Sticky spheres model

Chiarizia, R., Jensen, M.P., Borkowski, M., Thiyagarajan, P., Littrell, K.C. 2004. Interpretation of third phase formation in the ThdV )-HNO, TBP-n-octane system with Baxter s sticky spheres model. Solvent Extr. Ion Exch. 22 (3) 325-351. 

Chiarizia, R., Jensen, M.P., Rickert, P.G. et al. 2004. Extraction of zirconium nitrate by TBP in n-octanc Influence of cation type on third phase formation according to the sticky spheres model. Langmuir 20 (25) 10798-10808. 

Weak electrolytes in which dimerization (as opposed to ion pairing) is the result of chemical bonding between oppositely charged ions have been studied using a sticky electrolyte model (SEM). In this model, a delta fiinction interaction is introduced in the Mayer/-fiinction for the oppositely charged ions at a distance L = a, where a is the hard sphere diameter. The delta fiinction mimics bonding and tire Mayer /-function... 

Zhu J and Rasaiah J C 1989 Solvent effects in weak electrolytes II. Dipolar hard sphere solvent an the sticky electrolyte model with L = a J. Chem. Phys. 91 505... 

Structure factor of the sticky hard sphere model. The structure factor of a system cf interacting spheres is determined by the inter-particle interaction potential u(r). We consider a system of hard spheres with adhesive surfaces. The pair-wise inter-particle interaction potential is written as ... 

Figure 4. A series of SANS data and their fits for Pluronic P84 micellar solutions at 40 C. The fits in absolute intensity scale use an sticky hard sphere model for the inter-particle structure factor and a ctp-and-gown model for the form factor. The same stickiness and polymer segmental distribution are used to fit the entire series of SANS data in this graph simutaneously by varying only the polymer concentrations. This indicates that the microstructure of self-associated micelles is independent of the polymer concentration.

The model for weak electrolytes that I will discuss here is the sticky electrolyte model (SEM) in which a delta-function interaction is introduced into the Mayer f-function for the oppositely charged ions at a distance L < a, where a is the hard sphere diameter. This model was first introduced by Cummings and Stell (1984, 1985) to study association in uncharged systems and is closely related to Baxter s model (1968a) for... 

More modem approaches borrow ideas from the liquid state theory of small molecule fluids to develop a theory for polymers. The most popular of these is the polymer reference interaction site model (PRISM) theory " which is based on the RISM theory of Chandler and Andersen. More recent studies include the Kirkwood hierarchy, the Bom-Green-Yvon hierarchy, and the perturbation density functional theory of Kierlik and Rosinbeig. The latter is based on the thermodynamic perturbation theory of Wertheim " where the polymeric system is composed of very sticky spheres that assemble to form chains. For polymer melts all these liquid state approaches are in quantitative agreement with simulations for the pair correlation functions in short chain fluids. With the exception of the PRISM theory, these liquid state theories are in their infancy, and have not been applied to realistic models of polymers. 

Blum, L., Kalyuzhnyi, Yu.V., Bernard, O., and Herrera-Pacheco, J.N. Sticky charged spheres in the mean-spherical approximation A model for colloids and polyelectrolytes. Journal of Physics - Condensed Matter, 1996, 8, No. 25A, p. A143-A167. 

In conclusion, the material in this chapter is meant to give only an introduction to the subject of liquid structure. Much of what has been presented has dealt with systems which can be represented as point dipoles embedded in hard spheres. Very few liquid systems of chemical interest can be described in such simple terms. However, the simple models can often be modified to make them more realistic. For example, the effects of chemical interactions can be introduced by assuming that the hard sphere experiences sticky interactions in a given direction with respect to the central dipole. Other methods are available for dealing with the effects of non-sphericity. Thus, the simple models can often be made relevant to chemical systems after suitable modification. 

We start with the simplest model of the interface, which consists of a smooth charged hard wall near a ionic solution that is represented by a collection of charged hard spheres, all embedded in a continuum of dielectric constant c. This system is fairly well understood when the density and coupling parameters are low. Then we replace the continuum solvent by a molecular model of the solvent. The simplest of these is the hard sphere with a point dipole[32], which can be treated analytically in some simple cases. More elaborate models of the solvent introduce complications in the numerical discussions. A recently proposed model of ionic solutions uses a solvent model with tetrahedrally coordinated sticky sites. This model is still analytically solvable. More realistic models of the solvent, typically water, can be studied by computer simulations, which however is very difficult for charged interfaces. The full quantum mechanical treatment of the metal surface does not seem feasible at present. The jellium model is a simple alternative for the discussion of the thermodynamic and also kinetic properties of the smooth interface [33, 34, 35, 36, 37, 38, 39, 40]. 

Figure S. A SANS intensity distribution emd its model fit (solid line) as decomposed into the inter- and intro- particle structure factors S(k) and P ) for a 16 % Pluronic P84 solution in D2O at 40 C. The inter-particle structure factor is calculated by solving the OZ equation with an inter-mkellar potential of an sticky hard sphere system. The intra-particle structure factor is cdcidated using the ccp-and-gown model as the polymer segmental distribution in a micelle.

Figure 49 The small angle neutron scattering (SANS) scattering intensity vs. the scattering wave vector q for the silica/silicone oil at particle volume fraction (j)=0.055, under an electric field strength of IkV/mm. Two model curves are presented sticky hard-sphere (SI IS) and fractal model (a = 165 nm, X df = 1.6). Reproduced with permission from C. Gehin, J.

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