes into account many-body effects resembles closely this truncatedYukawa potential. We present a phase diagram for the truncated Yukawa system bycombining Helmholtz free energy calculations and the Kofke integration method. Comparedto the non-truncated Yukawa system we observe (i) a radical reduction of the stability ofthe body centred cubic (BCC) phase, (ii) a wider fluid region due to instability of the facecentred cubic (FCC) phase and due to a re-entrant fluid phase and (iii) hardly anyshift of the (FCC) melting line when compared with the (BCC) melting line forthe full Yukawa potential for sufficiently high salt concentrations, i.e. truncationof the potential does not affect the location of the solid–fluid line but replacesonly the BCC phase with the FCC at the melting line. We compare our resultswith earlier results on the truncated Yukawa potential and with results fromsimulations where the full many-body Poisson–Boltzmann problem is solved.

Phase diagrams of hard-core repulsive Yukawa particles.

We determine the phase behavior of hard spheres interacting with repulsive Yukawa (screened Coulomb) interaction using computer simulations. We study the effect of the hard-core diameter on the phase behavior of repulsive Yukawa particles by comparing our phase diagrams with that of repulsive point Yukawa particles. We show that for sufficiently high contact values of the pair potential (betaepsilon=20, 39, 81, and higher), the fluid-face-centered-cubic (fcc) solid, at high screening, the fluid-body-centered-cubic (bcc) solid and the bcc-fcc coexistence for packing fractions eta less, similar 0.5 are well described by the phase boundaries of point Yukawa particles, by employing a mapping of the point Yukawa system onto a hard-core repulsive Yukawa system. While the bcc-fcc coexistence is well described by the point Yukawa limit for eta<0.5, we find a deviation at higher eta as the hard-core repulsion favors the fcc solid for eta>/=0.5, independent of the screening. Consequently, a second triple point appears in the phase diagram in the weak screening regime. In addition, we find that all the phase coexistence regions in our phase diagrams for hard-core repulsive Yukawa system are very narrow, i.e., a small density jump in the coexisting phases.

A New Uniform Phase Bridge Functional: Test and Its Application to Non-uniform Phase Fluid**The project supported by National Natural Science Foundation of China (Grant No. 20206033), Scientific Research Fund of Education Department of Hunan Province of China (Grant No. 02B033)

A new bridge functional as a function of indirect correlation function was proposed, which was based on analysis on the asymptotic behavior of the Ornstein–Zernike (OZ) equation system and a series expansion whose renormalization resulted in an adjustable parameter determined by the thermodynamics consistency condition. The proposed bridge functional was tested by applying it to bulk hard sphere and hard core Yukawa fluid for the prediction of structure and thermodynamics properties based on the OZ equation. As an application, the present bridge functional was employed for non-uniform fluid of the above two kinds by means of the density functional theory methodology, the resulting density distribution profiles were in good agreement with the available computer simulation data.

Mixing Distributions within the Quasi-Gaussian Entropy Theory: Multistate Thermal Equations of State Valid for Large Temperature Ranges

In this article the quasi-Gaussian entropy (QGE) theory has been extended toward statistical-mechanical models that describe the temperature dependence of thermodynamic properties of fluids at fixed density over a very large temperature range, up to 15 times the critical temperature. The system's phase space is divided into multiple regions, each of which has a “potential” energy distribution that can be described by a simple model, e.g., a Gamma distribution. The overall “potential” energy distribution, which is directly related to the residual Helmholtz free energy of the system, then is a “mixture” of Gamma distributions, each with, for example, a different value of the “minimum” potential energy. Several such multistate models for the free energy were derived and tested on a series of small molecules at various densities: the hard core Yukawa fluid, the Lennard-Jones fluid, argon, methane, ammonia, water, and the extended simple point charge (SPC/E) water model. In almost all systems, a Gamma mixture of Gamma distributions provides a very accurate thermal model over a large temperature range starting at the coexistence line and applicable from ideal gas to dense liquid, even in the vicinity of the critical point. The shape of the “potential” energy distribution and its density dependence reflects the underlying molecular interactions, which are discussed by comparing different systems.

Surface tension of hard core Yukawa fluid

The statistical mechanical expression of Fowler for the surface tension is reformulated for a hard sphere Yukawa fluid within the mean field approximation to obtain an analytical solution. This allows an investigation of the role of long-range interactions and the effect of temperature on the surface tension. For the atomic density profile at the surfaces, the liquid–vapour coexistence curves were reproduced, and hence the rectilinear density was used as the average density at the surface. The results suggest that the surface tension and its temperature gradient are quite sensitive to the long-range interactions.

Microscopic theories of model macromolecular fluids and fullerenes: The role of thermodynamic consistency

We investigate the accuracy of two well-known integral equation theories (IETs) of the fluid state, namely, the modified hypernetted chain (MHNC) approximation and the hybridized mean spherical approximation (HMSA), as applied to systems characterized by short-range interactions. The theoretical approaches are implemented by enforcing their thermodynamic consistency according to two different strategies: in one case the equality of the isothermal compressibility, as calculated via the virial and fluctuation routes from structure to thermodynamics, is imposed [“local” consistency (LC)]; in the other case the equality of the pressure as calculated either via the two previous routes, or via the virial and the energy routes, is imposed [“global” consistency (GC)]. We show that for the class of potentials at issue the GC is in general considerably more accurate than the LC. We document this result by investigating the performances of the MHNC and the HMSA, as applied to the calculation of the thermodynamic and structural properties of the hard-core Yukawa (HCY) potential, the Derjaguin–Landau–Vervey–Overbeek (DLVO) potential and the Girifalco potential for fullerenes. The obtained results are then compared with Monte Carlo simulation data, that we also produce for the same model systems. As far as the HCY potential is concerned, the investigation covers a range of the Yukawa inverse decay length, z, spanning from z=1.8 when the interaction mimics the Lennard-Jones 12-6 potential, to z=7 when the potential mimics the “effective” short range interaction between globular proteins in a highly charge-screened aqueous solution. IETs are then applied to the DLVO potential with charge and Hamaker constant values which fit the dynamical interaction factor of lysozyme in a solution of high ionic strength, and to the Girifalco potential with parameters appropriate to model C60 and C70. It emerges from the present study that the GC is able to provide Helmholtz free energies and chemical potentials which compare quite favorably with the simulation data. As a consequence, we are able to show that the GC estimates of the phase coexistence densities for the HCY and Girifalco potential agree almost quantitatively with the Monte Carlo ones, by thus definitely improving upon previous results obtained within the LC. We also comment on the relevance of confident phase diagram determinations from IETs, in connection with the prediction of protein crystallization. Possible extensions of the present thermodynamic consistency strategies to more sophisticated theories and realistic models of protein solutions and fullerenes are finally suggested.