Adhesive Hard Sphere Research Articles

Analytic example of a free energy functional

We use the ideas of Percus for the construction of classical density functionals for two model interactions: simple hard spheres and adhesive hard spheres (AHSs). The required input, the properties of the uniform fluid, is taken from the analytic mean spherical solution for these two systems. For hard spheres we derive-via a bilinear decomposition of the direct correlation functions-a set of basis functions, which is the same as the one presented by Rosenfeld in his fundamental measure theory framework. For AHSs additional basis functions have to be considered to ensure the bilinear decomposition of the direct correlation functions; we present an expression for the free energy functional for the one-component case.

Characterizing nanoparticle interactions: Linking models to experiments

Self-assembly of nanoparticles involves manipulating particle interactions such that attractions are on the order of the average thermal energy in the system. If the self-assembly is to result in an ordered packing, an understanding of their phase behavior is necessary. Here we test the ability of simple pair potentials to characterize the interactions and phase behavior of silico tungstic acid (STA), a 1.2 nm particle. The strength of interaction is controlled by dispersing STA in different background salt concentrations. The experimental variables used in characterizing the interactions are the osmotic compressibility (dΠ/dρ), the second virial coefficient (B2), relative solution viscosity (η/ηc), and the solubility (ρσ3)sat. Various techniques are then developed to extract the parameters of square well, the adhesive hard sphere (AHS), and the Yukawa pair potentials that best describe the experimental data. The AHS model describes the solution thermodynamic behavior only where the system is weakly attractive but, as would be expected, fails when long range repulsions or nonmonotonic pair potentials become important. Model free representations are presented which offer the opportunity to extract pair potential parameters.

Molecular Mechanism of the Renneting Process of Casein Micelles in Skim Milk, Examined by Viscosity and Light-Scattering Experiments and Simulated by Model SCF Calculations

The properties of the outer hairy layer of casein micelles have been studied using several methods. Viscosity and light-scattering measurements have been performed on rennet-destabilized skim milk at different concentrations of calcium. We have reproduced the known result that calcium promotes destabilization at relatively high κ-casein surface coverages (even at only 50% of the κ-casein molecules cut off). This suggests that calcium bridges contribute to the casein micelle attraction. The effect may be direct by calcium-mediated bridging of β-casein or αs1-casein (phosphate/carboxylate) or κ-casein (carboxylate) or indirect by altering the interplay between several types of caseins in the micelle. In the analysis of the experiments, we make use of an adhesive hard sphere approximation. The casein hairy layer has further been modeled by a self-consistent field (SCF) theory in which coarse-grained molecular details were included. In these calculations the effect of calcium bridges cannot directly be accounted for. It is shown that a κ-casein layer induces repulsive interactions of mainly steric origin. Cutting the N-terminus part results in reduced (shorter-ranged) repulsive pair potentials, which still dominate over the van der Waals attraction. As a result of the altered interplay between the several types of casein molecules during the renneting process, the αs1-casein concentration in the outer hairy layer may increase. Again using the SCF model, but now using a less-detailed description of the system, we examine a possible result of this. We confirm that diblock copolymers (simple model for κ-casein) and diblock copolymer with shorter soluble block (simple model for para-κ-casein) give repulsion and triblock copolymers (simple model for αs1-casein) give attraction by means of bridging (adsorption). The renneting process is simulated by increasing the ratio of model para-κ-casein to κ-casein at a fixed αs1-casein concentration. As a result, the potential shifts from repulsive to attractive. We show that by reducing the range of repulsive interactions of the diblock copolymers, we can make room available to express the attractive contributions of the model αs1-casein. This comprises a novel molecular-based mechanism for the renneting process.

Connectedness-in-probability and continuum percolation of adhesive hard spheres: Integral equation theory

Integral equation theory was employed to study continuum percolation and clustering of adhesive hard spheres based on a “connectedness-in-probability” criterion. This differs from earlier studies in that an “all-or-nothing” direct connectivity criterion was used. The connectivity probability may be regarded as a “hopping probability” that describes excitation that passes from one particle to another in complex fluids and dispersions. The connectivity Ornstein–Zernike integral equation was solved for analytically in the Percus–Yevick approximation. Percolation transitions and mean size of particle clusters were obtained as a function of connectivity probability, stickiness parameter, and particle density. It was shown that the pair-connectedness function follows a delay-differential equation which yields analytical expressions in the Percus–Yevick theory.

Viscosity Anomaly and Charge Fluctuations in Dilute AOT Microemulsions with X < 20

An experimental investigation of the viscosity and microstructure of AOT/water/n-decane inverse microemulsions examines the maximum in the low shear viscosity as a function of the molar ratio of added water to surfactant, X, at a fixed volume fraction of droplets. Dilute viscometry and dynamic and static light scattering link the phenomena directly to a maximum in the attraction between droplets around X = 8. The strength of the attraction is quantified by static light scattering and compared with estimates via Baxter's adhesive hard sphere model from the viscometric and dynamic light scattering data. Together these confirm the connection, though the latter values exceed those from static light scattering, presumably due to an inaccurate treatment of hydrodynamic interactions. The exchange of ions between droplets is demonstrated to produce an attraction of sufficient strength and appropriate dependence on X with much less than one charge per droplet.

Nonergodicity transitions in colloidal suspensions with attractive interactions.

Colloidal gel and glass transitions are investigated using the idealized mode coupling theory (MCT) for model systems characterized by short-range attractive interactions. Results are presented for adhesive hard sphere and hard core attractive Yukawa systems. According to MCT, the former system shows a critical glass transition concentration that increases significantly with introduction of a weak attraction. For the latter attractive Yukawa system, MCT predicts low temperature nonergodic states that extend to the critical and subcritical region. Several features of the MCT nonergodicity transition in this system agree qualitatively with experimental observations on the colloidal gel transition, suggesting that the gel transition is caused by a low temperature extension of the glass transition. The range of the attraction is shown to govern the way the glass transition line traverses the phase diagram relative to the critical point, analogous to findings for the fluid-solid freezing transition.

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.

RESEARCH NOTE Analytical model for thermodynamic properties of an adhesive hard sphere Yukawa fluid

RESEARCH NOTE Analytical model for thermodynamic properties of an adhesive hard sphere Yukawa fluid

Ideal glass-glass transitions and logarithmic decay of correlations in a simple system

We calculate the ideal-glass-transition line for adhesive hard spheres in the temperature-volume-fraction plane within the framework of the mode-coupling theory. We find two intersecting lines, controlled by the hard-core and the adhesive part of the potential respectively, giving rise to two different mechanisms for structural arrest. In the glass region we identify the presence of a glass-glass-transition line ending in a cusp bifurcation which causes, even in the close by liquid region, a logarithmic decay of correlations.

Interactions in protein solutions near crystallisation: a colloid physics approach

I present a review of current ideas from colloid physics which might be relevant in order to understand the onset of crystallisation and amorphous aggregation processes in protein solution. In particular, the inadequacy of DLVO theory to account for all phenomenological aspects of crystallisation, such as salt-specificity, and for the basic features of the phase diagram, such as the presence of a metastable fluid–fluid separation, is discussed. The fundamental role of additional short-range attractive forces, microscopically orginating from the salting-out effect, is conversely stressed. In order to establish a simple model of protein interparticle interactions near crystallisation, I discuss some recent results obtained by our group for the osmotic compressibility of the metastable fluid phase of hen egg-white lysozyme. Light scattering measurements were performed in an extended volume fraction range at pH=4.7 as a function of temperature, adding NaC1 to screen the electrostatic interactions. The experimental compressibility up to particle volume fractions Φ≈0.23 is very successfully compared to the theoretical expression for a model of adhesive (“sticky”) hard spheres. This surprising quantitative agreement, obtained using a very simple form for the effective interparticle force, suggests that the thermodynamics of the system in the fluid phase is mainly determined by the very short-range nature of the interactions, and is rather insensitive to the detailed form of the potential.

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.

“Sticky hard spheres” model of proteins near crystallization: A test based on the osmotic compressibility of lysozyme solutions

In order to establish a simple model of protein interparticle interaction near crystallization conditions, we have measured the osmotic compressibility of the metastable fluid phase of hen egg-white lysozyme within the fluid-solid coexistence region. Light scattering measurements were performed in an extended volume fraction range at $\mathrm{pH}=4.7$ as a function of temperature. A sufficient amount of NaCl has been added in order to severely screen the electrostatic interactions. Below 0.2 M NaCl, the system can still be kept in a metastable fluid phase for a sufficiently long time before nucleation of the crystal phase takes place. The experimental compressibility up to particle volume fractions $\ensuremath{\Phi}\ensuremath{\approx}0.23$ is compared to the theoretical expression for adhesive (``sticky'') hard spheres, and the agreement is found satisfactory. We also determined the values of the Baxter \ensuremath{\tau} adhesiveness parameter, which shows an apparent linear temperature dependence within the investigated temperature range. These results suggest that the main temperature dependence of the interparticle interaction potential is simply through the Boltzmann ${k}_{B}T$ factor.

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...

Thermodynamic properties of an asymmetric fluid mixture with adhesive hard sphere Yukawa interaction in the mean spherical approximation

The analytical solution of the Ornstein-Zernike equation in the mean-spherical approximation for a Yukawa fluid and adhesive hard sphere Yukawa fluid with factorizable coefficients is used to derive a simple form for the thermodynamic properties for both mixtures. The second system contains an additional interaction. However, the general thermodynamic expressions were obtained in similar form for both systems. Some particular cases are discussed in terms of Γ, the principal parameter to be calculated for these systems. These results complement the recent work of Yasutomi, M., and Ginoza, M., 1996, Molec. Phys., 89, 1755.

Thermodynamic properties of an asymmetric fluid mixture with adhesive hard sphere Yukawa interaction in the mean spherical approximation

Thermodynamic properties of an asymmetric fluid mixture with adhesive hard sphere Yukawa interaction in the mean spherical approximation

Phase stability estimated for adhesive charged sphere fluids with an analytical solution

Phase separation, which can occur in a fluid mixture system (FMS) containing adhesive charged hard spheres (ACHSs) by the addition of a salt of pointlike ionic species, can be enhanced by increasing the absolute value of the charge z0e on the ACHS. Phase stability in the FMS can be revealed using an analytical solution based on the mean spherical approximation for the Ornstein–Zernike equation and can behave nonlinearly. For such a phenomenon, the effect given by the potential of mean force plays an important role. When the adhesive interaction between ACHSs is weak in an FMS containing ACHSs of low charge and low density, an increase in |z0e| can stabilize the dispersed state of the FMS. When each of the ACHSs reaches a highly charged state, the increase in |z0e| can enhance phase separation. When the adhesive interaction between ACHSs is strong in an FMS containing ACHSs of low charge, the stability of the dispersed state depends nonlinearly on the volume fraction φ of ACHSs. The clusters of the ACHSs resulting from phase separation can completely dissolve when φ reaches a certain value as φ increases.

Density functional theory of adhesive hard sphere fluids

We present a simple density functional approach to study the structure of homogeneous as well as inhomogeneous adhesive hard sphere fluid. Radial distribution function g(r) of the homogeneous adhesive hard sphere fluid is calculated by making use of the well known Percus identity which relates the density distribution of an inhomogeneous fluid to the g(r) of the corresponding homogeneous fluid when the external potential responsible for the inhomogeneity is the interparticle potential itself. We have also studied the local density distribution of the same fluid confined in a planar slit consisting of hard walls. The input required for the calculation is the two-particle direct correlation function of the bulk fluid, which is taken from the analytical results corresponding to the Percus Yevick approximation. Both perturbative and nonperturbative weighted density approaches are employed and the calculated radial distributions as well as the density profiles are shown on an average to compare quite well with results from computer simulation.

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.

Protein interactions and crystallization

A lysozyme phase boundary is successfully compared to theoretical predictions from simple fluid theory. The comparison is made using the adhesive hard sphere potential to model protein interactions. The lysozyme phase boundary is constructed from solubility and osmotic second virial coefficient measurements. The result suggests simple models for protein interactions can be useful for investigating protein crystallization. This study supports the use of osmotic second virial coefficient measurements to identify favorable protein crystallization conditions.

Percolation in fluid mixtures containing adhesive charged hard spheres.

Percolation in a fluid mixture system (FMS) containing adhesive charged hard spheres (ACHSs) and point charges can be analytically estimated by regarding the contact of an ACHS with another ACHS as the formation of directly connected bonds between the ACHSs. A feature of the percolation in the FMS can be determined via the phase diagrams displayed for the volume fraction \ensuremath{\varphi} of the ACHSs and the reciprocal ${\mathrm{\ensuremath{\zeta}}}_{00}^{\mathrm{\ensuremath{-}}1}$ of the strength of the adhesive interaction. In the \ensuremath{\varphi}-${\mathrm{\ensuremath{\zeta}}}_{00}^{\mathrm{\ensuremath{-}}1}$ plane, the phase diagrams indicate that the region in which percolation is induced is less dominant than that in which phase separation is induced if the ACHSs are highly charged spheres.