Hard sphere dispersions

The effective pair interactions measured with these techniques are the direct pair interactions between two colloidal particles plus the interactions mediated by the depletants. In practice depletants are poly disperse, for which there are sometimes theoretical results available. For the interaction potential between hard spheres we quote references for the depletion interaction in the presence of polydisperse penetrable hard spheres [74], poly disperse ideal chains [75], poly-disperse hard spheres [76] and polydisperse thin rods [77]. 

We discuss classical non-ideal liquids before treating solids. The strongly interacting fluid systems of interest are hard spheres characterized by their harsh repulsions, atoms and molecules with dispersion interactions responsible for the liquid-vapour transitions of the rare gases, ionic systems including strong and weak electrolytes, simple and not quite so simple polar fluids like water. The solid phase systems discussed are ferroniagnets and alloys. 

For dilute dispersions of hard spheres, Einstein s viscosity equation predicts... 

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... 

In Sec. 3 our presentation is focused on the most important results obtained by different authors in the framework of the rephca Ornstein-Zernike (ROZ) integral equations and by simulations of simple fluids in microporous matrices. For illustrative purposes, we discuss some original results obtained recently in our laboratory. Those allow us to show the application of the ROZ equations to the structure and thermodynamics of fluids adsorbed in disordered porous media. In particular, we present a solution of the ROZ equations for a hard sphere mixture that is highly asymmetric by size, adsorbed in a matrix of hard spheres. This example is relevant in describing the structure of colloidal dispersions in a disordered microporous medium. On the other hand, we present some of the results for the adsorption of a hard sphere fluid in a disordered medium of spherical permeable membranes. The theory developed for the description of this model agrees well with computer simulation data. Finally, in this section we demonstrate the applications of the ROZ theory and present simulation data for adsorption of a hard sphere fluid in a matrix of short chain molecules. This example serves to show the relevance of the theory of Wertheim to chemical association for a set of problems focused on adsorption of fluids and mixtures in disordered microporous matrices prepared by polymerization of species. 

First, we would like to eonsider a simple hard sphere model in a hard sphere matrix, similar to the one studied in Refs. 20, 21, 39. However, our foeus is on a very asymmetric hard sphere mixture adsorbed in a disordered matrix. Moreover, having assumed a large asymmetry of diameters of the eomponents and a very large differenee in the eoneentration of eomponents, here we restriet ourselves to the deseription of the struetural properties of the model. Our interest in this model is due, in part, to experimental findings eoneerning the potential of the mean foree aeting between eolloids in a eolloidal dispersion in the presenee of a matrix of obstaeles [12-14]. 

As in previous theoretical studies of the bulk dispersions of hard spheres we observe in Fig. 1(a) that the PMF exhibits oscillations that develop with increasing solvent density. The phase of the oscillations shifts to smaller intercolloidal separations with augmenting solvent density. Depletion-type attraction is observed close to the contact of two colloids. The structural barrier in the PMF for solvent-separated colloids, at the solvent densities in question, is not at cr /2 but at a larger distance between colloids. These general trends are well known in the theory of colloidal systems and do not require additional comments. 

Instead of the hard-sphere model, the Lennard-Jones (LJ) interaction pair potential can be used to describe soft-core repulsion and dispersion forces. The LJ interaction potential is... 

Lagues et al. [17] found that the percolation theory for hard spheres could be used to describe dramatic increases in electrical conductivity in reverse microemulsions as the volume fraction of water was increased. They also showed how certain scaling theoretical tools were applicable to the analysis of such percolation phenomena. Cazabat et al. [18] also examined percolation in reverse microemulsions with increasing disperse phase volume fraction. They reasoned the percolation came about as a result of formation of clusters of reverse microemulsion droplets. They envisioned increased transport as arising from a transformation of linear droplet clusters to tubular microstructures, to form wormlike reverse microemulsion tubules. 

The second term in equation (2.55) describes the long range Van der Waals or dispersive (attractive) forces. The first term describes the much shorter range repulsive forces experienced when the electron clouds on two atoms come into contact the repulsion increases rapidly with decreasing distance, with the atoms behaving almost as hard spheres. 

The foundations of the theory of flocculation kinetics were laid down early in this century by von Smoluchowski (33). He considered the rate of (irreversible) flocculation of a system of hard-sphere particles, i.e. in the absence of other interactions. With dispersions containing polymers, as we have seen, one is frequently dealing with reversible flocculation this is a much more difficult situation to analyse theoretically. Cowell and Vincent (34) have recently proposed the following semi-empirical equation for the effective flocculation rate constant, kg, ... 

We present an improved model for the flocculation of a dispersion of hard spheres in the presence of non-adsorbing polymer. The pair potential is derived from a recent theory for interacting polymer near a flat surface, and is a function of the depletion thickness. This thickness is of the order of the radius of gyration in dilute polymer solutions but decreases when the coils in solution begin to overlap. Flocculation occurs when the osmotic attraction energy, which is a consequence of the depletion, outweighs the loss in configurational entropy of the dispersed particles. Our analysis differs from that of De Hek and Vrij with respect to the dependence of the depletion thickness on the polymer concentration (i.e., we do not consider the polymer coils to be hard spheres) and to the stability criterion used (binodal, not spinodal phase separation conditions). 

The crucial question is at what value of <)> is the attraction high enough to induce phase separation De Hek and Vrij (6) assume that the critical flocculation concentration is equivalent to the phase separation condition defined by the spinodal point. From the pair potential between two hard spheres in a polymer solution they calculate the second virial coefficient B2 for the particles, and derive from the spinodal condition that if B2 = 1/2 (where is the volume fraction of particles in the dispersion) phase separation occurs. For a system in thermodynamic equilibrium, two phases coexist if the chemical potential of the hard spheres is the same in the dispersion and in the floe phase (i.e., the binodal condition). 

We consider the change in free energy if one hard-sphere particle is transferred from the dispersion to the floe phase. 

FLEER ETAL. Hard-Sphere Dispersion Stability... 

Vincent et al.(3) used a simplified configurational entropy term Ass = -k ln(4>f /4>.). For a dilute dispersion, the In 4>d term is probably correct, but for the floe phase, with of the order of 0.5, a term In 4>f certainly can overestimate the entropy in the floe, because hard spheres with finite volume have at high concentration much less translational freedom than (volumeless) point... 

An important conclusion of this discussion is the fact that at very high <)> thermodynamic stability is re-established. Restabilisation is not a kinetic effect, as suggested by Feigin and Napper (10, 11), but is a consequence of lower free energy of the dispersion as compared to the floe. This conclusion is supported by experimental evidence for soft spheres (3, 5, 23). We should add, however, that for hard spheres is so high that experimental verification is difficult for most polymer-solvent systems due to the high viscosity of the solution. 

Any fundamental study of the rheology of concentrated suspensions necessitates the use of simple systems of well-defined geometry and where the surface characteristics of the particles are well established. For that purpose well-characterized polymer particles of narrow size distribution are used in aqueous or non-aqueous systems. For interpretation of the rheological results, the inter-particle pair-potential must be well-defined and theories must be available for its calculation. The simplest system to consider is that where the pair potential may be represented by a hard sphere model. This, for example, is the case for polystyrene latex dispersions in organic solvents such as benzyl alcohol or cresol, whereby electrostatic interactions are well screened (1). Concentrated dispersions in non-polar media in which the particles are stabilized by a "built-in" stabilizer layer, may also be used, since the pair-potential can be represented by a hard-sphere interaction, where the hard sphere radius is given by the particles radius plus the adsorbed layer thickness. Systems of this type have been recently studied by Croucher and coworkers. (10,11) and Strivens (12). 

Figure 3.15 Calculated shear thinning response for a hard sphere dispersion at

Figure 5.5 The dynamic viscosity for a quasi-hard sphere dispersion from the data of Mellema et al.13 The frequency has been normalised to the diffusion time for two different particle radii. The volume fraction is

This is strong evidence for assuming that dispersions of ideal hard spheres would be expected to show a transition in the viscous behaviour between

short time selfdiffusion coefficient Z)s. This still shows a significant value after the order-disorder transition. The problem faced by the rheologist in interpreting hard sphere systems is that at high concentrations there is... 

Figure 5.6 The quasi-hard sphere diffusion data of Ottewill and Williams14 as a function of volume fraction. The long (DL) and short (Ds) time tracer diffusion coefficients are shown (symbols). The dotted line is representative of the relative fluidity of a hard sphere dispersion...

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