Adhesive Hard Sphere Model Research Articles

On the self-assembly of sodium caseinate

Abstract A detailed study of the concentration dependence of the osmotic pressure of sodium caseinate solutions encompassing the dilute, semi-dilute and highly concentrated regimes is presented. For the semi-dilute regime, assuming the model of sub-micelles forming `association polymers', the data is considered in relation to the adhesive hard sphere (AHS) model and an adapted f -functional polycondensation theory. The AHS model fits reasonably at higher temperatures but is unsatisfactory at 30°C. The polycondensation theory in general provides better fits and indicates low functionality ( f ca. 3) at 30°C suggesting non-centrally symmetric growth of aggregates (not within the scope of the AHS ( f =∞) model), a prediction supported by EM studies. Both models suggest small sub-micellar building blocks of diameter ca. 11 nm and aggregation number 4–5 with implicit voluminosity comparable to that of native casein sub-micelles. To represent the behaviour in the highly concentrated regime, above close packing of the sub-micelles, a rigid lattice model is proposed which suggests `soft sphere’ potentials for the sub-micelles ( V ( r )∼ r −6.75 ). Continuous shear rheology, contrasted with the Krieger–Dougherty hard sphere prediction, shows the expected sensitivity to aggregation at low phase volumes and to particle softness above close packing.

Supra-aggregates of Casein Micelles as a Prelude to Coagulation

Native casein micelles (i.e., the micelles in fresh milk) can be treated as a collection of polydisperse hard spheres. This follows from small angle neutron scattering, viscosity, diffusivity, and other measurements. Therefore, the equilibrium and transport properties of native casein micelles in an ultrafiltration permeate solvent can be described by theories developed for colloidal hard sphere dispersions because native casein micelles are sterically stabilized by a brush of κ-casein (CN) molecules (and perhaps β-CN). The κ-CN brush induces a very short-ranged or steep repulsion (it rises from zero to a large value) when two micelles meet each other. Electrostatic interactions are highly screened because of the high ionic strength (0.08 M) of skim milk. The colloidal properties of casein micelles change with the technological treatment applied to skim milk. This paper describes the consequences for the casein micelle properties of heating, renneting, and acidification.It appears that the properties of the micelles can be described generally by adopting the adhesive hard sphere model. In that model, the steep repulsive interaction of two micelles is preceded by a short-ranged Van der Waals attraction. By relating the strength of the attraction to the degree of technological treatment (e.g., renneting time or pH changes), the colloidal properties of the micelles can be described simply by using adhesive hard sphere theory. This theory also predicts the phase behavior of such a system. For instance, it predicts correctly the gelation of casein micelles under various conditions.The adhesive hard sphere model allows a general and consistent understanding of the colloidal properties of casein micelles caused by technological treatments. The practical relevance of the models is illustrated with a few examples.

Light-Scattering Studies of Concentrated Copolymer Micellar Solutions

We report the analyses of an extensive set of light scattering intensity distributions from triblock copolymer micelles in aqueous solutions. We investigated two Pluronics, P84 (PEO19-PPO43-PEO19) and P104 (PEO27-PPO61-PEO27) in an entire range of the disordered micellar phase. Our previous small-angle neutron scattering (SANS) study showed that the micelle is spherical and consists of a compact hydrophobic core and a hydrophilic corona region. In that paper (Liu, Y. C., et al. Macromolecules 1998, 31, 2236−2244), we proposed a “cap-and-gown” model for the microstructure of the micelle, taking into consideration the polymer segmental distribution and water penetration profile in the core and corona regions. We took into account the intermicellar correlation using the adhesive hard-sphere model of Baxter. With this combined model, we obtained consistent trends for important parameters such as the aggregation number, hydration number per polymer in both the core and corona, and surface stickiness. The struc...

Small-Angle Neutron Scattering Analysis of the Structure and Interaction of Triblock Copolymer Micelles in Aqueous Solution

We report analysis of an extensive set of small angle neutron scattering intensity distributions from triblock copolymer micelles in aqueous solutions. We investigated two Pluronics, P84 (PEO19 PPO43 PEO19) and P104 (PEO27 PPO61 PEO27), in an entire range of disordered spherical micellar phases both in temperature and in concentration. At room temperature, both poly(ethylene oxide) (PEO) and poly(propylene oxide) (PPO) are hydrophilic but at elevated temperature, PPO becomes significantly less hydrophilic, thus creating a thermodynamic driving force for micellization. It has been known that the resultant spherical micelle consists of a hydrophobic core and a hydrophilic corona region. In this paper, we propose a “cap-and-gown” model for the microstructure of the micelle, taking into consideration the polymer segmental distribution and water penetration profile in the core and corona regions. We take into account the intermicellar correlation using an adhesive hard sphere model of Baxter. With this combine...

Skim Milk Acidification

Skim milk was acidified slowly using glucono-δ-lactone and HCl. On lowering the pH the steric stabilization of the casein micelles diminishes and as a result the micelles exhibit attractive interactions. These interactions can be quantitatively described using an adhesive hard sphere (AHS) model. The strength of the attraction is found to be proportional to 1/(pH − pC); it is suggested that pC may be related to an effective dissociation constant of the carboxylic acid groups along the κ-casein backbone. If pH approaches pC, the polyelectrolyte-like κ-casein brush collapses onto the surface of the micelles. The AHS model is quantitatively tested using viscosity and dynamic light scattering measurements.

Adhesive-hard-sphere approximation for the structure of square-well fluids: application to colloidal dispersions

The accuracy of the adhesive-hard-sphere (AHS) model in representing structure factors of the short-range square-well potentials is examined by comparing theoretical predictions with Monte Carlo simulations for the square-well fluids over a range of conditions. We show that the analytical solution for the structure factors of the AHS model can be used to describe structure factors of the square-well potential adequately at low to moderate volume fractions (ϕ < 0·26) when the range of the square-well potential is less than 20% of the particle diameter and the magnitude of the contact potential is less than 3. At low values of the width of the square well (less than 5% of the particle diameter), the AHS model can be applied to even larger values of the contact potential and to higher volume fractions. We also show that the AHS model can be used to describe accurately structure factors of the non-ionic micro-emulsions and inverted micellar systems containing sodium bis 2-ethylhexyl-sulphosuccinate (aerosol-O...

Chemically modified proteins solubilized in AOT reverse micelles. Influence of proteins charges on intermicellar interactions

In previous papers it has been shown that solubilization of native cytochrome c in sodium bis (2-ethylhexyl) sulfosuccinate/water/isooctane reverse micelles induces a percolation phenomenon in dilute micellar solution compared to empty micelles. This has been attributed to an increase in the interaction potential between droplets. To confirm this hypothesis, structural studies of microemulsions filled by various cytochrome c differing by their net charge (+8, −3 and −12) have been performed by SAXS. The intermicellar interactions are analyzed with the adhesive hard-sphere model. It is shown that electrostatic interactions between the interfacial wall and the protein play an important role on the intermicellar interactions and on the percolation onset. This behavior has been confirmed by the change in the critical temperature for empty micelles and micelles filled by various cytochrome c derivatives.

Stability diagrams for colloidal dispersions with attractive interactions: An analytical approximation

We show that a single-parameter, adhesive-hard-sphere (AHS) model can be used to estimate, analytically, the stability diagrams for colloidal dispersions interacting through fairly long-range attractive forces. The use of the model is illustrated for polymer-induced depletion interactions. Analytical expressions are given for the static structure factors, pressures and chemical potentials in terms of the adhesiveness parameter of the AHS model. The adhesiveness parameter is obtained as a function of the relevant parameters appropriate for the actual interaction forces using the second virial coefficient (which can be measured experimentally using scattering techniques). It is shown that both upper-critical and lower-critical stability domains can be predicted, in addition to static structure factors. The results presented are consistent with the predictions based on more elaborate statistical mechanical theories.

The Turbidity of Renneted Skim Milk

The turbidity of skim milk dispersions was measured in a spectrophotometer at four different wavelengths and at various dilutions. After adding rennet to milk the transmission (∼exp(-turbidity)) of the samples decreases gradually. This decrease in transmission, which depends on the volume fraction of the micelles and the wavelength of the light used, is caused by the interactions between the micelles. During the renneting the interactions change from repulsive to attractive and can be characterized by the second osmotic virial coefficient. Before the rennet is added to the milk the casein micelles can be characterized as hard spheres for which B′ 2 = 4 V HS, where V HS is the (excluded) volume of a hard sphere. Renneting causes the casein micelles to lose their colloidal stability and B′ 2 decreases strongly. The micelle interaction can be described by the so-called adhesive hard sphere model, also called the "sticky" sphere model. A theory is presented which quantitatively describes the changes in transmission on renneting in the initial stages.

Solvation interactions in colloidal dispersions using the adhesive hard sphere model

Colloidal particles dispersed in a liquid are modelled by a binary mixture of large spheres in a “solvent” of small spheres; so the discrete nature of the solvent molecules is explicitly taken into account. It is shown in the Percus-Yevick approximation that, following the adhesive hard sphere model, the effective repulsion between the large particles decreases by increasing the stickness between the solvent molecules (simulation of a “poor” solvent). The isothermal osmotic compressibility goes to infinity when the adhesive strength between the solvent becomes very high and phase separation may occur. By increasing the stickness between the solvent and large solute particles the effective repulsion between the solute particles increases, which can be interpreted as a better solvation of the particles.

Sol-Gel transition in dense dispersions: Effect of particle bridging

Sol-gel transition in dense dispersions of particles is modeled through the use of statistical mechanical and continuum percolation theories. Specifically, we examine the effect of particle bridging on the gelation and clustering of a binary dispersion of particles which interact through a short-ranged potential represented by the adhesive hard-sphere model. The gelation thresholds for dense dispersions are determined analytically as a function of the particle concentrations, particle size ratios, and interparticle interaction within the context of the Percus-Yevick integral equation theory. It is found that depending on the temperature, particle concentration, and particle size ratio, the dispersion may undergo a single sol-to-gel transition, or a series of sol-to-gel, gel-to-sol, and sol-to-gel transitions. Pair-connectedness functions which account for the degree of aggregation in the dispersion are also determined analytically in the Percus-Yevick approximation.

A turbidity study on colloidal silica particles in concentrated suspensions using the polydisperse adhesive hard sphere model

A new type of analysis of turbidimetric measurements is proposed for concentrated sterically stabilized colloidal dispersions. It is applied to dispersions of silica spheres, coated with stearylalcohol in aromatic solvents up to high volume fractions of 0.30–0.40. Baxter’s polydisperse adhesive hard sphere model is used to explain the results. Influence of diameter, stickiness, and polydispersity in diameter on turbidity will be discussed. Model calculations fit the experimental curves very well up to high volume fractions and from the data it can be concluded that there exists a small attraction between the particles.

Continuum percolation and pair-connectedness function in binary mixtures of strongly interacting particles

The authors study continuum percolation and aggregation in binary mixtures of strongly interacting particles. The clustering and connectivity behaviour of the dispersed particles are determined using the Ornstein-Zernike integral equation in the Percus-Yevick (PY) integral equation. Specifically, they consider a binary mixture of spheres in which the interactions between like species are strongly attractive, while the interaction between unlike species is purely repulsive. They model this system through hard-core square-well (SW) potentials, which they approximate by the adhesive hard-sphere model, which yields analytic solutions in the PY theory.

Generalized mean spherical approximations.

A criterion for determining the coupling and the exponent [K(\ensuremath{\eta},\ensuremath{\beta}) and \ensuremath{\zeta}(\ensuremath{\eta},\ensuremath{\beta})] of the one-Yukawa generalized mean spherical approximation (GMSA), which better describes a given real fluid, is suggested. The relative spatial configuration of the surface (\ensuremath{\equiv}${\ensuremath{\Sigma}}_{\ensuremath{\chi}}$) where the compressibility diverges, with respect to the surface (\ensuremath{\equiv}${\ensuremath{\Sigma}}_{\ensuremath{\Delta}}$) bounding the region where physical solutions of the GMSA equations exist, is studied in terms of K, \ensuremath{\zeta}, and \ensuremath{\eta} (the packing fraction). ${\ensuremath{\Sigma}}_{\ensuremath{\chi}}$ is found to lie always in the physically accessible region and to become tangent to ${\ensuremath{\Sigma}}_{\ensuremath{\Delta}}$ along the \ensuremath{\eta} axis; that consequently will be the locus of the critical points of the real systems. By the aforesaid criterion we show that the Baxter solution of the adhesive hard-sphere model can be obtained from the limit of the GMSA solution as \ensuremath{\zeta}\ensuremath{\rightarrow}\ensuremath{\infty}. The study of the limit \ensuremath{\zeta}\ensuremath{\rightarrow}0 shows that the GMSA critical indices are generally different from the mean-field ones and that their values depend on the way \ensuremath{\zeta} approaches zero as one moves along the critical isochore and the critical isotherm. Attention is also called to the fact that, as \ensuremath{\zeta}\ensuremath{\rightarrow}0, K(\ensuremath{\eta},\ensuremath{\zeta})/${\ensuremath{\zeta}}^{2}$ has a minimum at \ensuremath{\eta}=0.128, a value rather close to the critical densities of the real fluids.

Competitive adsorption from binary mixtures: Adhesive hard sphere model

A model study is given of the phenomenon of competitive adsorption using the theories of liquid state physics. The adsorbate is modeled as a mixture of hard spheres with a short-range attraction. The composition dependence of the specific surface excess is studied as a function of molecular size ratio and intermolecular potentials.

Clustering phenomena in adhesive hard sphere fluids. Theory of freezing

We develop a version of the physical cluster theory of the equation of state based on a definition of a “cluster” which depends both on particle positions and relative momenta, namely we choose to call a group of i molecules a physical cluster if their total energy (kinetic energy plus interactions) does not exceed the translation energy of the centre-of-mass of the whole group. This definition in addition with the “microcrystal” model of a cluster allows us to calculate the properties of individual clusters, to consider the clustering phenomena and to write down an equation of state for a dilute system as well as for a system at moderate and high densities. The interaction between molecular aggregates is taken into consideration by using the adhesive hard sphere model and the Percus-Yevick approximation. Based on the relations obtained we develop the theory of freezing and perform a numerical investigation of the equilibrium behavior of the system at various temperatures and densities. In particular, the theory enables us to describe the regular phase diagram which includes (in coordinates pressure-temperature) three branches intersecting at the triple point. It is interesting that the coexistence curve of the fluid-solid as well as liquid-gas phase transition has an end point.

Analytic and Perturbation Theories of Fluid Mixtures

AbstractExact solutions to the Percus‐Yevick and Mean Spherical approximations for various model multicomponent systems are reviewed, with a view to their providing suitable reference states for perturbation theories of fluid mixtures. It is demonstrated that the formulation of the Ornstein‐Zernike equation due to Baxter [1] provides the most convenient approach to these problems, especially with respect to numerical computation of model correlation functions [2].The existence of these exact solutions has recently been shown to lead to a relatively simple theory of fluid mixtures at solid interfaces [3–5]. These results are reviewed in the light of the adhesive hard sphere model [6, 7], which although being a highly idealized representation of a fluid, can be shown to provide a convenient reference state for a more general perturbation theory than that based on a hard sphere reference system. This theory promises to be especially useful away from the dense liquid region, as the phase transitions are included in the reference part of the free energy.

Solution of the compressibility equation of the adhesive hard-sphere model for mixtures

The solution of the generalized equations of Percus and Yevick is obtained for mixtures of molecules interacting via the adhesive hard-sphere potential. It is shown that the possibility of a discontinuous phase or fluid-fluid transition appears only in such binary systems where the adhesion acts between like particles of at least one of the components. The found distribution functions are used to obtain from the compressibility equation the expression for the pressure of a mixture consisting of v components in arbitrary concentration. The pressure, chemical potential and other thermodynamic properties are calculated explicitly in the limit when several components (the solutes) are dilute and one component (the solvent) is in its “liquid” phase. The solubility of substances of small molecules is shown to increase when the temperature T rises. The same regularity is found in the case when the interaction between solvent and solute consists only of a hard-sphere potential. In the general case of the presence of adhesion between solvent and solute molecules and for arbitrary ratio of particles sizes a minimum appears on the solubility cuves versus T. If the attraction is sufficiently big the drop in the solubility may be very sharp and reach the value zero at a certain temperature. The results obtained are qualitatively supported by examples from experiment.