Adhesive hard sphere

Rouw P W and de Kruif C G 1989 Adhesive hard-sphere colloidal dispersions fractal structures and fractal growth in silica dispersions Phys. Rev. A 39 5399-408... 

Accounting for size differences can also be realized in terms of distribution functions, assuming certain interaction energies. Simply because of size differences between molecules preferential adsorption will take place, i.e. fractionation occurs near a phase boundary. In theories where molecular geometries are not constrained by a lattice, this distribution function is virtually determined by the repulsive part of the interaction. An example of this kind has been provided by Chan et al. who considered binary mixtures of adhesive hard spheres in the Percus-Yevick approximation. The theory incorporates a definition of the Gibbs dividing plane in terms of distribution functions. A more formal thermodynamic description for multicomponent mixtures has been given by Schlby and Ruckenstein ). 

For the adhesive hard-sphere model, the theoretical phase diagram in the Tg-0 plane has been partially calculated (Watts et al. 1971 Barboy 1974 Grant and Russel 1993). According to this model, there is a critical point r. c = 0.0976 below which the suspension is predicted to phase separate into a phase dilute in particles and one concentrated in them (see Fig. 7-4). The particle concentration at the critical point of this phase transition is 0c = 0.1213. This phase transition is analogous to the gas-liquid transition of ordinary... 

Figure 7.4 Phase diagram for adhesive hard spheres as a function of Baxter temperature rg. The solid line is the spinodal line for liquid-liquid phase separation (the dense liquid phase is probably metastable), the dot-dashed line is the freezing line for appearance of an ordered packing of spheres, and the dashed line is the percolation transition. (Adapted from Grant and Russel 1993, reprinted with permission from the American Physical Society.)...

The expression for the zero-shear viscosity for the adhesive hard-sphere model at modest particle volume fraction is (Woutersen and de Kruif 1991)------------------------... 

The interaction between ions of the same sign is assumed to be a pure hard sphere repulsion for r < cj. It follows from simple steric considerations that an exact solution will predict dimerization only if Z < all, but polymerization may occur for all < L = a. However, an approximate solution may not reveal the full extent of polymerization that occurs in a more accurate or exact theory. Cummings and Stell [69] used the model to study chemical association of uncharged atoms. It is closely related to the model for adhesive hard spheres studied by Baxter [70]. 

To study gelation phenomena in globular proteins, colloid dispersions, etc., Baxter s adhesive hard sphere (AHS) system [38] is often used as a model system. Particles in the AHS system interact with each other through strongly attractive short-range square well potentials. 

When the medium is a poor solvent for the attached polymers a rather different situation is encountered. The polymer chains then tend to assume collapsed configurations in order to minimize contact with solvent molecules and the polymer segments prefer to interact with each other. This results in (short-ranged) attraction between colloidal particles covered with polymer chains in a poor solvent (see Sect. 5.5 in [46]). The interaction of such sticky spheres (billiard balls with a thin layer of honey [47]) is often described in a simple manner using the adhesive hard sphere interaction (see right panel in Fig. 1.5)... 

Fig. 1.5 Hard-sphere (left) and adhesive hard-sphere (right) interaction...

Figure 6. Theoretically calculated relative viscosity (higher solid line) decomposed into the high frequency part (lower solid line) and the interaction part Tlj (dash line) as a Junction of (j) for cm adhesive hard sphere system with the fractional surface layer thickness e - 0.01 and surface potential 2 = —1.

An interesting result ofthe studies on protein solubility is the demonstration of existence of short-range attractions in protein and STA solutions as controlling the suspension phase behavior. Since the phase behavior of different proteins and inorganic particles can be modeled using the adhesive hard sphere model, the results suggest that the crystallization boundaries of these nanoparticle suspensions is relatively insensitive to the details of the interaction potential and that if two suspensions have the same then they would have the same solubility. 

Ramakrishnan and Zukoski (2000) extended the work of Rosenbaum et al. and tested the ability of different pair potentials to characterize the interactions and phase behavior of STA. The strength of interaction was controlled by dispersing STA in different salt concentrations. The experimental variables used in characterizing the interactions were the osmotic compressibility (dP/dp), the second virial coefficient(.82), relative solution viscosity and the solubility. Various techniques were then developed to extract the parameters ofthe square well, the adhesive hard sphere and the Yukawa pair potentials that best describe the experimental data. As mentioned before, the adhesive hard sphere potential describes the solution thermodynamics only where the system is weakly attractive but as would be expected fails when long range repulsions come into play at low salt concentrations. Model free representations were then presented which offer the opportunity to extract pair potential parameters (F/g. 19-8). 

Pontoni, D., Finet, S., Narayanan, T., and Rennie, A. (2003) Interactions and kinetic arrest in an adhesive hard-sphere colloidal system. / Chem. Phys., 119,... 

Wouterson ATJM, de Kruif CG (1991) The rheology of adhesive hard sphere dispersions. J Chem Phys 94 5739-5749... 

One approach of this category is to solve the integral equations using the Percus-Yevick closure for the system of adhesive hard sphere (AHS) mixtures (17-22). An adhesive hard sphere is a hard sphere that has attractive sites at surface. The attractive interaction on these attractive sites is infinitely strong and infinitesimally short ranged. The Percus-Yevick closime yields an analytical solution for such systems. The adhesive attraction, which resembles the chemical bonding, is used to build up chains by employing the proper connectivity constraints. 

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