Dispersions of Interacting Particles

The simplest colloidal dispersion is one of hard spheres, which interact with a potential 

Within the virial expansion, the presence of longer range repulsions and/or attractions can be accounted for by adding to the free energy of Eq. (7.2) a term 

It is important to note that the dependence of the virial coefficient on the particle diameter comes from the dependence of the interaction V on d. Generally speaking, one imagines, that the larger the particle size, the greater the number of contacts between the molecules on two adjacent particles. Thus, for example, for attractive interactions (where b 0) one expects V d) and hence the term, b d, T) to be an increasing function of d. In contrast to this dependence of the attractive interaction on d, the contribution to the total virial 

It is therefore of interest to estimate the dependence of the attractive interaction, b d,T) on the sphere size. For relatively short-range interactions between spheres, the attractive interaction per sphere can be approximately obtained from the interaction between two flat plates. As we now show, this approximation results in an effective attraction between spheres that increases linearly with the sphere radius. Once the interaction energy per sphere is known, the virial coefficient can be calculated from Eq. (7.4) and the stability of the system to phase separation is related to the magnitude of this coefficient. Thus, the general plan is to (i) find the effective interaction between two flat plates, (ii) relate this interaction energy to an interaction between spheres via the Derejaguin approximation, (iii) calculate the virial coefficient for the spheres in order to assess the stability of the single-phase colloidal dispersion. 

The effective interaction is calculated by noting tiiat when the distance between the spheres, D, is smaller than the range of the interaction (see Fig. 7.2 for a description of the geometry involved), one can evaluate the interaction potential by integrating the potential for flat plates over the range of distances 

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The effect of an applied pressure on the UCFT has been investigated for polymer particles that are sterically stabilized by polyisobutylene and dispersed in 2-methy1-butane. It was observed that the UCFT was shifted to a higher temperature as the hydrostatic pressure applied to the system increased. There was also a qualitative correlation between the UCFT as a function of applied pressure and the 6 conditions of PIB + 2-methylbutane in (P,T) space. These results can be rationalized by considering the effect of pressure on the free volume dissimilarity contribution to the free energy of close approach of interacting particles. Application of corresponding states concepts to the theory of steric stabilization enables a qualitative prediction of the observed stability behaviour as a function of temperature and pressure. 

Most recently, we have attempted to use this procedure to alter the dispersion of cobalt particles over the more strongly interacting 25% Co/A1203 catalyst. However, as shown in Table 8.5, the cluster size was not found to change significantly, and the TPR profiles (not shown for the sake of brevity) were observed... 

Batch equilibrium tests are conducted on solid phase suspensions, prepared with previously air-dried solids, ground to uniform powdery texture for mixing with various concentrations of the pollutants of interest in solution. The concentrations of these pollutants or the COMs leachate in the solution are designed to evaluate the capability of the suspended solids to adsorb all the pollutants possible with increasing amounts of available pollutants, consistent with interaction characteristics dictated by the surface properties of the solids and the pollutants [1,16,22-26,66,67,71]. For a successful and proper study of solid particle sorption of pollutants, the requirement for complete dispersion of solid particles in solution is absolute [143 -145]. Common practice is to use a solution to solid ratio of 10 1 [1], together with efficient sample agitation at a constant temperature (e.g.,48 h at 20 °C). 

Carbon support plays a vital role in the preparation and performance of catalysts since it influences the shape, size and dispersion of catalyst particles as well as the electronic interactions between catalyst and support [154,155]. 

Fig. 9.4.29 Comparison of stability of metallic nanoparticles in bulk liquid with a droplet on a metal surface, (a) Wetting of a droplet on a metal surface, (b) Coagulation and dispersion of metallic particles in liquid. Figures on the left-hand side stand for weak interaction in case A causing coagulation in case B. Those on the right-hand side are a strong interaction between metal and liquid, suggesting good dispersion and good contact.

We have so far focused our attention on dilute systems so that we could avoid dealing with interference of scattering from different particles. The interference effects considered until now are restricted to interference due to scattering centers from within the same particle. When we have a fairly concentrated dispersion or even a dilute dispersion of charged particles that influence the position of each other through their interactions, the scattering data may have to be corrected for interparticle interference effects. Extending the previous discussion to mte/particle interference is not difficult, but the subsequent analysis of the information obtained is not trivial. We shall not go into the details of these here, but just make some brief remarks to establish the connection between interparticle effects and what we have described so far for dilute systems. 

Although the analysis becomes complex for more concentrated dispersions (or even for dilute dispersions of charged particles, which can interact over very large distances), some general observations on two limiting cases are useful ... 

It has to be kept in mind that particulate materials are dispersions. In fact, the classical powder is a concentrated dispersion of solid particles in air. At a very low concentration, very Lne particles (micron, submicron size) can form an aerosol. In such a case-ewing to the large interparticulate distance-fhe particle-particle interactions can be neglected. In general, a particle can exhibit a substructure, that is, a particle may have external and internal pores. An external pore can be related to the roughness of the surface of a particle. 

The term dispersion in dispersion forces comes from an analogy to the refraction (dispersion) of light due to induced dipole interactions. Since London s induced dipole-induced dipole interactions resemble this, the term Dispersion Forces was coined which is unfortunate in that these dispersion forces act against the dispersion of colloidal particles. 

By this term two phenomena are understood, both referring to the Interaction between sound waves and dispersions of charged particles. These methods somewhat transcend the borders set for this section in that the inertia of the particles does play a role in this respect electroacoustics anticipates a.c. electrokinetics and dielectric dispersion (sec. 4.8). 

Pharmaceutical suspensions are dispersions of solid particles in a suspending medium or vehicle (usually aqueous in nature). When the suspended solids are less than 1 pm, the system is referred to as a colloidal suspension. When the particle sizes are greater than about 1 pm, the system is called a coarse suspension. The practical upper limit for particles in a coarse suspension is approximately 50-75 pm. Depending on the affinity or interaction between the dispersed phase and the dispersion medium, a colloidal dispersion can be classified as lyophilic (hydrophilic) or lyophobic (hydrophobic). ... 

A suspension is a dispersion of solid particles in a liquid. A colloidal suspension is a sol. Colloidal properties become significant when the size of the parhcles is of the order of a few micrometer or less. In suspensions of large particles, for example, of some 10 pm or higher, hydrod5mamic interactions dominate the suspension flow properties emd particle packing behaviour. In colloidal suspensions interaction forces between the particles as well as hydro-dynamic interactions play a role in determining the flow and particle packing properties. 

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