Sticky Hard Sphere Research Articles

Analysis of the hypernetted chain equation for ionic fluids

It is well known that the numerical solution of the hypernetted chain (HNC) equations yields satisfactory results for the pair correlation function of the primitive model of electrolytes and similar models of ionic particles over a considerable range of thermodynamic states. Despite this, it has become apparent that for low densities (or low ionic concentration in electrolytes) the numerical solution breaks down for temperatures well above the expected coexistence region between gas and liquid phases. Here we study the situation by analytic means, comparing it to a similar problem for sticky hard spheres in the Percus-Yevick (PY) approximation. On the basis of our analysis we conclude that the failure of the HNC is of the same nature and is connected to the existence of two possible solutions for low densities. When the temperature is lowered these solutions will merge into one at a particular temperature, below which a real solution is no longer possible. By extending our analysis to systems like the monoatomic Lennard-Jones fluid and comparing with previous results of Gallerani, Lo Vecchio and Reatto for two-Yukawa and Lennard-Jones systems in the PY approximation, we conclude that these are general common features of the HNC and PY approximations in the low-density regime. Numerically they appear to persist in the HNC closure at high densities as well (in contrast to the behaviour of the PY approximation) although our analysis is silent in this regard. Our analysis is consistent with the results of a recent comprehensive numerical study by Belloni.

Polydispersity in colloids: implications to static structure and scattering

Theoretical and experimental analyses of the structure of colloidal dispersions have traditionally relied on model systems of monodisperse particles. However, numerous theoretical studies aimed at incorporating the effects of size-polydispersity, charge-polydispersity, and polydispersity in attractive interactions on the equilibrium structure of dispersions of spherical particles have appeared in recent years. These studies can be divided into three different groups according to the nature of the interaction forces considered, namely, dispersions of neutral hard spheres, sticky hard spheres, and charged particles. Here, we review these studies and examine the major results and conclusions. Previously unreported new results obtained by us for dispersions of charged spheres in high-ionic-strength media are also included. The theoretical techniques for computing partial structure factors and average structure factors for the above-mentioned interaction potentials are summarized, and analytical expressions are presented wherever appropriate. Representative examples of applications of each class of interaction potentials identified above are presented in order to illustrate the use of the theoretical tools in interpreting the scattering data. The common features of the average structure factors for the above model potentials are emphasized, and issues such as the importance of correctly modeling intraparticle scattering properties of the particles and transformation of observed intensities to average structure factors are illustrated. An examination of the so-called decoupling approximation (which seeks to separate the intraparticle scattering effects from interparticle interaction effects) is also presented wherever the required data are available.

Clustering and relaxation in oil-continuous microemulsions

In the droplet phase of oil-continuous microemulsions clustering of the droplets has been observed by a number of experimental techniques such as dielectric constant and viscosity measurements. The clustering is found to be characterized by a droplet radius dependent binding enthalpy. Structure factors obtained using SAXS are found to fit those predicted by Baxter's sticky hard sphere model. An extension of this model to a positive value of the binding enthalpy is given. In dynamic Kerr measurements the slow relaxation due to rotation of clusters is observed.

The Percus-Yevick approximation for repulsive hard spheres with surface adhesion

We have adapted Baxter's sticky hard sphere (SHS) model so that it more adequately describes systems where the interparticle potential well is preceded by an energy barrier. In the original model the particles interact via a pair potential with a narrow attractive region next to a repulsive core. Colloidal particles, however, often exhibit long range repulsion whereas they do attract each other at shorter ranges. This motivated our effort to extend the model. It is shown that the Percus-Yevick equation can also be solved analytically for this particular potential, provided that the regions of attraction and repulsion are sufficiently small. It is found that the structure functions of both models are identical. The new feature of the repulsive sticky hard sphere (RSHS) model is that the fraction of aggregated particles may increase with temperature, a phenomenon which is experimentally observed in such systems. Structure functions from small angle X-ray studies on water/AOT/iso-octane microemulsions can be fit to those predicted by the RSHS model. The thus obtained binding enthalpy is comparable with earlier determinations from dielectric studies.

A model for repulsive hard spheres with surface adhesion

Baxter’s sticky hard sphere model has been extended so that it adequately describes systems where the attractive region of the interparticle potential is preceded by an energy barrier. The new feature of the model is that the fraction of aggregated particles may increase with temperature.

Adsorption of sticky hard spheres: relevance to protein competitive adsorption

The sticky hard sphere model provides a simple analytic theory of competitive adsorption which may be relevant to the modelling of protein competitive adsorption. Equilibrium monolayer compositions are calculated here for binary mixtures of equal sized molecules as a function of adsorbate concentration and difference in ‘stickiness’ at the adsorbent surface. These numerical results are compared with some of our recent experimental measurements on protein-stabilized emulsions.

A new interpretation of the sticky hard sphere model

The basic results of the sticky hard sphere model are derived using a perturbative solution of the factorized form of the Ornstein–Zernike equation and the Percus–Yevick closure relation. The perturbation parameter is Δ/(σ+Δ), where Δ and σ are, respectively, the width of the attractive square well pair potential and the hard core diameter. This derivation leads naturally to an expression for the stickiness parameter, different from the one conventionally used, without invoking the concept of an infinitely deep potential. The theoretical structure factor is compared with two sets of Monte Carlo simulation data and excellent agreement is observed in both cases without the scaling of the square well potential suggested in literature.

On the stability of charge-stabilized colloidal dispersions

Reversible 'liquid-vapour' like transitions and irreversible coagulation in charge-stabilized colloidal dispersions are studied using a one-component macroparticle framework with a conventional DLVO interaction potential. The spinodal and coexistence curves are calculated as a function of salt concentration using thermodynamic perturbation theory-specifically the random-phase approximation using firstly hard spheres and then sticky hard spheres as reference systems. With the hard spheres, the hard-core location varies between the first and second zero of the DLVO potential. In the case of the sticky hard spheres, the hard core is located at the first zero of the DLVO potential; this models the deep narrow primary minimum of this potential. The irreversible coagulation is distinguished from the reversible transition by the flattening of the coexistence curve at a value of salt concentration above the range over which the 'liquid-vapour' transition takes place.

Small angle neutron scattering from micellar solutions of triton X-100

Micellar solutions of non-ionic surfactant triton X-100 (8% by weight) show phase separation at cloud pointTcp ∼ 335 K. This paper reports results of small angle neutron scattering (SANS) experiments from this solution as a function of temperature between 298 and 332 K. The range of wave-vector transferQ, covered in these experiments is from 0.02 to 0.15 A−1. It is seen that as one approachesTcp, the neutron scattering cross section diverges in the region of lowQ ( 0.06 A−1). We believe that the divergence of scattering at lowQ with an increase in temperature is because of changes in the structure factorS(Q) of the solution. The measured distributions have been analyzed using four different models for inter-micellar potential. The models used to calculate the structure factorS(Q) are (1) mean spherical approximation (MSA) with Yukawa tail for attractive potential, (2) MSA with an attractive square well potential, (3) random phase approximation (RPA) with an attractive square-well potential and (4) Sticky hard sphere model (attractive square-well potential with Percus-Yevick approximation). The strengths of the attractive potential required to fit the SANS data are (−6.6 to − 14.4)/kt for model (1), (− 6.6 to − 15.0)/kt for model (2), (− 3.8 to − 7.3)/kB T for model (3) and (−2 to −2.7)/kt for model (4). On the basis of reasonableness of the derived strength of the potential near the phase separation temperature and its relative temperature dependence, it is concluded that present data favour the Sticky hard sphere model.

Percolation in binary mixtures with strong attraction between unlike particles

The percolation and clustering in binary mixtures with strong attraction between unlike particles is investigated. More specifically, the authors consider a binary mixture in which the interaction between unlike particles is described by Baxter's sticky hard sphere (SHS) potential, while the interaction amongst the same species is the hard sphere repulsion. The Ornstein-Zernike (OZ) equation for the pair connectedness is solved under the Percus-Yevick (PY) approximation. The percolation line always approaches zero as the density vanishes and also percolation only occurs above some lower limit of density. The percolation line is compared with the phase transition line and it is found that the existence of the phase transition line restricts the range of concentration for which percolation occurs. Moreover, the influences of density, stickiness, concentration and particle size on the pair connectedness, percolation line, mean cluster size and the coordination number are examined.

A solvable model of polydisperse charged particles with sticky interactions

The statistical mechanics of a polydisperse system of charged particles with sticky interactions is studied in the mean spherical/Percus–Yevick approximation. The excess thermodynamic properties are calculated, taking as a reference system the sticky hard spheres mixture and are given in terms of a single scaling parameter ΓT. When the sticking probability is of the form λij=λiλj, then all the equations are explicitly solvable for arbitrary mixtures of charged systems. We obtain explicit expressions for the thermodynamic properties, structure function, and the Laplace transform of the pair distribution function.

The effect of the composition of the oil phase on the microstructure and dielectric behavior of water-in-oil microemulsions

A comparison is made between the clustering process occurring the sodium-di-2-ethylhexyl-sulfosuccinate microemulsion in the four organic solvents: n-decane, iso-octane, n-heptane, and n-hexane. The clustering process is studied by small angle x-ray scattering (SAXS) and dielectric relaxation spectroscopy. The structural parameters obtained from the analysis of the SAXS experiments using the sticky hard sphere droplet model are used to calculate the dielectric permittivity and are compared with results from the dielectric relaxation experiments. It is found that an increase in the attraction between microemulsion droplets when the chain-length of the solvent molecule is increased.

Diffusion coefficients and effective viscosity of suspensions of sticky hard spheres with hydrodynamic interactions

We study the short-time and long-time diffusion coefficients and the high-frequency and zero-frequency effective viscosity of a dilute suspension of sticky hard spheres with hydrodynamic interactions. Due to the singular nature of the hydrodynamic interactions near touching the transport coefficients are strongly affected by the short-range attractive potential. This suggests that the transport coefficients may be used as a test of the interaction potential.

Cavity functions and association in models for weak electrolytes and sticky hard spheres

Weak electrolytes and other association reactions are modeled as sticky spheres. An analysis of the density expansion, including the bridge diagrams, of the cavity functions yAB(L) for sticky hard spheres (charged or uncharged spheres binding at a distance L) leads to an approximation which provides the degree of association α as the solution to a simple quadratic equation determined by the association constant K0 and the cavity function y0AB(L) for the reference system in which the chemical bonding between the reacting species has been turned off. Similar relations are assumed to hold when the bonding is directional and specific enough to lead only to the formation of dimers. Applications to the determination of the reference cavity functions for acetic acid and monochloro acetic acid from experimental data of the degree of association are discussed. In a discrete solvent, the approximation scheme for α remains the same, except that the reference cavity function is scaled differently. Solvent medium effects on the association constant are shown to be related to the cavity function of the undissociated dimer in a pure solvent. An exponential approximation for the reference cavity function y0AB(L) is derived when the associating species are of the same size and the bonding is spherically symmetric. Expressions for the changes in the thermodynamic functions due to association are obtained analytically in terms of the degree of association and the reference cavity functions. The magnitude of the degree of association, calculated from the exponential approximation for y0AB(L), and its effect on the thermodynamic properties are different from what was previously observed using the hypernetted chain (HNC) approximation. The thermodynamics of weak 1–1 electrolytes are discussed using the new method and a comparison is made between the new and old methods for 2–2 electrolytes.

Bridging flocculation of sticky hard spheres

Bridging flocculation of colloidal particles + polymer molecules is modelled as a binary mixture of spheres with a sticky hard-sphere particle–polymer pair potential. Predictions are made for the dependence of the average cluster size on system composition.

Solvent effects in weak electrolytes. I. Effect of a hard sphere solvent on the sticky electrolyte model with L=σ

The Ornstein–Zernike (OZ) equations are solved for the sticky electrolyte model (SEM) with a hard sphere solvent using the hypernetted chain (HNC) approximation for the stickiness and the mean spherical (MS) approximation for the electrical interactions. Relations among the coefficients of Baxter’s q functions and the equation for the excess internal energy are given in the MS approximation for L≤σ, where σ is the molecular diameter, and L is the distance at which the oppositely charged ions can stick. The analytical results for L=σ in the HNC/MS and PY/MS approximations are presented in detail. When the charges are switched off, the results automatically lead to those of the sticky hard sphere system; when the stickiness is turned off and the discrete solvent is changed to a continuum, the results lead correctly to those of the restricted primitive model (RPM). The thermodynamic properties of the SEM in a hard sphere solvent for L=σ are calculated and compared with the properties in a continuum solvent; special attention is paid to the derivation of the osmotic coefficient in the McMillan–Mayer system for the SEM and for the corresponding uncharged system. By switching off the charge and the stickiness, the osmotic coefficient of an isotopic solute–solvent system is also obtained. The numerical results show that the hard sphere solvent has a strong packing effect on the structural and thermodynamic properties of the electrolyte and the association of the oppositely charged ions is greatly enhanced by the hard sphere solvent. The influence of a discrete solvent on the osmotic coefficient is quite subtle: for the charged system, the solvent tends to raise the osmotic coefficient; for the sticky hard sphere system, the solvent has just the opposite effect.

Solution of the Percus–Yevick approximation of the multicomponent adhesive spheres system applied to the small angle x-ray scattering from microemulsions

This paper describes the application of the numerical solution of the Percus–Yevick approximation for a multicomponent system of sticky hard spheres to small angle x-ray scattering experiments. The effect of polydispersity on the pair correlation function is quantitatively described, thus enabling the analysis of experimental data. In this way quantitative estimates of the size distribution width, interaction radii and the measure of attraction between particles can be obtained. The attractive interaction in a system of sticky spheres is known to give rise to a transition analogous to a gas–liquid phase transition. The effect of polydispersity on this transition is discussed. Finally it is shown how the model can be used to describe small angle x-ray scattering from an AOT microemulsion system.

The second dielectric virial coefficient of a dipolar sticky hard sphere fluid

We calculate the second dielectric virial coefficient, B ϵ, of a fluid whose molecules are sticky hard spheres with embedded central point dipoles. The effect of stickiness is to increases B ϵ markedly.

The dielectric constant of dipolar sticky hard spheres

The dielectric constant and the correlation function of a system of sticky hard spheres with an embedded point dipole is calculated at different densities, temperatures and dipole moments using the LHNC and LIN approximations. Comparison between these theories and with the dielectric constant of hard dipolar spheres are presented. It is found that LIN values are lower than LHNC ones and that the theories predict opposite trends for the dielectric constant when the stickiness is included. An analysis of this discrepancy is given.

Critical behaviour of the lattice gas model in the Percus-Yevick approximation

The solution of the Percus-Yevick (P.Y.) equation for the lattice gas, obtained by Levesque and Verlet, is studied in the critical region. The behaviour of the equation ofstate is similar to the one found for a fluid of sticky hard spheres in the P.Y. approximation: the critical indices have the classical value but the scaling function is non-universal, is strongly asymmetric in density with respect to the critical value and there is a spinodal curve only for the liquid phase. This suggests that these features are generally valid for the P.Y. approximation and are not specific for sticky hard spheres.