Additive Hard-sphere Mixtures Research Articles

Coarse-grained Hamiltonian and effective one component theory of colloidal suspensions

We develop a theory to trace out the solvent degrees of freedom from the grand partition function of colloid-solvent mixtures. Our approach to coarse-graining is based on density functional formalism of density profile and the grand thermodynamic potential of solvent. The solvent-induced interaction which is many-body in character is expressed in terms of two functionals; one that couples the solvent to the colloidal density distribution and the second represents the density–density correlation function of the solvent. The nature, strength, and range of the potential depend on these functionals and therefore on the thermodynamic state of the solvent. The solvent-induced contribution to free energy functional is also derived. A self-consistent procedure is developed to calculate the effective potential between colloidal particles, colloid-solvent, and colloid-colloid correlation functions. The theory is used to investigate both additive and nonadditive binary hard-sphere mixtures. Results are reported for the two systems for several values of packing fractions ηb and ηs and particles diameter ratio q=σsσb where symbols b and s refer to colloid and solvent, respectively. Several interesting features are found: The short-range attractive part of the potential shows non-monotonic dependence on ηb; when ηb is increased from zero, initially the potential becomes more attractive but beyond a certain value of ηb that depends on q, the attraction starts weakening. The repulsive peaks formed at R∼(1+12nq) where R is a distance between centers of colloidal particles expressed in units of σb and n is an integer, become stronger on increasing ηb. These results show that many-body contribution to the effective potential depends in a subtle way on packing fractions ηb,ηs, size ratio q, and on nature of the interaction model and makes a non-negligible contribution to the coarse-grained Hamiltonian.

Structural properties of additive binary hard-sphere mixtures. III. Direct correlation functions.

An analysis of the direct correlation functions c_{ij}(r) of binary additive hard-sphere mixtures of diameters σ_{s} and σ_{b} (where the subscripts s and b refer to the "small" and "big" spheres, respectively), as obtained with the rational-function approximation method and the WM scheme introduced in previous work [S. Pieprzyk et al., Phys. Rev. E 101, 012117 (2020)2470-004510.1103/PhysRevE.101.012117], is performed. The results indicate that the functions c_{ss}(r<σ_{s}) and c_{bb}(r<σ_{b}) in both approaches are monotonic and can be well represented by a low-order polynomial, while the function c_{sb}(r<1/2(σ_{b}+σ_{s})) is not monotonic and exhibits a well-defined minimum near r=1/2(σ_{b}-σ_{s}), whose properties are studied in detail. Additionally, we show that the second derivative c_{sb}^{''}(r) presents a jump discontinuity at r=1/2(σ_{b}-σ_{s}) whose magnitude satisfies the same relationship with the contact values of the radial distribution function as in the Percus-Yevick theory.

Structural properties of additive binary hard-sphere mixtures. II. Asymptotic behavior and structural crossovers.

The structural properties of additive binary hard-sphere mixtures are addressed as a follow-up of a previous paper [S. Pieprzyk et al., Phys. Rev. E 101, 012117 (2020)]2470-004510.1103/PhysRevE.101.012117. The so-called rational-function approximation method and an approach combining accurate molecular dynamics simulation data, the pole structure representation of the total correlation functions, and the Ornstein-Zernike equation are considered. The density, composition, and size-ratio dependencies of the leading poles of the Fourier transforms of the total correlation functions h_{ij}(r) of such mixtures are presented, those poles accounting for the asymptotic decay of h_{ij}(r) for large r. Structural crossovers, in which the asymptotic wavelength of the oscillations of the total correlation functions changes discontinuously, are investigated. The behavior of the structural crossover lines as the size ratio and densities of the two species are changed is also discussed.

Structural properties of additive binary hard-sphere mixtures.

An approach to obtain the structural properties of additive binary hard-sphere mixtures is presented. Such an approach, which is a nontrivial generalization of the one recently used for monocomponent hard-sphere fluids [S. Pieprzyk, A. C. Brańka, and D. M. Heyes, Phys. Rev. E 95, 062104 (2017)2470-004510.1103/PhysRevE.95.062104], combines accurate molecular-dynamics simulation data, the pole structure representation of the total correlation functions, and the Ornstein-Zernike equation. A comparison of the direct correlation functions obtained with the present scheme with those derived from theoretical results stemming from the Percus-Yevick (PY) closure and the so-called rational-function approximation (RFA) is performed. The density dependence of the leading poles of the Fourier transforms of the total correlation functions and the decay of the pair correlation functions of the mixtures are also addressed and compared to the predictions of the two theoretical approximations. A very good overall agreement between the results of the present scheme and those of the RFA is found, thus suggesting that the latter (which is an improvement over the PY approximation) can safely be used to predict reasonably well the long-range behavior, including the structural crossover, of the correlation functions of additive binary hard-sphere mixtures.

Virial coefficients of the additive hard-sphere binary mixtures up to the eighth

ABSTRACTThe fifth to eighth virial coefficients of the additive hard-sphere binary mixtures are accurately calculated in the whole range of sphere diameter ratios. Calculations are based on an extension of the technique for topological analysis of the Ree–Hoover diagrams and the standard Metropolis Monte Carlo algorithm. The results for the fifth to the seventh partial virial coefficients are compared with literature data. The eighth virial coefficients are new. The results are also compared with a semi-empirical analytical equation due to Wheatley and discussed.

Fourth virial coefficient of additive hard-sphere mixtures in the Percus–Yevick and hypernetted-chain approximations

The fourth virial coefficient of additive hard-sphere mixtures, as predicted by the Percus-Yevick (PY) and hypernetted-chain (HNC) theories, is derived via the compressibility, virial, and chemical-potential routes, the outcomes being compared with exact results. Except in the case of the HNC compressibility route, the other five expressions exhibit a common structure involving the first three moments of the size distribution. In both theories, the chemical-potential route is slightly better than the virial one and the best behavior is generally presented by the compressibility route. Moreover, the PY results with any of the three routes are more accurate than any of the HNC results.

Chemical-potential route for multicomponent fluids

The chemical potentials of multicomponent fluids are derived in terms of the pair correlation functions for arbitrary number of components, interaction potentials, and dimensionality. The formally exact result is particularized to hard-sphere mixtures with zero or positive nonadditivity. As a simple application, the chemical potentials of three-dimensional additive hard-sphere mixtures are derived from the Percus-Yevick theory and the associated equation of state is obtained. This Percus-Yevick chemical-route equation of state is shown to be more accurate than the virial equation of state. An interpolation between the chemical-potential and compressibility routes exhibits a better performance than the well-known Boublík-Mansoori-Carnahan-Starling-Leland equation of state.

Communication: Virial coefficients and demixing in highly asymmetric binary additive hard-sphere mixtures

The problem of demixing in a binary fluid mixture of highly asymmetric additive hard spheres is revisited. A comparison is presented between the results derived previously using truncated virial expansions for three finite size ratios with those that one obtains with the same approach in the extreme case in which one of the components consists of point particles. Since this latter system is known not to exhibit fluid-fluid segregation, the similarity observed for the behavior of the critical constants arising in the truncated series in all instances, while not being conclusive, may cast serious doubts as to the actual existence of a demixing fluid-fluid transition in disparate-sized binary additive hard-sphere mixtures.

Universal cubic equation of state and contact values of the radial distribution functions for multi-component additive hard-sphere mixtures

A universal cubic equation of state (UC EOS) is proposed based on a modification of the virial Percus-Yevick (PY) integral equation EOS for hard-sphere fluid. The UC EOS is extended to multi-component hard-sphere mixtures based on a modification of Lebowitz solution of PY equation for hard-sphere mixtures. And expressions of the radial distribution functions at contact (RDFC) are improved with the form as simple as the original one. The numerical results for the compressibility factor and RDFC are in good agreement with the simulation results. The average errors of the compressibility factor relative to MC data are 3.40%, 1.84% and 0.92% for CP3P, BMCSL equations and UC EOS, respectively. The UC EOS is a unique cubic one with satisfactory precision among many EOSs in the literature both for pure and mixture fluids of hard spheres.

On the critical non-additivity driving segregation of asymmetric binary hard sphere fluids

A previously proposed version of thermodynamic perturbation theory, appropriate for singular pair interactions between particles, is applied to binary mixtures of hard spheres with non-additive diameters. The critical non-additivity ΔC required to drive fluid–fluid phase separation is determined as a function of the ratio ξ ≤ 1 of the diameters of the two species. ΔC(ξ) is found to decrease with ξ and to go through a minimum for ξ ≃ 0.015 before increasing sharply as ξ → 0, irrespective of the total packing-fraction η of the mixture. These results are the basis of an estimate of the range of size ratios for which a binary mixture of additive hard spheres exhibit a fluid–fluid miscibility gap. This range is conjectured to be 0.01 ≲ ξ ≲ 0.1.

Critical consolute point in hard-sphere binary mixtures: Effect of the value of the eighth and higher virial coefficients on its location

Various truncations for the virial series of a binary fluid mixture of additive hard spheres are used to analyze the location of the critical consolute point of this system for different size asymmetries. The effect of uncertainties in the values of the eighth virial coefficients on the resulting critical constants is assessed. It is also shown that a replacement of the exact virial coefficients in lieu of the corresponding coefficients in the virial expansion of the analytical Boublík–Mansoori–Carnahan–Starling–Leland equation of state, which still leads to an analytical equation of state, may lead to a critical consolute point in the system.

Exact analytical formula for a fifth virial coefficient of a hard-sphere mixture at infinite dilution for small diameter ratios

A new exact analytical formula for the fifth partial virial coefficient, B5[1], of the additive hard-sphere mixture (i.e., for four spheres having diameters of σ1 and one sphere having a diameter of σ2), for small diameter ratios (s = σ2/σ1), is derived. The derivation method is based on simple geometrical arguments and uses as the only input three first terms in the density expansion of the radial distribution function.

Analytical expressions for the fourth virial coefficient of a hard-sphere mixture

A method of numerical calculation of the fourth virial coefficients of the mixture of additive hard spheres is proposed. The results are compared with an exact analytical formula for the fourth partial virial coefficient B4[1] (i.e., three spheres of diameters sigma1 and one sphere of diameter sigma2) and a semiempirical expression for B4[2] (i.e., two spheres of each kind). It is shown that the first formula is nonanalytic and the implication to the equations of state for hard-sphere mixtures is discussed.

Contact values for disparate-size hard-sphere mixtures

A universality ansatz for the contact values of a multicomponent mixture of additive hard spheres is used to propose new formulae for the case of disparate-size binary mixtures. A comparison with simulation data and with a recent proposal by Alawneh and Henderson for binary mixtures shows reasonably good agreement with the predictions for the contact values of the large–large radial distribution functions. A discussion on the usefulness and limitations of the new proposals is also presented.

Alternative fundamental measure theory for additive hard sphere mixtures

The purpose of this short paper is to present an alternative fundamental measure theory (FMT) for hard sphere mixtures. Keeping the main features of the original Rosenfeld's FMT [Phys. Rev. Lett. 63, 980 (1989)] and using the dimensional and the low-density limit conditions a new functional is derived incorporating Boublik's multicomponent extension [Mol. Phys. 59, 371 (1986)] of highly accurate Kolafa's equation of state for pure hard spheres. We test the theory for pure hard spheres and hard sphere mixtures near a planar hard wall and compare the results with the original Rosenfeld's FMT and one of its modifications and with new very accurate simulation data. The test reveals an excellent agreement between the results based on the alternative FMT and simulation data for density profile near a contact and some improvement over the original Rosenfeld's FMT and its modification at the contact region.

A new generalization of the Carnahan-Starling equation of state to additive mixtures of hard spheres

We introduce an expansion of the equation of state for additive hard-sphere mixtures in powers of the total packing fraction with coefficients which depend on a set of weighted densities used in scaled particle theory and fundamental measure theory. We demand that the mixture equation of state recovers the quasiexact Carnahan-Starling [J. Chem. Phys. 51, 635 (1969)] result in the case of a one-component fluid and show from thermodynamic considerations and consistency with an exact scaled particle relation that the first and second orders of the expansion lead unambiguously to the Boublik-Mansoori-Carnahan-Starling-Leland [J. Chem. Phys. 53, 471 (1970); J. Chem. Phys. 54, 1523 (1971)] equation and the extended Carnahan-Starling equation introduced by Santos et al. [Mol. Phys. 96, 1 (1999)]. In the third order of the expansion, our approach allows us to define a new equation of state for hard-sphere mixtures which we find to be more accurate than the former equations when compared to available computer simulation data for binary and ternary mixtures. Using the new mixture equation of state, we calculate expressions for the surface tension and excess adsorption of the one-component fluid at a planar hard wall and compare its predictions to available simulation data.

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.

Rescaled density expansions and demixing in hard-sphere binary mixtures.

The demixing transition of a binary fluid mixture of additive hard spheres is analyzed for different size asymmetries by starting from the exact low-density expansion of the pressure. Already within the second virial approximation the fluid separates into two phases of different composition with a lower consolute critical point. By successively incorporating the third, fourth, and fifth virial coefficients, the critical consolute point moves to higher values of the pressure and to lower values of the partial number fraction of the large spheres. When the exact low-density expansion of the pressure is rescaled to higher densities as in the Percus-Yevick theory, by adding more exact virial coefficients a different qualitative movement of the critical consolute point in the phase diagram is found. It is argued that the Percus-Yevick factor appearing in many empirical equations of state for the mixture has a deep influence on the location of the critical consolute point, so that the resulting phase diagram for a prescribed equation has to be taken with caution.

Phase behavior of binary mixtures of sterically stabilized colloids with large size asymmetry.

Experimental phase diagrams of three types of mixtures of sterically stabilized colloids are presented. The size ratios are kept similar, 0.15< or =xi< or =0.17, while the thickness and the chemical nature of the steric layers are varied. For all particles their effective volume fractions are calculated from their hydrodynamic radii. When their phase behavior is expressed in this way, the experimental liquidus curves all lie slightly above recent computer simulation predictions for the fluid-solid binodal of additive hard sphere mixtures. No dramatic shift of the experimental liquidus curves due to nonadditive particle interactions is observed. The dense phase is in all cases solid, with crystallites of the large spheres visible in some samples.